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Theorem psgnunilem2 19284
Description: Lemma for psgnuni 19288. Induction step for moving a transposition as far to the right as possible. (Contributed by Stefan O'Rear, 24-Aug-2015.) (Revised by Mario Carneiro, 28-Feb-2016.)
Hypotheses
Ref Expression
psgnunilem2.g 𝐺 = (SymGrp‘𝐷)
psgnunilem2.t 𝑇 = ran (pmTrsp‘𝐷)
psgnunilem2.d (𝜑𝐷𝑉)
psgnunilem2.w (𝜑𝑊 ∈ Word 𝑇)
psgnunilem2.id (𝜑 → (𝐺 Σg 𝑊) = ( I ↾ 𝐷))
psgnunilem2.l (𝜑 → (♯‘𝑊) = 𝐿)
psgnunilem2.ix (𝜑𝐼 ∈ (0..^𝐿))
psgnunilem2.a (𝜑𝐴 ∈ dom ((𝑊𝐼) ∖ I ))
psgnunilem2.al (𝜑 → ∀𝑘 ∈ (0..^𝐼) ¬ 𝐴 ∈ dom ((𝑊𝑘) ∖ I ))
psgnunilem2.in (𝜑 → ¬ ∃𝑥 ∈ Word 𝑇((♯‘𝑥) = (𝐿 − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))
Assertion
Ref Expression
psgnunilem2 (𝜑 → ∃𝑤 ∈ Word 𝑇(((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ∧ ((𝐼 + 1) ∈ (0..^𝐿) ∧ 𝐴 ∈ dom ((𝑤‘(𝐼 + 1)) ∖ I ) ∧ ∀𝑗 ∈ (0..^(𝐼 + 1)) ¬ 𝐴 ∈ dom ((𝑤𝑗) ∖ I ))))
Distinct variable groups:   𝑗,𝑘,𝑤,𝐴   𝑥,𝑗,𝐷,𝑤   𝜑,𝑗   𝑗,𝐺   𝑥,𝑘,𝐺,𝑤   𝑗,𝐼,𝑘,𝑤,𝑥   𝑇,𝑗,𝑤,𝑥   𝑗,𝑊,𝑘,𝑤,𝑥   𝑤,𝐿,𝑥
Allowed substitution hints:   𝜑(𝑥,𝑤,𝑘)   𝐴(𝑥)   𝐷(𝑘)   𝑇(𝑘)   𝐿(𝑗,𝑘)   𝑉(𝑥,𝑤,𝑗,𝑘)

Proof of Theorem psgnunilem2
Dummy variables 𝑟 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 psgnunilem2.w . . . . . . 7 (𝜑𝑊 ∈ Word 𝑇)
2 wrd0 14434 . . . . . . 7 ∅ ∈ Word 𝑇
3 splcl 14647 . . . . . . 7 ((𝑊 ∈ Word 𝑇 ∧ ∅ ∈ Word 𝑇) → (𝑊 splice ⟨𝐼, (𝐼 + 2), ∅⟩) ∈ Word 𝑇)
41, 2, 3sylancl 587 . . . . . 6 (𝜑 → (𝑊 splice ⟨𝐼, (𝐼 + 2), ∅⟩) ∈ Word 𝑇)
54adantr 482 . . . . 5 ((𝜑 ∧ ((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → (𝑊 splice ⟨𝐼, (𝐼 + 2), ∅⟩) ∈ Word 𝑇)
6 fzossfz 13598 . . . . . . . . . . 11 (0..^𝐿) ⊆ (0...𝐿)
7 psgnunilem2.ix . . . . . . . . . . 11 (𝜑𝐼 ∈ (0..^𝐿))
86, 7sselid 3947 . . . . . . . . . 10 (𝜑𝐼 ∈ (0...𝐿))
9 elfznn0 13541 . . . . . . . . . 10 (𝐼 ∈ (0...𝐿) → 𝐼 ∈ ℕ0)
108, 9syl 17 . . . . . . . . 9 (𝜑𝐼 ∈ ℕ0)
11 2nn0 12437 . . . . . . . . . 10 2 ∈ ℕ0
12 nn0addcl 12455 . . . . . . . . . 10 ((𝐼 ∈ ℕ0 ∧ 2 ∈ ℕ0) → (𝐼 + 2) ∈ ℕ0)
1310, 11, 12sylancl 587 . . . . . . . . 9 (𝜑 → (𝐼 + 2) ∈ ℕ0)
1410nn0red 12481 . . . . . . . . . 10 (𝜑𝐼 ∈ ℝ)
15 nn0addge1 12466 . . . . . . . . . 10 ((𝐼 ∈ ℝ ∧ 2 ∈ ℕ0) → 𝐼 ≤ (𝐼 + 2))
1614, 11, 15sylancl 587 . . . . . . . . 9 (𝜑𝐼 ≤ (𝐼 + 2))
17 elfz2nn0 13539 . . . . . . . . 9 (𝐼 ∈ (0...(𝐼 + 2)) ↔ (𝐼 ∈ ℕ0 ∧ (𝐼 + 2) ∈ ℕ0𝐼 ≤ (𝐼 + 2)))
1810, 13, 16, 17syl3anbrc 1344 . . . . . . . 8 (𝜑𝐼 ∈ (0...(𝐼 + 2)))
19 psgnunilem2.g . . . . . . . . . . 11 𝐺 = (SymGrp‘𝐷)
20 psgnunilem2.t . . . . . . . . . . 11 𝑇 = ran (pmTrsp‘𝐷)
21 psgnunilem2.d . . . . . . . . . . 11 (𝜑𝐷𝑉)
22 psgnunilem2.id . . . . . . . . . . 11 (𝜑 → (𝐺 Σg 𝑊) = ( I ↾ 𝐷))
23 psgnunilem2.l . . . . . . . . . . 11 (𝜑 → (♯‘𝑊) = 𝐿)
24 psgnunilem2.a . . . . . . . . . . 11 (𝜑𝐴 ∈ dom ((𝑊𝐼) ∖ I ))
25 psgnunilem2.al . . . . . . . . . . 11 (𝜑 → ∀𝑘 ∈ (0..^𝐼) ¬ 𝐴 ∈ dom ((𝑊𝑘) ∖ I ))
2619, 20, 21, 1, 22, 23, 7, 24, 25psgnunilem5 19283 . . . . . . . . . 10 (𝜑 → (𝐼 + 1) ∈ (0..^𝐿))
27 fzofzp1 13676 . . . . . . . . . 10 ((𝐼 + 1) ∈ (0..^𝐿) → ((𝐼 + 1) + 1) ∈ (0...𝐿))
2826, 27syl 17 . . . . . . . . 9 (𝜑 → ((𝐼 + 1) + 1) ∈ (0...𝐿))
29 df-2 12223 . . . . . . . . . . 11 2 = (1 + 1)
3029oveq2i 7373 . . . . . . . . . 10 (𝐼 + 2) = (𝐼 + (1 + 1))
3110nn0cnd 12482 . . . . . . . . . . 11 (𝜑𝐼 ∈ ℂ)
32 1cnd 11157 . . . . . . . . . . 11 (𝜑 → 1 ∈ ℂ)
3331, 32, 32addassd 11184 . . . . . . . . . 10 (𝜑 → ((𝐼 + 1) + 1) = (𝐼 + (1 + 1)))
3430, 33eqtr4id 2796 . . . . . . . . 9 (𝜑 → (𝐼 + 2) = ((𝐼 + 1) + 1))
3523oveq2d 7378 . . . . . . . . 9 (𝜑 → (0...(♯‘𝑊)) = (0...𝐿))
3628, 34, 353eltr4d 2853 . . . . . . . 8 (𝜑 → (𝐼 + 2) ∈ (0...(♯‘𝑊)))
372a1i 11 . . . . . . . 8 (𝜑 → ∅ ∈ Word 𝑇)
381, 18, 36, 37spllen 14649 . . . . . . 7 (𝜑 → (♯‘(𝑊 splice ⟨𝐼, (𝐼 + 2), ∅⟩)) = ((♯‘𝑊) + ((♯‘∅) − ((𝐼 + 2) − 𝐼))))
39 hash0 14274 . . . . . . . . . . 11 (♯‘∅) = 0
4039oveq1i 7372 . . . . . . . . . 10 ((♯‘∅) − ((𝐼 + 2) − 𝐼)) = (0 − ((𝐼 + 2) − 𝐼))
41 df-neg 11395 . . . . . . . . . 10 -((𝐼 + 2) − 𝐼) = (0 − ((𝐼 + 2) − 𝐼))
4240, 41eqtr4i 2768 . . . . . . . . 9 ((♯‘∅) − ((𝐼 + 2) − 𝐼)) = -((𝐼 + 2) − 𝐼)
43 2cn 12235 . . . . . . . . . . 11 2 ∈ ℂ
44 pncan2 11415 . . . . . . . . . . 11 ((𝐼 ∈ ℂ ∧ 2 ∈ ℂ) → ((𝐼 + 2) − 𝐼) = 2)
4531, 43, 44sylancl 587 . . . . . . . . . 10 (𝜑 → ((𝐼 + 2) − 𝐼) = 2)
4645negeqd 11402 . . . . . . . . 9 (𝜑 → -((𝐼 + 2) − 𝐼) = -2)
4742, 46eqtrid 2789 . . . . . . . 8 (𝜑 → ((♯‘∅) − ((𝐼 + 2) − 𝐼)) = -2)
4823, 47oveq12d 7380 . . . . . . 7 (𝜑 → ((♯‘𝑊) + ((♯‘∅) − ((𝐼 + 2) − 𝐼))) = (𝐿 + -2))
49 elfzel2 13446 . . . . . . . . . 10 (𝐼 ∈ (0...𝐿) → 𝐿 ∈ ℤ)
