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Theorem sgrp2nmndlem4 17238
Description: Lemma 4 for sgrp2nmnd 17240: 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 13047 . . 3 ((#‘𝑆) = 2 → (𝐴𝑆𝐵𝑆𝐴𝐵))
3 3simpa 1051 . . 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 17233 . . 3 ((𝐴𝑆𝐵𝑆) → 𝑀 ∈ Mgm)
72, 3, 63syl 18 . 2 ((#‘𝑆) = 2 → 𝑀 ∈ Mgm)
8 eqid 2610 . . . . . . . . . . 11 (+g𝑀) = (+g𝑀)
91, 4, 5, 8sgrp2nmndlem2 17234 . . . . . . . . . 10 ((𝐴𝑆𝐴𝑆) → (𝐴(+g𝑀)𝐴) = 𝐴)
109oveq1d 6564 . . . . . . . . 9 ((𝐴𝑆𝐴𝑆) → ((𝐴(+g𝑀)𝐴)(+g𝑀)𝐴) = (𝐴(+g𝑀)𝐴))
119oveq2d 6565 . . . . . . . . 9 ((𝐴𝑆𝐴𝑆) → (𝐴(+g𝑀)(𝐴(+g𝑀)𝐴)) = (𝐴(+g𝑀)𝐴))
1210, 11eqtr4d 2647 . . . . . . . 8 ((𝐴𝑆𝐴𝑆) → ((𝐴(+g𝑀)𝐴)(+g𝑀)𝐴) = (𝐴(+g𝑀)(𝐴(+g𝑀)𝐴)))
1312anidms 675 . . . . . . 7 (𝐴𝑆 → ((𝐴(+g𝑀)𝐴)(+g𝑀)𝐴) = (𝐴(+g𝑀)(𝐴(+g𝑀)𝐴)))
14133ad2ant1 1075 . . . . . 6 ((𝐴𝑆𝐵𝑆𝐴𝐵) → ((𝐴(+g𝑀)𝐴)(+g𝑀)𝐴) = (𝐴(+g𝑀)(𝐴(+g𝑀)𝐴)))
159anidms 675 . . . . . . . . . 10 (𝐴𝑆 → (𝐴(+g𝑀)𝐴) = 𝐴)
1615adantr 480 . . . . . . . . 9 ((𝐴𝑆𝐵𝑆) → (𝐴(+g𝑀)𝐴) = 𝐴)
1716oveq1d 6564 . . . . . . . 8 ((𝐴𝑆𝐵𝑆) → ((𝐴(+g𝑀)𝐴)(+g𝑀)𝐵) = (𝐴(+g𝑀)𝐵))
181, 4, 5, 8sgrp2nmndlem2 17234 . . . . . . . . . 10 ((𝐴𝑆𝐵𝑆) → (𝐴(+g𝑀)𝐵) = 𝐴)
1918oveq2d 6565 . . . . . . . . 9 ((𝐴𝑆𝐵𝑆) → (𝐴(+g𝑀)(𝐴(+g𝑀)𝐵)) = (𝐴(+g𝑀)𝐴))
2016, 19, 183eqtr4rd 2655 . . . . . . . 8 ((𝐴𝑆𝐵𝑆) → (𝐴(+g𝑀)𝐵) = (𝐴(+g𝑀)(𝐴(+g𝑀)𝐵)))
2117, 20eqtrd 2644 . . . . . . 7 ((𝐴𝑆𝐵𝑆) → ((𝐴(+g𝑀)𝐴)(+g𝑀)𝐵) = (𝐴(+g𝑀)(𝐴(+g𝑀)𝐵)))
22213adant3 1074 . . . . . 6 ((𝐴𝑆𝐵𝑆𝐴𝐵) → ((𝐴(+g𝑀)𝐴)(+g𝑀)𝐵) = (𝐴(+g𝑀)(𝐴(+g𝑀)𝐵)))
2314, 22jca 553 . . . . 5 ((𝐴𝑆𝐵𝑆𝐴𝐵) → (((𝐴(+g𝑀)𝐴)(+g𝑀)𝐴) = (𝐴(+g𝑀)(𝐴(+g𝑀)𝐴)) ∧ ((𝐴(+g𝑀)𝐴)(+g𝑀)𝐵) = (𝐴(+g𝑀)(𝐴(+g𝑀)𝐵))))
24183adant3 1074 . . . . . 6 ((𝐴𝑆𝐵𝑆𝐴𝐵) → (𝐴(+g𝑀)𝐵) = 𝐴)
251, 4, 5, 8sgrp2nmndlem3 17235 . . . . . . 7 ((𝐴𝑆𝐵𝑆𝐴𝐵) → (𝐵(+g𝑀)𝐴) = 𝐵)
2625oveq2d 6565 . . . . . 6 ((𝐴𝑆𝐵𝑆𝐴𝐵) → (𝐴(+g𝑀)(𝐵(+g𝑀)𝐴)) = (𝐴(+g𝑀)𝐵))
