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Theorem psgnunilem2 19513
Description: Lemma for psgnuni 19517. 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 14577 . . . . . . 7 ∅ ∈ Word 𝑇
3 splcl 14790 . . . . . . 7 ((𝑊 ∈ Word 𝑇 ∧ ∅ ∈ Word 𝑇) → (𝑊 splice ⟨𝐼, (𝐼 + 2), ∅⟩) ∈ Word 𝑇)
41, 2, 3sylancl 586 . . . . . 6 (𝜑 → (𝑊 splice ⟨𝐼, (𝐼 + 2), ∅⟩) ∈ Word 𝑇)
54adantr 480 . . . . 5 ((𝜑 ∧ ((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → (𝑊 splice ⟨𝐼, (𝐼 + 2), ∅⟩) ∈ Word 𝑇)
6 fzossfz 13718 . . . . . . . . . . 11 (0..^𝐿) ⊆ (0...𝐿)
7 psgnunilem2.ix . . . . . . . . . . 11 (𝜑𝐼 ∈ (0..^𝐿))
86, 7sselid 3981 . . . . . . . . . 10 (𝜑𝐼 ∈ (0...𝐿))
9 elfznn0 13660 . . . . . . . . . 10 (𝐼 ∈ (0...𝐿) → 𝐼 ∈ ℕ0)
108, 9syl 17 . . . . . . . . 9 (𝜑𝐼 ∈ ℕ0)
11 2nn0 12543 . . . . . . . . . 10 2 ∈ ℕ0
12 nn0addcl 12561 . . . . . . . . . 10 ((𝐼 ∈ ℕ0 ∧ 2 ∈ ℕ0) → (𝐼 + 2) ∈ ℕ0)
1310, 11, 12sylancl 586 . . . . . . . . 9 (𝜑 → (𝐼 + 2) ∈ ℕ0)
1410nn0red 12588 . . . . . . . . . 10 (𝜑𝐼 ∈ ℝ)
15 nn0addge1 12572 . . . . . . . . . 10 ((𝐼 ∈ ℝ ∧ 2 ∈ ℕ0) → 𝐼 ≤ (𝐼 + 2))
1614, 11, 15sylancl 586 . . . . . . . . 9 (𝜑𝐼 ≤ (𝐼 + 2))
17 elfz2nn0 13658 . . . . . . . . 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 19512 . . . . . . . . . 10 (𝜑 → (𝐼 + 1) ∈ (0..^𝐿))
27 fzofzp1 13803 . . . . . . . . . 10 ((𝐼 + 1) ∈ (0..^𝐿) → ((𝐼 + 1) + 1) ∈ (0...𝐿))
2826, 27syl 17 . . . . . . . . 9 (𝜑 → ((𝐼 + 1) + 1) ∈ (0...𝐿))
29 df-2 12329 . . . . . . . . . . 11 2 = (1 + 1)
3029oveq2i 7442 . . . . . . . . . 10 (𝐼 + 2) = (𝐼 + (1 + 1))
3110nn0cnd 12589 . . . . . . . . . . 11 (𝜑𝐼 ∈ ℂ)
32 1cnd 11256 . . . . . . . . . . 11 (𝜑 → 1 ∈ ℂ)
3331, 32, 32addassd 11283 . . . . . . . . . 10 (𝜑 → ((𝐼 + 1) + 1) = (𝐼 + (1 + 1)))
3430, 33eqtr4id 2796 . . . . . . . . 9 (𝜑 → (𝐼 + 2) = ((𝐼 + 1) + 1))
3523oveq2d 7447 . . . . . . . . 9 (𝜑 → (0...(♯‘𝑊)) = (0...𝐿))
3628, 34, 353eltr4d 2856 . . . . . . . 8 (𝜑 → (𝐼 + 2) ∈ (0...(♯‘𝑊)))
372a1i 11 . . . . . . . 8 (𝜑 → ∅ ∈ Word 𝑇)
381, 18, 36, 37spllen 14792 . . . . . . 7 (𝜑 → (♯‘(𝑊 splice ⟨𝐼, (𝐼 + 2), ∅⟩)) = ((♯‘𝑊) + ((♯‘∅) − ((𝐼 + 2) − 𝐼))))
39 hash0 14406 . . . . . . . . . . 11 (♯‘∅) = 0
4039oveq1i 7441 . . . . . . . . . 10 ((♯‘∅) − ((𝐼 + 2) − 𝐼)) = (0 − ((𝐼 + 2) − 𝐼))
41 df-neg 11495 . . . . . . . . . 10 -((𝐼 + 2) − 𝐼) = (0 − ((𝐼 + 2) − 𝐼))
4240, 41eqtr4i 2768 . . . . . . . . 9 ((♯‘∅) − ((𝐼 + 2) − 𝐼)) = -((𝐼 + 2) − 𝐼)
43 2cn 12341 . . . . . . . . . . 11 2 ∈ ℂ
44 pncan2 11515 . . . . . . . . . . 11 ((𝐼 ∈ ℂ ∧ 2 ∈ ℂ) → ((𝐼 + 2) − 𝐼) = 2)
4531, 43, 44sylancl 586 . . . . . . . . . 10 (𝜑 → ((𝐼 + 2) − 𝐼) = 2)
4645negeqd 11502 . . . . . . . . 9 (𝜑 → -((𝐼 + 2) − 𝐼) = -2)
4742, 46eqtrid 2789 . . . . . . . 8 (𝜑 → ((♯‘∅) − ((𝐼 + 2) − 𝐼)) = -2)
4823, 47oveq12d 7449 . . . . . . 7 (𝜑 → ((♯‘𝑊) + ((♯‘∅) − ((𝐼 + 2) − 𝐼))) = (𝐿 + -2))
49 elfzel2 13562 . . . . . . . . . 10 (𝐼 ∈ (0...𝐿) → 𝐿 ∈ ℤ)
