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Theorem sgrp2nmndlem4 17636
 Description: Lemma 4 for sgrp2nmnd 17638: 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 13398 . . 3 ((♯‘𝑆) = 2 → (𝐴𝑆𝐵𝑆𝐴𝐵))
3 3simpa 1143 . . 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 17631 . . 3 ((𝐴𝑆𝐵𝑆) → 𝑀 ∈ Mgm)
72, 3, 63syl 18 . 2 ((♯‘𝑆) = 2 → 𝑀 ∈ Mgm)
8 eqid 2760 . . . . . . . . . . 11 (+g𝑀) = (+g𝑀)
91, 4, 5, 8sgrp2nmndlem2 17632 . . . . . . . . . 10 ((𝐴𝑆𝐴𝑆) → (𝐴(+g𝑀)𝐴) = 𝐴)
109oveq1d 6829 . . . . . . . . 9 ((𝐴𝑆𝐴𝑆) → ((𝐴(+g𝑀)𝐴)(+g𝑀)𝐴) = (𝐴(+g𝑀)𝐴))
119oveq2d 6830 . . . . . . . . 9 ((𝐴𝑆𝐴𝑆) → (𝐴(+g𝑀)(𝐴(+g𝑀)𝐴)) = (𝐴(+g𝑀)𝐴))
1210, 11eqtr4d 2797 . . . . . . . 8 ((𝐴𝑆𝐴𝑆) → ((𝐴(+g𝑀)𝐴)(+g𝑀)𝐴) = (𝐴(+g𝑀)(𝐴(+g𝑀)𝐴)))
1312anidms 680 . . . . . . 7 (𝐴𝑆 → ((𝐴(+g𝑀)𝐴)(+g𝑀)𝐴) = (𝐴(+g𝑀)(𝐴(+g𝑀)𝐴)))
14133ad2ant1 1128 . . . . . 6 ((𝐴𝑆𝐵𝑆𝐴𝐵) → ((𝐴(+g𝑀)𝐴)(+g𝑀)𝐴) = (𝐴(+g𝑀)(𝐴(+g𝑀)𝐴)))
159anidms 680 . . . . . . . . . 10 (𝐴𝑆 → (𝐴(+g𝑀)𝐴) = 𝐴)
1615adantr 472 . . . . . . . . 9 ((𝐴𝑆𝐵𝑆) → (𝐴(+g𝑀)𝐴) = 𝐴)
1716oveq1d 6829 . . . . . . . 8 ((𝐴𝑆𝐵𝑆) → ((𝐴(+g𝑀)𝐴)(+g𝑀)𝐵) = (𝐴(+g𝑀)𝐵))
181, 4, 5, 8sgrp2nmndlem2 17632 . . . . . . . . . 10 ((𝐴𝑆𝐵𝑆) → (𝐴(+g𝑀)𝐵) = 𝐴)
1918oveq2d 6830 . . . . . . . . 9 ((𝐴𝑆𝐵𝑆) → (𝐴(+g𝑀)(𝐴(+g𝑀)𝐵)) = (𝐴(+g𝑀)𝐴))
2016, 19, 183eqtr4rd 2805 . . . . . . . 8 ((𝐴𝑆𝐵𝑆) → (𝐴(+g𝑀)𝐵) = (𝐴(+g𝑀)(𝐴(+g𝑀)𝐵)))
2117, 20eqtrd 2794 . . . . . . 7 ((𝐴𝑆𝐵𝑆) → ((𝐴(+g𝑀)𝐴)(+g𝑀)𝐵) = (𝐴(+g𝑀)(𝐴(+g𝑀)𝐵)))
22213adant3 1127 . . . . . 6 ((𝐴𝑆𝐵𝑆𝐴𝐵) → ((𝐴(+g𝑀)𝐴)(+g𝑀)𝐵) = (𝐴(+g𝑀)(𝐴(+g𝑀)𝐵)))
2314, 22jca 555 . . . . 5 ((𝐴𝑆𝐵𝑆𝐴𝐵) → (((𝐴(+g𝑀)𝐴)(+g𝑀)𝐴) = (𝐴(+g𝑀)(𝐴(+g𝑀)𝐴)) ∧ ((𝐴(+g𝑀)𝐴)(+g𝑀)𝐵) = (𝐴(+g𝑀)(𝐴(+g𝑀)𝐵))))
24183adant3 1127 . . . . . 6 ((𝐴𝑆𝐵𝑆𝐴𝐵) → (𝐴(+g𝑀)𝐵) = 𝐴)
