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Theorem sgrp2nmndlem4 18028
Description: Lemma 4 for sgrp2nmnd 18030: 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 ∈ 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 13752 . . 3 ((♯‘𝑆) = 2 → (𝐴𝑆𝐵𝑆𝐴𝐵))
3 3simpa 1142 . . 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 18023 . . 3 ((𝐴𝑆𝐵𝑆) → 𝑀 ∈ Mgm)
72, 3, 63syl 18 . 2 ((♯‘𝑆) = 2 → 𝑀 ∈ Mgm)
8 eqid 2824 . . . . . . . . . . 11 (+g𝑀) = (+g𝑀)
91, 4, 5, 8sgrp2nmndlem2 18024 . . . . . . . . . 10 ((𝐴𝑆𝐴𝑆) → (𝐴(+g𝑀)𝐴) = 𝐴)
109oveq1d 7166 . . . . . . . . 9 ((𝐴𝑆𝐴𝑆) → ((𝐴(+g𝑀)𝐴)(+g𝑀)𝐴) = (𝐴(+g𝑀)𝐴))
119oveq2d 7167 . . . . . . . . 9 ((𝐴𝑆𝐴𝑆) → (𝐴(+g𝑀)(𝐴(+g𝑀)𝐴)) = (𝐴(+g𝑀)𝐴))
1210, 11eqtr4d 2863 . . . . . . . 8 ((𝐴𝑆𝐴𝑆) → ((𝐴(+g𝑀)𝐴)(+g𝑀)𝐴) = (𝐴(+g𝑀)(𝐴(+g𝑀)𝐴)))
1312anidms 567 . . . . . . 7 (𝐴𝑆 → ((𝐴(+g𝑀)𝐴)(+g𝑀)𝐴) = (𝐴(+g𝑀)(𝐴(+g𝑀)𝐴)))
14133ad2ant1 1127 . . . . . 6 ((𝐴𝑆𝐵𝑆𝐴𝐵) → ((𝐴(+g𝑀)𝐴)(+g𝑀)𝐴) = (𝐴(+g𝑀)(𝐴(+g𝑀)𝐴)))
159anidms 567 . . . . . . . . . 10 (𝐴𝑆 → (𝐴(+g𝑀)𝐴) = 𝐴)
1615adantr 481 . . . . . . . . 9 ((𝐴𝑆𝐵𝑆) → (𝐴(+g𝑀)𝐴) = 𝐴)
1716oveq1d 7166 . . . . . . . 8 ((𝐴𝑆𝐵𝑆) → ((𝐴(+g𝑀)𝐴)(+g𝑀)𝐵) = (𝐴(+g𝑀)𝐵))
181, 4, 5, 8sgrp2nmndlem2 18024 . . . . . . . . . 10 ((𝐴𝑆𝐵𝑆) → (𝐴(+g𝑀)𝐵) = 𝐴)
1918oveq2d 7167 . . . . . . . . 9 ((𝐴𝑆𝐵𝑆) → (𝐴(+g𝑀)(𝐴(+g𝑀)𝐵)) = (𝐴(+g𝑀)𝐴))
2016, 19, 183eqtr4rd 2871 . . . . . . . 8 ((𝐴𝑆𝐵𝑆) → (𝐴(+g𝑀)𝐵) = (𝐴(+g𝑀)(𝐴(+g𝑀)𝐵)))
2117, 20eqtrd 2860 . . . . . . 7 ((𝐴𝑆𝐵𝑆) → ((𝐴(+g𝑀)𝐴)(+g𝑀)𝐵) = (𝐴(+g𝑀)(𝐴(+g𝑀)𝐵)))
22213adant3 1126 . . . . . 6 ((𝐴𝑆𝐵𝑆𝐴𝐵) → ((𝐴(+g𝑀)𝐴)(+g𝑀)𝐵) = (𝐴(+g𝑀)(𝐴(+g𝑀)𝐵)))
2314, 22jca 512 . . . . 5 ((𝐴𝑆𝐵𝑆𝐴𝐵) → (((𝐴(+g𝑀)𝐴)(+g𝑀)𝐴) = (𝐴(+g𝑀)(𝐴(+g𝑀)𝐴)) ∧ ((𝐴(+g𝑀)𝐴)(+g𝑀)𝐵) = (𝐴(+g𝑀)(𝐴(+g𝑀)𝐵))))
24183adant3 1126 . . . . . 6 ((𝐴𝑆𝐵𝑆𝐴𝐵) → (𝐴(+g𝑀)𝐵) = 𝐴)
