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Theorem poimirlem15 35091
 Description: Lemma for poimir 35109, that the face in poimirlem22 35098 is a face. (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)
poimirlem15.3 (𝜑 → (2nd𝑇) ∈ (1...(𝑁 − 1)))
Assertion
Ref Expression
poimirlem15 (𝜑 → ⟨⟨(1st ‘(1st𝑇)), ((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))))⟩, (2nd𝑇)⟩ ∈ 𝑆)
Distinct variable groups:   𝑓,𝑗,𝑡,𝑦   𝜑,𝑗,𝑦   𝑗,𝐹,𝑦   𝑗,𝑁,𝑦   𝑇,𝑗,𝑦   𝜑,𝑡   𝑓,𝐾,𝑗,𝑡   𝑓,𝑁,𝑡   𝑇,𝑓   𝑓,𝐹,𝑡   𝑡,𝑇   𝑆,𝑗,𝑡,𝑦
Allowed substitution hints:   𝜑(𝑓)   𝑆(𝑓)   𝐾(𝑦)

Proof of Theorem poimirlem15
StepHypRef Expression
1 poimirlem22.2 . . . . . 6 (𝜑𝑇𝑆)
2 elrabi 3623 . . . . . . 7 (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)))
3 poimirlem22.s . . . . . . 7 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}))))}
42, 3eleq2s 2908 . . . . . 6 (𝑇𝑆𝑇 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
51, 4syl 17 . . . . 5 (𝜑𝑇 ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
6 xp1st 7706 . . . . 5 (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)}))
7 xp1st 7706 . . . . 5 ((1st𝑇) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (1st ‘(1st𝑇)) ∈ ((0..^𝐾) ↑m (1...𝑁)))
85, 6, 73syl 18 . . . 4 (𝜑 → (1st ‘(1st𝑇)) ∈ ((0..^𝐾) ↑m (1...𝑁)))
9 xp2nd 7707 . . . . . . . 8 ((1st𝑇) ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → (2nd ‘(1st𝑇)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
105, 6, 93syl 18 . . . . . . 7 (𝜑 → (2nd ‘(1st𝑇)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
11 fvex 6659 . . . . . . . 8 (2nd ‘(1st𝑇)) ∈ V
12 f1oeq1 6580 . . . . . . . 8 (𝑓 = (2nd ‘(1st𝑇)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁)))
1311, 12elab 3615 . . . . . . 7 ((2nd ‘(1st𝑇)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
1410, 13sylib 221 . . . . . 6 (𝜑 → (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
15 poimirlem15.3 . . . . . . . . . . . . 13 (𝜑 → (2nd𝑇) ∈ (1...(𝑁 − 1)))
16 elfznn 12934 . . . . . . . . . . . . 13 ((2nd𝑇) ∈ (1...(𝑁 − 1)) → (2nd𝑇) ∈ ℕ)
1715, 16syl 17 . . . . . . . . . . . 12 (𝜑 → (2nd𝑇) ∈ ℕ)
1817nnred 11643 . . . . . . . . . . 11 (𝜑 → (2nd𝑇) ∈ ℝ)
1918ltp1d 11562 . . . . . . . . . . 11 (𝜑 → (2nd𝑇) < ((2nd𝑇) + 1))
2018, 19ltned 10768 . . . . . . . . . 10 (𝜑 → (2nd𝑇) ≠ ((2nd𝑇) + 1))
2120necomd 3042 . . . . . . . . . 10 (𝜑 → ((2nd𝑇) + 1) ≠ (2nd𝑇))
22 fvex 6659 . . . . . . . . . . 11 (2nd𝑇) ∈ V
23 ovex 7169 . . . . . . . . . . 11 ((2nd𝑇) + 1) ∈ V
24 f1oprg 6635 . . . . . . . . . . 11 ((((2nd𝑇) ∈ V ∧ ((2nd𝑇) + 1) ∈ V) ∧ (((2nd𝑇) + 1) ∈ V ∧ (2nd𝑇) ∈ V)) → (((2nd𝑇) ≠ ((2nd𝑇) + 1) ∧ ((2nd𝑇) + 1) ≠ (2nd𝑇)) → {⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩}:{(2nd𝑇), ((2nd𝑇) + 1)}–1-1-onto→{((2nd𝑇) + 1), (2nd𝑇)}))
2522, 23, 23, 22, 24mp4an 692 . . . . . . . . . 10 (((2nd𝑇) ≠ ((2nd𝑇) + 1) ∧ ((2nd𝑇) + 1) ≠ (2nd𝑇)) → {⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩}:{(2nd𝑇), ((2nd𝑇) + 1)}–1-1-onto→{((2nd𝑇) + 1), (2nd𝑇)})
2620, 21, 25syl2anc 587 . . . . . . . . 9 (𝜑 → {⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩}:{(2nd𝑇), ((2nd𝑇) + 1)}–1-1-onto→{((2nd𝑇) + 1), (2nd𝑇)})
27 prcom 4628 . . . . . . . . . 10 {((2nd𝑇) + 1), (2nd𝑇)} = {(2nd𝑇), ((2nd𝑇) + 1)}
28 f1oeq3 6582 . . . . . . . . . 10 ({((2nd𝑇) + 1), (2nd𝑇)} = {(2nd𝑇), ((2nd𝑇) + 1)} → ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩}:{(2nd𝑇), ((2nd𝑇) + 1)}–1-1-onto→{((2nd𝑇) + 1), (2nd𝑇)} ↔ {⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩}:{(2nd𝑇), ((2nd𝑇) + 1)}–1-1-onto→{(2nd𝑇), ((2nd𝑇) + 1)}))
2927, 28ax-mp 5 . . . . . . . . 9 ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩}:{(2nd𝑇), ((2nd𝑇) + 1)}–1-1-onto→{((2nd𝑇) + 1), (2nd𝑇)} ↔ {⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩}:{(2nd𝑇), ((2nd𝑇) + 1)}–1-1-onto→{(2nd𝑇), ((2nd𝑇) + 1)})
3026, 29sylib 221 . . . . . . . 8 (𝜑 → {⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩}:{(2nd𝑇), ((2nd𝑇) + 1)}–1-1-onto→{(2nd𝑇), ((2nd𝑇) + 1)})
31 f1oi 6628 . . . . . . . 8 ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})):((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})–1-1-onto→((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})
32 disjdif 4379 . . . . . . . . 9 ({(2nd𝑇), ((2nd𝑇) + 1)} ∩ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) = ∅
33 f1oun 6610 . . . . . . . . 9 ((({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩}:{(2nd𝑇), ((2nd𝑇) + 1)}–1-1-onto→{(2nd𝑇), ((2nd𝑇) + 1)} ∧ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})):((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})–1-1-onto→((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) ∧ (({(2nd𝑇), ((2nd𝑇) + 1)} ∩ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) = ∅ ∧ ({(2nd𝑇), ((2nd𝑇) + 1)} ∩ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) = ∅)) → ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))):({(2nd𝑇), ((2nd𝑇) + 1)} ∪ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))–1-1-onto→({(2nd𝑇), ((2nd𝑇) + 1)} ∪ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))
3432, 32, 33mpanr12 704 . . . . . . . 8 (({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩}:{(2nd𝑇), ((2nd𝑇) + 1)}–1-1-onto→{(2nd𝑇), ((2nd𝑇) + 1)} ∧ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})):((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})–1-1-onto→((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) → ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))):({(2nd𝑇), ((2nd𝑇) + 1)} ∪ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))–1-1-onto→({(2nd𝑇), ((2nd𝑇) + 1)} ∪ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))
3530, 31, 34sylancl 589 . . . . . . 7 (𝜑 → ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))):({(2nd𝑇), ((2nd𝑇) + 1)} ∪ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))–1-1-onto→({(2nd𝑇), ((2nd𝑇) + 1)} ∪ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))
36 poimir.0 . . . . . . . . . . . . . . 15 (𝜑𝑁 ∈ ℕ)
3736nncnd 11644 . . . . . . . . . . . . . 14 (𝜑𝑁 ∈ ℂ)
38 npcan1 11057 . . . . . . . . . . . . . 14 (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁)
3937, 38syl 17 . . . . . . . . . . . . 13 (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
4036nnzd 12077 . . . . . . . . . . . . . . 15 (𝜑𝑁 ∈ ℤ)
41 peano2zm 12016 . . . . . . . . . . . . . . 15 (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ)
4240, 41syl 17 . . . . . . . . . . . . . 14 (𝜑 → (𝑁 − 1) ∈ ℤ)
43 uzid 12249 . . . . . . . . . . . . . 14 ((𝑁 − 1) ∈ ℤ → (𝑁 − 1) ∈ (ℤ‘(𝑁 − 1)))
44 peano2uz 12292 . . . . . . . . . . . . . 14 ((𝑁 − 1) ∈ (ℤ‘(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈ (ℤ‘(𝑁 − 1)))
4542, 43, 443syl 18 . . . . . . . . . . . . 13 (𝜑 → ((𝑁 − 1) + 1) ∈ (ℤ‘(𝑁 − 1)))
4639, 45eqeltrrd 2891 . . . . . . . . . . . 12 (𝜑𝑁 ∈ (ℤ‘(𝑁 − 1)))
47 fzss2 12945 . . . . . . . . . . . 12 (𝑁 ∈ (ℤ‘(𝑁 − 1)) → (1...(𝑁 − 1)) ⊆ (1...𝑁))
4846, 47syl 17 . . . . . . . . . . 11 (𝜑 → (1...(𝑁 − 1)) ⊆ (1...𝑁))
4948, 15sseldd 3916 . . . . . . . . . 10 (𝜑 → (2nd𝑇) ∈ (1...𝑁))
5017peano2nnd 11645 . . . . . . . . . . 11 (𝜑 → ((2nd𝑇) + 1) ∈ ℕ)
5142zred 12078 . . . . . . . . . . . . 13 (𝜑 → (𝑁 − 1) ∈ ℝ)
5236nnred 11643 . . . . . . . . . . . . 13 (𝜑𝑁 ∈ ℝ)
53 elfzle2 12909 . . . . . . . . . . . . . 14 ((2nd𝑇) ∈ (1...(𝑁 − 1)) → (2nd𝑇) ≤ (𝑁 − 1))
5415, 53syl 17 . . . . . . . . . . . . 13 (𝜑 → (2nd𝑇) ≤ (𝑁 − 1))
5552ltm1d 11564 . . . . . . . . . . . . 13 (𝜑 → (𝑁 − 1) < 𝑁)
5618, 51, 52, 54, 55lelttrd 10790 . . . . . . . . . . . 12 (𝜑 → (2nd𝑇) < 𝑁)
5717nnzd 12077 . . . . . . . . . . . . 13 (𝜑 → (2nd𝑇) ∈ ℤ)
58 zltp1le 12023 . . . . . . . . . . . . 13 (((2nd𝑇) ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((2nd𝑇) < 𝑁 ↔ ((2nd𝑇) + 1) ≤ 𝑁))
5957, 40, 58syl2anc 587 . . . . . . . . . . . 12 (𝜑 → ((2nd𝑇) < 𝑁 ↔ ((2nd𝑇) + 1) ≤ 𝑁))
6056, 59mpbid 235 . . . . . . . . . . 11 (𝜑 → ((2nd𝑇) + 1) ≤ 𝑁)
61 fznn 12973 . . . . . . . . . . . 12 (𝑁 ∈ ℤ → (((2nd𝑇) + 1) ∈ (1...𝑁) ↔ (((2nd𝑇) + 1) ∈ ℕ ∧ ((2nd𝑇) + 1) ≤ 𝑁)))
6240, 61syl 17 . . . . . . . . . . 11 (𝜑 → (((2nd𝑇) + 1) ∈ (1...𝑁) ↔ (((2nd𝑇) + 1) ∈ ℕ ∧ ((2nd𝑇) + 1) ≤ 𝑁)))
6350, 60, 62mpbir2and 712 . . . . . . . . . 10 (𝜑 → ((2nd𝑇) + 1) ∈ (1...𝑁))
64 prssi 4714 . . . . . . . . . 10 (((2nd𝑇) ∈ (1...𝑁) ∧ ((2nd𝑇) + 1) ∈ (1...𝑁)) → {(2nd𝑇), ((2nd𝑇) + 1)} ⊆ (1...𝑁))
6549, 63, 64syl2anc 587 . . . . . . . . 9 (𝜑 → {(2nd𝑇), ((2nd𝑇) + 1)} ⊆ (1...𝑁))
66 undif 4388 . . . . . . . . 9 ({(2nd𝑇), ((2nd𝑇) + 1)} ⊆ (1...𝑁) ↔ ({(2nd𝑇), ((2nd𝑇) + 1)} ∪ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) = (1...𝑁))
6765, 66sylib 221 . . . . . . . 8 (𝜑 → ({(2nd𝑇), ((2nd𝑇) + 1)} ∪ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) = (1...𝑁))
68 f1oeq23 6583 . . . . . . . 8 ((({(2nd𝑇), ((2nd𝑇) + 1)} ∪ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) = (1...𝑁) ∧ ({(2nd𝑇), ((2nd𝑇) + 1)} ∪ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) = (1...𝑁)) → (({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))):({(2nd𝑇), ((2nd𝑇) + 1)} ∪ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))–1-1-onto→({(2nd𝑇), ((2nd𝑇) + 1)} ∪ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) ↔ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))):(1...𝑁)–1-1-onto→(1...𝑁)))
