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Theorem gsmsymgreqlem2 17972
Description: Lemma 2 for gsmsymgreq 17973. (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 13512 . . . . . . . . . 10 (𝑋 ∈ Word 𝐵 → (♯‘(𝑋 ++ ⟨“𝐶”⟩)) = ((♯‘𝑋) + 1))
21adantr 472 . . . . . . . . 9 ((𝑋 ∈ Word 𝐵𝐶𝐵) → (♯‘(𝑋 ++ ⟨“𝐶”⟩)) = ((♯‘𝑋) + 1))
323ad2ant1 1125 . . . . . . . 8 (((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) → (♯‘(𝑋 ++ ⟨“𝐶”⟩)) = ((♯‘𝑋) + 1))
43oveq2d 6781 . . . . . . 7 (((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) → (0..^(♯‘(𝑋 ++ ⟨“𝐶”⟩))) = (0..^((♯‘𝑋) + 1)))
5 lencl 13431 . . . . . . . . . . 11 (𝑋 ∈ Word 𝐵 → (♯‘𝑋) ∈ ℕ0)
6 elnn0uz 11839 . . . . . . . . . . 11 ((♯‘𝑋) ∈ ℕ0 ↔ (♯‘𝑋) ∈ (ℤ‘0))
75, 6sylib 208 . . . . . . . . . 10 (𝑋 ∈ Word 𝐵 → (♯‘𝑋) ∈ (ℤ‘0))
87adantr 472 . . . . . . . . 9 ((𝑋 ∈ Word 𝐵𝐶𝐵) → (♯‘𝑋) ∈ (ℤ‘0))
9 fzosplitsn 12691 . . . . . . . . 9 ((♯‘𝑋) ∈ (ℤ‘0) → (0..^((♯‘𝑋) + 1)) = ((0..^(♯‘𝑋)) ∪ {(♯‘𝑋)}))
108, 9syl 17 . . . . . . . 8 ((𝑋 ∈ Word 𝐵𝐶𝐵) → (0..^((♯‘𝑋) + 1)) = ((0..^(♯‘𝑋)) ∪ {(♯‘𝑋)}))
11103ad2ant1 1125 . . . . . . 7 (((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) → (0..^((♯‘𝑋) + 1)) = ((0..^(♯‘𝑋)) ∪ {(♯‘𝑋)}))
124, 11eqtrd 2758 . . . . . 6 (((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) → (0..^(♯‘(𝑋 ++ ⟨“𝐶”⟩))) = ((0..^(♯‘𝑋)) ∪ {(♯‘𝑋)}))
1312raleqdv 3247 . . . . 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)))
145adantr 472 . . . . . . 7 ((𝑋 ∈ Word 𝐵𝐶𝐵) → (♯‘𝑋) ∈ ℕ0)
15143ad2ant1 1125 . . . . . 6 (((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) → (♯‘𝑋) ∈ ℕ0)
16 fveq2 6304 . . . . . . . . . 10 (𝑖 = (♯‘𝑋) → ((𝑋 ++ ⟨“𝐶”⟩)‘𝑖) = ((𝑋 ++ ⟨“𝐶”⟩)‘(♯‘𝑋)))
1716fveq1d 6306 . . . . . . . . 9 (𝑖 = (♯‘𝑋) → (((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = (((𝑋 ++ ⟨“𝐶”⟩)‘(♯‘𝑋))‘𝑛))
18 fveq2 6304 . . . . . . . . . 10 (𝑖 = (♯‘𝑋) → ((𝑌 ++ ⟨“𝑅”⟩)‘𝑖) = ((𝑌 ++ ⟨“𝑅”⟩)‘(♯‘𝑋)))
1918fveq1d 6306 . . . . . . . . 9 (𝑖 = (♯‘𝑋) → (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘(♯‘𝑋))‘𝑛))
2017, 19eqeq12d 2739 . . . . . . . 8 (𝑖 = (♯‘𝑋) → ((((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) ↔ (((𝑋 ++ ⟨“𝐶”⟩)‘(♯‘𝑋))‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘(♯‘𝑋))‘𝑛)))
2120ralbidv 3088 . . . . . . 7 (𝑖 = (♯‘𝑋) → (∀𝑛𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) ↔ ∀𝑛𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘(♯‘𝑋))‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘(♯‘𝑋))‘𝑛)))
2221ralunsn 4530 . . . . . 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))))
