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Theorem sgrp2nmndlem4 17355
Description: Lemma 4 for sgrp2nmnd 17357: 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 (+g𝑀) = (𝑥𝑆, 𝑦𝑆 ↦ if(𝑥 = 𝐴, 𝐴, 𝐵))
Assertion
Ref Expression
sgrp2nmndlem4 ((#‘𝑆) = 2 → 𝑀 ∈ SGrp)
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 13142 . . 3 ((#‘𝑆) = 2 → (𝐴𝑆𝐵𝑆𝐴𝐵))
3 3simpa 1056 . . 3 ((𝐴𝑆𝐵𝑆𝐴𝐵) → (𝐴𝑆𝐵𝑆))
4 mgm2nsgrp.b . . . 4 (Base‘𝑀) = 𝑆
5 sgrp2nmnd.o . . . 4 (+g𝑀) = (𝑥𝑆, 𝑦𝑆 ↦ if(𝑥 = 𝐴, 𝐴, 𝐵))
61, 4, 5sgrp2nmndlem1 17350 . . 3 ((𝐴𝑆𝐵𝑆) → 𝑀 ∈ Mgm)
72, 3, 63syl 18 . 2 ((#‘𝑆) = 2 → 𝑀 ∈ Mgm)
8 eqid 2621 . . . . . . . . . . 11 (+g𝑀) = (+g𝑀)
91, 4, 5, 8sgrp2nmndlem2 17351 . . . . . . . . . 10 ((𝐴𝑆𝐴𝑆) → (𝐴(+g𝑀)𝐴) = 𝐴)
109oveq1d 6630 . . . . . . . . 9 ((𝐴𝑆𝐴𝑆) → ((𝐴(+g𝑀)𝐴)(+g𝑀)𝐴) = (𝐴(+g𝑀)𝐴))
119oveq2d 6631 . . . . . . . . 9 ((𝐴𝑆𝐴𝑆) → (𝐴(+g𝑀)(𝐴(+g𝑀)𝐴)) = (𝐴(+g𝑀)𝐴))
1210, 11eqtr4d 2658 . . . . . . . 8 ((𝐴𝑆𝐴𝑆) → ((𝐴(+g𝑀)𝐴)(+g𝑀)𝐴) = (𝐴(+g𝑀)(𝐴(+g𝑀)𝐴)))
1312anidms 676 . . . . . . 7 (𝐴𝑆 → ((𝐴(+g𝑀)𝐴)(+g𝑀)𝐴) = (𝐴(+g𝑀)(𝐴(+g𝑀)𝐴)))
14133ad2ant1 1080 . . . . . 6 ((𝐴𝑆𝐵𝑆𝐴𝐵) → ((𝐴(+g𝑀)𝐴)(+g𝑀)𝐴) = (𝐴(+g𝑀)(𝐴(+g𝑀)𝐴)))
159anidms 676 . . . . . . . . . 10 (𝐴𝑆 → (𝐴(+g𝑀)𝐴) = 𝐴)
1615adantr 481 . . . . . . . . 9 ((𝐴𝑆𝐵𝑆) → (𝐴(+g𝑀)𝐴) = 𝐴)
1716oveq1d 6630 . . . . . . . 8 ((𝐴𝑆𝐵𝑆) → ((𝐴(+g𝑀)𝐴)(+g𝑀)𝐵) = (𝐴(+g𝑀)𝐵))
181, 4, 5, 8sgrp2nmndlem2 17351 . . . . . . . . . 10 ((𝐴𝑆𝐵𝑆) → (𝐴(+g𝑀)𝐵) = 𝐴)
1918oveq2d 6631 . . . . . . . . 9 ((𝐴𝑆𝐵𝑆) → (𝐴(+g𝑀)(𝐴(+g𝑀)𝐵)) = (𝐴(+g𝑀)𝐴))
2016, 19, 183eqtr4rd 2666 . . . . . . . 8 ((𝐴𝑆𝐵𝑆) → (𝐴(+g𝑀)𝐵) = (𝐴(+g𝑀)(𝐴(+g𝑀)𝐵)))
2117, 20eqtrd 2655 . . . . . . 7 ((𝐴𝑆𝐵𝑆) → ((𝐴(+g𝑀)𝐴)(+g𝑀)𝐵) = (𝐴(+g𝑀)(𝐴(+g𝑀)𝐵)))
22213adant3 1079 . . . . . 6 ((𝐴𝑆𝐵𝑆𝐴𝐵) → ((𝐴(+g𝑀)𝐴)(+g𝑀)𝐵) = (𝐴(+g𝑀)(𝐴(+g𝑀)𝐵)))
2314, 22jca 554 . . . . 5 ((𝐴𝑆𝐵𝑆𝐴𝐵) → (((𝐴(+g𝑀)𝐴)(+g𝑀)𝐴) = (𝐴(+g𝑀)(𝐴(+g𝑀)𝐴)) ∧ ((𝐴(+g𝑀)𝐴)(+g𝑀)𝐵) = (𝐴(+g𝑀)(𝐴(+g𝑀)𝐵))))
24183adant3 1079 . . . . . 6 ((𝐴𝑆𝐵𝑆𝐴𝐵) → (𝐴(+g𝑀)𝐵) = 𝐴)
251, 4, 5, 8sgrp2nmndlem3 17352 . . . . . . 7 ((𝐴𝑆𝐵𝑆𝐴𝐵) → (𝐵(+g𝑀)𝐴) = 𝐵)
