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Theorem poimirlem16 32595
Description: Lemma for poimir 32612 establishing the vertices of the simplex of poimirlem17 32596. (Contributed by Brendan Leahy, 21-Aug-2020.)
Hypotheses
Ref Expression
poimir.0 (𝜑𝑁 ∈ ℕ)
poimirlem22.s 𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘𝑓 + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}
poimirlem22.1 (𝜑𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑𝑚 (1...𝑁)))
poimirlem22.2 (𝜑𝑇𝑆)
poimirlem18.3 ((𝜑𝑛 ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝𝑛) ≠ 𝐾)
poimirlem18.4 (𝜑 → (2nd𝑇) = 0)
Assertion
Ref Expression
poimirlem16 (𝜑𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ((𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))) ∘𝑓 + (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})))))
Distinct variable groups:   𝑓,𝑗,𝑛,𝑝,𝑡,𝑦   𝜑,𝑗,𝑛,𝑦   𝑗,𝐹,𝑛,𝑦   𝑗,𝑁,𝑛,𝑦   𝑇,𝑗,𝑛,𝑦   𝜑,𝑝,𝑡   𝑓,𝐾,𝑗,𝑛,𝑝,𝑡   𝑓,𝑁,𝑝,𝑡   𝑇,𝑓,𝑝   𝑓,𝐹,𝑝,𝑡   𝑡,𝑇   𝑆,𝑗,𝑛,𝑝,𝑡,𝑦
Allowed substitution hints:   𝜑(𝑓)   𝑆(𝑓)   𝐾(𝑦)

Proof of Theorem poimirlem16
StepHypRef Expression
1 poimirlem22.2 . . 3 (𝜑𝑇𝑆)
2 fveq2 6103 . . . . . . . . . . 11 (𝑡 = 𝑇 → (2nd𝑡) = (2nd𝑇))
32breq2d 4595 . . . . . . . . . 10 (𝑡 = 𝑇 → (𝑦 < (2nd𝑡) ↔ 𝑦 < (2nd𝑇)))
43ifbid 4058 . . . . . . . . 9 (𝑡 = 𝑇 → if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)))
54csbeq1d 3506 . . . . . . . 8 (𝑡 = 𝑇if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘𝑓 + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘𝑓 + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))
6 fveq2 6103 . . . . . . . . . . 11 (𝑡 = 𝑇 → (1st𝑡) = (1st𝑇))
76fveq2d 6107 . . . . . . . . . 10 (𝑡 = 𝑇 → (1st ‘(1st𝑡)) = (1st ‘(1st𝑇)))
86fveq2d 6107 . . . . . . . . . . . . 13 (𝑡 = 𝑇 → (2nd ‘(1st𝑡)) = (2nd ‘(1st𝑇)))
98imaeq1d 5384 . . . . . . . . . . . 12 (𝑡 = 𝑇 → ((2nd ‘(1st𝑡)) “ (1...𝑗)) = ((2nd ‘(1st𝑇)) “ (1...𝑗)))
109xpeq1d 5062 . . . . . . . . . . 11 (𝑡 = 𝑇 → (((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}))
118imaeq1d 5384 . . . . . . . . . . . 12 (𝑡 = 𝑇 → ((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)))
1211xpeq1d 5062 . . . . . . . . . . 11 (𝑡 = 𝑇 → (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))
1310, 12uneq12d 3730 . . . . . . . . . 10 (𝑡 = 𝑇 → ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))
147, 13oveq12d 6567 . . . . . . . . 9 (𝑡 = 𝑇 → ((1st ‘(1st𝑡)) ∘𝑓 + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))
1514csbeq2dv 3944 . . . . . . . 8 (𝑡 = 𝑇if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘𝑓 + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))
165, 15eqtrd 2644 . . . . . . 7 (𝑡 = 𝑇if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘𝑓 + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))
1716mpteq2dv 4673 . . . . . 6 (𝑡 = 𝑇 → (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘𝑓 + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))))
1817eqeq2d 2620 . . . . 5 (𝑡 = 𝑇 → (𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘𝑓 + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})))) ↔ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))))
19 poimirlem22.s . . . . 5 𝑆 = {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘𝑓 + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))}
2018, 19elrab2 3333 . . . 4 (𝑇𝑆 ↔ (𝑇 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))))
2120simprbi 479 . . 3 (𝑇𝑆𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))))
221, 21syl 17 . 2 (𝜑𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))))
23 elrabi 3328 . . . . . . . . . . . 12 (𝑇 ∈ {𝑡 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∣ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘𝑓 + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))} → 𝑇 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
2423, 19eleq2s 2706 . . . . . . . . . . 11 (𝑇𝑆𝑇 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
251, 24syl 17 . . . . . . . . . 10 (𝜑𝑇 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
26 xp1st 7089 . . . . . . . . . 10 (𝑇 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) → (1st𝑇) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
2725, 26syl 17 . . . . . . . . 9 (𝜑 → (1st𝑇) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
28 xp1st 7089 . . . . . . . . 9 ((1st𝑇) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1st ‘(1st𝑇)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)))
2927, 28syl 17 . . . . . . . 8 (𝜑 → (1st ‘(1st𝑇)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)))
30 elmapfn 7766 . . . . . . . 8 ((1st ‘(1st𝑇)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)) → (1st ‘(1st𝑇)) Fn (1...𝑁))
3129, 30syl 17 . . . . . . 7 (𝜑 → (1st ‘(1st𝑇)) Fn (1...𝑁))
3231adantr 480 . . . . . 6 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (1st ‘(1st𝑇)) Fn (1...𝑁))
33 1ex 9914 . . . . . . . . . 10 1 ∈ V
34 fnconstg 6006 . . . . . . . . . 10 (1 ∈ V → (((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}) Fn ((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))))
3533, 34ax-mp 5 . . . . . . . . 9 (((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}) Fn ((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1)))
36 c0ex 9913 . . . . . . . . . 10 0 ∈ V
37 fnconstg 6006 . . . . . . . . . 10 (0 ∈ V → (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}) Fn ((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)))
3836, 37ax-mp 5 . . . . . . . . 9 (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}) Fn ((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁))
3935, 38pm3.2i 470 . . . . . . . 8 ((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}) Fn ((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) ∧ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}) Fn ((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)))
40 xp2nd 7090 . . . . . . . . . . . . 13 ((1st𝑇) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (2nd ‘(1st𝑇)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
4127, 40syl 17 . . . . . . . . . . . 12 (𝜑 → (2nd ‘(1st𝑇)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
42 fvex 6113 . . . . . . . . . . . . 13 (2nd ‘(1st𝑇)) ∈ V
43 f1oeq1 6040 . . . . . . . . . . . . 13 (𝑓 = (2nd ‘(1st𝑇)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁)))
4442, 43elab 3319 . . . . . . . . . . . 12 ((2nd ‘(1st𝑇)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
4541, 44sylib 207 . . . . . . . . . . 11 (𝜑 → (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
46 dff1o3 6056 . . . . . . . . . . . 12 ((2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) ↔ ((2nd ‘(1st𝑇)):(1...𝑁)–onto→(1...𝑁) ∧ Fun (2nd ‘(1st𝑇))))
4746simprbi 479 . . . . . . . . . . 11 ((2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → Fun (2nd ‘(1st𝑇)))
4845, 47syl 17 . . . . . . . . . 10 (𝜑 → Fun (2nd ‘(1st𝑇)))
49 imain 5888 . . . . . . . . . 10 (Fun (2nd ‘(1st𝑇)) → ((2nd ‘(1st𝑇)) “ ((1...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁))) = (((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) ∩ ((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁))))
5048, 49syl 17 . . . . . . . . 9 (𝜑 → ((2nd ‘(1st𝑇)) “ ((1...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁))) = (((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) ∩ ((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁))))
51 elfznn0 12302 . . . . . . . . . . . . . . 15 (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ∈ ℕ0)
52 nn0p1nn 11209 . . . . . . . . . . . . . . 15 (𝑦 ∈ ℕ0 → (𝑦 + 1) ∈ ℕ)
5351, 52syl 17 . . . . . . . . . . . . . 14 (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈ ℕ)
5453nnred 10912 . . . . . . . . . . . . 13 (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈ ℝ)
5554ltp1d 10833 . . . . . . . . . . . 12 (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) < ((𝑦 + 1) + 1))
56 fzdisj 12239 . . . . . . . . . . . 12 ((𝑦 + 1) < ((𝑦 + 1) + 1) → ((1...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁)) = ∅)
5755, 56syl 17 . . . . . . . . . . 11 (𝑦 ∈ (0...(𝑁 − 1)) → ((1...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁)) = ∅)
5857imaeq2d 5385 . . . . . . . . . 10 (𝑦 ∈ (0...(𝑁 − 1)) → ((2nd ‘(1st𝑇)) “ ((1...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁))) = ((2nd ‘(1st𝑇)) “ ∅))
59 ima0 5400 . . . . . . . . . 10 ((2nd ‘(1st𝑇)) “ ∅) = ∅
6058, 59syl6eq 2660 . . . . . . . . 9 (𝑦 ∈ (0...(𝑁 − 1)) → ((2nd ‘(1st𝑇)) “ ((1...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁))) = ∅)
6150, 60sylan9req 2665 . . . . . . . 8 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) ∩ ((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) = ∅)
62 fnun 5911 . . . . . . . 8 ((((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}) Fn ((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) ∧ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}) Fn ((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) ∧ (((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) ∩ ((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) = ∅) → ((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) Fn (((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) ∪ ((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁))))
6339, 61, 62sylancr 694 . . . . . . 7 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) Fn (((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) ∪ ((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁))))
64 imaundi 5464 . . . . . . . . 9 ((2nd ‘(1st𝑇)) “ ((1...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁))) = (((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) ∪ ((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)))
65 nnuz 11599 . . . . . . . . . . . . . . 15 ℕ = (ℤ‘1)
6653, 65syl6eleq 2698 . . . . . . . . . . . . . 14 (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈ (ℤ‘1))
67 peano2uz 11617 . . . . . . . . . . . . . 14 ((𝑦 + 1) ∈ (ℤ‘1) → ((𝑦 + 1) + 1) ∈ (ℤ‘1))
6866, 67syl 17 . . . . . . . . . . . . 13 (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑦 + 1) + 1) ∈ (ℤ‘1))
6968adantl 481 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((𝑦 + 1) + 1) ∈ (ℤ‘1))
70 poimir.0 . . . . . . . . . . . . . . . 16 (𝜑𝑁 ∈ ℕ)
7170nncnd 10913 . . . . . . . . . . . . . . 15 (𝜑𝑁 ∈ ℂ)
72 npcan1 10334 . . . . . . . . . . . . . . 15 (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁)
7371, 72syl 17 . . . . . . . . . . . . . 14 (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
7473adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((𝑁 − 1) + 1) = 𝑁)
75 elfzuz3 12210 . . . . . . . . . . . . . . 15 (𝑦 ∈ (0...(𝑁 − 1)) → (𝑁 − 1) ∈ (ℤ𝑦))
76 eluzp1p1 11589 . . . . . . . . . . . . . . 15 ((𝑁 − 1) ∈ (ℤ𝑦) → ((𝑁 − 1) + 1) ∈ (ℤ‘(𝑦 + 1)))
7775, 76syl 17 . . . . . . . . . . . . . 14 (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈ (ℤ‘(𝑦 + 1)))
7877adantl 481 . . . . . . . . . . . . 13 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((𝑁 − 1) + 1) ∈ (ℤ‘(𝑦 + 1)))
7974, 78eqeltrrd 2689 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ (ℤ‘(𝑦 + 1)))
80 fzsplit2 12237 . . . . . . . . . . . 12 ((((𝑦 + 1) + 1) ∈ (ℤ‘1) ∧ 𝑁 ∈ (ℤ‘(𝑦 + 1))) → (1...𝑁) = ((1...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁)))
8169, 79, 80syl2anc 691 . . . . . . . . . . 11 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (1...𝑁) = ((1...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁)))
8281imaeq2d 5385 . . . . . . . . . 10 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st𝑇)) “ (1...𝑁)) = ((2nd ‘(1st𝑇)) “ ((1...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁))))
83 f1ofo 6057 . . . . . . . . . . . 12 ((2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd ‘(1st𝑇)):(1...𝑁)–onto→(1...𝑁))
84 foima 6033 . . . . . . . . . . . 12 ((2nd ‘(1st𝑇)):(1...𝑁)–onto→(1...𝑁) → ((2nd ‘(1st𝑇)) “ (1...𝑁)) = (1...𝑁))
8545, 83, 843syl 18 . . . . . . . . . . 11 (𝜑 → ((2nd ‘(1st𝑇)) “ (1...𝑁)) = (1...𝑁))
8685adantr 480 . . . . . . . . . 10 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st𝑇)) “ (1...𝑁)) = (1...𝑁))
8782, 86eqtr3d 2646 . . . . . . . . 9 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st𝑇)) “ ((1...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁))) = (1...𝑁))
