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Theorem psgnunilem2 18602
 Description: Lemma for psgnuni 18606. 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 13871 . . . . . . 7 ∅ ∈ Word 𝑇
3 splcl 14094 . . . . . . 7 ((𝑊 ∈ Word 𝑇 ∧ ∅ ∈ Word 𝑇) → (𝑊 splice ⟨𝐼, (𝐼 + 2), ∅⟩) ∈ Word 𝑇)
41, 2, 3sylancl 588 . . . . . 6 (𝜑 → (𝑊 splice ⟨𝐼, (𝐼 + 2), ∅⟩) ∈ Word 𝑇)
54adantr 483 . . . . 5 ((𝜑 ∧ ((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → (𝑊 splice ⟨𝐼, (𝐼 + 2), ∅⟩) ∈ Word 𝑇)
6 fzossfz 13040 . . . . . . . . . . 11 (0..^𝐿) ⊆ (0...𝐿)
7 psgnunilem2.ix . . . . . . . . . . 11 (𝜑𝐼 ∈ (0..^𝐿))
86, 7sseldi 3944 . . . . . . . . . 10 (𝜑𝐼 ∈ (0...𝐿))
9 elfznn0 12984 . . . . . . . . . 10 (𝐼 ∈ (0...𝐿) → 𝐼 ∈ ℕ0)
108, 9syl 17 . . . . . . . . 9 (𝜑𝐼 ∈ ℕ0)
11 2nn0 11893 . . . . . . . . . 10 2 ∈ ℕ0
12 nn0addcl 11911 . . . . . . . . . 10 ((𝐼 ∈ ℕ0 ∧ 2 ∈ ℕ0) → (𝐼 + 2) ∈ ℕ0)
1310, 11, 12sylancl 588 . . . . . . . . 9 (𝜑 → (𝐼 + 2) ∈ ℕ0)
1410nn0red 11935 . . . . . . . . . 10 (𝜑𝐼 ∈ ℝ)
15 nn0addge1 11922 . . . . . . . . . 10 ((𝐼 ∈ ℝ ∧ 2 ∈ ℕ0) → 𝐼 ≤ (𝐼 + 2))
1614, 11, 15sylancl 588 . . . . . . . . 9 (𝜑𝐼 ≤ (𝐼 + 2))
17 elfz2nn0 12982 . . . . . . . . 9 (𝐼 ∈ (0...(𝐼 + 2)) ↔ (𝐼 ∈ ℕ0 ∧ (𝐼 + 2) ∈ ℕ0𝐼 ≤ (𝐼 + 2)))
1810, 13, 16, 17syl3anbrc 1339 . . . . . . . 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 18601 . . . . . . . . . 10 (𝜑 → (𝐼 + 1) ∈ (0..^𝐿))
27 fzofzp1 13118 . . . . . . . . . 10 ((𝐼 + 1) ∈ (0..^𝐿) → ((𝐼 + 1) + 1) ∈ (0...𝐿))
2826, 27syl 17 . . . . . . . . 9 (𝜑 → ((𝐼 + 1) + 1) ∈ (0...𝐿))
2910nn0cnd 11936 . . . . . . . . . . 11 (𝜑𝐼 ∈ ℂ)
30 1cnd 10614 . . . . . . . . . . 11 (𝜑 → 1 ∈ ℂ)
3129, 30, 30addassd 10641 . . . . . . . . . 10 (𝜑 → ((𝐼 + 1) + 1) = (𝐼 + (1 + 1)))
32 df-2 11679 . . . . . . . . . . 11 2 = (1 + 1)
3332oveq2i 7144 . . . . . . . . . 10 (𝐼 + 2) = (𝐼 + (1 + 1))
3431, 33syl6reqr 2874 . . . . . . . . 9 (𝜑 → (𝐼 + 2) = ((𝐼 + 1) + 1))
3523oveq2d 7149 . . . . . . . . 9 (𝜑 → (0...(♯‘𝑊)) = (0...𝐿))
3628, 34, 353eltr4d 2926 . . . . . . . 8 (𝜑 → (𝐼 + 2) ∈ (0...(♯‘𝑊)))
372a1i 11 . . . . . . . 8 (𝜑 → ∅ ∈ Word 𝑇)
381, 18, 36, 37spllen 14096 . . . . . . 7 (𝜑 → (♯‘(𝑊 splice ⟨𝐼, (𝐼 + 2), ∅⟩)) = ((♯‘𝑊) + ((♯‘∅) − ((𝐼 + 2) − 𝐼))))
39 hash0 13713 . . . . . . . . . . 11 (♯‘∅) = 0
4039oveq1i 7143 . . . . . . . . . 10 ((♯‘∅) − ((𝐼 + 2) − 𝐼)) = (0 − ((𝐼 + 2) − 𝐼))
41 df-neg 10851 . . . . . . . . . 10 -((𝐼 + 2) − 𝐼) = (0 − ((𝐼 + 2) − 𝐼))
4240, 41eqtr4i 2846 . . . . . . . . 9 ((♯‘∅) − ((𝐼 + 2) − 𝐼)) = -((𝐼 + 2) − 𝐼)
43 2cn 11691 . . . . . . . . . . 11 2 ∈ ℂ
44 pncan2 10871 . . . . . . . . . . 11 ((𝐼 ∈ ℂ ∧ 2 ∈ ℂ) → ((𝐼 + 2) − 𝐼) = 2)
4529, 43, 44sylancl 588 . . . . . . . . . 10 (𝜑 → ((𝐼 + 2) − 𝐼) = 2)
4645negeqd 10858 . . . . . . . . 9 (𝜑 → -((𝐼 + 2) − 𝐼) = -2)
4742, 46syl5eq 2867 . . . . . . . 8 (𝜑 → ((♯‘∅) − ((𝐼 + 2) − 𝐼)) = -2)
4823, 47oveq12d 7151 . . . . . . 7 (𝜑 → ((♯‘𝑊) + ((♯‘∅) − ((𝐼 + 2) − 𝐼))) = (𝐿 + -2))
49 elfzel2 12890 . . . . . . . . . 10 (𝐼 ∈ (0...𝐿) → 𝐿 ∈ ℤ)
508, 49syl 17 . . . . . . . . 9 (𝜑𝐿 ∈ ℤ)
5150zcnd 12067 . . . . . . . 8 (𝜑𝐿 ∈ ℂ)
52 negsub 10912 . . . . . . . 8 ((𝐿 ∈ ℂ ∧ 2 ∈ ℂ) → (𝐿 + -2) = (𝐿 − 2))
5351, 43, 52sylancl 588 . . . . . . 7 (𝜑 → (𝐿 + -2) = (𝐿 − 2))
5438, 48, 533eqtrd 2859 . . . . . 6 (𝜑 → (♯‘(𝑊 splice ⟨𝐼, (𝐼 + 2), ∅⟩)) = (𝐿 − 2))
5554adantr 483 . . . . 5 ((𝜑 ∧ ((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → (♯‘(𝑊 splice ⟨𝐼, (𝐼 + 2), ∅⟩)) = (𝐿 − 2))
56 splid 14095 . . . . . . . . 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 834 . . . . . . . 8 (𝜑 → (𝑊 splice ⟨𝐼, (𝐼 + 2), (𝑊 substr ⟨𝐼, (𝐼 + 2)⟩)⟩) = 𝑊)
5857oveq2d 7149 . . . . . . 7 (𝜑 → (𝐺 Σg (𝑊 splice ⟨𝐼, (𝐼 + 2), (𝑊 substr ⟨𝐼, (𝐼 + 2)⟩)⟩)) = (𝐺 Σg 𝑊))