2625oveq2d 6631 . . . . . 6 ((𝐴𝑆𝐵𝑆𝐴𝐵) → (𝐴(+g𝑀)(𝐵(+g𝑀)𝐴)) = (𝐴(+g𝑀)𝐵))
2724oveq1d 6630 . . . . . . 7 ((𝐴𝑆𝐵𝑆𝐴𝐵) → ((𝐴(+g𝑀)𝐵)(+g𝑀)𝐴) = (𝐴(+g𝑀)𝐴))
28153ad2ant1 1080 . . . . . . 7 ((𝐴𝑆𝐵𝑆𝐴𝐵) → (𝐴(+g𝑀)𝐴) = 𝐴)
2927, 28eqtrd 2655 . . . . . 6 ((𝐴𝑆𝐵𝑆𝐴𝐵) → ((𝐴(+g𝑀)𝐵)(+g𝑀)𝐴) = 𝐴)
3024, 26, 293eqtr4rd 2666 . . . . 5 ((𝐴𝑆𝐵𝑆𝐴𝐵) → ((𝐴(+g𝑀)𝐵)(+g𝑀)𝐴) = (𝐴(+g𝑀)(𝐵(+g𝑀)𝐴)))
31 simp2 1060 . . . . . . . 8 ((𝐴𝑆𝐵𝑆𝐴𝐵) → 𝐵𝑆)
321, 4, 5, 8sgrp2nmndlem3 17352 . . . . . . . 8 ((𝐵𝑆𝐵𝑆𝐴𝐵) → (𝐵(+g𝑀)𝐵) = 𝐵)
3331, 32syld3an1 1369 . . . . . . 7 ((𝐴𝑆𝐵𝑆𝐴𝐵) → (𝐵(+g𝑀)𝐵) = 𝐵)
3433oveq2d 6631 . . . . . 6 ((𝐴𝑆𝐵𝑆𝐴𝐵) → (𝐴(+g𝑀)(𝐵(+g𝑀)𝐵)) = (𝐴(+g𝑀)𝐵))
3518oveq1d 6630 . . . . . . . 8 ((𝐴𝑆𝐵𝑆) → ((𝐴(+g𝑀)𝐵)(+g𝑀)𝐵) = (𝐴(+g𝑀)𝐵))
3635, 18eqtrd 2655 . . . . . . 7 ((𝐴𝑆𝐵𝑆) → ((𝐴(+g𝑀)𝐵)(+g𝑀)𝐵) = 𝐴)
37363adant3 1079 . . . . . 6 ((𝐴𝑆𝐵𝑆𝐴𝐵) → ((𝐴(+g𝑀)𝐵)(+g𝑀)𝐵) = 𝐴)
3824, 34, 373eqtr4rd 2666 . . . . 5 ((𝐴𝑆𝐵𝑆𝐴𝐵) → ((𝐴(+g𝑀)𝐵)(+g𝑀)𝐵) = (𝐴(+g𝑀)(𝐵(+g𝑀)𝐵)))
3923, 30, 38jca32 557 . . . 4 ((𝐴𝑆𝐵𝑆𝐴𝐵) → ((((𝐴(+g𝑀)𝐴)(+g𝑀)𝐴) = (𝐴(+g𝑀)(𝐴(+g𝑀)𝐴)) ∧ ((𝐴(+g𝑀)𝐴)(+g𝑀)𝐵) = (𝐴(+g𝑀)(𝐴(+g𝑀)𝐵))) ∧ (((𝐴(+g𝑀)𝐵)(+g𝑀)𝐴) = (𝐴(+g𝑀)(𝐵(+g𝑀)𝐴)) ∧ ((𝐴(+g𝑀)𝐵)(+g𝑀)𝐵) = (𝐴(+g𝑀)(𝐵(+g𝑀)𝐵)))))
4025oveq1d 6630 . . . . . . 7 ((𝐴𝑆𝐵𝑆𝐴𝐵) → ((𝐵(+g𝑀)𝐴)(+g𝑀)𝐴) = (𝐵(+g𝑀)𝐴))
4128oveq2d 6631 . . . . . . 7 ((𝐴𝑆𝐵𝑆𝐴𝐵) → (𝐵(+g𝑀)(𝐴(+g𝑀)𝐴)) = (𝐵(+g𝑀)𝐴))
4240, 41eqtr4d 2658 . . . . . 6 ((𝐴𝑆𝐵𝑆𝐴𝐵) → ((𝐵(+g𝑀)𝐴)(+g𝑀)𝐴) = (𝐵(+g𝑀)(𝐴(+g𝑀)𝐴)))
4324oveq2d 6631 . . . . . . 7 ((𝐴𝑆𝐵𝑆𝐴𝐵) → (𝐵(+g𝑀)(𝐴(+g𝑀)𝐵)) = (𝐵(+g𝑀)𝐴))
4425oveq1d 6630 . . . . . . . 8 ((𝐴𝑆𝐵𝑆𝐴𝐵) → ((𝐵(+g𝑀)𝐴)(+g𝑀)𝐵) = (𝐵(+g𝑀)𝐵))
4544, 33eqtrd 2655 . . . . . . 7 ((𝐴𝑆𝐵𝑆𝐴𝐵) → ((𝐵(+g𝑀)𝐴)(+g𝑀)𝐵) = 𝐵)
4625, 43, 453eqtr4rd 2666 . . . . . 6 ((𝐴𝑆𝐵𝑆𝐴𝐵) → ((𝐵(+g𝑀)𝐴)(+g𝑀)𝐵) = (𝐵(+g𝑀)(𝐴(+g𝑀)𝐵)))
4742, 46jca 554 . . . . 5 ((𝐴𝑆𝐵𝑆𝐴𝐵) → (((𝐵(+g𝑀)𝐴)(+g𝑀)𝐴) = (𝐵(+g𝑀)(𝐴(+g𝑀)𝐴)) ∧ ((𝐵(+g𝑀)𝐴)(+g𝑀)𝐵) = (𝐵(+g𝑀)(𝐴(+g𝑀)𝐵))))
4825oveq2d 6631 . . . . . 6 ((𝐴𝑆𝐵𝑆𝐴𝐵) → (𝐵(+g𝑀)(𝐵(+g𝑀)𝐴)) = (𝐵(+g𝑀)𝐵))
4933oveq1d 6630 . . . . . . 7 ((𝐴𝑆𝐵𝑆𝐴𝐵) → ((𝐵(+g𝑀)𝐵)(+g𝑀)𝐴) = (𝐵(+g𝑀)𝐴))