8864, 87syl5eqr 2658 . . . . . . . 8 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) ∪ ((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) = (1...𝑁))
8988fneq2d 5896 . . . . . . 7 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) Fn (((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) ∪ ((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) ↔ ((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) Fn (1...𝑁)))
9063, 89mpbid 221 . . . . . 6 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) Fn (1...𝑁))
91 ovex 6577 . . . . . . 7 (1...𝑁) ∈ V
9291a1i 11 . . . . . 6 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (1...𝑁) ∈ V)
93 inidm 3784 . . . . . 6 ((1...𝑁) ∩ (1...𝑁)) = (1...𝑁)
94 eqidd 2611 . . . . . 6 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → ((1st ‘(1st𝑇))‘𝑛) = ((1st ‘(1st𝑇))‘𝑛))
95 eqidd 2611 . . . . . 6 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → (((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) = (((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛))
9632, 90, 92, 92, 93, 94, 95offval 6802 . . . . 5 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))) = (𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + (((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛))))
97 oveq1 6556 . . . . . . . . . 10 (1 = if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0) → (1 + ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛)) = (if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0) + ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛)))
9897eqeq2d 2620 . . . . . . . . 9 (1 = if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0) → ((((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) = (1 + ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛)) ↔ (((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) = (if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0) + ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛))))
99 oveq1 6556 . . . . . . . . . 10 (0 = if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0) → (0 + ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛)) = (if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0) + ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛)))
10099eqeq2d 2620 . . . . . . . . 9 (0 = if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0) → ((((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) = (0 + ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛)) ↔ (((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) = (if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0) + ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛))))
101 1p0e1 11010 . . . . . . . . . . . . . 14 (1 + 0) = 1
102101eqcomi 2619 . . . . . . . . . . . . 13 1 = (1 + 0)
103 f1ofn 6051 . . . . . . . . . . . . . . . . . 18 ((2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd ‘(1st𝑇)) Fn (1...𝑁))
10445, 103syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → (2nd ‘(1st𝑇)) Fn (1...𝑁))
105104adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (2nd ‘(1st𝑇)) Fn (1...𝑁))
106 fzss2 12252 . . . . . . . . . . . . . . . . 17 (𝑁 ∈ (ℤ‘(𝑦 + 1)) → (1...(𝑦 + 1)) ⊆ (1...𝑁))
10779, 106syl 17 . . . . . . . . . . . . . . . 16 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (1...(𝑦 + 1)) ⊆ (1...𝑁))
108 eluzfz1 12219 . . . . . . . . . . . . . . . . . 18 ((𝑦 + 1) ∈ (ℤ‘1) → 1 ∈ (1...(𝑦 + 1)))
10966, 108syl 17 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ (0...(𝑁 − 1)) → 1 ∈ (1...(𝑦 + 1)))
110109adantl 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → 1 ∈ (1...(𝑦 + 1)))
111 fnfvima 6400 . . . . . . . . . . . . . . . 16 (((2nd ‘(1st𝑇)) Fn (1...𝑁) ∧ (1...(𝑦 + 1)) ⊆ (1...𝑁) ∧ 1 ∈ (1...(𝑦 + 1))) → ((2nd ‘(1st𝑇))‘1) ∈ ((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))))
112105, 107, 110, 111syl3anc 1318 . . . . . . . . . . . . . . 15 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st𝑇))‘1) ∈ ((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))))
113 fvun1 6179 . . . . . . . . . . . . . . . 16 (((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}) Fn ((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) ∧ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}) Fn ((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) ∧ ((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) ∩ ((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) = ∅ ∧ ((2nd ‘(1st𝑇))‘1) ∈ ((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))))) → (((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘1)) = ((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1})‘((2nd ‘(1st𝑇))‘1)))
11435, 38, 113mp3an12 1406 . . . . . . . . . . . . . . 15 (((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) ∩ ((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) = ∅ ∧ ((2nd ‘(1st𝑇))‘1) ∈ ((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1)))) → (((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘1)) = ((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1})‘((2nd ‘(1st𝑇))‘1)))
11561, 112, 114syl2anc 691 . . . . . . . . . . . . . 14 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘1)) = ((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1})‘((2nd ‘(1st𝑇))‘1)))
11633fvconst2 6374 . . . . . . . . . . . . . . 15 (((2nd ‘(1st𝑇))‘1) ∈ ((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) → ((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1})‘((2nd ‘(1st𝑇))‘1)) = 1)
117112, 116syl 17 . . . . . . . . . . . . . 14 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1})‘((2nd ‘(1st𝑇))‘1)) = 1)
118115, 117eqtrd 2644 . . . . . . . . . . . . 13 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘1)) = 1)
119 simpr 476 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑛 ∈ (1...(𝑁 − 1))) → 𝑛 ∈ (1...(𝑁 − 1)))
12070nnzd 11357 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝜑𝑁 ∈ ℤ)
121 peano2zm 11297 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ)
122120, 121syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜑 → (𝑁 − 1) ∈ ℤ)
123 1z 11284 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1 ∈ ℤ
124122, 123jctil 558 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑 → (1 ∈ ℤ ∧ (𝑁 − 1) ∈ ℤ))
125 elfzelz 12213 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑛 ∈ (1...(𝑁 − 1)) → 𝑛 ∈ ℤ)
126125, 123jctir 559 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑛 ∈ (1...(𝑁 − 1)) → (𝑛 ∈ ℤ ∧ 1 ∈ ℤ))
127 fzaddel 12246 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((1 ∈ ℤ ∧ (𝑁 − 1) ∈ ℤ) ∧ (𝑛 ∈ ℤ ∧ 1 ∈ ℤ)) → (𝑛 ∈ (1...(𝑁 − 1)) ↔ (𝑛 + 1) ∈ ((1 + 1)...((𝑁 − 1) + 1))))
128124, 126, 127syl2an 493 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑛 ∈ (1...(𝑁 − 1))) → (𝑛 ∈ (1...(𝑁 − 1)) ↔ (𝑛 + 1) ∈ ((1 + 1)...((𝑁 − 1) + 1))))
129119, 128mpbid 221 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑛 ∈ (1...(𝑁 − 1))) → (𝑛 + 1) ∈ ((1 + 1)...((𝑁 − 1) + 1)))
13073oveq2d 6565 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → ((1 + 1)...((𝑁 − 1) + 1)) = ((1 + 1)...𝑁))
131130adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑛 ∈ (1...(𝑁 − 1))) → ((1 + 1)...((𝑁 − 1) + 1)) = ((1 + 1)...𝑁))
132129, 131eleqtrd 2690 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑛 ∈ (1...(𝑁 − 1))) → (𝑛 + 1) ∈ ((1 + 1)...𝑁))
133132ralrimiva 2949 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → ∀𝑛 ∈ (1...(𝑁 − 1))(𝑛 + 1) ∈ ((1 + 1)...𝑁))
134 simpr 476 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝑦 ∈ ((1 + 1)...𝑁)) → 𝑦 ∈ ((1 + 1)...𝑁))
135 peano2z 11295 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (1 ∈ ℤ → (1 + 1) ∈ ℤ)
136123, 135ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (1 + 1) ∈ ℤ
137120, 136jctil 558 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜑 → ((1 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ))
138 elfzelz 12213 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑦 ∈ ((1 + 1)...𝑁) → 𝑦 ∈ ℤ)
139138, 123jctir 559 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑦 ∈ ((1 + 1)...𝑁) → (𝑦 ∈ ℤ ∧ 1 ∈ ℤ))
140 fzsubel 12248 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((1 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑦 ∈ ℤ ∧ 1 ∈ ℤ)) → (𝑦 ∈ ((1 + 1)...𝑁) ↔ (𝑦 − 1) ∈ (((1 + 1) − 1)...(𝑁 − 1))))
141137, 139, 140syl2an 493 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝑦 ∈ ((1 + 1)...𝑁)) → (𝑦 ∈ ((1 + 1)...𝑁) ↔ (𝑦 − 1) ∈ (((1 + 1) − 1)...(𝑁 − 1))))
142134, 141mpbid 221 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑦 ∈ ((1 + 1)...𝑁)) → (𝑦 − 1) ∈ (((1 + 1) − 1)...(𝑁 − 1)))
143 ax-1cn 9873 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1 ∈ ℂ
144143, 143pncan3oi 10176 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((1 + 1) − 1) = 1
145144oveq1i 6559 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((1 + 1) − 1)...(𝑁 − 1)) = (1...(𝑁 − 1))
146142, 145syl6eleq 2698 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑦 ∈ ((1 + 1)...𝑁)) → (𝑦 − 1) ∈ (1...(𝑁 − 1)))
147138zcnd 11359 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑦 ∈ ((1 + 1)...𝑁) → 𝑦 ∈ ℂ)
148 elfznn 12241 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑛 ∈ (1...(𝑁 − 1)) → 𝑛 ∈ ℕ)
149148nncnd 10913 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑛 ∈ (1...(𝑁 − 1)) → 𝑛 ∈ ℂ)
150 subadd2 10164 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑦 ∈ ℂ ∧ 1 ∈ ℂ ∧ 𝑛 ∈ ℂ) → ((𝑦 − 1) = 𝑛 ↔ (𝑛 + 1) = 𝑦))
151143, 150mp3an2 1404 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑦 ∈ ℂ ∧ 𝑛 ∈ ℂ) → ((𝑦 − 1) = 𝑛 ↔ (𝑛 + 1) = 𝑦))
152151bicomd 212 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑦 ∈ ℂ ∧ 𝑛 ∈ ℂ) → ((𝑛 + 1) = 𝑦 ↔ (𝑦 − 1) = 𝑛))
153 eqcom 2617 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑛 + 1) = 𝑦𝑦 = (𝑛 + 1))
154 eqcom 2617 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑦 − 1) = 𝑛𝑛 = (𝑦 − 1))
155152, 153, 1543bitr3g 301 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑦 ∈ ℂ ∧ 𝑛 ∈ ℂ) → (𝑦 = (𝑛 + 1) ↔ 𝑛 = (𝑦 − 1)))
156147, 149, 155syl2an 493 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑦 ∈ ((1 + 1)...𝑁) ∧ 𝑛 ∈ (1...(𝑁 − 1))) → (𝑦 = (𝑛 + 1) ↔ 𝑛 = (𝑦 − 1)))
157156ralrimiva 2949 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 ∈ ((1 + 1)...𝑁) → ∀𝑛 ∈ (1...(𝑁 − 1))(𝑦 = (𝑛 + 1) ↔ 𝑛 = (𝑦 − 1)))
158157adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑦 ∈ ((1 + 1)...𝑁)) → ∀𝑛 ∈ (1...(𝑁 − 1))(𝑦 = (𝑛 + 1) ↔ 𝑛 = (𝑦 − 1)))
159 reu6i 3364 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑦 − 1) ∈ (1...(𝑁 − 1)) ∧ ∀𝑛 ∈ (1...(𝑁 − 1))(𝑦 = (𝑛 + 1) ↔ 𝑛 = (𝑦 − 1))) → ∃!𝑛 ∈ (1...(𝑁 − 1))𝑦 = (𝑛 + 1))
160146, 158, 159syl2anc 691 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑦 ∈ ((1 + 1)...𝑁)) → ∃!𝑛 ∈ (1...(𝑁 − 1))𝑦 = (𝑛 + 1))
161160ralrimiva 2949 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → ∀𝑦 ∈ ((1 + 1)...𝑁)∃!𝑛 ∈ (1...(𝑁 − 1))𝑦 = (𝑛 + 1))
162 eqid 2610 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1)) = (𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1))
163162f1ompt 6290 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1)):(1...(𝑁 − 1))–1-1-onto→((1 + 1)...𝑁) ↔ (∀𝑛 ∈ (1...(𝑁 − 1))(𝑛 + 1) ∈ ((1 + 1)...𝑁) ∧ ∀𝑦 ∈ ((1 + 1)...𝑁)∃!𝑛 ∈ (1...(𝑁 − 1))𝑦 = (𝑛 + 1)))
164133, 161, 163sylanbrc 695 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1)):(1...(𝑁 − 1))–1-1-onto→((1 + 1)...𝑁))
165 f1osng 6089 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑁 ∈ ℕ ∧ 1 ∈ V) → {⟨𝑁, 1⟩}:{𝑁}–1-1-onto→{1})
16670, 33, 165sylancl 693 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → {⟨𝑁, 1⟩}:{𝑁}–1-1-onto→{1})
16770nnred 10912 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑𝑁 ∈ ℝ)
168167ltm1d 10835 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → (𝑁 − 1) < 𝑁)
169122zred 11358 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → (𝑁 − 1) ∈ ℝ)
170169, 167ltnled 10063 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → ((𝑁 − 1) < 𝑁 ↔ ¬ 𝑁 ≤ (𝑁 − 1)))
171168, 170mpbid 221 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → ¬ 𝑁 ≤ (𝑁 − 1))
172 elfzle2 12216 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑁 ∈ (1...(𝑁 − 1)) → 𝑁 ≤ (𝑁 − 1))
173171, 172nsyl 134 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → ¬ 𝑁 ∈ (1...(𝑁 − 1)))