508, 49syl 17 . . . . . . . . 9 (𝜑𝐿 ∈ ℤ)
5150zcnd 12723 . . . . . . . 8 (𝜑𝐿 ∈ ℂ)
52 negsub 11557 . . . . . . . 8 ((𝐿 ∈ ℂ ∧ 2 ∈ ℂ) → (𝐿 + -2) = (𝐿 − 2))
5351, 43, 52sylancl 586 . . . . . . 7 (𝜑 → (𝐿 + -2) = (𝐿 − 2))
5438, 48, 533eqtrd 2781 . . . . . 6 (𝜑 → (♯‘(𝑊 splice ⟨𝐼, (𝐼 + 2), ∅⟩)) = (𝐿 − 2))
5554adantr 480 . . . . 5 ((𝜑 ∧ ((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → (♯‘(𝑊 splice ⟨𝐼, (𝐼 + 2), ∅⟩)) = (𝐿 − 2))
56 splid 14791 . . . . . . . . 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 837 . . . . . . . 8 (𝜑 → (𝑊 splice ⟨𝐼, (𝐼 + 2), (𝑊 substr ⟨𝐼, (𝐼 + 2)⟩)⟩) = 𝑊)
5857oveq2d 7447 . . . . . . 7 (𝜑 → (𝐺 Σg (𝑊 splice ⟨𝐼, (𝐼 + 2), (𝑊 substr ⟨𝐼, (𝐼 + 2)⟩)⟩)) = (𝐺 Σg 𝑊))
5958adantr 480 . . . . . 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 19418 . . . . . . . . . 10 (𝐷𝑉𝐺 ∈ Grp)
6221, 61syl 17 . . . . . . . . 9 (𝜑𝐺 ∈ Grp)
6362grpmndd 18964 . . . . . . . 8 (𝜑𝐺 ∈ Mnd)
6463adantr 480 . . . . . . 7 ((𝜑 ∧ ((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → 𝐺 ∈ Mnd)
6520, 19, 60symgtrf 19487 . . . . . . . . . 10 𝑇 ⊆ (Base‘𝐺)
66 sswrd 14560 . . . . . . . . . 10 (𝑇 ⊆ (Base‘𝐺) → Word 𝑇 ⊆ Word (Base‘𝐺))
6765, 66ax-mp 5 . . . . . . . . 9 Word 𝑇 ⊆ Word (Base‘𝐺)
6867, 1sselid 3981 . . . . . . . 8 (𝜑𝑊 ∈ Word (Base‘𝐺))
6968adantr 480 . . . . . . 7 ((𝜑 ∧ ((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → 𝑊 ∈ Word (Base‘𝐺))
7018adantr 480 . . . . . . 7 ((𝜑 ∧ ((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → 𝐼 ∈ (0...(𝐼 + 2)))
7136adantr 480 . . . . . . 7 ((𝜑 ∧ ((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → (𝐼 + 2) ∈ (0...(♯‘𝑊)))
72 swrdcl 14683 . . . . . . . . 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 480 . . . . . . 7 ((𝜑 ∧ ((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → (𝑊 substr ⟨𝐼, (𝐼 + 2)⟩) ∈ Word (Base‘𝐺))
75 wrd0 14577 . . . . . . . 8 ∅ ∈ Word (Base‘𝐺)
7675a1i 11 . . . . . . 7 ((𝜑 ∧ ((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → ∅ ∈ Word (Base‘𝐺))
7723oveq2d 7447 . . . . . . . . . . . . 13 (𝜑 → (0..^(♯‘𝑊)) = (0..^𝐿))
7826, 77eleqtrrd 2844 . . . . . . . . . . . 12 (𝜑 → (𝐼 + 1) ∈ (0..^(♯‘𝑊)))
79 swrds2 14979 . . . . . . . . . . . 12 ((𝑊 ∈ Word 𝑇𝐼 ∈ ℕ0 ∧ (𝐼 + 1) ∈ (0..^(♯‘𝑊))) → (𝑊 substr ⟨𝐼, (𝐼 + 2)⟩) = ⟨“(𝑊𝐼)(𝑊‘(𝐼 + 1))”⟩)
801, 10, 78, 79syl3anc 1373 . . . . . . . . . . 11 (𝜑 → (𝑊 substr ⟨𝐼, (𝐼 + 2)⟩) = ⟨“(𝑊𝐼)(𝑊‘(𝐼 + 1))”⟩)
8180oveq2d 7447 . . . . . . . . . 10 (𝜑 → (𝐺 Σg (𝑊 substr ⟨𝐼, (𝐼 + 2)⟩)) = (𝐺 Σg ⟨“(𝑊𝐼)(𝑊‘(𝐼 + 1))”⟩))
82 wrdf 14557 . . . . . . . . . . . . . . 15 (𝑊 ∈ Word 𝑇𝑊:(0..^(♯‘𝑊))⟶𝑇)
831, 82syl 17 . . . . . . . . . . . . . 14 (𝜑𝑊:(0..^(♯‘𝑊))⟶𝑇)
8477feq2d 6722 . . . . . . . . . . . . . 14 (𝜑 → (𝑊:(0..^(♯‘𝑊))⟶𝑇𝑊:(0..^𝐿)⟶𝑇))
8583, 84mpbid 232 . . . . . . . . . . . . 13 (𝜑𝑊:(0..^𝐿)⟶𝑇)
8685, 7ffvelcdmd 7105 . . . . . . . . . . . 12 (𝜑 → (𝑊𝐼) ∈ 𝑇)