251, 4, 5, 8sgrp2nmndlem3 18025 . . . . . . 7 ((𝐴𝑆𝐵𝑆𝐴𝐵) → (𝐵(+g𝑀)𝐴) = 𝐵)
2625oveq2d 7167 . . . . . 6 ((𝐴𝑆𝐵𝑆𝐴𝐵) → (𝐴(+g𝑀)(𝐵(+g𝑀)𝐴)) = (𝐴(+g𝑀)𝐵))
2724oveq1d 7166 . . . . . . 7 ((𝐴𝑆𝐵𝑆𝐴𝐵) → ((𝐴(+g𝑀)𝐵)(+g𝑀)𝐴) = (𝐴(+g𝑀)𝐴))
28153ad2ant1 1127 . . . . . . 7 ((𝐴𝑆𝐵𝑆𝐴𝐵) → (𝐴(+g𝑀)𝐴) = 𝐴)
2927, 28eqtrd 2860 . . . . . 6 ((𝐴𝑆𝐵𝑆𝐴𝐵) → ((𝐴(+g𝑀)𝐵)(+g𝑀)𝐴) = 𝐴)
3024, 26, 293eqtr4rd 2871 . . . . 5 ((𝐴𝑆𝐵𝑆𝐴𝐵) → ((𝐴(+g𝑀)𝐵)(+g𝑀)𝐴) = (𝐴(+g𝑀)(𝐵(+g𝑀)𝐴)))
31 simp2 1131 . . . . . . . 8 ((𝐴𝑆𝐵𝑆𝐴𝐵) → 𝐵𝑆)
321, 4, 5, 8sgrp2nmndlem3 18025 . . . . . . . 8 ((𝐵𝑆𝐵𝑆𝐴𝐵) → (𝐵(+g𝑀)𝐵) = 𝐵)
3331, 32syld3an1 1404 . . . . . . 7 ((𝐴𝑆𝐵𝑆𝐴𝐵) → (𝐵(+g𝑀)𝐵) = 𝐵)
3433oveq2d 7167 . . . . . 6 ((𝐴𝑆𝐵𝑆𝐴𝐵) → (𝐴(+g𝑀)(𝐵(+g𝑀)𝐵)) = (𝐴(+g𝑀)𝐵))
3518oveq1d 7166 . . . . . . . 8 ((𝐴𝑆𝐵𝑆) → ((𝐴(+g𝑀)𝐵)(+g𝑀)𝐵) = (𝐴(+g𝑀)𝐵))
3635, 18eqtrd 2860 . . . . . . 7 ((𝐴𝑆𝐵𝑆) → ((𝐴(+g𝑀)𝐵)(+g𝑀)𝐵) = 𝐴)
37363adant3 1126 . . . . . 6 ((𝐴𝑆𝐵𝑆𝐴𝐵) → ((𝐴(+g𝑀)𝐵)(+g𝑀)𝐵) = 𝐴)
3824, 34, 373eqtr4rd 2871 . . . . 5 ((𝐴𝑆𝐵𝑆𝐴𝐵) → ((𝐴(+g𝑀)𝐵)(+g𝑀)𝐵) = (𝐴(+g𝑀)(𝐵(+g𝑀)𝐵)))
3923, 30, 38jca32 516 . . . 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 7166 . . . . . . 7 ((𝐴𝑆𝐵𝑆𝐴𝐵) → ((𝐵(+g𝑀)𝐴)(+g𝑀)𝐴) = (𝐵(+g𝑀)𝐴))
4128oveq2d 7167 . . . . . . 7 ((𝐴𝑆𝐵𝑆𝐴𝐵) → (𝐵(+g𝑀)(𝐴(+g𝑀)𝐴)) = (𝐵(+g𝑀)𝐴))
4240, 41eqtr4d 2863 . . . . . 6 ((𝐴𝑆𝐵𝑆𝐴𝐵) → ((𝐵(+g𝑀)𝐴)(+g𝑀)𝐴) = (𝐵(+g𝑀)(𝐴(+g𝑀)𝐴)))
4324oveq2d 7167 . . . . . . 7 ((𝐴𝑆𝐵𝑆𝐴𝐵) → (𝐵(+g𝑀)(𝐴(+g𝑀)𝐵)) = (𝐵(+g𝑀)𝐴))
4425oveq1d 7166 . . . . . . . 8 ((𝐴𝑆𝐵𝑆𝐴𝐵) → ((𝐵(+g𝑀)𝐴)(+g𝑀)𝐵) = (𝐵(+g𝑀)𝐵))
4544, 33eqtrd 2860 . . . . . . 7 ((𝐴𝑆𝐵𝑆𝐴𝐵) → ((𝐵(+g𝑀)𝐴)(+g𝑀)𝐵) = 𝐵)
4625, 43, 453eqtr4rd 2871 . . . . . 6 ((𝐴𝑆𝐵𝑆𝐴𝐵) → ((𝐵(+g𝑀)𝐴)(+g𝑀)𝐵) = (𝐵(+g𝑀)(𝐴(+g𝑀)𝐵)))
4742, 46jca 512 . . . . 5 ((𝐴𝑆𝐵𝑆𝐴𝐵) → (((𝐵(+g𝑀)𝐴)(+g𝑀)𝐴) = (𝐵(+g𝑀)(𝐴(+g𝑀)𝐴)) ∧ ((𝐵(+g𝑀)𝐴)(+g𝑀)𝐵) = (𝐵(+g𝑀)(𝐴(+g𝑀)𝐵))))
