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Theorem poimirlem15 33091
Description: Lemma for poimir 33109, that the face in poimirlem22 33098 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 3346 . . . . . . 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 2716 . . . . . 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 7150 . . . . 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 7150 . . . . 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 7151 . . . . . . . 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 6163 . . . . . . . 8 (2nd ‘(1st𝑇)) ∈ V
12 f1oeq1 6089 . . . . . . . 8 (𝑓 = (2nd ‘(1st𝑇)) → (𝑓:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁)))
1311, 12elab 3337 . . . . . . 7 ((2nd ‘(1st𝑇)) ∈ {𝑓𝑓:(1...𝑁)–1-1-onto→(1...𝑁)} ↔ (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
1410, 13sylib 208 . . . . . 6 (𝜑 → (2nd ‘(1st𝑇)):(1...𝑁)–1-1-onto→(1...𝑁))
15 poimirlem15.3 . . . . . . . . . . . . 13 (𝜑 → (2nd𝑇) ∈ (1...(𝑁 − 1)))
16 elfznn 12320 . . . . . . . . . . . . 13 ((2nd𝑇) ∈ (1...(𝑁 − 1)) → (2nd𝑇) ∈ ℕ)
1715, 16syl 17 . . . . . . . . . . . 12 (𝜑 → (2nd𝑇) ∈ ℕ)
1817nnred 10987 . . . . . . . . . . 11 (𝜑 → (2nd𝑇) ∈ ℝ)
1918ltp1d 10906 . . . . . . . . . . 11 (𝜑 → (2nd𝑇) < ((2nd𝑇) + 1))
2018, 19ltned 10125 . . . . . . . . . 10 (𝜑 → (2nd𝑇) ≠ ((2nd𝑇) + 1))
2120necomd 2845 . . . . . . . . . 10 (𝜑 → ((2nd𝑇) + 1) ≠ (2nd𝑇))
22 fvex 6163 . . . . . . . . . . 11 (2nd𝑇) ∈ V
23 ovex 6638 . . . . . . . . . . 11 ((2nd𝑇) + 1) ∈ V
24 f1oprg 6143 . . . . . . . . . . 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 708 . . . . . . . . . 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 692 . . . . . . . . 9 (𝜑 → {⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩}:{(2nd𝑇), ((2nd𝑇) + 1)}–1-1-onto→{((2nd𝑇) + 1), (2nd𝑇)})
27 prcom 4242 . . . . . . . . . 10 {((2nd𝑇) + 1), (2nd𝑇)} = {(2nd𝑇), ((2nd𝑇) + 1)}
28 f1oeq3 6091 . . . . . . . . . 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 208 . . . . . . . 8 (𝜑 → {⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩}:{(2nd𝑇), ((2nd𝑇) + 1)}–1-1-onto→{(2nd𝑇), ((2nd𝑇) + 1)})
31 f1oi 6136 . . . . . . . 8 ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})):((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})–1-1-onto→((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})
32 disjdif 4017 . . . . . . . . 9 ({(2nd𝑇), ((2nd𝑇) + 1)} ∩ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) = ∅
33 f1oun 6118 . . . . . . . . 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 720 . . . . . . . 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 693 . . . . . . 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 10988 . . . . . . . . . . . . . 14 (𝜑𝑁 ∈ ℂ)
38 npcan1 10407 . . . . . . . . . . . . . 14 (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁)
3937, 38syl 17 . . . . . . . . . . . . 13 (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
4036nnzd 11433 . . . . . . . . . . . . . . 15 (𝜑𝑁 ∈ ℤ)
41 peano2zm 11372 . . . . . . . . . . . . . . 15 (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ)
4240, 41syl 17 . . . . . . . . . . . . . 14 (𝜑 → (𝑁 − 1) ∈ ℤ)
43 uzid 11654 . . . . . . . . . . . . . 14 ((𝑁 − 1) ∈ ℤ → (𝑁 − 1) ∈ (ℤ‘(𝑁 − 1)))
44 peano2uz 11693 . . . . . . . . . . . . . 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 2699 . . . . . . . . . . . 12 (𝜑𝑁 ∈ (ℤ‘(𝑁 − 1)))
47 fzss2 12331 . . . . . . . . . . . 12 (𝑁 ∈ (ℤ‘(𝑁 − 1)) → (1...(𝑁 − 1)) ⊆ (1...𝑁))
4846, 47syl 17 . . . . . . . . . . 11 (𝜑 → (1...(𝑁 − 1)) ⊆ (1...𝑁))
4948, 15sseldd 3588 . . . . . . . . . 10 (𝜑 → (2nd𝑇) ∈ (1...𝑁))
5017peano2nnd 10989 . . . . . . . . . . 11 (𝜑 → ((2nd𝑇) + 1) ∈ ℕ)
5142zred 11434 . . . . . . . . . . . . 13 (𝜑 → (𝑁 − 1) ∈ ℝ)
5236nnred 10987 . . . . . . . . . . . . 13 (𝜑𝑁 ∈ ℝ)
53 elfzle2 12295 . . . . . . . . . . . . . 14 ((2nd𝑇) ∈ (1...(𝑁 − 1)) → (2nd𝑇) ≤ (𝑁 − 1))
5415, 53syl 17 . . . . . . . . . . . . 13 (𝜑 → (2nd𝑇) ≤ (𝑁 − 1))
5552ltm1d 10908 . . . . . . . . . . . . 13 (𝜑 → (𝑁 − 1) < 𝑁)
5618, 51, 52, 54, 55lelttrd 10147 . . . . . . . . . . . 12 (𝜑 → (2nd𝑇) < 𝑁)
5717nnzd 11433 . . . . . . . . . . . . 13 (𝜑 → (2nd𝑇) ∈ ℤ)
58 zltp1le 11379 . . . . . . . . . . . . 13 (((2nd𝑇) ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((2nd𝑇) < 𝑁 ↔ ((2nd𝑇) + 1) ≤ 𝑁))
5957, 40, 58syl2anc 692 . . . . . . . . . . . 12 (𝜑 → ((2nd𝑇) < 𝑁 ↔ ((2nd𝑇) + 1) ≤ 𝑁))
6056, 59mpbid 222 . . . . . . . . . . 11 (𝜑 → ((2nd𝑇) + 1) ≤ 𝑁)
61 fznn 12358 . . . . . . . . . . . 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 956 . . . . . . . . . 10 (𝜑 → ((2nd𝑇) + 1) ∈ (1...𝑁))
64 prssi 4326 . . . . . . . . . 10 (((2nd𝑇) ∈ (1...𝑁) ∧ ((2nd𝑇) + 1) ∈ (1...𝑁)) → {(2nd𝑇), ((2nd𝑇) + 1)} ⊆ (1...𝑁))
6549, 63, 64syl2anc 692 . . . . . . . . 9 (𝜑 → {(2nd𝑇), ((2nd𝑇) + 1)} ⊆ (1...𝑁))
66 undif 4026 . . . . . . . . 9 ({(2nd𝑇), ((2nd𝑇) + 1)} ⊆ (1...𝑁) ↔ ({(2nd𝑇), ((2nd𝑇) + 1)} ∪ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) = (1...𝑁))
6765, 66sylib 208 . . . . . . . 8 (𝜑 → ({(2nd𝑇), ((2nd𝑇) + 1)} ∪ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) = (1...𝑁))
68 f1oeq23 6092 . . . . . . . 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 692 . . . . . . 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 222 . . . . . 6 (𝜑 → ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))):(1...𝑁)–1-1-onto→(1...𝑁))
71 f1oco 6121 . . . . . 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 692 . . . . 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 4875 . . . . . . . 8 {⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∈ V
74 ovex 6638 . . . . . . . . 9 (1...𝑁) ∈ V
75 difexg 4773 . . . . . . . . 9 ((1...𝑁) ∈ V → ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) ∈ V)
76 resiexg 7056 . . . . . . . . 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 6916 . . . . . . 7 ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))) ∈ V
7911, 78coex 7072 . . . . . 6 ((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))) ∈ V
80 f1oeq1 6089 . . . . . 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 3337 . . . . 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 224 . . . 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 5113 . . . 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 692 . . 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 1eluzge0 11684 . . . . . 6 1 ∈ (ℤ‘0)