8765, 86sselid 3981 . . . . . . . . . . 11 (𝜑 → (𝑊𝐼) ∈ (Base‘𝐺))
8885, 26ffvelcdmd 7105 . . . . . . . . . . . 12 (𝜑 → (𝑊‘(𝐼 + 1)) ∈ 𝑇)
8965, 88sselid 3981 . . . . . . . . . . 11 (𝜑 → (𝑊‘(𝐼 + 1)) ∈ (Base‘𝐺))
90 eqid 2737 . . . . . . . . . . . 12 (+g𝐺) = (+g𝐺)
9160, 90gsumws2 18855 . . . . . . . . . . 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 1373 . . . . . . . . . 10 (𝜑 → (𝐺 Σg ⟨“(𝑊𝐼)(𝑊‘(𝐼 + 1))”⟩) = ((𝑊𝐼)(+g𝐺)(𝑊‘(𝐼 + 1))))
9319, 60, 90symgov 19401 . . . . . . . . . . 11 (((𝑊𝐼) ∈ (Base‘𝐺) ∧ (𝑊‘(𝐼 + 1)) ∈ (Base‘𝐺)) → ((𝑊𝐼)(+g𝐺)(𝑊‘(𝐼 + 1))) = ((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))))
9487, 89, 93syl2anc 584 . . . . . . . . . 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 480 . . . . . . . 8 ((𝜑 ∧ ((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → (𝐺 Σg (𝑊 substr ⟨𝐼, (𝐼 + 2)⟩)) = ((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))))
97 simpr 484 . . . . . . . 8 ((𝜑 ∧ ((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → ((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷))
9819symgid 19419 . . . . . . . . . . 11 (𝐷𝑉 → ( I ↾ 𝐷) = (0g𝐺))
9921, 98syl 17 . . . . . . . . . 10 (𝜑 → ( I ↾ 𝐷) = (0g𝐺))
100 eqid 2737 . . . . . . . . . . 11 (0g𝐺) = (0g𝐺)
101100gsum0 18697 . . . . . . . . . 10 (𝐺 Σg ∅) = (0g𝐺)
10299, 101eqtr4di 2795 . . . . . . . . 9 (𝜑 → ( I ↾ 𝐷) = (𝐺 Σg ∅))
103102adantr 480 . . . . . . . 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 18857 . . . . . 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 480 . . . . . 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 6915 . . . . . . 7 (𝑥 = (𝑊 splice ⟨𝐼, (𝐼 + 2), ∅⟩) → ((♯‘𝑥) = (𝐿 − 2) ↔ (♯‘(𝑊 splice ⟨𝐼, (𝐼 + 2), ∅⟩)) = (𝐿 − 2)))
109 oveq2 7439 . . . . . . . 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 3622 . . . . 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 837 . . . 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 480 . . . 4 ((𝜑 ∧ ((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → ¬ ∃𝑥 ∈ Word 𝑇((♯‘𝑥) = (𝐿 − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))
116113, 115pm2.21dd 195 . . 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 412 . 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 480 . . . . . . 7 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → 𝑊 ∈ Word 𝑇)
119 simprl 771 . . . . . . . 8 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → 𝑟𝑇)
120 simprr 773 . . . . . . . 8 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → 𝑠𝑇)
121119, 120s2cld 14910 . . . . . . 7 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → ⟨“𝑟𝑠”⟩ ∈ Word 𝑇)
122 splcl 14790 . . . . . . 7 ((𝑊 ∈ Word 𝑇 ∧ ⟨“𝑟𝑠”⟩ ∈ Word 𝑇) → (𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩) ∈ Word 𝑇)
123118, 121, 122syl2anc 584 . . . . . 6 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → (𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩) ∈ Word 𝑇)