6967, 67, 68syl2anc 587 . . . . . . 7 (𝜑 → (({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))):({(2nd𝑇), ((2nd𝑇) + 1)} ∪ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))–1-1-onto→({(2nd𝑇), ((2nd𝑇) + 1)} ∪ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) ↔ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))):(1...𝑁)–1-1-onto→(1...𝑁)))
7035, 69mpbid 235 . . . . . 6 (𝜑 → ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))):(1...𝑁)–1-1-onto→(1...𝑁))
71 f1oco 6613 . . . . . 6 (((2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁) ∧ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))):(1...𝑁)–1-1-onto→(1...𝑁)) → ((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))):(1...𝑁)–1-1-onto→(1...𝑁))
7214, 70, 71syl2anc 587 . . . . 5 (𝜑 → ((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))):(1...𝑁)–1-1-onto→(1...𝑁))
73 prex 5299 . . . . . . . 8 {⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∈ V
74 ovex 7169 . . . . . . . . 9 (1...𝑁) ∈ V
75 difexg 5196 . . . . . . . . 9 ((1...𝑁) ∈ V → ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) ∈ V)
76 resiexg 7604 . . . . . . . . 9 (((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) ∈ V → ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) ∈ V)
7774, 75, 76mp2b 10 . . . . . . . 8 ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) ∈ V
7873, 77unex 7452 . . . . . . 7 ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))) ∈ V
7911, 78coex 7620 . . . . . 6 ((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) ∈ V
80 f1oeq1 6580 . . . . . 6 (𝑓 = ((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ ((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))):(1...𝑁)–1-1-onto→(1...𝑁)))
8179, 80elab 3615 . . . . 5 (((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ ((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))):(1...𝑁)–1-1-onto→(1...𝑁))
8272, 81sylibr 237 . . . 4 (𝜑 → ((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)})
83 opelxpi 5557 . . . 4 (((1st ‘(1st𝑇)) ∈ ((0..^𝐾) ↑m (1...𝑁)) ∧ ((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) → ⟨(1st ‘(1st𝑇)), ((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))))⟩ ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
848, 82, 83syl2anc 587 . . 3 (𝜑 → ⟨(1st ‘(1st𝑇)), ((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))))⟩ ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}))
85 fz1ssfz0 13001 . . . . 5 (1...𝑁) ⊆ (0...𝑁)
8648, 85sstrdi 3927 . . . 4 (𝜑 → (1...(𝑁 − 1)) ⊆ (0...𝑁))
8786, 15sseldd 3916 . . 3 (𝜑 → (2nd𝑇) ∈ (0...𝑁))
88 opelxpi 5557 . . 3 ((⟨(1st ‘(1st𝑇)), ((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))))⟩ ∈ (((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) ∧ (2nd𝑇) ∈ (0...𝑁)) → ⟨⟨(1st ‘(1st𝑇)), ((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))))⟩, (2nd𝑇)⟩ ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
8984, 87, 88syl2anc 587 . 2 (𝜑 → ⟨⟨(1st ‘(1st𝑇)), ((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))))⟩, (2nd𝑇)⟩ ∈ ((((0..^𝐾) ↑m (1...𝑁)) × {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)}) × (0...𝑁)))
90 fveq2 6646 . . . . . . . . . . . 12 (𝑡 = 𝑇 → (2nd𝑡) = (2nd𝑇))
9190breq2d 5043 . . . . . . . . . . 11 (𝑡 = 𝑇 → (𝑦 < (2nd𝑡) ↔ 𝑦 < (2nd𝑇)))
9291ifbid 4447 . . . . . . . . . 10 (𝑡 = 𝑇 → if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)))
9392csbeq1d 3832 . . . . . . . . 9 (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}))))
94 2fveq3 6651 . . . . . . . . . . 11 (𝑡 = 𝑇 → (1st ‘(1st𝑡)) = (1st ‘(1st𝑇)))
95 2fveq3 6651 . . . . . . . . . . . . . 14 (𝑡 = 𝑇 → (2nd ‘(1st𝑡)) = (2nd ‘(1st𝑇)))
9695imaeq1d 5896 . . . . . . . . . . . . 13 (𝑡 = 𝑇 → ((2nd ‘(1st𝑡)) “ (1...𝑗)) = ((2nd ‘(1st𝑇)) “ (1...𝑗)))
9796xpeq1d 5549 . . . . . . . . . . . 12 (𝑡 = 𝑇 → (((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}))
9895imaeq1d 5896 . . . . . . . . . . . . 13 (𝑡 = 𝑇 → ((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)))
9998xpeq1d 5549 . . . . . . . . . . . 12 (𝑡 = 𝑇 → (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))
10097, 99uneq12d 4091 . . . . . . . . . . 11 (𝑡 = 𝑇 → ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})))
10194, 100oveq12d 7154 . . . . . . . . . 10 (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}))))
102101csbeq2dv 3835 . . . . . . . . 9 (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}))))
10393, 102eqtrd 2833 . . . . . . . 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}))))
104103mpteq2dv 5127 . . . . . . 7 (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})))))
105104eqeq2d 2809 . . . . . 6 (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}))))))
106105, 3elrab2 3631 . . . . 5 (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}))))))
107106simprbi 500 . . . 4 (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})))))
1081, 107syl 17 . . 3 (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})))))
109 imaco 6072 . . . . . . . . . 10 (((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ (1...𝑦)) = ((2nd ‘(1st𝑇)) “ (({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))) “ (1...𝑦)))
110 f1ofn 6592 . . . . . . . . . . . . . . . 16 ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩}:{(2nd𝑇), ((2nd𝑇) + 1)}–1-1-onto→{((2nd𝑇) + 1), (2nd𝑇)} → {⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} Fn {(2nd𝑇), ((2nd𝑇) + 1)})
11126, 110syl 17 . . . . . . . . . . . . . . 15 (𝜑 → {⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} Fn {(2nd𝑇), ((2nd𝑇) + 1)})
112111ad2antrr 725 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → {⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} Fn {(2nd𝑇), ((2nd𝑇) + 1)})
113 incom 4128 . . . . . . . . . . . . . . 15 ({(2nd𝑇), ((2nd𝑇) + 1)} ∩ (1...𝑦)) = ((1...𝑦) ∩ {(2nd𝑇), ((2nd𝑇) + 1)})
114 elfznn0 12998 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ∈ ℕ0)
115114nn0red 11947 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ∈ ℝ)
116 ltnle 10712 . . . . . . . . . . . . . . . . . . . . 21 ((𝑦 ∈ ℝ ∧ (2nd𝑇) ∈ ℝ) → (𝑦 < (2nd𝑇) ↔ ¬ (2nd𝑇) ≤ 𝑦))
117115, 18, 116syl2anr 599 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (𝑦 < (2nd𝑇) ↔ ¬ (2nd𝑇) ≤ 𝑦))
118117biimpa 480 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → ¬ (2nd𝑇) ≤ 𝑦)
119 elfzle2 12909 . . . . . . . . . . . . . . . . . . 19 ((2nd𝑇) ∈ (1...𝑦) → (2nd𝑇) ≤ 𝑦)
120118, 119nsyl 142 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → ¬ (2nd𝑇) ∈ (1...𝑦))
121 disjsn 4607 . . . . . . . . . . . . . . . . . 18 (((1...𝑦) ∩ {(2nd𝑇)}) = ∅ ↔ ¬ (2nd𝑇) ∈ (1...𝑦))
122120, 121sylibr 237 . . . . . . . . . . . . . . . . 17 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → ((1...𝑦) ∩ {(2nd𝑇)}) = ∅)
123115ad2antlr 726 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → 𝑦 ∈ ℝ)
12418ad2antrr 725 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → (2nd𝑇) ∈ ℝ)
12550nnred 11643 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → ((2nd𝑇) + 1) ∈ ℝ)
126125ad2antrr 725 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → ((2nd𝑇) + 1) ∈ ℝ)
127 simpr 488 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → 𝑦 < (2nd𝑇))
12819ad2antrr 725 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → (2nd𝑇) < ((2nd𝑇) + 1))
129123, 124, 126, 127, 128lttrd 10793 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → 𝑦 < ((2nd𝑇) + 1))
130 ltnle 10712 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑦 ∈ ℝ ∧ ((2nd𝑇) + 1) ∈ ℝ) → (𝑦 < ((2nd𝑇) + 1) ↔ ¬ ((2nd𝑇) + 1) ≤ 𝑦))
131115, 125, 130syl2anr 599 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (𝑦 < ((2nd𝑇) + 1) ↔ ¬ ((2nd𝑇) + 1) ≤ 𝑦))
132131adantr 484 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → (𝑦 < ((2nd𝑇) + 1) ↔ ¬ ((2nd𝑇) + 1) ≤ 𝑦))
133129, 132mpbid 235 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → ¬ ((2nd𝑇) + 1) ≤ 𝑦)
134 elfzle2 12909 . . . . . . . . . . . . . . . . . . 19 (((2nd𝑇) + 1) ∈ (1...𝑦) → ((2nd𝑇) + 1) ≤ 𝑦)
135133, 134nsyl 142 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → ¬ ((2nd𝑇) + 1) ∈ (1...𝑦))
136 disjsn 4607 . . . . . . . . . . . . . . . . . 18 (((1...𝑦) ∩ {((2nd𝑇) + 1)}) = ∅ ↔ ¬ ((2nd𝑇) + 1) ∈ (1...𝑦))
137135, 136sylibr 237 . . . . . . . . . . . . . . . . 17 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → ((1...𝑦) ∩ {((2nd𝑇) + 1)}) = ∅)
138122, 137uneq12d 4091 . . . . . . . . . . . . . . . 16 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → (((1...𝑦) ∩ {(2nd𝑇)}) ∪ ((1...𝑦) ∩ {((2nd𝑇) + 1)})) = (∅ ∪ ∅))
139 df-pr 4528 . . . . . . . . . . . . . . . . . 18 {(2nd𝑇), ((2nd𝑇) + 1)} = ({(2nd𝑇)} ∪ {((2nd𝑇) + 1)})
140139ineq2i 4136 . . . . . . . . . . . . . . . . 17 ((1...𝑦) ∩ {(2nd𝑇), ((2nd𝑇) + 1)}) = ((1...𝑦) ∩ ({(2nd𝑇)} ∪ {((2nd𝑇) + 1)}))
141 indi 4200 . . . . . . . . . . . . . . . . 17 ((1...𝑦) ∩ ({(2nd𝑇)} ∪ {((2nd𝑇) + 1)})) = (((1...𝑦) ∩ {(2nd𝑇)}) ∪ ((1...𝑦) ∩ {((2nd𝑇) + 1)}))
142140, 141eqtr2i 2822 . . . . . . . . . . . . . . . 16 (((1...𝑦) ∩ {(2nd𝑇)}) ∪ ((1...𝑦) ∩ {((2nd𝑇) + 1)})) = ((1...𝑦) ∩ {(2nd𝑇), ((2nd𝑇) + 1)})
143 un0 4298 . . . . . . . . . . . . . . . 16 (∅ ∪ ∅) = ∅
144138, 142, 1433eqtr3g 2856 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → ((1...𝑦) ∩ {(2nd𝑇), ((2nd𝑇) + 1)}) = ∅)
145113, 144syl5eq 2845 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → ({(2nd𝑇), ((2nd𝑇) + 1)} ∩ (1...𝑦)) = ∅)
