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Theorem poimirlem20 32599
 Description: Lemma for poimir 32612 establishing existence for poimirlem21 32600. (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
poimirlem20 (𝜑 → ∃𝑧𝑆 𝑧𝑇)
Distinct variable groups:   𝑓,𝑗,𝑛,𝑝,𝑡,𝑦,𝑧   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   𝜑,𝑧   𝑓,𝐹,𝑝,𝑡,𝑧   𝑧,𝐾   𝑧,𝑁   𝑡,𝑇,𝑧   𝑆,𝑗,𝑛,𝑝,𝑡,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑓)   𝑆(𝑓)   𝐾(𝑦)

Proof of Theorem poimirlem20
StepHypRef Expression
1 oveq2 6557 . . . . . . . . 9 (1 = if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0) → (((1st ‘(1st𝑇))‘𝑛) − 1) = (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0)))
21eleq1d 2672 . . . . . . . 8 (1 = if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0) → ((((1st ‘(1st𝑇))‘𝑛) − 1) ∈ (0..^𝐾) ↔ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0)) ∈ (0..^𝐾)))
3 oveq2 6557 . . . . . . . . 9 (0 = if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0) → (((1st ‘(1st𝑇))‘𝑛) − 0) = (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0)))
43eleq1d 2672 . . . . . . . 8 (0 = if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0) → ((((1st ‘(1st𝑇))‘𝑛) − 0) ∈ (0..^𝐾) ↔ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0)) ∈ (0..^𝐾)))
5 fveq2 6103 . . . . . . . . . . . 12 (𝑛 = ((2nd ‘(1st𝑇))‘𝑁) → ((1st ‘(1st𝑇))‘𝑛) = ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)))
65oveq1d 6564 . . . . . . . . . . 11 (𝑛 = ((2nd ‘(1st𝑇))‘𝑁) → (((1st ‘(1st𝑇))‘𝑛) − 1) = (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)) − 1))
76adantl 481 . . . . . . . . . 10 ((𝜑𝑛 = ((2nd ‘(1st𝑇))‘𝑁)) → (((1st ‘(1st𝑇))‘𝑛) − 1) = (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)) − 1))
8 poimirlem22.2 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑇𝑆)
9 elrabi 3328 . . . . . . . . . . . . . . . . . . . . 21 (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)))
10 poimirlem22.s . . . . . . . . . . . . . . . . . . . . 21 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}))))}
119, 10eleq2s 2706 . . . . . . . . . . . . . . . . . . . 20 (𝑇𝑆𝑇 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
128, 11syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑇 ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
13 xp1st 7089 . . . . . . . . . . . . . . . . . . 19 (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)}))
1412, 13syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑 → (1st𝑇) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
15 xp1st 7089 . . . . . . . . . . . . . . . . . 18 ((1st𝑇) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1st ‘(1st𝑇)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)))
1614, 15syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → (1st ‘(1st𝑇)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)))
17 elmapi 7765 . . . . . . . . . . . . . . . . 17 ((1st ‘(1st𝑇)) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)) → (1st ‘(1st𝑇)):(1...𝑁)⟶(0..^𝐾))
1816, 17syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → (1st ‘(1st𝑇)):(1...𝑁)⟶(0..^𝐾))
19 xp2nd 7090 . . . . . . . . . . . . . . . . . . . 20 ((1st𝑇) ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (2nd ‘(1st𝑇)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
2014, 19syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (2nd ‘(1st𝑇)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
21 fvex 6113 . . . . . . . . . . . . . . . . . . . 20 (2nd ‘(1st𝑇)) ∈ V
22 f1oeq1 6040 . . . . . . . . . . . . . . . . . . . 20 (𝑓 = (2nd ‘(1st𝑇)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁)))
2321, 22elab 3319 . . . . . . . . . . . . . . . . . . 19 ((2nd ‘(1st𝑇)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
2420, 23sylib 207 . . . . . . . . . . . . . . . . . 18 (𝜑 → (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
25 f1of 6050 . . . . . . . . . . . . . . . . . 18 ((2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd ‘(1st𝑇)):(1...𝑁)⟶(1...𝑁))
2624, 25syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → (2nd ‘(1st𝑇)):(1...𝑁)⟶(1...𝑁))
27 poimir.0 . . . . . . . . . . . . . . . . . 18 (𝜑𝑁 ∈ ℕ)
28 elfz1end 12242 . . . . . . . . . . . . . . . . . 18 (𝑁 ∈ ℕ ↔ 𝑁 ∈ (1...𝑁))
2927, 28sylib 207 . . . . . . . . . . . . . . . . 17 (𝜑𝑁 ∈ (1...𝑁))
3026, 29ffvelrnd 6268 . . . . . . . . . . . . . . . 16 (𝜑 → ((2nd ‘(1st𝑇))‘𝑁) ∈ (1...𝑁))
3118, 30ffvelrnd 6268 . . . . . . . . . . . . . . 15 (𝜑 → ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)) ∈ (0..^𝐾))
32 elfzonn0 12380 . . . . . . . . . . . . . . 15 (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)) ∈ (0..^𝐾) → ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)) ∈ ℕ0)
3331, 32syl 17 . . . . . . . . . . . . . 14 (𝜑 → ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)) ∈ ℕ0)
34 fvex 6113 . . . . . . . . . . . . . . . . 17 ((2nd ‘(1st𝑇))‘𝑁) ∈ V
35 eleq1 2676 . . . . . . . . . . . . . . . . . . 19 (𝑛 = ((2nd ‘(1st𝑇))‘𝑁) → (𝑛 ∈ (1...𝑁) ↔ ((2nd ‘(1st𝑇))‘𝑁) ∈ (1...𝑁)))
3635anbi2d 736 . . . . . . . . . . . . . . . . . 18 (𝑛 = ((2nd ‘(1st𝑇))‘𝑁) → ((𝜑𝑛 ∈ (1...𝑁)) ↔ (𝜑 ∧ ((2nd ‘(1st𝑇))‘𝑁) ∈ (1...𝑁))))
37 fveq2 6103 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = ((2nd ‘(1st𝑇))‘𝑁) → (𝑝𝑛) = (𝑝‘((2nd ‘(1st𝑇))‘𝑁)))
3837neeq1d 2841 . . . . . . . . . . . . . . . . . . 19 (𝑛 = ((2nd ‘(1st𝑇))‘𝑁) → ((𝑝𝑛) ≠ 0 ↔ (𝑝‘((2nd ‘(1st𝑇))‘𝑁)) ≠ 0))
3938rexbidv 3034 . . . . . . . . . . . . . . . . . 18 (𝑛 = ((2nd ‘(1st𝑇))‘𝑁) → (∃𝑝 ∈ ran 𝐹(𝑝𝑛) ≠ 0 ↔ ∃𝑝 ∈ ran 𝐹(𝑝‘((2nd ‘(1st𝑇))‘𝑁)) ≠ 0))
4036, 39imbi12d 333 . . . . . . . . . . . . . . . . 17 (𝑛 = ((2nd ‘(1st𝑇))‘𝑁) → (((𝜑𝑛 ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝𝑛) ≠ 0) ↔ ((𝜑 ∧ ((2nd ‘(1st𝑇))‘𝑁) ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝‘((2nd ‘(1st𝑇))‘𝑁)) ≠ 0)))
41 poimirlem22.3 . . . . . . . . . . . . . . . . 17 ((𝜑𝑛 ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝𝑛) ≠ 0)
4234, 40, 41vtocl 3232 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ ((2nd ‘(1st𝑇))‘𝑁) ∈ (1...𝑁)) → ∃𝑝 ∈ ran 𝐹(𝑝‘((2nd ‘(1st𝑇))‘𝑁)) ≠ 0)
4330, 42mpdan 699 . . . . . . . . . . . . . . 15 (𝜑 → ∃𝑝 ∈ ran 𝐹(𝑝‘((2nd ‘(1st𝑇))‘𝑁)) ≠ 0)
44 fveq1 6102 . . . . . . . . . . . . . . . . . . . . 21 (𝑝 = ((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))) → (𝑝‘((2nd ‘(1st𝑇))‘𝑁)) = (((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})))‘((2nd ‘(1st𝑇))‘𝑁)))