508, 49syl 17 . . . . . . . . 9 (𝜑𝐿 ∈ ℤ)
5150zcnd 12615 . . . . . . . 8 (𝜑𝐿 ∈ ℂ)
52 negsub 11456 . . . . . . . 8 ((𝐿 ∈ ℂ ∧ 2 ∈ ℂ) → (𝐿 + -2) = (𝐿 − 2))
5351, 43, 52sylancl 587 . . . . . . 7 (𝜑 → (𝐿 + -2) = (𝐿 − 2))
5438, 48, 533eqtrd 2781 . . . . . 6 (𝜑 → (♯‘(𝑊 splice ⟨𝐼, (𝐼 + 2), ∅⟩)) = (𝐿 − 2))
5554adantr 482 . . . . 5 ((𝜑 ∧ ((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → (♯‘(𝑊 splice ⟨𝐼, (𝐼 + 2), ∅⟩)) = (𝐿 − 2))
56 splid 14648 . . . . . . . . 9 ((𝑊 ∈ Word 𝑇 ∧ (𝐼 ∈ (0...(𝐼 + 2)) ∧ (𝐼 + 2) ∈ (0...(♯‘𝑊)))) → (𝑊 splice ⟨𝐼, (𝐼 + 2), (𝑊 substr ⟨𝐼, (𝐼 + 2)⟩)⟩) = 𝑊)
571, 18, 36, 56syl12anc 836 . . . . . . . 8 (𝜑 → (𝑊 splice ⟨𝐼, (𝐼 + 2), (𝑊 substr ⟨𝐼, (𝐼 + 2)⟩)⟩) = 𝑊)
5857oveq2d 7378 . . . . . . 7 (𝜑 → (𝐺 Σg (𝑊 splice ⟨𝐼, (𝐼 + 2), (𝑊 substr ⟨𝐼, (𝐼 + 2)⟩)⟩)) = (𝐺 Σg 𝑊))
5958adantr 482 . . . . . 6 ((𝜑 ∧ ((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → (𝐺 Σg (𝑊 splice ⟨𝐼, (𝐼 + 2), (𝑊 substr ⟨𝐼, (𝐼 + 2)⟩)⟩)) = (𝐺 Σg 𝑊))
60 eqid 2737 . . . . . . 7 (Base‘𝐺) = (Base‘𝐺)
6119symggrp 19189 . . . . . . . . . 10 (𝐷𝑉𝐺 ∈ Grp)
6221, 61syl 17 . . . . . . . . 9 (𝜑𝐺 ∈ Grp)
6362grpmndd 18767 . . . . . . . 8 (𝜑𝐺 ∈ Mnd)
6463adantr 482 . . . . . . 7 ((𝜑 ∧ ((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → 𝐺 ∈ Mnd)
6520, 19, 60symgtrf 19258 . . . . . . . . . 10 𝑇 ⊆ (Base‘𝐺)
66 sswrd 14417 . . . . . . . . . 10 (𝑇 ⊆ (Base‘𝐺) → Word 𝑇 ⊆ Word (Base‘𝐺))
6765, 66ax-mp 5 . . . . . . . . 9 Word 𝑇 ⊆ Word (Base‘𝐺)
6867, 1sselid 3947 . . . . . . . 8 (𝜑𝑊 ∈ Word (Base‘𝐺))
6968adantr 482 . . . . . . 7 ((𝜑 ∧ ((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → 𝑊 ∈ Word (Base‘𝐺))
7018adantr 482 . . . . . . 7 ((𝜑 ∧ ((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → 𝐼 ∈ (0...(𝐼 + 2)))
7136adantr 482 . . . . . . 7 ((𝜑 ∧ ((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → (𝐼 + 2) ∈ (0...(♯‘𝑊)))
72 swrdcl 14540 . . . . . . . . 9 (𝑊 ∈ Word (Base‘𝐺) → (𝑊 substr ⟨𝐼, (𝐼 + 2)⟩) ∈ Word (Base‘𝐺))
7368, 72syl 17 . . . . . . . 8 (𝜑 → (𝑊 substr ⟨𝐼, (𝐼 + 2)⟩) ∈ Word (Base‘𝐺))
7473adantr 482 . . . . . . 7 ((𝜑 ∧ ((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → (𝑊 substr ⟨𝐼, (𝐼 + 2)⟩) ∈ Word (Base‘𝐺))
75 wrd0 14434 . . . . . . . 8 ∅ ∈ Word (Base‘𝐺)
7675a1i 11 . . . . . . 7 ((𝜑 ∧ ((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → ∅ ∈ Word (Base‘𝐺))
7723oveq2d 7378 . . . . . . . . . . . . 13 (𝜑 → (0..^(♯‘𝑊)) = (0..^𝐿))
7826, 77eleqtrrd 2841 . . . . . . . . . . . 12 (𝜑 → (𝐼 + 1) ∈ (0..^(♯‘𝑊)))
79 swrds2 14836 . . . . . . . . . . . 12 ((𝑊 ∈ Word 𝑇𝐼 ∈ ℕ0 ∧ (𝐼 + 1) ∈ (0..^(♯‘𝑊))) → (𝑊 substr ⟨𝐼, (𝐼 + 2)⟩) = ⟨“(𝑊𝐼)(𝑊‘(𝐼 + 1))”⟩)
801, 10, 78, 79syl3anc 1372 . . . . . . . . . . 11 (𝜑 → (𝑊 substr ⟨𝐼, (𝐼 + 2)⟩) = ⟨“(𝑊𝐼)(𝑊‘(𝐼 + 1))”⟩)
8180oveq2d 7378 . . . . . . . . . 10 (𝜑 → (𝐺 Σg (𝑊 substr ⟨𝐼, (𝐼 + 2)⟩)) = (𝐺 Σg ⟨“(𝑊𝐼)(𝑊‘(𝐼 + 1))”⟩))
82 wrdf 14414 . . . . . . . . . . . . . . 15 (𝑊 ∈ Word 𝑇𝑊:(0..^(♯‘𝑊))⟶𝑇)
831, 82syl 17 . . . . . . . . . . . . . 14 (𝜑𝑊:(0..^(♯‘𝑊))⟶𝑇)
8477feq2d 6659 . . . . . . . . . . . . . 14 (𝜑 → (𝑊:(0..^(♯‘𝑊))⟶𝑇𝑊:(0..^𝐿)⟶𝑇))
8583, 84mpbid 231 . . . . . . . . . . . . 13 (𝜑𝑊:(0..^𝐿)⟶𝑇)
8685, 7ffvelcdmd 7041 . . . . . . . . . . . 12 (𝜑 → (𝑊𝐼) ∈ 𝑇)
8765, 86sselid 3947 . . . . . . . . . . 11 (𝜑 → (𝑊𝐼) ∈ (Base‘𝐺))
8885, 26ffvelcdmd 7041 . . . . . . . . . . . 12 (𝜑 → (𝑊‘(𝐼 + 1)) ∈ 𝑇)
8965, 88sselid 3947 . . . . . . . . . . 11 (𝜑 → (𝑊‘(𝐼 + 1)) ∈ (Base‘𝐺))
90 eqid 2737 . . . . . . . . . . . 12 (+g𝐺) = (+g𝐺)
9160, 90gsumws2 18659 . . . . . . . . . . 11 ((𝐺 ∈ Mnd ∧ (𝑊𝐼) ∈ (Base‘𝐺) ∧ (𝑊‘(𝐼 + 1)) ∈ (Base‘𝐺)) → (𝐺 Σg ⟨“(𝑊𝐼)(𝑊‘(𝐼 + 1))”⟩) = ((𝑊𝐼)(+g𝐺)(𝑊‘(𝐼 + 1))))
9263, 87, 89, 91syl3anc 1372 . . . . . . . . . 10 (𝜑 → (𝐺 Σg ⟨“(𝑊𝐼)(𝑊‘(𝐼 + 1))”⟩) = ((𝑊𝐼)(+g𝐺)(𝑊‘(𝐼 + 1))))
9319, 60, 90symgov 19172 . . . . . . . . . . 11 (((𝑊𝐼) ∈ (Base‘𝐺) ∧ (𝑊‘(𝐼 + 1)) ∈ (Base‘𝐺)) → ((𝑊𝐼)(+g𝐺)(𝑊‘(𝐼 + 1))) = ((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))))
9487, 89, 93syl2anc 585 . . . . . . . . . 10 (𝜑 → ((𝑊𝐼)(+g𝐺)(𝑊‘(𝐼 + 1))) = ((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))))
9581, 92, 943eqtrd 2781 . . . . . . . . 9 (𝜑 → (𝐺 Σg (𝑊 substr ⟨𝐼, (𝐼 + 2)⟩)) = ((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))))
9695adantr 482 . . . . . . . 8 ((𝜑 ∧ ((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → (𝐺 Σg (𝑊 substr ⟨𝐼, (𝐼 + 2)⟩)) = ((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))))
97 simpr 486 . . . . . . . 8 ((𝜑 ∧ ((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → ((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷))
9819symgid 19190 . . . . . . . . . . 11 (𝐷𝑉 → ( I ↾ 𝐷) = (0g𝐺))
