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Theorem poimirlem19 32598
 Description: Lemma for poimir 32612 establishing the vertices of the simplex in poimirlem20 32599. (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)
poimirlem22.3 ((𝜑𝑛 ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝𝑛) ≠ 0)
poimirlem21.4 (𝜑 → (2nd𝑇) = 𝑁)
Assertion
Ref Expression
poimirlem19 (𝜑𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ((𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))) ∘𝑓 + (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 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 poimirlem19
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}) Fn ((2nd ‘(1st𝑇)) “ (1...𝑦)))
3533, 34ax-mp 5 . . . . . . . . 9 (((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) Fn ((2nd ‘(1st𝑇)) “ (1...𝑦))
36 c0ex 9913 . . . . . . . . . 10 0 ∈ V
37 fnconstg 6006 . . . . . . . . . 10 (0 ∈ V → (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}) Fn ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)))
3836, 37ax-mp 5 . . . . . . . . 9 (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}) Fn ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁))
3935, 38pm3.2i 470 . . . . . . . 8 ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) Fn ((2nd ‘(1st𝑇)) “ (1...𝑦)) ∧ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}) Fn ((2nd ‘(1st𝑇)) “ ((𝑦 + 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) ∩ ((𝑦 + 1)...𝑁))) = (((2nd ‘(1st𝑇)) “ (1...𝑦)) ∩ ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁))))
5048, 49syl 17 . . . . . . . . 9 (𝜑 → ((2nd ‘(1st𝑇)) “ ((1...𝑦) ∩ ((𝑦 + 1)...𝑁))) = (((2nd ‘(1st𝑇)) “ (1...𝑦)) ∩ ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁))))
51 elfznn0 12302 . . . . . . . . . . . . . 14 (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ∈ ℕ0)
5251nn0red 11229 . . . . . . . . . . . . 13 (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ∈ ℝ)
5352ltp1d 10833 . . . . . . . . . . . 12 (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 < (𝑦 + 1))
54 fzdisj 12239 . . . . . . . . . . . 12 (𝑦 < (𝑦 + 1) → ((1...𝑦) ∩ ((𝑦 + 1)...𝑁)) = ∅)
5553, 54syl 17 . . . . . . . . . . 11 (𝑦 ∈ (0...(𝑁 − 1)) → ((1...𝑦) ∩ ((𝑦 + 1)...𝑁)) = ∅)
5655imaeq2d 5385 . . . . . . . . . 10 (𝑦 ∈ (0...(𝑁 − 1)) → ((2nd ‘(1st𝑇)) “ ((1...𝑦) ∩ ((𝑦 + 1)...𝑁))) = ((2nd ‘(1st𝑇)) “ ∅))
57 ima0 5400 . . . . . . . . . 10 ((2nd ‘(1st𝑇)) “ ∅) = ∅
5856, 57syl6eq 2660 . . . . . . . . 9 (𝑦 ∈ (0...(𝑁 − 1)) → ((2nd ‘(1st𝑇)) “ ((1...𝑦) ∩ ((𝑦 + 1)...𝑁))) = ∅)
5950, 58sylan9req 2665 . . . . . . . 8 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((2nd ‘(1st𝑇)) “ (1...𝑦)) ∩ ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁))) = ∅)
60 fnun 5911 . . . . . . . 8 ((((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) Fn ((2nd ‘(1st𝑇)) “ (1...𝑦)) ∧ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}) Fn ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁))) ∧ (((2nd ‘(1st𝑇)) “ (1...𝑦)) ∩ ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁))) = ∅) → ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})) Fn (((2nd ‘(1st𝑇)) “ (1...𝑦)) ∪ ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁))))
6139, 59, 60sylancr 694 . . . . . . 7 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})) Fn (((2nd ‘(1st𝑇)) “ (1...𝑦)) ∪ ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁))))
62 imaundi 5464 . . . . . . . . 9 ((2nd ‘(1st𝑇)) “ ((1...𝑦) ∪ ((𝑦 + 1)...𝑁))) = (((2nd ‘(1st𝑇)) “ (1...𝑦)) ∪ ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)))
63 nn0p1nn 11209 . . . . . . . . . . . . . . 15 (𝑦 ∈ ℕ0 → (𝑦 + 1) ∈ ℕ)
6451, 63syl 17 . . . . . . . . . . . . . 14 (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈ ℕ)
65 nnuz 11599 . . . . . . . . . . . . . 14 ℕ = (ℤ‘1)
6664, 65syl6eleq 2698 . . . . . . . . . . . . 13 (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈ (ℤ‘1))
6766adantl 481 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (𝑦 + 1) ∈ (ℤ‘1))
68 poimir.0 . . . . . . . . . . . . . . . 16 (𝜑𝑁 ∈ ℕ)
6968nncnd 10913 . . . . . . . . . . . . . . 15 (𝜑𝑁 ∈ ℂ)
70 npcan1 10334 . . . . . . . . . . . . . . 15 (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁)
7169, 70syl 17 . . . . . . . . . . . . . 14 (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
7271adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((𝑁 − 1) + 1) = 𝑁)
73 elfzuz3 12210 . . . . . . . . . . . . . . 15 (𝑦 ∈ (0...(𝑁 − 1)) → (𝑁 − 1) ∈ (ℤ𝑦))
74 peano2uz 11617 . . . . . . . . . . . . . . 15 ((𝑁 − 1) ∈ (ℤ𝑦) → ((𝑁 − 1) + 1) ∈ (ℤ𝑦))
7573, 74syl 17 . . . . . . . . . . . . . 14 (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈ (ℤ𝑦))
7675adantl 481 . . . . . . . . . . . . 13 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((𝑁 − 1) + 1) ∈ (ℤ𝑦))
7772, 76eqeltrrd 2689 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ (ℤ𝑦))
78 fzsplit2 12237 . . . . . . . . . . . 12 (((𝑦 + 1) ∈ (ℤ‘1) ∧ 𝑁 ∈ (ℤ𝑦)) → (1...𝑁) = ((1...𝑦) ∪ ((𝑦 + 1)...𝑁)))
7967, 77, 78syl2anc 691 . . . . . . . . . . 11 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (1...𝑁) = ((1...𝑦) ∪ ((𝑦 + 1)...𝑁)))
8079imaeq2d 5385 . . . . . . . . . 10 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st𝑇)) “ (1...𝑁)) = ((2nd ‘(1st𝑇)) “ ((1...𝑦) ∪ ((𝑦 + 1)...𝑁))))
81 f1ofo 6057 . . . . . . . . . . . 12 ((2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd ‘(1st𝑇)):(1...𝑁)–onto→(1...𝑁))
82 foima 6033 . . . . . . . . . . . 12 ((2nd ‘(1st𝑇)):(1...𝑁)–onto→(1...𝑁) → ((2nd ‘(1st𝑇)) “ (1...𝑁)) = (1...𝑁))
8345, 81, 823syl 18 . . . . . . . . . . 11 (𝜑 → ((2nd ‘(1st𝑇)) “ (1...𝑁)) = (1...𝑁))
8483adantr 480 . . . . . . . . . 10 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st𝑇)) “ (1...𝑁)) = (1...𝑁))
8580, 84eqtr3d 2646 . . . . . . . . 9 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st𝑇)) “ ((1...𝑦) ∪ ((𝑦 + 1)...𝑁))) = (1...𝑁))
8662, 85syl5eqr 2658 . . . . . . . 8 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((2nd ‘(1st𝑇)) “ (1...𝑦)) ∪ ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁))) = (1...𝑁))
8786fneq2d 5896 . . . . . . 7 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})) Fn (((2nd ‘(1st𝑇)) “ (1...𝑦)) ∪ ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁))) ↔ ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})) Fn (1...𝑁)))
8861, 87mpbid 221 . . . . . 6 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})) Fn (1...𝑁))
89 ovex 6577 . . . . . . 7 (1...𝑁) ∈ V
9089a1i 11 . . . . . 6 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (1...𝑁) ∈ V)
91 inidm 3784 . . . . . 6 ((1...𝑁) ∩ (1...𝑁)) = (1...𝑁)
92 eqidd 2611 . . . . . 6 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → ((1st ‘(1st𝑇))‘𝑛) = ((1st ‘(1st𝑇))‘𝑛))
93 eqidd 2611 . . . . . 6 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → (((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛) = (((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛))
9432, 88, 90, 90, 91, 92, 93offval 6802 . . . . 5 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))) = (𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + (((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛))))
95 elmapi 7765 . . . . . . . . . . . . 13 ((1st ‘(1st𝑇)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)) → (1st ‘(1st𝑇)):(1...𝑁)⟶(0..^𝐾))
9629, 95syl 17 . . . . . . . . . . . 12 (𝜑 → (1st ‘(1st𝑇)):(1...𝑁)⟶(0..^𝐾))
9796ffvelrnda 6267 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (1...𝑁)) → ((1st ‘(1st𝑇))‘𝑛) ∈ (0..^𝐾))
98 elfzonn0 12380 . . . . . . . . . . 11 (((1st ‘(1st𝑇))‘𝑛) ∈ (0..^𝐾) → ((1st ‘(1st𝑇))‘𝑛) ∈ ℕ0)
9997, 98syl 17 . . . . . . . . . 10 ((𝜑𝑛 ∈ (1...𝑁)) → ((1st ‘(1st𝑇))‘𝑛) ∈ ℕ0)
10099nn0cnd 11230 . . . . . . . . 9 ((𝜑𝑛 ∈ (1...𝑁)) → ((1st ‘(1st𝑇))‘𝑛) ∈ ℂ)
101100adantlr 747 . . . . . . . 8 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → ((1st ‘(1st𝑇))‘𝑛) ∈ ℂ)
102 ax-1cn 9873 . . . . . . . . . 10 1 ∈ ℂ
103 0cn 9911 . . . . . . . . . 10 0 ∈ ℂ
104102, 103keepel 4105 . . . . . . . . 9 if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0) ∈ ℂ
105104a1i 11 . . . . . . . 8 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0) ∈ ℂ)
106 snssi 4280 . . . . . . . . . . 11 (1 ∈ ℂ → {1} ⊆ ℂ)
107102, 106ax-mp 5 . . . . . . . . . 10 {1} ⊆ ℂ
108 snssi 4280 . . . . . . . . . . 11 (0 ∈ ℂ → {0} ⊆ ℂ)
109103, 108ax-mp 5 . . . . . . . . . 10 {0} ⊆ ℂ
110107, 109unssi 3750 . . . . . . . . 9 ({1} ∪ {0}) ⊆ ℂ
11133fconst 6004 . . . . . . . . . . . . 13 ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}):(((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1)))⟶{1}
11236fconst 6004 . . . . . . . . . . . . 13 ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}):(((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁))⟶{0}
113111, 112pm3.2i 470 . . . . . . . . . . . 12 (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}):(((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1)))⟶{1} ∧ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}):(((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁))⟶{0})
114 simpr 476 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑛 ∈ ((1 + 1)...𝑁)) → 𝑛 ∈ ((1 + 1)...𝑁))
11568nnzd 11357 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝑁 ∈ ℤ)
116 1z 11284 . . . . . . . . . . . . . . . . . . . . . . . 24 1 ∈ ℤ
117 peano2z 11295 . . . . . . . . . . . . . . . . . . . . . . . 24 (1 ∈ ℤ → (1 + 1) ∈ ℤ)
118116, 117ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . 23 (1 + 1) ∈ ℤ
119115, 118jctil 558 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → ((1 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ))
120 elfzelz 12213 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 ∈ ((1 + 1)...𝑁) → 𝑛 ∈ ℤ)
121120, 116jctir 559 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 ∈ ((1 + 1)...𝑁) → (𝑛 ∈ ℤ ∧ 1 ∈ ℤ))
122 fzsubel 12248 . . . . . . . . . . . . . . . . . . . . . 22 ((((1 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑛 ∈ ℤ ∧ 1 ∈ ℤ)) → (𝑛 ∈ ((1 + 1)...𝑁) ↔ (𝑛 − 1) ∈ (((1 + 1) − 1)...(𝑁 − 1))))
123119, 121, 122syl2an 493 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑛 ∈ ((1 + 1)...𝑁)) → (𝑛 ∈ ((1 + 1)...𝑁) ↔ (𝑛 − 1) ∈ (((1 + 1) − 1)...(𝑁 − 1))))
124114, 123mpbid 221 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑛 ∈ ((1 + 1)...𝑁)) → (𝑛 − 1) ∈ (((1 + 1) − 1)...(𝑁 − 1)))
125102, 102pncan3oi 10176 . . . . . . . . . . . . . . . . . . . . 21 ((1 + 1) − 1) = 1
126125oveq1i 6559 . . . . . . . . . . . . . . . . . . . 20 (((1 + 1) − 1)...(𝑁 − 1)) = (1...(𝑁 − 1))
127124, 126syl6eleq 2698 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛 ∈ ((1 + 1)...𝑁)) → (𝑛 − 1) ∈ (1...(𝑁 − 1)))
128127ralrimiva 2949 . . . . . . . . . . . . . . . . . 18 (𝜑 → ∀𝑛 ∈ ((1 + 1)...𝑁)(𝑛 − 1) ∈ (1...(𝑁 − 1)))
129 simpr 476 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑦 ∈ (1...(𝑁 − 1))) → 𝑦 ∈ (1...(𝑁 − 1)))
130 peano2zm 11297 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ)
131115, 130syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (𝑁 − 1) ∈ ℤ)
132131, 116jctil 558 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (1 ∈ ℤ ∧ (𝑁 − 1) ∈ ℤ))
133 elfzelz 12213 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 ∈ (1...(𝑁 − 1)) → 𝑦 ∈ ℤ)