86 fzss1 12330 . . . . . 6 (1 ∈ (ℤ‘0) → (1...𝑁) ⊆ (0...𝑁))
8785, 86ax-mp 5 . . . . 5 (1...𝑁) ⊆ (0...𝑁)
8848, 87syl6ss 3599 . . . 4 (𝜑 → (1...(𝑁 − 1)) ⊆ (0...𝑁))
8988, 15sseldd 3588 . . 3 (𝜑 → (2nd𝑇) ∈ (0...𝑁))
90 opelxpi 5113 . . 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)))
9184, 89, 90syl2anc 692 . 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)))
92 fveq2 6153 . . . . . . . . . . . 12 (𝑡 = 𝑇 → (2nd𝑡) = (2nd𝑇))
9392breq2d 4630 . . . . . . . . . . 11 (𝑡 = 𝑇 → (𝑦 < (2nd𝑡) ↔ 𝑦 < (2nd𝑇)))
9493ifbid 4085 . . . . . . . . . 10 (𝑡 = 𝑇 → if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)))
9594csbeq1d 3525 . . . . . . . . 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}))))
96 fveq2 6153 . . . . . . . . . . . 12 (𝑡 = 𝑇 → (1st𝑡) = (1st𝑇))
9796fveq2d 6157 . . . . . . . . . . 11 (𝑡 = 𝑇 → (1st ‘(1st𝑡)) = (1st ‘(1st𝑇)))
9896fveq2d 6157 . . . . . . . . . . . . . 14 (𝑡 = 𝑇 → (2nd ‘(1st𝑡)) = (2nd ‘(1st𝑇)))
9998imaeq1d 5429 . . . . . . . . . . . . 13 (𝑡 = 𝑇 → ((2nd ‘(1st𝑡)) “ (1...𝑗)) = ((2nd ‘(1st𝑇)) “ (1...𝑗)))
10099xpeq1d 5103 . . . . . . . . . . . 12 (𝑡 = 𝑇 → (((2nd ‘(1st𝑡)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}))
10198imaeq1d 5429 . . . . . . . . . . . . 13 (𝑡 = 𝑇 → ((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)))
102101xpeq1d 5103 . . . . . . . . . . . 12 (𝑡 = 𝑇 → (((2nd ‘(1st𝑡)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}))
103100, 102uneq12d 3751 . . . . . . . . . . 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})))
10497, 103oveq12d 6628 . . . . . . . . . 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}))))
105104csbeq2dv 3969 . . . . . . . . 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}))))
10695, 105eqtrd 2655 . . . . . . . 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}))))
107106mpteq2dv 4710 . . . . . . 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})))))
108107eqeq2d 2631 . . . . . 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}))))))
109108, 3elrab2 3352 . . . . 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}))))))
110109simprbi 480 . . . 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})))))
1111, 110syl 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})))))
112 imaco 5604 . . . . . . . . . 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)))
113 f1ofn 6100 . . . . . . . . . . . . . . . 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)})
11426, 113syl 17 . . . . . . . . . . . . . . 15 (𝜑 → {⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} Fn {(2nd𝑇), ((2nd𝑇) + 1)})
115114ad2antrr 761 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → {⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} Fn {(2nd𝑇), ((2nd𝑇) + 1)})
116 incom 3788 . . . . . . . . . . . . . . 15 ({(2nd𝑇), ((2nd𝑇) + 1)} ∩ (1...𝑦)) = ((1...𝑦) ∩ {(2nd𝑇), ((2nd𝑇) + 1)})
117 elfznn0 12382 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ∈ ℕ0)
118117nn0red 11304 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ∈ ℝ)
119 ltnle 10069 . . . . . . . . . . . . . . . . . . . . 21 ((𝑦 ∈ ℝ ∧ (2nd𝑇) ∈ ℝ) → (𝑦 < (2nd𝑇) ↔ ¬ (2nd𝑇) ≤ 𝑦))
120118, 18, 119syl2anr 495 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (𝑦 < (2nd𝑇) ↔ ¬ (2nd𝑇) ≤ 𝑦))
121120biimpa 501 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → ¬ (2nd𝑇) ≤ 𝑦)
122 elfzle2 12295 . . . . . . . . . . . . . . . . . . 19 ((2nd𝑇) ∈ (1...𝑦) → (2nd𝑇) ≤ 𝑦)
123121, 122nsyl 135 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → ¬ (2nd𝑇) ∈ (1...𝑦))
124 disjsn 4221 . . . . . . . . . . . . . . . . . 18 (((1...𝑦) ∩ {(2nd𝑇)}) = ∅ ↔ ¬ (2nd𝑇) ∈ (1...𝑦))
125123, 124sylibr 224 . . . . . . . . . . . . . . . . 17 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → ((1...𝑦) ∩ {(2nd𝑇)}) = ∅)
126118ad2antlr 762 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → 𝑦 ∈ ℝ)
12718ad2antrr 761 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → (2nd𝑇) ∈ ℝ)
12850nnred 10987 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → ((2nd𝑇) + 1) ∈ ℝ)
129128ad2antrr 761 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → ((2nd𝑇) + 1) ∈ ℝ)
130 simpr 477 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → 𝑦 < (2nd𝑇))
13119ad2antrr 761 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → (2nd𝑇) < ((2nd𝑇) + 1))
132126, 127, 129, 130, 131lttrd 10150 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → 𝑦 < ((2nd𝑇) + 1))
133 ltnle 10069 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑦 ∈ ℝ ∧ ((2nd𝑇) + 1) ∈ ℝ) → (𝑦 < ((2nd𝑇) + 1) ↔ ¬ ((2nd𝑇) + 1) ≤ 𝑦))
134118, 128, 133syl2anr 495 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (𝑦 < ((2nd𝑇) + 1) ↔ ¬ ((2nd𝑇) + 1) ≤ 𝑦))
135134adantr 481 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → (𝑦 < ((2nd𝑇) + 1) ↔ ¬ ((2nd𝑇) + 1) ≤ 𝑦))
136132, 135mpbid 222 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → ¬ ((2nd𝑇) + 1) ≤ 𝑦)
137 elfzle2 12295 . . . . . . . . . . . . . . . . . . 19 (((2nd𝑇) + 1) ∈ (1...𝑦) → ((2nd𝑇) + 1) ≤ 𝑦)
138136, 137nsyl 135 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → ¬ ((2nd𝑇) + 1) ∈ (1...𝑦))
139 disjsn 4221 . . . . . . . . . . . . . . . . . 18 (((1...𝑦) ∩ {((2nd𝑇) + 1)}) = ∅ ↔ ¬ ((2nd𝑇) + 1) ∈ (1...𝑦))
140138, 139sylibr 224 . . . . . . . . . . . . . . . . 17 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → ((1...𝑦) ∩ {((2nd𝑇) + 1)}) = ∅)
141125, 140uneq12d 3751 . . . . . . . . . . . . . . . 16 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → (((1...𝑦) ∩ {(2nd𝑇)}) ∪ ((1...𝑦) ∩ {((2nd𝑇) + 1)})) = (∅ ∪ ∅))
142 df-pr 4156 . . . . . . . . . . . . . . . . . 18 {(2nd𝑇), ((2nd𝑇) + 1)} = ({(2nd𝑇)} ∪ {((2nd𝑇) + 1)})
143142ineq2i 3794 . . . . . . . . . . . . . . . . 17 ((1...𝑦) ∩ {(2nd𝑇), ((2nd𝑇) + 1)}) = ((1...𝑦) ∩ ({(2nd𝑇)} ∪ {((2nd𝑇) + 1)}))
144 indi 3854 . . . . . . . . . . . . . . . . 17 ((1...𝑦) ∩ ({(2nd𝑇)} ∪ {((2nd𝑇) + 1)})) = (((1...𝑦) ∩ {(2nd𝑇)}) ∪ ((1...𝑦) ∩ {((2nd𝑇) + 1)}))
145143, 144eqtr2i 2644 . . . . . . . . . . . . . . . 16 (((1...𝑦) ∩ {(2nd𝑇)}) ∪ ((1...𝑦) ∩ {((2nd𝑇) + 1)})) = ((1...𝑦) ∩ {(2nd𝑇), ((2nd𝑇) + 1)})
146 un0 3944 . . . . . . . . . . . . . . . 16 (∅ ∪ ∅) = ∅
147141, 145, 1463eqtr3g 2678 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → ((1...𝑦) ∩ {(2nd𝑇), ((2nd𝑇) + 1)}) = ∅)
148116, 147syl5eq 2667 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → ({(2nd𝑇), ((2nd𝑇) + 1)} ∩ (1...𝑦)) = ∅)
