Theorem gsmsymgreqlem2 18559

Description: Lemma 2 for gsmsymgreq 18560. (Contributed by AV, 26-Jan-2019.)

Hypotheses

Ref	Expression
gsmsymgrfix.s	⊢ 𝑆 = (SymGrp‘𝑁)
gsmsymgrfix.b	⊢ 𝐵 = (Base‘𝑆)
gsmsymgreq.z	⊢ 𝑍 = (SymGrp‘𝑀)
gsmsymgreq.p	⊢ 𝑃 = (Base‘𝑍)
gsmsymgreq.i	⊢ 𝐼 = (𝑁 ∩ 𝑀)

Assertion

Ref	Expression
gsmsymgreqlem2	⊢ (((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (♯‘𝑋) = (♯‘𝑌))) → ((∀𝑖 ∈ (0..^(♯‘𝑋))∀𝑛 ∈ 𝐼 ((𝑋‘𝑖)‘𝑛) = ((𝑌‘𝑖)‘𝑛) → ∀𝑛 ∈ 𝐼 ((𝑆 Σg 𝑋)‘𝑛) = ((𝑍 Σg 𝑌)‘𝑛)) → (∀𝑖 ∈ (0..^(♯‘(𝑋 ++ ⟨“𝐶”⟩)))∀𝑛 ∈ 𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) → ∀𝑛 ∈ 𝐼 ((𝑆 Σg (𝑋 ++ ⟨“𝐶”⟩))‘𝑛) = ((𝑍 Σg (𝑌 ++ ⟨“𝑅”⟩))‘𝑛))))

Distinct variable groups:   𝐵,𝑖   𝑖,𝑁   𝑃,𝑖   𝑛,𝐼   𝑛,𝑋   𝐶,𝑛   𝑅,𝑛   𝑆,𝑛   𝑛,𝑌   𝑛,𝑍   𝐵,𝑛   𝐶,𝑖,𝑛   𝑖,𝐼   𝑛,𝑀   𝑛,𝑁   𝑃,𝑛   𝑅,𝑖   𝑖,𝑋   𝑖,𝑌

Allowed substitution hints:   𝑆(𝑖)   𝑀(𝑖)   𝑍(𝑖)

Proof of Theorem gsmsymgreqlem2

Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ccatws1len 13974	. . . . . . . . . 10 ⊢ (𝑋 ∈ Word 𝐵 → (♯‘(𝑋 ++ ⟨“𝐶”⟩)) = ((♯‘𝑋) + 1))
2	1	oveq2d 7172	. . . . . . . . 9 ⊢ (𝑋 ∈ Word 𝐵 → (0..^(♯‘(𝑋 ++ ⟨“𝐶”⟩))) = (0..^((♯‘𝑋) + 1)))
3		lencl 13883	. . . . . . . . . . 11 ⊢ (𝑋 ∈ Word 𝐵 → (♯‘𝑋) ∈ ℕ0)
4		elnn0uz 12284	. . . . . . . . . . 11 ⊢ ((♯‘𝑋) ∈ ℕ0 ↔ (♯‘𝑋) ∈ (ℤ≥‘0))
5	3, 4	sylib 220	. . . . . . . . . 10 ⊢ (𝑋 ∈ Word 𝐵 → (♯‘𝑋) ∈ (ℤ≥‘0))
6		fzosplitsn 13146	. . . . . . . . . 10 ⊢ ((♯‘𝑋) ∈ (ℤ≥‘0) → (0..^((♯‘𝑋) + 1)) = ((0..^(♯‘𝑋)) ∪ {(♯‘𝑋)}))
7	5, 6	syl 17	. . . . . . . . 9 ⊢ (𝑋 ∈ Word 𝐵 → (0..^((♯‘𝑋) + 1)) = ((0..^(♯‘𝑋)) ∪ {(♯‘𝑋)}))
8	2, 7	eqtrd 2856	. . . . . . . 8 ⊢ (𝑋 ∈ Word 𝐵 → (0..^(♯‘(𝑋 ++ ⟨“𝐶”⟩))) = ((0..^(♯‘𝑋)) ∪ {(♯‘𝑋)}))
9	8	adantr 483	. . . . . . 7 ⊢ ((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) → (0..^(♯‘(𝑋 ++ ⟨“𝐶”⟩))) = ((0..^(♯‘𝑋)) ∪ {(♯‘𝑋)}))
10	9	3ad2ant1 1129	. . . . . 6 ⊢ (((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) → (0..^(♯‘(𝑋 ++ ⟨“𝐶”⟩))) = ((0..^(♯‘𝑋)) ∪ {(♯‘𝑋)}))
11	10	raleqdv 3415	. . . . 5 ⊢ (((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) → (∀𝑖 ∈ (0..^(♯‘(𝑋 ++ ⟨“𝐶”⟩)))∀𝑛 ∈ 𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) ↔ ∀𝑖 ∈ ((0..^(♯‘𝑋)) ∪ {(♯‘𝑋)})∀𝑛 ∈ 𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛)))