5958adantr 483 . . . . . 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 2820 . . . . . . 7 (Base‘𝐺) = (Base‘𝐺)
6119symggrp 18507 . . . . . . . . . 10 (𝐷𝑉𝐺 ∈ Grp)
6221, 61syl 17 . . . . . . . . 9 (𝜑𝐺 ∈ Grp)
63 grpmnd 18089 . . . . . . . . 9 (𝐺 ∈ Grp → 𝐺 ∈ Mnd)
6462, 63syl 17 . . . . . . . 8 (𝜑𝐺 ∈ Mnd)
6564adantr 483 . . . . . . 7 ((𝜑 ∧ ((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → 𝐺 ∈ Mnd)
6620, 19, 60symgtrf 18576 . . . . . . . . . 10 𝑇 ⊆ (Base‘𝐺)
67 sswrd 13854 . . . . . . . . . 10 (𝑇 ⊆ (Base‘𝐺) → Word 𝑇 ⊆ Word (Base‘𝐺))
6866, 67ax-mp 5 . . . . . . . . 9 Word 𝑇 ⊆ Word (Base‘𝐺)
6968, 1sseldi 3944 . . . . . . . 8 (𝜑𝑊 ∈ Word (Base‘𝐺))
7069adantr 483 . . . . . . 7 ((𝜑 ∧ ((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → 𝑊 ∈ Word (Base‘𝐺))
7118adantr 483 . . . . . . 7 ((𝜑 ∧ ((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → 𝐼 ∈ (0...(𝐼 + 2)))
7236adantr 483 . . . . . . 7 ((𝜑 ∧ ((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → (𝐼 + 2) ∈ (0...(♯‘𝑊)))
73 swrdcl 13987 . . . . . . . . 9 (𝑊 ∈ Word (Base‘𝐺) → (𝑊 substr ⟨𝐼, (𝐼 + 2)⟩) ∈ Word (Base‘𝐺))
7469, 73syl 17 . . . . . . . 8 (𝜑 → (𝑊 substr ⟨𝐼, (𝐼 + 2)⟩) ∈ Word (Base‘𝐺))
7574adantr 483 . . . . . . 7 ((𝜑 ∧ ((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → (𝑊 substr ⟨𝐼, (𝐼 + 2)⟩) ∈ Word (Base‘𝐺))
76 wrd0 13871 . . . . . . . 8 ∅ ∈ Word (Base‘𝐺)
7776a1i 11 . . . . . . 7 ((𝜑 ∧ ((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → ∅ ∈ Word (Base‘𝐺))
7823oveq2d 7149 . . . . . . . . . . . . 13 (𝜑 → (0..^(♯‘𝑊)) = (0..^𝐿))
7926, 78eleqtrrd 2914 . . . . . . . . . . . 12 (𝜑 → (𝐼 + 1) ∈ (0..^(♯‘𝑊)))
80 swrds2 14282 . . . . . . . . . . . 12 ((𝑊 ∈ Word 𝑇𝐼 ∈ ℕ0 ∧ (𝐼 + 1) ∈ (0..^(♯‘𝑊))) → (𝑊 substr ⟨𝐼, (𝐼 + 2)⟩) = ⟨“(𝑊𝐼)(𝑊‘(𝐼 + 1))”⟩)
811, 10, 79, 80syl3anc 1367 . . . . . . . . . . 11 (𝜑 → (𝑊 substr ⟨𝐼, (𝐼 + 2)⟩) = ⟨“(𝑊𝐼)(𝑊‘(𝐼 + 1))”⟩)
8281oveq2d 7149 . . . . . . . . . 10 (𝜑 → (𝐺 Σg (𝑊 substr ⟨𝐼, (𝐼 + 2)⟩)) = (𝐺 Σg ⟨“(𝑊𝐼)(𝑊‘(𝐼 + 1))”⟩))
83 wrdf 13851 . . . . . . . . . . . . . . 15 (𝑊 ∈ Word 𝑇𝑊:(0..^(♯‘𝑊))⟶𝑇)
841, 83syl 17 . . . . . . . . . . . . . 14 (𝜑𝑊:(0..^(♯‘𝑊))⟶𝑇)
8578feq2d 6476 . . . . . . . . . . . . . 14 (𝜑 → (𝑊:(0..^(♯‘𝑊))⟶𝑇𝑊:(0..^𝐿)⟶𝑇))
8684, 85mpbid 234 . . . . . . . . . . . . 13 (𝜑𝑊:(0..^𝐿)⟶𝑇)
8786, 7ffvelrnd 6828 . . . . . . . . . . . 12 (𝜑 → (𝑊𝐼) ∈ 𝑇)
8866, 87sseldi 3944 . . . . . . . . . . 11 (𝜑 → (𝑊𝐼) ∈ (Base‘𝐺))
8986, 26ffvelrnd 6828 . . . . . . . . . . . 12 (𝜑 → (𝑊‘(𝐼 + 1)) ∈ 𝑇)
9066, 89sseldi 3944 . . . . . . . . . . 11 (𝜑 → (𝑊‘(𝐼 + 1)) ∈ (Base‘𝐺))
91 eqid 2820 . . . . . . . . . . . 12 (+g𝐺) = (+g𝐺)
9260, 91gsumws2 17986 . . . . . . . . . . 11 ((𝐺 ∈ Mnd ∧ (𝑊𝐼) ∈ (Base‘𝐺) ∧ (𝑊‘(𝐼 + 1)) ∈ (Base‘𝐺)) → (𝐺 Σg ⟨“(𝑊𝐼)(𝑊‘(𝐼 + 1))”⟩) = ((𝑊𝐼)(+g𝐺)(𝑊‘(𝐼 + 1))))
9364, 88, 90, 92syl3anc 1367 . . . . . . . . . 10 (𝜑 → (𝐺 Σg ⟨“(𝑊𝐼)(𝑊‘(𝐼 + 1))”⟩) = ((𝑊𝐼)(+g𝐺)(𝑊‘(𝐼 + 1))))
9419, 60, 91symgov 18491 . . . . . . . . . . 11 (((𝑊𝐼) ∈ (Base‘𝐺) ∧ (𝑊‘(𝐼 + 1)) ∈ (Base‘𝐺)) → ((𝑊𝐼)(+g𝐺)(𝑊‘(𝐼 + 1))) = ((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))))
9588, 90, 94syl2anc 586 . . . . . . . . . 10 (𝜑 → ((𝑊𝐼)(+g𝐺)(𝑊‘(𝐼 + 1))) = ((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))))
9682, 93, 953eqtrd 2859 . . . . . . . . 9 (𝜑 → (𝐺 Σg (𝑊 substr ⟨𝐼, (𝐼 + 2)⟩)) = ((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))))
9796adantr 483 . . . . . . . 8 ((𝜑 ∧ ((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → (𝐺 Σg (𝑊 substr ⟨𝐼, (𝐼 + 2)⟩)) = ((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))))
98 simpr 487 . . . . . . . 8 ((𝜑 ∧ ((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → ((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷))
9919symgid 18508 . . . . . . . . . . 11 (𝐷𝑉 → ( I ↾ 𝐷) = (0g𝐺))