134133, 116jctir 559 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 ∈ (1...(𝑁 − 1)) → (𝑦 ∈ ℤ ∧ 1 ∈ ℤ))
135 fzaddel 12246 . . . . . . . . . . . . . . . . . . . . . . 23 (((1 ∈ ℤ ∧ (𝑁 − 1) ∈ ℤ) ∧ (𝑦 ∈ ℤ ∧ 1 ∈ ℤ)) → (𝑦 ∈ (1...(𝑁 − 1)) ↔ (𝑦 + 1) ∈ ((1 + 1)...((𝑁 − 1) + 1))))
136132, 134, 135syl2an 493 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑦 ∈ (1...(𝑁 − 1))) → (𝑦 ∈ (1...(𝑁 − 1)) ↔ (𝑦 + 1) ∈ ((1 + 1)...((𝑁 − 1) + 1))))
137129, 136mpbid 221 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑦 ∈ (1...(𝑁 − 1))) → (𝑦 + 1) ∈ ((1 + 1)...((𝑁 − 1) + 1)))
13871oveq2d 6565 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → ((1 + 1)...((𝑁 − 1) + 1)) = ((1 + 1)...𝑁))
139138adantr 480 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑦 ∈ (1...(𝑁 − 1))) → ((1 + 1)...((𝑁 − 1) + 1)) = ((1 + 1)...𝑁))
140137, 139eleqtrd 2690 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑦 ∈ (1...(𝑁 − 1))) → (𝑦 + 1) ∈ ((1 + 1)...𝑁))
141120zcnd 11359 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 ∈ ((1 + 1)...𝑁) → 𝑛 ∈ ℂ)
142133zcnd 11359 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 ∈ (1...(𝑁 − 1)) → 𝑦 ∈ ℂ)
143 subadd2 10164 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((𝑛 − 1) = 𝑦 ↔ (𝑦 + 1) = 𝑛))
144102, 143mp3an2 1404 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑛 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((𝑛 − 1) = 𝑦 ↔ (𝑦 + 1) = 𝑛))
145 eqcom 2617 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 = (𝑛 − 1) ↔ (𝑛 − 1) = 𝑦)
146 eqcom 2617 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 = (𝑦 + 1) ↔ (𝑦 + 1) = 𝑛)
147144, 145, 1463bitr4g 302 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑛 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑦 = (𝑛 − 1) ↔ 𝑛 = (𝑦 + 1)))
148141, 142, 147syl2anr 494 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑦 ∈ (1...(𝑁 − 1)) ∧ 𝑛 ∈ ((1 + 1)...𝑁)) → (𝑦 = (𝑛 − 1) ↔ 𝑛 = (𝑦 + 1)))
149148ralrimiva 2949 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 ∈ (1...(𝑁 − 1)) → ∀𝑛 ∈ ((1 + 1)...𝑁)(𝑦 = (𝑛 − 1) ↔ 𝑛 = (𝑦 + 1)))
150149adantl 481 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑦 ∈ (1...(𝑁 − 1))) → ∀𝑛 ∈ ((1 + 1)...𝑁)(𝑦 = (𝑛 − 1) ↔ 𝑛 = (𝑦 + 1)))
151 reu6i 3364 . . . . . . . . . . . . . . . . . . . 20 (((𝑦 + 1) ∈ ((1 + 1)...𝑁) ∧ ∀𝑛 ∈ ((1 + 1)...𝑁)(𝑦 = (𝑛 − 1) ↔ 𝑛 = (𝑦 + 1))) → ∃!𝑛 ∈ ((1 + 1)...𝑁)𝑦 = (𝑛 − 1))
152140, 150, 151syl2anc 691 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑦 ∈ (1...(𝑁 − 1))) → ∃!𝑛 ∈ ((1 + 1)...𝑁)𝑦 = (𝑛 − 1))
153152ralrimiva 2949 . . . . . . . . . . . . . . . . . 18 (𝜑 → ∀𝑦 ∈ (1...(𝑁 − 1))∃!𝑛 ∈ ((1 + 1)...𝑁)𝑦 = (𝑛 − 1))
154 eqid 2610 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1)) = (𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1))
155154f1ompt 6290 . . . . . . . . . . . . . . . . . 18 ((𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1)):((1 + 1)...𝑁)–1-1-onto→(1...(𝑁 − 1)) ↔ (∀𝑛 ∈ ((1 + 1)...𝑁)(𝑛 − 1) ∈ (1...(𝑁 − 1)) ∧ ∀𝑦 ∈ (1...(𝑁 − 1))∃!𝑛 ∈ ((1 + 1)...𝑁)𝑦 = (𝑛 − 1)))
156128, 153, 155sylanbrc 695 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1)):((1 + 1)...𝑁)–1-1-onto→(1...(𝑁 − 1)))
157 f1osng 6089 . . . . . . . . . . . . . . . . . 18 ((1 ∈ V ∧ 𝑁 ∈ ℕ) → {⟨1, 𝑁⟩}:{1}–1-1-onto→{𝑁})
15833, 68, 157sylancr 694 . . . . . . . . . . . . . . . . 17 (𝜑 → {⟨1, 𝑁⟩}:{1}–1-1-onto→{𝑁})
15968nnred 10912 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝑁 ∈ ℝ)
160159ltm1d 10835 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑁 − 1) < 𝑁)
161131zred 11358 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝑁 − 1) ∈ ℝ)
162161, 159ltnled 10063 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((𝑁 − 1) < 𝑁 ↔ ¬ 𝑁 ≤ (𝑁 − 1)))
163160, 162mpbid 221 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ¬ 𝑁 ≤ (𝑁 − 1))
164 elfzle2 12216 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ (1...(𝑁 − 1)) → 𝑁 ≤ (𝑁 − 1))
165163, 164nsyl 134 . . . . . . . . . . . . . . . . . 18 (𝜑 → ¬ 𝑁 ∈ (1...(𝑁 − 1)))
166 disjsn 4192 . . . . . . . . . . . . . . . . . 18 (((1...(𝑁 − 1)) ∩ {𝑁}) = ∅ ↔ ¬ 𝑁 ∈ (1...(𝑁 − 1)))
167165, 166sylibr 223 . . . . . . . . . . . . . . . . 17 (𝜑 → ((1...(𝑁 − 1)) ∩ {𝑁}) = ∅)
168 1re 9918 . . . . . . . . . . . . . . . . . . . . . 22 1 ∈ ℝ
169168ltp1i 10806 . . . . . . . . . . . . . . . . . . . . 21 1 < (1 + 1)
170118zrei 11260 . . . . . . . . . . . . . . . . . . . . . 22 (1 + 1) ∈ ℝ
171168, 170ltnlei 10037 . . . . . . . . . . . . . . . . . . . . 21 (1 < (1 + 1) ↔ ¬ (1 + 1) ≤ 1)
172169, 171mpbi 219 . . . . . . . . . . . . . . . . . . . 20 ¬ (1 + 1) ≤ 1
173 elfzle1 12215 . . . . . . . . . . . . . . . . . . . 20 (1 ∈ ((1 + 1)...𝑁) → (1 + 1) ≤ 1)
174172, 173mto 187 . . . . . . . . . . . . . . . . . . 19 ¬ 1 ∈ ((1 + 1)...𝑁)
175 disjsn 4192 . . . . . . . . . . . . . . . . . . 19 ((((1 + 1)...𝑁) ∩ {1}) = ∅ ↔ ¬ 1 ∈ ((1 + 1)...𝑁))
176174, 175mpbir 220 . . . . . . . . . . . . . . . . . 18 (((1 + 1)...𝑁) ∩ {1}) = ∅
177 f1oun 6069 . . . . . . . . . . . . . . . . . 18 ((((𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1)):((1 + 1)...𝑁)–1-1-onto→(1...(𝑁 − 1)) ∧ {⟨1, 𝑁⟩}:{1}–1-1-onto→{𝑁}) ∧ ((((1 + 1)...𝑁) ∩ {1}) = ∅ ∧ ((1...(𝑁 − 1)) ∩ {𝑁}) = ∅)) → ((𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1)) ∪ {⟨1, 𝑁⟩}):(((1 + 1)...𝑁) ∪ {1})–1-1-onto→((1...(𝑁 − 1)) ∪ {𝑁}))
178176, 177mpanr1 715 . . . . . . . . . . . . . . . . 17 ((((𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1)):((1 + 1)...𝑁)–1-1-onto→(1...(𝑁 − 1)) ∧ {⟨1, 𝑁⟩}:{1}–1-1-onto→{𝑁}) ∧ ((1...(𝑁 − 1)) ∩ {𝑁}) = ∅) → ((𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1)) ∪ {⟨1, 𝑁⟩}):(((1 + 1)...𝑁) ∪ {1})–1-1-onto→((1...(𝑁 − 1)) ∪ {𝑁}))
179156, 158, 167, 178syl21anc 1317 . . . . . . . . . . . . . . . 16 (𝜑 → ((𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1)) ∪ {⟨1, 𝑁⟩}):(((1 + 1)...𝑁) ∪ {1})–1-1-onto→((1...(𝑁 − 1)) ∪ {𝑁}))
180 eleq1 2676 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 = 1 → (𝑛 ∈ ((1 + 1)...𝑁) ↔ 1 ∈ ((1 + 1)...𝑁)))
181174, 180mtbiri 316 . . . . . . . . . . . . . . . . . . . . . 22 (𝑛 = 1 → ¬ 𝑛 ∈ ((1 + 1)...𝑁))
182181necon2ai 2811 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 ∈ ((1 + 1)...𝑁) → 𝑛 ≠ 1)
183 ifnefalse 4048 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 ≠ 1 → if(𝑛 = 1, 𝑁, (𝑛 − 1)) = (𝑛 − 1))
184182, 183syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ ((1 + 1)...𝑁) → if(𝑛 = 1, 𝑁, (𝑛 − 1)) = (𝑛 − 1))
185184mpteq2ia 4668 . . . . . . . . . . . . . . . . . . 19 (𝑛 ∈ ((1 + 1)...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) = (𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1))
186185uneq1i 3725 . . . . . . . . . . . . . . . . . 18 ((𝑛 ∈ ((1 + 1)...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) ∪ {⟨1, 𝑁⟩}) = ((𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1)) ∪ {⟨1, 𝑁⟩})
18733a1i 11 . . . . . . . . . . . . . . . . . . 19 (𝜑 → 1 ∈ V)
188 ssv 3588 . . . . . . . . . . . . . . . . . . . 20 ℕ ⊆ V
189188, 68sseldi 3566 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑁 ∈ V)
19068, 65syl6eleq 2698 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝑁 ∈ (ℤ‘1))
191 fzpred 12259 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ (ℤ‘1) → (1...𝑁) = ({1} ∪ ((1 + 1)...𝑁)))
192190, 191syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (1...𝑁) = ({1} ∪ ((1 + 1)...𝑁)))
193 uncom 3719 . . . . . . . . . . . . . . . . . . . 20 ({1} ∪ ((1 + 1)...𝑁)) = (((1 + 1)...𝑁) ∪ {1})
194192, 193syl6req 2661 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (((1 + 1)...𝑁) ∪ {1}) = (1...𝑁))
195 iftrue 4042 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 1 → if(𝑛 = 1, 𝑁, (𝑛 − 1)) = 𝑁)
196195adantl 481 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛 = 1) → if(𝑛 = 1, 𝑁, (𝑛 − 1)) = 𝑁)
197187, 189, 194, 196fmptapd 6342 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((𝑛 ∈ ((1 + 1)...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) ∪ {⟨1, 𝑁⟩}) = (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))
198186, 197syl5eqr 2658 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1)) ∪ {⟨1, 𝑁⟩}) = (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))
19971, 190eqeltrd 2688 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ((𝑁 − 1) + 1) ∈ (ℤ‘1))
200 uzid 11578 . . . . . . . . . . . . . . . . . . . . 21 ((𝑁 − 1) ∈ ℤ → (𝑁 − 1) ∈ (ℤ‘(𝑁 − 1)))
201 peano2uz 11617 . . . . . . . . . . . . . . . . . . . . 21 ((𝑁 − 1) ∈ (ℤ‘(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈ (ℤ‘(𝑁 − 1)))
202131, 200, 2013syl 18 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((𝑁 − 1) + 1) ∈ (ℤ‘(𝑁 − 1)))
20371, 202eqeltrrd 2689 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑁 ∈ (ℤ‘(𝑁 − 1)))
204 fzsplit2 12237 . . . . . . . . . . . . . . . . . . 19 ((((𝑁 − 1) + 1) ∈ (ℤ‘1) ∧ 𝑁 ∈ (ℤ‘(𝑁 − 1))) → (1...𝑁) = ((1...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)))
205199, 203, 204syl2anc 691 . . . . . . . . . . . . . . . . . 18 (𝜑 → (1...𝑁) = ((1...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)))
20671oveq1d 6564 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (((𝑁 − 1) + 1)...𝑁) = (𝑁...𝑁))
207 fzsn 12254 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℤ → (𝑁...𝑁) = {𝑁})
208115, 207syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑁...𝑁) = {𝑁})
209206, 208eqtrd 2644 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (((𝑁 − 1) + 1)...𝑁) = {𝑁})
210209uneq2d 3729 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((1...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)) = ((1...(𝑁 − 1)) ∪ {𝑁}))
211205, 210eqtr2d 2645 . . . . . . . . . . . . . . . . 17 (𝜑 → ((1...(𝑁 − 1)) ∪ {𝑁}) = (1...𝑁))
212198, 194, 211f1oeq123d 6046 . . . . . . . . . . . . . . . 16 (𝜑 → (((𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1)) ∪ {⟨1, 𝑁⟩}):(((1 + 1)...𝑁) ∪ {1})–1-1-onto→((1...(𝑁 − 1)) ∪ {𝑁}) ↔ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))):(1...𝑁)–1-1-onto→(1...𝑁)))
213179, 212mpbid 221 . . . . . . . . . . . . . . 15 (𝜑 → (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))):(1...𝑁)–1-1-onto→(1...𝑁))
214 f1oco 6072 . . . . . . . . . . . . . . 15 (((2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) ∧ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))):(1...𝑁)–1-1-onto→(1...𝑁)) → ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))):(1...𝑁)–1-1-onto→(1...𝑁))
21545, 213, 214syl2anc 691 . . . . . . . . . . . . . 14 (𝜑 → ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))):(1...𝑁)–1-1-onto→(1...𝑁))
216 dff1o3 6056 . . . . . . . . . . . . . . 15 (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))):(1...𝑁)–1-1-onto→(1...𝑁) ↔ (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))):(1...𝑁)–onto→(1...𝑁) ∧ Fun ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))))
217216simprbi 479 . . . . . . . . . . . . . 14 (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))):(1...𝑁)–1-1-onto→(1...𝑁) → Fun ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))))
218 imain 5888 . . . . . . . . . . . . . 14 (Fun ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) → (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((1...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁))) = ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) ∩ (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁))))