12	3	adantr 483	. . . . . . 7 ⊢ ((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) → (♯‘𝑋) ∈ ℕ0)
13	12	3ad2ant1 1129	. . . . . 6 ⊢ (((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) → (♯‘𝑋) ∈ ℕ0)
14		fveq2 6670	. . . . . . . . . 10 ⊢ (𝑖 = (♯‘𝑋) → ((𝑋 ++ ⟨“𝐶”⟩)‘𝑖) = ((𝑋 ++ ⟨“𝐶”⟩)‘(♯‘𝑋)))
15	14	fveq1d 6672	. . . . . . . . 9 ⊢ (𝑖 = (♯‘𝑋) → (((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = (((𝑋 ++ ⟨“𝐶”⟩)‘(♯‘𝑋))‘𝑛))
16		fveq2 6670	. . . . . . . . . 10 ⊢ (𝑖 = (♯‘𝑋) → ((𝑌 ++ ⟨“𝑅”⟩)‘𝑖) = ((𝑌 ++ ⟨“𝑅”⟩)‘(♯‘𝑋)))
17	16	fveq1d 6672	. . . . . . . . 9 ⊢ (𝑖 = (♯‘𝑋) → (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘(♯‘𝑋))‘𝑛))
18	15, 17	eqeq12d 2837	. . . . . . . 8 ⊢ (𝑖 = (♯‘𝑋) → ((((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) ↔ (((𝑋 ++ ⟨“𝐶”⟩)‘(♯‘𝑋))‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘(♯‘𝑋))‘𝑛)))
19	18	ralbidv 3197	. . . . . . 7 ⊢ (𝑖 = (♯‘𝑋) → (∀𝑛 ∈ 𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) ↔ ∀𝑛 ∈ 𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘(♯‘𝑋))‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘(♯‘𝑋))‘𝑛)))
20	19	ralunsn 4824	. . . . . 6 ⊢ ((♯‘𝑋) ∈ ℕ0 → (∀𝑖 ∈ ((0..^(♯‘𝑋)) ∪ {(♯‘𝑋)})∀𝑛 ∈ 𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) ↔ (∀𝑖 ∈ (0..^(♯‘𝑋))∀𝑛 ∈ 𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) ∧ ∀𝑛 ∈ 𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘(♯‘𝑋))‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘(♯‘𝑋))‘𝑛))))
21	13, 20	syl 17	. . . . 5 ⊢ (((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) → (∀𝑖 ∈ ((0..^(♯‘𝑋)) ∪ {(♯‘𝑋)})∀𝑛 ∈ 𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) ↔ (∀𝑖 ∈ (0..^(♯‘𝑋))∀𝑛 ∈ 𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) ∧ ∀𝑛 ∈ 𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘(♯‘𝑋))‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘(♯‘𝑋))‘𝑛))))
22		simp1l 1193	. . . . . . . . . . 11 ⊢ (((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) → 𝑋 ∈ Word 𝐵)
23		ccats1val1 13981	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ Word 𝐵 ∧ 𝑖 ∈ (0..^(♯‘𝑋))) → ((𝑋 ++ ⟨“𝐶”⟩)‘𝑖) = (𝑋‘𝑖))