5049, 25eqtrd 2655 . . . . . 6 ((𝐴𝑆𝐵𝑆𝐴𝐵) → ((𝐵(+g𝑀)𝐵)(+g𝑀)𝐴) = 𝐵)
5133, 48, 503eqtr4rd 2666 . . . . 5 ((𝐴𝑆𝐵𝑆𝐴𝐵) → ((𝐵(+g𝑀)𝐵)(+g𝑀)𝐴) = (𝐵(+g𝑀)(𝐵(+g𝑀)𝐴)))
5232oveq1d 6630 . . . . . . 7 ((𝐵𝑆𝐵𝑆𝐴𝐵) → ((𝐵(+g𝑀)𝐵)(+g𝑀)𝐵) = (𝐵(+g𝑀)𝐵))
5332oveq2d 6631 . . . . . . 7 ((𝐵𝑆𝐵𝑆𝐴𝐵) → (𝐵(+g𝑀)(𝐵(+g𝑀)𝐵)) = (𝐵(+g𝑀)𝐵))
5452, 53eqtr4d 2658 . . . . . 6 ((𝐵𝑆𝐵𝑆𝐴𝐵) → ((𝐵(+g𝑀)𝐵)(+g𝑀)𝐵) = (𝐵(+g𝑀)(𝐵(+g𝑀)𝐵)))
5531, 54syld3an1 1369 . . . . 5 ((𝐴𝑆𝐵𝑆𝐴𝐵) → ((𝐵(+g𝑀)𝐵)(+g𝑀)𝐵) = (𝐵(+g𝑀)(𝐵(+g𝑀)𝐵)))
5647, 51, 55jca32 557 . . . 4 ((𝐴𝑆𝐵𝑆𝐴𝐵) → ((((𝐵(+g𝑀)𝐴)(+g𝑀)𝐴) = (𝐵(+g𝑀)(𝐴(+g𝑀)𝐴)) ∧ ((𝐵(+g𝑀)𝐴)(+g𝑀)𝐵) = (𝐵(+g𝑀)(𝐴(+g𝑀)𝐵))) ∧ (((𝐵(+g𝑀)𝐵)(+g𝑀)𝐴) = (𝐵(+g𝑀)(𝐵(+g𝑀)𝐴)) ∧ ((𝐵(+g𝑀)𝐵)(+g𝑀)𝐵) = (𝐵(+g𝑀)(𝐵(+g𝑀)𝐵)))))
57 oveq1 6622 . . . . . . . . . 10 (𝑎 = 𝐴 → (𝑎(+g𝑀)𝑏) = (𝐴(+g𝑀)𝑏))
5857oveq1d 6630 . . . . . . . . 9 (𝑎 = 𝐴 → ((𝑎(+g𝑀)𝑏)(+g𝑀)𝑐) = ((𝐴(+g𝑀)𝑏)(+g𝑀)𝑐))
59 oveq1 6622 . . . . . . . . 9 (𝑎 = 𝐴 → (𝑎(+g𝑀)(𝑏(+g𝑀)𝑐)) = (𝐴(+g𝑀)(𝑏(+g𝑀)𝑐)))
6058, 59eqeq12d 2636 . . . . . . . 8 (𝑎 = 𝐴 → (((𝑎(+g𝑀)𝑏)(+g𝑀)𝑐) = (𝑎(+g𝑀)(𝑏(+g𝑀)𝑐)) ↔ ((𝐴(+g𝑀)𝑏)(+g𝑀)𝑐) = (𝐴(+g𝑀)(𝑏(+g𝑀)𝑐))))
61602ralbidv 2985 . . . . . . 7 (𝑎 = 𝐴 → (∀𝑏 ∈ {𝐴, 𝐵}∀𝑐 ∈ {𝐴, 𝐵} ((𝑎(+g𝑀)𝑏)(+g𝑀)𝑐) = (𝑎(+g𝑀)(𝑏(+g𝑀)𝑐)) ↔ ∀𝑏 ∈ {𝐴, 𝐵}∀𝑐 ∈ {𝐴, 𝐵} ((𝐴(+g𝑀)𝑏)(+g𝑀)𝑐) = (𝐴(+g𝑀)(𝑏(+g𝑀)𝑐))))
62 oveq1 6622 . . . . . . . . . 10 (𝑎 = 𝐵 → (𝑎(+g𝑀)𝑏) = (𝐵(+g𝑀)𝑏))
6362oveq1d 6630 . . . . . . . . 9 (𝑎 = 𝐵 → ((𝑎(+g𝑀)𝑏)(+g𝑀)𝑐) = ((𝐵(+g𝑀)𝑏)(+g𝑀)𝑐))
64 oveq1 6622 . . . . . . . . 9 (𝑎 = 𝐵 → (𝑎(+g𝑀)(𝑏(+g𝑀)𝑐)) = (𝐵(+g𝑀)(𝑏(+g𝑀)𝑐)))
6563, 64eqeq12d 2636 . . . . . . . 8 (𝑎 = 𝐵 → (((𝑎(+g𝑀)𝑏)(+g𝑀)𝑐) = (𝑎(+g𝑀)(𝑏(+g𝑀)𝑐)) ↔ ((𝐵(+g𝑀)𝑏)(+g𝑀)𝑐) = (𝐵(+g𝑀)(𝑏(+g𝑀)𝑐))))
66652ralbidv 2985 . . . . . . 7 (𝑎 = 𝐵 → (∀𝑏 ∈ {𝐴, 𝐵}∀𝑐 ∈ {𝐴, 𝐵} ((𝑎(+g𝑀)𝑏)(+g𝑀)𝑐) = (𝑎(+g𝑀)(𝑏(+g𝑀)𝑐)) ↔ ∀𝑏 ∈ {𝐴, 𝐵}∀𝑐 ∈ {𝐴, 𝐵} ((𝐵(+g𝑀)𝑏)(+g𝑀)𝑐) = (𝐵(+g𝑀)(𝑏(+g𝑀)𝑐))))
6761, 66ralprg 4212 . . . . . 6 ((𝐴𝑆𝐵𝑆) → (∀𝑎 ∈ {𝐴, 𝐵}∀𝑏 ∈ {𝐴, 𝐵}∀𝑐 ∈ {𝐴, 𝐵} ((𝑎(+g𝑀)𝑏)(+g𝑀)𝑐) = (𝑎(+g𝑀)(𝑏(+g𝑀)𝑐)) ↔ (∀𝑏 ∈ {𝐴, 𝐵}∀𝑐 ∈ {𝐴, 𝐵} ((𝐴(+g𝑀)𝑏)(+g𝑀)𝑐) = (𝐴(+g𝑀)(𝑏(+g𝑀)𝑐)) ∧ ∀𝑏 ∈ {𝐴, 𝐵}∀𝑐 ∈ {𝐴, 𝐵} ((𝐵(+g𝑀)𝑏)(+g𝑀)𝑐) = (𝐵(+g𝑀)(𝑏(+g𝑀)𝑐)))))