45 ffn 5958 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((1st ‘(1st𝑇)):(1...𝑁)⟶(0..^𝐾) → (1st ‘(1st𝑇)) Fn (1...𝑁))
4618, 45syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → (1st ‘(1st𝑇)) Fn (1...𝑁))
4746adantr 480 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (1st ‘(1st𝑇)) Fn (1...𝑁))
48 1ex 9914 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1 ∈ V
49 fnconstg 6006 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (1 ∈ V → (((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) Fn ((2nd ‘(1st𝑇)) “ (1...𝑦)))
5048, 49ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) Fn ((2nd ‘(1st𝑇)) “ (1...𝑦))
51 c0ex 9913 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 0 ∈ V
52 fnconstg 6006 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (0 ∈ V → (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}) Fn ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)))
5351, 52ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}) Fn ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁))
5450, 53pm3.2i 470 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) Fn ((2nd ‘(1st𝑇)) “ (1...𝑦)) ∧ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}) Fn ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)))
55 dff1o3 6056 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) ↔ ((2nd ‘(1st𝑇)):(1...𝑁)–onto→(1...𝑁) ∧ Fun (2nd ‘(1st𝑇))))
5655simprbi 479 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → Fun (2nd ‘(1st𝑇)))
5724, 56syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜑 → Fun (2nd ‘(1st𝑇)))
58 imain 5888 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (Fun (2nd ‘(1st𝑇)) → ((2nd ‘(1st𝑇)) “ ((1...𝑦) ∩ ((𝑦 + 1)...𝑁))) = (((2nd ‘(1st𝑇)) “ (1...𝑦)) ∩ ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁))))
5957, 58syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑 → ((2nd ‘(1st𝑇)) “ ((1...𝑦) ∩ ((𝑦 + 1)...𝑁))) = (((2nd ‘(1st𝑇)) “ (1...𝑦)) ∩ ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁))))
60 elfznn0 12302 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ∈ ℕ0)
6160nn0red 11229 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ∈ ℝ)
6261ltp1d 10833 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 < (𝑦 + 1))
63 fzdisj 12239 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑦 < (𝑦 + 1) → ((1...𝑦) ∩ ((𝑦 + 1)...𝑁)) = ∅)
6462, 63syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑦 ∈ (0...(𝑁 − 1)) → ((1...𝑦) ∩ ((𝑦 + 1)...𝑁)) = ∅)
6564imaeq2d 5385 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑦 ∈ (0...(𝑁 − 1)) → ((2nd ‘(1st𝑇)) “ ((1...𝑦) ∩ ((𝑦 + 1)...𝑁))) = ((2nd ‘(1st𝑇)) “ ∅))
66 ima0 5400 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((2nd ‘(1st𝑇)) “ ∅) = ∅
6765, 66syl6eq 2660 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑦 ∈ (0...(𝑁 − 1)) → ((2nd ‘(1st𝑇)) “ ((1...𝑦) ∩ ((𝑦 + 1)...𝑁))) = ∅)
6859, 67sylan9req 2665 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((2nd ‘(1st𝑇)) “ (1...𝑦)) ∩ ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁))) = ∅)
69 fnun 5911 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((((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))))
7054, 68, 69sylancr 694 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((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))))
71 imaundi 5464 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((2nd ‘(1st𝑇)) “ ((1...𝑦) ∪ ((𝑦 + 1)...𝑁))) = (((2nd ‘(1st𝑇)) “ (1...𝑦)) ∪ ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)))
72 nn0p1nn 11209 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑦 ∈ ℕ0 → (𝑦 + 1) ∈ ℕ)
73 nnuz 11599 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ℕ = (ℤ‘1)
7472, 73syl6eleq 2698 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑦 ∈ ℕ0 → (𝑦 + 1) ∈ (ℤ‘1))
7560, 74syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈ (ℤ‘1))
7675adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (𝑦 + 1) ∈ (ℤ‘1))
7727nncnd 10913 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝜑𝑁 ∈ ℂ)
78 npcan1 10334 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁)
7977, 78syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
8079adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((𝑁 − 1) + 1) = 𝑁)
81 elfzuz3 12210 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑦 ∈ (0...(𝑁 − 1)) → (𝑁 − 1) ∈ (ℤ𝑦))
82 peano2uz 11617 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑁 − 1) ∈ (ℤ𝑦) → ((𝑁 − 1) + 1) ∈ (ℤ𝑦))
8381, 82syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈ (ℤ𝑦))
8483adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((𝑁 − 1) + 1) ∈ (ℤ𝑦))
8580, 84eqeltrrd 2689 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ (ℤ𝑦))
86 fzsplit2 12237 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝑦 + 1) ∈ (ℤ‘1) ∧ 𝑁 ∈ (ℤ𝑦)) → (1...𝑁) = ((1...𝑦) ∪ ((𝑦 + 1)...𝑁)))
8776, 85, 86syl2anc 691 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (1...𝑁) = ((1...𝑦) ∪ ((𝑦 + 1)...𝑁)))
8887imaeq2d 5385 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st𝑇)) “ (1...𝑁)) = ((2nd ‘(1st𝑇)) “ ((1...𝑦) ∪ ((𝑦 + 1)...𝑁))))
89 f1ofo 6057 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd ‘(1st𝑇)):(1...𝑁)–onto→(1...𝑁))
90 foima 6033 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((2nd ‘(1st𝑇)):(1...𝑁)–onto→(1...𝑁) → ((2nd ‘(1st𝑇)) “ (1...𝑁)) = (1...𝑁))
9124, 89, 903syl 18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝜑 → ((2nd ‘(1st𝑇)) “ (1...𝑁)) = (1...𝑁))
9291adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st𝑇)) “ (1...𝑁)) = (1...𝑁))
9388, 92eqtr3d 2646 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st𝑇)) “ ((1...𝑦) ∪ ((𝑦 + 1)...𝑁))) = (1...𝑁))
9471, 93syl5eqr 2658 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((2nd ‘(1st𝑇)) “ (1...𝑦)) ∪ ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁))) = (1...𝑁))
9594fneq2d 5896 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((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)))
9670, 95mpbid 221 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})) Fn (1...𝑁))
97 ovex 6577 . . . . . . . . . . . . . . . . . . . . . . . . 25 (1...𝑁) ∈ V
9897a1i 11 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (1...𝑁) ∈ V)
99 inidm 3784 . . . . . . . . . . . . . . . . . . . . . . . 24 ((1...𝑁) ∩ (1...𝑁)) = (1...𝑁)
100 eqidd 2611 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ ((2nd ‘(1st𝑇))‘𝑁) ∈ (1...𝑁)) → ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)) = ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)))