149 fnimadisj 5974 . . . . . . . . . . . . . 14 (({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} Fn {(2nd𝑇), ((2nd𝑇) + 1)} ∧ ({(2nd𝑇), ((2nd𝑇) + 1)} ∩ (1...𝑦)) = ∅) → ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} “ (1...𝑦)) = ∅)
150115, 148, 149syl2anc 692 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} “ (1...𝑦)) = ∅)
15139adantr 481 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((𝑁 − 1) + 1) = 𝑁)
152 elfzuz3 12289 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ (0...(𝑁 − 1)) → (𝑁 − 1) ∈ (ℤ𝑦))
153 peano2uz 11693 . . . . . . . . . . . . . . . . . . . 20 ((𝑁 − 1) ∈ (ℤ𝑦) → ((𝑁 − 1) + 1) ∈ (ℤ𝑦))
154152, 153syl 17 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈ (ℤ𝑦))
155154adantl 482 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((𝑁 − 1) + 1) ∈ (ℤ𝑦))
156151, 155eqeltrrd 2699 . . . . . . . . . . . . . . . . 17 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ (ℤ𝑦))
157 fzss2 12331 . . . . . . . . . . . . . . . . 17 (𝑁 ∈ (ℤ𝑦) → (1...𝑦) ⊆ (1...𝑁))
158 reldisj 3997 . . . . . . . . . . . . . . . . 17 ((1...𝑦) ⊆ (1...𝑁) → (((1...𝑦) ∩ {(2nd𝑇), ((2nd𝑇) + 1)}) = ∅ ↔ (1...𝑦) ⊆ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))
159156, 157, 1583syl 18 . . . . . . . . . . . . . . . 16 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((1...𝑦) ∩ {(2nd𝑇), ((2nd𝑇) + 1)}) = ∅ ↔ (1...𝑦) ⊆ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))
160159adantr 481 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → (((1...𝑦) ∩ {(2nd𝑇), ((2nd𝑇) + 1)}) = ∅ ↔ (1...𝑦) ⊆ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))
161147, 160mpbid 222 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → (1...𝑦) ⊆ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))
162 resiima 5444 . . . . . . . . . . . . . 14 ((1...𝑦) ⊆ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) → (( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) “ (1...𝑦)) = (1...𝑦))
163161, 162syl 17 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → (( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) “ (1...𝑦)) = (1...𝑦))
164150, 163uneq12d 3751 . . . . . . . . . . . 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)))
165 imaundir 5510 . . . . . . . . . . . 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)))
166 uncom 3740 . . . . . . . . . . . . 13 (∅ ∪ (1...𝑦)) = ((1...𝑦) ∪ ∅)
167 un0 3944 . . . . . . . . . . . . 13 ((1...𝑦) ∪ ∅) = (1...𝑦)
168166, 167eqtr2i 2644 . . . . . . . . . . . 12 (1...𝑦) = (∅ ∪ (1...𝑦))
169164, 165, 1683eqtr4g 2680 . . . . . . . . . . 11 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → (({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))) “ (1...𝑦)) = (1...𝑦))
170169imaeq2d 5430 . . . . . . . . . 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)))
171112, 170syl5eq 2667 . . . . . . . . 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)))
172171xpeq1d 5103 . . . . . . . 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}))
173 imaco 5604 . . . . . . . . . 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)))
174 imaundir 5510 . . . . . . . . . . . . 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)))
175 imassrn 5441 . . . . . . . . . . . . . . . . 17 ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} “ ((𝑦 + 1)...𝑁)) ⊆ ran {⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩}
176175a1i 11 . . . . . . . . . . . . . . . 16 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} “ ((𝑦 + 1)...𝑁)) ⊆ ran {⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩})
177 fnima 5972 . . . . . . . . . . . . . . . . . . 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)⟩})
17826, 113, 1773syl 18 . . . . . . . . . . . . . . . . . 18 (𝜑 → ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} “ {(2nd𝑇), ((2nd𝑇) + 1)}) = ran {⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩})
179178ad2antrr 761 . . . . . . . . . . . . . . . . 17 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} “ {(2nd𝑇), ((2nd𝑇) + 1)}) = ran {⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩})
180 elfzelz 12292 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ∈ ℤ)
181 zltp1le 11379 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑦 ∈ ℤ ∧ (2nd𝑇) ∈ ℤ) → (𝑦 < (2nd𝑇) ↔ (𝑦 + 1) ≤ (2nd𝑇)))
182180, 57, 181syl2anr 495 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (𝑦 < (2nd𝑇) ↔ (𝑦 + 1) ≤ (2nd𝑇)))
183182biimpa 501 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → (𝑦 + 1) ≤ (2nd𝑇))
18418, 52, 56ltled 10137 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (2nd𝑇) ≤ 𝑁)
185184ad2antrr 761 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → (2nd𝑇) ≤ 𝑁)
18657adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (2nd𝑇) ∈ ℤ)
187 nn0p1nn 11284 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 ∈ ℕ0 → (𝑦 + 1) ∈ ℕ)
188117, 187syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈ ℕ)
189188nnzd 11433 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈ ℤ)
190189adantl 482 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (𝑦 + 1) ∈ ℤ)
19140adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ ℤ)
192 elfz 12282 . . . . . . . . . . . . . . . . . . . . . 22 (((2nd𝑇) ∈ ℤ ∧ (𝑦 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((2nd𝑇) ∈ ((𝑦 + 1)...𝑁) ↔ ((𝑦 + 1) ≤ (2nd𝑇) ∧ (2nd𝑇) ≤ 𝑁)))
193186, 190, 191, 192syl3anc 1323 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((2nd𝑇) ∈ ((𝑦 + 1)...𝑁) ↔ ((𝑦 + 1) ≤ (2nd𝑇) ∧ (2nd𝑇) ≤ 𝑁)))
194193adantr 481 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → ((2nd𝑇) ∈ ((𝑦 + 1)...𝑁) ↔ ((𝑦 + 1) ≤ (2nd𝑇) ∧ (2nd𝑇) ≤ 𝑁)))
195183, 185, 194mpbir2and 956 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → (2nd𝑇) ∈ ((𝑦 + 1)...𝑁))
196 1red 10007 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → 1 ∈ ℝ)
197 ltle 10078 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑦 ∈ ℝ ∧ (2nd𝑇) ∈ ℝ) → (𝑦 < (2nd𝑇) → 𝑦 ≤ (2nd𝑇)))
198118, 18, 197syl2anr 495 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (𝑦 < (2nd𝑇) → 𝑦 ≤ (2nd𝑇)))
199198imp 445 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → 𝑦 ≤ (2nd𝑇))
200126, 127, 196, 199leadd1dd 10593 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → (𝑦 + 1) ≤ ((2nd𝑇) + 1))
20160ad2antrr 761 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → ((2nd𝑇) + 1) ≤ 𝑁)
20257peano2zd 11437 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → ((2nd𝑇) + 1) ∈ ℤ)