124123adantrr 717 . . . . 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 480 . . . . . . . 8 ((𝜑 ∧ ((𝑟𝑇𝑠𝑇) ∧ (((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) → 𝐺 ∈ Mnd)
12668adantr 480 . . . . . . . 8 ((𝜑 ∧ ((𝑟𝑇𝑠𝑇) ∧ (((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) → 𝑊 ∈ Word (Base‘𝐺))
12718adantr 480 . . . . . . . 8 ((𝜑 ∧ ((𝑟𝑇𝑠𝑇) ∧ (((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) → 𝐼 ∈ (0...(𝐼 + 2)))
12836adantr 480 . . . . . . . 8 ((𝜑 ∧ ((𝑟𝑇𝑠𝑇) ∧ (((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) → (𝐼 + 2) ∈ (0...(♯‘𝑊)))
12967, 121sselid 3981 . . . . . . . . 9 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → ⟨“𝑟𝑠”⟩ ∈ Word (Base‘𝐺))
130129adantrr 717 . . . . . . . 8 ((𝜑 ∧ ((𝑟𝑇𝑠𝑇) ∧ (((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) → ⟨“𝑟𝑠”⟩ ∈ Word (Base‘𝐺))
13173adantr 480 . . . . . . . 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 480 . . . . . . . . 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 480 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → 𝐺 ∈ Mnd)
13565a1i 11 . . . . . . . . . . . . . 14 (𝜑𝑇 ⊆ (Base‘𝐺))
136135sselda 3983 . . . . . . . . . . . . 13 ((𝜑𝑟𝑇) → 𝑟 ∈ (Base‘𝐺))
137136adantrr 717 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → 𝑟 ∈ (Base‘𝐺))
138135sselda 3983 . . . . . . . . . . . . 13 ((𝜑𝑠𝑇) → 𝑠 ∈ (Base‘𝐺))
139138adantrl 716 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → 𝑠 ∈ (Base‘𝐺))
14060, 90gsumws2 18855 . . . . . . . . . . . 12 ((𝐺 ∈ Mnd ∧ 𝑟 ∈ (Base‘𝐺) ∧ 𝑠 ∈ (Base‘𝐺)) → (𝐺 Σg ⟨“𝑟𝑠”⟩) = (𝑟(+g𝐺)𝑠))
141134, 137, 139, 140syl3anc 1373 . . . . . . . . . . 11 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → (𝐺 Σg ⟨“𝑟𝑠”⟩) = (𝑟(+g𝐺)𝑠))
14219, 60, 90symgov 19401 . . . . . . . . . . . 12 ((𝑟 ∈ (Base‘𝐺) ∧ 𝑠 ∈ (Base‘𝐺)) → (𝑟(+g𝐺)𝑠) = (𝑟𝑠))
143137, 139, 142syl2anc 584 . . . . . . . . . . 11 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → (𝑟(+g𝐺)𝑠) = (𝑟𝑠))
144141, 143eqtrd 2777 . . . . . . . . . 10 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → (𝐺 Σg ⟨“𝑟𝑠”⟩) = (𝑟𝑠))
145144adantrr 717 . . . . . . . . 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 18857 . . . . . . 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 480 . . . . . . 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 480 . . . . . . 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 480 . . . . . . . . 9 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → 𝐼 ∈ (0...(𝐼 + 2)))
15236adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → (𝐼 + 2) ∈ (0...(♯‘𝑊)))
153118, 151, 152, 121spllen 14792 . . . . . . . 8 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → (♯‘(𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)) = ((♯‘𝑊) + ((♯‘⟨“𝑟𝑠”⟩) − ((𝐼 + 2) − 𝐼))))
154 s2len 14928 . . . . . . . . . . . . 13 (♯‘⟨“𝑟𝑠”⟩) = 2
