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Theorem sgrp2nmndlem4 18806
Description: Lemma 4 for sgrp2nmnd 18808: 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 14355 . . 3 ((♯‘𝑆) = 2 → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵))
3 3simpa 1149 . . 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 18801 . . 3 ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → 𝑀 ∈ Mgm)
72, 3, 63syl 18 . 2 ((♯‘𝑆) = 2 → 𝑀 ∈ Mgm)
8 eqid 2733 . . . . . . . . . . 11 (+g‘𝑀) = (+g‘𝑀)
91, 4, 5, 8sgrp2nmndlem2 18802 . . . . . . . . . 10 ((𝐴 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → (𝐴(+g‘𝑀)𝐴) = 𝐴)
109oveq1d 7421 . . . . . . . . 9 ((𝐴 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐴) = (𝐴(+g‘𝑀)𝐴))
119oveq2d 7422 . . . . . . . . 9 ((𝐴 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐴)) = (𝐴(+g‘𝑀)𝐴))
1210, 11eqtr4d 2776 . . . . . . . 8 ((𝐴 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐴) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐴)))
1312anidms 568 . . . . . . 7 (𝐴 ∈ 𝑆 → ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐴) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐴)))
14133ad2ant1 1134 . . . . . 6 ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐴) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐴)))
159anidms 568 . . . . . . . . . 10 (𝐴 ∈ 𝑆 → (𝐴(+g‘𝑀)𝐴) = 𝐴)
1615adantr 482 . . . . . . . . 9 ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴(+g‘𝑀)𝐴) = 𝐴)
1716oveq1d 7421 . . . . . . . 8 ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐵) = (𝐴(+g‘𝑀)𝐵))
181, 4, 5, 8sgrp2nmndlem2 18802 . . . . . . . . . 10 ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴(+g‘𝑀)𝐵) = 𝐴)
1918oveq2d 7422 . . . . . . . . 9 ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐵)) = (𝐴(+g‘𝑀)𝐴))
2016, 19, 183eqtr4rd 2784 . . . . . . . 8 ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴(+g‘𝑀)𝐵) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐵)))
2117, 20eqtrd 2773 . . . . . . 7 ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐵) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐵)))
22213adant3 1133 . . . . . 6 ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐵) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐵)))
2314, 22jca 513 . . . . 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 1133 . . . . . 6 ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → (𝐴(+g‘𝑀)𝐵) = 𝐴)
251, 4, 5, 8sgrp2nmndlem3 18803 . . . . . . 7 ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → (𝐵(+g‘𝑀)𝐴) = 𝐵)
2625oveq2d 7422 . . . . . 6 ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝐴)) = (𝐴(+g‘𝑀)𝐵))
2724oveq1d 7421 . . . . . . 7 ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝐴) = (𝐴(+g‘𝑀)𝐴))
28153ad2ant1 1134 . . . . . . 7 ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → (𝐴(+g‘𝑀)𝐴) = 𝐴)
2927, 28eqtrd 2773 . . . . . 6 ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝐴) = 𝐴)
3024, 26, 293eqtr4rd 2784 . . . . 5 ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝐴) = (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝐴)))
31 simp2 1138 . . . . . . . 8 ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → 𝐵 ∈ 𝑆)
321, 4, 5, 8sgrp2nmndlem3 18803 . . . . . . . 8 ((𝐵 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → (𝐵(+g‘𝑀)𝐵) = 𝐵)
3331, 32syld3an1 1411 . . . . . . 7 ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → (𝐵(+g‘𝑀)𝐵) = 𝐵)
3433oveq2d 7422 . . . . . 6 ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝐵)) = (𝐴(+g‘𝑀)𝐵))
3518oveq1d 7421 . . . . . . . 8 ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝐵) = (𝐴(+g‘𝑀)𝐵))
3635, 18eqtrd 2773 . . . . . . 7 ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝐵) = 𝐴)
37363adant3 1133 . . . . . 6 ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝐵) = 𝐴)
3824, 34, 373eqtr4rd 2784 . . . . 5 ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝐵) = (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝐵)))
3923, 30, 38jca32 517 . . . 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 7421 . . . . . . 7 ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝐴) = (𝐵(+g‘𝑀)𝐴))
4128oveq2d 7422 . . . . . . 7 ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝐴)) = (𝐵(+g‘𝑀)𝐴))
4240, 41eqtr4d 2776 . . . . . 6 ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝐴) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝐴)))
4324oveq2d 7422 . . . . . . 7 ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝐵)) = (𝐵(+g‘𝑀)𝐴))
4425oveq1d 7421 . . . . . . . 8 ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝐵) = (𝐵(+g‘𝑀)𝐵))
4544, 33eqtrd 2773 . . . . . . 7 ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝐵) = 𝐵)
4625, 43, 453eqtr4rd 2784 . . . . . 6 ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝐵) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝐵)))
4742, 46jca 513 . . . . 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 7422 . . . . . 6 ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝐴)) = (𝐵(+g‘𝑀)𝐵))
4933oveq1d 7421 . . . . . . 7 ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝐴) = (𝐵(+g‘𝑀)𝐴))