10021, 99syl 17 . . . . . . . . . 10 (𝜑 → ( I ↾ 𝐷) = (0g𝐺))
101 eqid 2820 . . . . . . . . . . 11 (0g𝐺) = (0g𝐺)
102101gsum0 17873 . . . . . . . . . 10 (𝐺 Σg ∅) = (0g𝐺)
103100, 102syl6eqr 2873 . . . . . . . . 9 (𝜑 → ( I ↾ 𝐷) = (𝐺 Σg ∅))
104103adantr 483 . . . . . . . 8 ((𝜑 ∧ ((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → ( I ↾ 𝐷) = (𝐺 Σg ∅))
10597, 98, 1043eqtrd 2859 . . . . . . 7 ((𝜑 ∧ ((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → (𝐺 Σg (𝑊 substr ⟨𝐼, (𝐼 + 2)⟩)) = (𝐺 Σg ∅))
10660, 65, 70, 71, 72, 75, 77, 105gsumspl 17988 . . . . . 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), ∅⟩)))
10722adantr 483 . . . . . 6 ((𝜑 ∧ ((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → (𝐺 Σg 𝑊) = ( I ↾ 𝐷))
10859, 106, 1073eqtr3d 2863 . . . . 5 ((𝜑 ∧ ((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → (𝐺 Σg (𝑊 splice ⟨𝐼, (𝐼 + 2), ∅⟩)) = ( I ↾ 𝐷))
109 fveqeq2 6655 . . . . . . 7 (𝑥 = (𝑊 splice ⟨𝐼, (𝐼 + 2), ∅⟩) → ((♯‘𝑥) = (𝐿 − 2) ↔ (♯‘(𝑊 splice ⟨𝐼, (𝐼 + 2), ∅⟩)) = (𝐿 − 2)))
110 oveq2 7141 . . . . . . . 8 (𝑥 = (𝑊 splice ⟨𝐼, (𝐼 + 2), ∅⟩) → (𝐺 Σg 𝑥) = (𝐺 Σg (𝑊 splice ⟨𝐼, (𝐼 + 2), ∅⟩)))
111110eqeq1d 2822 . . . . . . 7 (𝑥 = (𝑊 splice ⟨𝐼, (𝐼 + 2), ∅⟩) → ((𝐺 Σg 𝑥) = ( I ↾ 𝐷) ↔ (𝐺 Σg (𝑊 splice ⟨𝐼, (𝐼 + 2), ∅⟩)) = ( I ↾ 𝐷)))
112109, 111anbi12d 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))))
113112rspcev 3602 . . . . 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)))
1145, 55, 108, 113syl12anc 834 . . . 4 ((𝜑 ∧ ((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → ∃𝑥 ∈ Word 𝑇((♯‘𝑥) = (𝐿 − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))
115 psgnunilem2.in . . . . 5 (𝜑 → ¬ ∃𝑥 ∈ Word 𝑇((♯‘𝑥) = (𝐿 − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))
116115adantr 483 . . . 4 ((𝜑 ∧ ((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷)) → ¬ ∃𝑥 ∈ Word 𝑇((♯‘𝑥) = (𝐿 − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))
117114, 116pm2.21dd 197 . . 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 ))))
118117ex 415 . 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 )))))
1191adantr 483 . . . . . . 7 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → 𝑊 ∈ Word 𝑇)
120 simprl 769 . . . . . . . 8 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → 𝑟𝑇)
121 simprr 771 . . . . . . . 8 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → 𝑠𝑇)
122120, 121s2cld 14213 . . . . . . 7 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → ⟨“𝑟𝑠”⟩ ∈ Word 𝑇)
123 splcl 14094 . . . . . . 7 ((𝑊 ∈ Word 𝑇 ∧ ⟨“𝑟𝑠”⟩ ∈ Word 𝑇) → (𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩) ∈ Word 𝑇)
124119, 122, 123syl2anc 586 . . . . . 6 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → (𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩) ∈ Word 𝑇)
125124adantrr 715 . . . . 5 ((𝜑 ∧ ((𝑟𝑇𝑠𝑇) ∧ (((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) → (𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩) ∈ Word 𝑇)
12664adantr 483 . . . . . . . 8 ((𝜑 ∧ ((𝑟𝑇𝑠𝑇) ∧ (((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) → 𝐺 ∈ Mnd)
12769adantr 483 . . . . . . . 8 ((𝜑 ∧ ((𝑟𝑇𝑠𝑇) ∧ (((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) → 𝑊 ∈ Word (Base‘𝐺))
12818adantr 483 . . . . . . . 8 ((𝜑 ∧ ((𝑟𝑇𝑠𝑇) ∧ (((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) → 𝐼 ∈ (0...(𝐼 + 2)))
12936adantr 483 . . . . . . . 8 ((𝜑 ∧ ((𝑟𝑇𝑠𝑇) ∧ (((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) → (𝐼 + 2) ∈ (0...(♯‘𝑊)))
13068, 122sseldi 3944 . . . . . . . . 9 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → ⟨“𝑟𝑠”⟩ ∈ Word (Base‘𝐺))
131130adantrr 715 . . . . . . . 8 ((𝜑 ∧ ((𝑟𝑇𝑠𝑇) ∧ (((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) → ⟨“𝑟𝑠”⟩ ∈ Word (Base‘𝐺))