4825oveq2d 7167 . . . . . 6 ((𝐴𝑆𝐵𝑆𝐴𝐵) → (𝐵(+g𝑀)(𝐵(+g𝑀)𝐴)) = (𝐵(+g𝑀)𝐵))
4933oveq1d 7166 . . . . . . 7 ((𝐴𝑆𝐵𝑆𝐴𝐵) → ((𝐵(+g𝑀)𝐵)(+g𝑀)𝐴) = (𝐵(+g𝑀)𝐴))
5049, 25eqtrd 2860 . . . . . 6 ((𝐴𝑆𝐵𝑆𝐴𝐵) → ((𝐵(+g𝑀)𝐵)(+g𝑀)𝐴) = 𝐵)
5133, 48, 503eqtr4rd 2871 . . . . 5 ((𝐴𝑆𝐵𝑆𝐴𝐵) → ((𝐵(+g𝑀)𝐵)(+g𝑀)𝐴) = (𝐵(+g𝑀)(𝐵(+g𝑀)𝐴)))
5232oveq1d 7166 . . . . . . 7 ((𝐵𝑆𝐵𝑆𝐴𝐵) → ((𝐵(+g𝑀)𝐵)(+g𝑀)𝐵) = (𝐵(+g𝑀)𝐵))
5332oveq2d 7167 . . . . . . 7 ((𝐵𝑆𝐵𝑆𝐴𝐵) → (𝐵(+g𝑀)(𝐵(+g𝑀)𝐵)) = (𝐵(+g𝑀)𝐵))
5452, 53eqtr4d 2863 . . . . . 6 ((𝐵𝑆𝐵𝑆𝐴𝐵) → ((𝐵(+g𝑀)𝐵)(+g𝑀)𝐵) = (𝐵(+g𝑀)(𝐵(+g𝑀)𝐵)))
5531, 54syld3an1 1404 . . . . 5 ((𝐴𝑆𝐵𝑆𝐴𝐵) → ((𝐵(+g𝑀)𝐵)(+g𝑀)𝐵) = (𝐵(+g𝑀)(𝐵(+g𝑀)𝐵)))
5647, 51, 55jca32 516 . . . 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 7158 . . . . . . . . . 10 (𝑎 = 𝐴 → (𝑎(+g𝑀)𝑏) = (𝐴(+g𝑀)𝑏))
5857oveq1d 7166 . . . . . . . . 9 (𝑎 = 𝐴 → ((𝑎(+g𝑀)𝑏)(+g𝑀)𝑐) = ((𝐴(+g𝑀)𝑏)(+g𝑀)𝑐))
59 oveq1 7158 . . . . . . . . 9 (𝑎 = 𝐴 → (𝑎(+g𝑀)(𝑏(+g𝑀)𝑐)) = (𝐴(+g𝑀)(𝑏(+g𝑀)𝑐)))
6058, 59eqeq12d 2840 . . . . . . . 8 (𝑎 = 𝐴 → (((𝑎(+g𝑀)𝑏)(+g𝑀)𝑐) = (𝑎(+g𝑀)(𝑏(+g𝑀)𝑐)) ↔ ((𝐴(+g𝑀)𝑏)(+g𝑀)𝑐) = (𝐴(+g𝑀)(𝑏(+g𝑀)𝑐))))
61602ralbidv 3203 . . . . . . 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 7158 . . . . . . . . . 10 (𝑎 = 𝐵 → (𝑎(+g𝑀)𝑏) = (𝐵(+g𝑀)𝑏))
6362oveq1d 7166 . . . . . . . . 9 (𝑎 = 𝐵 → ((𝑎(+g𝑀)𝑏)(+g𝑀)𝑐) = ((𝐵(+g𝑀)𝑏)(+g𝑀)𝑐))
64 oveq1 7158 . . . . . . . . 9 (𝑎 = 𝐵 → (𝑎(+g𝑀)(𝑏(+g𝑀)𝑐)) = (𝐵(+g𝑀)(𝑏(+g𝑀)𝑐)))
6563, 64eqeq12d 2840 . . . . . . . 8 (𝑎 = 𝐵 → (((𝑎(+g𝑀)𝑏)(+g𝑀)𝑐) = (𝑎(+g𝑀)(𝑏(+g𝑀)𝑐)) ↔ ((𝐵(+g𝑀)𝑏)(+g𝑀)𝑐) = (𝐵(+g𝑀)(𝑏(+g𝑀)𝑐))))
66652ralbidv 3203 . . . . . . 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 4630 . . . . . 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 7159 . . . . . . . . . . 11 (𝑏 = 𝐴 → (𝐴(+g𝑀)𝑏) = (𝐴(+g𝑀)𝐴))