203202adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((2nd𝑇) + 1) ∈ ℤ)
204 elfz 12282 . . . . . . . . . . . . . . . . . . . . . 22 ((((2nd𝑇) + 1) ∈ ℤ ∧ (𝑦 + 1) ∈ ℤ ∧ 𝑁 ∈ ℤ) → (((2nd𝑇) + 1) ∈ ((𝑦 + 1)...𝑁) ↔ ((𝑦 + 1) ≤ ((2nd𝑇) + 1) ∧ ((2nd𝑇) + 1) ≤ 𝑁)))
205203, 190, 191, 204syl3anc 1323 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((2nd𝑇) + 1) ∈ ((𝑦 + 1)...𝑁) ↔ ((𝑦 + 1) ≤ ((2nd𝑇) + 1) ∧ ((2nd𝑇) + 1) ≤ 𝑁)))
206205adantr 481 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → (((2nd𝑇) + 1) ∈ ((𝑦 + 1)...𝑁) ↔ ((𝑦 + 1) ≤ ((2nd𝑇) + 1) ∧ ((2nd𝑇) + 1) ≤ 𝑁)))
207200, 201, 206mpbir2and 956 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → ((2nd𝑇) + 1) ∈ ((𝑦 + 1)...𝑁))
208 prssi 4326 . . . . . . . . . . . . . . . . . . 19 (((2nd𝑇) ∈ ((𝑦 + 1)...𝑁) ∧ ((2nd𝑇) + 1) ∈ ((𝑦 + 1)...𝑁)) → {(2nd𝑇), ((2nd𝑇) + 1)} ⊆ ((𝑦 + 1)...𝑁))
209195, 207, 208syl2anc 692 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → {(2nd𝑇), ((2nd𝑇) + 1)} ⊆ ((𝑦 + 1)...𝑁))
210 imass2 5465 . . . . . . . . . . . . . . . . . 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)))
211209, 210syl 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)))
212179, 211eqsstr3d 3624 . . . . . . . . . . . . . . . 16 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → ran {⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ⊆ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} “ ((𝑦 + 1)...𝑁)))
213176, 212eqssd 3604 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} “ ((𝑦 + 1)...𝑁)) = ran {⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩})
214 f1ofo 6106 . . . . . . . . . . . . . . . . . 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)})
215 forn 6080 . . . . . . . . . . . . . . . . . 18 ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩}:{(2nd𝑇), ((2nd𝑇) + 1)}–onto→{((2nd𝑇) + 1), (2nd𝑇)} → ran {⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} = {((2nd𝑇) + 1), (2nd𝑇)})
21626, 214, 2153syl 18 . . . . . . . . . . . . . . . . 17 (𝜑 → ran {⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} = {((2nd𝑇) + 1), (2nd𝑇)})
217216, 27syl6eq 2671 . . . . . . . . . . . . . . . 16 (𝜑 → ran {⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} = {(2nd𝑇), ((2nd𝑇) + 1)})
218217ad2antrr 761 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → ran {⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} = {(2nd𝑇), ((2nd𝑇) + 1)})
219213, 218eqtrd 2655 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} “ ((𝑦 + 1)...𝑁)) = {(2nd𝑇), ((2nd𝑇) + 1)})
220 undif 4026 . . . . . . . . . . . . . . . . 17 ({(2nd𝑇), ((2nd𝑇) + 1)} ⊆ ((𝑦 + 1)...𝑁) ↔ ({(2nd𝑇), ((2nd𝑇) + 1)} ∪ (((𝑦 + 1)...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) = ((𝑦 + 1)...𝑁))
221209, 220sylib 208 . . . . . . . . . . . . . . . 16 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → ({(2nd𝑇), ((2nd𝑇) + 1)} ∪ (((𝑦 + 1)...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) = ((𝑦 + 1)...𝑁))
222221imaeq2d 5430 . . . . . . . . . . . . . . 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)))
223 fnresi 5971 . . . . . . . . . . . . . . . . . . 19 ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) Fn ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})
224 incom 3788 . . . . . . . . . . . . . . . . . . . 20 (((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) ∩ {(2nd𝑇), ((2nd𝑇) + 1)}) = ({(2nd𝑇), ((2nd𝑇) + 1)} ∩ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))
225224, 32eqtri 2643 . . . . . . . . . . . . . . . . . . 19 (((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) ∩ {(2nd𝑇), ((2nd𝑇) + 1)}) = ∅
226 fnimadisj 5974 . . . . . . . . . . . . . . . . . . 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)}) = ∅)
227223, 225, 226mp2an 707 . . . . . . . . . . . . . . . . . 18 (( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) “ {(2nd𝑇), ((2nd𝑇) + 1)}) = ∅
228227a1i 11 . . . . . . . . . . . . . . . . 17 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → (( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) “ {(2nd𝑇), ((2nd𝑇) + 1)}) = ∅)
229 nnuz 11675 . . . . . . . . . . . . . . . . . . . . . 22 ℕ = (ℤ‘1)
230188, 229syl6eleq 2708 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈ (ℤ‘1))
231 fzss1 12330 . . . . . . . . . . . . . . . . . . . . 21 ((𝑦 + 1) ∈ (ℤ‘1) → ((𝑦 + 1)...𝑁) ⊆ (1...𝑁))
232230, 231syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑦 + 1)...𝑁) ⊆ (1...𝑁))
233232ssdifd 3729 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ (0...(𝑁 − 1)) → (((𝑦 + 1)...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) ⊆ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))
234 resiima 5444 . . . . . . . . . . . . . . . . . . 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)}))
235233, 234syl 17 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ (0...(𝑁 − 1)) → (( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) “ (((𝑦 + 1)...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) = (((𝑦 + 1)...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))
236235ad2antlr 762 . . . . . . . . . . . . . . . . 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)}))
237228, 236uneq12d 3751 . . . . . . . . . . . . . . . 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)})))
238 imaundi 5509 . . . . . . . . . . . . . . . 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)})))
239 uncom 3740 . . . . . . . . . . . . . . . . 17 (∅ ∪ (((𝑦 + 1)...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) = ((((𝑦 + 1)...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) ∪ ∅)
240 un0 3944 . . . . . . . . . . . . . . . . 17 ((((𝑦 + 1)...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) ∪ ∅) = (((𝑦 + 1)...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})
241239, 240eqtr2i 2644 . . . . . . . . . . . . . . . 16 (((𝑦 + 1)...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) = (∅ ∪ (((𝑦 + 1)...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))
242237, 238, 2413eqtr4g 2680 . . . . . . . . . . . . . . 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)}))
243222, 242eqtr3d 2657 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ 𝑦 < (2nd𝑇)) → (( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) “ ((𝑦 + 1)...𝑁)) = (((𝑦 + 1)...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))
244219, 243uneq12d 3751 . . . . . . . . . . . . 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)})))
