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Theorem sgrp2nmndlem4 18871
Description: Lemma 4 for sgrp2nmnd 18873: M is a semigroup. (Contributed by AV, 29-Jan-2020.)
Hypotheses
Ref Expression
mgm2nsgrp.s 𝑆 = {𝐴, 𝐵}
mgm2nsgrp.b (Base‘𝑀) = 𝑆
sgrp2nmnd.o (+g‘𝑀) = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ if(𝑥 = 𝐴, 𝐴, 𝐵))
Assertion
Ref Expression
sgrp2nmndlem4 ((♯‘𝑆) = 2 → 𝑀 ∈ Smgrp)
Distinct variable groups:   𝑥,𝑆,𝑦   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝑀
Allowed substitution hint:   𝑀(𝑦)
Proof of Theorem sgrp2nmndlem4
Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mgm2nsgrp.s . . . 4 𝑆 = {𝐴, 𝐵}
21hashprdifel 14381 . . 3 ((♯‘𝑆) = 2 → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵))
3 3simpa 1146 . . 3 ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆))
4 mgm2nsgrp.b . . . 4 (Base‘𝑀) = 𝑆
5 sgrp2nmnd.o . . . 4 (+g‘𝑀) = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ if(𝑥 = 𝐴, 𝐴, 𝐵))
61, 4, 5sgrp2nmndlem1 18866 . . 3 ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → 𝑀 ∈ Mgm)
72, 3, 63syl 18 . 2 ((♯‘𝑆) = 2 → 𝑀 ∈ Mgm)
8 eqid 2727 . . . . . . . . . . 11 (+g‘𝑀) = (+g‘𝑀)
91, 4, 5, 8sgrp2nmndlem2 18867 . . . . . . . . . 10 ((𝐴 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → (𝐴(+g‘𝑀)𝐴) = 𝐴)
109oveq1d 7429 . . . . . . . . 9 ((𝐴 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐴) = (𝐴(+g‘𝑀)𝐴))
119oveq2d 7430 . . . . . . . . 9 ((𝐴 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐴)) = (𝐴(+g‘𝑀)𝐴))
1210, 11eqtr4d 2770 . . . . . . . 8 ((𝐴 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐴) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐴)))
1312anidms 566 . . . . . . 7 (𝐴 ∈ 𝑆 → ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐴) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐴)))
14133ad2ant1 1131 . . . . . 6 ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐴) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐴)))
159anidms 566 . . . . . . . . . 10 (𝐴 ∈ 𝑆 → (𝐴(+g‘𝑀)𝐴) = 𝐴)
1615adantr 480 . . . . . . . . 9 ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴(+g‘𝑀)𝐴) = 𝐴)
1716oveq1d 7429 . . . . . . . 8 ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐵) = (𝐴(+g‘𝑀)𝐵))
181, 4, 5, 8sgrp2nmndlem2 18867 . . . . . . . . . 10 ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴(+g‘𝑀)𝐵) = 𝐴)
1918oveq2d 7430 . . . . . . . . 9 ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐵)) = (𝐴(+g‘𝑀)𝐴))
2016, 19, 183eqtr4rd 2778 . . . . . . . 8 ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴(+g‘𝑀)𝐵) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐵)))
2117, 20eqtrd 2767 . . . . . . 7 ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐵) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐵)))
22213adant3 1130 . . . . . 6 ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐵) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐵)))
2314, 22jca 511 . . . . 5 ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → (((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐴) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐴)) ∧ ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐵) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐵))))
24183adant3 1130 . . . . . 6 ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → (𝐴(+g‘𝑀)𝐵) = 𝐴)
251, 4, 5, 8sgrp2nmndlem3 18868 . . . . . . 7 ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → (𝐵(+g‘𝑀)𝐴) = 𝐵)
2625oveq2d 7430 . . . . . 6 ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝐴)) = (𝐴(+g‘𝑀)𝐵))
2724oveq1d 7429 . . . . . . 7 ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝐴) = (𝐴(+g‘𝑀)𝐴))
28153ad2ant1 1131 . . . . . . 7 ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → (𝐴(+g‘𝑀)𝐴) = 𝐴)
2927, 28eqtrd 2767 . . . . . 6 ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝐴) = 𝐴)
3024, 26, 293eqtr4rd 2778 . . . . 5 ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝐴) = (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝐴)))
31 simp2 1135 . . . . . . . 8 ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → 𝐵 ∈ 𝑆)
321, 4, 5, 8sgrp2nmndlem3 18868 . . . . . . . 8 ((𝐵 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → (𝐵(+g‘𝑀)𝐵) = 𝐵)
3331, 32syld3an1 1408 . . . . . . 7 ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → (𝐵(+g‘𝑀)𝐵) = 𝐵)
3433oveq2d 7430 . . . . . 6 ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝐵)) = (𝐴(+g‘𝑀)𝐵))
3518oveq1d 7429 . . . . . . . 8 ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝐵) = (𝐴(+g‘𝑀)𝐵))
3635, 18eqtrd 2767 . . . . . . 7 ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝐵) = 𝐴)
37363adant3 1130 . . . . . 6 ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝐵) = 𝐴)
3824, 34, 373eqtr4rd 2778 . . . . 5 ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝐵) = (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝐵)))
3923, 30, 38jca32 515 . . . 4 ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ((((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐴) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐴)) ∧ ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐵) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐵))) ∧ (((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝐴) = (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝐴)) ∧ ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝐵) = (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝐵)))))
4025oveq1d 7429 . . . . . . 7 ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝐴) = (𝐵(+g‘𝑀)𝐴))
4128oveq2d 7430 . . . . . . 7 ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝐴)) = (𝐵(+g‘𝑀)𝐴))
4240, 41eqtr4d 2770 . . . . . 6 ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝐴) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝐴)))
4324oveq2d 7430 . . . . . . 7 ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝐵)) = (𝐵(+g‘𝑀)𝐴))
4425oveq1d 7429 . . . . . . . 8 ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝐵) = (𝐵(+g‘𝑀)𝐵))
4544, 33eqtrd 2767 . . . . . . 7 ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝐵) = 𝐵)
4625, 43, 453eqtr4rd 2778 . . . . . 6 ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝐵) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝐵)))
4742, 46jca 511 . . . . 5 ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → (((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝐴) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝐴)) ∧ ((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝐵) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝐵))))
4825oveq2d 7430 . . . . . 6 ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝐴)) = (𝐵(+g‘𝑀)𝐵))
4933oveq1d 7429 . . . . . . 7 ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝐴) = (𝐵(+g‘𝑀)𝐴))
5049, 25eqtrd 2767 . . . . . 6 ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝐴) = 𝐵)
5133, 48, 503eqtr4rd 2778 . . . . 5 ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝐴) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝐴)))
5232oveq1d 7429 . . . . . . 7 ((𝐵 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝐵) = (𝐵(+g‘𝑀)𝐵))
5332oveq2d 7430 . . . . . . 7 ((𝐵 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝐵)) = (𝐵(+g‘𝑀)𝐵))
5452, 53eqtr4d 2770 . . . . . 6 ((𝐵 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝐵) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝐵)))
5531, 54syld3an1 1408 . . . . 5 ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝐵) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝐵)))
5647, 51, 55jca32 515 . . . 4 ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ((((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝐴) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝐴)) ∧ ((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝐵) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝐵))) ∧ (((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝐴) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝐴)) ∧ ((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝐵) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝐵)))))
57 oveq1 7421 . . . . . . . . . 10 (𝑎 = 𝐴 → (𝑎(+g‘𝑀)𝑏) = (𝐴(+g‘𝑀)𝑏))
5857oveq1d 7429 . . . . . . . . 9 (𝑎 = 𝐴 → ((𝑎(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = ((𝐴(+g‘𝑀)𝑏)(+g‘𝑀)𝑐))
59 oveq1 7421 . . . . . . . . 9 (𝑎 = 𝐴 → (𝑎(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)) = (𝐴(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)))
6058, 59eqeq12d 2743 . . . . . . . 8 (𝑎 = 𝐴 → (((𝑎(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝑎(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)) ↔ ((𝐴(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)(𝑏(+g‘𝑀)𝑐))))
61602ralbidv 3213 . . . . . . 7 (𝑎 = 𝐴 → (∀𝑏 ∈ {𝐴, 𝐵}∀𝑐 ∈ {𝐴, 𝐵} ((𝑎(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝑎(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)) ↔ ∀𝑏 ∈ {𝐴, 𝐵}∀𝑐 ∈ {𝐴, 𝐵} ((𝐴(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)(𝑏(+g‘𝑀)𝑐))))
62 oveq1 7421 . . . . . . . . . 10 (𝑎 = 𝐵 → (𝑎(+g‘𝑀)𝑏) = (𝐵(+g‘𝑀)𝑏))
6362oveq1d 7429 . . . . . . . . 9 (𝑎 = 𝐵 → ((𝑎(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = ((𝐵(+g‘𝑀)𝑏)(+g‘𝑀)𝑐))
64 oveq1 7421 . . . . . . . . 9 (𝑎 = 𝐵 → (𝑎(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)) = (𝐵(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)))