146 fnimadisj 6453 . . . . . . . . . . . . . 14 (({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} Fn {(2nd𝑇), ((2nd𝑇) + 1)} ∧ ({(2nd𝑇), ((2nd𝑇) + 1)} ∩ (1...𝑦)) = ∅) → ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} “ (1...𝑦)) = ∅)
147112, 145, 146syl2anc 587 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} “ (1...𝑦)) = ∅)
14839adantr 484 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((𝑁 − 1) + 1) = 𝑁)
149 elfzuz3 12902 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ (0...(𝑁 − 1)) → (𝑁 − 1) ∈ (ℤ𝑦))
150 peano2uz 12292 . . . . . . . . . . . . . . . . . . . 20 ((𝑁 − 1) ∈ (ℤ𝑦) → ((𝑁 − 1) + 1) ∈ (ℤ𝑦))
151149, 150syl 17 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈ (ℤ𝑦))
152151adantl 485 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((𝑁 − 1) + 1) ∈ (ℤ𝑦))
153148, 152eqeltrrd 2891 . . . . . . . . . . . . . . . . 17 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ (ℤ𝑦))
154 fzss2 12945 . . . . . . . . . . . . . . . . 17 (𝑁 ∈ (ℤ𝑦) → (1...𝑦) ⊆ (1...𝑁))
155 reldisj 4359 . . . . . . . . . . . . . . . . 17 ((1...𝑦) ⊆ (1...𝑁) → (((1...𝑦) ∩ {(2nd𝑇), ((2nd𝑇) + 1)}) = ∅ ↔ (1...𝑦) ⊆ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))
156153, 154, 1553syl 18 . . . . . . . . . . . . . . . 16 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((1...𝑦) ∩ {(2nd𝑇), ((2nd𝑇) + 1)}) = ∅ ↔ (1...𝑦) ⊆ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))
157156adantr 484 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → (((1...𝑦) ∩ {(2nd𝑇), ((2nd𝑇) + 1)}) = ∅ ↔ (1...𝑦) ⊆ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))
158144, 157mpbid 235 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → (1...𝑦) ⊆ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))
159 resiima 5912 . . . . . . . . . . . . . 14 ((1...𝑦) ⊆ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) → (( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) “ (1...𝑦)) = (1...𝑦))
160158, 159syl 17 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → (( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) “ (1...𝑦)) = (1...𝑦))
161147, 160uneq12d 4091 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → (({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} “ (1...𝑦)) ∪ (( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) “ (1...𝑦))) = (∅ ∪ (1...𝑦)))
162 imaundir 5977 . . . . . . . . . . . 12 (({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))) “ (1...𝑦)) = (({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} “ (1...𝑦)) ∪ (( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) “ (1...𝑦)))
163 uncom 4080 . . . . . . . . . . . . 13 (∅ ∪ (1...𝑦)) = ((1...𝑦) ∪ ∅)
164 un0 4298 . . . . . . . . . . . . 13 ((1...𝑦) ∪ ∅) = (1...𝑦)
165163, 164eqtr2i 2822 . . . . . . . . . . . 12 (1...𝑦) = (∅ ∪ (1...𝑦))
166161, 162, 1653eqtr4g 2858 . . . . . . . . . . 11 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → (({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))) “ (1...𝑦)) = (1...𝑦))
167166imaeq2d 5897 . . . . . . . . . 10 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → ((2nd ‘(1st𝑇)) “ (({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))) “ (1...𝑦))) = ((2nd ‘(1st𝑇)) “ (1...𝑦)))
168109, 167syl5eq 2845 . . . . . . . . 9 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → (((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ (1...𝑦)) = ((2nd ‘(1st𝑇)) “ (1...𝑦)))
169168xpeq1d 5549 . . . . . . . 8 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → ((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ (1...𝑦)) × {1}) = (((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}))
170 imaco 6072 . . . . . . . . . 10 (((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ ((𝑦 + 1)...𝑁)) = ((2nd ‘(1st𝑇)) “ (({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))) “ ((𝑦 + 1)...𝑁)))
171 imaundir 5977 . . . . . . . . . . . . 13 (({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))) “ ((𝑦 + 1)...𝑁)) = (({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} “ ((𝑦 + 1)...𝑁)) ∪ (( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) “ ((𝑦 + 1)...𝑁)))
172 imassrn 5908 . . . . . . . . . . . . . . . . 17 ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} “ ((𝑦 + 1)...𝑁)) ⊆ ran {⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩}
173172a1i 11 . . . . . . . . . . . . . . . 16 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} “ ((𝑦 + 1)...𝑁)) ⊆ ran {⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩})
174 fnima 6451 . . . . . . . . . . . . . . . . . . 19 ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} Fn {(2nd𝑇), ((2nd𝑇) + 1)} → ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} “ {(2nd𝑇), ((2nd𝑇) + 1)}) = ran {⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩})
17526, 110, 1743syl 18 . . . . . . . . . . . . . . . . . 18 (𝜑 → ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} “ {(2nd𝑇), ((2nd𝑇) + 1)}) = ran {⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩})
176175ad2antrr 725 . . . . . . . . . . . . . . . . 17 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} “ {(2nd𝑇), ((2nd𝑇) + 1)}) = ran {⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩})
177 elfzelz 12905 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ∈ ℤ)
178 zltp1le 12023 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑦 ∈ ℤ ∧ (2nd𝑇) ∈ ℤ) → (𝑦 < (2nd𝑇) ↔ (𝑦 + 1) ≤ (2nd𝑇)))
179177, 57, 178syl2anr 599 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (𝑦 < (2nd𝑇) ↔ (𝑦 + 1) ≤ (2nd𝑇)))
180179biimpa 480 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → (𝑦 + 1) ≤ (2nd𝑇))
18118, 52, 56ltled 10780 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (2nd𝑇) ≤ 𝑁)
182181ad2antrr 725 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → (2nd𝑇) ≤ 𝑁)
18357adantr 484 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (2nd𝑇) ∈ ℤ)
184 nn0p1nn 11927 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 ∈ ℕ0 → (𝑦 + 1) ∈ ℕ)
185114, 184syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈ ℕ)
186185nnzd 12077 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈ ℤ)
187186adantl 485 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (𝑦 + 1) ∈ ℤ)
18840adantr 484 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ ℤ)
189 elfz 12894 . . . . . . . . . . . . . . . . . . . . . 22 (((2nd𝑇) ∈ ℤ ∧ (𝑦 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((2nd𝑇) ∈ ((𝑦 + 1)...𝑁) ↔ ((𝑦 + 1) ≤ (2nd𝑇) ∧ (2nd𝑇) ≤ 𝑁)))
190183, 187, 188, 189syl3anc 1368 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((2nd𝑇) ∈ ((𝑦 + 1)...𝑁) ↔ ((𝑦 + 1) ≤ (2nd𝑇) ∧ (2nd𝑇) ≤ 𝑁)))
191190adantr 484 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → ((2nd𝑇) ∈ ((𝑦 + 1)...𝑁) ↔ ((𝑦 + 1) ≤ (2nd𝑇) ∧ (2nd𝑇) ≤ 𝑁)))
192180, 182, 191mpbir2and 712 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → (2nd𝑇) ∈ ((𝑦 + 1)...𝑁))
193 1red 10634 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → 1 ∈ ℝ)
194 ltle 10721 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑦 ∈ ℝ ∧ (2nd𝑇) ∈ ℝ) → (𝑦 < (2nd𝑇) → 𝑦 ≤ (2nd𝑇)))
195115, 18, 194syl2anr 599 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (𝑦 < (2nd𝑇) → 𝑦 ≤ (2nd𝑇)))
196195imp 410 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → 𝑦 ≤ (2nd𝑇))
197123, 124, 193, 196leadd1dd 11246 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → (𝑦 + 1) ≤ ((2nd𝑇) + 1))
19860ad2antrr 725 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → ((2nd𝑇) + 1) ≤ 𝑁)
19957peano2zd 12081 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → ((2nd𝑇) + 1) ∈ ℤ)
200199adantr 484 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((2nd𝑇) + 1) ∈ ℤ)
201 elfz 12894 . . . . . . . . . . . . . . . . . . . . . 22 ((((2nd𝑇) + 1) ∈ ℤ ∧ (𝑦 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) → (((2nd𝑇) + 1) ∈ ((𝑦 + 1)...𝑁) ↔ ((𝑦 + 1) ≤ ((2nd𝑇) + 1) ∧ ((2nd𝑇) + 1) ≤ 𝑁)))
202200, 187, 188, 201syl3anc 1368 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((2nd𝑇) + 1) ∈ ((𝑦 + 1)...𝑁) ↔ ((𝑦 + 1) ≤ ((2nd𝑇) + 1) ∧ ((2nd𝑇) + 1) ≤ 𝑁)))
203202adantr 484 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → (((2nd𝑇) + 1) ∈ ((𝑦 + 1)...𝑁) ↔ ((𝑦 + 1) ≤ ((2nd𝑇) + 1) ∧ ((2nd𝑇) + 1) ≤ 𝑁)))
204197, 198, 203mpbir2and 712 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → ((2nd𝑇) + 1) ∈ ((𝑦 + 1)...𝑁))
205 prssi 4714 . . . . . . . . . . . . . . . . . . 19 (((2nd𝑇) ∈ ((𝑦 + 1)...𝑁) ∧ ((2nd𝑇) + 1) ∈ ((𝑦 + 1)...𝑁)) → {(2nd𝑇), ((2nd𝑇) + 1)} ⊆ ((𝑦 + 1)...𝑁))
206192, 204, 205syl2anc 587 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → {(2nd𝑇), ((2nd𝑇) + 1)} ⊆ ((𝑦 + 1)...𝑁))
207 imass2 5933 . . . . . . . . . . . . . . . . . 18 ({(2nd𝑇), ((2nd𝑇) + 1)} ⊆ ((𝑦 + 1)...𝑁) → ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} “ {(2nd𝑇), ((2nd𝑇) + 1)}) ⊆ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} “ ((𝑦 + 1)...𝑁)))
208206, 207syl 17 . . . . . . . . . . . . . . . . 17 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} “ {(2nd𝑇), ((2nd𝑇) + 1)}) ⊆ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} “ ((𝑦 + 1)...𝑁)))
209176, 208eqsstrrd 3954 . . . . . . . . . . . . . . . 16 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → ran {⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ⊆ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} “ ((𝑦 + 1)...𝑁)))