174 disjsn 4192 . . . . . . . . . . . . . . . . . . . . . . 23 (((1...(𝑁 − 1)) ∩ {𝑁}) = ∅ ↔ ¬ 𝑁 ∈ (1...(𝑁 − 1)))
175173, 174sylibr 223 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → ((1...(𝑁 − 1)) ∩ {𝑁}) = ∅)
176 1re 9918 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 1 ∈ ℝ
177176ltp1i 10806 . . . . . . . . . . . . . . . . . . . . . . . . . 26 1 < (1 + 1)
178176, 176readdcli 9932 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (1 + 1) ∈ ℝ
179176, 178ltnlei 10037 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (1 < (1 + 1) ↔ ¬ (1 + 1) ≤ 1)
180177, 179mpbi 219 . . . . . . . . . . . . . . . . . . . . . . . . 25 ¬ (1 + 1) ≤ 1
181 elfzle1 12215 . . . . . . . . . . . . . . . . . . . . . . . . 25 (1 ∈ ((1 + 1)...𝑁) → (1 + 1) ≤ 1)
182180, 181mto 187 . . . . . . . . . . . . . . . . . . . . . . . 24 ¬ 1 ∈ ((1 + 1)...𝑁)
183 disjsn 4192 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((1 + 1)...𝑁) ∩ {1}) = ∅ ↔ ¬ 1 ∈ ((1 + 1)...𝑁))
184182, 183mpbir 220 . . . . . . . . . . . . . . . . . . . . . . 23 (((1 + 1)...𝑁) ∩ {1}) = ∅
185 f1oun 6069 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1)):(1...(𝑁 − 1))–1-1-onto→((1 + 1)...𝑁) ∧ {⟨𝑁, 1⟩}:{𝑁}–1-1-onto→{1}) ∧ (((1...(𝑁 − 1)) ∩ {𝑁}) = ∅ ∧ (((1 + 1)...𝑁) ∩ {1}) = ∅)) → ((𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1)) ∪ {⟨𝑁, 1⟩}):((1...(𝑁 − 1)) ∪ {𝑁})–1-1-onto→(((1 + 1)...𝑁) ∪ {1}))
186184, 185mpanr2 716 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1)):(1...(𝑁 − 1))–1-1-onto→((1 + 1)...𝑁) ∧ {⟨𝑁, 1⟩}:{𝑁}–1-1-onto→{1}) ∧ ((1...(𝑁 − 1)) ∩ {𝑁}) = ∅) → ((𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1)) ∪ {⟨𝑁, 1⟩}):((1...(𝑁 − 1)) ∪ {𝑁})–1-1-onto→(((1 + 1)...𝑁) ∪ {1}))
187164, 166, 175, 186syl21anc 1317 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → ((𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1)) ∪ {⟨𝑁, 1⟩}):((1...(𝑁 − 1)) ∪ {𝑁})–1-1-onto→(((1 + 1)...𝑁) ∪ {1}))
188 ssv 3588 . . . . . . . . . . . . . . . . . . . . . . . . 25 ℕ ⊆ V
189188, 70sseldi 3566 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑𝑁 ∈ V)
19033a1i 11 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → 1 ∈ V)
19170, 65syl6eleq 2698 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑𝑁 ∈ (ℤ‘1))
19273, 191eqeltrd 2688 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → ((𝑁 − 1) + 1) ∈ (ℤ‘1))
193 uzid 11578 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑁 − 1) ∈ ℤ → (𝑁 − 1) ∈ (ℤ‘(𝑁 − 1)))
194 peano2uz 11617 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑁 − 1) ∈ (ℤ‘(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈ (ℤ‘(𝑁 − 1)))
195122, 193, 1943syl 18 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑 → ((𝑁 − 1) + 1) ∈ (ℤ‘(𝑁 − 1)))
19673, 195eqeltrrd 2689 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑𝑁 ∈ (ℤ‘(𝑁 − 1)))
197 fzsplit2 12237 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝑁 − 1) + 1) ∈ (ℤ‘1) ∧ 𝑁 ∈ (ℤ‘(𝑁 − 1))) → (1...𝑁) = ((1...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)))
198192, 196, 197syl2anc 691 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → (1...𝑁) = ((1...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)))
19973oveq1d 6564 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑 → (((𝑁 − 1) + 1)...𝑁) = (𝑁...𝑁))
200 fzsn 12254 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑁 ∈ ℤ → (𝑁...𝑁) = {𝑁})
201120, 200syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑 → (𝑁...𝑁) = {𝑁})
202199, 201eqtrd 2644 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → (((𝑁 − 1) + 1)...𝑁) = {𝑁})
203202uneq2d 3729 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → ((1...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)) = ((1...(𝑁 − 1)) ∪ {𝑁}))
204198, 203eqtr2d 2645 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → ((1...(𝑁 − 1)) ∪ {𝑁}) = (1...𝑁))
205 iftrue 4042 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑛 = 𝑁 → if(𝑛 = 𝑁, 1, (𝑛 + 1)) = 1)
206205adantl 481 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑛 = 𝑁) → if(𝑛 = 𝑁, 1, (𝑛 + 1)) = 1)
207189, 190, 204, 206fmptapd 6342 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → ((𝑛 ∈ (1...(𝑁 − 1)) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) ∪ {⟨𝑁, 1⟩}) = (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))
208 eleq1 2676 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑛 = 𝑁 → (𝑛 ∈ (1...(𝑁 − 1)) ↔ 𝑁 ∈ (1...(𝑁 − 1))))
209208notbid 307 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑛 = 𝑁 → (¬ 𝑛 ∈ (1...(𝑁 − 1)) ↔ ¬ 𝑁 ∈ (1...(𝑁 − 1))))
210173, 209syl5ibrcom 236 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜑 → (𝑛 = 𝑁 → ¬ 𝑛 ∈ (1...(𝑁 − 1))))
211210necon2ad 2797 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑 → (𝑛 ∈ (1...(𝑁 − 1)) → 𝑛𝑁))
212211imp 444 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑛 ∈ (1...(𝑁 − 1))) → 𝑛𝑁)
213 ifnefalse 4048 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑛𝑁 → if(𝑛 = 𝑁, 1, (𝑛 + 1)) = (𝑛 + 1))
214212, 213syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑛 ∈ (1...(𝑁 − 1))) → if(𝑛 = 𝑁, 1, (𝑛 + 1)) = (𝑛 + 1))
215214mpteq2dva 4672 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (𝑛 ∈ (1...(𝑁 − 1)) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) = (𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1)))
216215uneq1d 3728 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → ((𝑛 ∈ (1...(𝑁 − 1)) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) ∪ {⟨𝑁, 1⟩}) = ((𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1)) ∪ {⟨𝑁, 1⟩}))
217207, 216eqtr3d 2646 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) = ((𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1)) ∪ {⟨𝑁, 1⟩}))
218198, 203eqtrd 2644 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (1...𝑁) = ((1...(𝑁 − 1)) ∪ {𝑁}))
219 uzid 11578 . . . . . . . . . . . . . . . . . . . . . . . . 25 (1 ∈ ℤ → 1 ∈ (ℤ‘1))
220 peano2uz 11617 . . . . . . . . . . . . . . . . . . . . . . . . 25 (1 ∈ (ℤ‘1) → (1 + 1) ∈ (ℤ‘1))
221123, 219, 220mp2b 10 . . . . . . . . . . . . . . . . . . . . . . . 24 (1 + 1) ∈ (ℤ‘1)
222 fzsplit2 12237 . . . . . . . . . . . . . . . . . . . . . . . 24 (((1 + 1) ∈ (ℤ‘1) ∧ 𝑁 ∈ (ℤ‘1)) → (1...𝑁) = ((1...1) ∪ ((1 + 1)...𝑁)))
223221, 191, 222sylancr 694 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (1...𝑁) = ((1...1) ∪ ((1 + 1)...𝑁)))
224 fzsn 12254 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (1 ∈ ℤ → (1...1) = {1})
225123, 224ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . . 25 (1...1) = {1}
226225uneq1i 3725 . . . . . . . . . . . . . . . . . . . . . . . 24 ((1...1) ∪ ((1 + 1)...𝑁)) = ({1} ∪ ((1 + 1)...𝑁))
227226equncomi 3721 . . . . . . . . . . . . . . . . . . . . . . 23 ((1...1) ∪ ((1 + 1)...𝑁)) = (((1 + 1)...𝑁) ∪ {1})
228223, 227syl6eq 2660 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (1...𝑁) = (((1 + 1)...𝑁) ∪ {1}))
229217, 218, 228f1oeq123d 6046 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))):(1...𝑁)–1-1-onto→(1...𝑁) ↔ ((𝑛 ∈ (1...(𝑁 − 1)) ↦ (𝑛 + 1)) ∪ {⟨𝑁, 1⟩}):((1...(𝑁 − 1)) ∪ {𝑁})–1-1-onto→(((1 + 1)...𝑁) ∪ {1})))
230187, 229mpbird 246 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))):(1...𝑁)–1-1-onto→(1...𝑁))
231 f1oco 6072 . . . . . . . . . . . . . . . . . . . 20 (((2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) ∧ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))):(1...𝑁)–1-1-onto→(1...𝑁)) → ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))):(1...𝑁)–1-1-onto→(1...𝑁))
23245, 230, 231syl2anc 691 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))):(1...𝑁)–1-1-onto→(1...𝑁))
233 dff1o3 6056 . . . . . . . . . . . . . . . . . . . 20 (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))):(1...𝑁)–1-1-onto→(1...𝑁) ↔ (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))):(1...𝑁)–onto→(1...𝑁) ∧ Fun ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))))
234233simprbi 479 . . . . . . . . . . . . . . . . . . 19 (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))):(1...𝑁)–1-1-onto→(1...𝑁) → Fun ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))))
235232, 234syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑 → Fun ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))))
236 imain 5888 . . . . . . . . . . . . . . . . . 18 (Fun ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) → (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((1...𝑦) ∩ ((𝑦 + 1)...𝑁))) = ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) ∩ (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁))))
237235, 236syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((1...𝑦) ∩ ((𝑦 + 1)...𝑁))) = ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) ∩ (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁))))
23851nn0red 11229 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ∈ ℝ)
239238ltp1d 10833 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 < (𝑦 + 1))
240 fzdisj 12239 . . . . . . . . . . . . . . . . . . . 20 (𝑦 < (𝑦 + 1) → ((1...𝑦) ∩ ((𝑦 + 1)...𝑁)) = ∅)
241239, 240syl 17 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ (0...(𝑁 − 1)) → ((1...𝑦) ∩ ((𝑦 + 1)...𝑁)) = ∅)
242241imaeq2d 5385 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ (0...(𝑁 − 1)) → (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((1...𝑦) ∩ ((𝑦 + 1)...𝑁))) = (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ∅))
243 ima0 5400 . . . . . . . . . . . . . . . . . 18 (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ∅) = ∅
244242, 243syl6eq 2660 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ (0...(𝑁 − 1)) → (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((1...𝑦) ∩ ((𝑦 + 1)...𝑁))) = ∅)
245237, 244sylan9req 2665 . . . . . . . . . . . . . . . 16 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) ∩ (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁))) = ∅)
246 imassrn 5396 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ ((𝑦 + 1)...𝑁)) ⊆ ran (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))
247 f1of 6050 . . . . . . . . . . . . . . . . . . . . 21 ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))):(1...𝑁)–1-1-onto→(1...𝑁) → (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))):(1...𝑁)⟶(1...𝑁))
248 frn 5966 . . . . . . . . . . . . . . . . . . . . 21 ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))):(1...𝑁)⟶(1...𝑁) → ran (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) ⊆ (1...𝑁))
249230, 247, 2483syl 18 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ran (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) ⊆ (1...𝑁))
250246, 249syl5ss 3579 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ ((𝑦 + 1)...𝑁)) ⊆ (1...𝑁))
251250adantr 480 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ ((𝑦 + 1)...𝑁)) ⊆ (1...𝑁))
252 elfz1end 12242 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 ∈ ℕ ↔ 𝑁 ∈ (1...𝑁))
25370, 252sylib 207 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝑁 ∈ (1...𝑁))
254 eqid 2610 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) = (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))
255205, 254, 33fvmpt 6191 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ (1...𝑁) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))‘𝑁) = 1)
256253, 255syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))‘𝑁) = 1)
257256adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))‘𝑁) = 1)
258 f1ofn 6051 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))):(1...𝑁)–1-1-onto→(1...𝑁) → (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) Fn (1...𝑁))
259230, 258syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) Fn (1...𝑁))
260259adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) Fn (1...𝑁))
261 fzss1 12251 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑦 + 1) ∈ (ℤ‘1) → ((𝑦 + 1)...𝑁) ⊆ (1...𝑁))
26266, 261syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑦 + 1)...𝑁) ⊆ (1...𝑁))
263262adantl 481 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((𝑦 + 1)...𝑁) ⊆ (1...𝑁))
264 eluzfz2 12220 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ (ℤ‘(𝑦 + 1)) → 𝑁 ∈ ((𝑦 + 1)...𝑁))
26579, 264syl 17 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ ((𝑦 + 1)...𝑁))