251, 4, 5, 8sgrp2nmndlem3 17633 . . . . . . 7 ((𝐴𝑆𝐵𝑆𝐴𝐵) → (𝐵(+g𝑀)𝐴) = 𝐵)
2625oveq2d 6830 . . . . . 6 ((𝐴𝑆𝐵𝑆𝐴𝐵) → (𝐴(+g𝑀)(𝐵(+g𝑀)𝐴)) = (𝐴(+g𝑀)𝐵))
2724oveq1d 6829 . . . . . . 7 ((𝐴𝑆𝐵𝑆𝐴𝐵) → ((𝐴(+g𝑀)𝐵)(+g𝑀)𝐴) = (𝐴(+g𝑀)𝐴))
28153ad2ant1 1128 . . . . . . 7 ((𝐴𝑆𝐵𝑆𝐴𝐵) → (𝐴(+g𝑀)𝐴) = 𝐴)
2927, 28eqtrd 2794 . . . . . 6 ((𝐴𝑆𝐵𝑆𝐴𝐵) → ((𝐴(+g𝑀)𝐵)(+g𝑀)𝐴) = 𝐴)
3024, 26, 293eqtr4rd 2805 . . . . 5 ((𝐴𝑆𝐵𝑆𝐴𝐵) → ((𝐴(+g𝑀)𝐵)(+g𝑀)𝐴) = (𝐴(+g𝑀)(𝐵(+g𝑀)𝐴)))
31 simp2 1132 . . . . . . . 8 ((𝐴𝑆𝐵𝑆𝐴𝐵) → 𝐵𝑆)
321, 4, 5, 8sgrp2nmndlem3 17633 . . . . . . . 8 ((𝐵𝑆𝐵𝑆𝐴𝐵) → (𝐵(+g𝑀)𝐵) = 𝐵)
3331, 32syld3an1 1517 . . . . . . 7 ((𝐴𝑆𝐵𝑆𝐴𝐵) → (𝐵(+g𝑀)𝐵) = 𝐵)
3433oveq2d 6830 . . . . . 6 ((𝐴𝑆𝐵𝑆𝐴𝐵) → (𝐴(+g𝑀)(𝐵(+g𝑀)𝐵)) = (𝐴(+g𝑀)𝐵))
3518oveq1d 6829 . . . . . . . 8 ((𝐴𝑆𝐵𝑆) → ((𝐴(+g𝑀)𝐵)(+g𝑀)𝐵) = (𝐴(+g𝑀)𝐵))
3635, 18eqtrd 2794 . . . . . . 7 ((𝐴𝑆𝐵𝑆) → ((𝐴(+g𝑀)𝐵)(+g𝑀)𝐵) = 𝐴)
37363adant3 1127 . . . . . 6 ((𝐴𝑆𝐵𝑆𝐴𝐵) → ((𝐴(+g𝑀)𝐵)(+g𝑀)𝐵) = 𝐴)
3824, 34, 373eqtr4rd 2805 . . . . 5 ((𝐴𝑆𝐵𝑆𝐴𝐵) → ((𝐴(+g𝑀)𝐵)(+g𝑀)𝐵) = (𝐴(+g𝑀)(𝐵(+g𝑀)𝐵)))
3923, 30, 38jca32 559 . . . 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 6829 . . . . . . 7 ((𝐴𝑆𝐵𝑆𝐴𝐵) → ((𝐵(+g𝑀)𝐴)(+g𝑀)𝐴) = (𝐵(+g𝑀)𝐴))
4128oveq2d 6830 . . . . . . 7 ((𝐴𝑆𝐵𝑆𝐴𝐵) → (𝐵(+g𝑀)(𝐴(+g𝑀)𝐴)) = (𝐵(+g𝑀)𝐴))
4240, 41eqtr4d 2797 . . . . . 6 ((𝐴𝑆𝐵𝑆𝐴𝐵) → ((𝐵(+g𝑀)𝐴)(+g𝑀)𝐴) = (𝐵(+g𝑀)(𝐴(+g𝑀)𝐴)))
4324oveq2d 6830 . . . . . . 7 ((𝐴𝑆𝐵𝑆𝐴𝐵) → (𝐵(+g𝑀)(𝐴(+g𝑀)𝐵)) = (𝐵(+g𝑀)𝐴))
4425oveq1d 6829 . . . . . . . 8 ((𝐴𝑆𝐵𝑆𝐴𝐵) → ((𝐵(+g𝑀)𝐴)(+g𝑀)𝐵) = (𝐵(+g𝑀)𝐵))
4544, 33eqtrd 2794 . . . . . . 7 ((𝐴𝑆𝐵𝑆𝐴𝐵) → ((𝐵(+g𝑀)𝐴)(+g𝑀)𝐵) = 𝐵)
4625, 43, 453eqtr4rd 2805 . . . . . 6 ((𝐴𝑆𝐵𝑆𝐴𝐵) → ((𝐵(+g𝑀)𝐴)(+g𝑀)𝐵) = (𝐵(+g𝑀)(𝐴(+g𝑀)𝐵)))