210173, 209eqssd 3932 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} “ ((𝑦 + 1)...𝑁)) = ran {⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩})
211 f1ofo 6598 . . . . . . . . . . . . . . . . . 18 ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩}:{(2nd𝑇), ((2nd𝑇) + 1)}–1-1-onto→{((2nd𝑇) + 1), (2nd𝑇)} → {⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩}:{(2nd𝑇), ((2nd𝑇) + 1)}–onto→{((2nd𝑇) + 1), (2nd𝑇)})
212 forn 6569 . . . . . . . . . . . . . . . . . 18 ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩}:{(2nd𝑇), ((2nd𝑇) + 1)}–onto→{((2nd𝑇) + 1), (2nd𝑇)} → ran {⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} = {((2nd𝑇) + 1), (2nd𝑇)})
21326, 211, 2123syl 18 . . . . . . . . . . . . . . . . 17 (𝜑 → ran {⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} = {((2nd𝑇) + 1), (2nd𝑇)})
214213, 27eqtrdi 2849 . . . . . . . . . . . . . . . 16 (𝜑 → ran {⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} = {(2nd𝑇), ((2nd𝑇) + 1)})
215214ad2antrr 725 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → ran {⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} = {(2nd𝑇), ((2nd𝑇) + 1)})
216210, 215eqtrd 2833 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} “ ((𝑦 + 1)...𝑁)) = {(2nd𝑇), ((2nd𝑇) + 1)})
217 undif 4388 . . . . . . . . . . . . . . . . 17 ({(2nd𝑇), ((2nd𝑇) + 1)} ⊆ ((𝑦 + 1)...𝑁) ↔ ({(2nd𝑇), ((2nd𝑇) + 1)} ∪ (((𝑦 + 1)...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) = ((𝑦 + 1)...𝑁))
218206, 217sylib 221 . . . . . . . . . . . . . . . 16 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → ({(2nd𝑇), ((2nd𝑇) + 1)} ∪ (((𝑦 + 1)...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) = ((𝑦 + 1)...𝑁))
219218imaeq2d 5897 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → (( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) “ ({(2nd𝑇), ((2nd𝑇) + 1)} ∪ (((𝑦 + 1)...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))) = (( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) “ ((𝑦 + 1)...𝑁)))
220 fnresi 6449 . . . . . . . . . . . . . . . . . . 19 ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) Fn ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})
221 incom 4128 . . . . . . . . . . . . . . . . . . . 20 (((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) ∩ {(2nd𝑇), ((2nd𝑇) + 1)}) = ({(2nd𝑇), ((2nd𝑇) + 1)} ∩ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))
222221, 32eqtri 2821 . . . . . . . . . . . . . . . . . . 19 (((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) ∩ {(2nd𝑇), ((2nd𝑇) + 1)}) = ∅
223 fnimadisj 6453 . . . . . . . . . . . . . . . . . . 19 ((( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) Fn ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) ∧ (((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) ∩ {(2nd𝑇), ((2nd𝑇) + 1)}) = ∅) → (( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) “ {(2nd𝑇), ((2nd𝑇) + 1)}) = ∅)
224220, 222, 223mp2an 691 . . . . . . . . . . . . . . . . . 18 (( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) “ {(2nd𝑇), ((2nd𝑇) + 1)}) = ∅
225224a1i 11 . . . . . . . . . . . . . . . . 17 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → (( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) “ {(2nd𝑇), ((2nd𝑇) + 1)}) = ∅)
226 nnuz 12272 . . . . . . . . . . . . . . . . . . . . . 22 ℕ = (ℤ‘1)
227185, 226eleqtrdi 2900 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈ (ℤ‘1))
228 fzss1 12944 . . . . . . . . . . . . . . . . . . . . 21 ((𝑦 + 1) ∈ (ℤ‘1) → ((𝑦 + 1)...𝑁) ⊆ (1...𝑁))
229227, 228syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑦 + 1)...𝑁) ⊆ (1...𝑁))
230229ssdifd 4068 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ (0...(𝑁 − 1)) → (((𝑦 + 1)...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) ⊆ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))
231 resiima 5912 . . . . . . . . . . . . . . . . . . 19 ((((𝑦 + 1)...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) ⊆ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) → (( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) “ (((𝑦 + 1)...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) = (((𝑦 + 1)...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))
232230, 231syl 17 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ (0...(𝑁 − 1)) → (( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) “ (((𝑦 + 1)...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) = (((𝑦 + 1)...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))
233232ad2antlr 726 . . . . . . . . . . . . . . . . 17 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → (( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) “ (((𝑦 + 1)...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) = (((𝑦 + 1)...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))
234225, 233uneq12d 4091 . . . . . . . . . . . . . . . 16 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → ((( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) “ {(2nd𝑇), ((2nd𝑇) + 1)}) ∪ (( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) “ (((𝑦 + 1)...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))) = (∅ ∪ (((𝑦 + 1)...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))
235 imaundi 5976 . . . . . . . . . . . . . . . 16 (( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) “ ({(2nd𝑇), ((2nd𝑇) + 1)} ∪ (((𝑦 + 1)...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))) = ((( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) “ {(2nd𝑇), ((2nd𝑇) + 1)}) ∪ (( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) “ (((𝑦 + 1)...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))
236 uncom 4080 . . . . . . . . . . . . . . . . 17 (∅ ∪ (((𝑦 + 1)...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) = ((((𝑦 + 1)...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) ∪ ∅)
237 un0 4298 . . . . . . . . . . . . . . . . 17 ((((𝑦 + 1)...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) ∪ ∅) = (((𝑦 + 1)...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})
238236, 237eqtr2i 2822 . . . . . . . . . . . . . . . 16 (((𝑦 + 1)...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) = (∅ ∪ (((𝑦 + 1)...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))
239234, 235, 2383eqtr4g 2858 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → (( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) “ ({(2nd𝑇), ((2nd𝑇) + 1)} ∪ (((𝑦 + 1)...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))) = (((𝑦 + 1)...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))
240219, 239eqtr3d 2835 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → (( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) “ ((𝑦 + 1)...𝑁)) = (((𝑦 + 1)...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))
241216, 240uneq12d 4091 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → (({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} “ ((𝑦 + 1)...𝑁)) ∪ (( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) “ ((𝑦 + 1)...𝑁))) = ({(2nd𝑇), ((2nd𝑇) + 1)} ∪ (((𝑦 + 1)...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))
242171, 241syl5eq 2845 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → (({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))) “ ((𝑦 + 1)...𝑁)) = ({(2nd𝑇), ((2nd𝑇) + 1)} ∪ (((𝑦 + 1)...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))
243242, 218eqtrd 2833 . . . . . . . . . . 11 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → (({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))) “ ((𝑦 + 1)...𝑁)) = ((𝑦 + 1)...𝑁))
244243imaeq2d 5897 . . . . . . . . . 10 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → ((2nd ‘(1st𝑇)) “ (({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))) “ ((𝑦 + 1)...𝑁))) = ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)))
245170, 244syl5eq 2845 . . . . . . . . 9 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → (((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ ((𝑦 + 1)...𝑁)) = ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)))
246245xpeq1d 5549 . . . . . . . 8 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → ((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ ((𝑦 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))
247169, 246uneq12d 4091 . . . . . . 7 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → (((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ ((𝑦 + 1)...𝑁)) × {0})) = ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})))
248247oveq2d 7152 . . . . . 6 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → ((1st ‘(1st𝑇)) ∘f + (((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ ((𝑦 + 1)...𝑁)) × {0}))) = ((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))))
249 iftrue 4431 . . . . . . . . 9 (𝑦 < (2nd𝑇) → if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) = 𝑦)
250249csbeq1d 3832 . . . . . . . 8 (𝑦 < (2nd𝑇) → if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑇)) ∘f + (((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ (1...𝑗)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ ((𝑗 + 1)...𝑁)) × {0}))) = 𝑦 / 𝑗((1st ‘(1st𝑇)) ∘f + (((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ (1...𝑗)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ ((𝑗 + 1)...𝑁)) × {0}))))
251 vex 3444 . . . . . . . . 9 𝑦 ∈ V