266 fnfvima 6400 . . . . . . . . . . . . . . . . . . . 20 (((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) Fn (1...𝑁) ∧ ((𝑦 + 1)...𝑁) ⊆ (1...𝑁) ∧ 𝑁 ∈ ((𝑦 + 1)...𝑁)) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))‘𝑁) ∈ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ ((𝑦 + 1)...𝑁)))
267260, 263, 265, 266syl3anc 1318 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))‘𝑁) ∈ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ ((𝑦 + 1)...𝑁)))
268257, 267eqeltrrd 2689 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → 1 ∈ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ ((𝑦 + 1)...𝑁)))
269 fnfvima 6400 . . . . . . . . . . . . . . . . . 18 (((2nd ‘(1st𝑇)) Fn (1...𝑁) ∧ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ ((𝑦 + 1)...𝑁)) ⊆ (1...𝑁) ∧ 1 ∈ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ ((𝑦 + 1)...𝑁))) → ((2nd ‘(1st𝑇))‘1) ∈ ((2nd ‘(1st𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ ((𝑦 + 1)...𝑁))))
270105, 251, 268, 269syl3anc 1318 . . . . . . . . . . . . . . . . 17 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st𝑇))‘1) ∈ ((2nd ‘(1st𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ ((𝑦 + 1)...𝑁))))
271 imaco 5557 . . . . . . . . . . . . . . . . 17 (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) = ((2nd ‘(1st𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ ((𝑦 + 1)...𝑁)))
272270, 271syl6eleqr 2699 . . . . . . . . . . . . . . . 16 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st𝑇))‘1) ∈ (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)))
273 fnconstg 6006 . . . . . . . . . . . . . . . . . 18 (1 ∈ V → ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) Fn (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)))
27433, 273ax-mp 5 . . . . . . . . . . . . . . . . 17 ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) Fn (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦))
275 fnconstg 6006 . . . . . . . . . . . . . . . . . 18 (0 ∈ V → ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}) Fn (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)))
27636, 275ax-mp 5 . . . . . . . . . . . . . . . . 17 ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}) Fn (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁))
277 fvun2 6180 . . . . . . . . . . . . . . . . 17 ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) Fn (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) ∧ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}) Fn (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) ∧ (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) ∩ (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁))) = ∅ ∧ ((2nd ‘(1st𝑇))‘1) ∈ (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)))) → ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘1)) = (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})‘((2nd ‘(1st𝑇))‘1)))
278274, 276, 277mp3an12 1406 . . . . . . . . . . . . . . . 16 ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) ∩ (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁))) = ∅ ∧ ((2nd ‘(1st𝑇))‘1) ∈ (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁))) → ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘1)) = (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})‘((2nd ‘(1st𝑇))‘1)))
279245, 272, 278syl2anc 691 . . . . . . . . . . . . . . 15 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘1)) = (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})‘((2nd ‘(1st𝑇))‘1)))
28036fvconst2 6374 . . . . . . . . . . . . . . . 16 (((2nd ‘(1st𝑇))‘1) ∈ (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) → (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})‘((2nd ‘(1st𝑇))‘1)) = 0)
281272, 280syl 17 . . . . . . . . . . . . . . 15 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})‘((2nd ‘(1st𝑇))‘1)) = 0)
282279, 281eqtrd 2644 . . . . . . . . . . . . . 14 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘1)) = 0)
283282oveq2d 6565 . . . . . . . . . . . . 13 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (1 + ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘1))) = (1 + 0))
284102, 118, 2833eqtr4a 2670 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘1)) = (1 + ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘1))))
285 fveq2 6103 . . . . . . . . . . . . 13 (𝑛 = ((2nd ‘(1st𝑇))‘1) → (((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) = (((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘1)))
286 fveq2 6103 . . . . . . . . . . . . . 14 (𝑛 = ((2nd ‘(1st𝑇))‘1) → ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛) = ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘1)))
287286oveq2d 6565 . . . . . . . . . . . . 13 (𝑛 = ((2nd ‘(1st𝑇))‘1) → (1 + ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛)) = (1 + ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘1))))
288285, 287eqeq12d 2625 . . . . . . . . . . . 12 (𝑛 = ((2nd ‘(1st𝑇))‘1) → ((((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) = (1 + ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛)) ↔ (((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘1)) = (1 + ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘1)))))
289284, 288syl5ibrcom 236 . . . . . . . . . . 11 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (𝑛 = ((2nd ‘(1st𝑇))‘1) → (((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) = (1 + ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛))))
290289imp 444 . . . . . . . . . 10 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 = ((2nd ‘(1st𝑇))‘1)) → (((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) = (1 + ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛)))
291290adantlr 747 . . . . . . . . 9 ((((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑛 = ((2nd ‘(1st𝑇))‘1)) → (((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) = (1 + ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛)))
292 eldifsn 4260 . . . . . . . . . . . . . 14 (𝑛 ∈ ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘1)}) ↔ (𝑛 ∈ (1...𝑁) ∧ 𝑛 ≠ ((2nd ‘(1st𝑇))‘1)))
293 df-ne 2782 . . . . . . . . . . . . . . 15 (𝑛 ≠ ((2nd ‘(1st𝑇))‘1) ↔ ¬ 𝑛 = ((2nd ‘(1st𝑇))‘1))
294293anbi2i 726 . . . . . . . . . . . . . 14 ((𝑛 ∈ (1...𝑁) ∧ 𝑛 ≠ ((2nd ‘(1st𝑇))‘1)) ↔ (𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = ((2nd ‘(1st𝑇))‘1)))
295292, 294bitri 263 . . . . . . . . . . . . 13 (𝑛 ∈ ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘1)}) ↔ (𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = ((2nd ‘(1st𝑇))‘1)))
296 fnconstg 6006 . . . . . . . . . . . . . . . . . 18 (1 ∈ V → (((2nd ‘(1st𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) Fn ((2nd ‘(1st𝑇)) “ ((1 + 1)...(𝑦 + 1))))
29733, 296ax-mp 5 . . . . . . . . . . . . . . . . 17 (((2nd ‘(1st𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) Fn ((2nd ‘(1st𝑇)) “ ((1 + 1)...(𝑦 + 1)))
298297, 38pm3.2i 470 . . . . . . . . . . . . . . . 16 ((((2nd ‘(1st𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) Fn ((2nd ‘(1st𝑇)) “ ((1 + 1)...(𝑦 + 1))) ∧ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}) Fn ((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)))
299 imain 5888 . . . . . . . . . . . . . . . . . 18 (Fun (2nd ‘(1st𝑇)) → ((2nd ‘(1st𝑇)) “ (((1 + 1)...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁))) = (((2nd ‘(1st𝑇)) “ ((1 + 1)...(𝑦 + 1))) ∩ ((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁))))
30048, 299syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → ((2nd ‘(1st𝑇)) “ (((1 + 1)...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁))) = (((2nd ‘(1st𝑇)) “ ((1 + 1)...(𝑦 + 1))) ∩ ((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁))))
301 fzdisj 12239 . . . . . . . . . . . . . . . . . . . 20 ((𝑦 + 1) < ((𝑦 + 1) + 1) → (((1 + 1)...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁)) = ∅)
30255, 301syl 17 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ (0...(𝑁 − 1)) → (((1 + 1)...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁)) = ∅)
303302imaeq2d 5385 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ (0...(𝑁 − 1)) → ((2nd ‘(1st𝑇)) “ (((1 + 1)...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁))) = ((2nd ‘(1st𝑇)) “ ∅))
304303, 59syl6eq 2660 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ (0...(𝑁 − 1)) → ((2nd ‘(1st𝑇)) “ (((1 + 1)...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁))) = ∅)
305300, 304sylan9req 2665 . . . . . . . . . . . . . . . 16 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((2nd ‘(1st𝑇)) “ ((1 + 1)...(𝑦 + 1))) ∩ ((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) = ∅)
306 fnun 5911 . . . . . . . . . . . . . . . 16 ((((((2nd ‘(1st𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) Fn ((2nd ‘(1st𝑇)) “ ((1 + 1)...(𝑦 + 1))) ∧ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}) Fn ((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) ∧ (((2nd ‘(1st𝑇)) “ ((1 + 1)...(𝑦 + 1))) ∩ ((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) = ∅) → ((((2nd ‘(1st𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) Fn (((2nd ‘(1st𝑇)) “ ((1 + 1)...(𝑦 + 1))) ∪ ((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁))))
307298, 305, 306sylancr 694 . . . . . . . . . . . . . . 15 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd ‘(1st𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) Fn (((2nd ‘(1st𝑇)) “ ((1 + 1)...(𝑦 + 1))) ∪ ((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁))))
308 imaundi 5464 . . . . . . . . . . . . . . . . 17 ((2nd ‘(1st𝑇)) “ (((1 + 1)...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁))) = (((2nd ‘(1st𝑇)) “ ((1 + 1)...(𝑦 + 1))) ∪ ((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)))
309 fzpred 12259 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑁 ∈ (ℤ‘1) → (1...𝑁) = ({1} ∪ ((1 + 1)...𝑁)))
310191, 309syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (1...𝑁) = ({1} ∪ ((1 + 1)...𝑁)))
311 uncom 3719 . . . . . . . . . . . . . . . . . . . . . . . 24 ({1} ∪ ((1 + 1)...𝑁)) = (((1 + 1)...𝑁) ∪ {1})
312310, 311syl6eq 2660 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (1...𝑁) = (((1 + 1)...𝑁) ∪ {1}))
313312difeq1d 3689 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → ((1...𝑁) ∖ {1}) = ((((1 + 1)...𝑁) ∪ {1}) ∖ {1}))
314 difun2 4000 . . . . . . . . . . . . . . . . . . . . . . 23 ((((1 + 1)...𝑁) ∪ {1}) ∖ {1}) = (((1 + 1)...𝑁) ∖ {1})
315 disj3 3973 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((1 + 1)...𝑁) ∩ {1}) = ∅ ↔ ((1 + 1)...𝑁) = (((1 + 1)...𝑁) ∖ {1}))
316184, 315mpbi 219 . . . . . . . . . . . . . . . . . . . . . . 23 ((1 + 1)...𝑁) = (((1 + 1)...𝑁) ∖ {1})
317314, 316eqtr4i 2635 . . . . . . . . . . . . . . . . . . . . . 22 ((((1 + 1)...𝑁) ∪ {1}) ∖ {1}) = ((1 + 1)...𝑁)
318313, 317syl6eq 2660 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → ((1...𝑁) ∖ {1}) = ((1 + 1)...𝑁))
319318adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((1...𝑁) ∖ {1}) = ((1 + 1)...𝑁))
320 eluzp1p1 11589 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑦 + 1) ∈ (ℤ‘1) → ((𝑦 + 1) + 1) ∈ (ℤ‘(1 + 1)))
32166, 320syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑦 + 1) + 1) ∈ (ℤ‘(1 + 1)))
322321adantl 481 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((𝑦 + 1) + 1) ∈ (ℤ‘(1 + 1)))
323 fzsplit2 12237 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑦 + 1) + 1) ∈ (ℤ‘(1 + 1)) ∧ 𝑁 ∈ (ℤ‘(𝑦 + 1))) → ((1 + 1)...𝑁) = (((1 + 1)...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁)))
324322, 79, 323syl2anc 691 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((1 + 1)...𝑁) = (((1 + 1)...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁)))
325319, 324eqtrd 2644 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((1...𝑁) ∖ {1}) = (((1 + 1)...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁)))
326325imaeq2d 5385 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st𝑇)) “ ((1...𝑁) ∖ {1})) = ((2nd ‘(1st𝑇)) “ (((1 + 1)...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁))))
327 imadif 5887 . . . . . . . . . . . . . . . . . . . . 21 (Fun (2nd ‘(1st𝑇)) → ((2nd ‘(1st𝑇)) “ ((1...𝑁) ∖ {1})) = (((2nd ‘(1st𝑇)) “ (1...𝑁)) ∖ ((2nd ‘(1st𝑇)) “ {1})))
32848, 327syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((2nd ‘(1st𝑇)) “ ((1...𝑁) ∖ {1})) = (((2nd ‘(1st𝑇)) “ (1...𝑁)) ∖ ((2nd ‘(1st𝑇)) “ {1})))