219215, 217, 2183syl 18 . . . . . . . . . . . . 13 (𝜑 → (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((1...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁))) = ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) ∩ (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁))))
22064nnred 10912 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈ ℝ)
221220ltp1d 10833 . . . . . . . . . . . . . . . 16 (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) < ((𝑦 + 1) + 1))
222 fzdisj 12239 . . . . . . . . . . . . . . . 16 ((𝑦 + 1) < ((𝑦 + 1) + 1) → ((1...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁)) = ∅)
223221, 222syl 17 . . . . . . . . . . . . . . 15 (𝑦 ∈ (0...(𝑁 − 1)) → ((1...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁)) = ∅)
224223imaeq2d 5385 . . . . . . . . . . . . . 14 (𝑦 ∈ (0...(𝑁 − 1)) → (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((1...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁))) = (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ∅))
225 ima0 5400 . . . . . . . . . . . . . 14 (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ∅) = ∅
226224, 225syl6eq 2660 . . . . . . . . . . . . 13 (𝑦 ∈ (0...(𝑁 − 1)) → (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((1...(𝑦 + 1)) ∩ (((𝑦 + 1) + 1)...𝑁))) = ∅)
227219, 226sylan9req 2665 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) ∩ (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁))) = ∅)
228 fun 5979 . . . . . . . . . . . 12 (((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}):(((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1)))⟶{1} ∧ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}):(((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁))⟶{0}) ∧ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) ∩ (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁))) = ∅) → (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0})):((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) ∪ (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)))⟶({1} ∪ {0}))
229113, 227, 228sylancr 694 . . . . . . . . . . 11 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0})):((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) ∪ (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)))⟶({1} ∪ {0}))
230 imaundi 5464 . . . . . . . . . . . . 13 (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((1...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁))) = ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) ∪ (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)))
23164peano2nnd 10914 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑦 + 1) + 1) ∈ ℕ)
232231, 65syl6eleq 2698 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑦 + 1) + 1) ∈ (ℤ‘1))
233232adantl 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((𝑦 + 1) + 1) ∈ (ℤ‘1))
234 eluzp1p1 11589 . . . . . . . . . . . . . . . . . . 19 ((𝑁 − 1) ∈ (ℤ𝑦) → ((𝑁 − 1) + 1) ∈ (ℤ‘(𝑦 + 1)))
23573, 234syl 17 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈ (ℤ‘(𝑦 + 1)))
236235adantl 481 . . . . . . . . . . . . . . . . 17 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((𝑁 − 1) + 1) ∈ (ℤ‘(𝑦 + 1)))
23772, 236eqeltrrd 2689 . . . . . . . . . . . . . . . 16 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ (ℤ‘(𝑦 + 1)))
238 fzsplit2 12237 . . . . . . . . . . . . . . . 16 ((((𝑦 + 1) + 1) ∈ (ℤ‘1) ∧ 𝑁 ∈ (ℤ‘(𝑦 + 1))) → (1...𝑁) = ((1...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁)))
239233, 237, 238syl2anc 691 . . . . . . . . . . . . . . 15 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (1...𝑁) = ((1...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁)))
240239imaeq2d 5385 . . . . . . . . . . . . . 14 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...𝑁)) = (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((1...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁))))
241 f1ofo 6057 . . . . . . . . . . . . . . . 16 (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))):(1...𝑁)–1-1-onto→(1...𝑁) → ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))):(1...𝑁)–onto→(1...𝑁))
242 foima 6033 . . . . . . . . . . . . . . . 16 (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))):(1...𝑁)–onto→(1...𝑁) → (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...𝑁)) = (1...𝑁))
243215, 241, 2423syl 18 . . . . . . . . . . . . . . 15 (𝜑 → (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...𝑁)) = (1...𝑁))
244243adantr 480 . . . . . . . . . . . . . 14 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...𝑁)) = (1...𝑁))
245240, 244eqtr3d 2646 . . . . . . . . . . . . 13 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((1...(𝑦 + 1)) ∪ (((𝑦 + 1) + 1)...𝑁))) = (1...𝑁))
246230, 245syl5eqr 2658 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) ∪ (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁))) = (1...𝑁))
247246feq2d 5944 . . . . . . . . . . 11 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0})):((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) ∪ (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)))⟶({1} ∪ {0}) ↔ (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0})):(1...𝑁)⟶({1} ∪ {0})))
248229, 247mpbid 221 . . . . . . . . . 10 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0})):(1...𝑁)⟶({1} ∪ {0}))
249248ffvelrnda 6267 . . . . . . . . 9 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) ∈ ({1} ∪ {0}))
250110, 249sseldi 3566 . . . . . . . 8 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) ∈ ℂ)
251101, 105, 250subadd23d 10293 . . . . . . 7 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → ((((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0)) + ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛)) = (((1st ‘(1st𝑇))‘𝑛) + (((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))))
252 oveq2 6557 . . . . . . . . . 10 (1 = if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0) → (((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) − 1) = (((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0)))
253252eqeq1d 2612 . . . . . . . . 9 (1 = if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0) → ((((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) − 1) = (((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛) ↔ (((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0)) = (((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛)))
254 oveq2 6557 . . . . . . . . . 10 (0 = if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0) → (((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) − 0) = (((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0)))
255254eqeq1d 2612 . . . . . . . . 9 (0 = if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0) → ((((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) − 0) = (((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛) ↔ (((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0)) = (((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛)))
256 1m1e0 10966 . . . . . . . . . . . . 13 (1 − 1) = 0
257 f1ofn 6051 . . . . . . . . . . . . . . . . . . . 20 ((2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd ‘(1st𝑇)) Fn (1...𝑁))
25845, 257syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (2nd ‘(1st𝑇)) Fn (1...𝑁))
259258adantr 480 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (2nd ‘(1st𝑇)) Fn (1...𝑁))
260 imassrn 5396 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ (1...(𝑦 + 1))) ⊆ ran (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))
261 f1of 6050 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))):(1...𝑁)–1-1-onto→(1...𝑁) → (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))):(1...𝑁)⟶(1...𝑁))
262213, 261syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))):(1...𝑁)⟶(1...𝑁))
263 frn 5966 . . . . . . . . . . . . . . . . . . . . 21 ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))):(1...𝑁)⟶(1...𝑁) → ran (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) ⊆ (1...𝑁))
264262, 263syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ran (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) ⊆ (1...𝑁))
265260, 264syl5ss 3579 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ (1...(𝑦 + 1))) ⊆ (1...𝑁))
266265adantr 480 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ (1...(𝑦 + 1))) ⊆ (1...𝑁))
267 eqidd 2611 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) = (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))
268 eluzfz1 12219 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 ∈ (ℤ‘1) → 1 ∈ (1...𝑁))
269190, 268syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → 1 ∈ (1...𝑁))
270267, 196, 269, 68fvmptd 6197 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))‘1) = 𝑁)
271270adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))‘1) = 𝑁)
272 f1ofn 6051 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))):(1...𝑁)–1-1-onto→(1...𝑁) → (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) Fn (1...𝑁))
273213, 272syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) Fn (1...𝑁))
274273adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) Fn (1...𝑁))
275 fzss2 12252 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ (ℤ‘(𝑦 + 1)) → (1...(𝑦 + 1)) ⊆ (1...𝑁))
276237, 275syl 17 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (1...(𝑦 + 1)) ⊆ (1...𝑁))
277 eluzfz1 12219 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑦 + 1) ∈ (ℤ‘1) → 1 ∈ (1...(𝑦 + 1)))
27866, 277syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 ∈ (0...(𝑁 − 1)) → 1 ∈ (1...(𝑦 + 1)))
279278adantl 481 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → 1 ∈ (1...(𝑦 + 1)))
280 fnfvima 6400 . . . . . . . . . . . . . . . . . . . 20 (((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) Fn (1...𝑁) ∧ (1...(𝑦 + 1)) ⊆ (1...𝑁) ∧ 1 ∈ (1...(𝑦 + 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))‘1) ∈ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ (1...(𝑦 + 1))))
281274, 276, 279, 280syl3anc 1318 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))‘1) ∈ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ (1...(𝑦 + 1))))
282271, 281eqeltrrd 2689 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ (1...(𝑦 + 1))))
283 fnfvima 6400 . . . . . . . . . . . . . . . . . 18 (((2nd ‘(1st𝑇)) Fn (1...𝑁) ∧ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ (1...(𝑦 + 1))) ⊆ (1...𝑁) ∧ 𝑁 ∈ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ (1...(𝑦 + 1)))) → ((2nd ‘(1st𝑇))‘𝑁) ∈ ((2nd ‘(1st𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ (1...(𝑦 + 1)))))
284259, 266, 282, 283syl3anc 1318 . . . . . . . . . . . . . . . . 17 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st𝑇))‘𝑁) ∈ ((2nd ‘(1st𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ (1...(𝑦 + 1)))))
285 imaco 5557 . . . . . . . . . . . . . . . . 17 (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) = ((2nd ‘(1st𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ (1...(𝑦 + 1))))
286284, 285syl6eleqr 2699 . . . . . . . . . . . . . . . 16 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st𝑇))‘𝑁) ∈ (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))))