252 oveq2 7144 . . . . . . . . . . . . 13 (𝑗 = 𝑦 → (1...𝑗) = (1...𝑦))
253252imaeq2d 5897 . . . . . . . . . . . 12 (𝑗 = 𝑦 → (((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ (1...𝑗)) = (((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ (1...𝑦)))
254253xpeq1d 5549 . . . . . . . . . . 11 (𝑗 = 𝑦 → ((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ (1...𝑗)) × {1}) = ((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ (1...𝑦)) × {1}))
255 oveq1 7143 . . . . . . . . . . . . . 14 (𝑗 = 𝑦 → (𝑗 + 1) = (𝑦 + 1))
256255oveq1d 7151 . . . . . . . . . . . . 13 (𝑗 = 𝑦 → ((𝑗 + 1)...𝑁) = ((𝑦 + 1)...𝑁))
257256imaeq2d 5897 . . . . . . . . . . . 12 (𝑗 = 𝑦 → (((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ ((𝑗 + 1)...𝑁)) = (((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ ((𝑦 + 1)...𝑁)))
258257xpeq1d 5549 . . . . . . . . . . 11 (𝑗 = 𝑦 → ((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ ((𝑗 + 1)...𝑁)) × {0}) = ((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ ((𝑦 + 1)...𝑁)) × {0}))
259254, 258uneq12d 4091 . . . . . . . . . 10 (𝑗 = 𝑦 → (((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ (1...𝑗)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ ((𝑗 + 1)...𝑁)) × {0})) = (((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ ((𝑦 + 1)...𝑁)) × {0})))
260259oveq2d 7152 . . . . . . . . 9 (𝑗 = 𝑦 → ((1st ‘(1st𝑇)) ∘f + (((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ (1...𝑗)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st𝑇)) ∘f + (((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ ((𝑦 + 1)...𝑁)) × {0}))))
261251, 260csbie 3863 . . . . . . . 8 𝑦 / 𝑗((1st ‘(1st𝑇)) ∘f + (((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ (1...𝑗)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st𝑇)) ∘f + (((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ ((𝑦 + 1)...𝑁)) × {0})))
262250, 261eqtrdi 2849 . . . . . . 7 (𝑦 < (2nd𝑇) → if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑇)) ∘f + (((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ (1...𝑗)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st𝑇)) ∘f + (((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ ((𝑦 + 1)...𝑁)) × {0}))))
263262adantl 485 . . . . . 6 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑇)) ∘f + (((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ (1...𝑗)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st𝑇)) ∘f + (((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ (1...𝑦)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ ((𝑦 + 1)...𝑁)) × {0}))))
264249csbeq1d 3832 . . . . . . . 8 (𝑦 < (2nd𝑇) → 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}))))
265252imaeq2d 5897 . . . . . . . . . . . 12 (𝑗 = 𝑦 → ((2nd ‘(1st𝑇)) “ (1...𝑗)) = ((2nd ‘(1st𝑇)) “ (1...𝑦)))
266265xpeq1d 5549 . . . . . . . . . . 11 (𝑗 = 𝑦 → (((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}))
267256imaeq2d 5897 . . . . . . . . . . . 12 (𝑗 = 𝑦 → ((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)))
268267xpeq1d 5549 . . . . . . . . . . 11 (𝑗 = 𝑦 → (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))
269266, 268uneq12d 4091 . . . . . . . . . 10 (𝑗 = 𝑦 → ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0})))
270269oveq2d 7152 . . . . . . . . 9 (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}))))
271251, 270csbie 3863 . . . . . . . 8 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})))
272264, 271eqtrdi 2849 . . . . . . 7 (𝑦 < (2nd𝑇) → 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}))))
273272adantl 485 . . . . . 6 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → 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}))))
274248, 263, 2733eqtr4d 2843 . . . . 5 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑇)) ∘f + (((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ (1...𝑗)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ ((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}))))
275 lenlt 10711 . . . . . . . . . 10 (((2nd𝑇) ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((2nd𝑇) ≤ 𝑦 ↔ ¬ 𝑦 < (2nd𝑇)))
27618, 115, 275syl2an 598 . . . . . . . . 9 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((2nd𝑇) ≤ 𝑦 ↔ ¬ 𝑦 < (2nd𝑇)))
277276biimpar 481 . . . . . . . 8 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ ¬ 𝑦 < (2nd𝑇)) → (2nd𝑇) ≤ 𝑦)
278 imaco 6072 . . . . . . . . . . 11 (((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ (1...(𝑦 + 1))) = ((2nd ‘(1st𝑇)) “ (({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))) “ (1...(𝑦 + 1))))
279 imaundir 5977 . . . . . . . . . . . . . 14 (({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))) “ (1...(𝑦 + 1))) = (({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} “ (1...(𝑦 + 1))) ∪ (( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) “ (1...(𝑦 + 1))))
280 imassrn 5908 . . . . . . . . . . . . . . . . . 18 ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} “ (1...(𝑦 + 1))) ⊆ ran {⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩}
281280a1i 11 . . . . . . . . . . . . . . . . 17 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} “ (1...(𝑦 + 1))) ⊆ ran {⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩})
282175ad2antrr 725 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} “ {(2nd𝑇), ((2nd𝑇) + 1)}) = ran {⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩})
28317ad2antrr 725 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → (2nd𝑇) ∈ ℕ)
28418ad2antrr 725 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → (2nd𝑇) ∈ ℝ)
285115ad2antlr 726 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → 𝑦 ∈ ℝ)
286185nnred 11643 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈ ℝ)
287286ad2antlr 726 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → (𝑦 + 1) ∈ ℝ)
288 simpr 488 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → (2nd𝑇) ≤ 𝑦)
289115lep1d 11563 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ≤ (𝑦 + 1))
290289ad2antlr 726 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → 𝑦 ≤ (𝑦 + 1))
291284, 285, 287, 288, 290letrd 10789 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → (2nd𝑇) ≤ (𝑦 + 1))
292 fznn 12973 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑦 + 1) ∈ ℤ → ((2nd𝑇) ∈ (1...(𝑦 + 1)) ↔ ((2nd𝑇) ∈ ℕ ∧ (2nd𝑇) ≤ (𝑦 + 1))))
293186, 292syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 ∈ (0...(𝑁 − 1)) → ((2nd𝑇) ∈ (1...(𝑦 + 1)) ↔ ((2nd𝑇) ∈ ℕ ∧ (2nd𝑇) ≤ (𝑦 + 1))))
294293ad2antlr 726 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → ((2nd𝑇) ∈ (1...(𝑦 + 1)) ↔ ((2nd𝑇) ∈ ℕ ∧ (2nd𝑇) ≤ (𝑦 + 1))))
295283, 291, 294mpbir2and 712 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → (2nd𝑇) ∈ (1...(𝑦 + 1)))
29650ad2antrr 725 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → ((2nd𝑇) + 1) ∈ ℕ)
297 1red 10634 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → 1 ∈ ℝ)
298284, 285, 297, 288leadd1dd 11246 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → ((2nd𝑇) + 1) ≤ (𝑦 + 1))
299 fznn 12973 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑦 + 1) ∈ ℤ → (((2nd𝑇) + 1) ∈ (1...(𝑦 + 1)) ↔ (((2nd𝑇) + 1) ∈ ℕ ∧ ((2nd𝑇) + 1) ≤ (𝑦 + 1))))
300186, 299syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 ∈ (0...(𝑁 − 1)) → (((2nd𝑇) + 1) ∈ (1...(𝑦 + 1)) ↔ (((2nd𝑇) + 1) ∈ ℕ ∧ ((2nd𝑇) + 1) ≤ (𝑦 + 1))))
301300ad2antlr 726 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → (((2nd𝑇) + 1) ∈ (1...(𝑦 + 1)) ↔ (((2nd𝑇) + 1) ∈ ℕ ∧ ((2nd𝑇) + 1) ≤ (𝑦 + 1))))
302296, 298, 301mpbir2and 712 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → ((2nd𝑇) + 1) ∈ (1...(𝑦 + 1)))
303 prssi 4714 . . . . . . . . . . . . . . . . . . . 20 (((2nd𝑇) ∈ (1...(𝑦 + 1)) ∧ ((2nd𝑇) + 1) ∈ (1...(𝑦 + 1))) → {(2nd𝑇), ((2nd𝑇) + 1)} ⊆ (1...(𝑦 + 1)))
304295, 302, 303syl2anc 587 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → {(2nd𝑇), ((2nd𝑇) + 1)} ⊆ (1...(𝑦 + 1)))
305 imass2 5933 . . . . . . . . . . . . . . . . . . 19 ({(2nd𝑇), ((2nd𝑇) + 1)} ⊆ (1...(𝑦 + 1)) → ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} “ {(2nd𝑇), ((2nd𝑇) + 1)}) ⊆ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} “ (1...(𝑦 + 1))))
306304, 305syl 17 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} “ {(2nd𝑇), ((2nd𝑇) + 1)}) ⊆ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} “ (1...(𝑦 + 1))))
307282, 306eqsstrrd 3954 . . . . . . . . . . . . . . . . 17 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → ran {⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ⊆ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} “ (1...(𝑦 + 1))))
308281, 307eqssd 3932 . . . . . . . . . . . . . . . 16 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} “ (1...(𝑦 + 1))) = ran {⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩})
309214ad2antrr 725 . . . . . . . . . . . . . . . 16 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → ran {⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} = {(2nd𝑇), ((2nd𝑇) + 1)})
310308, 309eqtrd 2833 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} “ (1...(𝑦 + 1))) = {(2nd𝑇), ((2nd𝑇) + 1)})
311 undif 4388 . . . . . . . . . . . . . . . . . 18 ({(2nd𝑇), ((2nd𝑇) + 1)} ⊆ (1...(𝑦 + 1)) ↔ ({(2nd𝑇), ((2nd𝑇) + 1)} ∪ ((1...(𝑦 + 1)) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) = (1...(𝑦 + 1)))