13274adantr 483 . . . . . . . 8 ((𝜑 ∧ ((𝑟𝑇𝑠𝑇) ∧ (((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) → (𝑊 substr ⟨𝐼, (𝐼 + 2)⟩) ∈ Word (Base‘𝐺))
133 simprr1 1217 . . . . . . . . 9 ((𝜑 ∧ ((𝑟𝑇𝑠𝑇) ∧ (((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) → ((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟𝑠))
13496adantr 483 . . . . . . . . 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))))
13564adantr 483 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → 𝐺 ∈ Mnd)
13666a1i 11 . . . . . . . . . . . . . 14 (𝜑𝑇 ⊆ (Base‘𝐺))
137136sselda 3946 . . . . . . . . . . . . 13 ((𝜑𝑟𝑇) → 𝑟 ∈ (Base‘𝐺))
138137adantrr 715 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → 𝑟 ∈ (Base‘𝐺))
139136sselda 3946 . . . . . . . . . . . . 13 ((𝜑𝑠𝑇) → 𝑠 ∈ (Base‘𝐺))
140139adantrl 714 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → 𝑠 ∈ (Base‘𝐺))
14160, 91gsumws2 17986 . . . . . . . . . . . 12 ((𝐺 ∈ Mnd ∧ 𝑟 ∈ (Base‘𝐺) ∧ 𝑠 ∈ (Base‘𝐺)) → (𝐺 Σg ⟨“𝑟𝑠”⟩) = (𝑟(+g𝐺)𝑠))
142135, 138, 140, 141syl3anc 1367 . . . . . . . . . . 11 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → (𝐺 Σg ⟨“𝑟𝑠”⟩) = (𝑟(+g𝐺)𝑠))
14319, 60, 91symgov 18491 . . . . . . . . . . . 12 ((𝑟 ∈ (Base‘𝐺) ∧ 𝑠 ∈ (Base‘𝐺)) → (𝑟(+g𝐺)𝑠) = (𝑟𝑠))
144138, 140, 143syl2anc 586 . . . . . . . . . . 11 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → (𝑟(+g𝐺)𝑠) = (𝑟𝑠))
145142, 144eqtrd 2855 . . . . . . . . . 10 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → (𝐺 Σg ⟨“𝑟𝑠”⟩) = (𝑟𝑠))
146145adantrr 715 . . . . . . . . 9 ((𝜑 ∧ ((𝑟𝑇𝑠𝑇) ∧ (((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) → (𝐺 Σg ⟨“𝑟𝑠”⟩) = (𝑟𝑠))
147133, 134, 1463eqtr4rd 2866 . . . . . . . 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)⟩)))
14860, 126, 127, 128, 129, 131, 132, 147gsumspl 17988 . . . . . . 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)⟩)⟩)))
14958adantr 483 . . . . . . 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))
15022adantr 483 . . . . . . 7 ((𝜑 ∧ ((𝑟𝑇𝑠𝑇) ∧ (((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) → (𝐺 Σg 𝑊) = ( I ↾ 𝐷))
151148, 149, 1503eqtrd 2859 . . . . . 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))
15218adantr 483 . . . . . . . . 9 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → 𝐼 ∈ (0...(𝐼 + 2)))
15336adantr 483 . . . . . . . . 9 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → (𝐼 + 2) ∈ (0...(♯‘𝑊)))
154119, 152, 153, 122spllen 14096 . . . . . . . 8 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → (♯‘(𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)) = ((♯‘𝑊) + ((♯‘⟨“𝑟𝑠”⟩) − ((𝐼 + 2) − 𝐼))))
155 s2len 14231 . . . . . . . . . . . . 13 (♯‘⟨“𝑟𝑠”⟩) = 2
156155oveq1i 7143 . . . . . . . . . . . 12 ((♯‘⟨“𝑟𝑠”⟩) − ((𝐼 + 2) − 𝐼)) = (2 − ((𝐼 + 2) − 𝐼))
15745oveq2d 7149 . . . . . . . . . . . . 13 (𝜑 → (2 − ((𝐼 + 2) − 𝐼)) = (2 − 2))
15843subidi 10935 . . . . . . . . . . . . 13 (2 − 2) = 0
159157, 158syl6eq 2871 . . . . . . . . . . . 12 (𝜑 → (2 − ((𝐼 + 2) − 𝐼)) = 0)
160156, 159syl5eq 2867 . . . . . . . . . . 11 (𝜑 → ((♯‘⟨“𝑟𝑠”⟩) − ((𝐼 + 2) − 𝐼)) = 0)
161160oveq2d 7149 . . . . . . . . . 10 (𝜑 → ((♯‘𝑊) + ((♯‘⟨“𝑟𝑠”⟩) − ((𝐼 + 2) − 𝐼))) = ((♯‘𝑊) + 0))
16223, 51eqeltrd 2911 . . . . . . . . . . 11 (𝜑 → (♯‘𝑊) ∈ ℂ)
163162addid1d 10818 . . . . . . . . . 10 (𝜑 → ((♯‘𝑊) + 0) = (♯‘𝑊))
164161, 163, 233eqtrd 2859 . . . . . . . . 9 (𝜑 → ((♯‘𝑊) + ((♯‘⟨“𝑟𝑠”⟩) − ((𝐼 + 2) − 𝐼))) = 𝐿)
165164adantr 483 . . . . . . . 8 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → ((♯‘𝑊) + ((♯‘⟨“𝑟𝑠”⟩) − ((𝐼 + 2) − 𝐼))) = 𝐿)
166154, 165eqtrd 2855 . . . . . . 7 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → (♯‘(𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)) = 𝐿)