287 fnconstg 6006 . . . . . . . . . . . . . . . . . 18 (1 ∈ V → ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) Fn (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))))
28833, 287ax-mp 5 . . . . . . . . . . . . . . . . 17 ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) Fn (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1)))
289 fnconstg 6006 . . . . . . . . . . . . . . . . . 18 (0 ∈ V → ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}) Fn (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)))
29036, 289ax-mp 5 . . . . . . . . . . . . . . . . 17 ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}) Fn (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁))
291 fvun1 6179 . . . . . . . . . . . . . . . . 17 ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) Fn (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) ∧ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}) Fn (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) ∧ (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) ∩ (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁))) = ∅ ∧ ((2nd ‘(1st𝑇))‘𝑁) ∈ (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))))) → ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘𝑁)) = (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1})‘((2nd ‘(1st𝑇))‘𝑁)))
292288, 290, 291mp3an12 1406 . . . . . . . . . . . . . . . 16 ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) ∩ (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁))) = ∅ ∧ ((2nd ‘(1st𝑇))‘𝑁) ∈ (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1)))) → ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘𝑁)) = (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1})‘((2nd ‘(1st𝑇))‘𝑁)))
293227, 286, 292syl2anc 691 . . . . . . . . . . . . . . 15 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘𝑁)) = (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1})‘((2nd ‘(1st𝑇))‘𝑁)))
29433fvconst2 6374 . . . . . . . . . . . . . . . 16 (((2nd ‘(1st𝑇))‘𝑁) ∈ (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) → (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1})‘((2nd ‘(1st𝑇))‘𝑁)) = 1)
295286, 294syl 17 . . . . . . . . . . . . . . 15 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1})‘((2nd ‘(1st𝑇))‘𝑁)) = 1)
296293, 295eqtrd 2644 . . . . . . . . . . . . . 14 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘𝑁)) = 1)
297296oveq1d 6564 . . . . . . . . . . . . 13 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘𝑁)) − 1) = (1 − 1))
298 fzss1 12251 . . . . . . . . . . . . . . . . . 18 ((𝑦 + 1) ∈ (ℤ‘1) → ((𝑦 + 1)...𝑁) ⊆ (1...𝑁))
29966, 298syl 17 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑦 + 1)...𝑁) ⊆ (1...𝑁))
300299adantl 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((𝑦 + 1)...𝑁) ⊆ (1...𝑁))
301 eluzfz2 12220 . . . . . . . . . . . . . . . . 17 (𝑁 ∈ (ℤ‘(𝑦 + 1)) → 𝑁 ∈ ((𝑦 + 1)...𝑁))
302237, 301syl 17 . . . . . . . . . . . . . . . 16 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ ((𝑦 + 1)...𝑁))
303 fnfvima 6400 . . . . . . . . . . . . . . . 16 (((2nd ‘(1st𝑇)) Fn (1...𝑁) ∧ ((𝑦 + 1)...𝑁) ⊆ (1...𝑁) ∧ 𝑁 ∈ ((𝑦 + 1)...𝑁)) → ((2nd ‘(1st𝑇))‘𝑁) ∈ ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)))
304259, 300, 302, 303syl3anc 1318 . . . . . . . . . . . . . . 15 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st𝑇))‘𝑁) ∈ ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)))
305 fvun2 6180 . . . . . . . . . . . . . . . 16 (((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) Fn ((2nd ‘(1st𝑇)) “ (1...𝑦)) ∧ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}) Fn ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) ∧ ((((2nd ‘(1st𝑇)) “ (1...𝑦)) ∩ ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁))) = ∅ ∧ ((2nd ‘(1st𝑇))‘𝑁) ∈ ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)))) → (((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘𝑁)) = ((((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})‘((2nd ‘(1st𝑇))‘𝑁)))
30635, 38, 305mp3an12 1406 . . . . . . . . . . . . . . 15 (((((2nd ‘(1st𝑇)) “ (1...𝑦)) ∩ ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁))) = ∅ ∧ ((2nd ‘(1st𝑇))‘𝑁) ∈ ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁))) → (((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘𝑁)) = ((((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})‘((2nd ‘(1st𝑇))‘𝑁)))
30759, 304, 306syl2anc 691 . . . . . . . . . . . . . 14 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘𝑁)) = ((((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})‘((2nd ‘(1st𝑇))‘𝑁)))
30836fvconst2 6374 . . . . . . . . . . . . . . 15 (((2nd ‘(1st𝑇))‘𝑁) ∈ ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) → ((((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})‘((2nd ‘(1st𝑇))‘𝑁)) = 0)
309304, 308syl 17 . . . . . . . . . . . . . 14 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})‘((2nd ‘(1st𝑇))‘𝑁)) = 0)
310307, 309eqtrd 2644 . . . . . . . . . . . . 13 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘𝑁)) = 0)
311256, 297, 3103eqtr4a 2670 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘𝑁)) − 1) = (((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘𝑁)))
312 fveq2 6103 . . . . . . . . . . . . . 14 (𝑛 = ((2nd ‘(1st𝑇))‘𝑁) → ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) = ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘𝑁)))
313312oveq1d 6564 . . . . . . . . . . . . 13 (𝑛 = ((2nd ‘(1st𝑇))‘𝑁) → (((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) − 1) = (((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘𝑁)) − 1))
314 fveq2 6103 . . . . . . . . . . . . 13 (𝑛 = ((2nd ‘(1st𝑇))‘𝑁) → (((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛) = (((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘𝑁)))
315313, 314eqeq12d 2625 . . . . . . . . . . . 12 (𝑛 = ((2nd ‘(1st𝑇))‘𝑁) → ((((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) − 1) = (((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛) ↔ (((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘𝑁)) − 1) = (((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘𝑁))))
316311, 315syl5ibrcom 236 . . . . . . . . . . 11 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (𝑛 = ((2nd ‘(1st𝑇))‘𝑁) → (((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) − 1) = (((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛)))
317316imp 444 . . . . . . . . . 10 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 = ((2nd ‘(1st𝑇))‘𝑁)) → (((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) − 1) = (((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛))
318317adantlr 747 . . . . . . . . 9 ((((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) ∧ 𝑛 = ((2nd ‘(1st𝑇))‘𝑁)) → (((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) − 1) = (((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛))
319250subid1d 10260 . . . . . . . . . . 11 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → (((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) − 0) = ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛))
320319adantr 480 . . . . . . . . . 10 ((((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) ∧ ¬ 𝑛 = ((2nd ‘(1st𝑇))‘𝑁)) → (((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) − 0) = ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛))
321 eldifsn 4260 . . . . . . . . . . . . . 14 (𝑛 ∈ ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑁)}) ↔ (𝑛 ∈ (1...𝑁) ∧ 𝑛 ≠ ((2nd ‘(1st𝑇))‘𝑁)))
322 df-ne 2782 . . . . . . . . . . . . . . 15 (𝑛 ≠ ((2nd ‘(1st𝑇))‘𝑁) ↔ ¬ 𝑛 = ((2nd ‘(1st𝑇))‘𝑁))
323322anbi2i 726 . . . . . . . . . . . . . 14 ((𝑛 ∈ (1...𝑁) ∧ 𝑛 ≠ ((2nd ‘(1st𝑇))‘𝑁)) ↔ (𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = ((2nd ‘(1st𝑇))‘𝑁)))
324321, 323bitri 263 . . . . . . . . . . . . 13 (𝑛 ∈ ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑁)}) ↔ (𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = ((2nd ‘(1st𝑇))‘𝑁)))
325 fnconstg 6006 . . . . . . . . . . . . . . . . . 18 (0 ∈ V → (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0}) Fn ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))))
32636, 325ax-mp 5 . . . . . . . . . . . . . . . . 17 (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0}) Fn ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1)))
32735, 326pm3.2i 470 . . . . . . . . . . . . . . . 16 ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) Fn ((2nd ‘(1st𝑇)) “ (1...𝑦)) ∧ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0}) Fn ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))))
328 imain 5888 . . . . . . . . . . . . . . . . . 18 (Fun (2nd ‘(1st𝑇)) → ((2nd ‘(1st𝑇)) “ ((1...𝑦) ∩ ((𝑦 + 1)...(𝑁 − 1)))) = (((2nd ‘(1st𝑇)) “ (1...𝑦)) ∩ ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1)))))
32948, 328syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → ((2nd ‘(1st𝑇)) “ ((1...𝑦) ∩ ((𝑦 + 1)...(𝑁 − 1)))) = (((2nd ‘(1st𝑇)) “ (1...𝑦)) ∩ ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1)))))
330 fzdisj 12239 . . . . . . . . . . . . . . . . . . . 20 (𝑦 < (𝑦 + 1) → ((1...𝑦) ∩ ((𝑦 + 1)...(𝑁 − 1))) = ∅)
33153, 330syl 17 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ (0...(𝑁 − 1)) → ((1...𝑦) ∩ ((𝑦 + 1)...(𝑁 − 1))) = ∅)
332331imaeq2d 5385 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ (0...(𝑁 − 1)) → ((2nd ‘(1st𝑇)) “ ((1...𝑦) ∩ ((𝑦 + 1)...(𝑁 − 1)))) = ((2nd ‘(1st𝑇)) “ ∅))
333332, 57syl6eq 2660 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ (0...(𝑁 − 1)) → ((2nd ‘(1st𝑇)) “ ((1...𝑦) ∩ ((𝑦 + 1)...(𝑁 − 1)))) = ∅)
334329, 333sylan9req 2665 . . . . . . . . . . . . . . . 16 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((2nd ‘(1st𝑇)) “ (1...𝑦)) ∩ ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1)))) = ∅)
335 fnun 5911 . . . . . . . . . . . . . . . 16 ((((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) Fn ((2nd ‘(1st𝑇)) “ (1...𝑦)) ∧ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0}) Fn ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1)))) ∧ (((2nd ‘(1st𝑇)) “ (1...𝑦)) ∩ ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1)))) = ∅) → ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) Fn (((2nd ‘(1st𝑇)) “ (1...𝑦)) ∪ ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1)))))
336327, 334, 335sylancr 694 . . . . . . . . . . . . . . 15 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) Fn (((2nd ‘(1st𝑇)) “ (1...𝑦)) ∪ ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1)))))
337 imaundi 5464 . . . . . . . . . . . . . . . . 17 ((2nd ‘(1st𝑇)) “ ((1...𝑦) ∪ ((𝑦 + 1)...(𝑁 − 1)))) = (((2nd ‘(1st𝑇)) “ (1...𝑦)) ∪ ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))))
338205, 210eqtrd 2644 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (1...𝑁) = ((1...(𝑁 − 1)) ∪ {𝑁}))
339338difeq1d 3689 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → ((1...𝑁) ∖ {𝑁}) = (((1...(𝑁 − 1)) ∪ {𝑁}) ∖ {𝑁}))
340 difun2 4000 . . . . . . . . . . . . . . . . . . . . . . 23 (((1...(𝑁 − 1)) ∪ {𝑁}) ∖ {𝑁}) = ((1...(𝑁 − 1)) ∖ {𝑁})
341339, 340syl6eq 2660 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → ((1...𝑁) ∖ {𝑁}) = ((1...(𝑁 − 1)) ∖ {𝑁}))
342 difsn 4269 . . . . . . . . . . . . . . . . . . . . . . 23 𝑁 ∈ (1...(𝑁 − 1)) → ((1...(𝑁 − 1)) ∖ {𝑁}) = (1...(𝑁 − 1)))
343165, 342syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → ((1...(𝑁 − 1)) ∖ {𝑁}) = (1...(𝑁 − 1)))
