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Theorem gsmsymgreqlem2 19292
Description: Lemma 2 for gsmsymgreq 19293. (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 = (𝑁𝑀)
Assertion
Ref Expression
gsmsymgreqlem2 (((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (♯‘𝑋) = (♯‘𝑌))) → ((∀𝑖 ∈ (0..^(♯‘𝑋))∀𝑛𝐼 ((𝑋𝑖)‘𝑛) = ((𝑌𝑖)‘𝑛) → ∀𝑛𝐼 ((𝑆 Σg 𝑋)‘𝑛) = ((𝑍 Σg 𝑌)‘𝑛)) → (∀𝑖 ∈ (0..^(♯‘(𝑋 ++ ⟨“𝐶”⟩)))∀𝑛𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) → ∀𝑛𝐼 ((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.
StepHypRef Expression
1 ccatws1len 14566 . . . . . . . . . 10 (𝑋 ∈ Word 𝐵 → (♯‘(𝑋 ++ ⟨“𝐶”⟩)) = ((♯‘𝑋) + 1))
21oveq2d 7420 . . . . . . . . 9 (𝑋 ∈ Word 𝐵 → (0..^(♯‘(𝑋 ++ ⟨“𝐶”⟩))) = (0..^((♯‘𝑋) + 1)))
3 lencl 14479 . . . . . . . . . . 11 (𝑋 ∈ Word 𝐵 → (♯‘𝑋) ∈ ℕ0)
4 elnn0uz 12863 . . . . . . . . . . 11 ((♯‘𝑋) ∈ ℕ0 ↔ (♯‘𝑋) ∈ (ℤ‘0))
53, 4sylib 217 . . . . . . . . . 10 (𝑋 ∈ Word 𝐵 → (♯‘𝑋) ∈ (ℤ‘0))
6 fzosplitsn 13736 . . . . . . . . . 10 ((♯‘𝑋) ∈ (ℤ‘0) → (0..^((♯‘𝑋) + 1)) = ((0..^(♯‘𝑋)) ∪ {(♯‘𝑋)}))
75, 6syl 17 . . . . . . . . 9 (𝑋 ∈ Word 𝐵 → (0..^((♯‘𝑋) + 1)) = ((0..^(♯‘𝑋)) ∪ {(♯‘𝑋)}))
82, 7eqtrd 2773 . . . . . . . 8 (𝑋 ∈ Word 𝐵 → (0..^(♯‘(𝑋 ++ ⟨“𝐶”⟩))) = ((0..^(♯‘𝑋)) ∪ {(♯‘𝑋)}))
98adantr 482 . . . . . . 7 ((𝑋 ∈ Word 𝐵𝐶𝐵) → (0..^(♯‘(𝑋 ++ ⟨“𝐶”⟩))) = ((0..^(♯‘𝑋)) ∪ {(♯‘𝑋)}))
1093ad2ant1 1134 . . . . . 6 (((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) → (0..^(♯‘(𝑋 ++ ⟨“𝐶”⟩))) = ((0..^(♯‘𝑋)) ∪ {(♯‘𝑋)}))
1110raleqdv 3326 . . . . 5 (((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) → (∀𝑖 ∈ (0..^(♯‘(𝑋 ++ ⟨“𝐶”⟩)))∀𝑛𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) ↔ ∀𝑖 ∈ ((0..^(♯‘𝑋)) ∪ {(♯‘𝑋)})∀𝑛𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛)))
123adantr 482 . . . . . . 7 ((𝑋 ∈ Word 𝐵𝐶𝐵) → (♯‘𝑋) ∈ ℕ0)
13123ad2ant1 1134 . . . . . 6 (((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) → (♯‘𝑋) ∈ ℕ0)