167166adantrr 715 . . . . . 6 ((𝜑 ∧ ((𝑟𝑇𝑠𝑇) ∧ (((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) → (♯‘(𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)) = 𝐿)
168151, 167jca 514 . . . . 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))
16926adantr 483 . . . . . 6 ((𝜑 ∧ ((𝑟𝑇𝑠𝑇) ∧ (((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) → (𝐼 + 1) ∈ (0..^𝐿))
170 simprr2 1218 . . . . . . 7 ((𝜑 ∧ ((𝑟𝑇𝑠𝑇) ∧ (((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) → 𝐴 ∈ dom (𝑠 ∖ I ))
171 1nn0 11892 . . . . . . . . . . . . . . 15 1 ∈ ℕ0
172 2nn 11689 . . . . . . . . . . . . . . 15 2 ∈ ℕ
173 1lt2 11787 . . . . . . . . . . . . . . 15 1 < 2
174 elfzo0 13062 . . . . . . . . . . . . . . 15 (1 ∈ (0..^2) ↔ (1 ∈ ℕ0 ∧ 2 ∈ ℕ ∧ 1 < 2))
175171, 172, 173, 174mpbir3an 1337 . . . . . . . . . . . . . 14 1 ∈ (0..^2)
176155oveq2i 7144 . . . . . . . . . . . . . 14 (0..^(♯‘⟨“𝑟𝑠”⟩)) = (0..^2)
177175, 176eleqtrri 2910 . . . . . . . . . . . . 13 1 ∈ (0..^(♯‘⟨“𝑟𝑠”⟩))
178177a1i 11 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → 1 ∈ (0..^(♯‘⟨“𝑟𝑠”⟩)))
179119, 152, 153, 122, 178splfv2a 14098 . . . . . . . . . . 11 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → ((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘(𝐼 + 1)) = (⟨“𝑟𝑠”⟩‘1))
180 s2fv1 14230 . . . . . . . . . . . 12 (𝑠𝑇 → (⟨“𝑟𝑠”⟩‘1) = 𝑠)
181180ad2antll 727 . . . . . . . . . . 11 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → (⟨“𝑟𝑠”⟩‘1) = 𝑠)
182179, 181eqtrd 2855 . . . . . . . . . 10 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → ((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘(𝐼 + 1)) = 𝑠)
183182adantrr 715 . . . . . . . . 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)
184183difeq1d 4077 . . . . . . . 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 ))
185184dmeqd 5750 . . . . . . 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 ))
186170, 185eleqtrrd 2914 . . . . . 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 ))
187 fzosplitsni 13132 . . . . . . . . . . 11 (𝐼 ∈ (ℤ‘0) → (𝑗 ∈ (0..^(𝐼 + 1)) ↔ (𝑗 ∈ (0..^𝐼) ∨ 𝑗 = 𝐼)))
188 nn0uz 12259 . . . . . . . . . . 11 0 = (ℤ‘0)
189187, 188eleq2s 2929 . . . . . . . . . 10 (𝐼 ∈ ℕ0 → (𝑗 ∈ (0..^(𝐼 + 1)) ↔ (𝑗 ∈ (0..^𝐼) ∨ 𝑗 = 𝐼)))
19010, 189syl 17 . . . . . . . . 9 (𝜑 → (𝑗 ∈ (0..^(𝐼 + 1)) ↔ (𝑗 ∈ (0..^𝐼) ∨ 𝑗 = 𝐼)))
191190adantr 483 . . . . . . . 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)))
192 fveq2 6646 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 𝑗 → (𝑊𝑘) = (𝑊𝑗))
193192difeq1d 4077 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑗 → ((𝑊𝑘) ∖ I ) = ((𝑊𝑗) ∖ I ))
194193dmeqd 5750 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑗 → dom ((𝑊𝑘) ∖ I ) = dom ((𝑊𝑗) ∖ I ))
195194eleq2d 2896 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑗 → (𝐴 ∈ dom ((𝑊𝑘) ∖ I ) ↔ 𝐴 ∈ dom ((𝑊𝑗) ∖ I )))
196195notbid 320 . . . . . . . . . . . . . . 15 (𝑘 = 𝑗 → (¬ 𝐴 ∈ dom ((𝑊𝑘) ∖ I ) ↔ ¬ 𝐴 ∈ dom ((𝑊𝑗) ∖ I )))
197196rspccva 3601 . . . . . . . . . . . . . 14 ((∀𝑘 ∈ (0..^𝐼) ¬ 𝐴 ∈ dom ((𝑊𝑘) ∖ I ) ∧ 𝑗 ∈ (0..^𝐼)) → ¬ 𝐴 ∈ dom ((𝑊𝑗) ∖ I ))
19825, 197sylan 582 . . . . . . . . . . . . 13 ((𝜑𝑗 ∈ (0..^𝐼)) → ¬ 𝐴 ∈ dom ((𝑊𝑗) ∖ I ))
199198adantlr 713 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑟𝑇𝑠𝑇)) ∧ 𝑗 ∈ (0..^𝐼)) → ¬ 𝐴 ∈ dom ((𝑊𝑗) ∖ I ))
2001ad2antrr 724 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑟𝑇𝑠𝑇)) ∧ 𝑗 ∈ (0..^𝐼)) → 𝑊 ∈ Word 𝑇)
20118ad2antrr 724 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑟𝑇𝑠𝑇)) ∧ 𝑗 ∈ (0..^𝐼)) → 𝐼 ∈ (0...(𝐼 + 2)))
20236ad2antrr 724 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑟𝑇𝑠𝑇)) ∧ 𝑗 ∈ (0..^𝐼)) → (𝐼 + 2) ∈ (0...(♯‘𝑊)))
203122adantr 483 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑟𝑇𝑠𝑇)) ∧ 𝑗 ∈ (0..^𝐼)) → ⟨“𝑟𝑠”⟩ ∈ Word 𝑇)