2724oveq1d 6564 . . . . . . 7 ((𝐴𝑆𝐵𝑆𝐴𝐵) → ((𝐴(+g𝑀)𝐵)(+g𝑀)𝐴) = (𝐴(+g𝑀)𝐴))
28153ad2ant1 1075 . . . . . . 7 ((𝐴𝑆𝐵𝑆𝐴𝐵) → (𝐴(+g𝑀)𝐴) = 𝐴)
2927, 28eqtrd 2644 . . . . . 6 ((𝐴𝑆𝐵𝑆𝐴𝐵) → ((𝐴(+g𝑀)𝐵)(+g𝑀)𝐴) = 𝐴)
3024, 26, 293eqtr4rd 2655 . . . . 5 ((𝐴𝑆𝐵𝑆𝐴𝐵) → ((𝐴(+g𝑀)𝐵)(+g𝑀)𝐴) = (𝐴(+g𝑀)(𝐵(+g𝑀)𝐴)))
31 simp2 1055 . . . . . . . 8 ((𝐴𝑆𝐵𝑆𝐴𝐵) → 𝐵𝑆)
321, 4, 5, 8sgrp2nmndlem3 17235 . . . . . . . 8 ((𝐵𝑆𝐵𝑆𝐴𝐵) → (𝐵(+g𝑀)𝐵) = 𝐵)
3331, 32syld3an1 1364 . . . . . . 7 ((𝐴𝑆𝐵𝑆𝐴𝐵) → (𝐵(+g𝑀)𝐵) = 𝐵)
3433oveq2d 6565 . . . . . 6 ((𝐴𝑆𝐵𝑆𝐴𝐵) → (𝐴(+g𝑀)(𝐵(+g𝑀)𝐵)) = (𝐴(+g𝑀)𝐵))
3518oveq1d 6564 . . . . . . . 8 ((𝐴𝑆𝐵𝑆) → ((𝐴(+g𝑀)𝐵)(+g𝑀)𝐵) = (𝐴(+g𝑀)𝐵))
3635, 18eqtrd 2644 . . . . . . 7 ((𝐴𝑆𝐵𝑆) → ((𝐴(+g𝑀)𝐵)(+g𝑀)𝐵) = 𝐴)
37363adant3 1074 . . . . . 6 ((𝐴𝑆𝐵𝑆𝐴𝐵) → ((𝐴(+g𝑀)𝐵)(+g𝑀)𝐵) = 𝐴)
3824, 34, 373eqtr4rd 2655 . . . . 5 ((𝐴𝑆𝐵𝑆𝐴𝐵) → ((𝐴(+g𝑀)𝐵)(+g𝑀)𝐵) = (𝐴(+g𝑀)(𝐵(+g𝑀)𝐵)))
3923, 30, 38jca32 556 . . . 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 6564 . . . . . . 7 ((𝐴𝑆𝐵𝑆𝐴𝐵) → ((𝐵(+g𝑀)𝐴)(+g𝑀)𝐴) = (𝐵(+g𝑀)𝐴))
4128oveq2d 6565 . . . . . . 7 ((𝐴𝑆𝐵𝑆𝐴𝐵) → (𝐵(+g𝑀)(𝐴(+g𝑀)𝐴)) = (𝐵(+g𝑀)𝐴))
4240, 41eqtr4d 2647 . . . . . 6 ((𝐴𝑆𝐵𝑆𝐴𝐵) → ((𝐵(+g𝑀)𝐴)(+g𝑀)𝐴) = (𝐵(+g𝑀)(𝐴(+g𝑀)𝐴)))
4324oveq2d 6565 . . . . . . 7 ((𝐴𝑆𝐵𝑆𝐴𝐵) → (𝐵(+g𝑀)(𝐴(+g𝑀)𝐵)) = (𝐵(+g𝑀)𝐴))
4425oveq1d 6564 . . . . . . . 8 ((𝐴𝑆𝐵𝑆𝐴𝐵) → ((𝐵(+g𝑀)𝐴)(+g𝑀)𝐵) = (𝐵(+g𝑀)𝐵))
4544, 33eqtrd 2644 . . . . . . 7 ((𝐴𝑆𝐵𝑆𝐴𝐵) → ((𝐵(+g𝑀)𝐴)(+g𝑀)𝐵) = 𝐵)
4625, 43, 453eqtr4rd 2655 . . . . . 6 ((𝐴𝑆𝐵𝑆𝐴𝐵) → ((𝐵(+g𝑀)𝐴)(+g𝑀)𝐵) = (𝐵(+g𝑀)(𝐴(+g𝑀)𝐵)))
4742, 46jca 553 . . . . 5 ((𝐴𝑆𝐵𝑆𝐴𝐵) → (((𝐵(+g𝑀)𝐴)(+g𝑀)𝐴) = (𝐵(+g𝑀)(𝐴(+g𝑀)𝐴)) ∧ ((𝐵(+g𝑀)𝐴)(+g𝑀)𝐵) = (𝐵(+g𝑀)(𝐴(+g𝑀)𝐵))))
4825oveq2d 6565 . . . . . 6 ((𝐴𝑆𝐵𝑆𝐴𝐵) → (𝐵(+g𝑀)(𝐵(+g𝑀)𝐴)) = (𝐵(+g𝑀)𝐵))
4933oveq1d 6564 . . . . . . 7 ((𝐴𝑆𝐵𝑆𝐴𝐵) → ((𝐵(+g𝑀)𝐵)(+g𝑀)𝐴) = (𝐵(+g𝑀)𝐴))
5049, 25eqtrd 2644 . . . . . 6 ((𝐴𝑆𝐵𝑆𝐴𝐵) → ((𝐵(+g𝑀)𝐵)(+g𝑀)𝐴) = 𝐵)