68 oveq2 6623 . . . . . . . . . . 11 (𝑏 = 𝐴 → (𝐴(+g𝑀)𝑏) = (𝐴(+g𝑀)𝐴))
6968oveq1d 6630 . . . . . . . . . 10 (𝑏 = 𝐴 → ((𝐴(+g𝑀)𝑏)(+g𝑀)𝑐) = ((𝐴(+g𝑀)𝐴)(+g𝑀)𝑐))
70 oveq1 6622 . . . . . . . . . . 11 (𝑏 = 𝐴 → (𝑏(+g𝑀)𝑐) = (𝐴(+g𝑀)𝑐))
7170oveq2d 6631 . . . . . . . . . 10 (𝑏 = 𝐴 → (𝐴(+g𝑀)(𝑏(+g𝑀)𝑐)) = (𝐴(+g𝑀)(𝐴(+g𝑀)𝑐)))
7269, 71eqeq12d 2636 . . . . . . . . 9 (𝑏 = 𝐴 → (((𝐴(+g𝑀)𝑏)(+g𝑀)𝑐) = (𝐴(+g𝑀)(𝑏(+g𝑀)𝑐)) ↔ ((𝐴(+g𝑀)𝐴)(+g𝑀)𝑐) = (𝐴(+g𝑀)(𝐴(+g𝑀)𝑐))))
7372ralbidv 2982 . . . . . . . 8 (𝑏 = 𝐴 → (∀𝑐 ∈ {𝐴, 𝐵} ((𝐴(+g𝑀)𝑏)(+g𝑀)𝑐) = (𝐴(+g𝑀)(𝑏(+g𝑀)𝑐)) ↔ ∀𝑐 ∈ {𝐴, 𝐵} ((𝐴(+g𝑀)𝐴)(+g𝑀)𝑐) = (𝐴(+g𝑀)(𝐴(+g𝑀)𝑐))))
74 oveq2 6623 . . . . . . . . . . 11 (𝑏 = 𝐵 → (𝐴(+g𝑀)𝑏) = (𝐴(+g𝑀)𝐵))
7574oveq1d 6630 . . . . . . . . . 10 (𝑏 = 𝐵 → ((𝐴(+g𝑀)𝑏)(+g𝑀)𝑐) = ((𝐴(+g𝑀)𝐵)(+g𝑀)𝑐))
76 oveq1 6622 . . . . . . . . . . 11 (𝑏 = 𝐵 → (𝑏(+g𝑀)𝑐) = (𝐵(+g𝑀)𝑐))
7776oveq2d 6631 . . . . . . . . . 10 (𝑏 = 𝐵 → (𝐴(+g𝑀)(𝑏(+g𝑀)𝑐)) = (𝐴(+g𝑀)(𝐵(+g𝑀)𝑐)))
7875, 77eqeq12d 2636 . . . . . . . . 9 (𝑏 = 𝐵 → (((𝐴(+g𝑀)𝑏)(+g𝑀)𝑐) = (𝐴(+g𝑀)(𝑏(+g𝑀)𝑐)) ↔ ((𝐴(+g𝑀)𝐵)(+g𝑀)𝑐) = (𝐴(+g𝑀)(𝐵(+g𝑀)𝑐))))
7978ralbidv 2982 . . . . . . . 8 (𝑏 = 𝐵 → (∀𝑐 ∈ {𝐴, 𝐵} ((𝐴(+g𝑀)𝑏)(+g𝑀)𝑐) = (𝐴(+g𝑀)(𝑏(+g𝑀)𝑐)) ↔ ∀𝑐 ∈ {𝐴, 𝐵} ((𝐴(+g𝑀)𝐵)(+g𝑀)𝑐) = (𝐴(+g𝑀)(𝐵(+g𝑀)𝑐))))
8073, 79ralprg 4212 . . . . . . 7 ((𝐴𝑆𝐵𝑆) → (∀𝑏 ∈ {𝐴, 𝐵}∀𝑐 ∈ {𝐴, 𝐵} ((𝐴(+g𝑀)𝑏)(+g𝑀)𝑐) = (𝐴(+g𝑀)(𝑏(+g𝑀)𝑐)) ↔ (∀𝑐 ∈ {𝐴, 𝐵} ((𝐴(+g𝑀)𝐴)(+g𝑀)𝑐) = (𝐴(+g𝑀)(𝐴(+g𝑀)𝑐)) ∧ ∀𝑐 ∈ {𝐴, 𝐵} ((𝐴(+g𝑀)𝐵)(+g𝑀)𝑐) = (𝐴(+g𝑀)(𝐵(+g𝑀)𝑐)))))
81 oveq2 6623 . . . . . . . . . . 11 (𝑏 = 𝐴 → (𝐵(+g𝑀)𝑏) = (𝐵(+g𝑀)𝐴))
8281oveq1d 6630 . . . . . . . . . 10 (𝑏 = 𝐴 → ((𝐵(+g𝑀)𝑏)(+g𝑀)𝑐) = ((𝐵(+g𝑀)𝐴)(+g𝑀)𝑐))
8370oveq2d 6631 . . . . . . . . . 10 (𝑏 = 𝐴 → (𝐵(+g𝑀)(𝑏(+g𝑀)𝑐)) = (𝐵(+g𝑀)(𝐴(+g𝑀)𝑐)))
8482, 83eqeq12d 2636 . . . . . . . . 9 (𝑏 = 𝐴 → (((𝐵(+g𝑀)𝑏)(+g𝑀)𝑐) = (𝐵(+g𝑀)(𝑏(+g𝑀)𝑐)) ↔ ((𝐵(+g𝑀)𝐴)(+g𝑀)𝑐) = (𝐵(+g𝑀)(𝐴(+g𝑀)𝑐))))
8584ralbidv 2982 . . . . . . . 8 (𝑏 = 𝐴 → (∀𝑐 ∈ {𝐴, 𝐵} ((𝐵(+g𝑀)𝑏)(+g𝑀)𝑐) = (𝐵(+g𝑀)(𝑏(+g𝑀)𝑐)) ↔ ∀𝑐 ∈ {𝐴, 𝐵} ((𝐵(+g𝑀)𝐴)(+g𝑀)𝑐) = (𝐵(+g𝑀)(𝐴(+g𝑀)𝑐))))
86 oveq2 6623 . . . . . . . . . . 11 (𝑏 = 𝐵 → (𝐵(+g𝑀)𝑏) = (𝐵(+g𝑀)𝐵))