2315, 22syl 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))))
24 simpl 474 . . . . . . . . . . . . 13 ((𝑋 ∈ Word 𝐵𝐶𝐵) → 𝑋 ∈ Word 𝐵)
25243ad2ant1 1125 . . . . . . . . . . . 12 (((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) → 𝑋 ∈ Word 𝐵)
2625adantr 472 . . . . . . . . . . 11 ((((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) ∧ 𝑖 ∈ (0..^(♯‘𝑋))) → 𝑋 ∈ Word 𝐵)
27 simpr 479 . . . . . . . . . . . . 13 ((𝑋 ∈ Word 𝐵𝐶𝐵) → 𝐶𝐵)
28273ad2ant1 1125 . . . . . . . . . . . 12 (((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) → 𝐶𝐵)
2928adantr 472 . . . . . . . . . . 11 ((((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) ∧ 𝑖 ∈ (0..^(♯‘𝑋))) → 𝐶𝐵)
30 simpr 479 . . . . . . . . . . 11 ((((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) ∧ 𝑖 ∈ (0..^(♯‘𝑋))) → 𝑖 ∈ (0..^(♯‘𝑋)))
31 ccats1val1 13521 . . . . . . . . . . 11 ((𝑋 ∈ Word 𝐵𝐶𝐵𝑖 ∈ (0..^(♯‘𝑋))) → ((𝑋 ++ ⟨“𝐶”⟩)‘𝑖) = (𝑋𝑖))
3226, 29, 30, 31syl3anc 1439 . . . . . . . . . 10 ((((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) ∧ 𝑖 ∈ (0..^(♯‘𝑋))) → ((𝑋 ++ ⟨“𝐶”⟩)‘𝑖) = (𝑋𝑖))
3332fveq1d 6306 . . . . . . . . 9 ((((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) ∧ 𝑖 ∈ (0..^(♯‘𝑋))) → (((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = ((𝑋𝑖)‘𝑛))
34 simpl2l 1259 . . . . . . . . . . 11 ((((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) ∧ 𝑖 ∈ (0..^(♯‘𝑋))) → 𝑌 ∈ Word 𝑃)
35 simpl2r 1261 . . . . . . . . . . 11 ((((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) ∧ 𝑖 ∈ (0..^(♯‘𝑋))) → 𝑅𝑃)
36 oveq2 6773 . . . . . . . . . . . . . . 15 ((♯‘𝑋) = (♯‘𝑌) → (0..^(♯‘𝑋)) = (0..^(♯‘𝑌)))
3736eleq2d 2789 . . . . . . . . . . . . . 14 ((♯‘𝑋) = (♯‘𝑌) → (𝑖 ∈ (0..^(♯‘𝑋)) ↔ 𝑖 ∈ (0..^(♯‘𝑌))))
3837biimpd 219 . . . . . . . . . . . . 13 ((♯‘𝑋) = (♯‘𝑌) → (𝑖 ∈ (0..^(♯‘𝑋)) → 𝑖 ∈ (0..^(♯‘𝑌))))
39383ad2ant3 1127 . . . . . . . . . . . 12 (((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) → (𝑖 ∈ (0..^(♯‘𝑋)) → 𝑖 ∈ (0..^(♯‘𝑌))))
4039imp 444 . . . . . . . . . . 11 ((((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) ∧ 𝑖 ∈ (0..^(♯‘𝑋))) → 𝑖 ∈ (0..^(♯‘𝑌)))
41 ccats1val1 13521 . . . . . . . . . . 11 ((𝑌 ∈ Word 𝑃𝑅𝑃𝑖 ∈ (0..^(♯‘𝑌))) → ((𝑌 ++ ⟨“𝑅”⟩)‘𝑖) = (𝑌𝑖))
4234, 35, 40, 41syl3anc 1439 . . . . . . . . . 10 ((((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) ∧ 𝑖 ∈ (0..^(♯‘𝑋))) → ((𝑌 ++ ⟨“𝑅”⟩)‘𝑖) = (𝑌𝑖))
4342fveq1d 6306 . . . . . . . . 9 ((((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) ∧ 𝑖 ∈ (0..^(♯‘𝑋))) → (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) = ((𝑌𝑖)‘𝑛))