5133, 48, 503eqtr4rd 2655 . . . . 5 ((𝐴𝑆𝐵𝑆𝐴𝐵) → ((𝐵(+g𝑀)𝐵)(+g𝑀)𝐴) = (𝐵(+g𝑀)(𝐵(+g𝑀)𝐴)))
5232oveq1d 6564 . . . . . . 7 ((𝐵𝑆𝐵𝑆𝐴𝐵) → ((𝐵(+g𝑀)𝐵)(+g𝑀)𝐵) = (𝐵(+g𝑀)𝐵))
5332oveq2d 6565 . . . . . . 7 ((𝐵𝑆𝐵𝑆𝐴𝐵) → (𝐵(+g𝑀)(𝐵(+g𝑀)𝐵)) = (𝐵(+g𝑀)𝐵))
5452, 53eqtr4d 2647 . . . . . 6 ((𝐵𝑆𝐵𝑆𝐴𝐵) → ((𝐵(+g𝑀)𝐵)(+g𝑀)𝐵) = (𝐵(+g𝑀)(𝐵(+g𝑀)𝐵)))
5531, 54syld3an1 1364 . . . . 5 ((𝐴𝑆𝐵𝑆𝐴𝐵) → ((𝐵(+g𝑀)𝐵)(+g𝑀)𝐵) = (𝐵(+g𝑀)(𝐵(+g𝑀)𝐵)))
5647, 51, 55jca32 556 . . . 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 6556 . . . . . . . . . 10 (𝑎 = 𝐴 → (𝑎(+g𝑀)𝑏) = (𝐴(+g𝑀)𝑏))
5857oveq1d 6564 . . . . . . . . 9 (𝑎 = 𝐴 → ((𝑎(+g𝑀)𝑏)(+g𝑀)𝑐) = ((𝐴(+g𝑀)𝑏)(+g𝑀)𝑐))
59 oveq1 6556 . . . . . . . . 9 (𝑎 = 𝐴 → (𝑎(+g𝑀)(𝑏(+g𝑀)𝑐)) = (𝐴(+g𝑀)(𝑏(+g𝑀)𝑐)))
6058, 59eqeq12d 2625 . . . . . . . 8 (𝑎 = 𝐴 → (((𝑎(+g𝑀)𝑏)(+g𝑀)𝑐) = (𝑎(+g𝑀)(𝑏(+g𝑀)𝑐)) ↔ ((𝐴(+g𝑀)𝑏)(+g𝑀)𝑐) = (𝐴(+g𝑀)(𝑏(+g𝑀)𝑐))))
61602ralbidv 2972 . . . . . . 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 6556 . . . . . . . . . 10 (𝑎 = 𝐵 → (𝑎(+g𝑀)𝑏) = (𝐵(+g𝑀)𝑏))
6362oveq1d 6564 . . . . . . . . 9 (𝑎 = 𝐵 → ((𝑎(+g𝑀)𝑏)(+g𝑀)𝑐) = ((𝐵(+g𝑀)𝑏)(+g𝑀)𝑐))
64 oveq1 6556 . . . . . . . . 9 (𝑎 = 𝐵 → (𝑎(+g𝑀)(𝑏(+g𝑀)𝑐)) = (𝐵(+g𝑀)(𝑏(+g𝑀)𝑐)))
6563, 64eqeq12d 2625 . . . . . . . 8 (𝑎 = 𝐵 → (((𝑎(+g𝑀)𝑏)(+g𝑀)𝑐) = (𝑎(+g𝑀)(𝑏(+g𝑀)𝑐)) ↔ ((𝐵(+g𝑀)𝑏)(+g𝑀)𝑐) = (𝐵(+g𝑀)(𝑏(+g𝑀)𝑐))))
66652ralbidv 2972 . . . . . . 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 4181 . . . . . 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 6557 . . . . . . . . . . 11 (𝑏 = 𝐴 → (𝐴(+g𝑀)𝑏) = (𝐴(+g𝑀)𝐴))
6968oveq1d 6564 . . . . . . . . . 10 (𝑏 = 𝐴 → ((𝐴(+g𝑀)𝑏)(+g𝑀)𝑐) = ((𝐴(+g𝑀)𝐴)(+g𝑀)𝑐))
70 oveq1 6556 . . . . . . . . . . 11 (𝑏 = 𝐴 → (𝑏(+g𝑀)𝑐) = (𝐴(+g𝑀)𝑐))
7170oveq2d 6565 . . . . . . . . . 10 (𝑏 = 𝐴 → (𝐴(+g𝑀)(𝑏(+g𝑀)𝑐)) = (𝐴(+g𝑀)(𝐴(+g𝑀)𝑐)))
7269, 71eqeq12d 2625 . . . . . . . . 9 (𝑏 = 𝐴 → (((𝐴(+g𝑀)𝑏)(+g𝑀)𝑐) = (𝐴(+g𝑀)(𝑏(+g𝑀)𝑐)) ↔ ((𝐴(+g𝑀)𝐴)(+g𝑀)𝑐) = (𝐴(+g𝑀)(𝐴(+g𝑀)𝑐))))