4742, 46jca 555 . . . . 5 ((𝐴𝑆𝐵𝑆𝐴𝐵) → (((𝐵(+g𝑀)𝐴)(+g𝑀)𝐴) = (𝐵(+g𝑀)(𝐴(+g𝑀)𝐴)) ∧ ((𝐵(+g𝑀)𝐴)(+g𝑀)𝐵) = (𝐵(+g𝑀)(𝐴(+g𝑀)𝐵))))
4825oveq2d 6830 . . . . . 6 ((𝐴𝑆𝐵𝑆𝐴𝐵) → (𝐵(+g𝑀)(𝐵(+g𝑀)𝐴)) = (𝐵(+g𝑀)𝐵))
4933oveq1d 6829 . . . . . . 7 ((𝐴𝑆𝐵𝑆𝐴𝐵) → ((𝐵(+g𝑀)𝐵)(+g𝑀)𝐴) = (𝐵(+g𝑀)𝐴))
5049, 25eqtrd 2794 . . . . . 6 ((𝐴𝑆𝐵𝑆𝐴𝐵) → ((𝐵(+g𝑀)𝐵)(+g𝑀)𝐴) = 𝐵)
5133, 48, 503eqtr4rd 2805 . . . . 5 ((𝐴𝑆𝐵𝑆𝐴𝐵) → ((𝐵(+g𝑀)𝐵)(+g𝑀)𝐴) = (𝐵(+g𝑀)(𝐵(+g𝑀)𝐴)))
5232oveq1d 6829 . . . . . . 7 ((𝐵𝑆𝐵𝑆𝐴𝐵) → ((𝐵(+g𝑀)𝐵)(+g𝑀)𝐵) = (𝐵(+g𝑀)𝐵))
5332oveq2d 6830 . . . . . . 7 ((𝐵𝑆𝐵𝑆𝐴𝐵) → (𝐵(+g𝑀)(𝐵(+g𝑀)𝐵)) = (𝐵(+g𝑀)𝐵))
5452, 53eqtr4d 2797 . . . . . 6 ((𝐵𝑆𝐵𝑆𝐴𝐵) → ((𝐵(+g𝑀)𝐵)(+g𝑀)𝐵) = (𝐵(+g𝑀)(𝐵(+g𝑀)𝐵)))
5531, 54syld3an1 1517 . . . . 5 ((𝐴𝑆𝐵𝑆𝐴𝐵) → ((𝐵(+g𝑀)𝐵)(+g𝑀)𝐵) = (𝐵(+g𝑀)(𝐵(+g𝑀)𝐵)))
5647, 51, 55jca32 559 . . . 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 6821 . . . . . . . . . 10 (𝑎 = 𝐴 → (𝑎(+g𝑀)𝑏) = (𝐴(+g𝑀)𝑏))
5857oveq1d 6829 . . . . . . . . 9 (𝑎 = 𝐴 → ((𝑎(+g𝑀)𝑏)(+g𝑀)𝑐) = ((𝐴(+g𝑀)𝑏)(+g𝑀)𝑐))
59 oveq1 6821 . . . . . . . . 9 (𝑎 = 𝐴 → (𝑎(+g𝑀)(𝑏(+g𝑀)𝑐)) = (𝐴(+g𝑀)(𝑏(+g𝑀)𝑐)))
6058, 59eqeq12d 2775 . . . . . . . 8 (𝑎 = 𝐴 → (((𝑎(+g𝑀)𝑏)(+g𝑀)𝑐) = (𝑎(+g𝑀)(𝑏(+g𝑀)𝑐)) ↔ ((𝐴(+g𝑀)𝑏)(+g𝑀)𝑐) = (𝐴(+g𝑀)(𝑏(+g𝑀)𝑐))))
61602ralbidv 3127 . . . . . . 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 6821 . . . . . . . . . 10 (𝑎 = 𝐵 → (𝑎(+g𝑀)𝑏) = (𝐵(+g𝑀)𝑏))
6362oveq1d 6829 . . . . . . . . 9 (𝑎 = 𝐵 → ((𝑎(+g𝑀)𝑏)(+g𝑀)𝑐) = ((𝐵(+g𝑀)𝑏)(+g𝑀)𝑐))
64 oveq1 6821 . . . . . . . . 9 (𝑎 = 𝐵 → (𝑎(+g𝑀)(𝑏(+g𝑀)𝑐)) = (𝐵(+g𝑀)(𝑏(+g𝑀)𝑐)))
6563, 64eqeq12d 2775 . . . . . . . 8 (𝑎 = 𝐵 → (((𝑎(+g𝑀)𝑏)(+g𝑀)𝑐) = (𝑎(+g𝑀)(𝑏(+g𝑀)𝑐)) ↔ ((𝐵(+g𝑀)𝑏)(+g𝑀)𝑐) = (𝐵(+g𝑀)(𝑏(+g𝑀)𝑐))))