4433, 43eqeq12d 2739 . . . . . . . 8 ((((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) ∧ 𝑖 ∈ (0..^(♯‘𝑋))) → ((((𝑋 ++ ⟨“𝐶”⟩)‘𝑖)‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘𝑖)‘𝑛) ↔ ((𝑋𝑖)‘𝑛) = ((𝑌𝑖)‘𝑛)))
4544ralbidv 3088 . . . . . . 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)))
4645ralbidva 3087 . . . . . 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)))
47 eqidd 2725 . . . . . . . . . . . 12 ((𝑋 ∈ Word 𝐵𝐶𝐵) → (♯‘𝑋) = (♯‘𝑋))
4824, 27, 473jca 1379 . . . . . . . . . . 11 ((𝑋 ∈ Word 𝐵𝐶𝐵) → (𝑋 ∈ Word 𝐵𝐶𝐵 ∧ (♯‘𝑋) = (♯‘𝑋)))
49483ad2ant1 1125 . . . . . . . . . 10 (((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) → (𝑋 ∈ Word 𝐵𝐶𝐵 ∧ (♯‘𝑋) = (♯‘𝑋)))
50 ccats1val2 13522 . . . . . . . . . 10 ((𝑋 ∈ Word 𝐵𝐶𝐵 ∧ (♯‘𝑋) = (♯‘𝑋)) → ((𝑋 ++ ⟨“𝐶”⟩)‘(♯‘𝑋)) = 𝐶)
5149, 50syl 17 . . . . . . . . 9 (((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) → ((𝑋 ++ ⟨“𝐶”⟩)‘(♯‘𝑋)) = 𝐶)
5251fveq1d 6306 . . . . . . . 8 (((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) → (((𝑋 ++ ⟨“𝐶”⟩)‘(♯‘𝑋))‘𝑛) = (𝐶𝑛))
53 df-3an 1074 . . . . . . . . . . 11 ((𝑌 ∈ Word 𝑃𝑅𝑃 ∧ (♯‘𝑋) = (♯‘𝑌)) ↔ ((𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)))
5453biimpri 218 . . . . . . . . . 10 (((𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) → (𝑌 ∈ Word 𝑃𝑅𝑃 ∧ (♯‘𝑋) = (♯‘𝑌)))
55543adant1 1122 . . . . . . . . 9 (((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) → (𝑌 ∈ Word 𝑃𝑅𝑃 ∧ (♯‘𝑋) = (♯‘𝑌)))
56 ccats1val2 13522 . . . . . . . . . 10 ((𝑌 ∈ Word 𝑃𝑅𝑃 ∧ (♯‘𝑋) = (♯‘𝑌)) → ((𝑌 ++ ⟨“𝑅”⟩)‘(♯‘𝑋)) = 𝑅)
5756fveq1d 6306 . . . . . . . . 9 ((𝑌 ∈ Word 𝑃𝑅𝑃 ∧ (♯‘𝑋) = (♯‘𝑌)) → (((𝑌 ++ ⟨“𝑅”⟩)‘(♯‘𝑋))‘𝑛) = (𝑅𝑛))
5855, 57syl 17 . . . . . . . 8 (((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) → (((𝑌 ++ ⟨“𝑅”⟩)‘(♯‘𝑋))‘𝑛) = (𝑅𝑛))
5952, 58eqeq12d 2739 . . . . . . 7 (((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) → ((((𝑋 ++ ⟨“𝐶”⟩)‘(♯‘𝑋))‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘(♯‘𝑋))‘𝑛) ↔ (𝐶𝑛) = (𝑅𝑛)))
6059ralbidv 3088 . . . . . 6 (((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (♯‘𝑋) = (♯‘𝑌)) → (∀𝑛𝐼 (((𝑋 ++ ⟨“𝐶”⟩)‘(♯‘𝑋))‘𝑛) = (((𝑌 ++ ⟨“𝑅”⟩)‘(♯‘𝑋))‘𝑛) ↔ ∀𝑛𝐼 (𝐶𝑛) = (𝑅𝑛)))
6146, 60anbi12d 749 . . . . 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))))
6213, 23, 613bitrd 294 . . . 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))))
6362ad2antlr 765 . . 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))))
64 pm3.35 612 . . . . . . 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))
65 fveq2 6304 . . . . . . . . . . 11 (𝑛 = 𝑗 → ((𝑆 Σg 𝑋)‘𝑛) = ((𝑆 Σg 𝑋)‘𝑗))
66 fveq2 6304 . . . . . . . . . . 11 (𝑛 = 𝑗 → ((𝑍 Σg 𝑌)‘𝑛) = ((𝑍 Σg 𝑌)‘𝑗))