101 f1ofn 6051 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) → (2nd ‘(1st𝑇)) Fn (1...𝑁))
10224, 101syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝜑 → (2nd ‘(1st𝑇)) Fn (1...𝑁))
103102adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (2nd ‘(1st𝑇)) Fn (1...𝑁))
104 fzss1 12251 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑦 + 1) ∈ (ℤ‘1) → ((𝑦 + 1)...𝑁) ⊆ (1...𝑁))
10575, 104syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑦 + 1)...𝑁) ⊆ (1...𝑁))
106105adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((𝑦 + 1)...𝑁) ⊆ (1...𝑁))
107 eluzp1p1 11589 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑁 − 1) ∈ (ℤ𝑦) → ((𝑁 − 1) + 1) ∈ (ℤ‘(𝑦 + 1)))
108 uzss 11584 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((𝑁 − 1) + 1) ∈ (ℤ‘(𝑦 + 1)) → (ℤ‘((𝑁 − 1) + 1)) ⊆ (ℤ‘(𝑦 + 1)))
10981, 107, 1083syl 18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑦 ∈ (0...(𝑁 − 1)) → (ℤ‘((𝑁 − 1) + 1)) ⊆ (ℤ‘(𝑦 + 1)))
110109adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (ℤ‘((𝑁 − 1) + 1)) ⊆ (ℤ‘(𝑦 + 1)))
11127nnzd 11357 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝜑𝑁 ∈ ℤ)
112 uzid 11578 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑁 ∈ ℤ → 𝑁 ∈ (ℤ𝑁))
113111, 112syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝜑𝑁 ∈ (ℤ𝑁))
11479fveq2d 6107 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝜑 → (ℤ‘((𝑁 − 1) + 1)) = (ℤ𝑁))
115113, 114eleqtrrd 2691 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝜑𝑁 ∈ (ℤ‘((𝑁 − 1) + 1)))
116115adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ (ℤ‘((𝑁 − 1) + 1)))
117110, 116sseldd 3569 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ (ℤ‘(𝑦 + 1)))
118 eluzfz2 12220 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑁 ∈ (ℤ‘(𝑦 + 1)) → 𝑁 ∈ ((𝑦 + 1)...𝑁))
119117, 118syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ ((𝑦 + 1)...𝑁))
120 fnfvima 6400 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((2nd ‘(1st𝑇)) Fn (1...𝑁) ∧ ((𝑦 + 1)...𝑁) ⊆ (1...𝑁) ∧ 𝑁 ∈ ((𝑦 + 1)...𝑁)) → ((2nd ‘(1st𝑇))‘𝑁) ∈ ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)))
121103, 106, 119, 120syl3anc 1318 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((2nd ‘(1st𝑇))‘𝑁) ∈ ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)))
122 fvun2 6180 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((((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)))
12350, 53, 122mp3an12 1406 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((((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)))
12468, 121, 123syl2anc 691 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((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)))
12551fvconst2 6374 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((2nd ‘(1st𝑇))‘𝑁) ∈ ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) → ((((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})‘((2nd ‘(1st𝑇))‘𝑁)) = 0)
126121, 125syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})‘((2nd ‘(1st𝑇))‘𝑁)) = 0)
127124, 126eqtrd 2644 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘𝑁)) = 0)
128127adantr 480 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ ((2nd ‘(1st𝑇))‘𝑁) ∈ (1...𝑁)) → (((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))‘((2nd ‘(1st𝑇))‘𝑁)) = 0)
12947, 96, 98, 98, 99, 100, 128ofval 6804 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ ((2nd ‘(1st𝑇))‘𝑁) ∈ (1...𝑁)) → (((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})))‘((2nd ‘(1st𝑇))‘𝑁)) = (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)) + 0))
13030, 129mpidan 701 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})))‘((2nd ‘(1st𝑇))‘𝑁)) = (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)) + 0))
13133nn0cnd 11230 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)) ∈ ℂ)
132131addid1d 10115 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)) + 0) = ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)))
133132adantr 480 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)) + 0) = ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)))
134130, 133eqtrd 2644 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})))‘((2nd ‘(1st𝑇))‘𝑁)) = ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)))
13544, 134sylan9eqr 2666 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑝 = ((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})))) → (𝑝‘((2nd ‘(1st𝑇))‘𝑁)) = ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)))
136135adantllr 751 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑝 ∈ ran 𝐹) ∧ 𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑝 = ((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})))) → (𝑝‘((2nd ‘(1st𝑇))‘𝑁)) = ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)))
137 fveq2 6103 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑡 = 𝑇 → (2nd𝑡) = (2nd𝑇))
138137breq2d 4595 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑡 = 𝑇 → (𝑦 < (2nd𝑡) ↔ 𝑦 < (2nd𝑇)))
139138ifbid 4058 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑡 = 𝑇 → if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)))
140139csbeq1d 3506 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (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}))))
141 fveq2 6103 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑡 = 𝑇 → (1st𝑡) = (1st𝑇))
142141fveq2d 6107 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑡 = 𝑇 → (1st ‘(1st𝑡)) = (1st ‘(1st𝑇)))
143141fveq2d 6107 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑡 = 𝑇 → (2nd ‘(1st𝑡)) = (2nd ‘(1st𝑇)))
144143imaeq1d 5384 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑡 = 𝑇 → ((2nd ‘(1st𝑡)) “ (1...𝑗)) = ((2nd ‘(1st𝑇)) “ (1...𝑗)))
145144xpeq1d 5062 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑡 = 𝑇 → (((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}))
146143imaeq1d 5384 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑡 = 𝑇 → ((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)))
147146xpeq1d 5062 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑡 = 𝑇 → (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))
148145, 147uneq12d 3730 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑡 = 𝑇 → ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))
149142, 148oveq12d 6567 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (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}))))