9921, 98syl 17 . . . . . . . . . 10 (𝜑 → ( I ↾ 𝐷) = (0g𝐺))
100 eqid 2737 . . . . . . . . . . 11 (0g𝐺) = (0g𝐺)
101100gsum0 18546 . . . . . . . . . 10 (𝐺 Σg ∅) = (0g𝐺)
10299, 101eqtr4di 2795 . . . . . . . . 9 (𝜑 → ( I ↾ 𝐷) = (𝐺 Σg ∅))
103102adantr 482 . . . . . . . 8 ((𝜑 ∧ ((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → ( I ↾ 𝐷) = (𝐺 Σg ∅))
10496, 97, 1033eqtrd 2781 . . . . . . 7 ((𝜑 ∧ ((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → (𝐺 Σg (𝑊 substr ⟨𝐼, (𝐼 + 2)⟩)) = (𝐺 Σg ∅))
10560, 64, 69, 70, 71, 74, 76, 104gsumspl 18661 . . . . . 6 ((𝜑 ∧ ((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → (𝐺 Σg (𝑊 splice ⟨𝐼, (𝐼 + 2), (𝑊 substr ⟨𝐼, (𝐼 + 2)⟩)⟩)) = (𝐺 Σg (𝑊 splice ⟨𝐼, (𝐼 + 2), ∅⟩)))
10622adantr 482 . . . . . 6 ((𝜑 ∧ ((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → (𝐺 Σg 𝑊) = ( I ↾ 𝐷))
10759, 105, 1063eqtr3d 2785 . . . . 5 ((𝜑 ∧ ((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → (𝐺 Σg (𝑊 splice ⟨𝐼, (𝐼 + 2), ∅⟩)) = ( I ↾ 𝐷))
108 fveqeq2 6856 . . . . . . 7 (𝑥 = (𝑊 splice ⟨𝐼, (𝐼 + 2), ∅⟩) → ((♯‘𝑥) = (𝐿 − 2) ↔ (♯‘(𝑊 splice ⟨𝐼, (𝐼 + 2), ∅⟩)) = (𝐿 − 2)))
109 oveq2 7370 . . . . . . . 8 (𝑥 = (𝑊 splice ⟨𝐼, (𝐼 + 2), ∅⟩) → (𝐺 Σg 𝑥) = (𝐺 Σg (𝑊 splice ⟨𝐼, (𝐼 + 2), ∅⟩)))
110109eqeq1d 2739 . . . . . . 7 (𝑥 = (𝑊 splice ⟨𝐼, (𝐼 + 2), ∅⟩) → ((𝐺 Σg 𝑥) = ( I ↾ 𝐷) ↔ (𝐺 Σg (𝑊 splice ⟨𝐼, (𝐼 + 2), ∅⟩)) = ( I ↾ 𝐷)))
111108, 110anbi12d 632 . . . . . 6 (𝑥 = (𝑊 splice ⟨𝐼, (𝐼 + 2), ∅⟩) → (((♯‘𝑥) = (𝐿 − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)) ↔ ((♯‘(𝑊 splice ⟨𝐼, (𝐼 + 2), ∅⟩)) = (𝐿 − 2) ∧ (𝐺 Σg (𝑊 splice ⟨𝐼, (𝐼 + 2), ∅⟩)) = ( I ↾ 𝐷))))
112111rspcev 3584 . . . . 5 (((𝑊 splice ⟨𝐼, (𝐼 + 2), ∅⟩) ∈ Word 𝑇 ∧ ((♯‘(𝑊 splice ⟨𝐼, (𝐼 + 2), ∅⟩)) = (𝐿 − 2) ∧ (𝐺 Σg (𝑊 splice ⟨𝐼, (𝐼 + 2), ∅⟩)) = ( I ↾ 𝐷))) → ∃𝑥 ∈ Word 𝑇((♯‘𝑥) = (𝐿 − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))
1135, 55, 107, 112syl12anc 836 . . . 4 ((𝜑 ∧ ((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → ∃𝑥 ∈ Word 𝑇((♯‘𝑥) = (𝐿 − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))
114 psgnunilem2.in . . . . 5 (𝜑 → ¬ ∃𝑥 ∈ Word 𝑇((♯‘𝑥) = (𝐿 − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))
115114adantr 482 . . . 4 ((𝜑 ∧ ((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → ¬ ∃𝑥 ∈ Word 𝑇((♯‘𝑥) = (𝐿 − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))
116113, 115pm2.21dd 194 . . 3 ((𝜑 ∧ ((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → ∃𝑤 ∈ Word 𝑇(((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ∧ ((𝐼 + 1) ∈ (0..^𝐿) ∧ 𝐴 ∈ dom ((𝑤‘(𝐼 + 1)) ∖ I ) ∧ ∀𝑗 ∈ (0..^(𝐼 + 1)) ¬ 𝐴 ∈ dom ((𝑤𝑗) ∖ I ))))
117116ex 414 . 2 (𝜑 → (((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷) → ∃𝑤 ∈ Word 𝑇(((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ∧ ((𝐼 + 1) ∈ (0..^𝐿) ∧ 𝐴 ∈ dom ((𝑤‘(𝐼 + 1)) ∖ I ) ∧ ∀𝑗 ∈ (0..^(𝐼 + 1)) ¬ 𝐴 ∈ dom ((𝑤𝑗) ∖ I )))))
1181adantr 482 . . . . . . 7 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → 𝑊 ∈ Word 𝑇)
119 simprl 770 . . . . . . . 8 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → 𝑟𝑇)
120 simprr 772 . . . . . . . 8 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → 𝑠𝑇)
121119, 120s2cld 14767 . . . . . . 7 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → ⟨“𝑟𝑠”⟩ ∈ Word 𝑇)
122 splcl 14647 . . . . . . 7 ((𝑊 ∈ Word 𝑇 ∧ ⟨“𝑟𝑠”⟩ ∈ Word 𝑇) → (𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩) ∈ Word 𝑇)
123118, 121, 122syl2anc 585 . . . . . 6 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → (𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩) ∈ Word 𝑇)
124123adantrr 716 . . . . 5 ((𝜑 ∧ ((𝑟𝑇𝑠𝑇) ∧ (((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) → (𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩) ∈ Word 𝑇)
12563adantr 482 . . . . . . . 8 ((𝜑 ∧ ((𝑟𝑇𝑠𝑇) ∧ (((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) → 𝐺 ∈ Mnd)
12668adantr 482 . . . . . . . 8 ((𝜑 ∧ ((𝑟𝑇𝑠𝑇) ∧ (((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) → 𝑊 ∈ Word (Base‘𝐺))
12718adantr 482 . . . . . . . 8 ((𝜑 ∧ ((𝑟𝑇𝑠𝑇) ∧ (((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) → 𝐼 ∈ (0...(𝐼 + 2)))
12836adantr 482 . . . . . . . 8 ((𝜑 ∧ ((𝑟𝑇𝑠𝑇) ∧ (((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) → (𝐼 + 2) ∈ (0...(♯‘𝑊)))
12967, 121sselid 3947 . . . . . . . . 9 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → ⟨“𝑟𝑠”⟩ ∈ Word (Base‘𝐺))
130129adantrr 716 . . . . . . . 8 ((𝜑 ∧ ((𝑟𝑇𝑠𝑇) ∧ (((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) → ⟨“𝑟𝑠”⟩ ∈ Word (Base‘𝐺))