14 fveq2 6888 . . . . . . . . . 10 (𝑖 = (♯‘𝑋) → ((𝑋 ++ ⟨“𝐶”⟩)‘𝑖) = ((𝑋 ++ ⟨“𝐶”⟩)‘(♯‘𝑋)))
1514fveq1d 6890 . . . . . . . . 9 (𝑖 = (♯‘𝑋) → (((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = (((𝑋 ++ ⟨“𝐶”⟩)‘(♯‘𝑋))‘𝑛))
16 fveq2 6888 . . . . . . . . . 10 (𝑖 = (♯‘𝑋) → ((𝑌 ++ ⟨“𝑅”⟩)‘𝑖) = ((𝑌 ++ ⟨“𝑅”⟩)‘(♯‘𝑋)))
1716fveq1d 6890 . . . . . . . . 9 (𝑖 = (♯‘𝑋) → (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘(♯‘𝑋))‘𝑛))
1815, 17eqeq12d 2749 . . . . . . . 8 (𝑖 = (♯‘𝑋) → ((((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) ↔ (((𝑋 ++ ⟨“𝐶”⟩)‘(♯‘𝑋))‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘(♯‘𝑋))‘𝑛)))
1918ralbidv 3178 . . . . . . 7 (𝑖 = (♯‘𝑋) → (∀𝑛𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) ↔ ∀𝑛𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘(♯‘𝑋))‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘(♯‘𝑋))‘𝑛)))
2019ralunsn 4893 . . . . . 6 ((♯‘𝑋) ∈ ℕ0 → (∀𝑖 ∈ ((0..^(♯‘𝑋)) ∪ {(♯‘𝑋)})∀𝑛𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) ↔ (∀𝑖 ∈ (0..^(♯‘𝑋))∀𝑛𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) ∧ ∀𝑛𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘(♯‘𝑋))‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘(♯‘𝑋))‘𝑛))))
2113, 20syl 17 . . . . 5 (((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) → (∀𝑖 ∈ ((0..^(♯‘𝑋)) ∪ {(♯‘𝑋)})∀𝑛𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) ↔ (∀𝑖 ∈ (0..^(♯‘𝑋))∀𝑛𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) ∧ ∀𝑛𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘(♯‘𝑋))‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘(♯‘𝑋))‘𝑛))))
22 simp1l 1198 . . . . . . . . . . 11 (((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) → 𝑋 ∈ Word 𝐵)
23 ccats1val1 14572 . . . . . . . . . . 11 ((𝑋 ∈ Word 𝐵𝑖 ∈ (0..^(♯‘𝑋))) → ((𝑋 ++ ⟨“𝐶”⟩)‘𝑖) = (𝑋𝑖))
2422, 23sylan 581 . . . . . . . . . 10 ((((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) ∧ 𝑖 ∈ (0..^(♯‘𝑋))) → ((𝑋 ++ ⟨“𝐶”⟩)‘𝑖) = (𝑋𝑖))