66652ralbidv 3127 . . . . . . 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 4378 . . . . . 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 6822 . . . . . . . . . . 11 (𝑏 = 𝐴 → (𝐴(+g𝑀)𝑏) = (𝐴(+g𝑀)𝐴))
6968oveq1d 6829 . . . . . . . . . 10 (𝑏 = 𝐴 → ((𝐴(+g𝑀)𝑏)(+g𝑀)𝑐) = ((𝐴(+g𝑀)𝐴)(+g𝑀)𝑐))
70 oveq1 6821 . . . . . . . . . . 11 (𝑏 = 𝐴 → (𝑏(+g𝑀)𝑐) = (𝐴(+g𝑀)𝑐))
7170oveq2d 6830 . . . . . . . . . 10 (𝑏 = 𝐴 → (𝐴(+g𝑀)(𝑏(+g𝑀)𝑐)) = (𝐴(+g𝑀)(𝐴(+g𝑀)𝑐)))
7269, 71eqeq12d 2775 . . . . . . . . 9 (𝑏 = 𝐴 → (((𝐴(+g𝑀)𝑏)(+g𝑀)𝑐) = (𝐴(+g𝑀)(𝑏(+g𝑀)𝑐)) ↔ ((𝐴(+g𝑀)𝐴)(+g𝑀)𝑐) = (𝐴(+g𝑀)(𝐴(+g𝑀)𝑐))))
7372ralbidv 3124 . . . . . . . 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 6822 . . . . . . . . . . 11 (𝑏 = 𝐵 → (𝐴(+g𝑀)𝑏) = (𝐴(+g𝑀)𝐵))
7574oveq1d 6829 . . . . . . . . . 10 (𝑏 = 𝐵 → ((𝐴(+g𝑀)𝑏)(+g𝑀)𝑐) = ((𝐴(+g𝑀)𝐵)(+g𝑀)𝑐))
76 oveq1 6821 . . . . . . . . . . 11 (𝑏 = 𝐵 → (𝑏(+g𝑀)𝑐) = (𝐵(+g𝑀)𝑐))
7776oveq2d 6830 . . . . . . . . . 10 (𝑏 = 𝐵 → (𝐴(+g𝑀)(𝑏(+g𝑀)𝑐)) = (𝐴(+g𝑀)(𝐵(+g𝑀)𝑐)))
7875, 77eqeq12d 2775 . . . . . . . . 9 (𝑏 = 𝐵 → (((𝐴(+g𝑀)𝑏)(+g𝑀)𝑐) = (𝐴(+g𝑀)(𝑏(+g𝑀)𝑐)) ↔ ((𝐴(+g𝑀)𝐵)(+g𝑀)𝑐) = (𝐴(+g𝑀)(𝐵(+g𝑀)𝑐))))
7978ralbidv 3124 . . . . . . . 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 4378 . . . . . . 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 6822 . . . . . . . . . . 11 (𝑏 = 𝐴 → (𝐵(+g𝑀)𝑏) = (𝐵(+g𝑀)𝐴))
8281oveq1d 6829 . . . . . . . . . 10 (𝑏 = 𝐴 → ((𝐵(+g𝑀)𝑏)(+g𝑀)𝑐) = ((𝐵(+g𝑀)𝐴)(+g𝑀)𝑐))
8370oveq2d 6830 . . . . . . . . . 10 (𝑏 = 𝐴 → (𝐵(+g𝑀)(𝑏(+g𝑀)𝑐)) = (𝐵(+g𝑀)(𝐴(+g𝑀)𝑐)))
8482, 83eqeq12d 2775 . . . . . . . . 9 (𝑏 = 𝐴 → (((𝐵(+g𝑀)𝑏)(+g𝑀)𝑐) = (𝐵(+g𝑀)(𝑏(+g𝑀)𝑐)) ↔ ((𝐵(+g𝑀)𝐴)(+g𝑀)𝑐) = (𝐵(+g𝑀)(𝐴(+g𝑀)𝑐))))
8584ralbidv 3124 . . . . . . . 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 6822 . . . . . . . . . . 11 (𝑏 = 𝐵 → (𝐵(+g𝑀)𝑏) = (𝐵(+g𝑀)𝐵))