344341, 343eqtrd 2644 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → ((1...𝑁) ∖ {𝑁}) = (1...(𝑁 − 1)))
345344adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((1...𝑁) ∖ {𝑁}) = (1...(𝑁 − 1)))
34673adantl 481 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (𝑁 − 1) ∈ (ℤ𝑦))
347 fzsplit2 12237 . . . . . . . . . . . . . . . . . . . . 21 (((𝑦 + 1) ∈ (ℤ‘1) ∧ (𝑁 − 1) ∈ (ℤ𝑦)) → (1...(𝑁 − 1)) = ((1...𝑦) ∪ ((𝑦 + 1)...(𝑁 − 1))))
34867, 346, 347syl2anc 691 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (1...(𝑁 − 1)) = ((1...𝑦) ∪ ((𝑦 + 1)...(𝑁 − 1))))
349345, 348eqtrd 2644 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((1...𝑁) ∖ {𝑁}) = ((1...𝑦) ∪ ((𝑦 + 1)...(𝑁 − 1))))
350349imaeq2d 5385 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st𝑇)) “ ((1...𝑁) ∖ {𝑁})) = ((2nd ‘(1st𝑇)) “ ((1...𝑦) ∪ ((𝑦 + 1)...(𝑁 − 1)))))
351 imadif 5887 . . . . . . . . . . . . . . . . . . . . 21 (Fun (2nd ‘(1st𝑇)) → ((2nd ‘(1st𝑇)) “ ((1...𝑁) ∖ {𝑁})) = (((2nd ‘(1st𝑇)) “ (1...𝑁)) ∖ ((2nd ‘(1st𝑇)) “ {𝑁})))
35248, 351syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((2nd ‘(1st𝑇)) “ ((1...𝑁) ∖ {𝑁})) = (((2nd ‘(1st𝑇)) “ (1...𝑁)) ∖ ((2nd ‘(1st𝑇)) “ {𝑁})))
353 elfz1end 12242 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑁 ∈ ℕ ↔ 𝑁 ∈ (1...𝑁))
35468, 353sylib 207 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝑁 ∈ (1...𝑁))
355 fnsnfv 6168 . . . . . . . . . . . . . . . . . . . . . . 23 (((2nd ‘(1st𝑇)) Fn (1...𝑁) ∧ 𝑁 ∈ (1...𝑁)) → {((2nd ‘(1st𝑇))‘𝑁)} = ((2nd ‘(1st𝑇)) “ {𝑁}))
356258, 354, 355syl2anc 691 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → {((2nd ‘(1st𝑇))‘𝑁)} = ((2nd ‘(1st𝑇)) “ {𝑁}))
357356eqcomd 2616 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → ((2nd ‘(1st𝑇)) “ {𝑁}) = {((2nd ‘(1st𝑇))‘𝑁)})
35883, 357difeq12d 3691 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (((2nd ‘(1st𝑇)) “ (1...𝑁)) ∖ ((2nd ‘(1st𝑇)) “ {𝑁})) = ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑁)}))
359352, 358eqtrd 2644 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ((2nd ‘(1st𝑇)) “ ((1...𝑁) ∖ {𝑁})) = ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑁)}))
360359adantr 480 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st𝑇)) “ ((1...𝑁) ∖ {𝑁})) = ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑁)}))
361350, 360eqtr3d 2646 . . . . . . . . . . . . . . . . 17 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st𝑇)) “ ((1...𝑦) ∪ ((𝑦 + 1)...(𝑁 − 1)))) = ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑁)}))
362337, 361syl5eqr 2658 . . . . . . . . . . . . . . . 16 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((2nd ‘(1st𝑇)) “ (1...𝑦)) ∪ ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1)))) = ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑁)}))
363362fneq2d 5896 . . . . . . . . . . . . . . 15 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) Fn (((2nd ‘(1st𝑇)) “ (1...𝑦)) ∪ ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1)))) ↔ ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) Fn ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑁)})))
364336, 363mpbid 221 . . . . . . . . . . . . . 14 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) Fn ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑁)}))
365 incom 3767 . . . . . . . . . . . . . . . 16 (((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑁)}) ∩ {((2nd ‘(1st𝑇))‘𝑁)}) = ({((2nd ‘(1st𝑇))‘𝑁)} ∩ ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑁)}))
366 disjdif 3992 . . . . . . . . . . . . . . . 16 ({((2nd ‘(1st𝑇))‘𝑁)} ∩ ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑁)})) = ∅
367365, 366eqtri 2632 . . . . . . . . . . . . . . 15 (((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑁)}) ∩ {((2nd ‘(1st𝑇))‘𝑁)}) = ∅
368 fnconstg 6006 . . . . . . . . . . . . . . . . . 18 (1 ∈ V → ({((2nd ‘(1st𝑇))‘𝑁)} × {1}) Fn {((2nd ‘(1st𝑇))‘𝑁)})
36933, 368ax-mp 5 . . . . . . . . . . . . . . . . 17 ({((2nd ‘(1st𝑇))‘𝑁)} × {1}) Fn {((2nd ‘(1st𝑇))‘𝑁)}
370 fvun1 6179 . . . . . . . . . . . . . . . . 17 ((((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) Fn ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑁)}) ∧ ({((2nd ‘(1st𝑇))‘𝑁)} × {1}) Fn {((2nd ‘(1st𝑇))‘𝑁)} ∧ ((((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑁)}) ∩ {((2nd ‘(1st𝑇))‘𝑁)}) = ∅ ∧ 𝑛 ∈ ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑁)}))) → ((((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) ∪ ({((2nd ‘(1st𝑇))‘𝑁)} × {1}))‘𝑛) = (((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0}))‘𝑛))
371369, 370mp3an2 1404 . . . . . . . . . . . . . . . 16 ((((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) Fn ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑁)}) ∧ ((((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑁)}) ∩ {((2nd ‘(1st𝑇))‘𝑁)}) = ∅ ∧ 𝑛 ∈ ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑁)}))) → ((((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) ∪ ({((2nd ‘(1st𝑇))‘𝑁)} × {1}))‘𝑛) = (((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0}))‘𝑛))
372 fnconstg 6006 . . . . . . . . . . . . . . . . . 18 (0 ∈ V → ({((2nd ‘(1st𝑇))‘𝑁)} × {0}) Fn {((2nd ‘(1st𝑇))‘𝑁)})
37336, 372ax-mp 5 . . . . . . . . . . . . . . . . 17 ({((2nd ‘(1st𝑇))‘𝑁)} × {0}) Fn {((2nd ‘(1st𝑇))‘𝑁)}
374 fvun1 6179 . . . . . . . . . . . . . . . . 17 ((((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) Fn ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑁)}) ∧ ({((2nd ‘(1st𝑇))‘𝑁)} × {0}) Fn {((2nd ‘(1st𝑇))‘𝑁)} ∧ ((((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑁)}) ∩ {((2nd ‘(1st𝑇))‘𝑁)}) = ∅ ∧ 𝑛 ∈ ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑁)}))) → ((((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) ∪ ({((2nd ‘(1st𝑇))‘𝑁)} × {0}))‘𝑛) = (((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0}))‘𝑛))
375373, 374mp3an2 1404 . . . . . . . . . . . . . . . 16 ((((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) Fn ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑁)}) ∧ ((((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑁)}) ∩ {((2nd ‘(1st𝑇))‘𝑁)}) = ∅ ∧ 𝑛 ∈ ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑁)}))) → ((((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) ∪ ({((2nd ‘(1st𝑇))‘𝑁)} × {0}))‘𝑛) = (((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0}))‘𝑛))
376371, 375eqtr4d 2647 . . . . . . . . . . . . . . 15 ((((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) Fn ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑁)}) ∧ ((((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑁)}) ∩ {((2nd ‘(1st𝑇))‘𝑁)}) = ∅ ∧ 𝑛 ∈ ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑁)}))) → ((((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) ∪ ({((2nd ‘(1st𝑇))‘𝑁)} × {1}))‘𝑛) = ((((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) ∪ ({((2nd ‘(1st𝑇))‘𝑁)} × {0}))‘𝑛))
377367, 376mpanr1 715 . . . . . . . . . . . . . 14 ((((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) Fn ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑁)}) ∧ 𝑛 ∈ ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑁)})) → ((((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) ∪ ({((2nd ‘(1st𝑇))‘𝑁)} × {1}))‘𝑛) = ((((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) ∪ ({((2nd ‘(1st𝑇))‘𝑁)} × {0}))‘𝑛))
378364, 377sylan 487 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ ((1...𝑁) ∖ {((2nd ‘(1st𝑇))‘𝑁)})) → ((((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) ∪ ({((2nd ‘(1st𝑇))‘𝑁)} × {1}))‘𝑛) = ((((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) ∪ ({((2nd ‘(1st𝑇))‘𝑁)} × {0}))‘𝑛))
379324, 378sylan2br 492 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (𝑛 ∈ (1...𝑁) ∧ ¬ 𝑛 = ((2nd ‘(1st𝑇))‘𝑁))) → ((((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) ∪ ({((2nd ‘(1st𝑇))‘𝑁)} × {1}))‘𝑛) = ((((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) ∪ ({((2nd ‘(1st𝑇))‘𝑁)} × {0}))‘𝑛))
380379anassrs 678 . . . . . . . . . . 11 ((((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) ∧ ¬ 𝑛 = ((2nd ‘(1st𝑇))‘𝑁)) → ((((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) ∪ ({((2nd ‘(1st𝑇))‘𝑁)} × {1}))‘𝑛) = ((((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) ∪ ({((2nd ‘(1st𝑇))‘𝑁)} × {0}))‘𝑛))
381 imaundi 5464 . . . . . . . . . . . . . . . . . . 19 (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((1 + 1)...(𝑦 + 1)) ∪ {1})) = ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((1 + 1)...(𝑦 + 1))) ∪ (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ {1}))
382 imaco 5557 . . . . . . . . . . . . . . . . . . . 20 (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((1 + 1)...(𝑦 + 1))) = ((2nd ‘(1st𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ ((1 + 1)...(𝑦 + 1))))
383 imaco 5557 . . . . . . . . . . . . . . . . . . . 20 (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ {1}) = ((2nd ‘(1st𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ {1}))
384382, 383uneq12i 3727 . . . . . . . . . . . . . . . . . . 19 ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((1 + 1)...(𝑦 + 1))) ∪ (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ {1})) = (((2nd ‘(1st𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ ((1 + 1)...(𝑦 + 1)))) ∪ ((2nd ‘(1st𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ {1})))
385381, 384eqtri 2632 . . . . . . . . . . . . . . . . . 18 (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((1 + 1)...(𝑦 + 1)) ∪ {1})) = (((2nd ‘(1st𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ ((1 + 1)...(𝑦 + 1)))) ∪ ((2nd ‘(1st𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ {1})))
386 fzpred 12259 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑦 + 1) ∈ (ℤ‘1) → (1...(𝑦 + 1)) = ({1} ∪ ((1 + 1)...(𝑦 + 1))))
38766, 386syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 ∈ (0...(𝑁 − 1)) → (1...(𝑦 + 1)) = ({1} ∪ ((1 + 1)...(𝑦 + 1))))
388 uncom 3719 . . . . . . . . . . . . . . . . . . . . 21 ({1} ∪ ((1 + 1)...(𝑦 + 1))) = (((1 + 1)...(𝑦 + 1)) ∪ {1})
389387, 388syl6eq 2660 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ (0...(𝑁 − 1)) → (1...(𝑦 + 1)) = (((1 + 1)...(𝑦 + 1)) ∪ {1}))
390389imaeq2d 5385 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ (0...(𝑁 − 1)) → (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) = (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((1 + 1)...(𝑦 + 1)) ∪ {1})))
391390adantl 481 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) = (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((1 + 1)...(𝑦 + 1)) ∪ {1})))
392 elfzelz 12213 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ∈ ℤ)
393125a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 ∈ ℤ → ((1 + 1) − 1) = 1)
394 zcn 11259 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 ∈ ℤ → 𝑦 ∈ ℂ)
395 pncan1 10333 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 ∈ ℂ → ((𝑦 + 1) − 1) = 𝑦)
396394, 395syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 ∈ ℤ → ((𝑦 + 1) − 1) = 𝑦)
397393, 396oveq12d 6567 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 ∈ ℤ → (((1 + 1) − 1)...((𝑦 + 1) − 1)) = (1...𝑦))
398 elfzelz 12213 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑗 ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1)) → 𝑗 ∈ ℤ)