150149csbeq2dv 3944 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (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}))))
151140, 150eqtrd 2644 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (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}))))
152151mpteq2dv 4673 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (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})))))
153152eqeq2d 2620 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (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}))))))
154153, 10elrab2 3333 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (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}))))))
155154simprbi 479 . . . . . . . . . . . . . . . . . . . . . . . . 25 (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})))))
1568, 155syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (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})))))
15761adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → 𝑦 ∈ ℝ)
158 peano2zm 11297 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ)
159111, 158syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝜑 → (𝑁 − 1) ∈ ℤ)
160159zred 11358 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝜑 → (𝑁 − 1) ∈ ℝ)
161160adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (𝑁 − 1) ∈ ℝ)
16227nnred 10912 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝜑𝑁 ∈ ℝ)
163162adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ ℝ)
164 elfzle2 12216 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ≤ (𝑁 − 1))
165164adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → 𝑦 ≤ (𝑁 − 1))
166162ltm1d 10835 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝜑 → (𝑁 − 1) < 𝑁)
167166adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (𝑁 − 1) < 𝑁)
168157, 161, 163, 165, 167lelttrd 10074 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → 𝑦 < 𝑁)
169 poimirlem21.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝜑 → (2nd𝑇) = 𝑁)
170169adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (2nd𝑇) = 𝑁)
171168, 170breqtrrd 4611 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → 𝑦 < (2nd𝑇))
172171iftrued 4044 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) = 𝑦)
173172csbeq1d 3506 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((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}))))
174 vex 3176 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑦 ∈ V
175 oveq2 6557 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑗 = 𝑦 → (1...𝑗) = (1...𝑦))
176175imaeq2d 5385 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑗 = 𝑦 → ((2nd ‘(1st𝑇)) “ (1...𝑗)) = ((2nd ‘(1st𝑇)) “ (1...𝑦)))
177176xpeq1d 5062 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑗 = 𝑦 → (((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}))
178 oveq1 6556 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑗 = 𝑦 → (𝑗 + 1) = (𝑦 + 1))
179178oveq1d 6564 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑗 = 𝑦 → ((𝑗 + 1)...𝑁) = ((𝑦 + 1)...𝑁))
180179imaeq2d 5385 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑗 = 𝑦 → ((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)))
181180xpeq1d 5062 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑗 = 𝑦 → (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))
182177, 181uneq12d 3730 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑗 = 𝑦 → ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})))
183182oveq2d 6565 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (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}))))
184174, 183csbie 3525 . . . . . . . . . . . . . . . . . . . . . . . . . 26 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})))
185173, 184syl6eq 2660 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((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}))))
186185mpteq2dva 4672 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (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)) ↦ ((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})))))
187156, 186eqtrd 2644 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})))))
188187rneqd 5274 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → ran 𝐹 = ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ ((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})))))
189188eleq2d 2673 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝑝 ∈ ran 𝐹𝑝 ∈ ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ ((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))))))
190 eqid 2610 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 ∈ (0...(𝑁 − 1)) ↦ ((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})))) = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))))
191 ovex 6577 . . . . . . . . . . . . . . . . . . . . . 22 ((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))) ∈ V
192190, 191elrnmpti 5297 . . . . . . . . . . . . . . . . . . . . 21 (𝑝 ∈ ran (𝑦 ∈ (0...(𝑁 − 1)) ↦ ((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})))) ↔ ∃𝑦 ∈ (0...(𝑁 − 1))𝑝 = ((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))))
193189, 192syl6bb 275 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑝 ∈ ran 𝐹 ↔ ∃𝑦 ∈ (0...(𝑁 − 1))𝑝 = ((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})))))
194193biimpa 500 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑝 ∈ ran 𝐹) → ∃𝑦 ∈ (0...(𝑁 − 1))𝑝 = ((1st ‘(1st𝑇)) ∘𝑓 + ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))))
195136, 194r19.29a 3060 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑝 ∈ ran 𝐹) → (𝑝‘((2nd ‘(1st𝑇))‘𝑁)) = ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)))
196195neeq1d 2841 . . . . . . . . . . . . . . . . 17 ((𝜑𝑝 ∈ ran 𝐹) → ((𝑝‘((2nd ‘(1st𝑇))‘𝑁)) ≠ 0 ↔ ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)) ≠ 0))
197196biimpd 218 . . . . . . . . . . . . . . . 16 ((𝜑𝑝 ∈ ran 𝐹) → ((𝑝‘((2nd ‘(1st𝑇))‘𝑁)) ≠ 0 → ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)) ≠ 0))
198197rexlimdva 3013 . . . . . . . . . . . . . . 15 (𝜑 → (∃𝑝 ∈ ran 𝐹(𝑝‘((2nd ‘(1st𝑇))‘𝑁)) ≠ 0 → ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)) ≠ 0))
19943, 198mpd 15 . . . . . . . . . . . . . 14 (𝜑 → ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)) ≠ 0)
200 elnnne0 11183 . . . . . . . . . . . . . 14 (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)) ∈ ℕ ↔ (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)) ∈ ℕ0 ∧ ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)) ≠ 0))