329 eluzfz1 12219 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑁 ∈ (ℤ‘1) → 1 ∈ (1...𝑁))
330191, 329syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → 1 ∈ (1...𝑁))
331 fnsnfv 6168 . . . . . . . . . . . . . . . . . . . . . . 23 (((2nd ‘(1st𝑇)) Fn (1...𝑁) ∧ 1 ∈ (1...𝑁)) → {((2nd ‘(1st𝑇))‘1)} = ((2nd ‘(1st𝑇)) “ {1}))
332104, 330, 331syl2anc 691 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → {((2nd ‘(1st𝑇))‘1)} = ((2nd ‘(1st𝑇)) “ {1}))
333332eqcomd 2616 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → ((2nd ‘(1st𝑇)) “ {1}) = {((2nd ‘(1st𝑇))‘1)})
33485, 333difeq12d 3691 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (((2nd ‘(1st𝑇)) “ (1...𝑁)) ∖ ((2nd ‘(1st𝑇)) “ {1})) = ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘1)}))
335328, 334eqtrd 2644 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ((2nd ‘(1st𝑇)) “ ((1...𝑁) ∖ {1})) = ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘1)}))
336335adantr 480 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st𝑇)) “ ((1...𝑁) ∖ {1})) = ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘1)}))
337326, 336eqtr3d 2646 . . . . . . . . . . . . . . . . 17 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st𝑇)) “ (((1 + 1)...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁))) = ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘1)}))
338308, 337syl5eqr 2658 . . . . . . . . . . . . . . . 16 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((2nd ‘(1st𝑇)) “ ((1 + 1)...(𝑦 + 1))) ∪ ((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) = ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘1)}))
339338fneq2d 5896 . . . . . . . . . . . . . . 15 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd ‘(1st𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) Fn (((2nd ‘(1st𝑇)) “ ((1 + 1)...(𝑦 + 1))) ∪ ((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) ↔ ((((2nd ‘(1st𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) Fn ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘1)})))
340307, 339mpbid 221 . . . . . . . . . . . . . 14 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd ‘(1st𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) Fn ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘1)}))
341 incom 3767 . . . . . . . . . . . . . . . 16 (((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘1)}) ∩ {((2nd ‘(1st𝑇))‘1)}) = ({((2nd ‘(1st𝑇))‘1)} ∩ ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘1)}))
342 disjdif 3992 . . . . . . . . . . . . . . . 16 ({((2nd ‘(1st𝑇))‘1)} ∩ ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘1)})) = ∅
343341, 342eqtri 2632 . . . . . . . . . . . . . . 15 (((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘1)}) ∩ {((2nd ‘(1st𝑇))‘1)}) = ∅
344 fnconstg 6006 . . . . . . . . . . . . . . . . . 18 (1 ∈ V → ({((2nd ‘(1st𝑇))‘1)} × {1}) Fn {((2nd ‘(1st𝑇))‘1)})
34533, 344ax-mp 5 . . . . . . . . . . . . . . . . 17 ({((2nd ‘(1st𝑇))‘1)} × {1}) Fn {((2nd ‘(1st𝑇))‘1)}
346 fvun1 6179 . . . . . . . . . . . . . . . . 17 ((((((2nd ‘(1st𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) Fn ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘1)}) ∧ ({((2nd ‘(1st𝑇))‘1)} × {1}) Fn {((2nd ‘(1st𝑇))‘1)} ∧ ((((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘1)}) ∩ {((2nd ‘(1st𝑇))‘1)}) = ∅ ∧ 𝑛 ∈ ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘1)}))) → ((((((2nd ‘(1st𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2nd ‘(1st𝑇))‘1)} × {1}))‘𝑛) = (((((2nd ‘(1st𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛))
347345, 346mp3an2 1404 . . . . . . . . . . . . . . . 16 ((((((2nd ‘(1st𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) Fn ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘1)}) ∧ ((((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘1)}) ∩ {((2nd ‘(1st𝑇))‘1)}) = ∅ ∧ 𝑛 ∈ ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘1)}))) → ((((((2nd ‘(1st𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2nd ‘(1st𝑇))‘1)} × {1}))‘𝑛) = (((((2nd ‘(1st𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛))
348 fnconstg 6006 . . . . . . . . . . . . . . . . . 18 (0 ∈ V → ({((2nd ‘(1st𝑇))‘1)} × {0}) Fn {((2nd ‘(1st𝑇))‘1)})
34936, 348ax-mp 5 . . . . . . . . . . . . . . . . 17 ({((2nd ‘(1st𝑇))‘1)} × {0}) Fn {((2nd ‘(1st𝑇))‘1)}
350 fvun1 6179 . . . . . . . . . . . . . . . . 17 ((((((2nd ‘(1st𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) Fn ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘1)}) ∧ ({((2nd ‘(1st𝑇))‘1)} × {0}) Fn {((2nd ‘(1st𝑇))‘1)} ∧ ((((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘1)}) ∩ {((2nd ‘(1st𝑇))‘1)}) = ∅ ∧ 𝑛 ∈ ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘1)}))) → ((((((2nd ‘(1st𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2nd ‘(1st𝑇))‘1)} × {0}))‘𝑛) = (((((2nd ‘(1st𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛))
351349, 350mp3an2 1404 . . . . . . . . . . . . . . . 16 ((((((2nd ‘(1st𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) Fn ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘1)}) ∧ ((((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘1)}) ∩ {((2nd ‘(1st𝑇))‘1)}) = ∅ ∧ 𝑛 ∈ ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘1)}))) → ((((((2nd ‘(1st𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2nd ‘(1st𝑇))‘1)} × {0}))‘𝑛) = (((((2nd ‘(1st𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛))
352347, 351eqtr4d 2647 . . . . . . . . . . . . . . 15 ((((((2nd ‘(1st𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) Fn ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘1)}) ∧ ((((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘1)}) ∩ {((2nd ‘(1st𝑇))‘1)}) = ∅ ∧ 𝑛 ∈ ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘1)}))) → ((((((2nd ‘(1st𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2nd ‘(1st𝑇))‘1)} × {1}))‘𝑛) = ((((((2nd ‘(1st𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2nd ‘(1st𝑇))‘1)} × {0}))‘𝑛))
353343, 352mpanr1 715 . . . . . . . . . . . . . 14 ((((((2nd ‘(1st𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) Fn ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘1)}) ∧ 𝑛 ∈ ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘1)})) → ((((((2nd ‘(1st𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2nd ‘(1st𝑇))‘1)} × {1}))‘𝑛) = ((((((2nd ‘(1st𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2nd ‘(1st𝑇))‘1)} × {0}))‘𝑛))
354340, 353sylan 487 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘1)})) → ((((((2nd ‘(1st𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2nd ‘(1st𝑇))‘1)} × {1}))‘𝑛) = ((((((2nd ‘(1st𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2nd ‘(1st𝑇))‘1)} × {0}))‘𝑛))
355295, 354sylan2br 492 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = ((2nd ‘(1st𝑇))‘1))) → ((((((2nd ‘(1st𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2nd ‘(1st𝑇))‘1)} × {1}))‘𝑛) = ((((((2nd ‘(1st𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2nd ‘(1st𝑇))‘1)} × {0}))‘𝑛))
356355anassrs 678 . . . . . . . . . . 11 ((((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) ∧ ¬ 𝑛 = ((2nd ‘(1st𝑇))‘1)) → ((((((2nd ‘(1st𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2nd ‘(1st𝑇))‘1)} × {1}))‘𝑛) = ((((((2nd ‘(1st𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2nd ‘(1st𝑇))‘1)} × {0}))‘𝑛))
357 fzpred 12259 . . . . . . . . . . . . . . . . . . . . 21 ((𝑦 + 1) ∈ (ℤ‘1) → (1...(𝑦 + 1)) = ({1} ∪ ((1 + 1)...(𝑦 + 1))))
35866, 357syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ (0...(𝑁 − 1)) → (1...(𝑦 + 1)) = ({1} ∪ ((1 + 1)...(𝑦 + 1))))
359358imaeq2d 5385 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ (0...(𝑁 − 1)) → ((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) = ((2nd ‘(1st𝑇)) “ ({1} ∪ ((1 + 1)...(𝑦 + 1)))))
360359adantl 481 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) = ((2nd ‘(1st𝑇)) “ ({1} ∪ ((1 + 1)...(𝑦 + 1)))))
361332uneq1d 3728 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ({((2nd ‘(1st𝑇))‘1)} ∪ ((2nd ‘(1st𝑇)) “ ((1 + 1)...(𝑦 + 1)))) = (((2nd ‘(1st𝑇)) “ {1}) ∪ ((2nd ‘(1st𝑇)) “ ((1 + 1)...(𝑦 + 1)))))
362 uncom 3719 . . . . . . . . . . . . . . . . . . . 20 (((2nd ‘(1st𝑇)) “ ((1 + 1)...(𝑦 + 1))) ∪ {((2nd ‘(1st𝑇))‘1)}) = ({((2nd ‘(1st𝑇))‘1)} ∪ ((2nd ‘(1st𝑇)) “ ((1 + 1)...(𝑦 + 1))))
363 imaundi 5464 . . . . . . . . . . . . . . . . . . . 20 ((2nd ‘(1st𝑇)) “ ({1} ∪ ((1 + 1)...(𝑦 + 1)))) = (((2nd ‘(1st𝑇)) “ {1}) ∪ ((2nd ‘(1st𝑇)) “ ((1 + 1)...(𝑦 + 1))))
364361, 362, 3633eqtr4g 2669 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (((2nd ‘(1st𝑇)) “ ((1 + 1)...(𝑦 + 1))) ∪ {((2nd ‘(1st𝑇))‘1)}) = ((2nd ‘(1st𝑇)) “ ({1} ∪ ((1 + 1)...(𝑦 + 1)))))
365364adantr 480 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((2nd ‘(1st𝑇)) “ ((1 + 1)...(𝑦 + 1))) ∪ {((2nd ‘(1st𝑇))‘1)}) = ((2nd ‘(1st𝑇)) “ ({1} ∪ ((1 + 1)...(𝑦 + 1)))))
366360, 365eqtr4d 2647 . . . . . . . . . . . . . . . . 17 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) = (((2nd ‘(1st𝑇)) “ ((1 + 1)...(𝑦 + 1))) ∪ {((2nd ‘(1st𝑇))‘1)}))
367366xpeq1d 5062 . . . . . . . . . . . . . . . 16 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}) = ((((2nd ‘(1st𝑇)) “ ((1 + 1)...(𝑦 + 1))) ∪ {((2nd ‘(1st𝑇))‘1)}) × {1}))
368 xpundir 5095 . . . . . . . . . . . . . . . 16 ((((2nd ‘(1st𝑇)) “ ((1 + 1)...(𝑦 + 1))) ∪ {((2nd ‘(1st𝑇))‘1)}) × {1}) = ((((2nd ‘(1st𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ ({((2nd ‘(1st𝑇))‘1)} × {1}))
369367, 368syl6eq 2660 . . . . . . . . . . . . . . 15 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}) = ((((2nd ‘(1st𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ ({((2nd ‘(1st𝑇))‘1)} × {1})))
370369uneq1d 3728 . . . . . . . . . . . . . 14 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) = (((((2nd ‘(1st𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ ({((2nd ‘(1st𝑇))‘1)} × {1})) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})))
371 un23 3734 . . . . . . . . . . . . . 14 (((((2nd ‘(1st𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ ({((2nd ‘(1st𝑇))‘1)} × {1})) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) = (((((2nd ‘(1st𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2nd ‘(1st𝑇))‘1)} × {1}))
372370, 371syl6eq 2660 . . . . . . . . . . . . 13 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) = (((((2nd ‘(1st𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2nd ‘(1st𝑇))‘1)} × {1})))
373372fveq1d 6105 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) = ((((((2nd ‘(1st𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2nd ‘(1st𝑇))‘1)} × {1}))‘𝑛))
374373ad2antrr 758 . . . . . . . . . . 11 ((((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) ∧ ¬ 𝑛 = ((2nd ‘(1st𝑇))‘1)) → (((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) = ((((((2nd ‘(1st𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2nd ‘(1st𝑇))‘1)} × {1}))‘𝑛))
375 imaco 5557 . . . . . . . . . . . . . . . . 17 (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) = ((2nd ‘(1st𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ (1...𝑦)))
376 df-ima 5051 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ (1...𝑦)) = ran ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) ↾ (1...𝑦))
377 peano2uz 11617 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑁 − 1) ∈ (ℤ𝑦) → ((𝑁 − 1) + 1) ∈ (ℤ𝑦))
37875, 377syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈ (ℤ𝑦))
379378adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((𝑁 − 1) + 1) ∈ (ℤ𝑦))
38074, 379eqeltrrd 2689 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ (ℤ𝑦))
381 fzss2 12252 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑁 ∈ (ℤ𝑦) → (1...𝑦) ⊆ (1...𝑁))
382380, 381syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (1...𝑦) ⊆ (1...𝑁))
383382resmptd 5371 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) ↾ (1...𝑦)) = (𝑛 ∈ (1...𝑦) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))
384173adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ¬ 𝑁 ∈ (1...(𝑁 − 1)))
385 fzss2 12252 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑁 − 1) ∈ (ℤ𝑦) → (1...𝑦) ⊆ (1...(𝑁 − 1)))