204 simpr 487 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑟𝑇𝑠𝑇)) ∧ 𝑗 ∈ (0..^𝐼)) → 𝑗 ∈ (0..^𝐼))
205200, 201, 202, 203, 204splfv1 14097 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑟𝑇𝑠𝑇)) ∧ 𝑗 ∈ (0..^𝐼)) → ((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘𝑗) = (𝑊𝑗))
206205difeq1d 4077 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑟𝑇𝑠𝑇)) ∧ 𝑗 ∈ (0..^𝐼)) → (((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘𝑗) ∖ I ) = ((𝑊𝑗) ∖ I ))
207206dmeqd 5750 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑟𝑇𝑠𝑇)) ∧ 𝑗 ∈ (0..^𝐼)) → dom (((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘𝑗) ∖ I ) = dom ((𝑊𝑗) ∖ I ))
208199, 207neleqtrrd 2933 . . . . . . . . . . 11 (((𝜑 ∧ (𝑟𝑇𝑠𝑇)) ∧ 𝑗 ∈ (0..^𝐼)) → ¬ 𝐴 ∈ dom (((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘𝑗) ∖ I ))
209208ex 415 . . . . . . . . . 10 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → (𝑗 ∈ (0..^𝐼) → ¬ 𝐴 ∈ dom (((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘𝑗) ∖ I )))
210209adantrr 715 . . . . . . . . 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 )))
211 simprr3 1219 . . . . . . . . . . 11 ((𝜑 ∧ ((𝑟𝑇𝑠𝑇) ∧ (((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))) → ¬ 𝐴 ∈ dom (𝑟 ∖ I ))
212 0nn0 11891 . . . . . . . . . . . . . . . . . . . 20 0 ∈ ℕ0
213 2pos 11719 . . . . . . . . . . . . . . . . . . . 20 0 < 2
214 elfzo0 13062 . . . . . . . . . . . . . . . . . . . 20 (0 ∈ (0..^2) ↔ (0 ∈ ℕ0 ∧ 2 ∈ ℕ ∧ 0 < 2))
215212, 172, 213, 214mpbir3an 1337 . . . . . . . . . . . . . . . . . . 19 0 ∈ (0..^2)
216215, 176eleqtrri 2910 . . . . . . . . . . . . . . . . . 18 0 ∈ (0..^(♯‘⟨“𝑟𝑠”⟩))
217216a1i 11 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → 0 ∈ (0..^(♯‘⟨“𝑟𝑠”⟩)))
218119, 152, 153, 122, 217splfv2a 14098 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → ((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘(𝐼 + 0)) = (⟨“𝑟𝑠”⟩‘0))
21929addid1d 10818 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝐼 + 0) = 𝐼)
220219adantr 483 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → (𝐼 + 0) = 𝐼)
221220fveq2d 6650 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → ((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘(𝐼 + 0)) = ((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘𝐼))
222 s2fv0 14229 . . . . . . . . . . . . . . . . 17 (𝑟𝑇 → (⟨“𝑟𝑠”⟩‘0) = 𝑟)
223222ad2antrl 726 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → (⟨“𝑟𝑠”⟩‘0) = 𝑟)
224218, 221, 2233eqtr3d 2863 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → ((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘𝐼) = 𝑟)
225224difeq1d 4077 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → (((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘𝐼) ∖ I ) = (𝑟 ∖ I ))
226225dmeqd 5750 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → dom (((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘𝐼) ∖ I ) = dom (𝑟 ∖ I ))
227226eleq2d 2896 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑟𝑇𝑠𝑇)) → (𝐴 ∈ dom (((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘𝐼) ∖ I ) ↔ 𝐴 ∈ dom (𝑟 ∖ I )))
228227adantrr 715 . . . . . . . . . . 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 )))
229211, 228mtbird 327 . . . . . . . . . 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 ))
230 fveq2 6646 . . . . . . . . . . . . . 14 (𝑗 = 𝐼 → ((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘𝑗) = ((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘𝐼))
231230difeq1d 4077 . . . . . . . . . . . . 13 (𝑗 = 𝐼 → (((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘𝑗) ∖ I ) = (((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘𝐼) ∖ I ))