13173adantr 482 . . . . . . . 8 ((𝜑 ∧ ((𝑟𝑇𝑠𝑇) ∧ (((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) → (𝑊 substr ⟨𝐼, (𝐼 + 2)⟩) ∈ Word (Base‘𝐺))
132 simprr1 1222 . . . . . . . . 9 ((𝜑 ∧ ((𝑟𝑇𝑠𝑇) ∧ (((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) → ((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟𝑠))
13395adantr 482 . . . . . . . . 9 ((𝜑 ∧ ((𝑟𝑇𝑠𝑇) ∧ (((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) → (𝐺 Σg (𝑊 substr ⟨𝐼, (𝐼 + 2)⟩)) = ((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))))
13463adantr 482 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → 𝐺 ∈ Mnd)
13565a1i 11 . . . . . . . . . . . . . 14 (𝜑𝑇 ⊆ (Base‘𝐺))
136135sselda 3949 . . . . . . . . . . . . 13 ((𝜑𝑟𝑇) → 𝑟 ∈ (Base‘𝐺))
137136adantrr 716 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → 𝑟 ∈ (Base‘𝐺))
138135sselda 3949 . . . . . . . . . . . . 13 ((𝜑𝑠𝑇) → 𝑠 ∈ (Base‘𝐺))
139138adantrl 715 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → 𝑠 ∈ (Base‘𝐺))
14060, 90gsumws2 18659 . . . . . . . . . . . 12 ((𝐺 ∈ Mnd ∧ 𝑟 ∈ (Base‘𝐺) ∧ 𝑠 ∈ (Base‘𝐺)) → (𝐺 Σg ⟨“𝑟𝑠”⟩) = (𝑟(+g𝐺)𝑠))
141134, 137, 139, 140syl3anc 1372 . . . . . . . . . . 11 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → (𝐺 Σg ⟨“𝑟𝑠”⟩) = (𝑟(+g𝐺)𝑠))
14219, 60, 90symgov 19172 . . . . . . . . . . . 12 ((𝑟 ∈ (Base‘𝐺) ∧ 𝑠 ∈ (Base‘𝐺)) → (𝑟(+g𝐺)𝑠) = (𝑟𝑠))
143137, 139, 142syl2anc 585 . . . . . . . . . . 11 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → (𝑟(+g𝐺)𝑠) = (𝑟𝑠))
144141, 143eqtrd 2777 . . . . . . . . . 10 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → (𝐺 Σg ⟨“𝑟𝑠”⟩) = (𝑟𝑠))
145144adantrr 716 . . . . . . . . 9 ((𝜑 ∧ ((𝑟𝑇𝑠𝑇) ∧ (((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) → (𝐺 Σg ⟨“𝑟𝑠”⟩) = (𝑟𝑠))
146132, 133, 1453eqtr4rd 2788 . . . . . . . 8 ((𝜑 ∧ ((𝑟𝑇𝑠𝑇) ∧ (((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) → (𝐺 Σg ⟨“𝑟𝑠”⟩) = (𝐺 Σg (𝑊 substr ⟨𝐼, (𝐼 + 2)⟩)))
14760, 125, 126, 127, 128, 130, 131, 146gsumspl 18661 . . . . . . 7 ((𝜑 ∧ ((𝑟𝑇𝑠𝑇) ∧ (((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) → (𝐺 Σg (𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)) = (𝐺 Σg (𝑊 splice ⟨𝐼, (𝐼 + 2), (𝑊 substr ⟨𝐼, (𝐼 + 2)⟩)⟩)))
14858adantr 482 . . . . . . 7 ((𝜑 ∧ ((𝑟𝑇𝑠𝑇) ∧ (((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) → (𝐺 Σg (𝑊 splice ⟨𝐼, (𝐼 + 2), (𝑊 substr ⟨𝐼, (𝐼 + 2)⟩)⟩)) = (𝐺 Σg 𝑊))
14922adantr 482 . . . . . . 7 ((𝜑 ∧ ((𝑟𝑇𝑠𝑇) ∧ (((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) → (𝐺 Σg 𝑊) = ( I ↾ 𝐷))
150147, 148, 1493eqtrd 2781 . . . . . 6 ((𝜑 ∧ ((𝑟𝑇𝑠𝑇) ∧ (((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) → (𝐺 Σg (𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)) = ( I ↾ 𝐷))
15118adantr 482 . . . . . . . . 9 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → 𝐼 ∈ (0...(𝐼 + 2)))
15236adantr 482 . . . . . . . . 9 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → (𝐼 + 2) ∈ (0...(♯‘𝑊)))
153118, 151, 152, 121spllen 14649 . . . . . . . 8 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → (♯‘(𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)) = ((♯‘𝑊) + ((♯‘⟨“𝑟𝑠”⟩) − ((𝐼 + 2) − 𝐼))))
154 s2len 14785 . . . . . . . . . . . . 13 (♯‘⟨“𝑟𝑠”⟩) = 2
155154oveq1i 7372 . . . . . . . . . . . 12 ((♯‘⟨“𝑟𝑠”⟩) − ((𝐼 + 2) − 𝐼)) = (2 − ((𝐼 + 2) − 𝐼))
15645oveq2d 7378 . . . . . . . . . . . . 13 (𝜑 → (2 − ((𝐼 + 2) − 𝐼)) = (2 − 2))
15743subidi 11479 . . . . . . . . . . . . 13 (2 − 2) = 0
158156, 157eqtrdi 2793 . . . . . . . . . . . 12 (𝜑 → (2 − ((𝐼 + 2) − 𝐼)) = 0)
159155, 158eqtrid 2789 . . . . . . . . . . 11 (𝜑 → ((♯‘⟨“𝑟𝑠”⟩) − ((𝐼 + 2) − 𝐼)) = 0)
160159oveq2d 7378 . . . . . . . . . 10 (𝜑 → ((♯‘𝑊) + ((♯‘⟨“𝑟𝑠”⟩) − ((𝐼 + 2) − 𝐼))) = ((♯‘𝑊) + 0))
16123, 51eqeltrd 2838 . . . . . . . . . . 11 (𝜑 → (♯‘𝑊) ∈ ℂ)
162161addid1d 11362 . . . . . . . . . 10 (𝜑 → ((♯‘𝑊) + 0) = (♯‘𝑊))
163160, 162, 233eqtrd 2781 . . . . . . . . 9 (𝜑 → ((♯‘𝑊) + ((♯‘⟨“𝑟𝑠”⟩) − ((𝐼 + 2) − 𝐼))) = 𝐿)
164163adantr 482 . . . . . . . 8 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → ((♯‘𝑊) + ((♯‘⟨“𝑟𝑠”⟩) − ((𝐼 + 2) − 𝐼))) = 𝐿)
165153, 164eqtrd 2777 . . . . . . 7 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → (♯‘(𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)) = 𝐿)
166165adantrr 716 . . . . . 6 ((𝜑 ∧ ((𝑟𝑇𝑠𝑇) ∧ (((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) → (♯‘(𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)) = 𝐿)
167150, 166jca 513 . . . . 5 ((𝜑 ∧ ((𝑟𝑇𝑠𝑇) ∧ (((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) → ((𝐺 Σg (𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)) = ( I ↾ 𝐷) ∧ (♯‘(𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)) = 𝐿))