24	22, 23	sylan 582	. . . . . . . . . 10 ⊢ ((((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) ∧ 𝑖 ∈ (0..^(♯‘𝑋))) → ((𝑋 ++ ⟨“𝐶”⟩)‘𝑖) = (𝑋‘𝑖))
25	24	fveq1d 6672	. . . . . . . . 9 ⊢ ((((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) ∧ 𝑖 ∈ (0..^(♯‘𝑋))) → (((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = ((𝑋‘𝑖)‘𝑛))
26		simp2l 1195	. . . . . . . . . . 11 ⊢ (((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) → 𝑌 ∈ Word 𝑃)
27		oveq2 7164	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝑋) = (♯‘𝑌) → (0..^(♯‘𝑋)) = (0..^(♯‘𝑌)))
28	27	eleq2d 2898	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝑋) = (♯‘𝑌) → (𝑖 ∈ (0..^(♯‘𝑋)) ↔ 𝑖 ∈ (0..^(♯‘𝑌))))
29	28	biimpd 231	. . . . . . . . . . . . 13 ⊢ ((♯‘𝑋) = (♯‘𝑌) → (𝑖 ∈ (0..^(♯‘𝑋)) → 𝑖 ∈ (0..^(♯‘𝑌))))
30	29	3ad2ant3 1131	. . . . . . . . . . . 12 ⊢ (((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) → (𝑖 ∈ (0..^(♯‘𝑋)) → 𝑖 ∈ (0..^(♯‘𝑌))))
31	30	imp 409	. . . . . . . . . . 11 ⊢ ((((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) ∧ 𝑖 ∈ (0..^(♯‘𝑋))) → 𝑖 ∈ (0..^(♯‘𝑌)))
32		ccats1val1 13981	. . . . . . . . . . 11 ⊢ ((𝑌 ∈ Word 𝑃 ∧ 𝑖 ∈ (0..^(♯‘𝑌))) → ((𝑌 ++ ⟨“𝑅”⟩)‘𝑖) = (𝑌‘𝑖))
33	26, 31, 32	syl2an2r 683	. . . . . . . . . 10 ⊢ ((((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) ∧ 𝑖 ∈ (0..^(♯‘𝑋))) → ((𝑌 ++ ⟨“𝑅”⟩)‘𝑖) = (𝑌‘𝑖))
34	33	fveq1d 6672	. . . . . . . . 9 ⊢ ((((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) ∧ 𝑖 ∈ (0..^(♯‘𝑋))) → (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) = ((𝑌‘𝑖)‘𝑛))
35	25, 34	eqeq12d 2837	. . . . . . . 8 ⊢ ((((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) ∧ 𝑖 ∈ (0..^(♯‘𝑋))) → ((((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) ↔ ((𝑋‘𝑖)‘𝑛) = ((𝑌‘𝑖)‘𝑛)))
36	35	ralbidv 3197	. . . . . . 7 ⊢ ((((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) ∧ 𝑖 ∈ (0..^(♯‘𝑋))) → (∀𝑛 ∈ 𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) ↔ ∀𝑛 ∈ 𝐼 ((𝑋‘𝑖)‘𝑛) = ((𝑌‘𝑖)‘𝑛)))
37	36	ralbidva 3196	. . . . . 6 ⊢ (((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) → (∀𝑖 ∈ (0..^(♯‘𝑋))∀𝑛 ∈ 𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) ↔ ∀𝑖 ∈ (0..^(♯‘𝑋))∀𝑛 ∈ 𝐼 ((𝑋‘𝑖)‘𝑛) = ((𝑌‘𝑖)‘𝑛)))
38		eqidd 2822	. . . . . . . . . 10 ⊢ ((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) → (♯‘𝑋) = (♯‘𝑋))
