Theorem gsmsymgreqlem2 19558

Description: Lemma 2 for gsmsymgreq 19559. (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 14688	. . . . . . . . . 10 ⊢ (𝑋 ∈ Word 𝐵 → (♯‘(𝑋 ++ ⟨“𝐶”⟩)) = ((♯‘𝑋) + 1))
2	1	oveq2d 7429	. . . . . . . . 9 ⊢ (𝑋 ∈ Word 𝐵 → (0..^(♯‘(𝑋 ++ ⟨“𝐶”⟩))) = (0..^((♯‘𝑋) + 1)))
3		lencl 14598	. . . . . . . . . . 11 ⊢ (𝑋 ∈ Word 𝐵 → (♯‘𝑋) ∈ ℕ0)
4		elnn0uz 12928	. . . . . . . . . . 11 ⊢ ((♯‘𝑋) ∈ ℕ0 ↔ (♯‘𝑋) ∈ (ℤ≥‘0))
5	3, 4	sylib 221	. . . . . . . . . 10 ⊢ (𝑋 ∈ Word 𝐵 → (♯‘𝑋) ∈ (ℤ≥‘0))
6		fzosplitsn 13832	. . . . . . . . . 10 ⊢ ((♯‘𝑋) ∈ (ℤ≥‘0) → (0..^((♯‘𝑋) + 1)) = ((0..^(♯‘𝑋)) ∪ {(♯‘𝑋)}))
7	5, 6	syl 18	. . . . . . . . 9 ⊢ (𝑋 ∈ Word 𝐵 → (0..^((♯‘𝑋) + 1)) = ((0..^(♯‘𝑋)) ∪ {(♯‘𝑋)}))
8	2, 7	eqtrd 2795	. . . . . . . 8 ⊢ (𝑋 ∈ Word 𝐵 → (0..^(♯‘(𝑋 ++ ⟨“𝐶”⟩))) = ((0..^(♯‘𝑋)) ∪ {(♯‘𝑋)}))
9	8	adantr 486	. . . . . . 7 ⊢ ((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) → (0..^(♯‘(𝑋 ++ ⟨“𝐶”⟩))) = ((0..^(♯‘𝑋)) ∪ {(♯‘𝑋)}))
10	9	3ad2ant1 1151	. . . . . 6 ⊢ (((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) → (0..^(♯‘(𝑋 ++ ⟨“𝐶”⟩))) = ((0..^(♯‘𝑋)) ∪ {(♯‘𝑋)}))
11	10	raleqdv 3319	. . . . 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 486	. . . . . . 7 ⊢ ((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) → (♯‘𝑋) ∈ ℕ0)
13	12	3ad2ant1 1151	. . . . . 6 ⊢ (((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) → (♯‘𝑋) ∈ ℕ0)
14		fveq2 6878	. . . . . . . . . 10 ⊢ (𝑖 = (♯‘𝑋) → ((𝑋 ++ ⟨“𝐶”⟩)‘𝑖) = ((𝑋 ++ ⟨“𝐶”⟩)‘(♯‘𝑋)))
15	14	fveq1d 6880	. . . . . . . . 9 ⊢ (𝑖 = (♯‘𝑋) → (((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = (((𝑋 ++ ⟨“𝐶”⟩)‘(♯‘𝑋))‘𝑛))
16		fveq2 6878	. . . . . . . . . 10 ⊢ (𝑖 = (♯‘𝑋) → ((𝑌 ++ ⟨“𝑅”⟩)‘𝑖) = ((𝑌 ++ ⟨“𝑅”⟩)‘(♯‘𝑋)))
17	16	fveq1d 6880	. . . . . . . . 9 ⊢ (𝑖 = (♯‘𝑋) → (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘(♯‘𝑋))‘𝑛))
18	15, 17	eqeq12d 2776	. . . . . . . 8 ⊢ (𝑖 = (♯‘𝑋) → ((((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) ↔ (((𝑋 ++ ⟨“𝐶”⟩)‘(♯‘𝑋))‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘(♯‘𝑋))‘𝑛)))
19	18	ralbidv 3185	. . . . . . 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 4854	. . . . . 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 18	. . . . 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 1216	. . . . . . . . . . 11 ⊢ (((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) → 𝑋 ∈ Word 𝐵)
23		ccats1val1 14694	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ Word 𝐵 ∧ 𝑖 ∈ (0..^(♯‘𝑋))) → ((𝑋 ++ ⟨“𝐶”⟩)‘𝑖) = (𝑋‘𝑖))
24	22, 23	sylan 592	. . . . . . . . . 10 ⊢ ((((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) ∧ 𝑖 ∈ (0..^(♯‘𝑋))) → ((𝑋 ++ ⟨“𝐶”⟩)‘𝑖) = (𝑋‘𝑖))
25	24	fveq1d 6880	. . . . . . . . 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 1218	. . . . . . . . . . 11 ⊢ (((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) → 𝑌 ∈ Word 𝑃)
27		oveq2 7421	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝑋) = (♯‘𝑌) → (0..^(♯‘𝑋)) = (0..^(♯‘𝑌)))
28	27	eleq2d 2846	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝑋) = (♯‘𝑌) → (𝑖 ∈ (0..^(♯‘𝑋)) ↔ 𝑖 ∈ (0..^(♯‘𝑌))))
29	28	biimpd 232	. . . . . . . . . . . . 13 ⊢ ((♯‘𝑋) = (♯‘𝑌) → (𝑖 ∈ (0..^(♯‘𝑋)) → 𝑖 ∈ (0..^(♯‘𝑌))))