8786oveq1d 6630 . . . . . . . . . 10 (𝑏 = 𝐵 → ((𝐵(+g𝑀)𝑏)(+g𝑀)𝑐) = ((𝐵(+g𝑀)𝐵)(+g𝑀)𝑐))
8876oveq2d 6631 . . . . . . . . . 10 (𝑏 = 𝐵 → (𝐵(+g𝑀)(𝑏(+g𝑀)𝑐)) = (𝐵(+g𝑀)(𝐵(+g𝑀)𝑐)))
8987, 88eqeq12d 2636 . . . . . . . . 9 (𝑏 = 𝐵 → (((𝐵(+g𝑀)𝑏)(+g𝑀)𝑐) = (𝐵(+g𝑀)(𝑏(+g𝑀)𝑐)) ↔ ((𝐵(+g𝑀)𝐵)(+g𝑀)𝑐) = (𝐵(+g𝑀)(𝐵(+g𝑀)𝑐))))
9089ralbidv 2982 . . . . . . . 8 (𝑏 = 𝐵 → (∀𝑐 ∈ {𝐴, 𝐵} ((𝐵(+g𝑀)𝑏)(+g𝑀)𝑐) = (𝐵(+g𝑀)(𝑏(+g𝑀)𝑐)) ↔ ∀𝑐 ∈ {𝐴, 𝐵} ((𝐵(+g𝑀)𝐵)(+g𝑀)𝑐) = (𝐵(+g𝑀)(𝐵(+g𝑀)𝑐))))
9185, 90ralprg 4212 . . . . . . 7 ((𝐴𝑆𝐵𝑆) → (∀𝑏 ∈ {𝐴, 𝐵}∀𝑐 ∈ {𝐴, 𝐵} ((𝐵(+g𝑀)𝑏)(+g𝑀)𝑐) = (𝐵(+g𝑀)(𝑏(+g𝑀)𝑐)) ↔ (∀𝑐 ∈ {𝐴, 𝐵} ((𝐵(+g𝑀)𝐴)(+g𝑀)𝑐) = (𝐵(+g𝑀)(𝐴(+g𝑀)𝑐)) ∧ ∀𝑐 ∈ {𝐴, 𝐵} ((𝐵(+g𝑀)𝐵)(+g𝑀)𝑐) = (𝐵(+g𝑀)(𝐵(+g𝑀)𝑐)))))
9280, 91anbi12d 746 . . . . . 6 ((𝐴𝑆𝐵𝑆) → ((∀𝑏 ∈ {𝐴, 𝐵}∀𝑐 ∈ {𝐴, 𝐵} ((𝐴(+g𝑀)𝑏)(+g𝑀)𝑐) = (𝐴(+g𝑀)(𝑏(+g𝑀)𝑐)) ∧ ∀𝑏 ∈ {𝐴, 𝐵}∀𝑐 ∈ {𝐴, 𝐵} ((𝐵(+g𝑀)𝑏)(+g𝑀)𝑐) = (𝐵(+g𝑀)(𝑏(+g𝑀)𝑐))) ↔ ((∀𝑐 ∈ {𝐴, 𝐵} ((𝐴(+g𝑀)𝐴)(+g𝑀)𝑐) = (𝐴(+g𝑀)(𝐴(+g𝑀)𝑐)) ∧ ∀𝑐 ∈ {𝐴, 𝐵} ((𝐴(+g𝑀)𝐵)(+g𝑀)𝑐) = (𝐴(+g𝑀)(𝐵(+g𝑀)𝑐))) ∧ (∀𝑐 ∈ {𝐴, 𝐵} ((𝐵(+g𝑀)𝐴)(+g𝑀)𝑐) = (𝐵(+g𝑀)(𝐴(+g𝑀)𝑐)) ∧ ∀𝑐 ∈ {𝐴, 𝐵} ((𝐵(+g𝑀)𝐵)(+g𝑀)𝑐) = (𝐵(+g𝑀)(𝐵(+g𝑀)𝑐))))))
93 oveq2 6623 . . . . . . . . . 10 (𝑐 = 𝐴 → ((𝐴(+g𝑀)𝐴)(+g𝑀)𝑐) = ((𝐴(+g𝑀)𝐴)(+g𝑀)𝐴))
94 oveq2 6623 . . . . . . . . . . 11 (𝑐 = 𝐴 → (𝐴(+g𝑀)𝑐) = (𝐴(+g𝑀)𝐴))
9594oveq2d 6631 . . . . . . . . . 10 (𝑐 = 𝐴 → (𝐴(+g𝑀)(𝐴(+g𝑀)𝑐)) = (𝐴(+g𝑀)(𝐴(+g𝑀)𝐴)))
9693, 95eqeq12d 2636 . . . . . . . . 9 (𝑐 = 𝐴 → (((𝐴(+g𝑀)𝐴)(+g𝑀)𝑐) = (𝐴(+g𝑀)(𝐴(+g𝑀)𝑐)) ↔ ((𝐴(+g𝑀)𝐴)(+g𝑀)𝐴) = (𝐴(+g𝑀)(𝐴(+g𝑀)𝐴))))
97 oveq2 6623 . . . . . . . . . 10 (𝑐 = 𝐵 → ((𝐴(+g𝑀)𝐴)(+g𝑀)𝑐) = ((𝐴(+g𝑀)𝐴)(+g𝑀)𝐵))
98 oveq2 6623 . . . . . . . . . . 11 (𝑐 = 𝐵 → (𝐴(+g𝑀)𝑐) = (𝐴(+g𝑀)𝐵))
9998oveq2d 6631 . . . . . . . . . 10 (𝑐 = 𝐵 → (𝐴(+g𝑀)(𝐴(+g𝑀)𝑐)) = (𝐴(+g𝑀)(𝐴(+g𝑀)𝐵)))
10097, 99eqeq12d 2636 . . . . . . . . 9 (𝑐 = 𝐵 → (((𝐴(+g𝑀)𝐴)(+g𝑀)𝑐) = (𝐴(+g𝑀)(𝐴(+g𝑀)𝑐)) ↔ ((𝐴(+g𝑀)𝐴)(+g𝑀)𝐵) = (𝐴(+g𝑀)(𝐴(+g𝑀)𝐵))))
10196, 100ralprg 4212 . . . . . . . 8 ((𝐴𝑆𝐵𝑆) → (∀𝑐 ∈ {𝐴, 𝐵} ((𝐴(+g𝑀)𝐴)(+g𝑀)𝑐) = (𝐴(+g𝑀)(𝐴(+g𝑀)𝑐)) ↔ (((𝐴(+g𝑀)𝐴)(+g𝑀)𝐴) = (𝐴(+g𝑀)(𝐴(+g𝑀)𝐴)) ∧ ((𝐴(+g𝑀)𝐴)(+g𝑀)𝐵) = (𝐴(+g𝑀)(𝐴(+g𝑀)𝐵)))))