312304, 311sylib 221 . . . . . . . . . . . . . . . . 17 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → ({(2nd𝑇), ((2nd𝑇) + 1)} ∪ ((1...(𝑦 + 1)) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) = (1...(𝑦 + 1)))
313312imaeq2d 5897 . . . . . . . . . . . . . . . 16 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → (( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) “ ({(2nd𝑇), ((2nd𝑇) + 1)} ∪ ((1...(𝑦 + 1)) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))) = (( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) “ (1...(𝑦 + 1))))
314224a1i 11 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → (( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) “ {(2nd𝑇), ((2nd𝑇) + 1)}) = ∅)
315 eluzp1p1 12261 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑁 − 1) ∈ (ℤ𝑦) → ((𝑁 − 1) + 1) ∈ (ℤ‘(𝑦 + 1)))
316149, 315syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈ (ℤ‘(𝑦 + 1)))
317316adantl 485 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((𝑁 − 1) + 1) ∈ (ℤ‘(𝑦 + 1)))
318148, 317eqeltrrd 2891 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ (ℤ‘(𝑦 + 1)))
319 fzss2 12945 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 ∈ (ℤ‘(𝑦 + 1)) → (1...(𝑦 + 1)) ⊆ (1...𝑁))
320318, 319syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (1...(𝑦 + 1)) ⊆ (1...𝑁))
321320ssdifd 4068 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((1...(𝑦 + 1)) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) ⊆ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))
322321adantr 484 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → ((1...(𝑦 + 1)) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) ⊆ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))
323 resiima 5912 . . . . . . . . . . . . . . . . . . 19 (((1...(𝑦 + 1)) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) ⊆ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) → (( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) “ ((1...(𝑦 + 1)) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) = ((1...(𝑦 + 1)) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))
324322, 323syl 17 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → (( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) “ ((1...(𝑦 + 1)) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) = ((1...(𝑦 + 1)) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))
325314, 324uneq12d 4091 . . . . . . . . . . . . . . . . 17 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → ((( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) “ {(2nd𝑇), ((2nd𝑇) + 1)}) ∪ (( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) “ ((1...(𝑦 + 1)) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))) = (∅ ∪ ((1...(𝑦 + 1)) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))
326 imaundi 5976 . . . . . . . . . . . . . . . . 17 (( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) “ ({(2nd𝑇), ((2nd𝑇) + 1)} ∪ ((1...(𝑦 + 1)) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))) = ((( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) “ {(2nd𝑇), ((2nd𝑇) + 1)}) ∪ (( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) “ ((1...(𝑦 + 1)) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))
327 uncom 4080 . . . . . . . . . . . . . . . . . 18 (∅ ∪ ((1...(𝑦 + 1)) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) = (((1...(𝑦 + 1)) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) ∪ ∅)
328 un0 4298 . . . . . . . . . . . . . . . . . 18 (((1...(𝑦 + 1)) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) ∪ ∅) = ((1...(𝑦 + 1)) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})
329327, 328eqtr2i 2822 . . . . . . . . . . . . . . . . 17 ((1...(𝑦 + 1)) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) = (∅ ∪ ((1...(𝑦 + 1)) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))
330325, 326, 3293eqtr4g 2858 . . . . . . . . . . . . . . . 16 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → (( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) “ ({(2nd𝑇), ((2nd𝑇) + 1)} ∪ ((1...(𝑦 + 1)) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))) = ((1...(𝑦 + 1)) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))
331313, 330eqtr3d 2835 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → (( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) “ (1...(𝑦 + 1))) = ((1...(𝑦 + 1)) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))
332310, 331uneq12d 4091 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → (({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} “ (1...(𝑦 + 1))) ∪ (( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) “ (1...(𝑦 + 1)))) = ({(2nd𝑇), ((2nd𝑇) + 1)} ∪ ((1...(𝑦 + 1)) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))
333279, 332syl5eq 2845 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → (({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))) “ (1...(𝑦 + 1))) = ({(2nd𝑇), ((2nd𝑇) + 1)} ∪ ((1...(𝑦 + 1)) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))
334333, 312eqtrd 2833 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → (({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))) “ (1...(𝑦 + 1))) = (1...(𝑦 + 1)))
335334imaeq2d 5897 . . . . . . . . . . 11 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → ((2nd ‘(1st𝑇)) “ (({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))) “ (1...(𝑦 + 1)))) = ((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))))
336278, 335syl5eq 2845 . . . . . . . . . 10 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → (((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ (1...(𝑦 + 1))) = ((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))))
337336xpeq1d 5549 . . . . . . . . 9 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → ((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ (1...(𝑦 + 1))) × {1}) = (((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}))
338 imaco 6072 . . . . . . . . . . 11 (((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ (((𝑦 + 1) + 1)...𝑁)) = ((2nd ‘(1st𝑇)) “ (({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))) “ (((𝑦 + 1) + 1)...𝑁)))
339111ad2antrr 725 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → {⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} Fn {(2nd𝑇), ((2nd𝑇) + 1)})
340 incom 4128 . . . . . . . . . . . . . . . 16 ({(2nd𝑇), ((2nd𝑇) + 1)} ∩ (((𝑦 + 1) + 1)...𝑁)) = ((((𝑦 + 1) + 1)...𝑁) ∩ {(2nd𝑇), ((2nd𝑇) + 1)})
341125ad2antrr 725 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → ((2nd𝑇) + 1) ∈ ℝ)
342185peano2nnd 11645 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑦 + 1) + 1) ∈ ℕ)
343342nnred 11643 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑦 + 1) + 1) ∈ ℝ)
344343ad2antlr 726 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → ((𝑦 + 1) + 1) ∈ ℝ)
34519ad2antrr 725 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → (2nd𝑇) < ((2nd𝑇) + 1))
346115ltp1d 11562 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 < (𝑦 + 1))
347346ad2antlr 726 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → 𝑦 < (𝑦 + 1))
348284, 285, 287, 288, 347lelttrd 10790 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → (2nd𝑇) < (𝑦 + 1))
349284, 287, 297, 348ltadd1dd 11243 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → ((2nd𝑇) + 1) < ((𝑦 + 1) + 1))
350284, 341, 344, 345, 349lttrd 10793 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → (2nd𝑇) < ((𝑦 + 1) + 1))
351 ltnle 10712 . . . . . . . . . . . . . . . . . . . . . . 23 (((2nd𝑇) ∈ ℝ ∧ ((𝑦 + 1) + 1) ∈ ℝ) → ((2nd𝑇) < ((𝑦 + 1) + 1) ↔ ¬ ((𝑦 + 1) + 1) ≤ (2nd𝑇)))
35218, 343, 351syl2an 598 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((2nd𝑇) < ((𝑦 + 1) + 1) ↔ ¬ ((𝑦 + 1) + 1) ≤ (2nd𝑇)))
353352adantr 484 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → ((2nd𝑇) < ((𝑦 + 1) + 1) ↔ ¬ ((𝑦 + 1) + 1) ≤ (2nd𝑇)))
354350, 353mpbid 235 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → ¬ ((𝑦 + 1) + 1) ≤ (2nd𝑇))
355 elfzle1 12908 . . . . . . . . . . . . . . . . . . . 20 ((2nd𝑇) ∈ (((𝑦 + 1) + 1)...𝑁) → ((𝑦 + 1) + 1) ≤ (2nd𝑇))
356354, 355nsyl 142 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → ¬ (2nd𝑇) ∈ (((𝑦 + 1) + 1)...𝑁))
357 disjsn 4607 . . . . . . . . . . . . . . . . . . 19 (((((𝑦 + 1) + 1)...𝑁) ∩ {(2nd𝑇)}) = ∅ ↔ ¬ (2nd𝑇) ∈ (((𝑦 + 1) + 1)...𝑁))
358356, 357sylibr 237 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → ((((𝑦 + 1) + 1)...𝑁) ∩ {(2nd𝑇)}) = ∅)
359 ltnle 10712 . . . . . . . . . . . . . . . . . . . . . . 23 ((((2nd𝑇) + 1) ∈ ℝ ∧ ((𝑦 + 1) + 1) ∈ ℝ) → (((2nd𝑇) + 1) < ((𝑦 + 1) + 1) ↔ ¬ ((𝑦 + 1) + 1) ≤ ((2nd𝑇) + 1)))
360125, 343, 359syl2an 598 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((2nd𝑇) + 1) < ((𝑦 + 1) + 1) ↔ ¬ ((𝑦 + 1) + 1) ≤ ((2nd𝑇) + 1)))
361360adantr 484 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → (((2nd𝑇) + 1) < ((𝑦 + 1) + 1) ↔ ¬ ((𝑦 + 1) + 1) ≤ ((2nd𝑇) + 1)))
362349, 361mpbid 235 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → ¬ ((𝑦 + 1) + 1) ≤ ((2nd𝑇) + 1))
363 elfzle1 12908 . . . . . . . . . . . . . . . . . . . 20 (((2nd𝑇) + 1) ∈ (((𝑦 + 1) + 1)...𝑁) → ((𝑦 + 1) + 1) ≤ ((2nd𝑇) + 1))
364362, 363nsyl 142 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → ¬ ((2nd𝑇) + 1) ∈ (((𝑦 + 1) + 1)...𝑁))