8786oveq1d 6829 . . . . . . . . . 10 (𝑏 = 𝐵 → ((𝐵(+g𝑀)𝑏)(+g𝑀)𝑐) = ((𝐵(+g𝑀)𝐵)(+g𝑀)𝑐))
8876oveq2d 6830 . . . . . . . . . 10 (𝑏 = 𝐵 → (𝐵(+g𝑀)(𝑏(+g𝑀)𝑐)) = (𝐵(+g𝑀)(𝐵(+g𝑀)𝑐)))
8987, 88eqeq12d 2775 . . . . . . . . 9 (𝑏 = 𝐵 → (((𝐵(+g𝑀)𝑏)(+g𝑀)𝑐) = (𝐵(+g𝑀)(𝑏(+g𝑀)𝑐)) ↔ ((𝐵(+g𝑀)𝐵)(+g𝑀)𝑐) = (𝐵(+g𝑀)(𝐵(+g𝑀)𝑐))))
9089ralbidv 3124 . . . . . . . 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 4378 . . . . . . 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 749 . . . . . 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 6822 . . . . . . . . . 10 (𝑐 = 𝐴 → ((𝐴(+g𝑀)𝐴)(+g𝑀)𝑐) = ((𝐴(+g𝑀)𝐴)(+g𝑀)𝐴))
94 oveq2 6822 . . . . . . . . . . 11 (𝑐 = 𝐴 → (𝐴(+g𝑀)𝑐) = (𝐴(+g𝑀)𝐴))
9594oveq2d 6830 . . . . . . . . . 10 (𝑐 = 𝐴 → (𝐴(+g𝑀)(𝐴(+g𝑀)𝑐)) = (𝐴(+g𝑀)(𝐴(+g𝑀)𝐴)))
9693, 95eqeq12d 2775 . . . . . . . . 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 6822 . . . . . . . . . 10 (𝑐 = 𝐵 → ((𝐴(+g𝑀)𝐴)(+g𝑀)𝑐) = ((𝐴(+g𝑀)𝐴)(+g𝑀)𝐵))
98 oveq2 6822 . . . . . . . . . . 11 (𝑐 = 𝐵 → (𝐴(+g𝑀)𝑐) = (𝐴(+g𝑀)𝐵))
9998oveq2d 6830 . . . . . . . . . 10 (𝑐 = 𝐵 → (𝐴(+g𝑀)(𝐴(+g𝑀)𝑐)) = (𝐴(+g𝑀)(𝐴(+g𝑀)𝐵)))
10097, 99eqeq12d 2775 . . . . . . . . 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 4378 . . . . . . . 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 6822 . . . . . . . . . 10 (𝑐 = 𝐴 → ((𝐴(+g𝑀)𝐵)(+g𝑀)𝑐) = ((𝐴(+g𝑀)𝐵)(+g𝑀)𝐴))
103 oveq2 6822 . . . . . . . . . . 11 (𝑐 = 𝐴 → (𝐵(+g𝑀)𝑐) = (𝐵(+g𝑀)𝐴))
104103oveq2d 6830 . . . . . . . . . 10 (𝑐 = 𝐴 → (𝐴(+g𝑀)(𝐵(+g𝑀)𝑐)) = (𝐴(+g𝑀)(𝐵(+g𝑀)𝐴)))
105102, 104eqeq12d 2775 . . . . . . . . 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 6822 . . . . . . . . . 10 (𝑐 = 𝐵 → ((𝐴(+g𝑀)𝐵)(+g𝑀)𝑐) = ((𝐴(+g𝑀)𝐵)(+g𝑀)𝐵))