232231dmeqd 5750 . . . . . . . . . . . 12 (𝑗 = 𝐼 → dom (((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘𝑗) ∖ I ) = dom (((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘𝐼) ∖ I ))
233232eleq2d 2896 . . . . . . . . . . 11 (𝑗 = 𝐼 → (𝐴 ∈ dom (((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘𝑗) ∖ I ) ↔ 𝐴 ∈ dom (((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘𝐼) ∖ I )))
234233notbid 320 . . . . . . . . . 10 (𝑗 = 𝐼 → (¬ 𝐴 ∈ dom (((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘𝑗) ∖ I ) ↔ ¬ 𝐴 ∈ dom (((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘𝐼) ∖ I )))
235229, 234syl5ibrcom 249 . . . . . . . . 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 )))
236210, 235jaod 855 . . . . . . . 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 )))
237191, 236sylbid 242 . . . . . . 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 )))
238237ralrimiv 3168 . . . . . 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 ))
239169, 186, 2383jca 1124 . . . . 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 )))
240 oveq2 7141 . . . . . . . . 9 (𝑤 = (𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩) → (𝐺 Σg 𝑤) = (𝐺 Σg (𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)))
241240eqeq1d 2822 . . . . . . . 8 (𝑤 = (𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩) → ((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ↔ (𝐺 Σg (𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)) = ( I ↾ 𝐷)))
242 fveqeq2 6655 . . . . . . . 8 (𝑤 = (𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩) → ((♯‘𝑤) = 𝐿 ↔ (♯‘(𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)) = 𝐿))
243241, 242anbi12d 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)))
244 fveq1 6645 . . . . . . . . . . 11 (𝑤 = (𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩) → (𝑤‘(𝐼 + 1)) = ((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘(𝐼 + 1)))
245244difeq1d 4077 . . . . . . . . . 10 (𝑤 = (𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩) → ((𝑤‘(𝐼 + 1)) ∖ I ) = (((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘(𝐼 + 1)) ∖ I ))
246245dmeqd 5750 . . . . . . . . 9 (𝑤 = (𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩) → dom ((𝑤‘(𝐼 + 1)) ∖ I ) = dom (((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘(𝐼 + 1)) ∖ I ))
247246eleq2d 2896 . . . . . . . 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 )))
248 fveq1 6645 . . . . . . . . . . . . 13 (𝑤 = (𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩) → (𝑤𝑗) = ((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘𝑗))
249248difeq1d 4077 . . . . . . . . . . . 12 (𝑤 = (𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩) → ((𝑤𝑗) ∖ I ) = (((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘𝑗) ∖ I ))
250249dmeqd 5750 . . . . . . . . . . 11 (𝑤 = (𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩) → dom ((𝑤𝑗) ∖ I ) = dom (((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘𝑗) ∖ I ))
251250eleq2d 2896 . . . . . . . . . 10 (𝑤 = (𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩) → (𝐴 ∈ dom ((𝑤𝑗) ∖ I ) ↔ 𝐴 ∈ dom (((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘𝑗) ∖ I )))
252251notbid 320 . . . . . . . . 9 (𝑤 = (𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩) → (¬ 𝐴 ∈ dom ((𝑤𝑗) ∖ I ) ↔ ¬ 𝐴 ∈ dom (((𝑊 splice ⟨𝐼, (𝐼 + 2), ⟨“𝑟𝑠”⟩⟩)‘𝑗) ∖ I )))
253252ralbidv 3184 . . . . . . . 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 )))
254247, 2533anbi23d 1435 . . . . . . 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 ))))
255243, 254anbi12d 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 )))))