399398zcnd 11359 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑗 ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1)) → 𝑗 ∈ ℂ)
400 pncan1 10333 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑗 ∈ ℂ → ((𝑗 + 1) − 1) = 𝑗)
401399, 400syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑗 ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1)) → ((𝑗 + 1) − 1) = 𝑗)
402401eleq1d 2672 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑗 ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1)) → (((𝑗 + 1) − 1) ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1)) ↔ 𝑗 ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1))))
403402ibir 256 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑗 ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1)) → ((𝑗 + 1) − 1) ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1)))
404403adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑦 ∈ ℤ ∧ 𝑗 ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1))) → ((𝑗 + 1) − 1) ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1)))
405 peano2z 11295 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑦 ∈ ℤ → (𝑦 + 1) ∈ ℤ)
406405, 118jctil 558 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑦 ∈ ℤ → ((1 + 1) ∈ ℤ ∧ (𝑦 + 1) ∈ ℤ))
407398peano2zd 11361 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑗 ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1)) → (𝑗 + 1) ∈ ℤ)
408407, 116jctir 559 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑗 ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1)) → ((𝑗 + 1) ∈ ℤ ∧ 1 ∈ ℤ))
409 fzsubel 12248 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((1 + 1) ∈ ℤ ∧ (𝑦 + 1) ∈ ℤ) ∧ ((𝑗 + 1) ∈ ℤ ∧ 1 ∈ ℤ)) → ((𝑗 + 1) ∈ ((1 + 1)...(𝑦 + 1)) ↔ ((𝑗 + 1) − 1) ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1))))
410406, 408, 409syl2an 493 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑦 ∈ ℤ ∧ 𝑗 ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1))) → ((𝑗 + 1) ∈ ((1 + 1)...(𝑦 + 1)) ↔ ((𝑗 + 1) − 1) ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1))))
411404, 410mpbird 246 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑦 ∈ ℤ ∧ 𝑗 ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1))) → (𝑗 + 1) ∈ ((1 + 1)...(𝑦 + 1)))
412401eqcomd 2616 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑗 ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1)) → 𝑗 = ((𝑗 + 1) − 1))
413412adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑦 ∈ ℤ ∧ 𝑗 ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1))) → 𝑗 = ((𝑗 + 1) − 1))
414 oveq1 6556 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑛 = (𝑗 + 1) → (𝑛 − 1) = ((𝑗 + 1) − 1))
415414eqeq2d 2620 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑛 = (𝑗 + 1) → (𝑗 = (𝑛 − 1) ↔ 𝑗 = ((𝑗 + 1) − 1)))
416415rspcev 3282 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑗 + 1) ∈ ((1 + 1)...(𝑦 + 1)) ∧ 𝑗 = ((𝑗 + 1) − 1)) → ∃𝑛 ∈ ((1 + 1)...(𝑦 + 1))𝑗 = (𝑛 − 1))
417411, 413, 416syl2anc 691 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑦 ∈ ℤ ∧ 𝑗 ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1))) → ∃𝑛 ∈ ((1 + 1)...(𝑦 + 1))𝑗 = (𝑛 − 1))
418417ex 449 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑦 ∈ ℤ → (𝑗 ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1)) → ∃𝑛 ∈ ((1 + 1)...(𝑦 + 1))𝑗 = (𝑛 − 1)))
419 simpr 476 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑦 ∈ ℤ ∧ 𝑛 ∈ ((1 + 1)...(𝑦 + 1))) → 𝑛 ∈ ((1 + 1)...(𝑦 + 1)))
420 elfzelz 12213 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) → 𝑛 ∈ ℤ)
421420, 116jctir 559 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) → (𝑛 ∈ ℤ ∧ 1 ∈ ℤ))
422 fzsubel 12248 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((1 + 1) ∈ ℤ ∧ (𝑦 + 1) ∈ ℤ) ∧ (𝑛 ∈ ℤ ∧ 1 ∈ ℤ)) → (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) ↔ (𝑛 − 1) ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1))))
423406, 421, 422syl2an 493 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑦 ∈ ℤ ∧ 𝑛 ∈ ((1 + 1)...(𝑦 + 1))) → (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) ↔ (𝑛 − 1) ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1))))
424419, 423mpbid 221 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑦 ∈ ℤ ∧ 𝑛 ∈ ((1 + 1)...(𝑦 + 1))) → (𝑛 − 1) ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1)))
425 eleq1 2676 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑗 = (𝑛 − 1) → (𝑗 ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1)) ↔ (𝑛 − 1) ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1))))
426424, 425syl5ibrcom 236 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑦 ∈ ℤ ∧ 𝑛 ∈ ((1 + 1)...(𝑦 + 1))) → (𝑗 = (𝑛 − 1) → 𝑗 ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1))))
427426rexlimdva 3013 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑦 ∈ ℤ → (∃𝑛 ∈ ((1 + 1)...(𝑦 + 1))𝑗 = (𝑛 − 1) → 𝑗 ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1))))
428418, 427impbid 201 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 ∈ ℤ → (𝑗 ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1)) ↔ ∃𝑛 ∈ ((1 + 1)...(𝑦 + 1))𝑗 = (𝑛 − 1)))
429 vex 3176 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑗 ∈ V
430 eqid 2610 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) ↦ (𝑛 − 1)) = (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) ↦ (𝑛 − 1))
431430elrnmpt 5293 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ V → (𝑗 ∈ ran (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) ↦ (𝑛 − 1)) ↔ ∃𝑛 ∈ ((1 + 1)...(𝑦 + 1))𝑗 = (𝑛 − 1)))
432429, 431ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑗 ∈ ran (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) ↦ (𝑛 − 1)) ↔ ∃𝑛 ∈ ((1 + 1)...(𝑦 + 1))𝑗 = (𝑛 − 1))
433428, 432syl6bbr 277 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 ∈ ℤ → (𝑗 ∈ (((1 + 1) − 1)...((𝑦 + 1) − 1)) ↔ 𝑗 ∈ ran (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) ↦ (𝑛 − 1))))
434433eqrdv 2608 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 ∈ ℤ → (((1 + 1) − 1)...((𝑦 + 1) − 1)) = ran (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) ↦ (𝑛 − 1)))
435397, 434eqtr3d 2646 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 ∈ ℤ → (1...𝑦) = ran (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) ↦ (𝑛 − 1)))
436392, 435syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 ∈ (0...(𝑁 − 1)) → (1...𝑦) = ran (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) ↦ (𝑛 − 1)))
437436adantl 481 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (1...𝑦) = ran (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) ↦ (𝑛 − 1)))
438 df-ima 5051 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ ((1 + 1)...(𝑦 + 1))) = ran ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) ↾ ((1 + 1)...(𝑦 + 1)))
439 uzid 11578 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (1 ∈ ℤ → 1 ∈ (ℤ‘1))
440 peano2uz 11617 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (1 ∈ (ℤ‘1) → (1 + 1) ∈ (ℤ‘1))
441116, 439, 440mp2b 10 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (1 + 1) ∈ (ℤ‘1)
442 fzss1 12251 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((1 + 1) ∈ (ℤ‘1) → ((1 + 1)...(𝑦 + 1)) ⊆ (1...(𝑦 + 1)))
443441, 442ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((1 + 1)...(𝑦 + 1)) ⊆ (1...(𝑦 + 1))
444443, 276syl5ss 3579 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((1 + 1)...(𝑦 + 1)) ⊆ (1...𝑁))
445444resmptd 5371 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) ↾ ((1 + 1)...(𝑦 + 1))) = (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))
446 elfzle1 12215 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (1 ∈ ((1 + 1)...(𝑦 + 1)) → (1 + 1) ≤ 1)
447172, 446mto 187 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ¬ 1 ∈ ((1 + 1)...(𝑦 + 1))
448 eleq1 2676 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑛 = 1 → (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) ↔ 1 ∈ ((1 + 1)...(𝑦 + 1))))
449447, 448mtbiri 316 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑛 = 1 → ¬ 𝑛 ∈ ((1 + 1)...(𝑦 + 1)))
450449necon2ai 2811 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) → 𝑛 ≠ 1)
451450, 183syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) → if(𝑛 = 1, 𝑁, (𝑛 − 1)) = (𝑛 − 1))
452451mpteq2ia 4668 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) = (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) ↦ (𝑛 − 1))
453445, 452syl6eq 2660 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) ↾ ((1 + 1)...(𝑦 + 1))) = (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) ↦ (𝑛 − 1)))
454453rneqd 5274 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ran ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) ↾ ((1 + 1)...(𝑦 + 1))) = ran (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) ↦ (𝑛 − 1)))
455438, 454syl5eq 2656 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ ((1 + 1)...(𝑦 + 1))) = ran (𝑛 ∈ ((1 + 1)...(𝑦 + 1)) ↦ (𝑛 − 1)))
456437, 455eqtr4d 2647 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (1...𝑦) = ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ ((1 + 1)...(𝑦 + 1))))
457456imaeq2d 5385 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st𝑇)) “ (1...𝑦)) = ((2nd ‘(1st𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ ((1 + 1)...(𝑦 + 1)))))
458270sneqd 4137 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → {((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))‘1)} = {𝑁})
459 fnsnfv 6168 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) Fn (1...𝑁) ∧ 1 ∈ (1...𝑁)) → {((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))‘1)} = ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ {1}))
460273, 269, 459syl2anc 691 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → {((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))‘1)} = ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ {1}))
461458, 460eqtr3d 2646 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → {𝑁} = ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ {1}))
462461imaeq2d 5385 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → ((2nd ‘(1st𝑇)) “ {𝑁}) = ((2nd ‘(1st𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ {1})))
463356, 462eqtrd 2644 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → {((2nd ‘(1st𝑇))‘𝑁)} = ((2nd ‘(1st𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ {1})))
464463adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → {((2nd ‘(1st𝑇))‘𝑁)} = ((2nd ‘(1st𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ {1})))
465457, 464uneq12d 3730 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((2nd ‘(1st𝑇)) “ (1...𝑦)) ∪ {((2nd ‘(1st𝑇))‘𝑁)}) = (((2nd ‘(1st𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ ((1 + 1)...(𝑦 + 1)))) ∪ ((2nd ‘(1st𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ {1}))))
466385, 391, 4653eqtr4a 2670 . . . . . . . . . . . . . . . . 17 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) = (((2nd ‘(1st𝑇)) “ (1...𝑦)) ∪ {((2nd ‘(1st𝑇))‘𝑁)}))
467466xpeq1d 5062 . . . . . . . . . . . . . . . 16 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) = ((((2nd ‘(1st𝑇)) “ (1...𝑦)) ∪ {((2nd ‘(1st𝑇))‘𝑁)}) × {1}))
468 xpundir 5095 . . . . . . . . . . . . . . . 16 ((((2nd ‘(1st𝑇)) “ (1...𝑦)) ∪ {((2nd ‘(1st𝑇))‘𝑁)}) × {1}) = ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ ({((2nd ‘(1st𝑇))‘𝑁)} × {1}))