39		ccats1val2 13983	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵 ∧ (♯‘𝑋) = (♯‘𝑋)) → ((𝑋 ++ ⟨“𝐶”⟩)‘(♯‘𝑋)) = 𝐶)
40	39	fveq1d 6672	. . . . . . . . . 10 ⊢ ((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵 ∧ (♯‘𝑋) = (♯‘𝑋)) → (((𝑋 ++ ⟨“𝐶”⟩)‘(♯‘𝑋))‘𝑛) = (𝐶‘𝑛))
41	38, 40	mpd3an3 1458	. . . . . . . . 9 ⊢ ((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) → (((𝑋 ++ ⟨“𝐶”⟩)‘(♯‘𝑋))‘𝑛) = (𝐶‘𝑛))
42	41	3ad2ant1 1129	. . . . . . . 8 ⊢ (((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) → (((𝑋 ++ ⟨“𝐶”⟩)‘(♯‘𝑋))‘𝑛) = (𝐶‘𝑛))
43		ccats1val2 13983	. . . . . . . . . . 11 ⊢ ((𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃 ∧ (♯‘𝑋) = (♯‘𝑌)) → ((𝑌 ++ ⟨“𝑅”⟩)‘(♯‘𝑋)) = 𝑅)
44	43	fveq1d 6672	. . . . . . . . . 10 ⊢ ((𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃 ∧ (♯‘𝑋) = (♯‘𝑌)) → (((𝑌 ++ ⟨“𝑅”⟩)‘(♯‘𝑋))‘𝑛) = (𝑅‘𝑛))
45	44	3expa 1114	. . . . . . . . 9 ⊢ (((𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) → (((𝑌 ++ ⟨“𝑅”⟩)‘(♯‘𝑋))‘𝑛) = (𝑅‘𝑛))
46	45	3adant1 1126	. . . . . . . 8 ⊢ (((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) → (((𝑌 ++ ⟨“𝑅”⟩)‘(♯‘𝑋))‘𝑛) = (𝑅‘𝑛))
47	42, 46	eqeq12d 2837	. . . . . . 7 ⊢ (((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) → ((((𝑋 ++ ⟨“𝐶”⟩)‘(♯‘𝑋))‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘(♯‘𝑋))‘𝑛) ↔ (𝐶‘𝑛) = (𝑅‘𝑛)))
48	47	ralbidv 3197	. . . . . 6 ⊢ (((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) → (∀𝑛 ∈ 𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘(♯‘𝑋))‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘(♯‘𝑋))‘𝑛) ↔ ∀𝑛 ∈ 𝐼 (𝐶‘𝑛) = (𝑅‘𝑛)))
49	37, 48	anbi12d 632	. . . . 5 ⊢ (((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) → ((∀𝑖 ∈ (0..^(♯‘𝑋))∀𝑛 ∈ 𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) ∧ ∀𝑛 ∈ 𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘(♯‘𝑋))‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘(♯‘𝑋))‘𝑛)) ↔ (∀𝑖 ∈ (0..^(♯‘𝑋))∀𝑛 ∈ 𝐼 ((𝑋‘𝑖)‘𝑛) = ((𝑌‘𝑖)‘𝑛) ∧ ∀𝑛 ∈ 𝐼 (𝐶‘𝑛) = (𝑅‘𝑛))))
50	11, 21, 49	3bitrd 307	. . . 4 ⊢ (((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) → (∀𝑖 ∈ (0..^(♯‘(𝑋 ++ ⟨“𝐶”⟩)))∀𝑛 ∈ 𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) ↔ (∀𝑖 ∈ (0..^(♯‘𝑋))∀𝑛 ∈ 𝐼 ((𝑋‘𝑖)‘𝑛) = ((𝑌‘𝑖)‘𝑛) ∧ ∀𝑛 ∈ 𝐼 (𝐶‘𝑛) = (𝑅‘𝑛))))