102 oveq2 6623 . . . . . . . . . 10 (𝑐 = 𝐴 → ((𝐴(+g𝑀)𝐵)(+g𝑀)𝑐) = ((𝐴(+g𝑀)𝐵)(+g𝑀)𝐴))
103 oveq2 6623 . . . . . . . . . . 11 (𝑐 = 𝐴 → (𝐵(+g𝑀)𝑐) = (𝐵(+g𝑀)𝐴))
104103oveq2d 6631 . . . . . . . . . 10 (𝑐 = 𝐴 → (𝐴(+g𝑀)(𝐵(+g𝑀)𝑐)) = (𝐴(+g𝑀)(𝐵(+g𝑀)𝐴)))
105102, 104eqeq12d 2636 . . . . . . . . 9 (𝑐 = 𝐴 → (((𝐴(+g𝑀)𝐵)(+g𝑀)𝑐) = (𝐴(+g𝑀)(𝐵(+g𝑀)𝑐)) ↔ ((𝐴(+g𝑀)𝐵)(+g𝑀)𝐴) = (𝐴(+g𝑀)(𝐵(+g𝑀)𝐴))))
106 oveq2 6623 . . . . . . . . . 10 (𝑐 = 𝐵 → ((𝐴(+g𝑀)𝐵)(+g𝑀)𝑐) = ((𝐴(+g𝑀)𝐵)(+g𝑀)𝐵))
107 oveq2 6623 . . . . . . . . . . 11 (𝑐 = 𝐵 → (𝐵(+g𝑀)𝑐) = (𝐵(+g𝑀)𝐵))
108107oveq2d 6631 . . . . . . . . . 10 (𝑐 = 𝐵 → (𝐴(+g𝑀)(𝐵(+g𝑀)𝑐)) = (𝐴(+g𝑀)(𝐵(+g𝑀)𝐵)))
109106, 108eqeq12d 2636 . . . . . . . . 9 (𝑐 = 𝐵 → (((𝐴(+g𝑀)𝐵)(+g𝑀)𝑐) = (𝐴(+g𝑀)(𝐵(+g𝑀)𝑐)) ↔ ((𝐴(+g𝑀)𝐵)(+g𝑀)𝐵) = (𝐴(+g𝑀)(𝐵(+g𝑀)𝐵))))
110105, 109ralprg 4212 . . . . . . . 8 ((𝐴𝑆𝐵𝑆) → (∀𝑐 ∈ {𝐴, 𝐵} ((𝐴(+g𝑀)𝐵)(+g𝑀)𝑐) = (𝐴(+g𝑀)(𝐵(+g𝑀)𝑐)) ↔ (((𝐴(+g𝑀)𝐵)(+g𝑀)𝐴) = (𝐴(+g𝑀)(𝐵(+g𝑀)𝐴)) ∧ ((𝐴(+g𝑀)𝐵)(+g𝑀)𝐵) = (𝐴(+g𝑀)(𝐵(+g𝑀)𝐵)))))
111101, 110anbi12d 746 . . . . . . 7 ((𝐴𝑆𝐵𝑆) → ((∀𝑐 ∈ {𝐴, 𝐵} ((𝐴(+g𝑀)𝐴)(+g𝑀)𝑐) = (𝐴(+g𝑀)(𝐴(+g𝑀)𝑐)) ∧ ∀𝑐 ∈ {𝐴, 𝐵} ((𝐴(+g𝑀)𝐵)(+g𝑀)𝑐) = (𝐴(+g𝑀)(𝐵(+g𝑀)𝑐))) ↔ ((((𝐴(+g𝑀)𝐴)(+g𝑀)𝐴) = (𝐴(+g𝑀)(𝐴(+g𝑀)𝐴)) ∧ ((𝐴(+g𝑀)𝐴)(+g𝑀)𝐵) = (𝐴(+g𝑀)(𝐴(+g𝑀)𝐵))) ∧ (((𝐴(+g𝑀)𝐵)(+g𝑀)𝐴) = (𝐴(+g𝑀)(𝐵(+g𝑀)𝐴)) ∧ ((𝐴(+g𝑀)𝐵)(+g𝑀)𝐵) = (𝐴(+g𝑀)(𝐵(+g𝑀)𝐵))))))
112 oveq2 6623 . . . . . . . . . 10 (𝑐 = 𝐴 → ((𝐵(+g𝑀)𝐴)(+g𝑀)𝑐) = ((𝐵(+g𝑀)𝐴)(+g𝑀)𝐴))
11394oveq2d 6631 . . . . . . . . . 10 (𝑐 = 𝐴 → (𝐵(+g𝑀)(𝐴(+g𝑀)𝑐)) = (𝐵(+g𝑀)(𝐴(+g𝑀)𝐴)))
114112, 113eqeq12d 2636 . . . . . . . . 9 (𝑐 = 𝐴 → (((𝐵(+g𝑀)𝐴)(+g𝑀)𝑐) = (𝐵(+g𝑀)(𝐴(+g𝑀)𝑐)) ↔ ((𝐵(+g𝑀)𝐴)(+g𝑀)𝐴) = (𝐵(+g𝑀)(𝐴(+g𝑀)𝐴))))
115 oveq2 6623 . . . . . . . . . 10 (𝑐 = 𝐵 → ((𝐵(+g𝑀)𝐴)(+g𝑀)𝑐) = ((𝐵(+g𝑀)𝐴)(+g𝑀)𝐵))
11698oveq2d 6631 . . . . . . . . . 10 (𝑐 = 𝐵 → (𝐵(+g𝑀)(𝐴(+g𝑀)𝑐)) = (𝐵(+g𝑀)(𝐴(+g𝑀)𝐵)))
117115, 116eqeq12d 2636 . . . . . . . . 9 (𝑐 = 𝐵 → (((𝐵(+g𝑀)𝐴)(+g𝑀)𝑐) = (𝐵(+g𝑀)(𝐴(+g𝑀)𝑐)) ↔ ((𝐵(+g𝑀)𝐴)(+g𝑀)𝐵) = (𝐵(+g𝑀)(𝐴(+g𝑀)𝐵))))