469467, 468syl6eq 2660 . . . . . . . . . . . . . . 15 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) = ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ ({((2nd ‘(1st𝑇))‘𝑁)} × {1})))
470 imaco 5557 . . . . . . . . . . . . . . . . 17 (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) = ((2nd ‘(1st𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ (((𝑦 + 1) + 1)...𝑁)))
471 df-ima 5051 . . . . . . . . . . . . . . . . . . . 20 ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ (((𝑦 + 1) + 1)...𝑁)) = ran ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) ↾ (((𝑦 + 1) + 1)...𝑁))
472 fzss1 12251 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑦 + 1) + 1) ∈ (ℤ‘1) → (((𝑦 + 1) + 1)...𝑁) ⊆ (1...𝑁))
473233, 472syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((𝑦 + 1) + 1)...𝑁) ⊆ (1...𝑁))
474473resmptd 5371 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) ↾ (((𝑦 + 1) + 1)...𝑁)) = (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))
475 1red 9934 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑦 ∈ (0...(𝑁 − 1)) → 1 ∈ ℝ)
47664nnzd 11357 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈ ℤ)
477476peano2zd 11361 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑦 + 1) + 1) ∈ ℤ)
478477zred 11358 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑦 + 1) + 1) ∈ ℝ)
47964nnge1d 10940 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑦 ∈ (0...(𝑁 − 1)) → 1 ≤ (𝑦 + 1))
480475, 220, 478, 479, 221lelttrd 10074 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑦 ∈ (0...(𝑁 − 1)) → 1 < ((𝑦 + 1) + 1))
481475, 478ltnled 10063 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑦 ∈ (0...(𝑁 − 1)) → (1 < ((𝑦 + 1) + 1) ↔ ¬ ((𝑦 + 1) + 1) ≤ 1))
482480, 481mpbid 221 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑦 ∈ (0...(𝑁 − 1)) → ¬ ((𝑦 + 1) + 1) ≤ 1)
483 elfzle1 12215 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (1 ∈ (((𝑦 + 1) + 1)...𝑁) → ((𝑦 + 1) + 1) ≤ 1)
484482, 483nsyl 134 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑦 ∈ (0...(𝑁 − 1)) → ¬ 1 ∈ (((𝑦 + 1) + 1)...𝑁))
485 eleq1 2676 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑛 = 1 → (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) ↔ 1 ∈ (((𝑦 + 1) + 1)...𝑁)))
486485notbid 307 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑛 = 1 → (¬ 𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) ↔ ¬ 1 ∈ (((𝑦 + 1) + 1)...𝑁)))
487484, 486syl5ibrcom 236 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑦 ∈ (0...(𝑁 − 1)) → (𝑛 = 1 → ¬ 𝑛 ∈ (((𝑦 + 1) + 1)...𝑁)))
488487necon2ad 2797 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 ∈ (0...(𝑁 − 1)) → (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) → 𝑛 ≠ 1))
489488imp 444 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑦 ∈ (0...(𝑁 − 1)) ∧ 𝑛 ∈ (((𝑦 + 1) + 1)...𝑁)) → 𝑛 ≠ 1)
490489, 183syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑦 ∈ (0...(𝑁 − 1)) ∧ 𝑛 ∈ (((𝑦 + 1) + 1)...𝑁)) → if(𝑛 = 1, 𝑁, (𝑛 − 1)) = (𝑛 − 1))
491490mpteq2dva 4672 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 ∈ (0...(𝑁 − 1)) → (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) = (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) ↦ (𝑛 − 1)))
492491adantl 481 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) = (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) ↦ (𝑛 − 1)))
493474, 492eqtrd 2644 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) ↾ (((𝑦 + 1) + 1)...𝑁)) = (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) ↦ (𝑛 − 1)))
494493rneqd 5274 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ran ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) ↾ (((𝑦 + 1) + 1)...𝑁)) = ran (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) ↦ (𝑛 − 1)))
495471, 494syl5eq 2656 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ (((𝑦 + 1) + 1)...𝑁)) = ran (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) ↦ (𝑛 − 1)))
496 eqid 2610 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) ↦ (𝑛 − 1)) = (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) ↦ (𝑛 − 1))
497496elrnmpt 5293 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 ∈ V → (𝑗 ∈ ran (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) ↦ (𝑛 − 1)) ↔ ∃𝑛 ∈ (((𝑦 + 1) + 1)...𝑁)𝑗 = (𝑛 − 1)))
498429, 497ax-mp 5 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 ∈ ran (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) ↦ (𝑛 − 1)) ↔ ∃𝑛 ∈ (((𝑦 + 1) + 1)...𝑁)𝑗 = (𝑛 − 1))
499 simpr 476 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (((𝑦 + 1) + 1)...𝑁)) → 𝑛 ∈ (((𝑦 + 1) + 1)...𝑁))
500115, 477anim12ci 589 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((𝑦 + 1) + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ))
501 elfzelz 12213 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) → 𝑛 ∈ ℤ)
502501, 116jctir 559 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) → (𝑛 ∈ ℤ ∧ 1 ∈ ℤ))
503 fzsubel 12248 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝑦 + 1) + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑛 ∈ ℤ ∧ 1 ∈ ℤ)) → (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) ↔ (𝑛 − 1) ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1))))
504500, 502, 503syl2an 493 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (((𝑦 + 1) + 1)...𝑁)) → (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) ↔ (𝑛 − 1) ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1))))
505499, 504mpbid 221 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (((𝑦 + 1) + 1)...𝑁)) → (𝑛 − 1) ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)))
506 eleq1 2676 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 = (𝑛 − 1) → (𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)) ↔ (𝑛 − 1) ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1))))
507505, 506syl5ibrcom 236 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (((𝑦 + 1) + 1)...𝑁)) → (𝑗 = (𝑛 − 1) → 𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1))))
508507rexlimdva 3013 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (∃𝑛 ∈ (((𝑦 + 1) + 1)...𝑁)𝑗 = (𝑛 − 1) → 𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1))))
509 elfzelz 12213 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)) → 𝑗 ∈ ℤ)
510509zcnd 11359 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)) → 𝑗 ∈ ℂ)
511510, 400syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)) → ((𝑗 + 1) − 1) = 𝑗)
512511eleq1d 2672 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)) → (((𝑗 + 1) − 1) ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)) ↔ 𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1))))
513512ibir 256 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)) → ((𝑗 + 1) − 1) ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)))
514513adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1))) → ((𝑗 + 1) − 1) ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)))
515509peano2zd 11361 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)) → (𝑗 + 1) ∈ ℤ)
516515, 116jctir 559 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)) → ((𝑗 + 1) ∈ ℤ ∧ 1 ∈ ℤ))
517 fzsubel 12248 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝑦 + 1) + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ((𝑗 + 1) ∈ ℤ ∧ 1 ∈ ℤ)) → ((𝑗 + 1) ∈ (((𝑦 + 1) + 1)...𝑁) ↔ ((𝑗 + 1) − 1) ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1))))
518500, 516, 517syl2an 493 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1))) → ((𝑗 + 1) ∈ (((𝑦 + 1) + 1)...𝑁) ↔ ((𝑗 + 1) − 1) ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1))))
519514, 518mpbird 246 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1))) → (𝑗 + 1) ∈ (((𝑦 + 1) + 1)...𝑁))
520511eqcomd 2616 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)) → 𝑗 = ((𝑗 + 1) − 1))
521520adantl 481 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1))) → 𝑗 = ((𝑗 + 1) − 1))
522415rspcev 3282 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑗 + 1) ∈ (((𝑦 + 1) + 1)...𝑁) ∧ 𝑗 = ((𝑗 + 1) − 1)) → ∃𝑛 ∈ (((𝑦 + 1) + 1)...𝑁)𝑗 = (𝑛 − 1))
523519, 521, 522syl2anc 691 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1))) → ∃𝑛 ∈ (((𝑦 + 1) + 1)...𝑁)𝑗 = (𝑛 − 1))
524523ex 449 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)) → ∃𝑛 ∈ (((𝑦 + 1) + 1)...𝑁)𝑗 = (𝑛 − 1)))
525508, 524impbid 201 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (∃𝑛 ∈ (((𝑦 + 1) + 1)...𝑁)𝑗 = (𝑛 − 1) ↔ 𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1))))
526498, 525syl5bb 271 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (𝑗 ∈ ran (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) ↦ (𝑛 − 1)) ↔ 𝑗 ∈ ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1))))
527526eqrdv 2608 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ran (𝑛 ∈ (((𝑦 + 1) + 1)...𝑁) ↦ (𝑛 − 1)) = ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)))
52864nncnd 10913 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈ ℂ)
529 pncan1 10333 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑦 + 1) ∈ ℂ → (((𝑦 + 1) + 1) − 1) = (𝑦 + 1))
530528, 529syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 ∈ (0...(𝑁 − 1)) → (((𝑦 + 1) + 1) − 1) = (𝑦 + 1))
531530oveq1d 6564 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ (0...(𝑁 − 1)) → ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)) = ((𝑦 + 1)...(𝑁 − 1)))
532531adantl 481 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((((𝑦 + 1) + 1) − 1)...(𝑁 − 1)) = ((𝑦 + 1)...(𝑁 − 1)))
533495, 527, 5323eqtrd 2648 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ (((𝑦 + 1) + 1)...𝑁)) = ((𝑦 + 1)...(𝑁 − 1)))
534533imaeq2d 5385 . . . . . . . . . . . . . . . . 17 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st𝑇)) “ ((𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) “ (((𝑦 + 1) + 1)...𝑁))) = ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))))
535470, 534syl5eq 2656 . . . . . . . . . . . . . . . 16 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) = ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))))
536535xpeq1d 5062 . . . . . . . . . . . . . . 15 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0}))
537469, 536uneq12d 3730 . . . . . . . . . . . . . 14 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) = (((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ ({((2nd ‘(1st𝑇))‘𝑁)} × {1})) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})))
538 un23 3734 . . . . . . . . . . . . . 14 (((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ ({((2nd ‘(1st𝑇))‘𝑁)} × {1})) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) = (((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) ∪ ({((2nd ‘(1st𝑇))‘𝑁)} × {1}))
539537, 538syl6eq 2660 . . . . . . . . . . . . 13 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) = (((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) ∪ ({((2nd ‘(1st𝑇))‘𝑁)} × {1})))
540539fveq1d 6105 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) = ((((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) ∪ ({((2nd ‘(1st𝑇))‘𝑁)} × {1}))‘𝑛))