20133, 199, 200sylanbrc 695 . . . . . . . . . . . . 13 (𝜑 → ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)) ∈ ℕ)
202 nnm1nn0 11211 . . . . . . . . . . . . 13 (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)) ∈ ℕ → (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)) − 1) ∈ ℕ0)
203201, 202syl 17 . . . . . . . . . . . 12 (𝜑 → (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)) − 1) ∈ ℕ0)
204 elfzo0 12376 . . . . . . . . . . . . . 14 (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)) ∈ (0..^𝐾) ↔ (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)) ∈ ℕ0𝐾 ∈ ℕ ∧ ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)) < 𝐾))
20531, 204sylib 207 . . . . . . . . . . . . 13 (𝜑 → (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)) ∈ ℕ0𝐾 ∈ ℕ ∧ ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)) < 𝐾))
206205simp2d 1067 . . . . . . . . . . . 12 (𝜑𝐾 ∈ ℕ)
207203nn0red 11229 . . . . . . . . . . . . 13 (𝜑 → (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)) − 1) ∈ ℝ)
20833nn0red 11229 . . . . . . . . . . . . 13 (𝜑 → ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)) ∈ ℝ)
209206nnred 10912 . . . . . . . . . . . . 13 (𝜑𝐾 ∈ ℝ)
210208ltm1d 10835 . . . . . . . . . . . . 13 (𝜑 → (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)) − 1) < ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)))
211 elfzolt2 12348 . . . . . . . . . . . . . 14 (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)) ∈ (0..^𝐾) → ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)) < 𝐾)
21231, 211syl 17 . . . . . . . . . . . . 13 (𝜑 → ((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)) < 𝐾)
213207, 208, 209, 210, 212lttrd 10077 . . . . . . . . . . . 12 (𝜑 → (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)) − 1) < 𝐾)
214 elfzo0 12376 . . . . . . . . . . . 12 ((((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)) − 1) ∈ (0..^𝐾) ↔ ((((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)) − 1) ∈ ℕ0𝐾 ∈ ℕ ∧ (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)) − 1) < 𝐾))
215203, 206, 213, 214syl3anbrc 1239 . . . . . . . . . . 11 (𝜑 → (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)) − 1) ∈ (0..^𝐾))
216215adantr 480 . . . . . . . . . 10 ((𝜑𝑛 = ((2nd ‘(1st𝑇))‘𝑁)) → (((1st ‘(1st𝑇))‘((2nd ‘(1st𝑇))‘𝑁)) − 1) ∈ (0..^𝐾))
2177, 216eqeltrd 2688 . . . . . . . . 9 ((𝜑𝑛 = ((2nd ‘(1st𝑇))‘𝑁)) → (((1st ‘(1st𝑇))‘𝑛) − 1) ∈ (0..^𝐾))
218217adantlr 747 . . . . . . . 8 (((𝜑𝑛 ∈ (1...𝑁)) ∧ 𝑛 = ((2nd ‘(1st𝑇))‘𝑁)) → (((1st ‘(1st𝑇))‘𝑛) − 1) ∈ (0..^𝐾))
21918ffvelrnda 6267 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (1...𝑁)) → ((1st ‘(1st𝑇))‘𝑛) ∈ (0..^𝐾))
220 elfzonn0 12380 . . . . . . . . . . . . 13 (((1st ‘(1st𝑇))‘𝑛) ∈ (0..^𝐾) → ((1st ‘(1st𝑇))‘𝑛) ∈ ℕ0)
221219, 220syl 17 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (1...𝑁)) → ((1st ‘(1st𝑇))‘𝑛) ∈ ℕ0)
222221nn0cnd 11230 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (1...𝑁)) → ((1st ‘(1st𝑇))‘𝑛) ∈ ℂ)
223222subid1d 10260 . . . . . . . . . 10 ((𝜑𝑛 ∈ (1...𝑁)) → (((1st ‘(1st𝑇))‘𝑛) − 0) = ((1st ‘(1st𝑇))‘𝑛))
224223, 219eqeltrd 2688 . . . . . . . . 9 ((𝜑𝑛 ∈ (1...𝑁)) → (((1st ‘(1st𝑇))‘𝑛) − 0) ∈ (0..^𝐾))
225224adantr 480 . . . . . . . 8 (((𝜑𝑛 ∈ (1...𝑁)) ∧ ¬ 𝑛 = ((2nd ‘(1st𝑇))‘𝑁)) → (((1st ‘(1st𝑇))‘𝑛) − 0) ∈ (0..^𝐾))
2262, 4, 218, 225ifbothda 4073 . . . . . . 7 ((𝜑𝑛 ∈ (1...𝑁)) → (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0)) ∈ (0..^𝐾))
227 eqid 2610 . . . . . . 7 (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)))
228226, 227fmptd 6292 . . . . . 6 (𝜑 → (𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))):(1...𝑁)⟶(0..^𝐾))
229 ovex 6577 . . . . . . 7 (0..^𝐾) ∈ V
230229, 97elmap 7772 . . . . . 6 ((𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)) ↔ (𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))):(1...𝑁)⟶(0..^𝐾))
231228, 230sylibr 223 . . . . 5 (𝜑 → (𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)))
232 simpr 476 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ((1 + 1)...𝑁)) → 𝑛 ∈ ((1 + 1)...𝑁))
233 1z 11284 . . . . . . . . . . . . . . . 16 1 ∈ ℤ
234 peano2z 11295 . . . . . . . . . . . . . . . 16 (1 ∈ ℤ → (1 + 1) ∈ ℤ)
235233, 234ax-mp 5 . . . . . . . . . . . . . . 15 (1 + 1) ∈ ℤ
236111, 235jctil 558 . . . . . . . . . . . . . 14 (𝜑 → ((1 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ))
237 elfzelz 12213 . . . . . . . . . . . . . . 15 (𝑛 ∈ ((1 + 1)...𝑁) → 𝑛 ∈ ℤ)
238237, 233jctir 559 . . . . . . . . . . . . . 14 (𝑛 ∈ ((1 + 1)...𝑁) → (𝑛 ∈ ℤ ∧ 1 ∈ ℤ))
239 fzsubel 12248 . . . . . . . . . . . . . 14 ((((1 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑛 ∈ ℤ ∧ 1 ∈ ℤ)) → (𝑛 ∈ ((1 + 1)...𝑁) ↔ (𝑛 − 1) ∈ (((1 + 1) − 1)...(𝑁 − 1))))
240236, 238, 239syl2an 493 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ((1 + 1)...𝑁)) → (𝑛 ∈ ((1 + 1)...𝑁) ↔ (𝑛 − 1) ∈ (((1 + 1) − 1)...(𝑁 − 1))))
241232, 240mpbid 221 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ((1 + 1)...𝑁)) → (𝑛 − 1) ∈ (((1 + 1) − 1)...(𝑁 − 1)))
242 ax-1cn 9873 . . . . . . . . . . . . . 14 1 ∈ ℂ
243242, 242pncan3oi 10176 . . . . . . . . . . . . 13 ((1 + 1) − 1) = 1
244243oveq1i 6559 . . . . . . . . . . . 12 (((1 + 1) − 1)...(𝑁 − 1)) = (1...(𝑁 − 1))
245241, 244syl6eleq 2698 . . . . . . . . . . 11 ((𝜑𝑛 ∈ ((1 + 1)...𝑁)) → (𝑛 − 1) ∈ (1...(𝑁 − 1)))
246245ralrimiva 2949 . . . . . . . . . 10 (𝜑 → ∀𝑛 ∈ ((1 + 1)...𝑁)(𝑛 − 1) ∈ (1...(𝑁 − 1)))
247 simpr 476 . . . . . . . . . . . . . 14 ((𝜑𝑦 ∈ (1...(𝑁 − 1))) → 𝑦 ∈ (1...(𝑁 − 1)))
248159, 233jctil 558 . . . . . . . . . . . . . . 15 (𝜑 → (1 ∈ ℤ ∧ (𝑁 − 1) ∈ ℤ))
249 elfzelz 12213 . . . . . . . . . . . . . . . 16 (𝑦 ∈ (1...(𝑁 − 1)) → 𝑦 ∈ ℤ)
250249, 233jctir 559 . . . . . . . . . . . . . . 15 (𝑦 ∈ (1...(𝑁 − 1)) → (𝑦 ∈ ℤ ∧ 1 ∈ ℤ))
251 fzaddel 12246 . . . . . . . . . . . . . . 15 (((1 ∈ ℤ ∧ (𝑁 − 1) ∈ ℤ) ∧ (𝑦 ∈ ℤ ∧ 1 ∈ ℤ)) → (𝑦 ∈ (1...(𝑁 − 1)) ↔ (𝑦 + 1) ∈ ((1 + 1)...((𝑁 − 1) + 1))))
252248, 250, 251syl2an 493 . . . . . . . . . . . . . 14 ((𝜑𝑦 ∈ (1...(𝑁 − 1))) → (𝑦 ∈ (1...(𝑁 − 1)) ↔ (𝑦 + 1) ∈ ((1 + 1)...((𝑁 − 1) + 1))))
253247, 252mpbid 221 . . . . . . . . . . . . 13 ((𝜑𝑦 ∈ (1...(𝑁 − 1))) → (𝑦 + 1) ∈ ((1 + 1)...((𝑁 − 1) + 1)))
25479oveq2d 6565 . . . . . . . . . . . . . 14 (𝜑 → ((1 + 1)...((𝑁 − 1) + 1)) = ((1 + 1)...𝑁))
255254adantr 480 . . . . . . . . . . . . 13 ((𝜑𝑦 ∈ (1...(𝑁 − 1))) → ((1 + 1)...((𝑁 − 1) + 1)) = ((1 + 1)...𝑁))