51	50	ad2antlr 725	. . 3 ⊢ ((((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (♯‘𝑋) = (♯‘𝑌))) ∧ (∀𝑖 ∈ (0..^(♯‘𝑋))∀𝑛 ∈ 𝐼 ((𝑋‘𝑖)‘𝑛) = ((𝑌‘𝑖)‘𝑛) → ∀𝑛 ∈ 𝐼 ((𝑆 Σg 𝑋)‘𝑛) = ((𝑍 Σg 𝑌)‘𝑛))) → (∀𝑖 ∈ (0..^(♯‘(𝑋 ++ ⟨“𝐶”⟩)))∀𝑛 ∈ 𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) ↔ (∀𝑖 ∈ (0..^(♯‘𝑋))∀𝑛 ∈ 𝐼 ((𝑋‘𝑖)‘𝑛) = ((𝑌‘𝑖)‘𝑛) ∧ ∀𝑛 ∈ 𝐼 (𝐶‘𝑛) = (𝑅‘𝑛))))
52		pm3.35 801	. . . . . . 7 ⊢ ((∀𝑖 ∈ (0..^(♯‘𝑋))∀𝑛 ∈ 𝐼 ((𝑋‘𝑖)‘𝑛) = ((𝑌‘𝑖)‘𝑛) ∧ (∀𝑖 ∈ (0..^(♯‘𝑋))∀𝑛 ∈ 𝐼 ((𝑋‘𝑖)‘𝑛) = ((𝑌‘𝑖)‘𝑛) → ∀𝑛 ∈ 𝐼 ((𝑆 Σg 𝑋)‘𝑛) = ((𝑍 Σg 𝑌)‘𝑛))) → ∀𝑛 ∈ 𝐼 ((𝑆 Σg 𝑋)‘𝑛) = ((𝑍 Σg 𝑌)‘𝑛))
53		fveq2 6670	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑗 → ((𝑆 Σg 𝑋)‘𝑛) = ((𝑆 Σg 𝑋)‘𝑗))
54		fveq2 6670	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑗 → ((𝑍 Σg 𝑌)‘𝑛) = ((𝑍 Σg 𝑌)‘𝑗))
55	53, 54	eqeq12d 2837	. . . . . . . . . 10 ⊢ (𝑛 = 𝑗 → (((𝑆 Σg 𝑋)‘𝑛) = ((𝑍 Σg 𝑌)‘𝑛) ↔ ((𝑆 Σg 𝑋)‘𝑗) = ((𝑍 Σg 𝑌)‘𝑗)))
56	55	cbvralvw 3449	. . . . . . . . 9 ⊢ (∀𝑛 ∈ 𝐼 ((𝑆 Σg 𝑋)‘𝑛) = ((𝑍 Σg 𝑌)‘𝑛) ↔ ∀𝑗 ∈ 𝐼 ((𝑆 Σg 𝑋)‘𝑗) = ((𝑍 Σg 𝑌)‘𝑗))
57		simp-4l 781	. . . . . . . . . . . . . . 15 ⊢ (((((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (♯‘𝑋) = (♯‘𝑌))) ∧ ∀𝑗 ∈ 𝐼 ((𝑆 Σg 𝑋)‘𝑗) = ((𝑍 Σg 𝑌)‘𝑗)) ∧ 𝑛 ∈ 𝐼) → 𝑁 ∈ Fin)
58		simp-4r 782	. . . . . . . . . . . . . . 15 ⊢ (((((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (♯‘𝑋) = (♯‘𝑌))) ∧ ∀𝑗 ∈ 𝐼 ((𝑆 Σg 𝑋)‘𝑗) = ((𝑍 Σg 𝑌)‘𝑗)) ∧ 𝑛 ∈ 𝐼) → 𝑀 ∈ Fin)
59		simpr 487	. . . . . . . . . . . . . . 15 ⊢ (((((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (♯‘𝑋) = (♯‘𝑌))) ∧ ∀𝑗 ∈ 𝐼 ((𝑆 Σg 𝑋)‘𝑗) = ((𝑍 Σg 𝑌)‘𝑗)) ∧ 𝑛 ∈ 𝐼) → 𝑛 ∈ 𝐼)
60	57, 58, 59	3jca 1124	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (♯‘𝑋) = (♯‘𝑌))) ∧ ∀𝑗 ∈ 𝐼 ((𝑆 Σg 𝑋)‘𝑗) = ((𝑍 Σg 𝑌)‘𝑗)) ∧ 𝑛 ∈ 𝐼) → (𝑁 ∈ Fin ∧ 𝑀 ∈ Fin ∧ 𝑛 ∈ 𝐼))
61	60	adantr 483	. . . . . . . . . . . . 13 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (♯‘𝑋) = (♯‘𝑌))) ∧ ∀𝑗 ∈ 𝐼 ((𝑆 Σg 𝑋)‘𝑗) = ((𝑍 Σg 𝑌)‘𝑗)) ∧ 𝑛 ∈ 𝐼) ∧ (𝐶‘𝑛) = (𝑅‘𝑛)) → (𝑁 ∈ Fin ∧ 𝑀 ∈ Fin ∧ 𝑛 ∈ 𝐼))
62		simp-4r 782	. . . . . . . . . . . . 13 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (♯‘𝑋) = (♯‘𝑌))) ∧ ∀𝑗 ∈ 𝐼 ((𝑆 Σg 𝑋)‘𝑗) = ((𝑍 Σg 𝑌)‘𝑗)) ∧ 𝑛 ∈ 𝐼) ∧ (𝐶‘𝑛) = (𝑅‘𝑛)) → ((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)))