256255rspcev 3602 . . . . 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 ))))
257125, 168, 239, 256syl12anc 834 . . . 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 ))))
258257expr 459 . . 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 )))))
259258rexlimdvva 3281 . 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 )))))
26020, 21, 87, 89, 24psgnunilem1 18600 . 2 (𝜑 → (((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = ( I ↾ 𝐷) ∨ ∃𝑟𝑇𝑠𝑇 (((𝑊𝐼) ∘ (𝑊‘(𝐼 + 1))) = (𝑟𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I ))))
261118, 259, 260mpjaod 856 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 208   ∧ wa 398   ∨ wo 843   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  ∀wral 3125  ∃wrex 3126   ∖ cdif 3910   ⊆ wss 3913  ∅c0 4269  ⟨cop 4549  ⟨cotp 4551   class class class wbr 5042   I cid 5435  dom cdm 5531  ran crn 5532   ↾ cres 5533   ∘ ccom 5535  ⟶wf 6327  ‘cfv 6331  (class class class)co 7133  ℂcc 10513  ℝcr 10514  0cc0 10515  1c1 10516   + caddc 10518   < clt 10653   ≤ cle 10654   − cmin 10848  -cneg 10849  ℕcn 11616  2c2 11671  ℕ0cn0 11876  ℤcz 11960  ℤ≥cuz 12222  ...cfz 12876  ..^cfzo 13017  ♯chash 13675  Word cword 13846   substr csubstr 13982   splice csplice 14091  ⟨“cs2 14183  Basecbs 16462  +gcplusg 16544  0gc0g 16692   Σg cgsu 16693  Mndcmnd 17890  Grpcgrp 18082  SymGrpcsymg 18474  pmTrspcpmtr 18548 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306  ax-un 7439  ax-cnex 10571  ax-resscn 10572  ax-1cn 10573  ax-icn 10574  ax-addcl 10575  ax-addrcl 10576  ax-mulcl 10577  ax-mulrcl 10578  ax-mulcom 10579  ax-addass 10580  ax-mulass 10581  ax-distr 10582  ax-i2m1 10583  ax-1ne0 10584  ax-1rid 10585  ax-rnegex 10586  ax-rrecex 10587  ax-cnre 10588  ax-pre-lttri 10589  ax-pre-lttrn 10590  ax-pre-ltadd 10591  ax-pre-mulgt0 10592 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-xor 1502  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-nel 3111  df-ral 3130  df-rex 3131  df-reu 3132  df-rmo 3133  df-rab 3134  df-v 3475  df-sbc 3753  df-csb 3861  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3932  df-nul 4270  df-if 4444  df-pw 4517  df-sn 4544  df-pr 4546  df-tp 4548  df-op 4550  df-ot 4552  df-uni 4815  df-int 4853  df-iun 4897  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5436  df-eprel 5441  df-po 5450  df-so 5451  df-fr 5490  df-we 5492  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-pred 6124  df-ord 6170  df-on 6171  df-lim 6172  df-suc 6173  df-iota 6290  df-fun 6333  df-fn 6334  df-f 6335  df-f1 6336  df-fo 6337  df-f1o 6338  df-fv 6339  df-riota 7091  df-ov 7136  df-oprab 7137  df-mpo 7138  df-om 7559  df-1st 7667  df-2nd 7668  df-wrecs 7925  df-recs 7986  df-rdg 8024  df-1o 8080  df-2o 8081  df-oadd 8084  df-er 8267  df-map 8386  df-en 8488  df-dom 8489  df-sdom 8490  df-fin 8491  df-card 9346  df-pnf 10655  df-mnf 10656  df-xr 10657  df-ltxr 10658  df-le 10659  df-sub 10850  df-neg 10851  df-nn 11617  df-2 11679  df-3 11680  df-4 11681  df-5 11682  df-6 11683  df-7 11684  df-8 11685  df-9 11686  df-n0 11877  df-xnn0 11947  df-z 11961  df-uz 12223  df-fz 12877  df-fzo 13018  df-seq 13354  df-hash 13676  df-word 13847  df-lsw 13895  df-concat 13903  df-s1 13930  df-substr 13983  df-pfx 14013  df-splice 14092  df-s2 14190  df-struct 16464  df-ndx 16465  df-slot 16466  df-base 16468  df-sets 16469  df-ress 16470  df-plusg 16557  df-tset 16563  df-0g 16694  df-gsum 16695  df-mgm 17831  df-sgrp 17880  df-mnd 17891  df-submnd 17936  df-efmnd 18013  df-grp 18085  df-minusg 18086  df-subg 18255  df-symg 18475  df-pmtr 18549 This theorem is referenced by:  psgnunilem3  18603
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