107 oveq2 6822 . . . . . . . . . . 11 (𝑐 = 𝐵 → (𝐵(+g𝑀)𝑐) = (𝐵(+g𝑀)𝐵))
108107oveq2d 6830 . . . . . . . . . 10 (𝑐 = 𝐵 → (𝐴(+g𝑀)(𝐵(+g𝑀)𝑐)) = (𝐴(+g𝑀)(𝐵(+g𝑀)𝐵)))
109106, 108eqeq12d 2775 . . . . . . . . 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 4378 . . . . . . . 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 749 . . . . . . 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 6822 . . . . . . . . . 10 (𝑐 = 𝐴 → ((𝐵(+g𝑀)𝐴)(+g𝑀)𝑐) = ((𝐵(+g𝑀)𝐴)(+g𝑀)𝐴))
11394oveq2d 6830 . . . . . . . . . 10 (𝑐 = 𝐴 → (𝐵(+g𝑀)(𝐴(+g𝑀)𝑐)) = (𝐵(+g𝑀)(𝐴(+g𝑀)𝐴)))
114112, 113eqeq12d 2775 . . . . . . . . 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 6822 . . . . . . . . . 10 (𝑐 = 𝐵 → ((𝐵(+g𝑀)𝐴)(+g𝑀)𝑐) = ((𝐵(+g𝑀)𝐴)(+g𝑀)𝐵))
11698oveq2d 6830 . . . . . . . . . 10 (𝑐 = 𝐵 → (𝐵(+g𝑀)(𝐴(+g𝑀)𝑐)) = (𝐵(+g𝑀)(𝐴(+g𝑀)𝐵)))
117115, 116eqeq12d 2775 . . . . . . . . 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 4378 . . . . . . . 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 6822 . . . . . . . . . 10 (𝑐 = 𝐴 → ((𝐵(+g𝑀)𝐵)(+g𝑀)𝑐) = ((𝐵(+g𝑀)𝐵)(+g𝑀)𝐴))
120103oveq2d 6830 . . . . . . . . . 10 (𝑐 = 𝐴 → (𝐵(+g𝑀)(𝐵(+g𝑀)𝑐)) = (𝐵(+g𝑀)(𝐵(+g𝑀)𝐴)))
121119, 120eqeq12d 2775 . . . . . . . . 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 6822 . . . . . . . . . 10 (𝑐 = 𝐵 → ((𝐵(+g𝑀)𝐵)(+g𝑀)𝑐) = ((𝐵(+g𝑀)𝐵)(+g𝑀)𝐵))
123107oveq2d 6830 . . . . . . . . . 10 (𝑐 = 𝐵 → (𝐵(+g𝑀)(𝐵(+g𝑀)𝑐)) = (𝐵(+g𝑀)(𝐵(+g𝑀)𝐵)))