38675, 385syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑦 ∈ (0...(𝑁 − 1)) → (1...𝑦) ⊆ (1...(𝑁 − 1)))
387386adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (1...𝑦) ⊆ (1...(𝑁 − 1)))
388387sseld 3567 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (𝑁 ∈ (1...𝑦) → 𝑁 ∈ (1...(𝑁 − 1))))
389384, 388mtod 188 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ¬ 𝑁 ∈ (1...𝑦))
390 eleq1 2676 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑛 = 𝑁 → (𝑛 ∈ (1...𝑦) ↔ 𝑁 ∈ (1...𝑦)))
391390notbid 307 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑛 = 𝑁 → (¬ 𝑛 ∈ (1...𝑦) ↔ ¬ 𝑁 ∈ (1...𝑦)))
392389, 391syl5ibrcom 236 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (𝑛 = 𝑁 → ¬ 𝑛 ∈ (1...𝑦)))
393392necon2ad 2797 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (𝑛 ∈ (1...𝑦) → 𝑛𝑁))
394393imp 444 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑦)) → 𝑛𝑁)
395394, 213syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑦)) → if(𝑛 = 𝑁, 1, (𝑛 + 1)) = (𝑛 + 1))
396395mpteq2dva 4672 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (𝑛 ∈ (1...𝑦) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) = (𝑛 ∈ (1...𝑦) ↦ (𝑛 + 1)))
397383, 396eqtrd 2644 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) ↾ (1...𝑦)) = (𝑛 ∈ (1...𝑦) ↦ (𝑛 + 1)))
398397rneqd 5274 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ran ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) ↾ (1...𝑦)) = ran (𝑛 ∈ (1...𝑦) ↦ (𝑛 + 1)))
399376, 398syl5eq 2656 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ (1...𝑦)) = ran (𝑛 ∈ (1...𝑦) ↦ (𝑛 + 1)))
400 vex 3176 . . . . . . . . . . . . . . . . . . . . . 22 𝑗 ∈ V
401 eqid 2610 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 ∈ (1...𝑦) ↦ (𝑛 + 1)) = (𝑛 ∈ (1...𝑦) ↦ (𝑛 + 1))
402401elrnmpt 5293 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 ∈ V → (𝑗 ∈ ran (𝑛 ∈ (1...𝑦) ↦ (𝑛 + 1)) ↔ ∃𝑛 ∈ (1...𝑦)𝑗 = (𝑛 + 1)))
403400, 402ax-mp 5 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 ∈ ran (𝑛 ∈ (1...𝑦) ↦ (𝑛 + 1)) ↔ ∃𝑛 ∈ (1...𝑦)𝑗 = (𝑛 + 1))
404 elfzelz 12213 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ∈ ℤ)
405404adantl 481 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → 𝑦 ∈ ℤ)
406 simpr 476 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑦 ∈ ℤ ∧ 𝑛 ∈ (1...𝑦)) → 𝑛 ∈ (1...𝑦))
407123jctl 562 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑦 ∈ ℤ → (1 ∈ ℤ ∧ 𝑦 ∈ ℤ))
408 elfzelz 12213 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑛 ∈ (1...𝑦) → 𝑛 ∈ ℤ)
409408, 123jctir 559 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑛 ∈ (1...𝑦) → (𝑛 ∈ ℤ ∧ 1 ∈ ℤ))
410 fzaddel 12246 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((1 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (𝑛 ∈ ℤ ∧ 1 ∈ ℤ)) → (𝑛 ∈ (1...𝑦) ↔ (𝑛 + 1) ∈ ((1 + 1)...(𝑦 + 1))))
411407, 409, 410syl2an 493 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑦 ∈ ℤ ∧ 𝑛 ∈ (1...𝑦)) → (𝑛 ∈ (1...𝑦) ↔ (𝑛 + 1) ∈ ((1 + 1)...(𝑦 + 1))))
412406, 411mpbid 221 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑦 ∈ ℤ ∧ 𝑛 ∈ (1...𝑦)) → (𝑛 + 1) ∈ ((1 + 1)...(𝑦 + 1)))
413 eleq1 2676 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 = (𝑛 + 1) → (𝑗 ∈ ((1 + 1)...(𝑦 + 1)) ↔ (𝑛 + 1) ∈ ((1 + 1)...(𝑦 + 1))))
414412, 413syl5ibrcom 236 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑦 ∈ ℤ ∧ 𝑛 ∈ (1...𝑦)) → (𝑗 = (𝑛 + 1) → 𝑗 ∈ ((1 + 1)...(𝑦 + 1))))
415414rexlimdva 3013 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 ∈ ℤ → (∃𝑛 ∈ (1...𝑦)𝑗 = (𝑛 + 1) → 𝑗 ∈ ((1 + 1)...(𝑦 + 1))))
416 elfzelz 12213 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑗 ∈ ((1 + 1)...(𝑦 + 1)) → 𝑗 ∈ ℤ)
417416zcnd 11359 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑗 ∈ ((1 + 1)...(𝑦 + 1)) → 𝑗 ∈ ℂ)
418 npcan1 10334 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑗 ∈ ℂ → ((𝑗 − 1) + 1) = 𝑗)
419417, 418syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑗 ∈ ((1 + 1)...(𝑦 + 1)) → ((𝑗 − 1) + 1) = 𝑗)
420419eleq1d 2672 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑗 ∈ ((1 + 1)...(𝑦 + 1)) → (((𝑗 − 1) + 1) ∈ ((1 + 1)...(𝑦 + 1)) ↔ 𝑗 ∈ ((1 + 1)...(𝑦 + 1))))
421420ibir 256 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ ((1 + 1)...(𝑦 + 1)) → ((𝑗 − 1) + 1) ∈ ((1 + 1)...(𝑦 + 1)))
422421adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑦 ∈ ℤ ∧ 𝑗 ∈ ((1 + 1)...(𝑦 + 1))) → ((𝑗 − 1) + 1) ∈ ((1 + 1)...(𝑦 + 1)))
423 peano2zm 11297 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑗 ∈ ℤ → (𝑗 − 1) ∈ ℤ)
424416, 423syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑗 ∈ ((1 + 1)...(𝑦 + 1)) → (𝑗 − 1) ∈ ℤ)
425424, 123jctir 559 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ ((1 + 1)...(𝑦 + 1)) → ((𝑗 − 1) ∈ ℤ ∧ 1 ∈ ℤ))
426 fzaddel 12246 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((1 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ ((𝑗 − 1) ∈ ℤ ∧ 1 ∈ ℤ)) → ((𝑗 − 1) ∈ (1...𝑦) ↔ ((𝑗 − 1) + 1) ∈ ((1 + 1)...(𝑦 + 1))))
427407, 425, 426syl2an 493 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑦 ∈ ℤ ∧ 𝑗 ∈ ((1 + 1)...(𝑦 + 1))) → ((𝑗 − 1) ∈ (1...𝑦) ↔ ((𝑗 − 1) + 1) ∈ ((1 + 1)...(𝑦 + 1))))
428422, 427mpbird 246 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑦 ∈ ℤ ∧ 𝑗 ∈ ((1 + 1)...(𝑦 + 1))) → (𝑗 − 1) ∈ (1...𝑦))
429417adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑦 ∈ ℤ ∧ 𝑗 ∈ ((1 + 1)...(𝑦 + 1))) → 𝑗 ∈ ℂ)
430418eqcomd 2616 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑗 ∈ ℂ → 𝑗 = ((𝑗 − 1) + 1))
431429, 430syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑦 ∈ ℤ ∧ 𝑗 ∈ ((1 + 1)...(𝑦 + 1))) → 𝑗 = ((𝑗 − 1) + 1))
432 oveq1 6556 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑛 = (𝑗 − 1) → (𝑛 + 1) = ((𝑗 − 1) + 1))
433432eqeq2d 2620 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑛 = (𝑗 − 1) → (𝑗 = (𝑛 + 1) ↔ 𝑗 = ((𝑗 − 1) + 1)))
434433rspcev 3282 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑗 − 1) ∈ (1...𝑦) ∧ 𝑗 = ((𝑗 − 1) + 1)) → ∃𝑛 ∈ (1...𝑦)𝑗 = (𝑛 + 1))
435428, 431, 434syl2anc 691 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑦 ∈ ℤ ∧ 𝑗 ∈ ((1 + 1)...(𝑦 + 1))) → ∃𝑛 ∈ (1...𝑦)𝑗 = (𝑛 + 1))
436435ex 449 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 ∈ ℤ → (𝑗 ∈ ((1 + 1)...(𝑦 + 1)) → ∃𝑛 ∈ (1...𝑦)𝑗 = (𝑛 + 1)))
437415, 436impbid 201 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 ∈ ℤ → (∃𝑛 ∈ (1...𝑦)𝑗 = (𝑛 + 1) ↔ 𝑗 ∈ ((1 + 1)...(𝑦 + 1))))
438405, 437syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (∃𝑛 ∈ (1...𝑦)𝑗 = (𝑛 + 1) ↔ 𝑗 ∈ ((1 + 1)...(𝑦 + 1))))
439403, 438syl5bb 271 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (𝑗 ∈ ran (𝑛 ∈ (1...𝑦) ↦ (𝑛 + 1)) ↔ 𝑗 ∈ ((1 + 1)...(𝑦 + 1))))
440439eqrdv 2608 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ran (𝑛 ∈ (1...𝑦) ↦ (𝑛 + 1)) = ((1 + 1)...(𝑦 + 1)))
441399, 440eqtrd 2644 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ (1...𝑦)) = ((1 + 1)...(𝑦 + 1)))
442441imaeq2d 5385 . . . . . . . . . . . . . . . . 17 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ (1...𝑦))) = ((2nd ‘(1st𝑇)) “ ((1 + 1)...(𝑦 + 1))))
443375, 442syl5eq 2656 . . . . . . . . . . . . . . . 16 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) = ((2nd ‘(1st𝑇)) “ ((1 + 1)...(𝑦 + 1))))
444443xpeq1d 5062 . . . . . . . . . . . . . . 15 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) = (((2nd ‘(1st𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}))
445 imaundi 5464 . . . . . . . . . . . . . . . . . . 19 (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ({𝑁} ∪ ((𝑦 + 1)...(𝑁 − 1)))) = ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ {𝑁}) ∪ (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...(𝑁 − 1))))
446 imaco 5557 . . . . . . . . . . . . . . . . . . . 20 (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ {𝑁}) = ((2nd ‘(1st𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ {𝑁}))
447 imaco 5557 . . . . . . . . . . . . . . . . . . . 20 (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...(𝑁 − 1))) = ((2nd ‘(1st𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ ((𝑦 + 1)...(𝑁 − 1))))
448446, 447uneq12i 3727 . . . . . . . . . . . . . . . . . . 19 ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ {𝑁}) ∪ (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...(𝑁 − 1)))) = (((2nd ‘(1st𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ {𝑁})) ∪ ((2nd ‘(1st𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ ((𝑦 + 1)...(𝑁 − 1)))))
449445, 448eqtri 2632 . . . . . . . . . . . . . . . . . 18 (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ({𝑁} ∪ ((𝑦 + 1)...(𝑁 − 1)))) = (((2nd ‘(1st𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ {𝑁})) ∪ ((2nd ‘(1st𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ ((𝑦 + 1)...(𝑁 − 1)))))
450196adantr 480 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ (ℤ‘(𝑁 − 1)))
451 fzsplit2 12237 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑁 − 1) + 1) ∈ (ℤ‘(𝑦 + 1)) ∧ 𝑁 ∈ (ℤ‘(𝑁 − 1))) → ((𝑦 + 1)...𝑁) = (((𝑦 + 1)...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)))
45278, 450, 451syl2anc 691 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((𝑦 + 1)...𝑁) = (((𝑦 + 1)...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)))
453202uneq2d 3729 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (((𝑦 + 1)...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)) = (((𝑦 + 1)...(𝑁 − 1)) ∪ {𝑁}))
454453adantr 480 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((𝑦 + 1)...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)) = (((𝑦 + 1)...(𝑁 − 1)) ∪ {𝑁}))
455452, 454eqtrd 2644 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((𝑦 + 1)...𝑁) = (((𝑦 + 1)...(𝑁 − 1)) ∪ {𝑁}))
456 uncom 3719 . . . . . . . . . . . . . . . . . . . 20 (((𝑦 + 1)...(𝑁 − 1)) ∪ {𝑁}) = ({𝑁} ∪ ((𝑦 + 1)...(𝑁 − 1)))
457455, 456syl6eq 2660 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((𝑦 + 1)...𝑁) = ({𝑁} ∪ ((𝑦 + 1)...(𝑁 − 1))))
458457imaeq2d 5385 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) = (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ({𝑁} ∪ ((𝑦 + 1)...(𝑁 − 1)))))
459256sneqd 4137 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → {((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))‘𝑁)} = {1})
460 fnsnfv 6168 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) Fn (1...𝑁) ∧ 𝑁 ∈ (1...𝑁)) → {((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))‘𝑁)} = ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ {𝑁}))
461259, 253, 460syl2anc 691 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → {((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))‘𝑁)} = ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ {𝑁}))
462459, 461eqtr3d 2646 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → {1} = ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ {𝑁}))
463462imaeq2d 5385 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → ((2nd ‘(1st𝑇)) “ {1}) = ((2nd ‘(1st𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ {𝑁})))
464332, 463eqtrd 2644 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → {((2nd ‘(1st𝑇))‘1)} = ((2nd ‘(1st𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ {𝑁})))
465464adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → {((2nd ‘(1st𝑇))‘1)} = ((2nd ‘(1st𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ {𝑁})))
466 df-ima 5051 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ ((𝑦 + 1)...(𝑁 − 1))) = ran ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) ↾ ((𝑦 + 1)...(𝑁 − 1)))
467 fzss1 12251 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑦 + 1) ∈ (ℤ‘1) → ((𝑦 + 1)...(𝑁 − 1)) ⊆ (1...(𝑁 − 1)))
46866, 467syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑦 + 1)...(𝑁 − 1)) ⊆ (1...(𝑁 − 1)))
469 fzss2 12252 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑁 ∈ (ℤ‘(𝑁 − 1)) → (1...(𝑁 − 1)) ⊆ (1...𝑁))
470196, 469syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → (1...(𝑁 − 1)) ⊆ (1...𝑁))
471468, 470sylan9ssr 3582 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((𝑦 + 1)...(𝑁 − 1)) ⊆ (1...𝑁))
472471resmptd 5371 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) ↾ ((𝑦 + 1)...(𝑁 − 1))) = (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))
473 elfzle2 12216 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑁 ∈ ((𝑦 + 1)...(𝑁 − 1)) → 𝑁 ≤ (𝑁 − 1))
474171, 473nsyl 134 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝜑 → ¬ 𝑁 ∈ ((𝑦 + 1)...(𝑁 − 1)))