6563, 64eqeq12d 2743 . . . . . . . 8 (𝑎 = 𝐵 → (((𝑎(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝑎(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)) ↔ ((𝐵(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)(𝑏(+g‘𝑀)𝑐))))
66652ralbidv 3213 . . . . . . 7 (𝑎 = 𝐵 → (∀𝑏 ∈ {𝐴, 𝐵}∀𝑐 ∈ {𝐴, 𝐵} ((𝑎(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝑎(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)) ↔ ∀𝑏 ∈ {𝐴, 𝐵}∀𝑐 ∈ {𝐴, 𝐵} ((𝐵(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)(𝑏(+g‘𝑀)𝑐))))
6761, 66ralprg 4694 . . . . . 6 ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (∀𝑎 ∈ {𝐴, 𝐵}∀𝑏 ∈ {𝐴, 𝐵}∀𝑐 ∈ {𝐴, 𝐵} ((𝑎(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝑎(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)) ↔ (∀𝑏 ∈ {𝐴, 𝐵}∀𝑐 ∈ {𝐴, 𝐵} ((𝐴(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)) ∧ ∀𝑏 ∈ {𝐴, 𝐵}∀𝑐 ∈ {𝐴, 𝐵} ((𝐵(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)))))
68 oveq2 7422 . . . . . . . . . . 11 (𝑏 = 𝐴 → (𝐴(+g‘𝑀)𝑏) = (𝐴(+g‘𝑀)𝐴))
6968oveq1d 7429 . . . . . . . . . 10 (𝑏 = 𝐴 → ((𝐴(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝑐))
70 oveq1 7421 . . . . . . . . . . 11 (𝑏 = 𝐴 → (𝑏(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)𝑐))
7170oveq2d 7430 . . . . . . . . . 10 (𝑏 = 𝐴 → (𝐴(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝑐)))
7269, 71eqeq12d 2743 . . . . . . . . 9 (𝑏 = 𝐴 → (((𝐴(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)) ↔ ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝑐))))
7372ralbidv 3172 . . . . . . . 8 (𝑏 = 𝐴 → (∀𝑐 ∈ {𝐴, 𝐵} ((𝐴(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)) ↔ ∀𝑐 ∈ {𝐴, 𝐵} ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝑐))))
74 oveq2 7422 . . . . . . . . . . 11 (𝑏 = 𝐵 → (𝐴(+g‘𝑀)𝑏) = (𝐴(+g‘𝑀)𝐵))
7574oveq1d 7429 . . . . . . . . . 10 (𝑏 = 𝐵 → ((𝐴(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝑐))
76 oveq1 7421 . . . . . . . . . . 11 (𝑏 = 𝐵 → (𝑏(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)𝑐))
7776oveq2d 7430 . . . . . . . . . 10 (𝑏 = 𝐵 → (𝐴(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)) = (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝑐)))
7875, 77eqeq12d 2743 . . . . . . . . 9 (𝑏 = 𝐵 → (((𝐴(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)) ↔ ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝑐))))
7978ralbidv 3172 . . . . . . . 8 (𝑏 = 𝐵 → (∀𝑐 ∈ {𝐴, 𝐵} ((𝐴(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)) ↔ ∀𝑐 ∈ {𝐴, 𝐵} ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝑐))))
8073, 79ralprg 4694 . . . . . . 7 ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (∀𝑏 ∈ {𝐴, 𝐵}∀𝑐 ∈ {𝐴, 𝐵} ((𝐴(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)) ↔ (∀𝑐 ∈ {𝐴, 𝐵} ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝑐)) ∧ ∀𝑐 ∈ {𝐴, 𝐵} ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝑐)))))
81 oveq2 7422 . . . . . . . . . . 11 (𝑏 = 𝐴 → (𝐵(+g‘𝑀)𝑏) = (𝐵(+g‘𝑀)𝐴))
8281oveq1d 7429 . . . . . . . . . 10 (𝑏 = 𝐴 → ((𝐵(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = ((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝑐))
8370oveq2d 7430 . . . . . . . . . 10 (𝑏 = 𝐴 → (𝐵(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝑐)))
8482, 83eqeq12d 2743 . . . . . . . . 9 (𝑏 = 𝐴 → (((𝐵(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)) ↔ ((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝑐))))
8584ralbidv 3172 . . . . . . . 8 (𝑏 = 𝐴 → (∀𝑐 ∈ {𝐴, 𝐵} ((𝐵(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)) ↔ ∀𝑐 ∈ {𝐴, 𝐵} ((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝑐))))
86 oveq2 7422 . . . . . . . . . . 11 (𝑏 = 𝐵 → (𝐵(+g‘𝑀)𝑏) = (𝐵(+g‘𝑀)𝐵))
8786oveq1d 7429 . . . . . . . . . 10 (𝑏 = 𝐵 → ((𝐵(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = ((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝑐))
8876oveq2d 7430 . . . . . . . . . 10 (𝑏 = 𝐵 → (𝐵(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝑐)))
8987, 88eqeq12d 2743 . . . . . . . . 9 (𝑏 = 𝐵 → (((𝐵(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)) ↔ ((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝑐))))
9089ralbidv 3172 . . . . . . . 8 (𝑏 = 𝐵 → (∀𝑐 ∈ {𝐴, 𝐵} ((𝐵(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)) ↔ ∀𝑐 ∈ {𝐴, 𝐵} ((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝑐))))
9185, 90ralprg 4694 . . . . . . 7 ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (∀𝑏 ∈ {𝐴, 𝐵}∀𝑐 ∈ {𝐴, 𝐵} ((𝐵(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)) ↔ (∀𝑐 ∈ {𝐴, 𝐵} ((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝑐)) ∧ ∀𝑐 ∈ {𝐴, 𝐵} ((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝑐)))))