63		simplr 767	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (♯‘𝑋) = (♯‘𝑌))) ∧ ∀𝑗 ∈ 𝐼 ((𝑆 Σg 𝑋)‘𝑗) = ((𝑍 Σg 𝑌)‘𝑗)) ∧ 𝑛 ∈ 𝐼) → ∀𝑗 ∈ 𝐼 ((𝑆 Σg 𝑋)‘𝑗) = ((𝑍 Σg 𝑌)‘𝑗))
64	63	anim1i 616	. . . . . . . . . . . . 13 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (♯‘𝑋) = (♯‘𝑌))) ∧ ∀𝑗 ∈ 𝐼 ((𝑆 Σg 𝑋)‘𝑗) = ((𝑍 Σg 𝑌)‘𝑗)) ∧ 𝑛 ∈ 𝐼) ∧ (𝐶‘𝑛) = (𝑅‘𝑛)) → (∀𝑗 ∈ 𝐼 ((𝑆 Σg 𝑋)‘𝑗) = ((𝑍 Σg 𝑌)‘𝑗) ∧ (𝐶‘𝑛) = (𝑅‘𝑛)))
65		gsmsymgrfix.s	. . . . . . . . . . . . . . 15 ⊢ 𝑆 = (SymGrp‘𝑁)
66		gsmsymgrfix.b	. . . . . . . . . . . . . . 15 ⊢ 𝐵 = (Base‘𝑆)
67		gsmsymgreq.z	. . . . . . . . . . . . . . 15 ⊢ 𝑍 = (SymGrp‘𝑀)
68		gsmsymgreq.p	. . . . . . . . . . . . . . 15 ⊢ 𝑃 = (Base‘𝑍)
69		gsmsymgreq.i	. . . . . . . . . . . . . . 15 ⊢ 𝐼 = (𝑁 ∩ 𝑀)
70	65, 66, 67, 68, 69	gsmsymgreqlem1 18558	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin ∧ 𝑛 ∈ 𝐼) ∧ ((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (♯‘𝑋) = (♯‘𝑌))) → ((∀𝑗 ∈ 𝐼 ((𝑆 Σg 𝑋)‘𝑗) = ((𝑍 Σg 𝑌)‘𝑗) ∧ (𝐶‘𝑛) = (𝑅‘𝑛)) → ((𝑆 Σg (𝑋 ++ ⟨“𝐶”⟩))‘𝑛) = ((𝑍 Σg (𝑌 ++ ⟨“𝑅”⟩))‘𝑛)))
71	70	imp 409	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin ∧ 𝑛 ∈ 𝐼) ∧ ((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (♯‘𝑋) = (♯‘𝑌))) ∧ (∀𝑗 ∈ 𝐼 ((𝑆 Σg 𝑋)‘𝑗) = ((𝑍 Σg 𝑌)‘𝑗) ∧ (𝐶‘𝑛) = (𝑅‘𝑛))) → ((𝑆 Σg (𝑋 ++ ⟨“𝐶”⟩))‘𝑛) = ((𝑍 Σg (𝑌 ++ ⟨“𝑅”⟩))‘𝑛))
72	61, 62, 64, 71	syl21anc 835	. . . . . . . . . . . 12 ⊢ ((((((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (♯‘𝑋) = (♯‘𝑌))) ∧ ∀𝑗 ∈ 𝐼 ((𝑆 Σg 𝑋)‘𝑗) = ((𝑍 Σg 𝑌)‘𝑗)) ∧ 𝑛 ∈ 𝐼) ∧ (𝐶‘𝑛) = (𝑅‘𝑛)) → ((𝑆 Σg (𝑋 ++ ⟨“𝐶”⟩))‘𝑛) = ((𝑍 Σg (𝑌 ++ ⟨“𝑅”⟩))‘𝑛))
73	72	ex 415	. . . . . . . . . . 11 ⊢ (((((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (♯‘𝑋) = (♯‘𝑌))) ∧ ∀𝑗 ∈ 𝐼 ((𝑆 Σg 𝑋)‘𝑗) = ((𝑍 Σg 𝑌)‘𝑗)) ∧ 𝑛 ∈ 𝐼) → ((𝐶‘𝑛) = (𝑅‘𝑛) → ((𝑆 Σg (𝑋 ++ ⟨“𝐶”⟩))‘𝑛) = ((𝑍 Σg (𝑌 ++ ⟨“𝑅”⟩))‘𝑛)))
74	73	ralimdva 3177	. . . . . . . . . 10 ⊢ ((((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (♯‘𝑋) = (♯‘𝑌))) ∧ ∀𝑗 ∈ 𝐼 ((𝑆 Σg 𝑋)‘𝑗) = ((𝑍 Σg 𝑌)‘𝑗)) → (∀𝑛 ∈ 𝐼 (𝐶‘𝑛) = (𝑅‘𝑛) → ∀𝑛 ∈ 𝐼 ((𝑆 Σg (𝑋 ++ ⟨“𝐶”⟩))‘𝑛) = ((𝑍 Σg (𝑌 ++ ⟨“𝑅”⟩))‘𝑛)))