256253, 255eleqtrd 2690 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ (1...(𝑁 − 1))) → (𝑦 + 1) ∈ ((1 + 1)...𝑁))
257237zcnd 11359 . . . . . . . . . . . . . . 15 (𝑛 ∈ ((1 + 1)...𝑁) → 𝑛 ∈ ℂ)
258249zcnd 11359 . . . . . . . . . . . . . . 15 (𝑦 ∈ (1...(𝑁 − 1)) → 𝑦 ∈ ℂ)
259 subadd2 10164 . . . . . . . . . . . . . . . . 17 ((𝑛 ∈ ℂ ∧ 1 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((𝑛 − 1) = 𝑦 ↔ (𝑦 + 1) = 𝑛))
260242, 259mp3an2 1404 . . . . . . . . . . . . . . . 16 ((𝑛 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((𝑛 − 1) = 𝑦 ↔ (𝑦 + 1) = 𝑛))
261 eqcom 2617 . . . . . . . . . . . . . . . 16 (𝑦 = (𝑛 − 1) ↔ (𝑛 − 1) = 𝑦)
262 eqcom 2617 . . . . . . . . . . . . . . . 16 (𝑛 = (𝑦 + 1) ↔ (𝑦 + 1) = 𝑛)
263260, 261, 2623bitr4g 302 . . . . . . . . . . . . . . 15 ((𝑛 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑦 = (𝑛 − 1) ↔ 𝑛 = (𝑦 + 1)))
264257, 258, 263syl2anr 494 . . . . . . . . . . . . . 14 ((𝑦 ∈ (1...(𝑁 − 1)) ∧ 𝑛 ∈ ((1 + 1)...𝑁)) → (𝑦 = (𝑛 − 1) ↔ 𝑛 = (𝑦 + 1)))
265264ralrimiva 2949 . . . . . . . . . . . . 13 (𝑦 ∈ (1...(𝑁 − 1)) → ∀𝑛 ∈ ((1 + 1)...𝑁)(𝑦 = (𝑛 − 1) ↔ 𝑛 = (𝑦 + 1)))
266265adantl 481 . . . . . . . . . . . 12 ((𝜑𝑦 ∈ (1...(𝑁 − 1))) → ∀𝑛 ∈ ((1 + 1)...𝑁)(𝑦 = (𝑛 − 1) ↔ 𝑛 = (𝑦 + 1)))
267 reu6i 3364 . . . . . . . . . . . 12 (((𝑦 + 1) ∈ ((1 + 1)...𝑁) ∧ ∀𝑛 ∈ ((1 + 1)...𝑁)(𝑦 = (𝑛 − 1) ↔ 𝑛 = (𝑦 + 1))) → ∃!𝑛 ∈ ((1 + 1)...𝑁)𝑦 = (𝑛 − 1))
268256, 266, 267syl2anc 691 . . . . . . . . . . 11 ((𝜑𝑦 ∈ (1...(𝑁 − 1))) → ∃!𝑛 ∈ ((1 + 1)...𝑁)𝑦 = (𝑛 − 1))
269268ralrimiva 2949 . . . . . . . . . 10 (𝜑 → ∀𝑦 ∈ (1...(𝑁 − 1))∃!𝑛 ∈ ((1 + 1)...𝑁)𝑦 = (𝑛 − 1))
270 eqid 2610 . . . . . . . . . . 11 (𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1)) = (𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1))
271270f1ompt 6290 . . . . . . . . . 10 ((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)))
272246, 269, 271sylanbrc 695 . . . . . . . . 9 (𝜑 → (𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1)):((1 + 1)...𝑁)–1-1-onto→(1...(𝑁 − 1)))
273 f1osng 6089 . . . . . . . . . 10 ((1 ∈ V ∧ 𝑁 ∈ ℕ) → {⟨1, 𝑁⟩}:{1}–1-1-onto→{𝑁})
27448, 27, 273sylancr 694 . . . . . . . . 9 (𝜑 → {⟨1, 𝑁⟩}:{1}–1-1-onto→{𝑁})
275160, 162ltnled 10063 . . . . . . . . . . . 12 (𝜑 → ((𝑁 − 1) < 𝑁 ↔ ¬ 𝑁 ≤ (𝑁 − 1)))
276166, 275mpbid 221 . . . . . . . . . . 11 (𝜑 → ¬ 𝑁 ≤ (𝑁 − 1))
277 elfzle2 12216 . . . . . . . . . . 11 (𝑁 ∈ (1...(𝑁 − 1)) → 𝑁 ≤ (𝑁 − 1))
278276, 277nsyl 134 . . . . . . . . . 10 (𝜑 → ¬ 𝑁 ∈ (1...(𝑁 − 1)))
279 disjsn 4192 . . . . . . . . . 10 (((1...(𝑁 − 1)) ∩ {𝑁}) = ∅ ↔ ¬ 𝑁 ∈ (1...(𝑁 − 1)))
280278, 279sylibr 223 . . . . . . . . 9 (𝜑 → ((1...(𝑁 − 1)) ∩ {𝑁}) = ∅)
281 1re 9918 . . . . . . . . . . . . . 14 1 ∈ ℝ
282281ltp1i 10806 . . . . . . . . . . . . 13 1 < (1 + 1)
283235zrei 11260 . . . . . . . . . . . . . 14 (1 + 1) ∈ ℝ
284281, 283ltnlei 10037 . . . . . . . . . . . . 13 (1 < (1 + 1) ↔ ¬ (1 + 1) ≤ 1)
285282, 284mpbi 219 . . . . . . . . . . . 12 ¬ (1 + 1) ≤ 1
286 elfzle1 12215 . . . . . . . . . . . 12 (1 ∈ ((1 + 1)...𝑁) → (1 + 1) ≤ 1)
287285, 286mto 187 . . . . . . . . . . 11 ¬ 1 ∈ ((1 + 1)...𝑁)
288 disjsn 4192 . . . . . . . . . . 11 ((((1 + 1)...𝑁) ∩ {1}) = ∅ ↔ ¬ 1 ∈ ((1 + 1)...𝑁))
289287, 288mpbir 220 . . . . . . . . . 10 (((1 + 1)...𝑁) ∩ {1}) = ∅
290 f1oun 6069 . . . . . . . . . 10 ((((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}))
291289, 290mpanr1 715 . . . . . . . . 9 ((((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}))
292272, 274, 280, 291syl21anc 1317 . . . . . . . 8 (𝜑 → ((𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1)) ∪ {⟨1, 𝑁⟩}):(((1 + 1)...𝑁) ∪ {1})–1-1-onto→((1...(𝑁 − 1)) ∪ {𝑁}))
293 eleq1 2676 . . . . . . . . . . . . . . 15 (𝑛 = 1 → (𝑛 ∈ ((1 + 1)...𝑁) ↔ 1 ∈ ((1 + 1)...𝑁)))
294287, 293mtbiri 316 . . . . . . . . . . . . . 14 (𝑛 = 1 → ¬ 𝑛 ∈ ((1 + 1)...𝑁))
295294necon2ai 2811 . . . . . . . . . . . . 13 (𝑛 ∈ ((1 + 1)...𝑁) → 𝑛 ≠ 1)
296 ifnefalse 4048 . . . . . . . . . . . . 13 (𝑛 ≠ 1 → if(𝑛 = 1, 𝑁, (𝑛 − 1)) = (𝑛 − 1))
297295, 296syl 17 . . . . . . . . . . . 12 (𝑛 ∈ ((1 + 1)...𝑁) → if(𝑛 = 1, 𝑁, (𝑛 − 1)) = (𝑛 − 1))
298297mpteq2ia 4668 . . . . . . . . . . 11 (𝑛 ∈ ((1 + 1)...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) = (𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1))
299298uneq1i 3725 . . . . . . . . . 10 ((𝑛 ∈ ((1 + 1)...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) ∪ {⟨1, 𝑁⟩}) = ((𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1)) ∪ {⟨1, 𝑁⟩})
30048a1i 11 . . . . . . . . . . 11 (𝜑 → 1 ∈ V)
301 ssv 3588 . . . . . . . . . . . 12 ℕ ⊆ V
302301, 27sseldi 3566 . . . . . . . . . . 11 (𝜑𝑁 ∈ V)
30327, 73syl6eleq 2698 . . . . . . . . . . . . 13 (𝜑𝑁 ∈ (ℤ‘1))
304 fzpred 12259 . . . . . . . . . . . . 13 (𝑁 ∈ (ℤ‘1) → (1...𝑁) = ({1} ∪ ((1 + 1)...𝑁)))
305303, 304syl 17 . . . . . . . . . . . 12 (𝜑 → (1...𝑁) = ({1} ∪ ((1 + 1)...𝑁)))
306 uncom 3719 . . . . . . . . . . . 12 ({1} ∪ ((1 + 1)...𝑁)) = (((1 + 1)...𝑁) ∪ {1})
307305, 306syl6req 2661 . . . . . . . . . . 11 (𝜑 → (((1 + 1)...𝑁) ∪ {1}) = (1...𝑁))
308 iftrue 4042 . . . . . . . . . . . 12 (𝑛 = 1 → if(𝑛 = 1, 𝑁, (𝑛 − 1)) = 𝑁)
309308adantl 481 . . . . . . . . . . 11 ((𝜑𝑛 = 1) → if(𝑛 = 1, 𝑁, (𝑛 − 1)) = 𝑁)
310300, 302, 307, 309fmptapd 6342 . . . . . . . . . 10 (𝜑 → ((𝑛 ∈ ((1 + 1)...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) ∪ {⟨1, 𝑁⟩}) = (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))
311299, 310syl5eqr 2658 . . . . . . . . 9 (𝜑 → ((𝑛 ∈ ((1 + 1)...𝑁) ↦ (𝑛 − 1)) ∪ {⟨1, 𝑁⟩}) = (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))
31279, 303eqeltrd 2688 . . . . . . . . . . 11 (𝜑 → ((𝑁 − 1) + 1) ∈ (ℤ‘1))
313 uzid 11578 . . . . . . . . . . . . 13 ((𝑁 − 1) ∈ ℤ → (𝑁 − 1) ∈ (ℤ‘(𝑁 − 1)))
314 peano2uz 11617 . . . . . . . . . . . . 13 ((𝑁 − 1) ∈ (ℤ‘(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈ (ℤ‘(𝑁 − 1)))
315159, 313, 3143syl 18 . . . . . . . . . . . 12 (𝜑 → ((𝑁 − 1) + 1) ∈ (ℤ‘(𝑁 − 1)))
31679, 315eqeltrrd 2689 . . . . . . . . . . 11 (𝜑𝑁 ∈ (ℤ‘(𝑁 − 1)))
317 fzsplit2 12237 . . . . . . . . . . 11 ((((𝑁 − 1) + 1) ∈ (ℤ‘1) ∧ 𝑁 ∈ (ℤ‘(𝑁 − 1))) → (1...𝑁) = ((1...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)))
318312, 316, 317syl2anc 691 . . . . . . . . . 10 (𝜑 → (1...𝑁) = ((1...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)))
31979oveq1d 6564 . . . . . . . . . . . 12 (𝜑 → (((𝑁 − 1) + 1)...𝑁) = (𝑁...𝑁))
320 fzsn 12254 . . . . . . . . . . . . 13 (𝑁 ∈ ℤ → (𝑁...𝑁) = {𝑁})