2524fveq1d 6890 . . . . . . . . 9 ((((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) ∧ 𝑖 ∈ (0..^(♯‘𝑋))) → (((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = ((𝑋𝑖)‘𝑛))
26 simp2l 1200 . . . . . . . . . . 11 (((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) → 𝑌 ∈ Word 𝑃)
27 oveq2 7412 . . . . . . . . . . . . . . 15 ((♯‘𝑋) = (♯‘𝑌) → (0..^(♯‘𝑋)) = (0..^(♯‘𝑌)))
2827eleq2d 2820 . . . . . . . . . . . . . 14 ((♯‘𝑋) = (♯‘𝑌) → (𝑖 ∈ (0..^(♯‘𝑋)) ↔ 𝑖 ∈ (0..^(♯‘𝑌))))
2928biimpd 228 . . . . . . . . . . . . 13 ((♯‘𝑋) = (♯‘𝑌) → (𝑖 ∈ (0..^(♯‘𝑋)) → 𝑖 ∈ (0..^(♯‘𝑌))))
30293ad2ant3 1136 . . . . . . . . . . . 12 (((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) → (𝑖 ∈ (0..^(♯‘𝑋)) → 𝑖 ∈ (0..^(♯‘𝑌))))
3130imp 408 . . . . . . . . . . 11 ((((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) ∧ 𝑖 ∈ (0..^(♯‘𝑋))) → 𝑖 ∈ (0..^(♯‘𝑌)))
32 ccats1val1 14572 . . . . . . . . . . 11 ((𝑌 ∈ Word 𝑃𝑖 ∈ (0..^(♯‘𝑌))) → ((𝑌 ++ ⟨“𝑅”⟩)‘𝑖) = (𝑌𝑖))
3326, 31, 32syl2an2r 684 . . . . . . . . . 10 ((((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) ∧ 𝑖 ∈ (0..^(♯‘𝑋))) → ((𝑌 ++ ⟨“𝑅”⟩)‘𝑖) = (𝑌𝑖))
3433fveq1d 6890 . . . . . . . . 9 ((((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) ∧ 𝑖 ∈ (0..^(♯‘𝑋))) → (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) = ((𝑌𝑖)‘𝑛))
3525, 34eqeq12d 2749 . . . . . . . 8 ((((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) ∧ 𝑖 ∈ (0..^(♯‘𝑋))) → ((((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) ↔ ((𝑋𝑖)‘𝑛) = ((𝑌𝑖)‘𝑛)))
3635ralbidv 3178 . . . . . . 7 ((((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) ∧ 𝑖 ∈ (0..^(♯‘𝑋))) → (∀𝑛𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) ↔ ∀𝑛𝐼 ((𝑋𝑖)‘𝑛) = ((𝑌𝑖)‘𝑛)))
3736ralbidva 3176 . . . . . 6 (((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) → (∀𝑖 ∈ (0..^(♯‘𝑋))∀𝑛𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) ↔ ∀𝑖 ∈ (0..^(♯‘𝑋))∀𝑛𝐼 ((𝑋𝑖)‘𝑛) = ((𝑌𝑖)‘𝑛)))