245174, 244syl5eq 2667 . . . . . . . . . . . 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)})))
246245, 221eqtrd 2655 . . . . . . . . . . 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))
247246imaeq2d 5430 . . . . . . . . . 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)))
248173, 247syl5eq 2667 . . . . . . . . 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)))
249248xpeq1d 5103 . . . . . . . 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}))
250172, 249uneq12d 3751 . . . . . . 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})))
251250oveq2d 6626 . . . . . 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}))))
252 iftrue 4069 . . . . . . . . 9 (𝑦 < (2nd𝑇) → if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) = 𝑦)
253252csbeq1d 3525 . . . . . . . 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}))))
254 vex 3192 . . . . . . . . 9 𝑦 ∈ V
255 oveq2 6618 . . . . . . . . . . . . 13 (𝑗 = 𝑦 → (1...𝑗) = (1...𝑦))
256255imaeq2d 5430 . . . . . . . . . . . 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)))
257256xpeq1d 5103 . . . . . . . . . . 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}))
258 oveq1 6617 . . . . . . . . . . . . . 14 (𝑗 = 𝑦 → (𝑗 + 1) = (𝑦 + 1))
259258oveq1d 6625 . . . . . . . . . . . . 13 (𝑗 = 𝑦 → ((𝑗 + 1)...𝑁) = ((𝑦 + 1)...𝑁))
260259imaeq2d 5430 . . . . . . . . . . . 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)))
261260xpeq1d 5103 . . . . . . . . . . 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}))
262257, 261uneq12d 3751 . . . . . . . . . 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})))
263262oveq2d 6626 . . . . . . . . 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}))))
264254, 263csbie 3544 . . . . . . . 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})))
265253, 264syl6eq 2671 . . . . . . 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}))))
266265adantl 482 . . . . . 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}))))
267252csbeq1d 3525 . . . . . . . 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}))))
268255imaeq2d 5430 . . . . . . . . . . . 12 (𝑗 = 𝑦 → ((2nd ‘(1st𝑇)) “ (1...𝑗)) = ((2nd ‘(1st𝑇)) “ (1...𝑦)))
269268xpeq1d 5103 . . . . . . . . . . 11 (𝑗 = 𝑦 → (((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑇)) “ (1...𝑦)) × {1}))
270259imaeq2d 5430 . . . . . . . . . . . 12 (𝑗 = 𝑦 → ((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)))
271270xpeq1d 5103 . . . . . . . . . . 11 (𝑗 = 𝑦 → (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑇)) “ ((𝑦 + 1)...𝑁)) × {0}))
272269, 271uneq12d 3751 . . . . . . . . . 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})))
273272oveq2d 6626 . . . . . . . . 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}))))
274254, 273csbie 3544 . . . . . . . 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})))
275267, 274syl6eq 2671 . . . . . . 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}))))
276275adantl 482 . . . . . 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}))))
277251, 266, 2763eqtr4d 2665 . . . . 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}))))
278 lenlt 10068 . . . . . . . . . 10 (((2nd𝑇) ∈ ℝ ∧ 𝑦 ∈ ℝ) → ((2nd𝑇) ≤ 𝑦 ↔ ¬ 𝑦 < (2nd𝑇)))
27918, 118, 278syl2an 494 . . . . . . . . 9 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((2nd𝑇) ≤ 𝑦 ↔ ¬ 𝑦 < (2nd𝑇)))
280279biimpar 502 . . . . . . . 8 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ ¬ 𝑦 < (2nd𝑇)) → (2nd𝑇) ≤ 𝑦)
281 imaco 5604 . . . . . . . . . . 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))))
282 imaundir 5510 . . . . . . . . . . . . . 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))))
283 imassrn 5441 . . . . . . . . . . . . . . . . . 18 ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} “ (1...(𝑦 + 1))) ⊆ ran {⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩}
284283a1i 11 . . . . . . . . . . . . . . . . 17 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} “ (1...(𝑦 + 1))) ⊆ ran {⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩})
285178ad2antrr 761 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} “ {(2nd𝑇), ((2nd𝑇) + 1)}) = ran {⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩})
28617ad2antrr 761 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → (2nd𝑇) ∈ ℕ)
28718ad2antrr 761 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → (2nd𝑇) ∈ ℝ)
288118ad2antlr 762 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → 𝑦 ∈ ℝ)
289188nnred 10987 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 ∈ (0...(𝑁 − 1)) → (𝑦 + 1) ∈ ℝ)
290289ad2antlr 762 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → (𝑦 + 1) ∈ ℝ)
291 simpr 477 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → (2nd𝑇) ≤ 𝑦)
292118lep1d 10907 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 ≤ (𝑦 + 1))
293292ad2antlr 762 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → 𝑦 ≤ (𝑦 + 1))
294287, 288, 290, 291, 293letrd 10146 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → (2nd𝑇) ≤ (𝑦 + 1))
295 fznn 12358 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑦 + 1) ∈ ℤ → ((2nd𝑇) ∈ (1...(𝑦 + 1)) ↔ ((2nd𝑇) ∈ ℕ ∧ (2nd𝑇) ≤ (𝑦 + 1))))
296189, 295syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 ∈ (0...(𝑁 − 1)) → ((2nd𝑇) ∈ (1...(𝑦 + 1)) ↔ ((2nd𝑇) ∈ ℕ ∧ (2nd𝑇) ≤ (𝑦 + 1))))
297296ad2antlr 762 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → ((2nd𝑇) ∈ (1...(𝑦 + 1)) ↔ ((2nd𝑇) ∈ ℕ ∧ (2nd𝑇) ≤ (𝑦 + 1))))
298286, 294, 297mpbir2and 956 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → (2nd𝑇) ∈ (1...(𝑦 + 1)))
29950ad2antrr 761 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → ((2nd𝑇) + 1) ∈ ℕ)
300 1red 10007 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → 1 ∈ ℝ)
301287, 288, 300, 291leadd1dd 10593 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → ((2nd𝑇) + 1) ≤ (𝑦 + 1))
302 fznn 12358 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑦 + 1) ∈ ℤ → (((2nd𝑇) + 1) ∈ (1...(𝑦 + 1)) ↔ (((2nd𝑇) + 1) ∈ ℕ ∧ ((2nd𝑇) + 1) ≤ (𝑦 + 1))))
303189, 302syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 ∈ (0...(𝑁 − 1)) → (((2nd𝑇) + 1) ∈ (1...(𝑦 + 1)) ↔ (((2nd𝑇) + 1) ∈ ℕ ∧ ((2nd𝑇) + 1) ≤ (𝑦 + 1))))
304303ad2antlr 762 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → (((2nd𝑇) + 1) ∈ (1...(𝑦 + 1)) ↔ (((2nd𝑇) + 1) ∈ ℕ ∧ ((2nd𝑇) + 1) ≤ (𝑦 + 1))))