475 eleq1 2676 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑛 = 𝑁 → (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↔ 𝑁 ∈ ((𝑦 + 1)...(𝑁 − 1))))
476475notbid 307 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑛 = 𝑁 → (¬ 𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↔ ¬ 𝑁 ∈ ((𝑦 + 1)...(𝑁 − 1))))
477474, 476syl5ibrcom 236 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝜑 → (𝑛 = 𝑁 → ¬ 𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1))))
478477con2d 128 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜑 → (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) → ¬ 𝑛 = 𝑁))
479478imp 444 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1))) → ¬ 𝑛 = 𝑁)
480479iffalsed 4047 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1))) → if(𝑛 = 𝑁, 1, (𝑛 + 1)) = (𝑛 + 1))
481480mpteq2dva 4672 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) = (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↦ (𝑛 + 1)))
482481adantr 480 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) = (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↦ (𝑛 + 1)))
483472, 482eqtrd 2644 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) ↾ ((𝑦 + 1)...(𝑁 − 1))) = (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↦ (𝑛 + 1)))
484483rneqd 5274 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ran ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) ↾ ((𝑦 + 1)...(𝑁 − 1))) = ran (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↦ (𝑛 + 1)))
485466, 484syl5eq 2656 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ ((𝑦 + 1)...(𝑁 − 1))) = ran (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↦ (𝑛 + 1)))
486 elfzelz 12213 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)) → 𝑗 ∈ ℤ)
487486zcnd 11359 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)) → 𝑗 ∈ ℂ)
488487, 418syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)) → ((𝑗 − 1) + 1) = 𝑗)
489488eleq1d 2672 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)) → (((𝑗 − 1) + 1) ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)) ↔ 𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1))))
490489ibir 256 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)) → ((𝑗 − 1) + 1) ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)))
491490adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1))) → ((𝑗 − 1) + 1) ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)))
49253nnzd 11357 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈ ℤ)
493122, 492anim12ci 589 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((𝑦 + 1) ∈ ℤ ∧ (𝑁 − 1) ∈ ℤ))
494486, 423syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)) → (𝑗 − 1) ∈ ℤ)
495494, 123jctir 559 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)) → ((𝑗 − 1) ∈ ℤ ∧ 1 ∈ ℤ))
496 fzaddel 12246 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝑦 + 1) ∈ ℤ ∧ (𝑁 − 1) ∈ ℤ) ∧ ((𝑗 − 1) ∈ ℤ ∧ 1 ∈ ℤ)) → ((𝑗 − 1) ∈ ((𝑦 + 1)...(𝑁 − 1)) ↔ ((𝑗 − 1) + 1) ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1))))
497493, 495, 496syl2an 493 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1))) → ((𝑗 − 1) ∈ ((𝑦 + 1)...(𝑁 − 1)) ↔ ((𝑗 − 1) + 1) ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1))))
498491, 497mpbird 246 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1))) → (𝑗 − 1) ∈ ((𝑦 + 1)...(𝑁 − 1)))
499487, 430syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)) → 𝑗 = ((𝑗 − 1) + 1))
500499adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1))) → 𝑗 = ((𝑗 − 1) + 1))
501433rspcev 3282 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝑗 − 1) ∈ ((𝑦 + 1)...(𝑁 − 1)) ∧ 𝑗 = ((𝑗 − 1) + 1)) → ∃𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1))𝑗 = (𝑛 + 1))
502498, 500, 501syl2anc 691 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1))) → ∃𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1))𝑗 = (𝑛 + 1))
503502ex 449 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)) → ∃𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1))𝑗 = (𝑛 + 1)))
504 simpr 476 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1))) → 𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)))
505 elfzelz 12213 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) → 𝑛 ∈ ℤ)
506505, 123jctir 559 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) → (𝑛 ∈ ℤ ∧ 1 ∈ ℤ))
507 fzaddel 12246 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝑦 + 1) ∈ ℤ ∧ (𝑁 − 1) ∈ ℤ) ∧ (𝑛 ∈ ℤ ∧ 1 ∈ ℤ)) → (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↔ (𝑛 + 1) ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1))))
508493, 506, 507syl2an 493 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1))) → (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↔ (𝑛 + 1) ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1))))
509504, 508mpbid 221 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1))) → (𝑛 + 1) ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)))
510 eleq1 2676 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑗 = (𝑛 + 1) → (𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)) ↔ (𝑛 + 1) ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1))))
511509, 510syl5ibrcom 236 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1))) → (𝑗 = (𝑛 + 1) → 𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1))))
512511rexlimdva 3013 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (∃𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1))𝑗 = (𝑛 + 1) → 𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1))))
513503, 512impbid 201 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)) ↔ ∃𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1))𝑗 = (𝑛 + 1)))
514 eqid 2610 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↦ (𝑛 + 1)) = (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↦ (𝑛 + 1))
515514elrnmpt 5293 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 ∈ V → (𝑗 ∈ ran (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↦ (𝑛 + 1)) ↔ ∃𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1))𝑗 = (𝑛 + 1)))
516400, 515ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑗 ∈ ran (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↦ (𝑛 + 1)) ↔ ∃𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1))𝑗 = (𝑛 + 1))
517513, 516syl6bbr 277 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (𝑗 ∈ (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)) ↔ 𝑗 ∈ ran (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↦ (𝑛 + 1))))
518517eqrdv 2608 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)) = ran (𝑛 ∈ ((𝑦 + 1)...(𝑁 − 1)) ↦ (𝑛 + 1)))
51973oveq2d 6565 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)) = (((𝑦 + 1) + 1)...𝑁))
520519adantr 480 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((𝑦 + 1) + 1)...((𝑁 − 1) + 1)) = (((𝑦 + 1) + 1)...𝑁))
521485, 518, 5203eqtr2rd 2651 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((𝑦 + 1) + 1)...𝑁) = ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ ((𝑦 + 1)...(𝑁 − 1))))
522521imaeq2d 5385 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) = ((2nd ‘(1st𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ ((𝑦 + 1)...(𝑁 − 1)))))
523465, 522uneq12d 3730 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ({((2nd ‘(1st𝑇))‘1)} ∪ ((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) = (((2nd ‘(1st𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ {𝑁})) ∪ ((2nd ‘(1st𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))) “ ((𝑦 + 1)...(𝑁 − 1))))))
524449, 458, 5233eqtr4a 2670 . . . . . . . . . . . . . . . . 17 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) = ({((2nd ‘(1st𝑇))‘1)} ∪ ((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁))))
525524xpeq1d 5062 . . . . . . . . . . . . . . . 16 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}) = (({((2nd ‘(1st𝑇))‘1)} ∪ ((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) × {0}))
526 xpundir 5095 . . . . . . . . . . . . . . . 16 (({((2nd ‘(1st𝑇))‘1)} ∪ ((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁))) × {0}) = (({((2nd ‘(1st𝑇))‘1)} × {0}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))
527525, 526syl6eq 2660 . . . . . . . . . . . . . . 15 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}) = (({((2nd ‘(1st𝑇))‘1)} × {0}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})))
528444, 527uneq12d 3730 . . . . . . . . . . . . . 14 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})) = ((((2nd ‘(1st𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (({((2nd ‘(1st𝑇))‘1)} × {0}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))))
529 unass 3732 . . . . . . . . . . . . . . 15 (((((2nd ‘(1st𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ ({((2nd ‘(1st𝑇))‘1)} × {0})) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) = ((((2nd ‘(1st𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (({((2nd ‘(1st𝑇))‘1)} × {0}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})))
530 un23 3734 . . . . . . . . . . . . . . 15 (((((2nd ‘(1st𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ ({((2nd ‘(1st𝑇))‘1)} × {0})) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) = (((((2nd ‘(1st𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2nd ‘(1st𝑇))‘1)} × {0}))
531529, 530eqtr3i 2634 . . . . . . . . . . . . . 14 ((((2nd ‘(1st𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (({((2nd ‘(1st𝑇))‘1)} × {0}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))) = (((((2nd ‘(1st𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2nd ‘(1st𝑇))‘1)} × {0}))
532528, 531syl6eq 2660 . . . . . . . . . . . . 13 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})) = (((((2nd ‘(1st𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2nd ‘(1st𝑇))‘1)} × {0})))
533532fveq1d 6105 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛) = ((((((2nd ‘(1st𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2nd ‘(1st𝑇))‘1)} × {0}))‘𝑛))
534533ad2antrr 758 . . . . . . . . . . 11 ((((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) ∧ ¬ 𝑛 = ((2nd ‘(1st𝑇))‘1)) → ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛) = ((((((2nd ‘(1st𝑇)) “ ((1 + 1)...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) ∪ ({((2nd ‘(1st𝑇))‘1)} × {0}))‘𝑛))
535356, 374, 5343eqtr4d 2654 . . . . . . . . . 10 ((((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) ∧ ¬ 𝑛 = ((2nd ‘(1st𝑇))‘1)) → (((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) = ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛))
536 snssi 4280 . . . . . . . . . . . . . . 15 (1 ∈ ℂ → {1} ⊆ ℂ)
537143, 536ax-mp 5 . . . . . . . . . . . . . 14 {1} ⊆ ℂ
538 0cn 9911 . . . . . . . . . . . . . . 15 0 ∈ ℂ
539 snssi 4280 . . . . . . . . . . . . . . 15 (0 ∈ ℂ → {0} ⊆ ℂ)
540538, 539ax-mp 5 . . . . . . . . . . . . . 14 {0} ⊆ ℂ
541537, 540unssi 3750 . . . . . . . . . . . . 13 ({1} ∪ {0}) ⊆ ℂ
54233fconst 6004 . . . . . . . . . . . . . . . . 17 ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}):(((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦))⟶{1}
54336fconst 6004 . . . . . . . . . . . . . . . . 17 ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}):(((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁))⟶{0}
544542, 543pm3.2i 470 . . . . . . . . . . . . . . . 16 (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}):(((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦))⟶{1} ∧ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}):(((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁))⟶{0})
545 fun 5979 . . . . . . . . . . . . . . . 16 (((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}):(((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦))⟶{1} ∧ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}):(((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁))⟶{0}) ∧ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) ∩ (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁))) = ∅) → (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})):((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) ∪ (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)))⟶({1} ∪ {0}))