7372ralbidv 2969 . . . . . . . 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 6557 . . . . . . . . . . 11 (𝑏 = 𝐵 → (𝐴(+g𝑀)𝑏) = (𝐴(+g𝑀)𝐵))
7574oveq1d 6564 . . . . . . . . . 10 (𝑏 = 𝐵 → ((𝐴(+g𝑀)𝑏)(+g𝑀)𝑐) = ((𝐴(+g𝑀)𝐵)(+g𝑀)𝑐))
76 oveq1 6556 . . . . . . . . . . 11 (𝑏 = 𝐵 → (𝑏(+g𝑀)𝑐) = (𝐵(+g𝑀)𝑐))
7776oveq2d 6565 . . . . . . . . . 10 (𝑏 = 𝐵 → (𝐴(+g𝑀)(𝑏(+g𝑀)𝑐)) = (𝐴(+g𝑀)(𝐵(+g𝑀)𝑐)))
7875, 77eqeq12d 2625 . . . . . . . . 9 (𝑏 = 𝐵 → (((𝐴(+g𝑀)𝑏)(+g𝑀)𝑐) = (𝐴(+g𝑀)(𝑏(+g𝑀)𝑐)) ↔ ((𝐴(+g𝑀)𝐵)(+g𝑀)𝑐) = (𝐴(+g𝑀)(𝐵(+g𝑀)𝑐))))
7978ralbidv 2969 . . . . . . . 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 4181 . . . . . . 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 6557 . . . . . . . . . . 11 (𝑏 = 𝐴 → (𝐵(+g𝑀)𝑏) = (𝐵(+g𝑀)𝐴))
8281oveq1d 6564 . . . . . . . . . 10 (𝑏 = 𝐴 → ((𝐵(+g𝑀)𝑏)(+g𝑀)𝑐) = ((𝐵(+g𝑀)𝐴)(+g𝑀)𝑐))
8370oveq2d 6565 . . . . . . . . . 10 (𝑏 = 𝐴 → (𝐵(+g𝑀)(𝑏(+g𝑀)𝑐)) = (𝐵(+g𝑀)(𝐴(+g𝑀)𝑐)))
8482, 83eqeq12d 2625 . . . . . . . . 9 (𝑏 = 𝐴 → (((𝐵(+g𝑀)𝑏)(+g𝑀)𝑐) = (𝐵(+g𝑀)(𝑏(+g𝑀)𝑐)) ↔ ((𝐵(+g𝑀)𝐴)(+g𝑀)𝑐) = (𝐵(+g𝑀)(𝐴(+g𝑀)𝑐))))
8584ralbidv 2969 . . . . . . . 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 6557 . . . . . . . . . . 11 (𝑏 = 𝐵 → (𝐵(+g𝑀)𝑏) = (𝐵(+g𝑀)𝐵))
8786oveq1d 6564 . . . . . . . . . 10 (𝑏 = 𝐵 → ((𝐵(+g𝑀)𝑏)(+g𝑀)𝑐) = ((𝐵(+g𝑀)𝐵)(+g𝑀)𝑐))
8876oveq2d 6565 . . . . . . . . . 10 (𝑏 = 𝐵 → (𝐵(+g𝑀)(𝑏(+g𝑀)𝑐)) = (𝐵(+g𝑀)(𝐵(+g𝑀)𝑐)))
8987, 88eqeq12d 2625 . . . . . . . . 9 (𝑏 = 𝐵 → (((𝐵(+g𝑀)𝑏)(+g𝑀)𝑐) = (𝐵(+g𝑀)(𝑏(+g𝑀)𝑐)) ↔ ((𝐵(+g𝑀)𝐵)(+g𝑀)𝑐) = (𝐵(+g𝑀)(𝐵(+g𝑀)𝑐))))
9089ralbidv 2969 . . . . . . . 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 4181 . . . . . . 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 743 . . . . . 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 6557 . . . . . . . . . 10 (𝑐 = 𝐴 → ((𝐴(+g𝑀)𝐴)(+g𝑀)𝑐) = ((𝐴(+g𝑀)𝐴)(+g𝑀)𝐴))