305299, 301, 304mpbir2and 956 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → ((2nd𝑇) + 1) ∈ (1...(𝑦 + 1)))
306 prssi 4326 . . . . . . . . . . . . . . . . . . . 20 (((2nd𝑇) ∈ (1...(𝑦 + 1)) ∧ ((2nd𝑇) + 1) ∈ (1...(𝑦 + 1))) → {(2nd𝑇), ((2nd𝑇) + 1)} ⊆ (1...(𝑦 + 1)))
307298, 305, 306syl2anc 692 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → {(2nd𝑇), ((2nd𝑇) + 1)} ⊆ (1...(𝑦 + 1)))
308 imass2 5465 . . . . . . . . . . . . . . . . . . 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))))
309307, 308syl 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))))
310285, 309eqsstr3d 3624 . . . . . . . . . . . . . . . . 17 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → ran {⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ⊆ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} “ (1...(𝑦 + 1))))
311284, 310eqssd 3604 . . . . . . . . . . . . . . . 16 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} “ (1...(𝑦 + 1))) = ran {⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩})
312217ad2antrr 761 . . . . . . . . . . . . . . . 16 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → ran {⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} = {(2nd𝑇), ((2nd𝑇) + 1)})
313311, 312eqtrd 2655 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} “ (1...(𝑦 + 1))) = {(2nd𝑇), ((2nd𝑇) + 1)})
314 undif 4026 . . . . . . . . . . . . . . . . . 18 ({(2nd𝑇), ((2nd𝑇) + 1)} ⊆ (1...(𝑦 + 1)) ↔ ({(2nd𝑇), ((2nd𝑇) + 1)} ∪ ((1...(𝑦 + 1)) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) = (1...(𝑦 + 1)))
315307, 314sylib 208 . . . . . . . . . . . . . . . . 17 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → ({(2nd𝑇), ((2nd𝑇) + 1)} ∪ ((1...(𝑦 + 1)) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) = (1...(𝑦 + 1)))
316315imaeq2d 5430 . . . . . . . . . . . . . . . 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))))
317227a1i 11 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → (( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) “ {(2nd𝑇), ((2nd𝑇) + 1)}) = ∅)
318 eluzp1p1 11665 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑁 − 1) ∈ (ℤ𝑦) → ((𝑁 − 1) + 1) ∈ (ℤ‘(𝑦 + 1)))
319152, 318syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈ (ℤ‘(𝑦 + 1)))
320319adantl 482 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((𝑁 − 1) + 1) ∈ (ℤ‘(𝑦 + 1)))
321151, 320eqeltrrd 2699 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → 𝑁 ∈ (ℤ‘(𝑦 + 1)))
322 fzss2 12331 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 ∈ (ℤ‘(𝑦 + 1)) → (1...(𝑦 + 1)) ⊆ (1...𝑁))
323321, 322syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (1...(𝑦 + 1)) ⊆ (1...𝑁))
324323ssdifd 3729 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((1...(𝑦 + 1)) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) ⊆ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))
325324adantr 481 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → ((1...(𝑦 + 1)) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) ⊆ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))
326 resiima 5444 . . . . . . . . . . . . . . . . . . 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)}))
327325, 326syl 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)}))
328317, 327uneq12d 3751 . . . . . . . . . . . . . . . . 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)})))
329 imaundi 5509 . . . . . . . . . . . . . . . . 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)})))
330 uncom 3740 . . . . . . . . . . . . . . . . . 18 (∅ ∪ ((1...(𝑦 + 1)) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) = (((1...(𝑦 + 1)) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) ∪ ∅)
331 un0 3944 . . . . . . . . . . . . . . . . . 18 (((1...(𝑦 + 1)) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) ∪ ∅) = ((1...(𝑦 + 1)) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})
332330, 331eqtr2i 2644 . . . . . . . . . . . . . . . . 17 ((1...(𝑦 + 1)) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) = (∅ ∪ ((1...(𝑦 + 1)) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))
333328, 329, 3323eqtr4g 2680 . . . . . . . . . . . . . . . 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)}))
334316, 333eqtr3d 2657 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → (( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) “ (1...(𝑦 + 1))) = ((1...(𝑦 + 1)) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))
335313, 334uneq12d 3751 . . . . . . . . . . . . . 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)})))
336282, 335syl5eq 2667 . . . . . . . . . . . . 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)})))
337336, 315eqtrd 2655 . . . . . . . . . . . 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)))
338337imaeq2d 5430 . . . . . . . . . . 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))))
339281, 338syl5eq 2667 . . . . . . . . . 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))))
340339xpeq1d 5103 . . . . . . . . 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}))
341 imaco 5604 . . . . . . . . . . 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)))
342114ad2antrr 761 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → {⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} Fn {(2nd𝑇), ((2nd𝑇) + 1)})
343 incom 3788 . . . . . . . . . . . . . . . 16 ({(2nd𝑇), ((2nd𝑇) + 1)} ∩ (((𝑦 + 1) + 1)...𝑁)) = ((((𝑦 + 1) + 1)...𝑁) ∩ {(2nd𝑇), ((2nd𝑇) + 1)})
344128ad2antrr 761 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → ((2nd𝑇) + 1) ∈ ℝ)
345188peano2nnd 10989 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑦 + 1) + 1) ∈ ℕ)
346345nnred 10987 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑦 + 1) + 1) ∈ ℝ)
347346ad2antlr 762 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → ((𝑦 + 1) + 1) ∈ ℝ)
34819ad2antrr 761 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → (2nd𝑇) < ((2nd𝑇) + 1))
349118ltp1d 10906 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 ∈ (0...(𝑁 − 1)) → 𝑦 < (𝑦 + 1))
350349ad2antlr 762 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → 𝑦 < (𝑦 + 1))
351287, 288, 290, 291, 350lelttrd 10147 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → (2nd𝑇) < (𝑦 + 1))
352287, 290, 300, 351ltadd1dd 10590 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → ((2nd𝑇) + 1) < ((𝑦 + 1) + 1))
353287, 344, 347, 348, 352lttrd 10150 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → (2nd𝑇) < ((𝑦 + 1) + 1))