365 disjsn 4607 . . . . . . . . . . . . . . . . . . 19 (((((𝑦 + 1) + 1)...𝑁) ∩ {((2nd𝑇) + 1)}) = ∅ ↔ ¬ ((2nd𝑇) + 1) ∈ (((𝑦 + 1) + 1)...𝑁))
366364, 365sylibr 237 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → ((((𝑦 + 1) + 1)...𝑁) ∩ {((2nd𝑇) + 1)}) = ∅)
367358, 366uneq12d 4091 . . . . . . . . . . . . . . . . 17 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → (((((𝑦 + 1) + 1)...𝑁) ∩ {(2nd𝑇)}) ∪ ((((𝑦 + 1) + 1)...𝑁) ∩ {((2nd𝑇) + 1)})) = (∅ ∪ ∅))
368139ineq2i 4136 . . . . . . . . . . . . . . . . . 18 ((((𝑦 + 1) + 1)...𝑁) ∩ {(2nd𝑇), ((2nd𝑇) + 1)}) = ((((𝑦 + 1) + 1)...𝑁) ∩ ({(2nd𝑇)} ∪ {((2nd𝑇) + 1)}))
369 indi 4200 . . . . . . . . . . . . . . . . . 18 ((((𝑦 + 1) + 1)...𝑁) ∩ ({(2nd𝑇)} ∪ {((2nd𝑇) + 1)})) = (((((𝑦 + 1) + 1)...𝑁) ∩ {(2nd𝑇)}) ∪ ((((𝑦 + 1) + 1)...𝑁) ∩ {((2nd𝑇) + 1)}))
370368, 369eqtr2i 2822 . . . . . . . . . . . . . . . . 17 (((((𝑦 + 1) + 1)...𝑁) ∩ {(2nd𝑇)}) ∪ ((((𝑦 + 1) + 1)...𝑁) ∩ {((2nd𝑇) + 1)})) = ((((𝑦 + 1) + 1)...𝑁) ∩ {(2nd𝑇), ((2nd𝑇) + 1)})
371367, 370, 1433eqtr3g 2856 . . . . . . . . . . . . . . . 16 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → ((((𝑦 + 1) + 1)...𝑁) ∩ {(2nd𝑇), ((2nd𝑇) + 1)}) = ∅)
372340, 371syl5eq 2845 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → ({(2nd𝑇), ((2nd𝑇) + 1)} ∩ (((𝑦 + 1) + 1)...𝑁)) = ∅)
373 fnimadisj 6453 . . . . . . . . . . . . . . 15 (({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} Fn {(2nd𝑇), ((2nd𝑇) + 1)} ∧ ({(2nd𝑇), ((2nd𝑇) + 1)} ∩ (((𝑦 + 1) + 1)...𝑁)) = ∅) → ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} “ (((𝑦 + 1) + 1)...𝑁)) = ∅)
374339, 372, 373syl2anc 587 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} “ (((𝑦 + 1) + 1)...𝑁)) = ∅)
375342, 226eleqtrdi 2900 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑦 + 1) + 1) ∈ (ℤ‘1))
376 fzss1 12944 . . . . . . . . . . . . . . . . . 18 (((𝑦 + 1) + 1) ∈ (ℤ‘1) → (((𝑦 + 1) + 1)...𝑁) ⊆ (1...𝑁))
377 reldisj 4359 . . . . . . . . . . . . . . . . . 18 ((((𝑦 + 1) + 1)...𝑁) ⊆ (1...𝑁) → (((((𝑦 + 1) + 1)...𝑁) ∩ {(2nd𝑇), ((2nd𝑇) + 1)}) = ∅ ↔ (((𝑦 + 1) + 1)...𝑁) ⊆ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))
378375, 376, 3773syl 18 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ (0...(𝑁 − 1)) → (((((𝑦 + 1) + 1)...𝑁) ∩ {(2nd𝑇), ((2nd𝑇) + 1)}) = ∅ ↔ (((𝑦 + 1) + 1)...𝑁) ⊆ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))
379378ad2antlr 726 . . . . . . . . . . . . . . . 16 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → (((((𝑦 + 1) + 1)...𝑁) ∩ {(2nd𝑇), ((2nd𝑇) + 1)}) = ∅ ↔ (((𝑦 + 1) + 1)...𝑁) ⊆ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))
380371, 379mpbid 235 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → (((𝑦 + 1) + 1)...𝑁) ⊆ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))
381 resiima 5912 . . . . . . . . . . . . . . 15 ((((𝑦 + 1) + 1)...𝑁) ⊆ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) → (( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) “ (((𝑦 + 1) + 1)...𝑁)) = (((𝑦 + 1) + 1)...𝑁))
382380, 381syl 17 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → (( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) “ (((𝑦 + 1) + 1)...𝑁)) = (((𝑦 + 1) + 1)...𝑁))
383374, 382uneq12d 4091 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → (({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} “ (((𝑦 + 1) + 1)...𝑁)) ∪ (( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) “ (((𝑦 + 1) + 1)...𝑁))) = (∅ ∪ (((𝑦 + 1) + 1)...𝑁)))
384 imaundir 5977 . . . . . . . . . . . . 13 (({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))) “ (((𝑦 + 1) + 1)...𝑁)) = (({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} “ (((𝑦 + 1) + 1)...𝑁)) ∪ (( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) “ (((𝑦 + 1) + 1)...𝑁)))
385 uncom 4080 . . . . . . . . . . . . . 14 (∅ ∪ (((𝑦 + 1) + 1)...𝑁)) = ((((𝑦 + 1) + 1)...𝑁) ∪ ∅)
386 un0 4298 . . . . . . . . . . . . . 14 ((((𝑦 + 1) + 1)...𝑁) ∪ ∅) = (((𝑦 + 1) + 1)...𝑁)
387385, 386eqtr2i 2822 . . . . . . . . . . . . 13 (((𝑦 + 1) + 1)...𝑁) = (∅ ∪ (((𝑦 + 1) + 1)...𝑁))
388383, 384, 3873eqtr4g 2858 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → (({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))) “ (((𝑦 + 1) + 1)...𝑁)) = (((𝑦 + 1) + 1)...𝑁))
389388imaeq2d 5897 . . . . . . . . . . 11 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → ((2nd ‘(1st𝑇)) “ (({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))) “ (((𝑦 + 1) + 1)...𝑁))) = ((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)))
390338, 389syl5eq 2845 . . . . . . . . . 10 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → (((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ (((𝑦 + 1) + 1)...𝑁)) = ((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)))
391390xpeq1d 5549 . . . . . . . . 9 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → ((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))
392337, 391uneq12d 4091 . . . . . . . 8 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → (((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) = ((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})))
393277, 392syldan 594 . . . . . . 7 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ ¬ 𝑦 < (2nd𝑇)) → (((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ (((𝑦 + 1) + 1)...𝑁)) × {0})) = ((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})))
394393oveq2d 7152 . . . . . 6 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ ¬ 𝑦 < (2nd𝑇)) → ((1st ‘(1st𝑇)) ∘f + (((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))) = ((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))))
395 iffalse 4434 . . . . . . . . 9 𝑦 < (2nd𝑇) → if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) = (𝑦 + 1))
396395csbeq1d 3832 . . . . . . . 8 𝑦 < (2nd𝑇) → if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑇)) ∘f + (((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ (1...𝑗)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ ((𝑗 + 1)...𝑁)) × {0}))) = (𝑦 + 1) / 𝑗((1st ‘(1st𝑇)) ∘f + (((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ (1...𝑗)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ ((𝑗 + 1)...𝑁)) × {0}))))
397 ovex 7169 . . . . . . . . 9 (𝑦 + 1) ∈ V
398 oveq2 7144 . . . . . . . . . . . . 13 (𝑗 = (𝑦 + 1) → (1...𝑗) = (1...(𝑦 + 1)))
399398imaeq2d 5897 . . . . . . . . . . . 12 (𝑗 = (𝑦 + 1) → (((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ (1...𝑗)) = (((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ (1...(𝑦 + 1))))
400399xpeq1d 5549 . . . . . . . . . . 11 (𝑗 = (𝑦 + 1) → ((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ (1...𝑗)) × {1}) = ((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ (1...(𝑦 + 1))) × {1}))
401 oveq1 7143 . . . . . . . . . . . . . 14 (𝑗 = (𝑦 + 1) → (𝑗 + 1) = ((𝑦 + 1) + 1))
402401oveq1d 7151 . . . . . . . . . . . . 13 (𝑗 = (𝑦 + 1) → ((𝑗 + 1)...𝑁) = (((𝑦 + 1) + 1)...𝑁))
403402imaeq2d 5897 . . . . . . . . . . . 12 (𝑗 = (𝑦 + 1) → (((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ ((𝑗 + 1)...𝑁)) = (((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ (((𝑦 + 1) + 1)...𝑁)))
404403xpeq1d 5549 . . . . . . . . . . 11 (𝑗 = (𝑦 + 1) → ((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ ((𝑗 + 1)...𝑁)) × {0}) = ((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))
405400, 404uneq12d 4091 . . . . . . . . . 10 (𝑗 = (𝑦 + 1) → (((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ (1...𝑗)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ ((𝑗 + 1)...𝑁)) × {0})) = (((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ (((𝑦 + 1) + 1)...𝑁)) × {0})))
406405oveq2d 7152 . . . . . . . . 9 (𝑗 = (𝑦 + 1) → ((1st ‘(1st𝑇)) ∘f + (((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ (1...𝑗)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st𝑇)) ∘f + (((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))))
407397, 406csbie 3863 . . . . . . . 8 (𝑦 + 1) / 𝑗((1st ‘(1st𝑇)) ∘f + (((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ (1...𝑗)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st𝑇)) ∘f + (((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ (((𝑦 + 1) + 1)...𝑁)) × {0})))
408396, 407eqtrdi 2849 . . . . . . 7 𝑦 < (2nd𝑇) → if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑇)) ∘f + (((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ (1...𝑗)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st𝑇)) ∘f + (((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))))
409408adantl 485 . . . . . 6 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ ¬ 𝑦 < (2nd𝑇)) → if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑇)) ∘f + (((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ (1...𝑗)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st𝑇)) ∘f + (((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ (1...(𝑦 + 1))) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))))
410395csbeq1d 3832 . . . . . . . 8 𝑦 < (2nd𝑇) → if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = (𝑦 + 1) / 𝑗((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))))
411398imaeq2d 5897 . . . . . . . . . . . 12 (𝑗 = (𝑦 + 1) → ((2nd ‘(1st𝑇)) “ (1...𝑗)) = ((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))))
412411xpeq1d 5549 . . . . . . . . . . 11 (𝑗 = (𝑦 + 1) → (((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}))