16826adantr 482 . . . . . 6 ((𝜑 ∧ ((𝑟𝑇𝑠𝑇) ∧ (((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) → (𝐼 + 1) ∈ (0..^𝐿))
169 simprr2 1223 . . . . . . 7 ((𝜑 ∧ ((𝑟𝑇𝑠𝑇) ∧ (((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) → 𝐴 ∈ dom (𝑠 ∖ I ))
170 1nn0 12436 . . . . . . . . . . . . . . 15 1 ∈ ℕ0
171 2nn 12233 . . . . . . . . . . . . . . 15 2 ∈ ℕ
172 1lt2 12331 . . . . . . . . . . . . . . 15 1 < 2
173 elfzo0 13620 . . . . . . . . . . . . . . 15 (1 ∈ (0..^2) ↔ (1 ∈ ℕ0 ∧ 2 ∈ ℕ ∧ 1 < 2))
174170, 171, 172, 173mpbir3an 1342 . . . . . . . . . . . . . 14 1 ∈ (0..^2)
175154oveq2i 7373 . . . . . . . . . . . . . 14 (0..^(♯‘⟨“𝑟𝑠”⟩)) = (0..^2)
176174, 175eleqtrri 2837 . . . . . . . . . . . . 13 1 ∈ (0..^(♯‘⟨“𝑟𝑠”⟩))
177176a1i 11 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → 1 ∈ (0..^(♯‘⟨“𝑟𝑠”⟩)))
178118, 151, 152, 121, 177splfv2a 14651 . . . . . . . . . . 11 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → ((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘(𝐼 + 1)) = (⟨“𝑟𝑠”⟩‘1))
179 s2fv1 14784 . . . . . . . . . . . 12 (𝑠𝑇 → (⟨“𝑟𝑠”⟩‘1) = 𝑠)
180179ad2antll 728 . . . . . . . . . . 11 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → (⟨“𝑟𝑠”⟩‘1) = 𝑠)
181178, 180eqtrd 2777 . . . . . . . . . 10 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → ((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘(𝐼 + 1)) = 𝑠)
182181adantrr 716 . . . . . . . . 9 ((𝜑 ∧ ((𝑟𝑇𝑠𝑇) ∧ (((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) → ((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘(𝐼 + 1)) = 𝑠)
183182difeq1d 4086 . . . . . . . 8 ((𝜑 ∧ ((𝑟𝑇𝑠𝑇) ∧ (((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) → (((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘(𝐼 + 1)) ∖ I ) = (𝑠 ∖ I ))
184183dmeqd 5866 . . . . . . 7 ((𝜑 ∧ ((𝑟𝑇𝑠𝑇) ∧ (((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) → dom (((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘(𝐼 + 1)) ∖ I ) = dom (𝑠 ∖ I ))
185169, 184eleqtrrd 2841 . . . . . 6 ((𝜑 ∧ ((𝑟𝑇𝑠𝑇) ∧ (((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) → 𝐴 ∈ dom (((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘(𝐼 + 1)) ∖ I ))
186 fzosplitsni 13690 . . . . . . . . . . 11 (𝐼 ∈ (ℤ‘0) → (𝑗 ∈ (0..^(𝐼 + 1)) ↔ (𝑗 ∈ (0..^𝐼) ∨ 𝑗 = 𝐼)))
187 nn0uz 12812 . . . . . . . . . . 11 0 = (ℤ‘0)
188186, 187eleq2s 2856 . . . . . . . . . 10 (𝐼 ∈ ℕ0 → (𝑗 ∈ (0..^(𝐼 + 1)) ↔ (𝑗 ∈ (0..^𝐼) ∨ 𝑗 = 𝐼)))
18910, 188syl 17 . . . . . . . . 9 (𝜑 → (𝑗 ∈ (0..^(𝐼 + 1)) ↔ (𝑗 ∈ (0..^𝐼) ∨ 𝑗 = 𝐼)))
190189adantr 482 . . . . . . . 8 ((𝜑 ∧ ((𝑟𝑇𝑠𝑇) ∧ (((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) → (𝑗 ∈ (0..^(𝐼 + 1)) ↔ (𝑗 ∈ (0..^𝐼) ∨ 𝑗 = 𝐼)))
191 fveq2 6847 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 𝑗 → (𝑊𝑘) = (𝑊𝑗))
192191difeq1d 4086 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑗 → ((𝑊𝑘) ∖ I ) = ((𝑊𝑗) ∖ I ))
193192dmeqd 5866 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑗 → dom ((𝑊𝑘) ∖ I ) = dom ((𝑊𝑗) ∖ I ))
194193eleq2d 2824 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑗 → (𝐴 ∈ dom ((𝑊𝑘) ∖ I ) ↔ 𝐴 ∈ dom ((𝑊𝑗) ∖ I )))
195194notbid 318 . . . . . . . . . . . . . . 15 (𝑘 = 𝑗 → (¬ 𝐴 ∈ dom ((𝑊𝑘) ∖ I ) ↔ ¬ 𝐴 ∈ dom ((𝑊𝑗) ∖ I )))
196195rspccva 3583 . . . . . . . . . . . . . 14 ((∀𝑘 ∈ (0..^𝐼) ¬ 𝐴 ∈ dom ((𝑊𝑘) ∖ I ) ∧ 𝑗 ∈ (0..^𝐼)) → ¬ 𝐴 ∈ dom ((𝑊𝑗) ∖ I ))
19725, 196sylan 581 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (0..^𝐼)) → ¬ 𝐴 ∈ dom ((𝑊𝑗) ∖ I ))
198197adantlr 714 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑟𝑇𝑠𝑇)) ∧ 𝑗 ∈ (0..^𝐼)) → ¬ 𝐴 ∈ dom ((𝑊𝑗) ∖ I ))
1991ad2antrr 725 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑟𝑇𝑠𝑇)) ∧ 𝑗 ∈ (0..^𝐼)) → 𝑊 ∈ Word 𝑇)
20018ad2antrr 725 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑟𝑇𝑠𝑇)) ∧ 𝑗 ∈ (0..^𝐼)) → 𝐼 ∈ (0...(𝐼 + 2)))
20136ad2antrr 725 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑟𝑇𝑠𝑇)) ∧ 𝑗 ∈ (0..^𝐼)) → (𝐼 + 2) ∈ (0...(♯‘𝑊)))
202121adantr 482 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑟𝑇𝑠𝑇)) ∧ 𝑗 ∈ (0..^𝐼)) → ⟨“𝑟𝑠”⟩ ∈ Word 𝑇)
203 simpr 486 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑟𝑇𝑠𝑇)) ∧ 𝑗 ∈ (0..^𝐼)) → 𝑗 ∈ (0..^𝐼))
204199, 200, 201, 202, 203splfv1 14650 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑟𝑇𝑠𝑇)) ∧ 𝑗 ∈ (0..^𝐼)) → ((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘𝑗) = (𝑊𝑗))
205204difeq1d 4086 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑟𝑇𝑠𝑇)) ∧ 𝑗 ∈ (0..^𝐼)) → (((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘𝑗) ∖ I ) = ((𝑊𝑗) ∖ I ))