6968oveq1d 7166 . . . . . . . . . 10 (𝑏 = 𝐴 → ((𝐴(+g𝑀)𝑏)(+g𝑀)𝑐) = ((𝐴(+g𝑀)𝐴)(+g𝑀)𝑐))
70 oveq1 7158 . . . . . . . . . . 11 (𝑏 = 𝐴 → (𝑏(+g𝑀)𝑐) = (𝐴(+g𝑀)𝑐))
7170oveq2d 7167 . . . . . . . . . 10 (𝑏 = 𝐴 → (𝐴(+g𝑀)(𝑏(+g𝑀)𝑐)) = (𝐴(+g𝑀)(𝐴(+g𝑀)𝑐)))
7269, 71eqeq12d 2840 . . . . . . . . 9 (𝑏 = 𝐴 → (((𝐴(+g𝑀)𝑏)(+g𝑀)𝑐) = (𝐴(+g𝑀)(𝑏(+g𝑀)𝑐)) ↔ ((𝐴(+g𝑀)𝐴)(+g𝑀)𝑐) = (𝐴(+g𝑀)(𝐴(+g𝑀)𝑐))))
7372ralbidv 3201 . . . . . . . 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 7159 . . . . . . . . . . 11 (𝑏 = 𝐵 → (𝐴(+g𝑀)𝑏) = (𝐴(+g𝑀)𝐵))
7574oveq1d 7166 . . . . . . . . . 10 (𝑏 = 𝐵 → ((𝐴(+g𝑀)𝑏)(+g𝑀)𝑐) = ((𝐴(+g𝑀)𝐵)(+g𝑀)𝑐))
76 oveq1 7158 . . . . . . . . . . 11 (𝑏 = 𝐵 → (𝑏(+g𝑀)𝑐) = (𝐵(+g𝑀)𝑐))
7776oveq2d 7167 . . . . . . . . . 10 (𝑏 = 𝐵 → (𝐴(+g𝑀)(𝑏(+g𝑀)𝑐)) = (𝐴(+g𝑀)(𝐵(+g𝑀)𝑐)))
7875, 77eqeq12d 2840 . . . . . . . . 9 (𝑏 = 𝐵 → (((𝐴(+g𝑀)𝑏)(+g𝑀)𝑐) = (𝐴(+g𝑀)(𝑏(+g𝑀)𝑐)) ↔ ((𝐴(+g𝑀)𝐵)(+g𝑀)𝑐) = (𝐴(+g𝑀)(𝐵(+g𝑀)𝑐))))
7978ralbidv 3201 . . . . . . . 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 4630 . . . . . . 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 7159 . . . . . . . . . . 11 (𝑏 = 𝐴 → (𝐵(+g𝑀)𝑏) = (𝐵(+g𝑀)𝐴))
8281oveq1d 7166 . . . . . . . . . 10 (𝑏 = 𝐴 → ((𝐵(+g𝑀)𝑏)(+g𝑀)𝑐) = ((𝐵(+g𝑀)𝐴)(+g𝑀)𝑐))
8370oveq2d 7167 . . . . . . . . . 10 (𝑏 = 𝐴 → (𝐵(+g𝑀)(𝑏(+g𝑀)𝑐)) = (𝐵(+g𝑀)(𝐴(+g𝑀)𝑐)))
8482, 83eqeq12d 2840 . . . . . . . . 9 (𝑏 = 𝐴 → (((𝐵(+g𝑀)𝑏)(+g𝑀)𝑐) = (𝐵(+g𝑀)(𝑏(+g𝑀)𝑐)) ↔ ((𝐵(+g𝑀)𝐴)(+g𝑀)𝑐) = (𝐵(+g𝑀)(𝐴(+g𝑀)𝑐))))
8584ralbidv 3201 . . . . . . . 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 7159 . . . . . . . . . . 11 (𝑏 = 𝐵 → (𝐵(+g𝑀)𝑏) = (𝐵(+g𝑀)𝐵))
8786oveq1d 7166 . . . . . . . . . 10 (𝑏 = 𝐵 → ((𝐵(+g𝑀)𝑏)(+g𝑀)𝑐) = ((𝐵(+g𝑀)𝐵)(+g𝑀)𝑐))
8876oveq2d 7167 . . . . . . . . . 10 (𝑏 = 𝐵 → (𝐵(+g𝑀)(𝑏(+g𝑀)𝑐)) = (𝐵(+g𝑀)(𝐵(+g𝑀)𝑐)))