413402imaeq2d 5897 . . . . . . . . . . . 12 (𝑗 = (𝑦 + 1) → ((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)))
414413xpeq1d 5549 . . . . . . . . . . 11 (𝑗 = (𝑦 + 1) → (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))
415412, 414uneq12d 4091 . . . . . . . . . 10 (𝑗 = (𝑦 + 1) → ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0})) = ((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})))
416415oveq2d 7152 . . . . . . . . 9 (𝑗 = (𝑦 + 1) → ((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))))
417397, 416csbie 3863 . . . . . . . 8 (𝑦 + 1) / 𝑗((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0})))
418410, 417eqtrdi 2849 . . . . . . 7 𝑦 < (2nd𝑇) → if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))))
419418adantl 485 . . . . . 6 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ ¬ 𝑦 < (2nd𝑇)) → if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st𝑇)) ∘f + ((((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}) ∪ (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))))
420394, 409, 4193eqtr4d 2843 . . . . 5 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ ¬ 𝑦 < (2nd𝑇)) → if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑇)) ∘f + (((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ (1...𝑗)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ ((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}))))
421274, 420pm2.61dan 812 . . . 4 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑇)) ∘f + (((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ (1...𝑗)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ ((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}))))
422421mpteq2dva 5126 . . 3 (𝜑 → (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑇)) ∘f + (((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ (1...𝑗)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ ((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})))))
423108, 422eqtr4d 2836 . 2 (𝜑𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) / 𝑗((1st ‘(1st𝑇)) ∘f + (((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ (1...𝑗)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ ((𝑗 + 1)...𝑁)) × {0})))))
424 opex 5322 . . . . . . 7 ⟨(1st ‘(1st𝑇)), ((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))))⟩ ∈ V
425424, 22op1std 7684 . . . . . 6 (𝑡 = ⟨⟨(1st ‘(1st𝑇)), ((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))))⟩, (2nd𝑇)⟩ → (1st𝑡) = ⟨(1st ‘(1st𝑇)), ((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))))⟩)
426424, 22op2ndd 7685 . . . . . 6 (𝑡 = ⟨⟨(1st ‘(1st𝑇)), ((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))))⟩, (2nd𝑇)⟩ → (2nd𝑡) = (2nd𝑇))
427 breq2 5035 . . . . . . . . 9 ((2nd𝑡) = (2nd𝑇) → (𝑦 < (2nd𝑡) ↔ 𝑦 < (2nd𝑇)))
428427ifbid 4447 . . . . . . . 8 ((2nd𝑡) = (2nd𝑇) → if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)))
429428csbeq1d 3832 . . . . . . 7 ((2nd𝑡) = (2nd𝑇) → if(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}))))
430 fvex 6659 . . . . . . . . . 10 (1st ‘(1st𝑇)) ∈ V
431430, 79op1std 7684 . . . . . . . . 9 ((1st𝑡) = ⟨(1st ‘(1st𝑇)), ((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))))⟩ → (1st ‘(1st𝑡)) = (1st ‘(1st𝑇)))
432430, 79op2ndd 7685 . . . . . . . . 9 ((1st𝑡) = ⟨(1st ‘(1st𝑇)), ((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))))⟩ → (2nd ‘(1st𝑡)) = ((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))))
433 id 22 . . . . . . . . . 10 ((1st ‘(1st𝑡)) = (1st ‘(1st𝑇)) → (1st ‘(1st𝑡)) = (1st ‘(1st𝑇)))
434 imaeq1 5892 . . . . . . . . . . . 12 ((2nd ‘(1st𝑡)) = ((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) → ((2nd ‘(1st𝑡)) “ (1...𝑗)) = (((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ (1...𝑗)))
435434xpeq1d 5549 . . . . . . . . . . 11 ((2nd ‘(1st𝑡)) = ((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) → (((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) = ((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ (1...𝑗)) × {1}))
436 imaeq1 5892 . . . . . . . . . . . 12 ((2nd ‘(1st𝑡)) = ((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) → ((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) = (((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ ((𝑗 + 1)...𝑁)))
437436xpeq1d 5549 . . . . . . . . . . 11 ((2nd ‘(1st𝑡)) = ((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) → (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = ((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ ((𝑗 + 1)...𝑁)) × {0}))
438435, 437uneq12d 4091 . . . . . . . . . 10 ((2nd ‘(1st𝑡)) = ((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) → ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0})) = (((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ (1...𝑗)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ ((𝑗 + 1)...𝑁)) × {0})))
439433, 438oveqan12d 7155 . . . . . . . . 9 (((1st ‘(1st𝑡)) = (1st ‘(1st𝑇)) ∧ (2nd ‘(1st𝑡)) = ((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))))) → ((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st𝑇)) ∘f + (((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ (1...𝑗)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ ((𝑗 + 1)...𝑁)) × {0}))))
440431, 432, 439syl2anc 587 . . . . . . . 8 ((1st𝑡) = ⟨(1st ‘(1st𝑇)), ((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))))⟩ → ((1st ‘(1st𝑡)) ∘f + ((((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) ∪ (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}))) = ((1st ‘(1st𝑇)) ∘f + (((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ (1...𝑗)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ ((𝑗 + 1)...𝑁)) × {0}))))
441440csbeq2dv 3835 . . . . . . 7 ((1st𝑡) = ⟨(1st ‘(1st𝑇)), ((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))))⟩ → if(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)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ (1...𝑗)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ ((𝑗 + 1)...𝑁)) × {0}))))
442429, 441sylan9eqr 2855 . . . . . 6 (((1st𝑡) = ⟨(1st ‘(1st𝑇)), ((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))))⟩ ∧ (2nd𝑡) = (2nd𝑇)) → if(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)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ (1...𝑗)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ ((𝑗 + 1)...𝑁)) × {0}))))
443425, 426, 442syl2anc 587 . . . . 5 (𝑡 = ⟨⟨(1st ‘(1st𝑇)), ((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))))⟩, (2nd𝑇)⟩ → if(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)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ (1...𝑗)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ ((𝑗 + 1)...𝑁)) × {0}))))
444443mpteq2dv 5127 . . . 4 (𝑡 = ⟨⟨(1st ‘(1st𝑇)), ((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))))⟩, (2nd𝑇)⟩ → (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)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ (1...𝑗)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ ((𝑗 + 1)...𝑁)) × {0})))))
445444eqeq2d 2809 . . 3 (𝑡 = ⟨⟨(1st ‘(1st𝑇)), ((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))))⟩, (2nd𝑇)⟩ → (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)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ (1...𝑗)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ ((𝑗 + 1)...𝑁)) × {0}))))))
446445, 3elrab2 3631 . 2 (⟨⟨(1st ‘(1st𝑇)), ((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))))⟩, (2nd𝑇)⟩ ∈ 𝑆 ↔ (⟨⟨(1st ‘(1st𝑇)), ((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))))⟩, (2nd𝑇)⟩ ∈ ((((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)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ (1...𝑗)) × {1}) ∪ ((((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) “ ((𝑗 + 1)...𝑁)) × {0}))))))
44789, 423, 446sylanbrc 586 1 (𝜑 → ⟨⟨(1st ‘(1st𝑇)), ((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))))⟩, (2nd𝑇)⟩ ∈ 𝑆)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  {cab 2776   ≠ wne 2987  {crab 3110  Vcvv 3441  ⦋csb 3828   ∖ cdif 3878   ∪ cun 3879   ∩ cin 3880   ⊆ wss 3881  ∅c0 4243  ifcif 4425  {csn 4525  {cpr 4527  ⟨cop 4531   class class class wbr 5031   ↦ cmpt 5111   I cid 5425   × cxp 5518  ran crn 5521   ↾ cres 5522   “ cima 5523   ∘ ccom 5524   Fn wfn 6320  ⟶wf 6321  –onto→wfo 6323  –1-1-onto→wf1o 6324  ‘cfv 6325  (class class class)co 7136   ∘f cof 7389  1st c1st 7672  2nd c2nd 7673   ↑m cmap 8392  ℂcc 10527  ℝcr 10528  0cc0 10529  1c1 10530   + caddc 10532   < clt 10667   ≤ cle 10668   − cmin 10862  ℕcn 11628  ℕ0cn0 11888  ℤcz 11972  ℤ≥cuz 12234  ...cfz 12888  ..^cfzo 13031 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-tr 5138  df-id 5426  df-eprel 5431  df-po 5439  df-so 5440  df-fr 5479  df-we 5481  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-pred 6117  df-ord 6163  df-on 6164  df-lim 6165  df-suc 6166  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-f1 6330  df-fo 6331  df-f1o 6332  df-fv 6333  df-riota 7094  df-ov 7139  df-oprab 7140  df-mpo 7141  df-om 7564  df-1st 7674  df-2nd 7675  df-wrecs 7933  df-recs 7994  df-rdg 8032  df-er 8275  df-en 8496  df-dom 8497  df-sdom 8498  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11629  df-n0 11889  df-z 11973  df-uz 12235  df-fz 12889 This theorem is referenced by:  poimirlem22  35098
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