541540ad2antrr 758 . . . . . . . . . . 11 ((((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) ∧ ¬ 𝑛 = ((2nd ‘(1st𝑇))‘𝑁)) → ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) = ((((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) ∪ ({((2nd ‘(1st𝑇))‘𝑁)} × {1}))‘𝑛))
542 imaundi 5464 . . . . . . . . . . . . . . . . . 18 ((2nd ‘(1st𝑇)) “ (((𝑦 + 1)...(𝑁 − 1)) ∪ {𝑁})) = (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) ∪ ((2nd ‘(1st𝑇)) “ {𝑁}))
543 fzsplit2 12237 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑁 − 1) + 1) ∈ (ℤ‘(𝑦 + 1)) ∧ 𝑁 ∈ (ℤ‘(𝑁 − 1))) → ((𝑦 + 1)...𝑁) = (((𝑦 + 1)...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)))
544235, 203, 543syl2anr 494 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((𝑦 + 1)...𝑁) = (((𝑦 + 1)...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)))
545209uneq2d 3729 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (((𝑦 + 1)...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)) = (((𝑦 + 1)...(𝑁 − 1)) ∪ {𝑁}))
546545adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((𝑦 + 1)...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)) = (((𝑦 + 1)...(𝑁 − 1)) ∪ {𝑁}))
547544, 546eqtrd 2644 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((𝑦 + 1)...𝑁) = (((𝑦 + 1)...(𝑁 − 1)) ∪ {𝑁}))
548547imaeq2d 5385 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) = ((2nd ‘(1st𝑇)) “ (((𝑦 + 1)...(𝑁 − 1)) ∪ {𝑁})))
549356adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → {((2nd ‘(1st𝑇))‘𝑁)} = ((2nd ‘(1st𝑇)) “ {𝑁}))
550549uneq2d 3729 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) ∪ {((2nd ‘(1st𝑇))‘𝑁)}) = (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) ∪ ((2nd ‘(1st𝑇)) “ {𝑁})))
551542, 548, 5503eqtr4a 2670 . . . . . . . . . . . . . . . . 17 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) = (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) ∪ {((2nd ‘(1st𝑇))‘𝑁)}))
552551xpeq1d 5062 . . . . . . . . . . . . . . . 16 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}) = ((((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) ∪ {((2nd ‘(1st𝑇))‘𝑁)}) × {0}))
553 xpundir 5095 . . . . . . . . . . . . . . . 16 ((((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) ∪ {((2nd ‘(1st𝑇))‘𝑁)}) × {0}) = ((((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0}) ∪ ({((2nd ‘(1st𝑇))‘𝑁)} × {0}))
554552, 553syl6eq 2660 . . . . . . . . . . . . . . 15 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}) = ((((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0}) ∪ ({((2nd ‘(1st𝑇))‘𝑁)} × {0})))
555554uneq2d 3729 . . . . . . . . . . . . . 14 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})) = ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0}) ∪ ({((2nd ‘(1st𝑇))‘𝑁)} × {0}))))
556 unass 3732 . . . . . . . . . . . . . 14 (((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) ∪ ({((2nd ‘(1st𝑇))‘𝑁)} × {0})) = ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0}) ∪ ({((2nd ‘(1st𝑇))‘𝑁)} × {0})))
557555, 556syl6eqr 2662 . . . . . . . . . . . . 13 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})) = (((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) ∪ ({((2nd ‘(1st𝑇))‘𝑁)} × {0})))
558557fveq1d 6105 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛) = ((((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) ∪ ({((2nd ‘(1st𝑇))‘𝑁)} × {0}))‘𝑛))
559558ad2antrr 758 . . . . . . . . . . 11 ((((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) ∧ ¬ 𝑛 = ((2nd ‘(1st𝑇))‘𝑁)) → (((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛) = ((((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...(𝑁 − 1))) × {0})) ∪ ({((2nd ‘(1st𝑇))‘𝑁)} × {0}))‘𝑛))
560380, 541, 5593eqtr4d 2654 . . . . . . . . . 10 ((((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) ∧ ¬ 𝑛 = ((2nd ‘(1st𝑇))‘𝑁)) → ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) = (((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛))
561320, 560eqtrd 2644 . . . . . . . . 9 ((((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) ∧ ¬ 𝑛 = ((2nd ‘(1st𝑇))‘𝑁)) → (((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) − 0) = (((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛))
562253, 255, 318, 561ifbothda 4073 . . . . . . . 8 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → (((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0)) = (((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛))
563562oveq2d 6565 . . . . . . 7 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → (((1st ‘(1st𝑇))‘𝑛) + (((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))) = (((1st ‘(1st𝑇))‘𝑛) + (((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛)))
564251, 563eqtr2d 2645 . . . . . 6 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → (((1st ‘(1st𝑇))‘𝑛) + (((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛)) = ((((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0)) + ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛)))
565564mpteq2dva 4672 . . . . 5 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) + (((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘𝑛))) = (𝑛 ∈ (1...𝑁) ↦ ((((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0)) + ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛))))
56694, 565eqtrd 2644 . . . 4 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))) = (𝑛 ∈ (1...𝑁) ↦ ((((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0)) + ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛))))
56752adantl 481 . . . . . . . . 9 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → 𝑦 ∈ ℝ)
568161adantr 480 . . . . . . . . 9 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (𝑁 − 1) ∈ ℝ)
569159adantr 480 . . . . . . . . 9 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ ℝ)
570 elfzle2 12216 . . . . . . . . . 10 (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ≤ (𝑁 − 1))
571570adantl 481 . . . . . . . . 9 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → 𝑦 ≤ (𝑁 − 1))
572160adantr 480 . . . . . . . . 9 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (𝑁 − 1) < 𝑁)
573567, 568, 569, 571, 572lelttrd 10074 . . . . . . . 8 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → 𝑦 < 𝑁)
574 poimirlem21.4 . . . . . . . . 9 (𝜑 → (2nd𝑇) = 𝑁)
575574adantr 480 . . . . . . . 8 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (2nd𝑇) = 𝑁)
576573, 575breqtrrd 4611 . . . . . . 7 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → 𝑦 < (2nd𝑇))
577576iftrued 4044 . . . . . 6 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) = 𝑦)
578577csbeq1d 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 / 𝑗((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))
579 vex 3176 . . . . . 6 𝑦 ∈ V
580 oveq2 6557 . . . . . . . . . 10 (𝑗 = 𝑦 → (1...𝑗) = (1...𝑦))
581580imaeq2d 5385 . . . . . . . . 9 (𝑗 = 𝑦 → ((2nd ‘(1st𝑇)) “ (1...𝑗)) = ((2nd ‘(1st𝑇)) “ (1...𝑦)))
582581xpeq1d 5062 . . . . . . . 8 (𝑗 = 𝑦 → (((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}))
583 oveq1 6556 . . . . . . . . . . 11 (𝑗 = 𝑦 → (𝑗 + 1) = (𝑦 + 1))
584583oveq1d 6564 . . . . . . . . . 10 (𝑗 = 𝑦 → ((𝑗 + 1)...𝑁) = ((𝑦 + 1)...𝑁))
585584imaeq2d 5385 . . . . . . . . 9 (𝑗 = 𝑦 → ((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)))
586585xpeq1d 5062 . . . . . . . 8 (𝑗 = 𝑦 → (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))
587582, 586uneq12d 3730 . . . . . . 7 (𝑗 = 𝑦 → ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})))
588587oveq2d 6565 . . . . . 6 (𝑗 = 𝑦 → ((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))))
589579, 588csbie 3525 . . . . 5 𝑦 / 𝑗((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})))
590578, 589syl6eq 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}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))))
591 ovex 6577 . . . . . 6 (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0)) ∈ V
592591a1i 11 . . . . 5 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0)) ∈ V)
593 fvex 6113 . . . . . 6 ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) ∈ V
594593a1i 11 . . . . 5 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑛 ∈ (1...𝑁)) → ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛) ∈ V)
595 eqidd 2611 . . . . 5 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))) = (𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))))
596 ffn 5958 . . . . . . 7 ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0})):(1...𝑁)⟶({1} ∪ {0}) → (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) Fn (1...𝑁))
597248, 596syl 17 . . . . . 6 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) Fn (1...𝑁))
598 nfcv 2751 . . . . . . . . . . 11 𝑛(2nd ‘(1st𝑇))
599 nfmpt1 4675 . . . . . . . . . . 11 𝑛(𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))
600598, 599nfco 5209 . . . . . . . . . 10 𝑛((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))
601 nfcv 2751 . . . . . . . . . 10 𝑛(1...(𝑦 + 1))
602600, 601nfima 5393 . . . . . . . . 9 𝑛(((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1)))
603 nfcv 2751 . . . . . . . . 9 𝑛{1}
604602, 603nfxp 5066 . . . . . . . 8 𝑛((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1})
605 nfcv 2751 . . . . . . . . . 10 𝑛(((𝑦 + 1) + 1)...𝑁)
606600, 605nfima 5393 . . . . . . . . 9 𝑛(((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁))
607 nfcv 2751 . . . . . . . . 9 𝑛{0}
608606, 607nfxp 5066 . . . . . . . 8 𝑛((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0})
609604, 608nfun 3731 . . . . . . 7 𝑛(((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))
610609dffn5f 6162 . . . . . 6 ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) Fn (1...𝑁) ↔ (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) = (𝑛 ∈ (1...𝑁) ↦ ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛)))
611597, 610sylib 207 . . . . 5 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) = (𝑛 ∈ (1...𝑁) ↦ ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛)))
61290, 592, 594, 595, 611offval2 6812 . . . 4 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))) ∘𝑓 + (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))) = (𝑛 ∈ (1...𝑁) ↦ ((((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0)) + ((((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))‘𝑛))))
613566, 590, 6123eqtr4rd 2655 . . 3 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))) ∘𝑓 + (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))) = if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))
614613mpteq2dva 4672 . 2 (𝜑 → (𝑦 ∈ (0...(𝑁 − 1)) ↦ ((𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))) ∘𝑓 + (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 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})))))
61522, 614eqtr4d 2647 1 (𝜑𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ((𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))) ∘𝑓 + (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 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  –onto→wfo 5802  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549   ∘𝑓 cof 6793  1st c1st 7057  2nd c2nd 7058   ↑𝑚 cmap 7744  ℂcc 9813  ℝcr 9814  0cc0 9815  1c1 9816   + caddc 9818   < clt 9953   ≤ cle 9954   − cmin 10145  ℕcn 10897  ℕ0cn0 11169  ℤcz 11254  ℤ≥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:  poimirlem20  32599
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