155154oveq1i 7441 . . . . . . . . . . . 12 ((♯‘⟨“𝑟𝑠”⟩) − ((𝐼 + 2) − 𝐼)) = (2 − ((𝐼 + 2) − 𝐼))
15645oveq2d 7447 . . . . . . . . . . . . 13 (𝜑 → (2 − ((𝐼 + 2) − 𝐼)) = (2 − 2))
15743subidi 11580 . . . . . . . . . . . . 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 7447 . . . . . . . . . 10 (𝜑 → ((♯‘𝑊) + ((♯‘⟨“𝑟𝑠”⟩) − ((𝐼 + 2) − 𝐼))) = ((♯‘𝑊) + 0))
16123, 51eqeltrd 2841 . . . . . . . . . . 11 (𝜑 → (♯‘𝑊) ∈ ℂ)
162161addridd 11461 . . . . . . . . . 10 (𝜑 → ((♯‘𝑊) + 0) = (♯‘𝑊))
163160, 162, 233eqtrd 2781 . . . . . . . . 9 (𝜑 → ((♯‘𝑊) + ((♯‘⟨“𝑟𝑠”⟩) − ((𝐼 + 2) − 𝐼))) = 𝐿)
164163adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → ((♯‘𝑊) + ((♯‘⟨“𝑟𝑠”⟩) − ((𝐼 + 2) − 𝐼))) = 𝐿)
165153, 164eqtrd 2777 . . . . . . 7 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → (♯‘(𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)) = 𝐿)
166165adantrr 717 . . . . . 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 511 . . . . 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 480 . . . . . 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 12542 . . . . . . . . . . . . . . 15 1 ∈ ℕ0
171 2nn 12339 . . . . . . . . . . . . . . 15 2 ∈ ℕ
172 1lt2 12437 . . . . . . . . . . . . . . 15 1 < 2
173 elfzo0 13740 . . . . . . . . . . . . . . 15 (1 ∈ (0..^2) ↔ (1 ∈ ℕ0 ∧ 2 ∈ ℕ ∧ 1 < 2))
174170, 171, 172, 173mpbir3an 1342 . . . . . . . . . . . . . 14 1 ∈ (0..^2)
175154oveq2i 7442 . . . . . . . . . . . . . 14 (0..^(♯‘⟨“𝑟𝑠”⟩)) = (0..^2)
176174, 175eleqtrri 2840 . . . . . . . . . . . . 13 1 ∈ (0..^(♯‘⟨“𝑟𝑠”⟩))
177176a1i 11 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → 1 ∈ (0..^(♯‘⟨“𝑟𝑠”⟩)))
178118, 151, 152, 121, 177splfv2a 14794 . . . . . . . . . . 11 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → ((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘(𝐼 + 1)) = (⟨“𝑟𝑠”⟩‘1))
179 s2fv1 14927 . . . . . . . . . . . 12 (𝑠𝑇 → (⟨“𝑟𝑠”⟩‘1) = 𝑠)
180179ad2antll 729 . . . . . . . . . . 11 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → (⟨“𝑟𝑠”⟩‘1) = 𝑠)
181178, 180eqtrd 2777 . . . . . . . . . 10 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → ((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘(𝐼 + 1)) = 𝑠)
182181adantrr 717 . . . . . . . . 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 4125 . . . . . . . 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 5916 . . . . . . 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 2844 . . . . . 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 13817 . . . . . . . . . . 11 (𝐼 ∈ (ℤ‘0) → (𝑗 ∈ (0..^(𝐼 + 1)) ↔ (𝑗 ∈ (0..^𝐼) ∨ 𝑗 = 𝐼)))
187 nn0uz 12920 . . . . . . . . . . 11 0 = (ℤ‘0)
188186, 187eleq2s 2859 . . . . . . . . . 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 480 . . . . . . . 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 6906 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 𝑗 → (𝑊𝑘) = (𝑊𝑗))