321111, 320syl 17 . . . . . . . . . . . 12 (𝜑 → (𝑁...𝑁) = {𝑁})
322319, 321eqtrd 2644 . . . . . . . . . . 11 (𝜑 → (((𝑁 − 1) + 1)...𝑁) = {𝑁})
323322uneq2d 3729 . . . . . . . . . 10 (𝜑 → ((1...(𝑁 − 1)) ∪ (((𝑁 − 1) + 1)...𝑁)) = ((1...(𝑁 − 1)) ∪ {𝑁}))
324318, 323eqtr2d 2645 . . . . . . . . 9 (𝜑 → ((1...(𝑁 − 1)) ∪ {𝑁}) = (1...𝑁))
325311, 307, 324f1oeq123d 6046 . . . . . . . 8 (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)))
326292, 325mpbid 221 . . . . . . 7 (𝜑 → (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))):(1...𝑁)–1-1-onto→(1...𝑁))
327 f1oco 6072 . . . . . . 7 (((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))
32824, 326, 327syl2anc 691 . . . . . 6 (𝜑 → ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))):(1...𝑁)–1-1-onto→(1...𝑁))
32997mptex 6390 . . . . . . . 8 (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))) ∈ V
33021, 329coex 7011 . . . . . . 7 ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) ∈ V
331 f1oeq1 6040 . . . . . . 7 (𝑓 = ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) → (𝑓:(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)))
332330, 331elab 3319 . . . . . 6 (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) ∈ {𝑓𝑓:(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))
333328, 332sylibr 223 . . . . 5 (𝜑 → ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
334 opelxpi 5072 . . . . 5 (((𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))) ∈ ((0..^𝐾) ↑𝑚 (1...𝑁)) ∧ ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...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))))⟩ ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
335231, 333, 334syl2anc 691 . . . 4 (𝜑 → ⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩ ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
33627nnnn0d 11228 . . . . 5 (𝜑𝑁 ∈ ℕ0)
337 0elfz 12305 . . . . 5 (𝑁 ∈ ℕ0 → 0 ∈ (0...𝑁))
338336, 337syl 17 . . . 4 (𝜑 → 0 ∈ (0...𝑁))
339 opelxpi 5072 . . . 4 ((⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩ ∈ (((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ 0 ∈ (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))))⟩, 0⟩ ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
340335, 338, 339syl2anc 691 . . 3 (𝜑 → ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩, 0⟩ ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
341 poimirlem22.1 . . . . 5 (𝜑𝐹:(0...(𝑁 − 1))⟶((0...𝐾) ↑𝑚 (1...𝑁)))
34227, 10, 341, 8, 41, 169poimirlem19 32598 . . . 4 (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})))))
343 elfzle1 12215 . . . . . . . . 9 (𝑦 ∈ (0...(𝑁 − 1)) → 0 ≤ 𝑦)
344 0re 9919 . . . . . . . . . 10 0 ∈ ℝ
345 lenlt 9995 . . . . . . . . . 10 ((0 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (0 ≤ 𝑦 ↔ ¬ 𝑦 < 0))
346344, 61, 345sylancr 694 . . . . . . . . 9 (𝑦 ∈ (0...(𝑁 − 1)) → (0 ≤ 𝑦 ↔ ¬ 𝑦 < 0))
347343, 346mpbid 221 . . . . . . . 8 (𝑦 ∈ (0...(𝑁 − 1)) → ¬ 𝑦 < 0)
348347iffalsed 4047 . . . . . . 7 (𝑦 ∈ (0...(𝑁 − 1)) → if(𝑦 < 0, 𝑦, (𝑦 + 1)) = (𝑦 + 1))
349348csbeq1d 3506 . . . . . 6 (𝑦 ∈ (0...(𝑁 − 1)) → if(𝑦 < 0, 𝑦, (𝑦 + 1)) / 𝑗((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...𝑗)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((𝑗 + 1)...𝑁)) × {0}))) = (𝑦 + 1) / 𝑗((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...𝑗)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((𝑗 + 1)...𝑁)) × {0}))))
350 ovex 6577 . . . . . . 7 (𝑦 + 1) ∈ V
351 oveq2 6557 . . . . . . . . . . 11 (𝑗 = (𝑦 + 1) → (1...𝑗) = (1...(𝑦 + 1)))
352351imaeq2d 5385 . . . . . . . . . 10 (𝑗 = (𝑦 + 1) → (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...𝑗)) = (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))))
353352xpeq1d 5062 . . . . . . . . 9 (𝑗 = (𝑦 + 1) → ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...𝑗)) × {1}) = ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...(𝑦 + 1))) × {1}))
354 oveq1 6556 . . . . . . . . . . . 12 (𝑗 = (𝑦 + 1) → (𝑗 + 1) = ((𝑦 + 1) + 1))
355354oveq1d 6564 . . . . . . . . . . 11 (𝑗 = (𝑦 + 1) → ((𝑗 + 1)...𝑁) = (((𝑦 + 1) + 1)...𝑁))
356355imaeq2d 5385 . . . . . . . . . 10 (𝑗 = (𝑦 + 1) → (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((𝑗 + 1)...𝑁)) = (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)))
357356xpeq1d 5062 . . . . . . . . 9 (𝑗 = (𝑦 + 1) → ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((𝑗 + 1)...𝑁)) × {0}) = ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))
358353, 357uneq12d 3730 . . . . . . . 8 (𝑗 = (𝑦 + 1) → (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...𝑗)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((𝑗 + 1)...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})))
359358oveq2d 6565 . . . . . . 7 (𝑗 = (𝑦 + 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...𝑗)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((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}))))
360350, 359csbie 3525 . . . . . 6 (𝑦 + 1) / 𝑗((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...𝑗)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((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})))
361349, 360syl6eq 2660 . . . . 5 (𝑦 ∈ (0...(𝑁 − 1)) → if(𝑦 < 0, 𝑦, (𝑦 + 1)) / 𝑗((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...𝑗)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((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}))))
362361mpteq2ia 4668 . . . 4 (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < 0, 𝑦, (𝑦 + 1)) / 𝑗((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...𝑗)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((𝑗 + 1)...𝑁)) × {0})))) = (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}))))
363342, 362syl6eqr 2662 . . 3 (𝜑𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < 0, 𝑦, (𝑦 + 1)) / 𝑗((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...𝑗)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((𝑗 + 1)...𝑁)) × {0})))))
364 opex 4859 . . . . . . . . . . 11 ⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩ ∈ V
365364, 51op2ndd 7070 . . . . . . . . . 10 (𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩, 0⟩ → (2nd𝑡) = 0)
366365breq2d 4595 . . . . . . . . 9 (𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩, 0⟩ → (𝑦 < (2nd𝑡) ↔ 𝑦 < 0))
367366ifbid 4058 . . . . . . . 8 (𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩, 0⟩ → if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < 0, 𝑦, (𝑦 + 1)))