8987, 88eqeq12d 2840 . . . . . . . . 9 (𝑏 = 𝐵 → (((𝐵(+g𝑀)𝑏)(+g𝑀)𝑐) = (𝐵(+g𝑀)(𝑏(+g𝑀)𝑐)) ↔ ((𝐵(+g𝑀)𝐵)(+g𝑀)𝑐) = (𝐵(+g𝑀)(𝐵(+g𝑀)𝑐))))
9089ralbidv 3201 . . . . . . . 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 4630 . . . . . . 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 630 . . . . . 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 7159 . . . . . . . . . 10 (𝑐 = 𝐴 → ((𝐴(+g𝑀)𝐴)(+g𝑀)𝑐) = ((𝐴(+g𝑀)𝐴)(+g𝑀)𝐴))
94 oveq2 7159 . . . . . . . . . . 11 (𝑐 = 𝐴 → (𝐴(+g𝑀)𝑐) = (𝐴(+g𝑀)𝐴))
9594oveq2d 7167 . . . . . . . . . 10 (𝑐 = 𝐴 → (𝐴(+g𝑀)(𝐴(+g𝑀)𝑐)) = (𝐴(+g𝑀)(𝐴(+g𝑀)𝐴)))
9693, 95eqeq12d 2840 . . . . . . . . 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 7159 . . . . . . . . . 10 (𝑐 = 𝐵 → ((𝐴(+g𝑀)𝐴)(+g𝑀)𝑐) = ((𝐴(+g𝑀)𝐴)(+g𝑀)𝐵))
98 oveq2 7159 . . . . . . . . . . 11 (𝑐 = 𝐵 → (𝐴(+g𝑀)𝑐) = (𝐴(+g𝑀)𝐵))
9998oveq2d 7167 . . . . . . . . . 10 (𝑐 = 𝐵 → (𝐴(+g𝑀)(𝐴(+g𝑀)𝑐)) = (𝐴(+g𝑀)(𝐴(+g𝑀)𝐵)))
10097, 99eqeq12d 2840 . . . . . . . . 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 4630 . . . . . . . 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 7159 . . . . . . . . . 10 (𝑐 = 𝐴 → ((𝐴(+g𝑀)𝐵)(+g𝑀)𝑐) = ((𝐴(+g𝑀)𝐵)(+g𝑀)𝐴))
103 oveq2 7159 . . . . . . . . . . 11 (𝑐 = 𝐴 → (𝐵(+g𝑀)𝑐) = (𝐵(+g𝑀)𝐴))
104103oveq2d 7167 . . . . . . . . . 10 (𝑐 = 𝐴 → (𝐴(+g𝑀)(𝐵(+g𝑀)𝑐)) = (𝐴(+g𝑀)(𝐵(+g𝑀)𝐴)))
105102, 104eqeq12d 2840 . . . . . . . . 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 7159 . . . . . . . . . 10 (𝑐 = 𝐵 → ((𝐴(+g𝑀)𝐵)(+g𝑀)𝑐) = ((𝐴(+g𝑀)𝐵)(+g𝑀)𝐵))
107 oveq2 7159 . . . . . . . . . . 11 (𝑐 = 𝐵 → (𝐵(+g𝑀)𝑐) = (𝐵(+g𝑀)𝐵))
108107oveq2d 7167 . . . . . . . . . 10 (𝑐 = 𝐵 → (𝐴(+g𝑀)(𝐵(+g𝑀)𝑐)) = (𝐴(+g𝑀)(𝐵(+g𝑀)𝐵)))