94 oveq2 6557 . . . . . . . . . . 11 (𝑐 = 𝐴 → (𝐴(+g𝑀)𝑐) = (𝐴(+g𝑀)𝐴))
9594oveq2d 6565 . . . . . . . . . 10 (𝑐 = 𝐴 → (𝐴(+g𝑀)(𝐴(+g𝑀)𝑐)) = (𝐴(+g𝑀)(𝐴(+g𝑀)𝐴)))
9693, 95eqeq12d 2625 . . . . . . . . 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 6557 . . . . . . . . . 10 (𝑐 = 𝐵 → ((𝐴(+g𝑀)𝐴)(+g𝑀)𝑐) = ((𝐴(+g𝑀)𝐴)(+g𝑀)𝐵))
98 oveq2 6557 . . . . . . . . . . 11 (𝑐 = 𝐵 → (𝐴(+g𝑀)𝑐) = (𝐴(+g𝑀)𝐵))
9998oveq2d 6565 . . . . . . . . . 10 (𝑐 = 𝐵 → (𝐴(+g𝑀)(𝐴(+g𝑀)𝑐)) = (𝐴(+g𝑀)(𝐴(+g𝑀)𝐵)))
10097, 99eqeq12d 2625 . . . . . . . . 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 4181 . . . . . . . 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 6557 . . . . . . . . . 10 (𝑐 = 𝐴 → ((𝐴(+g𝑀)𝐵)(+g𝑀)𝑐) = ((𝐴(+g𝑀)𝐵)(+g𝑀)𝐴))
103 oveq2 6557 . . . . . . . . . . 11 (𝑐 = 𝐴 → (𝐵(+g𝑀)𝑐) = (𝐵(+g𝑀)𝐴))
104103oveq2d 6565 . . . . . . . . . 10 (𝑐 = 𝐴 → (𝐴(+g𝑀)(𝐵(+g𝑀)𝑐)) = (𝐴(+g𝑀)(𝐵(+g𝑀)𝐴)))
105102, 104eqeq12d 2625 . . . . . . . . 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 6557 . . . . . . . . . 10 (𝑐 = 𝐵 → ((𝐴(+g𝑀)𝐵)(+g𝑀)𝑐) = ((𝐴(+g𝑀)𝐵)(+g𝑀)𝐵))
107 oveq2 6557 . . . . . . . . . . 11 (𝑐 = 𝐵 → (𝐵(+g𝑀)𝑐) = (𝐵(+g𝑀)𝐵))
108107oveq2d 6565 . . . . . . . . . 10 (𝑐 = 𝐵 → (𝐴(+g𝑀)(𝐵(+g𝑀)𝑐)) = (𝐴(+g𝑀)(𝐵(+g𝑀)𝐵)))
109106, 108eqeq12d 2625 . . . . . . . . 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 4181 . . . . . . . 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 743 . . . . . . 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 6557 . . . . . . . . . 10 (𝑐 = 𝐴 → ((𝐵(+g𝑀)𝐴)(+g𝑀)𝑐) = ((𝐵(+g𝑀)𝐴)(+g𝑀)𝐴))
11394oveq2d 6565 . . . . . . . . . 10 (𝑐 = 𝐴 → (𝐵(+g𝑀)(𝐴(+g𝑀)𝑐)) = (𝐵(+g𝑀)(𝐴(+g𝑀)𝐴)))
114112, 113eqeq12d 2625 . . . . . . . . 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 6557 . . . . . . . . . 10 (𝑐 = 𝐵 → ((𝐵(+g𝑀)𝐴)(+g𝑀)𝑐) = ((𝐵(+g𝑀)𝐴)(+g𝑀)𝐵))