38 eqidd 2734 . . . . . . . . . 10 ((𝑋 ∈ Word 𝐵𝐶𝐵) → (♯‘𝑋) = (♯‘𝑋))
39 ccats1val2 14573 . . . . . . . . . . 11 ((𝑋 ∈ Word 𝐵𝐶𝐵 ∧ (♯‘𝑋) = (♯‘𝑋)) → ((𝑋 ++ ⟨“𝐶”⟩)‘(♯‘𝑋)) = 𝐶)
4039fveq1d 6890 . . . . . . . . . 10 ((𝑋 ∈ Word 𝐵𝐶𝐵 ∧ (♯‘𝑋) = (♯‘𝑋)) → (((𝑋 ++ ⟨“𝐶”⟩)‘(♯‘𝑋))‘𝑛) = (𝐶𝑛))
4138, 40mpd3an3 1463 . . . . . . . . 9 ((𝑋 ∈ Word 𝐵𝐶𝐵) → (((𝑋 ++ ⟨“𝐶”⟩)‘(♯‘𝑋))‘𝑛) = (𝐶𝑛))
42413ad2ant1 1134 . . . . . . . 8 (((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) → (((𝑋 ++ ⟨“𝐶”⟩)‘(♯‘𝑋))‘𝑛) = (𝐶𝑛))
43 ccats1val2 14573 . . . . . . . . . . 11 ((𝑌 ∈ Word 𝑃𝑅𝑃 ∧ (♯‘𝑋) = (♯‘𝑌)) → ((𝑌 ++ ⟨“𝑅”⟩)‘(♯‘𝑋)) = 𝑅)
4443fveq1d 6890 . . . . . . . . . 10 ((𝑌 ∈ Word 𝑃𝑅𝑃 ∧ (♯‘𝑋) = (♯‘𝑌)) → (((𝑌 ++ ⟨“𝑅”⟩)‘(♯‘𝑋))‘𝑛) = (𝑅𝑛))
45443expa 1119 . . . . . . . . 9 (((𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) → (((𝑌 ++ ⟨“𝑅”⟩)‘(♯‘𝑋))‘𝑛) = (𝑅𝑛))
46453adant1 1131 . . . . . . . 8 (((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) → (((𝑌 ++ ⟨“𝑅”⟩)‘(♯‘𝑋))‘𝑛) = (𝑅𝑛))
4742, 46eqeq12d 2749 . . . . . . 7 (((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) → ((((𝑋 ++ ⟨“𝐶”⟩)‘(♯‘𝑋))‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘(♯‘𝑋))‘𝑛) ↔ (𝐶𝑛) = (𝑅𝑛)))
4847ralbidv 3178 . . . . . 6 (((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) → (∀𝑛𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘(♯‘𝑋))‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘(♯‘𝑋))‘𝑛) ↔ ∀𝑛𝐼 (𝐶𝑛) = (𝑅𝑛)))
4937, 48anbi12d 632 . . . . 5 (((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) → ((∀𝑖 ∈ (0..^(♯‘𝑋))∀𝑛𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) ∧ ∀𝑛𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘(♯‘𝑋))‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘(♯‘𝑋))‘𝑛)) ↔ (∀𝑖 ∈ (0..^(♯‘𝑋))∀𝑛𝐼 ((𝑋𝑖)‘𝑛) = ((𝑌𝑖)‘𝑛) ∧ ∀𝑛𝐼 (𝐶𝑛) = (𝑅𝑛))))
5011, 21, 493bitrd 305 . . . 4 (((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) → (∀𝑖 ∈ (0..^(♯‘(𝑋 ++ ⟨“𝐶”⟩)))∀𝑛𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) ↔ (∀𝑖 ∈ (0..^(♯‘𝑋))∀𝑛𝐼 ((𝑋𝑖)‘𝑛) = ((𝑌𝑖)‘𝑛) ∧ ∀𝑛𝐼 (𝐶𝑛) = (𝑅𝑛))))