30	29	3ad2ant3 1153	. . . . . . . . . . . 12 ⊢ (((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) → (𝑖 ∈ (0..^(♯‘𝑋)) → 𝑖 ∈ (0..^(♯‘𝑌))))
31	30	imp 412	. . . . . . . . . . 11 ⊢ ((((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) ∧ 𝑖 ∈ (0..^(♯‘𝑋))) → 𝑖 ∈ (0..^(♯‘𝑌)))
32		ccats1val1 14694	. . . . . . . . . . 11 ⊢ ((𝑌 ∈ Word 𝑃 ∧ 𝑖 ∈ (0..^(♯‘𝑌))) → ((𝑌 ++ ⟨“𝑅”⟩)‘𝑖) = (𝑌‘𝑖))
33	26, 31, 32	syl2an2r 698	. . . . . . . . . 10 ⊢ ((((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) ∧ 𝑖 ∈ (0..^(♯‘𝑋))) → ((𝑌 ++ ⟨“𝑅”⟩)‘𝑖) = (𝑌‘𝑖))
34	33	fveq1d 6880	. . . . . . . . 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 2776	. . . . . . . 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 3185	. . . . . . 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 3183	. . . . . 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 2761	. . . . . . . . . 10 ⊢ ((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) → (♯‘𝑋) = (♯‘𝑋))
39		ccats1val2 14695	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵 ∧ (♯‘𝑋) = (♯‘𝑋)) → ((𝑋 ++ ⟨“𝐶”⟩)‘(♯‘𝑋)) = 𝐶)
40	39	fveq1d 6880	. . . . . . . . . 10 ⊢ ((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵 ∧ (♯‘𝑋) = (♯‘𝑋)) → (((𝑋 ++ ⟨“𝐶”⟩)‘(♯‘𝑋))‘𝑛) = (𝐶‘𝑛))
41	38, 40	mpd3an3 1491	. . . . . . . . 9 ⊢ ((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) → (((𝑋 ++ ⟨“𝐶”⟩)‘(♯‘𝑋))‘𝑛) = (𝐶‘𝑛))
42	41	3ad2ant1 1151	. . . . . . . 8 ⊢ (((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) → (((𝑋 ++ ⟨“𝐶”⟩)‘(♯‘𝑋))‘𝑛) = (𝐶‘𝑛))
43		ccats1val2 14695	. . . . . . . . . . 11 ⊢ ((𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃 ∧ (♯‘𝑋) = (♯‘𝑌)) → ((𝑌 ++ ⟨“𝑅”⟩)‘(♯‘𝑋)) = 𝑅)
44	43	fveq1d 6880	. . . . . . . . . 10 ⊢ ((𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃 ∧ (♯‘𝑋) = (♯‘𝑌)) → (((𝑌 ++ ⟨“𝑅”⟩)‘(♯‘𝑋))‘𝑛) = (𝑅‘𝑛))
45	44	3expa 1136	. . . . . . . . 9 ⊢ (((𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) → (((𝑌 ++ ⟨“𝑅”⟩)‘(♯‘𝑋))‘𝑛) = (𝑅‘𝑛))
46	45	3adant1 1148	. . . . . . . 8 ⊢ (((𝑋 ∈ Word 𝐵 ∧ 𝐶 ∈ 𝐵) ∧ (𝑌 ∈ Word 𝑃 ∧ 𝑅 ∈ 𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) → (((𝑌 ++ ⟨“𝑅”⟩)‘(♯‘𝑋))‘𝑛) = (𝑅‘𝑛))
47	42, 46	eqeq12d 2776	. . . . . . 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 3185	. . . . . 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 644	. . . . 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 308	. . . 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 740	. . 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		pm3.35 815	. . . . . . 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 6878	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑗 → ((𝑆 Σg 𝑋)‘𝑛) = ((𝑆 Σg 𝑋)‘𝑗))
54		fveq2 6878	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑗 → ((𝑍 Σg 𝑌)‘𝑛) = ((𝑍 Σg 𝑌)‘𝑗))
55	53, 54	eqeq12d 2776	. . . . . . . . . 10 ⊢ (𝑛 = 𝑗 → (((𝑆 Σg 𝑋)‘𝑛) = ((𝑍 Σg 𝑌)‘𝑛) ↔ ((𝑆 Σg 𝑋)‘𝑗) = ((𝑍 Σg 𝑌)‘𝑗)))
56	55	cbvralvw 3240	. . . . . . . . 9 ⊢ (∀𝑛 ∈ 𝐼 ((𝑆 Σg 𝑋)‘𝑛) = ((𝑍 Σg 𝑌)‘𝑛) ↔ ∀𝑗 ∈ 𝐼 ((𝑆 Σg 𝑋)‘𝑗) = ((𝑍 Σg 𝑌)‘𝑗))