75	74	expcom 416	. . . . . . . . 9 ⊢ (∀𝑗 ∈ 𝐼 ((𝑆 Σg 𝑋)‘𝑗) = ((𝑍 Σg 𝑌)‘𝑗) → (((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (♯‘𝑋) = (♯‘𝑌))) → (∀𝑛 ∈ 𝐼 (𝐶‘𝑛) = (𝑅‘𝑛) → ∀𝑛 ∈ 𝐼 ((𝑆 Σg (𝑋 ++ ⟨“𝐶”⟩))‘𝑛) = ((𝑍 Σg (𝑌 ++ ⟨“𝑅”⟩))‘𝑛))))
76	56, 75	sylbi 219	. . . . . . . 8 ⊢ (∀𝑛 ∈ 𝐼 ((𝑆 Σg 𝑋)‘𝑛) = ((𝑍 Σg 𝑌)‘𝑛) → (((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (♯‘𝑋) = (♯‘𝑌))) → (∀𝑛 ∈ 𝐼 (𝐶‘𝑛) = (𝑅‘𝑛) → ∀𝑛 ∈ 𝐼 ((𝑆 Σg (𝑋 ++ ⟨“𝐶”⟩))‘𝑛) = ((𝑍 Σg (𝑌 ++ ⟨“𝑅”⟩))‘𝑛))))
77	76	com23 86	. . . . . . 7 ⊢ (∀𝑛 ∈ 𝐼 ((𝑆 Σg 𝑋)‘𝑛) = ((𝑍 Σg 𝑌)‘𝑛) → (∀𝑛 ∈ 𝐼 (𝐶‘𝑛) = (𝑅‘𝑛) → (((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (♯‘𝑋) = (♯‘𝑌))) → ∀𝑛 ∈ 𝐼 ((𝑆 Σg (𝑋 ++ ⟨“𝐶”⟩))‘𝑛) = ((𝑍 Σg (𝑌 ++ ⟨“𝑅”⟩))‘𝑛))))
78	52, 77	syl 17	. . . . . 6 ⊢ ((∀𝑖 ∈ (0..^(♯‘𝑋))∀𝑛 ∈ 𝐼 ((𝑋‘𝑖)‘𝑛) = ((𝑌‘𝑖)‘𝑛) ∧ (∀𝑖 ∈ (0..^(♯‘𝑋))∀𝑛 ∈ 𝐼 ((𝑋‘𝑖)‘𝑛) = ((𝑌‘𝑖)‘𝑛) → ∀𝑛 ∈ 𝐼 ((𝑆 Σg 𝑋)‘𝑛) = ((𝑍 Σg 𝑌)‘𝑛))) → (∀𝑛 ∈ 𝐼 (𝐶‘𝑛) = (𝑅‘𝑛) → (((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (♯‘𝑋) = (♯‘𝑌))) → ∀𝑛 ∈ 𝐼 ((𝑆 Σg (𝑋 ++ ⟨“𝐶”⟩))‘𝑛) = ((𝑍 Σg (𝑌 ++ ⟨“𝑅”⟩))‘𝑛))))
79	78	impancom 454	. . . . 5 ⊢ ((∀𝑖 ∈ (0..^(♯‘𝑋))∀𝑛 ∈ 𝐼 ((𝑋‘𝑖)‘𝑛) = ((𝑌‘𝑖)‘𝑛) ∧ ∀𝑛 ∈ 𝐼 (𝐶‘𝑛) = (𝑅‘𝑛)) → ((∀𝑖 ∈ (0..^(♯‘𝑋))∀𝑛 ∈ 𝐼 ((𝑋‘𝑖)‘𝑛) = ((𝑌‘𝑖)‘𝑛) → ∀𝑛 ∈ 𝐼 ((𝑆 Σg 𝑋)‘𝑛) = ((𝑍 Σg 𝑌)‘𝑛)) → (((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (♯‘𝑋) = (♯‘𝑌))) → ∀𝑛 ∈ 𝐼 ((𝑆 Σg (𝑋 ++ ⟨“𝐶”⟩))‘𝑛) = ((𝑍 Σg (𝑌 ++ ⟨“𝑅”⟩))‘𝑛))))
80	79	com13 88	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (♯‘𝑋) = (♯‘𝑌))) → ((∀𝑖 ∈ (0..^(♯‘𝑋))∀𝑛 ∈ 𝐼 ((𝑋‘𝑖)‘𝑛) = ((𝑌‘𝑖)‘𝑛) → ∀𝑛 ∈ 𝐼 ((𝑆 Σg 𝑋)‘𝑛) = ((𝑍 Σg 𝑌)‘𝑛)) → ((∀𝑖 ∈ (0..^(♯‘𝑋))∀𝑛 ∈ 𝐼 ((𝑋‘𝑖)‘𝑛) = ((𝑌‘𝑖)‘𝑛) ∧ ∀𝑛 ∈ 𝐼 (𝐶‘𝑛) = (𝑅‘𝑛)) → ∀𝑛 ∈ 𝐼 ((𝑆 Σg (𝑋 ++ ⟨“𝐶”⟩))‘𝑛) = ((𝑍 Σg (𝑌 ++ ⟨“𝑅”⟩))‘𝑛))))
81	80	imp 409	. . 3 ⊢ ((((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (♯‘𝑋) = (♯‘𝑌))) ∧ (∀𝑖 ∈ (0..^(♯‘𝑋))∀𝑛 ∈ 𝐼 ((𝑋‘𝑖)‘𝑛) = ((𝑌‘𝑖)‘𝑛) → ∀𝑛 ∈ 𝐼 ((𝑆 Σg 𝑋)‘𝑛) = ((𝑍 Σg 𝑌)‘𝑛))) → ((∀𝑖 ∈ (0..^(♯‘𝑋))∀𝑛 ∈ 𝐼 ((𝑋‘𝑖)‘𝑛) = ((𝑌‘𝑖)‘𝑛) ∧ ∀𝑛 ∈ 𝐼 (𝐶‘𝑛) = (𝑅‘𝑛)) → ∀𝑛 ∈ 𝐼 ((𝑆 Σg (𝑋 ++ ⟨“𝐶”⟩))‘𝑛) = ((𝑍 Σg (𝑌 ++ ⟨“𝑅”⟩))‘𝑛)))
82	51, 81	sylbid 242	. 2 ⊢ ((((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (♯‘𝑋) = (♯‘𝑌))) ∧ (∀𝑖 ∈ (0..^(♯‘𝑋))∀𝑛 ∈ 𝐼 ((𝑋‘𝑖)‘𝑛) = ((𝑌‘𝑖)‘𝑛) → ∀𝑛 ∈ 𝐼 ((𝑆 Σg 𝑋)‘𝑛) = ((𝑍 Σg 𝑌)‘𝑛))) → (∀𝑖 ∈ (0..^(♯‘(𝑋 ++ ⟨“𝐶”⟩)))∀𝑛 ∈ 𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) → ∀𝑛 ∈ 𝐼 ((𝑆 Σg (𝑋 ++ ⟨“𝐶”⟩))‘𝑛) = ((𝑍 Σg (𝑌 ++ ⟨“𝑅”⟩))‘𝑛)))