546544, 245, 545sylancr 694 . . . . . . . . . . . . . . 15 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})):((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) ∪ (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)))⟶({1} ∪ {0}))
547 imaundi 5464 . . . . . . . . . . . . . . . . 17 (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((1...𝑦) ∪ ((𝑦 + 1)...𝑁))) = ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) ∪ (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)))
54866adantl 481 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (𝑦 + 1) ∈ (ℤ‘1))
549 fzsplit2 12237 . . . . . . . . . . . . . . . . . . . 20 (((𝑦 + 1) ∈ (ℤ‘1) ∧ 𝑁 ∈ (ℤ𝑦)) → (1...𝑁) = ((1...𝑦) ∪ ((𝑦 + 1)...𝑁)))
550548, 380, 549syl2anc 691 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (1...𝑁) = ((1...𝑦) ∪ ((𝑦 + 1)...𝑁)))
551550imaeq2d 5385 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑁)) = (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((1...𝑦) ∪ ((𝑦 + 1)...𝑁))))
552 f1ofo 6057 . . . . . . . . . . . . . . . . . . . 20 (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))):(1...𝑁)–1-1-onto→(1...𝑁) → ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))):(1...𝑁)–onto→(1...𝑁))
553 foima 6033 . . . . . . . . . . . . . . . . . . . 20 (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))):(1...𝑁)–onto→(1...𝑁) → (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑁)) = (1...𝑁))
554232, 552, 5533syl 18 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑁)) = (1...𝑁))
555554adantr 480 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑁)) = (1...𝑁))
556551, 555eqtr3d 2646 . . . . . . . . . . . . . . . . 17 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((1...𝑦) ∪ ((𝑦 + 1)...𝑁))) = (1...𝑁))
557547, 556syl5eqr 2658 . . . . . . . . . . . . . . . 16 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) ∪ (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁))) = (1...𝑁))
558557feq2d 5944 . . . . . . . . . . . . . . 15 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})):((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) ∪ (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)))⟶({1} ∪ {0}) ↔ (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})):(1...𝑁)⟶({1} ∪ {0})))
559546, 558mpbid 221 . . . . . . . . . . . . . 14 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})):(1...𝑁)⟶({1} ∪ {0}))
560559ffvelrnda 6267 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛) ∈ ({1} ∪ {0}))
561541, 560sseldi 3566 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛) ∈ ℂ)
562561addid2d 10116 . . . . . . . . . . 11 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → (0 + ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛)) = ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛))
563562adantr 480 . . . . . . . . . 10 ((((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) ∧ ¬ 𝑛 = ((2nd ‘(1st𝑇))‘1)) → (0 + ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛)) = ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛))
564535, 563eqtr4d 2647 . . . . . . . . 9 ((((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) ∧ ¬ 𝑛 = ((2nd ‘(1st𝑇))‘1)) → (((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) = (0 + ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛)))
56598, 100, 291, 564ifbothda 4073 . . . . . . . 8 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → (((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) = (if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0) + ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛)))
566565oveq2d 6565 . . . . . . 7 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → (((1st ‘(1st𝑇))‘𝑛) + (((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛)) = (((1st ‘(1st𝑇))‘𝑛) + (if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0) + ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛))))
567 elmapi 7765 . . . . . . . . . . . . 13 ((1st ‘(1st𝑇)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)) → (1st ‘(1st𝑇)):(1...𝑁)⟶(0..^𝐾))
56829, 567syl 17 . . . . . . . . . . . 12 (𝜑 → (1st ‘(1st𝑇)):(1...𝑁)⟶(0..^𝐾))
569568ffvelrnda 6267 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (1...𝑁)) → ((1st ‘(1st𝑇))‘𝑛) ∈ (0..^𝐾))
570 elfzonn0 12380 . . . . . . . . . . 11 (((1st ‘(1st𝑇))‘𝑛) ∈ (0..^𝐾) → ((1st ‘(1st𝑇))‘𝑛) ∈ ℕ0)
571569, 570syl 17 . . . . . . . . . 10 ((𝜑𝑛 ∈ (1...𝑁)) → ((1st ‘(1st𝑇))‘𝑛) ∈ ℕ0)
572571nn0cnd 11230 . . . . . . . . 9 ((𝜑𝑛 ∈ (1...𝑁)) → ((1st ‘(1st𝑇))‘𝑛) ∈ ℂ)
573572adantlr 747 . . . . . . . 8 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → ((1st ‘(1st𝑇))‘𝑛) ∈ ℂ)
574143, 538keepel 4105 . . . . . . . . 9 if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0) ∈ ℂ
575574a1i 11 . . . . . . . 8 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0) ∈ ℂ)
576573, 575, 561addassd 9941 . . . . . . 7 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → ((((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0)) + ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛)) = (((1st ‘(1st𝑇))‘𝑛) + (if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0) + ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛))))
577566, 576eqtr4d 2647 . . . . . 6 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → (((1st ‘(1st𝑇))‘𝑛) + (((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛)) = ((((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0)) + ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛)))
578577mpteq2dva 4672 . . . . 5 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + (((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛))) = (𝑛 ∈ (1...𝑁) ↦ ((((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0)) + ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛))))
57996, 578eqtrd 2644 . . . 4 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))) = (𝑛 ∈ (1...𝑁) ↦ ((((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0)) + ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛))))
580 poimirlem18.4 . . . . . . . . . 10 (𝜑 → (2nd𝑇) = 0)
581580adantr 480 . . . . . . . . 9 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (2nd𝑇) = 0)
582 elfzle1 12215 . . . . . . . . . 10 (𝑦 ∈ (0...(𝑁 − 1)) → 0 ≤ 𝑦)
583582adantl 481 . . . . . . . . 9 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → 0 ≤ 𝑦)
584581, 583eqbrtrd 4605 . . . . . . . 8 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (2nd𝑇) ≤ 𝑦)
585 0re 9919 . . . . . . . . . 10 0 ∈ ℝ
586580, 585syl6eqel 2696 . . . . . . . . 9 (𝜑 → (2nd𝑇) ∈ ℝ)
587 lenlt 9995 . . . . . . . . 9 (((2nd𝑇) ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((2nd𝑇) ≤ 𝑦 ↔ ¬ 𝑦 < (2nd𝑇)))
588586, 238, 587syl2an 493 . . . . . . . 8 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((2nd𝑇) ≤ 𝑦 ↔ ¬ 𝑦 < (2nd𝑇)))
589584, 588mpbid 221 . . . . . . 7 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ¬ 𝑦 < (2nd𝑇))
590589iffalsed 4047 . . . . . 6 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) = (𝑦 + 1))
591590csbeq1d 3506 . . . . 5 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = (𝑦 + 1) / 𝑗((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))
592 ovex 6577 . . . . . 6 (𝑦 + 1) ∈ V
593 oveq2 6557 . . . . . . . . . 10 (𝑗 = (𝑦 + 1) → (1...𝑗) = (1...(𝑦 + 1)))
594593imaeq2d 5385 . . . . . . . . 9 (𝑗 = (𝑦 + 1) → ((2nd ‘(1st𝑇)) “ (1...𝑗)) = ((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))))
595594xpeq1d 5062 . . . . . . . 8 (𝑗 = (𝑦 + 1) → (((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}))
596 oveq1 6556 . . . . . . . . . . 11 (𝑗 = (𝑦 + 1) → (𝑗 + 1) = ((𝑦 + 1) + 1))
597596oveq1d 6564 . . . . . . . . . 10 (𝑗 = (𝑦 + 1) → ((𝑗 + 1)...𝑁) = (((𝑦 + 1) + 1)...𝑁))
598597imaeq2d 5385 . . . . . . . . 9 (𝑗 = (𝑦 + 1) → ((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)))
599598xpeq1d 5062 . . . . . . . 8 (𝑗 = (𝑦 + 1) → (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))
600595, 599uneq12d 3730 . . . . . . 7 (𝑗 = (𝑦 + 1) → ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})))
601600oveq2d 6565 . . . . . 6 (𝑗 = (𝑦 + 1) → ((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))))
602592, 601csbie 3525 . . . . 5 (𝑦 + 1) / 𝑗((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})))
603591, 602syl6eq 2660 . . . 4 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))))
604 ovex 6577 . . . . . 6 (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0)) ∈ V
605604a1i 11 . . . . 5 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0)) ∈ V)
606 fvex 6113 . . . . . 6 ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛) ∈ V
607606a1i 11 . . . . 5 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛) ∈ V)
608 eqidd 2611 . . . . 5 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))) = (𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))))
609 ffn 5958 . . . . . . 7 ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})):(1...𝑁)⟶({1} ∪ {0}) → (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})) Fn (1...𝑁))
610559, 609syl 17 . . . . . 6 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})) Fn (1...𝑁))
611 nfcv 2751 . . . . . . . . . . 11 𝑛(2nd ‘(1st𝑇))
612 nfmpt1 4675 . . . . . . . . . . 11 𝑛(𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))
613611, 612nfco 5209 . . . . . . . . . 10 𝑛((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1))))
614 nfcv 2751 . . . . . . . . . 10 𝑛(1...𝑦)
615613, 614nfima 5393 . . . . . . . . 9 𝑛(((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦))
616 nfcv 2751 . . . . . . . . 9 𝑛{1}
617615, 616nfxp 5066 . . . . . . . 8 𝑛((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1})
618 nfcv 2751 . . . . . . . . . 10 𝑛((𝑦 + 1)...𝑁)
619613, 618nfima 5393 . . . . . . . . 9 𝑛(((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁))
620 nfcv 2751 . . . . . . . . 9 𝑛{0}
621619, 620nfxp 5066 . . . . . . . 8 𝑛((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})
622617, 621nfun 3731 . . . . . . 7 𝑛(((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))
623622dffn5f 6162 . . . . . 6 ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})) Fn (1...𝑁) ↔ (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})) = (𝑛 ∈ (1...𝑁) ↦ ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛)))
624610, 623sylib 207 . . . . 5 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})) = (𝑛 ∈ (1...𝑁) ↦ ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛)))
62592, 605, 607, 608, 624offval2 6812 . . . 4 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))) ∘𝑓 + (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))) = (𝑛 ∈ (1...𝑁) ↦ ((((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0)) + ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛))))
626579, 603, 6253eqtr4rd 2655 . . 3 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))) ∘𝑓 + (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0}))) = if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))
627626mpteq2dva 4672 . 2 (𝜑 → (𝑦 ∈ (0...(𝑁 − 1)) ↦ ((𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))) ∘𝑓 + (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})))) = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))))
62822, 627eqtr4d 2647 1 (𝜑𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ((𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + if(𝑛 = ((2nd ‘(1st𝑇))‘1), 1, 0))) ∘𝑓 + (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 𝑁, 1, (𝑛 + 1)))) “ ((𝑦 + 1)...𝑁)) × {0})))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  {cab 2596  wne 2780  wral 2896  wrex 2897  ∃!wreu 2898  {crab 2900  Vcvv 3173  csb 3499  cdif 3537  cun 3538  cin 3539  wss 3540  c0 3874  ifcif 4036  {csn 4125  cop 4131   class class class wbr 4583  cmpt 4643   × cxp 5036  ccnv 5037  ran crn 5039  cres 5040  cima 5041  ccom 5042  Fun wfun 5798   Fn wfn 5799  wf 5800  ontowfo 5802  1-1-ontowf1o 5803  cfv 5804  (class class class)co 6549  𝑓 cof 6793  1st c1st 7057  2nd c2nd 7058  𝑚 cmap 7744  cc 9813  cr 9814  0cc0 9815  1c1 9816   + caddc 9818   < clt 9953  cle 9954  cmin 10145  cn 10897  0cn0 11169  cz 11254  cuz 11563  ...cfz 12197  ..^cfzo 12334
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335
This theorem is referenced by:  poimirlem17  32596
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