9280, 91anbi12d 630 . . . . . 6 ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((∀𝑏 ∈ {𝐴, 𝐵}∀𝑐 ∈ {𝐴, 𝐵} ((𝐴(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)) ∧ ∀𝑏 ∈ {𝐴, 𝐵}∀𝑐 ∈ {𝐴, 𝐵} ((𝐵(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)(𝑏(+g‘𝑀)𝑐))) ↔ ((∀𝑐 ∈ {𝐴, 𝐵} ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝑐)) ∧ ∀𝑐 ∈ {𝐴, 𝐵} ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝑐))) ∧ (∀𝑐 ∈ {𝐴, 𝐵} ((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝑐)) ∧ ∀𝑐 ∈ {𝐴, 𝐵} ((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝑐))))))
93 oveq2 7422 . . . . . . . . . 10 (𝑐 = 𝐴 → ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝑐) = ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐴))
94 oveq2 7422 . . . . . . . . . . 11 (𝑐 = 𝐴 → (𝐴(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)𝐴))
9594oveq2d 7430 . . . . . . . . . 10 (𝑐 = 𝐴 → (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝑐)) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐴)))
9693, 95eqeq12d 2743 . . . . . . . . 9 (𝑐 = 𝐴 → (((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝑐)) ↔ ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐴) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐴))))
97 oveq2 7422 . . . . . . . . . 10 (𝑐 = 𝐵 → ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝑐) = ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐵))
98 oveq2 7422 . . . . . . . . . . 11 (𝑐 = 𝐵 → (𝐴(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)𝐵))
9998oveq2d 7430 . . . . . . . . . 10 (𝑐 = 𝐵 → (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝑐)) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐵)))
10097, 99eqeq12d 2743 . . . . . . . . 9 (𝑐 = 𝐵 → (((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝑐)) ↔ ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐵) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐵))))
10196, 100ralprg 4694 . . . . . . . 8 ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (∀𝑐 ∈ {𝐴, 𝐵} ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝑐)) ↔ (((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐴) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐴)) ∧ ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐵) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐵)))))
102 oveq2 7422 . . . . . . . . . 10 (𝑐 = 𝐴 → ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝑐) = ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝐴))
103 oveq2 7422 . . . . . . . . . . 11 (𝑐 = 𝐴 → (𝐵(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)𝐴))
104103oveq2d 7430 . . . . . . . . . 10 (𝑐 = 𝐴 → (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝑐)) = (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝐴)))
105102, 104eqeq12d 2743 . . . . . . . . 9 (𝑐 = 𝐴 → (((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝑐)) ↔ ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝐴) = (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝐴))))
106 oveq2 7422 . . . . . . . . . 10 (𝑐 = 𝐵 → ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝑐) = ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝐵))
107 oveq2 7422 . . . . . . . . . . 11 (𝑐 = 𝐵 → (𝐵(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)𝐵))
108107oveq2d 7430 . . . . . . . . . 10 (𝑐 = 𝐵 → (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝑐)) = (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝐵)))
109106, 108eqeq12d 2743 . . . . . . . . 9 (𝑐 = 𝐵 → (((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝑐)) ↔ ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝐵) = (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝐵))))
110105, 109ralprg 4694 . . . . . . . 8 ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (∀𝑐 ∈ {𝐴, 𝐵} ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝑐)) ↔ (((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝐴) = (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝐴)) ∧ ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝐵) = (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝐵)))))
111101, 110anbi12d 630 . . . . . . 7 ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((∀𝑐 ∈ {𝐴, 𝐵} ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝑐)) ∧ ∀𝑐 ∈ {𝐴, 𝐵} ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝑐))) ↔ ((((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐴) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐴)) ∧ ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐵) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐵))) ∧ (((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝐴) = (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝐴)) ∧ ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝐵) = (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝐵))))))
112 oveq2 7422 . . . . . . . . . 10 (𝑐 = 𝐴 → ((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝑐) = ((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝐴))