5049, 25eqtrd 2773 . . . . . 6 ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝐴) = 𝐵)
5133, 48, 503eqtr4rd 2784 . . . . 5 ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝐴) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝐴)))
5232oveq1d 7421 . . . . . . 7 ((𝐵 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝐵) = (𝐵(+g‘𝑀)𝐵))
5332oveq2d 7422 . . . . . . 7 ((𝐵 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝐵)) = (𝐵(+g‘𝑀)𝐵))
5452, 53eqtr4d 2776 . . . . . 6 ((𝐵 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝐵) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝐵)))
5531, 54syld3an1 1411 . . . . 5 ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝐵) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝐵)))
5647, 51, 55jca32 517 . . . 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 7413 . . . . . . . . . 10 (𝑎 = 𝐴 → (𝑎(+g‘𝑀)𝑏) = (𝐴(+g‘𝑀)𝑏))
5857oveq1d 7421 . . . . . . . . 9 (𝑎 = 𝐴 → ((𝑎(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = ((𝐴(+g‘𝑀)𝑏)(+g‘𝑀)𝑐))
59 oveq1 7413 . . . . . . . . 9 (𝑎 = 𝐴 → (𝑎(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)) = (𝐴(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)))
6058, 59eqeq12d 2749 . . . . . . . 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 3219 . . . . . . 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 7413 . . . . . . . . . 10 (𝑎 = 𝐵 → (𝑎(+g‘𝑀)𝑏) = (𝐵(+g‘𝑀)𝑏))
6362oveq1d 7421 . . . . . . . . 9 (𝑎 = 𝐵 → ((𝑎(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = ((𝐵(+g‘𝑀)𝑏)(+g‘𝑀)𝑐))
64 oveq1 7413 . . . . . . . . 9 (𝑎 = 𝐵 → (𝑎(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)) = (𝐵(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)))
6563, 64eqeq12d 2749 . . . . . . . 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 3219 . . . . . . 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 4698 . . . . . 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 7414 . . . . . . . . . . 11 (𝑏 = 𝐴 → (𝐴(+g‘𝑀)𝑏) = (𝐴(+g‘𝑀)𝐴))
6968oveq1d 7421 . . . . . . . . . 10 (𝑏 = 𝐴 → ((𝐴(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝑐))
70 oveq1 7413 . . . . . . . . . . 11 (𝑏 = 𝐴 → (𝑏(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)𝑐))
7170oveq2d 7422 . . . . . . . . . 10 (𝑏 = 𝐴 → (𝐴(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝑐)))
7269, 71eqeq12d 2749 . . . . . . . . 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 3178 . . . . . . . 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 7414 . . . . . . . . . . 11 (𝑏 = 𝐵 → (𝐴(+g‘𝑀)𝑏) = (𝐴(+g‘𝑀)𝐵))
7574oveq1d 7421 . . . . . . . . . 10 (𝑏 = 𝐵 → ((𝐴(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝑐))
76 oveq1 7413 . . . . . . . . . . 11 (𝑏 = 𝐵 → (𝑏(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)𝑐))
7776oveq2d 7422 . . . . . . . . . 10 (𝑏 = 𝐵 → (𝐴(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)) = (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝑐)))
7875, 77eqeq12d 2749 . . . . . . . . 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 3178 . . . . . . . 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 4698 . . . . . . 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 7414 . . . . . . . . . . 11 (𝑏 = 𝐴 → (𝐵(+g‘𝑀)𝑏) = (𝐵(+g‘𝑀)𝐴))
8281oveq1d 7421 . . . . . . . . . 10 (𝑏 = 𝐴 → ((𝐵(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = ((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝑐))
8370oveq2d 7422 . . . . . . . . . 10 (𝑏 = 𝐴 → (𝐵(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝑐)))
8482, 83eqeq12d 2749 . . . . . . . . 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 3178 . . . . . . . 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 7414 . . . . . . . . . . 11 (𝑏 = 𝐵 → (𝐵(+g‘𝑀)𝑏) = (𝐵(+g‘𝑀)𝐵))
8786oveq1d 7421 . . . . . . . . . 10 (𝑏 = 𝐵 → ((𝐵(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = ((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝑐))
8876oveq2d 7422 . . . . . . . . . 10 (𝑏 = 𝐵 → (𝐵(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝑐)))
8987, 88eqeq12d 2749 . . . . . . . . 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 3178 . . . . . . . 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 4698 . . . . . . 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 632 . . . . . 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 7414 . . . . . . . . . 10 (𝑐 = 𝐴 → ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝑐) = ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐴))
94 oveq2 7414 . . . . . . . . . . 11 (𝑐 = 𝐴 → (𝐴(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)𝐴))
9594oveq2d 7422 . . . . . . . . . 10 (𝑐 = 𝐴 → (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝑐)) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐴)))
9693, 95eqeq12d 2749 . . . . . . . . 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 7414 . . . . . . . . . 10 (𝑐 = 𝐵 → ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝑐) = ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐵))
98 oveq2 7414 . . . . . . . . . . 11 (𝑐 = 𝐵 → (𝐴(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)𝐵))