354 ltnle 10069 . . . . . . . . . . . . . . . . . . . . . . 23 (((2nd𝑇) ∈ ℝ ∧ ((𝑦 + 1) + 1) ∈ ℝ) → ((2nd𝑇) < ((𝑦 + 1) + 1) ↔ ¬ ((𝑦 + 1) + 1) ≤ (2nd𝑇)))
35518, 346, 354syl2an 494 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → ((2nd𝑇) < ((𝑦 + 1) + 1) ↔ ¬ ((𝑦 + 1) + 1) ≤ (2nd𝑇)))
356355adantr 481 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → ((2nd𝑇) < ((𝑦 + 1) + 1) ↔ ¬ ((𝑦 + 1) + 1) ≤ (2nd𝑇)))
357353, 356mpbid 222 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → ¬ ((𝑦 + 1) + 1) ≤ (2nd𝑇))
358 elfzle1 12294 . . . . . . . . . . . . . . . . . . . 20 ((2nd𝑇) ∈ (((𝑦 + 1) + 1)...𝑁) → ((𝑦 + 1) + 1) ≤ (2nd𝑇))
359357, 358nsyl 135 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → ¬ (2nd𝑇) ∈ (((𝑦 + 1) + 1)...𝑁))
360 disjsn 4221 . . . . . . . . . . . . . . . . . . 19 (((((𝑦 + 1) + 1)...𝑁) ∩ {(2nd𝑇)}) = ∅ ↔ ¬ (2nd𝑇) ∈ (((𝑦 + 1) + 1)...𝑁))
361359, 360sylibr 224 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → ((((𝑦 + 1) + 1)...𝑁) ∩ {(2nd𝑇)}) = ∅)
362 ltnle 10069 . . . . . . . . . . . . . . . . . . . . . . 23 ((((2nd𝑇) + 1) ∈ ℝ ∧ ((𝑦 + 1) + 1) ∈ ℝ) → (((2nd𝑇) + 1) < ((𝑦 + 1) + 1) ↔ ¬ ((𝑦 + 1) + 1) ≤ ((2nd𝑇) + 1)))
363128, 346, 362syl2an 494 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑦 ∈ (0...(𝑁 − 1))) → (((2nd𝑇) + 1) < ((𝑦 + 1) + 1) ↔ ¬ ((𝑦 + 1) + 1) ≤ ((2nd𝑇) + 1)))
364363adantr 481 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → (((2nd𝑇) + 1) < ((𝑦 + 1) + 1) ↔ ¬ ((𝑦 + 1) + 1) ≤ ((2nd𝑇) + 1)))
365352, 364mpbid 222 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → ¬ ((𝑦 + 1) + 1) ≤ ((2nd𝑇) + 1))
366 elfzle1 12294 . . . . . . . . . . . . . . . . . . . 20 (((2nd𝑇) + 1) ∈ (((𝑦 + 1) + 1)...𝑁) → ((𝑦 + 1) + 1) ≤ ((2nd𝑇) + 1))
367365, 366nsyl 135 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → ¬ ((2nd𝑇) + 1) ∈ (((𝑦 + 1) + 1)...𝑁))
368 disjsn 4221 . . . . . . . . . . . . . . . . . . 19 (((((𝑦 + 1) + 1)...𝑁) ∩ {((2nd𝑇) + 1)}) = ∅ ↔ ¬ ((2nd𝑇) + 1) ∈ (((𝑦 + 1) + 1)...𝑁))
369367, 368sylibr 224 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → ((((𝑦 + 1) + 1)...𝑁) ∩ {((2nd𝑇) + 1)}) = ∅)
370361, 369uneq12d 3751 . . . . . . . . . . . . . . . . 17 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → (((((𝑦 + 1) + 1)...𝑁) ∩ {(2nd𝑇)}) ∪ ((((𝑦 + 1) + 1)...𝑁) ∩ {((2nd𝑇) + 1)})) = (∅ ∪ ∅))
371142ineq2i 3794 . . . . . . . . . . . . . . . . . 18 ((((𝑦 + 1) + 1)...𝑁) ∩ {(2nd𝑇), ((2nd𝑇) + 1)}) = ((((𝑦 + 1) + 1)...𝑁) ∩ ({(2nd𝑇)} ∪ {((2nd𝑇) + 1)}))
372 indi 3854 . . . . . . . . . . . . . . . . . 18 ((((𝑦 + 1) + 1)...𝑁) ∩ ({(2nd𝑇)} ∪ {((2nd𝑇) + 1)})) = (((((𝑦 + 1) + 1)...𝑁) ∩ {(2nd𝑇)}) ∪ ((((𝑦 + 1) + 1)...𝑁) ∩ {((2nd𝑇) + 1)}))
373371, 372eqtr2i 2644 . . . . . . . . . . . . . . . . 17 (((((𝑦 + 1) + 1)...𝑁) ∩ {(2nd𝑇)}) ∪ ((((𝑦 + 1) + 1)...𝑁) ∩ {((2nd𝑇) + 1)})) = ((((𝑦 + 1) + 1)...𝑁) ∩ {(2nd𝑇), ((2nd𝑇) + 1)})
374370, 373, 1463eqtr3g 2678 . . . . . . . . . . . . . . . 16 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → ((((𝑦 + 1) + 1)...𝑁) ∩ {(2nd𝑇), ((2nd𝑇) + 1)}) = ∅)
375343, 374syl5eq 2667 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → ({(2nd𝑇), ((2nd𝑇) + 1)} ∩ (((𝑦 + 1) + 1)...𝑁)) = ∅)
376 fnimadisj 5974 . . . . . . . . . . . . . . 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)) = ∅)
377342, 375, 376syl2anc 692 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} “ (((𝑦 + 1) + 1)...𝑁)) = ∅)
378345, 229syl6eleq 2708 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ (0...(𝑁 − 1)) → ((𝑦 + 1) + 1) ∈ (ℤ‘1))
379 fzss1 12330 . . . . . . . . . . . . . . . . . 18 (((𝑦 + 1) + 1) ∈ (ℤ‘1) → (((𝑦 + 1) + 1)...𝑁) ⊆ (1...𝑁))
380 reldisj 3997 . . . . . . . . . . . . . . . . . 18 ((((𝑦 + 1) + 1)...𝑁) ⊆ (1...𝑁) → (((((𝑦 + 1) + 1)...𝑁) ∩ {(2nd𝑇), ((2nd𝑇) + 1)}) = ∅ ↔ (((𝑦 + 1) + 1)...𝑁) ⊆ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))
381378, 379, 3803syl 18 . . . . . . . . . . . . . . . . 17 (𝑦 ∈ (0...(𝑁 − 1)) → (((((𝑦 + 1) + 1)...𝑁) ∩ {(2nd𝑇), ((2nd𝑇) + 1)}) = ∅ ↔ (((𝑦 + 1) + 1)...𝑁) ⊆ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))
382381ad2antlr 762 . . . . . . . . . . . . . . . 16 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → (((((𝑦 + 1) + 1)...𝑁) ∩ {(2nd𝑇), ((2nd𝑇) + 1)}) = ∅ ↔ (((𝑦 + 1) + 1)...𝑁) ⊆ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})))
383374, 382mpbid 222 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → (((𝑦 + 1) + 1)...𝑁) ⊆ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))
384 resiima 5444 . . . . . . . . . . . . . . 15 ((((𝑦 + 1) + 1)...𝑁) ⊆ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}) → (( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) “ (((𝑦 + 1) + 1)...𝑁)) = (((𝑦 + 1) + 1)...𝑁))
385383, 384syl 17 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ (0...(𝑁 − 1))) ∧ (2nd𝑇) ≤ 𝑦) → (( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)})) “ (((𝑦 + 1) + 1)...𝑁)) = (((𝑦 + 1) + 1)...𝑁))
386377, 385uneq12d 3751 . . . . . . . . . . . . 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)))
387 imaundir 5510 . . . . . . . . . . . . 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)))
388 uncom 3740 . . . . . . . . . . . . . 14 (∅ ∪ (((𝑦 + 1) + 1)...𝑁)) = ((((𝑦 + 1) + 1)...𝑁) ∪ ∅)
389 un0 3944 . . . . . . . . . . . . . 14 ((((𝑦 + 1) + 1)...𝑁) ∪ ∅) = (((𝑦 + 1) + 1)...𝑁)
390388, 389eqtr2i 2644 . . . . . . . . . . . . 13 (((𝑦 + 1) + 1)...𝑁) = (∅ ∪ (((𝑦 + 1) + 1)...𝑁))
391386, 387, 3903eqtr4g 2680 . . . . . . . . . . . 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))
392391imaeq2d 5430 . . . . . . . . . . 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)))
393341, 392syl5eq 2667 . . . . . . . . . 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)))
394393xpeq1d 5103 . . . . . . . . 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}))
395340, 394uneq12d 3751 . . . . . . . 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})))
396280, 395syldan 487 . . . . . . 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})))
397396oveq2d 6626 . . . . . 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}))))
398 iffalse 4072 . . . . . . . . 9 𝑦 < (2nd𝑇) → if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)) = (𝑦 + 1))
399398csbeq1d 3525 . . . . . . . 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}))))
400 ovex 6638 . . . . . . . . 9 (𝑦 + 1) ∈ V
401 oveq2 6618 . . . . . . . . . . . . 13 (𝑗 = (𝑦 + 1) → (1...𝑗) = (1...(𝑦 + 1)))