11394oveq2d 7430 . . . . . . . . . 10 (𝑐 = 𝐴 → (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝑐)) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝐴)))
114112, 113eqeq12d 2743 . . . . . . . . 9 (𝑐 = 𝐴 → (((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝑐)) ↔ ((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝐴) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝐴))))
115 oveq2 7422 . . . . . . . . . 10 (𝑐 = 𝐵 → ((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝑐) = ((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝐵))
11698oveq2d 7430 . . . . . . . . . 10 (𝑐 = 𝐵 → (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝑐)) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝐵)))
117115, 116eqeq12d 2743 . . . . . . . . 9 (𝑐 = 𝐵 → (((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝑐)) ↔ ((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝐵) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝐵))))
118114, 117ralprg 4694 . . . . . . . 8 ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (∀𝑐 ∈ {𝐴, 𝐵} ((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝑐)) ↔ (((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝐴) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝐴)) ∧ ((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝐵) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝐵)))))
119 oveq2 7422 . . . . . . . . . 10 (𝑐 = 𝐴 → ((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝑐) = ((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝐴))
120103oveq2d 7430 . . . . . . . . . 10 (𝑐 = 𝐴 → (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝑐)) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝐴)))
121119, 120eqeq12d 2743 . . . . . . . . 9 (𝑐 = 𝐴 → (((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝑐)) ↔ ((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝐴) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝐴))))
122 oveq2 7422 . . . . . . . . . 10 (𝑐 = 𝐵 → ((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝑐) = ((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝐵))
123107oveq2d 7430 . . . . . . . . . 10 (𝑐 = 𝐵 → (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝑐)) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝐵)))
124122, 123eqeq12d 2743 . . . . . . . . 9 (𝑐 = 𝐵 → (((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝑐)) ↔ ((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝐵) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝐵))))
125121, 124ralprg 4694 . . . . . . . 8 ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (∀𝑐 ∈ {𝐴, 𝐵} ((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝑐)) ↔ (((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝐴) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝐴)) ∧ ((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝐵) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝐵)))))
126118, 125anbi12d 630 . . . . . . 7 ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((∀𝑐 ∈ {𝐴, 𝐵} ((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝑐)) ∧ ∀𝑐 ∈ {𝐴, 𝐵} ((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝑐))) ↔ ((((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝐴) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝐴)) ∧ ((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝐵) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝐵))) ∧ (((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝐴) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝐴)) ∧ ((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝐵) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝐵))))))
127111, 126anbi12d 630 . . . . . 6 ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (((∀𝑐 ∈ {𝐴, 𝐵} ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝑐)) ∧ ∀𝑐 ∈ {𝐴, 𝐵} ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝑐))) ∧ (∀𝑐 ∈ {𝐴, 𝐵} ((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝑐)) ∧ ∀𝑐 ∈ {𝐴, 𝐵} ((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝑐)))) ↔ (((((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐴) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐴)) ∧ ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐵) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐵))) ∧ (((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝐴) = (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝐴)) ∧ ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝐵) = (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝐵)))) ∧ ((((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝐴) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝐴)) ∧ ((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝐵) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝐵))) ∧ (((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝐴) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝐴)) ∧ ((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝐵) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝐵)))))))