124122, 123eqeq12d 2775 . . . . . . . . 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 4378 . . . . . . . 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 749 . . . . . . 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 749 . . . . . 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 1127 . . . 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 995 . . 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 2783 . . 3 {𝐴, 𝐵} = (Base‘𝑀)
133132, 8issgrp 17506 . 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 701 1 ((♯‘𝑆) = 2 → 𝑀 ∈ SGrp)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   ∧ w3a 1072   = wceq 1632   ∈ wcel 2139   ≠ wne 2932  ∀wral 3050  ifcif 4230  {cpr 4323  ‘cfv 6049  (class class class)co 6814   ↦ cmpt2 6816  2c2 11282  ♯chash 13331  Basecbs 16079  +gcplusg 16163  Mgmcmgm 17461  SGrpcsgrp 17504 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-cnex 10204  ax-resscn 10205  ax-1cn 10206  ax-icn 10207  ax-addcl 10208  ax-addrcl 10209  ax-mulcl 10210  ax-mulrcl 10211  ax-mulcom 10212  ax-addass 10213  ax-mulass 10214  ax-distr 10215  ax-i2m1 10216  ax-1ne0 10217  ax-1rid 10218  ax-rnegex 10219  ax-rrecex 10220  ax-cnre 10221  ax-pre-lttri 10222  ax-pre-lttrn 10223  ax-pre-ltadd 10224  ax-pre-mulgt0 10225 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-om 7232  df-1st 7334  df-2nd 7335  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-1o 7730  df-oadd 7734  df-er 7913  df-en 8124  df-dom 8125  df-sdom 8126  df-fin 8127  df-card 8975  df-cda 9202  df-pnf 10288  df-mnf 10289  df-xr 10290  df-ltxr 10291  df-le 10292  df-sub 10480  df-neg 10481  df-nn 11233  df-2 11291  df-n0 11505  df-z 11590  df-uz 11900  df-fz 12540  df-hash 13332  df-mgm 17463  df-sgrp 17505 This theorem is referenced by:  sgrp2nmnd  17638  sgrpnmndex  17640
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