83	82	ex 415	1 ⊢ (((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (♯‘𝑋) = (♯‘𝑌))) → ((∀𝑖 ∈ (0..^(♯‘𝑋))∀𝑛 ∈ 𝐼 ((𝑋‘𝑖)‘𝑛) = ((𝑌‘𝑖)‘𝑛) → ∀𝑛 ∈ 𝐼 ((𝑆 Σg 𝑋)‘𝑛) = ((𝑍 Σg 𝑌)‘𝑛)) → (∀𝑖 ∈ (0..^(♯‘(𝑋 ++ ⟨“𝐶”⟩)))∀𝑛 ∈ 𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) → ∀𝑛 ∈ 𝐼 ((𝑆 Σg (𝑋 ++ ⟨“𝐶”⟩))‘𝑛) = ((𝑍 Σg (𝑌 ++ ⟨“𝑅”⟩))‘𝑛))))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  ∀wral 3138   ∪ cun 3934   ∩ cin 3935  {csn 4567  ‘cfv 6355  (class class class)co 7156  Fincfn 8509  0cc0 10537  1c1 10538   + caddc 10540  ℕ₀cn0 11898  ℤ_≥cuz 12244  ..^cfzo 13034  ♯chash 13691  Word cword 13862   ++ cconcat 13922  ⟨“cs1 13949  Basecbs 16483   Σ_g cgsu 16714  SymGrpcsymg 18495

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 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-cnex 10593  ax-resscn 10594  ax-1cn 10595  ax-icn 10596  ax-addcl 10597  ax-addrcl 10598  ax-mulcl 10599  ax-mulrcl 10600  ax-mulcom 10601  ax-addass 10602  ax-mulass 10603  ax-distr 10604  ax-i2m1 10605  ax-1ne0 10606  ax-1rid 10607  ax-rnegex 10608  ax-rrecex 10609  ax-cnre 10610  ax-pre-lttri 10611  ax-pre-lttrn 10612  ax-pre-ltadd 10613  ax-pre-mulgt0 10614

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-int 4877  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7581  df-1st 7689  df-2nd 7690  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-1o 8102  df-oadd 8106  df-er 8289  df-map 8408  df-en 8510  df-dom 8511  df-sdom 8512  df-fin 8513  df-card 9368  df-pnf 10677  df-mnf 10678  df-xr 10679  df-ltxr 10680  df-le 10681  df-sub 10872  df-neg 10873  df-nn 11639  df-2 11701  df-3 11702  df-4 11703  df-5 11704  df-6 11705  df-7 11706  df-8 11707  df-9 11708  df-n0 11899  df-z 11983  df-uz 12245  df-fz 12894  df-fzo 13035  df-seq 13371  df-hash 13692  df-word 13863  df-concat 13923  df-s1 13950  df-struct 16485  df-ndx 16486  df-slot 16487  df-base 16489  df-sets 16490  df-ress 16491  df-plusg 16578  df-tset 16584  df-0g 16715  df-gsum 16716  df-mgm 17852  df-sgrp 17901  df-mnd 17912  df-submnd 17957  df-efmnd 18034  df-grp 18106  df-symg 18496

This theorem is referenced by:  gsmsymgreq  18560