402401imaeq2d 5430 . . . . . . . . . . . 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))))
403402xpeq1d 5103 . . . . . . . . . . 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}))
404 oveq1 6617 . . . . . . . . . . . . . 14 (𝑗 = (𝑦 + 1) → (𝑗 + 1) = ((𝑦 + 1) + 1))
405404oveq1d 6625 . . . . . . . . . . . . 13 (𝑗 = (𝑦 + 1) → ((𝑗 + 1)...𝑁) = (((𝑦 + 1) + 1)...𝑁))
406405imaeq2d 5430 . . . . . . . . . . . 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)))
407406xpeq1d 5103 . . . . . . . . . . 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}))
408403, 407uneq12d 3751 . . . . . . . . . 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})))
409408oveq2d 6626 . . . . . . . . 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}))))
410400, 409csbie 3544 . . . . . . . 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})))
411399, 410syl6eq 2671 . . . . . . 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}))))
412411adantl 482 . . . . . 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}))))
413398csbeq1d 3525 . . . . . . . 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}))))
414401imaeq2d 5430 . . . . . . . . . . . 12 (𝑗 = (𝑦 + 1) → ((2nd ‘(1st𝑇)) “ (1...𝑗)) = ((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))))
415414xpeq1d 5103 . . . . . . . . . . 11 (𝑗 = (𝑦 + 1) → (((2nd ‘(1st𝑇)) “ (1...𝑗)) × {1}) = (((2nd ‘(1st𝑇)) “ (1...(𝑦 + 1))) × {1}))
416405imaeq2d 5430 . . . . . . . . . . . 12 (𝑗 = (𝑦 + 1) → ((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) = ((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)))
417416xpeq1d 5103 . . . . . . . . . . 11 (𝑗 = (𝑦 + 1) → (((2nd ‘(1st𝑇)) “ ((𝑗 + 1)...𝑁)) × {0}) = (((2nd ‘(1st𝑇)) “ (((𝑦 + 1) + 1)...𝑁)) × {0}))
418415, 417uneq12d 3751 . . . . . . . . . 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})))
419418oveq2d 6626 . . . . . . . . 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}))))
420400, 419csbie 3544 . . . . . . . 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})))
421413, 420syl6eq 2671 . . . . . . 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}))))
422421adantl 482 . . . . . 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}))))
423397, 412, 4223eqtr4d 2665 . . . . 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}))))
424277, 423pm2.61dan 831 . . . 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}))))
425424mpteq2dva 4709 . . 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})))))
426111, 425eqtr4d 2658 . 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})))))
427 opex 4898 . . . . . . 7 ⟨(1st ‘(1st𝑇)), ((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))))⟩ ∈ V
428427, 22op1std 7130 . . . . . 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)}))))⟩)
429427, 22op2ndd 7131 . . . . . 6 (𝑡 = ⟨⟨(1st ‘(1st𝑇)), ((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))))⟩, (2nd𝑇)⟩ → (2nd𝑡) = (2nd𝑇))
430 breq2 4622 . . . . . . . . 9 ((2nd𝑡) = (2nd𝑇) → (𝑦 < (2nd𝑡) ↔ 𝑦 < (2nd𝑇)))
431430ifbid 4085 . . . . . . . 8 ((2nd𝑡) = (2nd𝑇) → if(𝑦 < (2nd𝑡), 𝑦, (𝑦 + 1)) = if(𝑦 < (2nd𝑇), 𝑦, (𝑦 + 1)))
432431csbeq1d 3525 . . . . . . 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}))))
433 fvex 6163 . . . . . . . . . 10 (1st ‘(1st𝑇)) ∈ V
434433, 79op1std 7130 . . . . . . . . 9 ((1st𝑡) = ⟨(1st ‘(1st𝑇)), ((2nd ‘(1st𝑇)) ∘ ({⟨(2nd𝑇), ((2nd𝑇) + 1)⟩, ⟨((2nd𝑇) + 1), (2nd𝑇)⟩} ∪ ( I ↾ ((1...𝑁) ∖ {(2nd𝑇), ((2nd𝑇) + 1)}))))⟩ → (1st ‘(1st𝑡)) = (1st ‘(1st𝑇)))
435433, 79op2ndd 7131 . . . . . . . . 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)})))))
436 id 22 . . . . . . . . . 10 ((1st ‘(1st𝑡)) = (1st ‘(1st𝑇)) → (1st ‘(1st𝑡)) = (1st ‘(1st𝑇)))
437 imaeq1 5425 . . . . . . . . . . . 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)))
438437xpeq1d 5103 . . . . . . . . . . 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}))
439 imaeq1 5425 . . . . . . . . . . . 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)))
440439xpeq1d 5103 . . . . . . . . . . 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}))
441438, 440uneq12d 3751 . . . . . . . . . 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})))
442436, 441oveqan12d 6629 . . . . . . . . 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}))))
443434, 435, 442syl2anc 692 . . . . . . . 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}))))
444443csbeq2dv 3969 . . . . . . 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}))))
445432, 444sylan9eqr 2677 . . . . . 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}))))
446428, 429, 445syl2anc 692 . . . . 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}))))
447446mpteq2dv 4710 . . . 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})))))
448447eqeq2d 2631 . . 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}))))))
449448, 3elrab2 3352 . 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}))))))
45091, 426, 449sylanbrc 697 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 196  wa 384   = wceq 1480  wcel 1987  {cab 2607  wne 2790  {crab 2911  Vcvv 3189  csb 3518  cdif 3556  cun 3557  cin 3558  wss 3559  c0 3896  ifcif 4063  {csn 4153  {cpr 4155  cop 4159   class class class wbr 4618  cmpt 4678   I cid 4989   × cxp 5077  ran crn 5080  cres 5081  cima 5082  ccom 5083   Fn wfn 5847  wf 5848  ontowfo 5850  1-1-ontowf1o 5851  cfv 5852  (class class class)co 6610  𝑓 cof 6855  1st c1st 7118  2nd c2nd 7119  𝑚 cmap 7809  cc 9886  cr 9887  0cc0 9888  1c1 9889   + caddc 9891   < clt 10026  cle 10027  cmin 10218  cn 10972  0cn0 11244  cz 11329  cuz 11639  ...cfz 12276  ..^cfzo 12414
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9944  ax-resscn 9945  ax-1cn 9946  ax-icn 9947  ax-addcl 9948  ax-addrcl 9949  ax-mulcl 9950  ax-mulrcl 9951  ax-mulcom 9952  ax-addass 9953  ax-mulass 9954  ax-distr 9955  ax-i2m1 9956  ax-1ne0 9957  ax-1rid 9958  ax-rnegex 9959  ax-rrecex 9960  ax-cnre 9961  ax-pre-lttri 9962  ax-pre-lttrn 9963  ax-pre-ltadd 9964  ax-pre-mulgt0 9965
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-er 7694  df-en 7908  df-dom 7909  df-sdom 7910  df-pnf 10028  df-mnf 10029  df-xr 10030  df-ltxr 10031  df-le 10032  df-sub 10220  df-neg 10221  df-nn 10973  df-n0 11245  df-z 11330  df-uz 11640  df-fz 12277
This theorem is referenced by:  poimirlem22  33098
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