5150ad2antlr 726 . . 3 ((((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (♯‘𝑋) = (♯‘𝑌))) ∧ (∀𝑖 ∈ (0..^(♯‘𝑋))∀𝑛𝐼 ((𝑋𝑖)‘𝑛) = ((𝑌𝑖)‘𝑛) → ∀𝑛𝐼 ((𝑆 Σg 𝑋)‘𝑛) = ((𝑍 Σg 𝑌)‘𝑛))) → (∀𝑖 ∈ (0..^(♯‘(𝑋 ++ ⟨“𝐶”⟩)))∀𝑛𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) ↔ (∀𝑖 ∈ (0..^(♯‘𝑋))∀𝑛𝐼 ((𝑋𝑖)‘𝑛) = ((𝑌𝑖)‘𝑛) ∧ ∀𝑛𝐼 (𝐶𝑛) = (𝑅𝑛))))
52 pm3.35 802 . . . . . . 7 ((∀𝑖 ∈ (0..^(♯‘𝑋))∀𝑛𝐼 ((𝑋𝑖)‘𝑛) = ((𝑌𝑖)‘𝑛) ∧ (∀𝑖 ∈ (0..^(♯‘𝑋))∀𝑛𝐼 ((𝑋𝑖)‘𝑛) = ((𝑌𝑖)‘𝑛) → ∀𝑛𝐼 ((𝑆 Σg 𝑋)‘𝑛) = ((𝑍 Σg 𝑌)‘𝑛))) → ∀𝑛𝐼 ((𝑆 Σg 𝑋)‘𝑛) = ((𝑍 Σg 𝑌)‘𝑛))
53 fveq2 6888 . . . . . . . . . . 11 (𝑛 = 𝑗 → ((𝑆 Σg 𝑋)‘𝑛) = ((𝑆 Σg 𝑋)‘𝑗))
54 fveq2 6888 . . . . . . . . . . 11 (𝑛 = 𝑗 → ((𝑍 Σg 𝑌)‘𝑛) = ((𝑍 Σg 𝑌)‘𝑗))
5553, 54eqeq12d 2749 . . . . . . . . . 10 (𝑛 = 𝑗 → (((𝑆 Σg 𝑋)‘𝑛) = ((𝑍 Σg 𝑌)‘𝑛) ↔ ((𝑆 Σg 𝑋)‘𝑗) = ((𝑍 Σg 𝑌)‘𝑗)))
5655cbvralvw 3235 . . . . . . . . 9 (∀𝑛𝐼 ((𝑆 Σg 𝑋)‘𝑛) = ((𝑍 Σg 𝑌)‘𝑛) ↔ ∀𝑗𝐼 ((𝑆 Σg 𝑋)‘𝑗) = ((𝑍 Σg 𝑌)‘𝑗))
57 simp-4l 782 . . . . . . . . . . . . . . 15 (((((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (♯‘𝑋) = (♯‘𝑌))) ∧ ∀𝑗𝐼 ((𝑆 Σg 𝑋)‘𝑗) = ((𝑍 Σg 𝑌)‘𝑗)) ∧ 𝑛𝐼) → 𝑁 ∈ Fin)
58 simp-4r 783 . . . . . . . . . . . . . . 15 (((((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (♯‘𝑋) = (♯‘𝑌))) ∧ ∀𝑗𝐼 ((𝑆 Σg 𝑋)‘𝑗) = ((𝑍 Σg 𝑌)‘𝑗)) ∧ 𝑛𝐼) → 𝑀 ∈ Fin)
59 simpr 486 . . . . . . . . . . . . . . 15 (((((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (♯‘𝑋) = (♯‘𝑌))) ∧ ∀𝑗𝐼 ((𝑆 Σg 𝑋)‘𝑗) = ((𝑍 Σg 𝑌)‘𝑗)) ∧ 𝑛𝐼) → 𝑛𝐼)
6057, 58, 593jca 1129 . . . . . . . . . . . . . 14 (((((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (♯‘𝑋) = (♯‘𝑌))) ∧ ∀𝑗𝐼 ((𝑆 Σg 𝑋)‘𝑗) = ((𝑍 Σg 𝑌)‘𝑗)) ∧ 𝑛𝐼) → (𝑁 ∈ Fin ∧ 𝑀 ∈ Fin ∧ 𝑛𝐼))
6160adantr 482 . . . . . . . . . . . . 13 ((((((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (♯‘𝑋) = (♯‘𝑌))) ∧ ∀𝑗𝐼 ((𝑆 Σg 𝑋)‘𝑗) = ((𝑍 Σg 𝑌)‘𝑗)) ∧ 𝑛𝐼) ∧ (𝐶𝑛) = (𝑅𝑛)) → (𝑁 ∈ Fin ∧ 𝑀 ∈ Fin ∧ 𝑛𝐼))