118114, 117ralprg 4212 . . . . . . . 8 ((𝐴𝑆𝐵𝑆) → (∀𝑐 ∈ {𝐴, 𝐵} ((𝐵(+g𝑀)𝐴)(+g𝑀)𝑐) = (𝐵(+g𝑀)(𝐴(+g𝑀)𝑐)) ↔ (((𝐵(+g𝑀)𝐴)(+g𝑀)𝐴) = (𝐵(+g𝑀)(𝐴(+g𝑀)𝐴)) ∧ ((𝐵(+g𝑀)𝐴)(+g𝑀)𝐵) = (𝐵(+g𝑀)(𝐴(+g𝑀)𝐵)))))
119 oveq2 6623 . . . . . . . . . 10 (𝑐 = 𝐴 → ((𝐵(+g𝑀)𝐵)(+g𝑀)𝑐) = ((𝐵(+g𝑀)𝐵)(+g𝑀)𝐴))
120103oveq2d 6631 . . . . . . . . . 10 (𝑐 = 𝐴 → (𝐵(+g𝑀)(𝐵(+g𝑀)𝑐)) = (𝐵(+g𝑀)(𝐵(+g𝑀)𝐴)))
121119, 120eqeq12d 2636 . . . . . . . . 9 (𝑐 = 𝐴 → (((𝐵(+g𝑀)𝐵)(+g𝑀)𝑐) = (𝐵(+g𝑀)(𝐵(+g𝑀)𝑐)) ↔ ((𝐵(+g𝑀)𝐵)(+g𝑀)𝐴) = (𝐵(+g𝑀)(𝐵(+g𝑀)𝐴))))
122 oveq2 6623 . . . . . . . . . 10 (𝑐 = 𝐵 → ((𝐵(+g𝑀)𝐵)(+g𝑀)𝑐) = ((𝐵(+g𝑀)𝐵)(+g𝑀)𝐵))
123107oveq2d 6631 . . . . . . . . . 10 (𝑐 = 𝐵 → (𝐵(+g𝑀)(𝐵(+g𝑀)𝑐)) = (𝐵(+g𝑀)(𝐵(+g𝑀)𝐵)))
124122, 123eqeq12d 2636 . . . . . . . . 9 (𝑐 = 𝐵 → (((𝐵(+g𝑀)𝐵)(+g𝑀)𝑐) = (𝐵(+g𝑀)(𝐵(+g𝑀)𝑐)) ↔ ((𝐵(+g𝑀)𝐵)(+g𝑀)𝐵) = (𝐵(+g𝑀)(𝐵(+g𝑀)𝐵))))
125121, 124ralprg 4212 . . . . . . . 8 ((𝐴𝑆𝐵𝑆) → (∀𝑐 ∈ {𝐴, 𝐵} ((𝐵(+g𝑀)𝐵)(+g𝑀)𝑐) = (𝐵(+g𝑀)(𝐵(+g𝑀)𝑐)) ↔ (((𝐵(+g𝑀)𝐵)(+g𝑀)𝐴) = (𝐵(+g𝑀)(𝐵(+g𝑀)𝐴)) ∧ ((𝐵(+g𝑀)𝐵)(+g𝑀)𝐵) = (𝐵(+g𝑀)(𝐵(+g𝑀)𝐵)))))
126118, 125anbi12d 746 . . . . . . 7 ((𝐴𝑆𝐵𝑆) → ((∀𝑐 ∈ {𝐴, 𝐵} ((𝐵(+g𝑀)𝐴)(+g𝑀)𝑐) = (𝐵(+g𝑀)(𝐴(+g𝑀)𝑐)) ∧ ∀𝑐 ∈ {𝐴, 𝐵} ((𝐵(+g𝑀)𝐵)(+g𝑀)𝑐) = (𝐵(+g𝑀)(𝐵(+g𝑀)𝑐))) ↔ ((((𝐵(+g𝑀)𝐴)(+g𝑀)𝐴) = (𝐵(+g𝑀)(𝐴(+g𝑀)𝐴)) ∧ ((𝐵(+g𝑀)𝐴)(+g𝑀)𝐵) = (𝐵(+g𝑀)(𝐴(+g𝑀)𝐵))) ∧ (((𝐵(+g𝑀)𝐵)(+g𝑀)𝐴) = (𝐵(+g𝑀)(𝐵(+g𝑀)𝐴)) ∧ ((𝐵(+g𝑀)𝐵)(+g𝑀)𝐵) = (𝐵(+g𝑀)(𝐵(+g𝑀)𝐵))))))
127111, 126anbi12d 746 . . . . . 6 ((𝐴𝑆𝐵𝑆) → (((∀𝑐 ∈ {𝐴, 𝐵} ((𝐴(+g𝑀)𝐴)(+g𝑀)𝑐) = (𝐴(+g𝑀)(𝐴(+g𝑀)𝑐)) ∧ ∀𝑐 ∈ {𝐴, 𝐵} ((𝐴(+g𝑀)𝐵)(+g𝑀)𝑐) = (𝐴(+g𝑀)(𝐵(+g𝑀)𝑐))) ∧ (∀𝑐 ∈ {𝐴, 𝐵} ((𝐵(+g𝑀)𝐴)(+g𝑀)𝑐) = (𝐵(+g𝑀)(𝐴(+g𝑀)𝑐)) ∧ ∀𝑐 ∈ {𝐴, 𝐵} ((𝐵(+g𝑀)𝐵)(+g𝑀)𝑐) = (𝐵(+g𝑀)(𝐵(+g𝑀)𝑐)))) ↔ (((((𝐴(+g𝑀)𝐴)(+g𝑀)𝐴) = (𝐴(+g𝑀)(𝐴(+g𝑀)𝐴)) ∧ ((𝐴(+g𝑀)𝐴)(+g𝑀)𝐵) = (𝐴(+g𝑀)(𝐴(+g𝑀)𝐵))) ∧ (((𝐴(+g𝑀)𝐵)(+g𝑀)𝐴) = (𝐴(+g𝑀)(𝐵(+g𝑀)𝐴)) ∧ ((𝐴(+g𝑀)𝐵)(+g𝑀)𝐵) = (𝐴(+g𝑀)(𝐵(+g𝑀)𝐵)))) ∧ ((((𝐵(+g𝑀)𝐴)(+g𝑀)𝐴) = (𝐵(+g𝑀)(𝐴(+g𝑀)𝐴)) ∧ ((𝐵(+g𝑀)𝐴)(+g𝑀)𝐵) = (𝐵(+g𝑀)(𝐴(+g𝑀)𝐵))) ∧ (((𝐵(+g𝑀)𝐵)(+g𝑀)𝐴) = (𝐵(+g𝑀)(𝐵(+g𝑀)𝐴)) ∧ ((𝐵(+g𝑀)𝐵)(+g𝑀)𝐵) = (𝐵(+g𝑀)(𝐵(+g𝑀)𝐵)))))))