192191difeq1d 4125 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑗 → ((𝑊𝑘) ∖ I ) = ((𝑊𝑗) ∖ I ))
193192dmeqd 5916 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑗 → dom ((𝑊𝑘) ∖ I ) = dom ((𝑊𝑗) ∖ I ))
194193eleq2d 2827 . . . . . . . . . . . . . . . 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 3621 . . . . . . . . . . . . . 14 ((∀𝑘 ∈ (0..^𝐼) ¬ 𝐴 ∈ dom ((𝑊𝑘) ∖ I ) ∧ 𝑗 ∈ (0..^𝐼)) → ¬ 𝐴 ∈ dom ((𝑊𝑗) ∖ I ))
19725, 196sylan 580 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (0..^𝐼)) → ¬ 𝐴 ∈ dom ((𝑊𝑗) ∖ I ))
198197adantlr 715 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑟𝑇𝑠𝑇)) ∧ 𝑗 ∈ (0..^𝐼)) → ¬ 𝐴 ∈ dom ((𝑊𝑗) ∖ I ))
1991ad2antrr 726 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑟𝑇𝑠𝑇)) ∧ 𝑗 ∈ (0..^𝐼)) → 𝑊 ∈ Word 𝑇)
20018ad2antrr 726 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑟𝑇𝑠𝑇)) ∧ 𝑗 ∈ (0..^𝐼)) → 𝐼 ∈ (0...(𝐼 + 2)))
20136ad2antrr 726 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑟𝑇𝑠𝑇)) ∧ 𝑗 ∈ (0..^𝐼)) → (𝐼 + 2) ∈ (0...(♯‘𝑊)))
202121adantr 480 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑟𝑇𝑠𝑇)) ∧ 𝑗 ∈ (0..^𝐼)) → ⟨“𝑟𝑠”⟩ ∈ Word 𝑇)
203 simpr 484 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑟𝑇𝑠𝑇)) ∧ 𝑗 ∈ (0..^𝐼)) → 𝑗 ∈ (0..^𝐼))
204199, 200, 201, 202, 203splfv1 14793 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑟𝑇𝑠𝑇)) ∧ 𝑗 ∈ (0..^𝐼)) → ((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘𝑗) = (𝑊𝑗))
205204difeq1d 4125 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑟𝑇𝑠𝑇)) ∧ 𝑗 ∈ (0..^𝐼)) → (((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘𝑗) ∖ I ) = ((𝑊𝑗) ∖ I ))
206205dmeqd 5916 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑟𝑇𝑠𝑇)) ∧ 𝑗 ∈ (0..^𝐼)) → dom (((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘𝑗) ∖ I ) = dom ((𝑊𝑗) ∖ I ))
207198, 206neleqtrrd 2864 . . . . . . . . . . 11 (((𝜑 ∧ (𝑟𝑇𝑠𝑇)) ∧ 𝑗 ∈ (0..^𝐼)) → ¬ 𝐴 ∈ dom (((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘𝑗) ∖ I ))
208207ex 412 . . . . . . . . . 10 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → (𝑗 ∈ (0..^𝐼) → ¬ 𝐴 ∈ dom (((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘𝑗) ∖ I )))
209208adantrr 717 . . . . . . . . 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 12541 . . . . . . . . . . . . . . . . . . . 20 0 ∈ ℕ0
212 2pos 12369 . . . . . . . . . . . . . . . . . . . 20 0 < 2
213 elfzo0 13740 . . . . . . . . . . . . . . . . . . . 20 (0 ∈ (0..^2) ↔ (0 ∈ ℕ0 ∧ 2 ∈ ℕ ∧ 0 < 2))
214211, 171, 212, 213mpbir3an 1342 . . . . . . . . . . . . . . . . . . 19 0 ∈ (0..^2)
215214, 175eleqtrri 2840 . . . . . . . . . . . . . . . . . 18 0 ∈ (0..^(♯‘⟨“𝑟𝑠”⟩))
216215a1i 11 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → 0 ∈ (0..^(♯‘⟨“𝑟𝑠”⟩)))