206205dmeqd 5866 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑟𝑇𝑠𝑇)) ∧ 𝑗 ∈ (0..^𝐼)) → dom (((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘𝑗) ∖ I ) = dom ((𝑊𝑗) ∖ I ))
207198, 206neleqtrrd 2861 . . . . . . . . . . 11 (((𝜑 ∧ (𝑟𝑇𝑠𝑇)) ∧ 𝑗 ∈ (0..^𝐼)) → ¬ 𝐴 ∈ dom (((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘𝑗) ∖ I ))
208207ex 414 . . . . . . . . . 10 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → (𝑗 ∈ (0..^𝐼) → ¬ 𝐴 ∈ dom (((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘𝑗) ∖ I )))
209208adantrr 716 . . . . . . . . 9 ((𝜑 ∧ ((𝑟𝑇𝑠𝑇) ∧ (((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) → (𝑗 ∈ (0..^𝐼) → ¬ 𝐴 ∈ dom (((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘𝑗) ∖ I )))
210 simprr3 1224 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑟𝑇𝑠𝑇) ∧ (((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) → ¬ 𝐴 ∈ dom (𝑟 ∖ I ))
211 0nn0 12435 . . . . . . . . . . . . . . . . . . . 20 0 ∈ ℕ0
212 2pos 12263 . . . . . . . . . . . . . . . . . . . 20 0 < 2
213 elfzo0 13620 . . . . . . . . . . . . . . . . . . . 20 (0 ∈ (0..^2) ↔ (0 ∈ ℕ0 ∧ 2 ∈ ℕ ∧ 0 < 2))
214211, 171, 212, 213mpbir3an 1342 . . . . . . . . . . . . . . . . . . 19 0 ∈ (0..^2)
215214, 175eleqtrri 2837 . . . . . . . . . . . . . . . . . 18 0 ∈ (0..^(♯‘⟨“𝑟𝑠”⟩))
216215a1i 11 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → 0 ∈ (0..^(♯‘⟨“𝑟𝑠”⟩)))
217118, 151, 152, 121, 216splfv2a 14651 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → ((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘(𝐼 + 0)) = (⟨“𝑟𝑠”⟩‘0))
21831addid1d 11362 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐼 + 0) = 𝐼)
219218adantr 482 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → (𝐼 + 0) = 𝐼)
220219fveq2d 6851 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → ((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘(𝐼 + 0)) = ((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘𝐼))
221 s2fv0 14783 . . . . . . . . . . . . . . . . 17 (𝑟𝑇 → (⟨“𝑟𝑠”⟩‘0) = 𝑟)
222221ad2antrl 727 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → (⟨“𝑟𝑠”⟩‘0) = 𝑟)
223217, 220, 2223eqtr3d 2785 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → ((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘𝐼) = 𝑟)
224223difeq1d 4086 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → (((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘𝐼) ∖ I ) = (𝑟 ∖ I ))
225224dmeqd 5866 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → dom (((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘𝐼) ∖ I ) = dom (𝑟 ∖ I ))
226225eleq2d 2824 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → (𝐴 ∈ dom (((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘𝐼) ∖ I ) ↔ 𝐴 ∈ dom (𝑟 ∖ I )))
227226adantrr 716 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑟𝑇𝑠𝑇) ∧ (((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) → (𝐴 ∈ dom (((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘𝐼) ∖ I ) ↔ 𝐴 ∈ dom (𝑟 ∖ I )))
228210, 227mtbird 325 . . . . . . . . . 10 ((𝜑 ∧ ((𝑟𝑇𝑠𝑇) ∧ (((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) → ¬ 𝐴 ∈ dom (((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘𝐼) ∖ I ))
229 fveq2 6847 . . . . . . . . . . . . . 14 (𝑗 = 𝐼 → ((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘𝑗) = ((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘𝐼))
230229difeq1d 4086 . . . . . . . . . . . . 13 (𝑗 = 𝐼 → (((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘𝑗) ∖ I ) = (((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘𝐼) ∖ I ))
231230dmeqd 5866 . . . . . . . . . . . 12 (𝑗 = 𝐼 → dom (((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘𝑗) ∖ I ) = dom (((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘𝐼) ∖ I ))
232231eleq2d 2824 . . . . . . . . . . 11 (𝑗 = 𝐼 → (𝐴 ∈ dom (((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘𝑗) ∖ I ) ↔ 𝐴 ∈ dom (((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘𝐼) ∖ I )))
233232notbid 318 . . . . . . . . . 10 (𝑗 = 𝐼 → (¬ 𝐴 ∈ dom (((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘𝑗) ∖ I ) ↔ ¬ 𝐴 ∈ dom (((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘𝐼) ∖ I )))
234228, 233syl5ibrcom 247 . . . . . . . . 9 ((𝜑 ∧ ((𝑟𝑇𝑠𝑇) ∧ (((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) → (𝑗 = 𝐼 → ¬ 𝐴 ∈ dom (((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘𝑗) ∖ I )))