12867, 92, 1273bitrd 305 . . . . 5 ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (∀𝑎 ∈ {𝐴, 𝐵}∀𝑏 ∈ {𝐴, 𝐵}∀𝑐 ∈ {𝐴, 𝐵} ((𝑎(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝑎(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)) ↔ (((((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐴) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐴)) ∧ ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐵) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐵))) ∧ (((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝐴) = (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝐴)) ∧ ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝐵) = (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝐵)))) ∧ ((((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝐴) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝐴)) ∧ ((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝐵) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝐵))) ∧ (((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝐴) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝐴)) ∧ ((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝐵) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝐵)))))))
1291283adant3 1130 . . . 4 ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → (∀𝑎 ∈ {𝐴, 𝐵}∀𝑏 ∈ {𝐴, 𝐵}∀𝑐 ∈ {𝐴, 𝐵} ((𝑎(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝑎(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)) ↔ (((((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐴) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐴)) ∧ ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐵) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐵))) ∧ (((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝐴) = (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝐴)) ∧ ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝐵) = (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝐵)))) ∧ ((((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝐴) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝐴)) ∧ ((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝐵) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝐵))) ∧ (((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝐴) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝐴)) ∧ ((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝐵) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝐵)))))))
13039, 56, 129mpbir2and 712 . . 3 ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ∀𝑎 ∈ {𝐴, 𝐵}∀𝑏 ∈ {𝐴, 𝐵}∀𝑐 ∈ {𝐴, 𝐵} ((𝑎(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝑎(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)))
1312, 130syl 17 . 2 ((♯‘𝑆) = 2 → ∀𝑎 ∈ {𝐴, 𝐵}∀𝑏 ∈ {𝐴, 𝐵}∀𝑐 ∈ {𝐴, 𝐵} ((𝑎(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝑎(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)))
1324, 1eqtr2i 2756 . . 3 {𝐴, 𝐵} = (Base‘𝑀)
133132, 8issgrp 18671 . 2 (𝑀 ∈ Smgrp ↔ (𝑀 ∈ Mgm ∧ ∀𝑎 ∈ {𝐴, 𝐵}∀𝑏 ∈ {𝐴, 𝐵}∀𝑐 ∈ {𝐴, 𝐵} ((𝑎(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝑎(+g‘𝑀)(𝑏(+g‘𝑀)𝑐))))
1347, 131, 133sylanbrc 582 1 ((♯‘𝑆) = 2 → 𝑀 ∈ Smgrp)
Colors of variables: wff setvar class
Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099   ≠ wne 2935  ∀wral 3056  ifcif 4524  {cpr 4626  ‘cfv 6542  (class class class)co 7414   ∈ cmpo 7416  2c2 12289  ♯chash 14313  Basecbs 17171  +gcplusg 17224  Mgmcmgm 18589  Smgrpcsgrp 18669
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-cnex 11186  ax-resscn 11187  ax-1cn 11188  ax-icn 11189  ax-addcl 11190  ax-addrcl 11191  ax-mulcl 11192  ax-mulrcl 11193  ax-mulcom 11194  ax-addass 11195  ax-mulass 11196  ax-distr 11197  ax-i2m1 11198  ax-1ne0 11199  ax-1rid 11200  ax-rnegex 11201  ax-rrecex 11202  ax-cnre 11203  ax-pre-lttri 11204  ax-pre-lttrn 11205  ax-pre-ltadd 11206  ax-pre-mulgt0 11207
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-oadd 8484  df-er 8718  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-dju 9916  df-card 9954  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11468  df-neg 11469  df-nn 12235  df-2 12297  df-n0 12495  df-z 12581  df-uz 12845  df-fz 13509  df-hash 14314  df-mgm 18591  df-sgrp 18670
This theorem is referenced by:  sgrp2nmnd  18873  sgrpnmndex  18875
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