62 simp-4r 783 . . . . . . . . . . . . 13 ((((((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (♯‘𝑋) = (♯‘𝑌))) ∧ ∀𝑗𝐼 ((𝑆 Σg 𝑋)‘𝑗) = ((𝑍 Σg 𝑌)‘𝑗)) ∧ 𝑛𝐼) ∧ (𝐶𝑛) = (𝑅𝑛)) → ((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)))
63 simplr 768 . . . . . . . . . . . . . 14 (((((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (♯‘𝑋) = (♯‘𝑌))) ∧ ∀𝑗𝐼 ((𝑆 Σg 𝑋)‘𝑗) = ((𝑍 Σg 𝑌)‘𝑗)) ∧ 𝑛𝐼) → ∀𝑗𝐼 ((𝑆 Σg 𝑋)‘𝑗) = ((𝑍 Σg 𝑌)‘𝑗))
6463anim1i 616 . . . . . . . . . . . . 13 ((((((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (♯‘𝑋) = (♯‘𝑌))) ∧ ∀𝑗𝐼 ((𝑆 Σg 𝑋)‘𝑗) = ((𝑍 Σg 𝑌)‘𝑗)) ∧ 𝑛𝐼) ∧ (𝐶𝑛) = (𝑅𝑛)) → (∀𝑗𝐼 ((𝑆 Σg 𝑋)‘𝑗) = ((𝑍 Σg 𝑌)‘𝑗) ∧ (𝐶𝑛) = (𝑅𝑛)))
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 = (𝑁𝑀)
7065, 66, 67, 68, 69gsmsymgreqlem1 19291 . . . . . . . . . . . . . 14 (((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin ∧ 𝑛𝐼) ∧ ((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (♯‘𝑋) = (♯‘𝑌))) → ((∀𝑗𝐼 ((𝑆 Σg 𝑋)‘𝑗) = ((𝑍 Σg 𝑌)‘𝑗) ∧ (𝐶𝑛) = (𝑅𝑛)) → ((𝑆 Σg (𝑋 ++ ⟨“𝐶”⟩))‘𝑛) = ((𝑍 Σg (𝑌 ++ ⟨“𝑅”⟩))‘𝑛)))
7170imp 408 . . . . . . . . . . . . 13 ((((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin ∧ 𝑛𝐼) ∧ ((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (♯‘𝑋) = (♯‘𝑌))) ∧ (∀𝑗𝐼 ((𝑆 Σg 𝑋)‘𝑗) = ((𝑍 Σg 𝑌)‘𝑗) ∧ (𝐶𝑛) = (𝑅𝑛))) → ((𝑆 Σg (𝑋 ++ ⟨“𝐶”⟩))‘𝑛) = ((𝑍 Σg (𝑌 ++ ⟨“𝑅”⟩))‘𝑛))
7261, 62, 64, 71syl21anc 837 . . . . . . . . . . . 12 ((((((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (♯‘𝑋) = (♯‘𝑌))) ∧ ∀𝑗𝐼 ((𝑆 Σg 𝑋)‘𝑗) = ((𝑍 Σg 𝑌)‘𝑗)) ∧ 𝑛𝐼) ∧ (𝐶𝑛) = (𝑅𝑛)) → ((𝑆 Σg (𝑋 ++ ⟨“𝐶”⟩))‘𝑛) = ((𝑍 Σg (𝑌 ++ ⟨“𝑅”⟩))‘𝑛))
7372ex 414 . . . . . . . . . . 11 (((((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (♯‘𝑋) = (♯‘𝑌))) ∧ ∀𝑗𝐼 ((𝑆 Σg 𝑋)‘𝑗) = ((𝑍 Σg 𝑌)‘𝑗)) ∧ 𝑛𝐼) → ((𝐶𝑛) = (𝑅𝑛) → ((𝑆 Σg (𝑋 ++ ⟨“𝐶”⟩))‘𝑛) = ((𝑍 Σg (𝑌 ++ ⟨“𝑅”⟩))‘𝑛)))
7473ralimdva 3168 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (♯‘𝑋) = (♯‘𝑌))) ∧ ∀𝑗𝐼 ((𝑆 Σg 𝑋)‘𝑗) = ((𝑍 Σg 𝑌)‘𝑗)) → (∀𝑛𝐼 (𝐶𝑛) = (𝑅𝑛) → ∀𝑛𝐼 ((𝑆 Σg (𝑋 ++ ⟨“𝐶”⟩))‘𝑛) = ((𝑍 Σg (𝑌 ++ ⟨“𝑅”⟩))‘𝑛)))