235209, 234jaod 858 . . . . . . . 8 ((𝜑 ∧ ((𝑟𝑇𝑠𝑇) ∧ (((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) → ((𝑗 ∈ (0..^𝐼) ∨ 𝑗 = 𝐼) → ¬ 𝐴 ∈ dom (((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘𝑗) ∖ I )))
236190, 235sylbid 239 . . . . . . 7 ((𝜑 ∧ ((𝑟𝑇𝑠𝑇) ∧ (((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) → (𝑗 ∈ (0..^(𝐼 + 1)) → ¬ 𝐴 ∈ dom (((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘𝑗) ∖ I )))
237236ralrimiv 3143 . . . . . 6 ((𝜑 ∧ ((𝑟𝑇𝑠𝑇) ∧ (((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) → ∀𝑗 ∈ (0..^(𝐼 + 1)) ¬ 𝐴 ∈ dom (((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘𝑗) ∖ I ))
238168, 185, 2373jca 1129 . . . . 5 ((𝜑 ∧ ((𝑟𝑇𝑠𝑇) ∧ (((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) → ((𝐼 + 1) ∈ (0..^𝐿) ∧ 𝐴 ∈ dom (((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘(𝐼 + 1)) ∖ I ) ∧ ∀𝑗 ∈ (0..^(𝐼 + 1)) ¬ 𝐴 ∈ dom (((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘𝑗) ∖ I )))
239 oveq2 7370 . . . . . . . . 9 (𝑤 = (𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩) → (𝐺 Σg 𝑤) = (𝐺 Σg (𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)))
240239eqeq1d 2739 . . . . . . . 8 (𝑤 = (𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩) → ((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ↔ (𝐺 Σg (𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)) = ( I ↾ 𝐷)))
241 fveqeq2 6856 . . . . . . . 8 (𝑤 = (𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩) → ((♯‘𝑤) = 𝐿 ↔ (♯‘(𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)) = 𝐿))
242240, 241anbi12d 632 . . . . . . 7 (𝑤 = (𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩) → (((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ↔ ((𝐺 Σg (𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)) = ( I ↾ 𝐷) ∧ (♯‘(𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)) = 𝐿)))
243 fveq1 6846 . . . . . . . . . . 11 (𝑤 = (𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩) → (𝑤‘(𝐼 + 1)) = ((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘(𝐼 + 1)))
244243difeq1d 4086 . . . . . . . . . 10 (𝑤 = (𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩) → ((𝑤‘(𝐼 + 1)) ∖ I ) = (((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘(𝐼 + 1)) ∖ I ))
245244dmeqd 5866 . . . . . . . . 9 (𝑤 = (𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩) → dom ((𝑤‘(𝐼 + 1)) ∖ I ) = dom (((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘(𝐼 + 1)) ∖ I ))
246245eleq2d 2824 . . . . . . . 8 (𝑤 = (𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩) → (𝐴 ∈ dom ((𝑤‘(𝐼 + 1)) ∖ I ) ↔ 𝐴 ∈ dom (((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘(𝐼 + 1)) ∖ I )))
247 fveq1 6846 . . . . . . . . . . . . 13 (𝑤 = (𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩) → (𝑤𝑗) = ((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘𝑗))
248247difeq1d 4086 . . . . . . . . . . . 12 (𝑤 = (𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩) → ((𝑤𝑗) ∖ I ) = (((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘𝑗) ∖ I ))
249248dmeqd 5866 . . . . . . . . . . 11 (𝑤 = (𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩) → dom ((𝑤𝑗) ∖ I ) = dom (((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘𝑗) ∖ I ))
250249eleq2d 2824 . . . . . . . . . 10 (𝑤 = (𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩) → (𝐴 ∈ dom ((𝑤𝑗) ∖ I ) ↔ 𝐴 ∈ dom (((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘𝑗) ∖ I )))
251250notbid 318 . . . . . . . . 9 (𝑤 = (𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩) → (¬ 𝐴 ∈ dom ((𝑤𝑗) ∖ I ) ↔ ¬ 𝐴 ∈ dom (((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘𝑗) ∖ I )))
252251ralbidv 3175 . . . . . . . 8 (𝑤 = (𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩) → (∀𝑗 ∈ (0..^(𝐼 + 1)) ¬ 𝐴 ∈ dom ((𝑤𝑗) ∖ I ) ↔ ∀𝑗 ∈ (0..^(𝐼 + 1)) ¬ 𝐴 ∈ dom (((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘𝑗) ∖ I )))
253246, 2523anbi23d 1440 . . . . . . 7 (𝑤 = (𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩) → (((𝐼 + 1) ∈ (0..^𝐿) ∧ 𝐴 ∈ dom ((𝑤‘(𝐼 + 1)) ∖ I ) ∧ ∀𝑗 ∈ (0..^(𝐼 + 1)) ¬ 𝐴 ∈ dom ((𝑤𝑗) ∖ I )) ↔ ((𝐼 + 1) ∈ (0..^𝐿) ∧ 𝐴 ∈ dom (((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘(𝐼 + 1)) ∖ I ) ∧ ∀𝑗 ∈ (0..^(𝐼 + 1)) ¬ 𝐴 ∈ dom (((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘𝑗) ∖ I ))))