368367csbeq1d 3506 . . . . . . 7 (𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩, 0⟩ → if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘𝑓 + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = if(𝑦 < 0, 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘𝑓 + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))))
369364, 51op1std 7069 . . . . . . . . . 10 (𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩, 0⟩ → (1st𝑡) = ⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩)
37097mptex 6390 . . . . . . . . . . 11 (𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))) ∈ V
371370, 330op1std 7069 . . . . . . . . . 10 ((1st𝑡) = ⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩ → (1st ‘(1st𝑡)) = (𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))))
372369, 371syl 17 . . . . . . . . 9 (𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩, 0⟩ → (1st ‘(1st𝑡)) = (𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))))
373370, 330op2ndd 7070 . . . . . . . . . . . . 13 ((1st𝑡) = ⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩ → (2nd ‘(1st𝑡)) = ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))))
374369, 373syl 17 . . . . . . . . . . . 12 (𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩, 0⟩ → (2nd ‘(1st𝑡)) = ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))))
375374imaeq1d 5384 . . . . . . . . . . 11 (𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩, 0⟩ → ((2nd ‘(1st𝑡)) “ (1...𝑗)) = (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...𝑗)))
376375xpeq1d 5062 . . . . . . . . . 10 (𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩, 0⟩ → (((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) = ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...𝑗)) × {1}))
377374imaeq1d 5384 . . . . . . . . . . 11 (𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩, 0⟩ → ((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) = (((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((𝑗 + 1)...𝑁)))
378377xpeq1d 5062 . . . . . . . . . 10 (𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩, 0⟩ → (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((𝑗 + 1)...𝑁)) × {0}))
379376, 378uneq12d 3730 . . . . . . . . 9 (𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩, 0⟩ → ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})) = (((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ (1...𝑗)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((𝑗 + 1)...𝑁)) × {0})))
380372, 379oveq12d 6567 . . . . . . . 8 (𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩, 0⟩ → ((1st ‘(1st𝑡)) ∘𝑓 + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((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...𝑗)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((𝑗 + 1)...𝑁)) × {0}))))
381380csbeq2dv 3944 . . . . . . 7 (𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩, 0⟩ → if(𝑦 < 0, 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘𝑓 + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = if(𝑦 < 0, 𝑦, (𝑦 + 1)) / 𝑗((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...𝑗)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((𝑗 + 1)...𝑁)) × {0}))))
382368, 381eqtrd 2644 . . . . . 6 (𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩, 0⟩ → if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑡)) ∘𝑓 + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = if(𝑦 < 0, 𝑦, (𝑦 + 1)) / 𝑗((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...𝑗)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((𝑗 + 1)...𝑁)) × {0}))))
383382mpteq2dv 4673 . . . . 5 (𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩, 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})))) = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < 0, 𝑦, (𝑦 + 1)) / 𝑗((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...𝑗)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((𝑗 + 1)...𝑁)) × {0})))))
384383eqeq2d 2620 . . . 4 (𝑡 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩, 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})))) ↔ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < 0, 𝑦, (𝑦 + 1)) / 𝑗((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...𝑗)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((𝑗 + 1)...𝑁)) × {0}))))))
385384, 10elrab2 3333 . . 3 (⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩, 0⟩ ∈ 𝑆 ↔ (⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩, 0⟩ ∈ ((((0..^𝐾) ↑𝑚 (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)) ∧ 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < 0, 𝑦, (𝑦 + 1)) / 𝑗((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...𝑗)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1)))) “ ((𝑗 + 1)...𝑁)) × {0}))))))
386340, 363, 385sylanbrc 695 . 2 (𝜑 → ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩, 0⟩ ∈ 𝑆)
387364, 51op2ndd 7070 . . . . . 6 (𝑇 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩, 0⟩ → (2nd𝑇) = 0)
388387eqcoms 2618 . . . . 5 (⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩, 0⟩ = 𝑇 → (2nd𝑇) = 0)
38927nnne0d 10942 . . . . . . 7 (𝜑𝑁 ≠ 0)
390389necomd 2837 . . . . . 6 (𝜑 → 0 ≠ 𝑁)
391 neeq1 2844 . . . . . 6 ((2nd𝑇) = 0 → ((2nd𝑇) ≠ 𝑁 ↔ 0 ≠ 𝑁))
392390, 391syl5ibrcom 236 . . . . 5 (𝜑 → ((2nd𝑇) = 0 → (2nd𝑇) ≠ 𝑁))
393388, 392syl5 33 . . . 4 (𝜑 → (⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩, 0⟩ = 𝑇 → (2nd𝑇) ≠ 𝑁))
394393necon2d 2805 . . 3 (𝜑 → ((2nd𝑇) = 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))))⟩, 0⟩ ≠ 𝑇))
395169, 394mpd 15 . 2 (𝜑 → ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩, 0⟩ ≠ 𝑇)
396 neeq1 2844 . . 3 (𝑧 = ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩, 0⟩ → (𝑧𝑇 ↔ ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩, 0⟩ ≠ 𝑇))
397396rspcev 3282 . 2 ((⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩, 0⟩ ∈ 𝑆 ∧ ⟨⟨(𝑛 ∈ (1...𝑁) ↦ (((1st ‘(1st𝑇))‘𝑛) − if(𝑛 = ((2nd ‘(1st𝑇))‘𝑁), 1, 0))), ((2nd ‘(1st𝑇)) ∘ (𝑛 ∈ (1...𝑁) ↦ if(𝑛 = 1, 𝑁, (𝑛 − 1))))⟩, 0⟩ ≠ 𝑇) → ∃𝑧𝑆 𝑧𝑇)
398386, 395, 397syl2anc 691 1 (𝜑 → ∃𝑧𝑆 𝑧𝑇)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  {cab 2596   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  ∃!wreu 2898  {crab 2900  Vcvv 3173  ⦋csb 3499   ∪ 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   “ 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:  poimirlem21  32600
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