9998oveq2d 7422 . . . . . . . . . 10 (𝑐 = 𝐵 → (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝑐)) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐵)))
10097, 99eqeq12d 2749 . . . . . . . . 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 4698 . . . . . . . 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 7414 . . . . . . . . . 10 (𝑐 = 𝐴 → ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝑐) = ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝐴))
103 oveq2 7414 . . . . . . . . . . 11 (𝑐 = 𝐴 → (𝐵(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)𝐴))
104103oveq2d 7422 . . . . . . . . . 10 (𝑐 = 𝐴 → (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝑐)) = (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝐴)))
105102, 104eqeq12d 2749 . . . . . . . . 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 7414 . . . . . . . . . 10 (𝑐 = 𝐵 → ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝑐) = ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝐵))
107 oveq2 7414 . . . . . . . . . . 11 (𝑐 = 𝐵 → (𝐵(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)𝐵))
108107oveq2d 7422 . . . . . . . . . 10 (𝑐 = 𝐵 → (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝑐)) = (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝐵)))
109106, 108eqeq12d 2749 . . . . . . . . 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 4698 . . . . . . . 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 632 . . . . . . 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 7414 . . . . . . . . . 10 (𝑐 = 𝐴 → ((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝑐) = ((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝐴))
11394oveq2d 7422 . . . . . . . . . 10 (𝑐 = 𝐴 → (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝑐)) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝐴)))
114112, 113eqeq12d 2749 . . . . . . . . 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 7414 . . . . . . . . . 10 (𝑐 = 𝐵 → ((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝑐) = ((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝐵))
11698oveq2d 7422 . . . . . . . . . 10 (𝑐 = 𝐵 → (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝑐)) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝐵)))
117115, 116eqeq12d 2749 . . . . . . . . 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 4698 . . . . . . . 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 7414 . . . . . . . . . 10 (𝑐 = 𝐴 → ((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝑐) = ((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝐴))
120103oveq2d 7422 . . . . . . . . . 10 (𝑐 = 𝐴 → (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝑐)) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝐴)))
121119, 120eqeq12d 2749 . . . . . . . . 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 7414 . . . . . . . . . 10 (𝑐 = 𝐵 → ((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝑐) = ((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝐵))
123107oveq2d 7422 . . . . . . . . . 10 (𝑐 = 𝐵 → (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝑐)) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝐵)))
124122, 123eqeq12d 2749 . . . . . . . . 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 4698 . . . . . . . 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 632 . . . . . . 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 632 . . . . . 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 1133 . . . 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 2762 . . 3 {𝐴, 𝐵} = (Base‘𝑀)
133132, 8issgrp 18608 . 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 584 1 ((♯‘𝑆) = 2 → 𝑀 ∈ Smgrp)
Colors of variables: wff setvar class
Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   ≠ wne 2941  ∀wral 3062  ifcif 4528  {cpr 4630  ‘cfv 6541  (class class class)co 7406   ∈ cmpo 7408  2c2 12264  ♯chash 14287  Basecbs 17141  +gcplusg 17194  Mgmcmgm 18556  Smgrpcsgrp 18606
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-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722  ax-cnex 11163  ax-resscn 11164  ax-1cn 11165  ax-icn 11166  ax-addcl 11167  ax-addrcl 11168  ax-mulcl 11169  ax-mulrcl 11170  ax-mulcom 11171  ax-addass 11172  ax-mulass 11173  ax-distr 11174  ax-i2m1 11175  ax-1ne0 11176  ax-1rid 11177  ax-rnegex 11178  ax-rrecex 11179  ax-cnre 11180  ax-pre-lttri 11181  ax-pre-lttrn 11182  ax-pre-ltadd 11183  ax-pre-mulgt0 11184
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-reu 3378  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6298  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7362  df-ov 7409  df-oprab 7410  df-mpo 7411  df-om 7853  df-1st 7972  df-2nd 7973  df-frecs 8263  df-wrecs 8294  df-recs 8368  df-rdg 8407  df-1o 8463  df-oadd 8467  df-er 8700  df-en 8937  df-dom 8938  df-sdom 8939  df-fin 8940  df-dju 9893  df-card 9931  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11443  df-neg 11444  df-nn 12210  df-2 12272  df-n0 12470  df-z 12556  df-uz 12820  df-fz 13482  df-hash 14288  df-mgm 18558  df-sgrp 18607
This theorem is referenced by:  sgrp2nmnd  18808  sgrpnmndex  18810
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