254242, 253anbi12d 632 . . . . . 6 (𝑤 = (𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩) → ((((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ∧ ((𝐼 + 1) ∈ (0..^𝐿) ∧ 𝐴 ∈ dom ((𝑤‘(𝐼 + 1)) ∖ I ) ∧ ∀𝑗 ∈ (0..^(𝐼 + 1)) ¬ 𝐴 ∈ dom ((𝑤𝑗) ∖ I ))) ↔ (((𝐺 Σg (𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)) = ( I ↾ 𝐷) ∧ (♯‘(𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)) = 𝐿) ∧ ((𝐼 + 1) ∈ (0..^𝐿) ∧ 𝐴 ∈ dom (((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘(𝐼 + 1)) ∖ I ) ∧ ∀𝑗 ∈ (0..^(𝐼 + 1)) ¬ 𝐴 ∈ dom (((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘𝑗) ∖ I )))))
255254rspcev 3584 . . . . 5 (((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩) ∈ Word 𝑇 ∧ (((𝐺 Σg (𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)) = ( I ↾ 𝐷) ∧ (♯‘(𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)) = 𝐿) ∧ ((𝐼 + 1) ∈ (0..^𝐿) ∧ 𝐴 ∈ dom (((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘(𝐼 + 1)) ∖ I ) ∧ ∀𝑗 ∈ (0..^(𝐼 + 1)) ¬ 𝐴 ∈ dom (((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘𝑗) ∖ I )))) → ∃𝑤 ∈ Word 𝑇(((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ∧ ((𝐼 + 1) ∈ (0..^𝐿) ∧ 𝐴 ∈ dom ((𝑤‘(𝐼 + 1)) ∖ I ) ∧ ∀𝑗 ∈ (0..^(𝐼 + 1)) ¬ 𝐴 ∈ dom ((𝑤𝑗) ∖ I ))))
256124, 167, 238, 255syl12anc 836 . . . 4 ((𝜑 ∧ ((𝑟𝑇𝑠𝑇) ∧ (((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) → ∃𝑤 ∈ Word 𝑇(((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ∧ ((𝐼 + 1) ∈ (0..^𝐿) ∧ 𝐴 ∈ dom ((𝑤‘(𝐼 + 1)) ∖ I ) ∧ ∀𝑗 ∈ (0..^(𝐼 + 1)) ¬ 𝐴 ∈ dom ((𝑤𝑗) ∖ I ))))
257256expr 458 . . 3 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → ((((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )) → ∃𝑤 ∈ Word 𝑇(((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ∧ ((𝐼 + 1) ∈ (0..^𝐿) ∧ 𝐴 ∈ dom ((𝑤‘(𝐼 + 1)) ∖ I ) ∧ ∀𝑗 ∈ (0..^(𝐼 + 1)) ¬ 𝐴 ∈ dom ((𝑤𝑗) ∖ I )))))
258257rexlimdvva 3206 . 2 (𝜑 → (∃𝑟𝑇𝑠𝑇 (((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )) → ∃𝑤 ∈ Word 𝑇(((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ∧ ((𝐼 + 1) ∈ (0..^𝐿) ∧ 𝐴 ∈ dom ((𝑤‘(𝐼 + 1)) ∖ I ) ∧ ∀𝑗 ∈ (0..^(𝐼 + 1)) ¬ 𝐴 ∈ dom ((𝑤𝑗) ∖ I )))))
25920, 21, 86, 88, 24psgnunilem1 19282 . 2 (𝜑 → (((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷) ∨ ∃𝑟𝑇𝑠𝑇 (((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I ))))
260117, 258, 259mpjaod 859 1 (𝜑 → ∃𝑤 ∈ Word 𝑇(((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ∧ ((𝐼 + 1) ∈ (0..^𝐿) ∧ 𝐴 ∈ dom ((𝑤‘(𝐼 + 1)) ∖ I ) ∧ ∀𝑗 ∈ (0..^(𝐼 + 1)) ¬ 𝐴 ∈ dom ((𝑤𝑗) ∖ I ))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397  wo 846  w3a 1088   = wceq 1542  wcel 2107  wral 3065  wrex 3074  cdif 3912  wss 3915  c0 4287  cop 4597  cotp 4599   class class class wbr 5110   I cid 5535  dom cdm 5638  ran crn 5639  cres 5640  ccom 5642  wf 6497  cfv 6501  (class class class)co 7362  cc 11056  cr 11057  0cc0 11058  1c1 11059   + caddc 11061   < clt 11196  cle 11197  cmin 11392  -cneg 11393  cn 12160  2c2 12215  0cn0 12420  cz 12506  cuz 12770  ...cfz 13431  ..^cfzo 13574  chash 14237  Word cword 14409   substr csubstr 14535   splice csplice 14644  ⟨“cs2 14737  Basecbs 17090  +gcplusg 17140  0gc0g 17328   Σg cgsu 17329  Mndcmnd 18563  Grpcgrp 18755  SymGrpcsymg 19155  pmTrspcpmtr 19230
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-cnex 11114  ax-resscn 11115  ax-1cn 11116  ax-icn 11117  ax-addcl 11118  ax-addrcl 11119  ax-mulcl 11120  ax-mulrcl 11121  ax-mulcom 11122  ax-addass 11123  ax-mulass 11124  ax-distr 11125  ax-i2m1 11126  ax-1ne0 11127  ax-1rid 11128  ax-rnegex 11129  ax-rrecex 11130  ax-cnre 11131  ax-pre-lttri 11132  ax-pre-lttrn 11133  ax-pre-ltadd 11134  ax-pre-mulgt0 11135
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-xor 1511  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-ot 4600  df-uni 4871  df-int 4913  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-2o 8418  df-er 8655  df-map 8774  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-card 9882  df-pnf 11198  df-mnf 11199  df-xr 11200  df-ltxr 11201  df-le 11202  df-sub 11394  df-neg 11395  df-nn 12161  df-2 12223  df-3 12224  df-4 12225  df-5 12226  df-6 12227  df-7 12228  df-8 12229  df-9 12230  df-n0 12421  df-xnn0 12493  df-z 12507  df-uz 12771  df-fz 13432  df-fzo 13575  df-seq 13914  df-hash 14238  df-word 14410  df-lsw 14458  df-concat 14466  df-s1 14491  df-substr 14536  df-pfx 14566  df-splice 14645  df-s2 14744  df-struct 17026  df-sets 17043  df-slot 17061  df-ndx 17073  df-base 17091  df-ress 17120  df-plusg 17153  df-tset 17159  df-0g 17330  df-gsum 17331  df-mgm 18504  df-sgrp 18553  df-mnd 18564  df-submnd 18609  df-efmnd 18686  df-grp 18758  df-minusg 18759  df-subg 18932  df-symg 19156  df-pmtr 19231
This theorem is referenced by:  psgnunilem3  19285
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