7574expcom 415 . . . . . . . . 9 (∀𝑗𝐼 ((𝑆 Σg 𝑋)‘𝑗) = ((𝑍 Σg 𝑌)‘𝑗) → (((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (♯‘𝑋) = (♯‘𝑌))) → (∀𝑛𝐼 (𝐶𝑛) = (𝑅𝑛) → ∀𝑛𝐼 ((𝑆 Σg (𝑋 ++ ⟨“𝐶”⟩))‘𝑛) = ((𝑍 Σg (𝑌 ++ ⟨“𝑅”⟩))‘𝑛))))
7656, 75sylbi 216 . . . . . . . 8 (∀𝑛𝐼 ((𝑆 Σg 𝑋)‘𝑛) = ((𝑍 Σg 𝑌)‘𝑛) → (((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (♯‘𝑋) = (♯‘𝑌))) → (∀𝑛𝐼 (𝐶𝑛) = (𝑅𝑛) → ∀𝑛𝐼 ((𝑆 Σg (𝑋 ++ ⟨“𝐶”⟩))‘𝑛) = ((𝑍 Σg (𝑌 ++ ⟨“𝑅”⟩))‘𝑛))))
7776com23 86 . . . . . . 7 (∀𝑛𝐼 ((𝑆 Σg 𝑋)‘𝑛) = ((𝑍 Σg 𝑌)‘𝑛) → (∀𝑛𝐼 (𝐶𝑛) = (𝑅𝑛) → (((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (♯‘𝑋) = (♯‘𝑌))) → ∀𝑛𝐼 ((𝑆 Σg (𝑋 ++ ⟨“𝐶”⟩))‘𝑛) = ((𝑍 Σg (𝑌 ++ ⟨“𝑅”⟩))‘𝑛))))
7852, 77syl 17 . . . . . 6 ((∀𝑖 ∈ (0..^(♯‘𝑋))∀𝑛𝐼 ((𝑋𝑖)‘𝑛) = ((𝑌𝑖)‘𝑛) ∧ (∀𝑖 ∈ (0..^(♯‘𝑋))∀𝑛𝐼 ((𝑋𝑖)‘𝑛) = ((𝑌𝑖)‘𝑛) → ∀𝑛𝐼 ((𝑆 Σg 𝑋)‘𝑛) = ((𝑍 Σg 𝑌)‘𝑛))) → (∀𝑛𝐼 (𝐶𝑛) = (𝑅𝑛) → (((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (♯‘𝑋) = (♯‘𝑌))) → ∀𝑛𝐼 ((𝑆 Σg (𝑋 ++ ⟨“𝐶”⟩))‘𝑛) = ((𝑍 Σg (𝑌 ++ ⟨“𝑅”⟩))‘𝑛))))
7978impancom 453 . . . . 5 ((∀𝑖 ∈ (0..^(♯‘𝑋))∀𝑛𝐼 ((𝑋𝑖)‘𝑛) = ((𝑌𝑖)‘𝑛) ∧ ∀𝑛𝐼 (𝐶𝑛) = (𝑅𝑛)) → ((∀𝑖 ∈ (0..^(♯‘𝑋))∀𝑛𝐼 ((𝑋𝑖)‘𝑛) = ((𝑌𝑖)‘𝑛) → ∀𝑛𝐼 ((𝑆 Σg 𝑋)‘𝑛) = ((𝑍 Σg 𝑌)‘𝑛)) → (((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (♯‘𝑋) = (♯‘𝑌))) → ∀𝑛𝐼 ((𝑆 Σg (𝑋 ++ ⟨“𝐶”⟩))‘𝑛) = ((𝑍 Σg (𝑌 ++ ⟨“𝑅”⟩))‘𝑛))))
8079com13 88 . . . 4 (((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (♯‘𝑋) = (♯‘𝑌))) → ((∀𝑖 ∈ (0..^(♯‘𝑋))∀𝑛𝐼 ((𝑋𝑖)‘𝑛) = ((𝑌𝑖)‘𝑛) → ∀𝑛𝐼 ((𝑆 Σg 𝑋)‘𝑛) = ((𝑍 Σg 𝑌)‘𝑛)) → ((∀𝑖 ∈ (0..^(♯‘𝑋))∀𝑛𝐼 ((𝑋𝑖)‘𝑛) = ((𝑌𝑖)‘𝑛) ∧ ∀𝑛𝐼 (𝐶𝑛) = (𝑅𝑛)) → ∀𝑛𝐼 ((𝑆 Σg (𝑋 ++ ⟨“𝐶”⟩))‘𝑛) = ((𝑍 Σg (𝑌 ++ ⟨“𝑅”⟩))‘𝑛))))
8180imp 408 . . 3 ((((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (♯‘𝑋) = (♯‘𝑌))) ∧ (∀𝑖 ∈ (0..^(♯‘𝑋))∀𝑛𝐼 ((𝑋𝑖)‘𝑛) = ((𝑌𝑖)‘𝑛) → ∀𝑛𝐼 ((𝑆 Σg 𝑋)‘𝑛) = ((𝑍 Σg 𝑌)‘𝑛))) → ((∀𝑖 ∈ (0..^(♯‘𝑋))∀𝑛𝐼 ((𝑋𝑖)‘𝑛) = ((𝑌𝑖)‘𝑛) ∧ ∀𝑛𝐼 (𝐶𝑛) = (𝑅𝑛)) → ∀𝑛𝐼 ((𝑆 Σg (𝑋 ++ ⟨“𝐶”⟩))‘𝑛) = ((𝑍 Σg (𝑌 ++ ⟨“𝑅”⟩))‘𝑛)))