11698oveq2d 6565 . . . . . . . . . 10 (𝑐 = 𝐵 → (𝐵(+g𝑀)(𝐴(+g𝑀)𝑐)) = (𝐵(+g𝑀)(𝐴(+g𝑀)𝐵)))
117115, 116eqeq12d 2625 . . . . . . . . 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 4181 . . . . . . . 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 6557 . . . . . . . . . 10 (𝑐 = 𝐴 → ((𝐵(+g𝑀)𝐵)(+g𝑀)𝑐) = ((𝐵(+g𝑀)𝐵)(+g𝑀)𝐴))
120103oveq2d 6565 . . . . . . . . . 10 (𝑐 = 𝐴 → (𝐵(+g𝑀)(𝐵(+g𝑀)𝑐)) = (𝐵(+g𝑀)(𝐵(+g𝑀)𝐴)))
121119, 120eqeq12d 2625 . . . . . . . . 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 6557 . . . . . . . . . 10 (𝑐 = 𝐵 → ((𝐵(+g𝑀)𝐵)(+g𝑀)𝑐) = ((𝐵(+g𝑀)𝐵)(+g𝑀)𝐵))
123107oveq2d 6565 . . . . . . . . . 10 (𝑐 = 𝐵 → (𝐵(+g𝑀)(𝐵(+g𝑀)𝑐)) = (𝐵(+g𝑀)(𝐵(+g𝑀)𝐵)))
124122, 123eqeq12d 2625 . . . . . . . . 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 4181 . . . . . . . 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 743 . . . . . . 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 743 . . . . . 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 293 . . . . 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 1074 . . . 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 959 . . 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 2633 . . 3 {𝐴, 𝐵} = (Base‘𝑀)
133132, 8issgrp 17108 . 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 695 1 ((#‘𝑆) = 2 → 𝑀 ∈ SGrp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383  w3a 1031   = wceq 1475  wcel 1977  wne 2780  wral 2896  ifcif 4036  {cpr 4127  cfv 5804  (class class class)co 6549  cmpt2 6551  2c2 10947  #chash 12979  Basecbs 15695  +gcplusg 15768  Mgmcmgm 17063  SGrpcsgrp 17106
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-hash 12980  df-mgm 17065  df-sgrp 17107
This theorem is referenced by:  sgrp2nmnd  17240  sgrpnmndex  17242
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