6765, 66eqeq12d 2739 . . . . . . . . . 10 (𝑛 = 𝑗 → (((𝑆 Σg 𝑋)‘𝑛) = ((𝑍 Σg 𝑌)‘𝑛) ↔ ((𝑆 Σg 𝑋)‘𝑗) = ((𝑍 Σg 𝑌)‘𝑗)))
6867cbvralv 3274 . . . . . . . . 9 (∀𝑛𝐼 ((𝑆 Σg 𝑋)‘𝑛) = ((𝑍 Σg 𝑌)‘𝑛) ↔ ∀𝑗𝐼 ((𝑆 Σg 𝑋)‘𝑗) = ((𝑍 Σg 𝑌)‘𝑗))
69 simp-4l 825 . . . . . . . . . . . . . . 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)
70 simp-4r 827 . . . . . . . . . . . . . . 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)
71 simpr 479 . . . . . . . . . . . . . . 15 (((((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (♯‘𝑋) = (♯‘𝑌))) ∧ ∀𝑗𝐼 ((𝑆 Σg 𝑋)‘𝑗) = ((𝑍 Σg 𝑌)‘𝑗)) ∧ 𝑛𝐼) → 𝑛𝐼)
7269, 70, 713jca 1379 . . . . . . . . . . . . . 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))
7372adantr 472 . . . . . . . . . . . . 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))
74 simp-4r 827 . . . . . . . . . . . . 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)))
75 simpr 479 . . . . . . . . . . . . . . 15 ((((𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ ((𝑋 ∈ Word 𝐵𝐶𝐵) ∧ (𝑌 ∈ Word 𝑃𝑅𝑃) ∧ (♯‘𝑋) = (♯‘𝑌))) ∧ ∀𝑗𝐼 ((𝑆 Σg 𝑋)‘𝑗) = ((𝑍 Σg 𝑌)‘𝑗)) → ∀𝑗𝐼 ((𝑆 Σg 𝑋)‘𝑗) = ((𝑍 Σg 𝑌)‘𝑗))
7675adantr 472 . . . . . . . . . . . . . 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))
7776anim1i 593 . . . . . . . . . . . . 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)))
78 gsmsymgrfix.s . . . . . . . . . . . . . . 15 𝑆 = (SymGrp‘𝑁)
79 gsmsymgrfix.b . . . . . . . . . . . . . . 15 𝐵 = (Base‘𝑆)
80 gsmsymgreq.z . . . . . . . . . . . . . . 15 𝑍 = (SymGrp‘𝑀)
81 gsmsymgreq.p . . . . . . . . . . . . . . 15 𝑃 = (Base‘𝑍)
82 gsmsymgreq.i . . . . . . . . . . . . . . 15 𝐼 = (𝑁𝑀)
8378, 79, 80, 81, 82gsmsymgreqlem1 17971 . . . . . . . . . . . . . 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)))
8483imp 444 . . . . . . . . . . . . 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))
8573, 74, 77, 84syl21anc 1438 . . . . . . . . . . . 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))
8685ex 449 . . . . . . . . . . 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)))
8786ralimdva 3064 . . . . . . . . . 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)))
8887expcom 450 . . . . . . . . 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))))
8968, 88sylbi 207 . . . . . . . 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))))
9089com23 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))))
9164, 90syl 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))))
9291impancom 455 . . . . 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))))
9392com13 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))))