8251, 81sylbid 239 . 2 ((((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (♯‘𝑋) = (♯‘𝑌))) ∧ (∀𝑖 ∈ (0..^(♯‘𝑋))∀𝑛𝐼 ((𝑋𝑖)‘𝑛) = ((𝑌𝑖)‘𝑛) → ∀𝑛𝐼 ((𝑆 Σg 𝑋)‘𝑛) = ((𝑍 Σg 𝑌)‘𝑛))) → (∀𝑖 ∈ (0..^(♯‘(𝑋 ++ ⟨“𝐶”⟩)))∀𝑛𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) → ∀𝑛𝐼 ((𝑆 Σg (𝑋 ++ ⟨“𝐶”⟩))‘𝑛) = ((𝑍 Σg (𝑌 ++ ⟨“𝑅”⟩))‘𝑛)))
8382ex 414 1 (((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (♯‘𝑋) = (♯‘𝑌))) → ((∀𝑖 ∈ (0..^(♯‘𝑋))∀𝑛𝐼 ((𝑋𝑖)‘𝑛) = ((𝑌𝑖)‘𝑛) → ∀𝑛𝐼 ((𝑆 Σg 𝑋)‘𝑛) = ((𝑍 Σg 𝑌)‘𝑛)) → (∀𝑖 ∈ (0..^(♯‘(𝑋 ++ ⟨“𝐶”⟩)))∀𝑛𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) → ∀𝑛𝐼 ((𝑆 Σg (𝑋 ++ ⟨“𝐶”⟩))‘𝑛) = ((𝑍 Σg (𝑌 ++ ⟨“𝑅”⟩))‘𝑛))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  wral 3062  cun 3945  cin 3946  {csn 4627  cfv 6540  (class class class)co 7404  Fincfn 8935  0cc0 11106  1c1 11107   + caddc 11109  0cn0 12468  cuz 12818  ..^cfzo 13623  chash 14286  Word cword 14460   ++ cconcat 14516  ⟨“cs1 14541  Basecbs 17140   Σg cgsu 17382  SymGrpcsymg 19227
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7720  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-tp 4632  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-om 7851  df-1st 7970  df-2nd 7971  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-1o 8461  df-er 8699  df-map 8818  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-card 9930  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-2 12271  df-3 12272  df-4 12273  df-5 12274  df-6 12275  df-7 12276  df-8 12277  df-9 12278  df-n0 12469  df-z 12555  df-uz 12819  df-fz 13481  df-fzo 13624  df-seq 13963  df-hash 14287  df-word 14461  df-concat 14517  df-s1 14542  df-struct 17076  df-sets 17093  df-slot 17111  df-ndx 17123  df-base 17141  df-ress 17170  df-plusg 17206  df-tset 17212  df-0g 17383  df-gsum 17384  df-mgm 18557  df-sgrp 18606  df-mnd 18622  df-submnd 18668  df-efmnd 18746  df-grp 18818  df-symg 19228
This theorem is referenced by:  gsmsymgreq  19293
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