217118, 151, 152, 121, 216splfv2a 14794 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → ((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘(𝐼 + 0)) = (⟨“𝑟𝑠”⟩‘0))
21831addridd 11461 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐼 + 0) = 𝐼)
219218adantr 480 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → (𝐼 + 0) = 𝐼)
220219fveq2d 6910 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → ((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘(𝐼 + 0)) = ((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘𝐼))
221 s2fv0 14926 . . . . . . . . . . . . . . . . 17 (𝑟𝑇 → (⟨“𝑟𝑠”⟩‘0) = 𝑟)
222221ad2antrl 728 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → (⟨“𝑟𝑠”⟩‘0) = 𝑟)
223217, 220, 2223eqtr3d 2785 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → ((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘𝐼) = 𝑟)
224223difeq1d 4125 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → (((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘𝐼) ∖ I ) = (𝑟 ∖ I ))
225224dmeqd 5916 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → dom (((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘𝐼) ∖ I ) = dom (𝑟 ∖ I ))
226225eleq2d 2827 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → (𝐴 ∈ dom (((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘𝐼) ∖ I ) ↔ 𝐴 ∈ dom (𝑟 ∖ I )))
227226adantrr 717 . . . . . . . . . . 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 6906 . . . . . . . . . . . . . 14 (𝑗 = 𝐼 → ((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘𝑗) = ((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘𝐼))
230229difeq1d 4125 . . . . . . . . . . . . 13 (𝑗 = 𝐼 → (((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘𝑗) ∖ I ) = (((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘𝐼) ∖ I ))
231230dmeqd 5916 . . . . . . . . . . . 12 (𝑗 = 𝐼 → dom (((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘𝑗) ∖ I ) = dom (((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘𝐼) ∖ I ))
232231eleq2d 2827 . . . . . . . . . . 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 860 . . . . . . . 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 240 . . . . . . 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 3145 . . . . . 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 7439 . . . . . . . . 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 6915 . . . . . . . 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 6905 . . . . . . . . . . 11 (𝑤 = (𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩) → (𝑤‘(𝐼 + 1)) = ((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘(𝐼 + 1)))
244243difeq1d 4125 . . . . . . . . . 10 (𝑤 = (𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩) → ((𝑤‘(𝐼 + 1)) ∖ I ) = (((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘(𝐼 + 1)) ∖ I ))