57		simp-4l 795	. . . . . . . . . . . . . . 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 796	. . . . . . . . . . . . . . 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 490	. . . . . . . . . . . . . . 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 1146	. . . . . . . . . . . . . 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 486	. . . . . . . . . . . . 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 796	. . . . . . . . . . . . 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 781	. . . . . . . . . . . . . 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 627	. . . . . . . . . . . . 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 19557	. . . . . . . . . . . . . 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 412	. . . . . . . . . . . . 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 851	. . . . . . . . . . . 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 418	. . . . . . . . . . 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 3174	. . . . . . . . . 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 419	. . . . . . . . 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 220	. . . . . . . 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 87	. . . . . . 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 18	. . . . . 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 457	. . . . 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 89	. . . 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 412	. . 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 243	. 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 418	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))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3076   ∪ cun 3897   ∩ cin 3898  {csn 4584  ‘cfv 6533  (class class class)co 7413  Fincfn 8952  0cc0 11124  1c1 11125   + caddc 11127  ℕ₀cn0 12528  ℤ_≥cuz 12887  ..^cfzo 13709  ♯chash 14394  Word cword 14578   ++ cconcat 14635  ⟨“cs1 14662  Basecbs 17301   Σ_g cgsu 17525  SymGrpcsymg 19496

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736  ax-cnex 11180  ax-resscn 11181  ax-1cn 11182  ax-icn 11183  ax-addcl 11184  ax-addrcl 11185  ax-mulcl 11186  ax-mulrcl 11187  ax-mulcom 11188  ax-addass 11189  ax-mulass 11190  ax-distr 11191  ax-i2m1 11192  ax-1ne0 11193  ax-1rid 11194  ax-rnegex 11195  ax-rrecex 11196  ax-cnre 11197  ax-pre-lttri 11198  ax-pre-lttrn 11199  ax-pre-ltadd 11200  ax-pre-mulgt0 11201

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7863  df-1st 7986  df-2nd 7987  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-1o 8455  df-er 8696  df-map 8828  df-en 8953  df-dom 8954  df-sdom 8955  df-fin 8956  df-card 9944  df-pnf 11269  df-mnf 11270  df-xr 11271  df-ltxr 11272  df-le 11273  df-sub 11467  df-neg 11468  df-nn 12258  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12529  df-z 12616  df-uz 12888  df-fz 13562  df-fzo 13710  df-seq 14066  df-hash 14395  df-word 14579  df-concat 14636  df-s1 14663  df-struct 17239  df-sets 17256  df-slot 17274  df-ndx 17286  df-base 17302  df-ress 17323  df-plusg 17355  df-tset 17361  df-0g 17526  df-gsum 17527  df-mgm 18730  df-sgrp 18821  df-mnd 18837  df-submnd 18892  df-efmnd 18978  df-grp 19060  df-symg 19497

This theorem is used by:  gsmsymgreq  19559