12867, 92, 1273bitrd 294 . . . . 5 ((𝐴𝑆𝐵𝑆) → (∀𝑎 ∈ {𝐴, 𝐵}∀𝑏 ∈ {𝐴, 𝐵}∀𝑐 ∈ {𝐴, 𝐵} ((𝑎(+g𝑀)𝑏)(+g𝑀)𝑐) = (𝑎(+g𝑀)(𝑏(+g𝑀)𝑐)) ↔ (((((𝐴(+g𝑀)𝐴)(+g𝑀)𝐴) = (𝐴(+g𝑀)(𝐴(+g𝑀)𝐴)) ∧ ((𝐴(+g𝑀)𝐴)(+g𝑀)𝐵) = (𝐴(+g𝑀)(𝐴(+g𝑀)𝐵))) ∧ (((𝐴(+g𝑀)𝐵)(+g𝑀)𝐴) = (𝐴(+g𝑀)(𝐵(+g𝑀)𝐴)) ∧ ((𝐴(+g𝑀)𝐵)(+g𝑀)𝐵) = (𝐴(+g𝑀)(𝐵(+g𝑀)𝐵)))) ∧ ((((𝐵(+g𝑀)𝐴)(+g𝑀)𝐴) = (𝐵(+g𝑀)(𝐴(+g𝑀)𝐴)) ∧ ((𝐵(+g𝑀)𝐴)(+g𝑀)𝐵) = (𝐵(+g𝑀)(𝐴(+g𝑀)𝐵))) ∧ (((𝐵(+g𝑀)𝐵)(+g𝑀)𝐴) = (𝐵(+g𝑀)(𝐵(+g𝑀)𝐴)) ∧ ((𝐵(+g𝑀)𝐵)(+g𝑀)𝐵) = (𝐵(+g𝑀)(𝐵(+g𝑀)𝐵)))))))
1291283adant3 1079 . . . 4 ((𝐴𝑆𝐵𝑆𝐴𝐵) → (∀𝑎 ∈ {𝐴, 𝐵}∀𝑏 ∈ {𝐴, 𝐵}∀𝑐 ∈ {𝐴, 𝐵} ((𝑎(+g𝑀)𝑏)(+g𝑀)𝑐) = (𝑎(+g𝑀)(𝑏(+g𝑀)𝑐)) ↔ (((((𝐴(+g𝑀)𝐴)(+g𝑀)𝐴) = (𝐴(+g𝑀)(𝐴(+g𝑀)𝐴)) ∧ ((𝐴(+g𝑀)𝐴)(+g𝑀)𝐵) = (𝐴(+g𝑀)(𝐴(+g𝑀)𝐵))) ∧ (((𝐴(+g𝑀)𝐵)(+g𝑀)𝐴) = (𝐴(+g𝑀)(𝐵(+g𝑀)𝐴)) ∧ ((𝐴(+g𝑀)𝐵)(+g𝑀)𝐵) = (𝐴(+g𝑀)(𝐵(+g𝑀)𝐵)))) ∧ ((((𝐵(+g𝑀)𝐴)(+g𝑀)𝐴) = (𝐵(+g𝑀)(𝐴(+g𝑀)𝐴)) ∧ ((𝐵(+g𝑀)𝐴)(+g𝑀)𝐵) = (𝐵(+g𝑀)(𝐴(+g𝑀)𝐵))) ∧ (((𝐵(+g𝑀)𝐵)(+g𝑀)𝐴) = (𝐵(+g𝑀)(𝐵(+g𝑀)𝐴)) ∧ ((𝐵(+g𝑀)𝐵)(+g𝑀)𝐵) = (𝐵(+g𝑀)(𝐵(+g𝑀)𝐵)))))))
13039, 56, 129mpbir2and 956 . . 3 ((𝐴𝑆𝐵𝑆𝐴𝐵) → ∀𝑎 ∈ {𝐴, 𝐵}∀𝑏 ∈ {𝐴, 𝐵}∀𝑐 ∈ {𝐴, 𝐵} ((𝑎(+g𝑀)𝑏)(+g𝑀)𝑐) = (𝑎(+g𝑀)(𝑏(+g𝑀)𝑐)))
1312, 130syl 17 . 2 ((#‘𝑆) = 2 → ∀𝑎 ∈ {𝐴, 𝐵}∀𝑏 ∈ {𝐴, 𝐵}∀𝑐 ∈ {𝐴, 𝐵} ((𝑎(+g𝑀)𝑏)(+g𝑀)𝑐) = (𝑎(+g𝑀)(𝑏(+g𝑀)𝑐)))
1324, 1eqtr2i 2644 . . 3 {𝐴, 𝐵} = (Base‘𝑀)
133132, 8issgrp 17225 . 2 (𝑀 ∈ SGrp ↔ (𝑀 ∈ Mgm ∧ ∀𝑎 ∈ {𝐴, 𝐵}∀𝑏 ∈ {𝐴, 𝐵}∀𝑐 ∈ {𝐴, 𝐵} ((𝑎(+g𝑀)𝑏)(+g𝑀)𝑐) = (𝑎(+g𝑀)(𝑏(+g𝑀)𝑐))))
1347, 131, 133sylanbrc 697 1 ((#‘𝑆) = 2 → 𝑀 ∈ SGrp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wral 2908  ifcif 4064  {cpr 4157  cfv 5857  (class class class)co 6615  cmpt2 6617  2c2 11030  #chash 13073  Basecbs 15800  +gcplusg 15881  Mgmcmgm 17180  SGrpcsgrp 17223
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-oadd 7524  df-er 7702  df-en 7916  df-dom 7917  df-sdom 7918  df-fin 7919  df-card 8725  df-cda 8950  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-nn 10981  df-2 11039  df-n0 11253  df-z 11338  df-uz 11648  df-fz 12285  df-hash 13074  df-mgm 17182  df-sgrp 17224
This theorem is referenced by:  sgrp2nmnd  17357  sgrpnmndex  17359
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