245244dmeqd 5916 . . . . . . . . 9 (𝑤 = (𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩) → dom ((𝑤‘(𝐼 + 1)) ∖ I ) = dom (((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘(𝐼 + 1)) ∖ I ))
246245eleq2d 2827 . . . . . . . 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 6905 . . . . . . . . . . . . 13 (𝑤 = (𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩) → (𝑤𝑗) = ((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘𝑗))
248247difeq1d 4125 . . . . . . . . . . . 12 (𝑤 = (𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩) → ((𝑤𝑗) ∖ I ) = (((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘𝑗) ∖ I ))
249248dmeqd 5916 . . . . . . . . . . 11 (𝑤 = (𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩) → dom ((𝑤𝑗) ∖ I ) = dom (((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘𝑗) ∖ I ))
250249eleq2d 2827 . . . . . . . . . 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 3178 . . . . . . . 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 1441 . . . . . . 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 3622 . . . . 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 837 . . . 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 456 . . 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 3213 . 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 19511 . 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 861 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 206  wa 395  wo 848  w3a 1087   = wceq 1540  wcel 2108  wral 3061  wrex 3070  cdif 3948  wss 3951  c0 4333  cop 4632  cotp 4634   class class class wbr 5143   I cid 5577  dom cdm 5685  ran crn 5686  cres 5687  ccom 5689  wf 6557  cfv 6561  (class class class)co 7431  cc 11153  cr 11154  0cc0 11155  1c1 11156   + caddc 11158   < clt 11295  cle 11296  cmin 11492  -cneg 11493  cn 12266  2c2 12321  0cn0 12526  cz 12613  cuz 12878  ...cfz 13547  ..^cfzo 13694  chash 14369  Word cword 14552   substr csubstr 14678   splice csplice 14787  ⟨“cs2 14880  Basecbs 17247  +gcplusg 17297  0gc0g 17484   Σg cgsu 17485  Mndcmnd 18747  Grpcgrp 18951  SymGrpcsymg 19386  pmTrspcpmtr 19459
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-xor 1512  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-ot 4635  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-xnn0 12600  df-z 12614  df-uz 12879  df-fz 13548  df-fzo 13695  df-seq 14043  df-hash 14370  df-word 14553  df-lsw 14601  df-concat 14609  df-s1 14634  df-substr 14679  df-pfx 14709  df-splice 14788  df-s2 14887  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-tset 17316  df-0g 17486  df-gsum 17487  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-submnd 18797  df-efmnd 18882  df-grp 18954  df-minusg 18955  df-subg 19141  df-symg 19387  df-pmtr 19460
This theorem is referenced by:  psgnunilem3  19514
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