9493imp 444 . . 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)))
9563, 94sylbid 230 . 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)))
9695ex 449 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 196  wa 383  w3a 1072   = wceq 1596  wcel 2103  wral 3014  cun 3678  cin 3679  {csn 4285  cfv 6001  (class class class)co 6765  Fincfn 8072  0cc0 10049  1c1 10050   + caddc 10052  0cn0 11405  cuz 11800  ..^cfzo 12580  chash 13232  Word cword 13398   ++ cconcat 13400  ⟨“cs1 13401  Basecbs 15980   Σg cgsu 16224  SymGrpcsymg 17918
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066  ax-cnex 10105  ax-resscn 10106  ax-1cn 10107  ax-icn 10108  ax-addcl 10109  ax-addrcl 10110  ax-mulcl 10111  ax-mulrcl 10112  ax-mulcom 10113  ax-addass 10114  ax-mulass 10115  ax-distr 10116  ax-i2m1 10117  ax-1ne0 10118  ax-1rid 10119  ax-rnegex 10120  ax-rrecex 10121  ax-cnre 10122  ax-pre-lttri 10123  ax-pre-lttrn 10124  ax-pre-ltadd 10125  ax-pre-mulgt0 10126
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-nel 3000  df-ral 3019  df-rex 3020  df-reu 3021  df-rmo 3022  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-pss 3696  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-tp 4290  df-op 4292  df-uni 4545  df-int 4584  df-iun 4630  df-br 4761  df-opab 4821  df-mpt 4838  df-tr 4861  df-id 5128  df-eprel 5133  df-po 5139  df-so 5140  df-fr 5177  df-we 5179  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-pred 5793  df-ord 5839  df-on 5840  df-lim 5841  df-suc 5842  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-riota 6726  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-om 7183  df-1st 7285  df-2nd 7286  df-wrecs 7527  df-recs 7588  df-rdg 7626  df-1o 7680  df-oadd 7684  df-er 7862  df-map 7976  df-en 8073  df-dom 8074  df-sdom 8075  df-fin 8076  df-card 8878  df-pnf 10189  df-mnf 10190  df-xr 10191  df-ltxr 10192  df-le 10193  df-sub 10381  df-neg 10382  df-nn 11134  df-2 11192  df-3 11193  df-4 11194  df-5 11195  df-6 11196  df-7 11197  df-8 11198  df-9 11199  df-n0 11406  df-z 11491  df-uz 11801  df-fz 12441  df-fzo 12581  df-seq 12917  df-hash 13233  df-word 13406  df-concat 13408  df-s1 13409  df-struct 15982  df-ndx 15983  df-slot 15984  df-base 15986  df-sets 15987  df-ress 15988  df-plusg 16077  df-tset 16083  df-0g 16225  df-gsum 16226  df-mgm 17364  df-sgrp 17406  df-mnd 17417  df-submnd 17458  df-grp 17547  df-symg 17919
This theorem is referenced by:  gsmsymgreq  17973
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