109106, 108eqeq12d 2840 . . . . . . . . 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 4630 . . . . . . . 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 630 . . . . . . 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 7159 . . . . . . . . . 10 (𝑐 = 𝐴 → ((𝐵(+g𝑀)𝐴)(+g𝑀)𝑐) = ((𝐵(+g𝑀)𝐴)(+g𝑀)𝐴))
11394oveq2d 7167 . . . . . . . . . 10 (𝑐 = 𝐴 → (𝐵(+g𝑀)(𝐴(+g𝑀)𝑐)) = (𝐵(+g𝑀)(𝐴(+g𝑀)𝐴)))
114112, 113eqeq12d 2840 . . . . . . . . 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 7159 . . . . . . . . . 10 (𝑐 = 𝐵 → ((𝐵(+g𝑀)𝐴)(+g𝑀)𝑐) = ((𝐵(+g𝑀)𝐴)(+g𝑀)𝐵))
11698oveq2d 7167 . . . . . . . . . 10 (𝑐 = 𝐵 → (𝐵(+g𝑀)(𝐴(+g𝑀)𝑐)) = (𝐵(+g𝑀)(𝐴(+g𝑀)𝐵)))
117115, 116eqeq12d 2840 . . . . . . . . 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 4630 . . . . . . . 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 7159 . . . . . . . . . 10 (𝑐 = 𝐴 → ((𝐵(+g𝑀)𝐵)(+g𝑀)𝑐) = ((𝐵(+g𝑀)𝐵)(+g𝑀)𝐴))
120103oveq2d 7167 . . . . . . . . . 10 (𝑐 = 𝐴 → (𝐵(+g𝑀)(𝐵(+g𝑀)𝑐)) = (𝐵(+g𝑀)(𝐵(+g𝑀)𝐴)))
121119, 120eqeq12d 2840 . . . . . . . . 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 7159 . . . . . . . . . 10 (𝑐 = 𝐵 → ((𝐵(+g𝑀)𝐵)(+g𝑀)𝑐) = ((𝐵(+g𝑀)𝐵)(+g𝑀)𝐵))
123107oveq2d 7167 . . . . . . . . . 10 (𝑐 = 𝐵 → (𝐵(+g𝑀)(𝐵(+g𝑀)𝑐)) = (𝐵(+g𝑀)(𝐵(+g𝑀)𝐵)))
124122, 123eqeq12d 2840 . . . . . . . . 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 4630 . . . . . . . 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 630 . . . . . . 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 630 . . . . . 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 306 . . . . 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 1126 . . . 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 709 . . 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 2849 . . 3 {𝐴, 𝐵} = (Base‘𝑀)
133132, 8issgrp 17893 . 2 (𝑀 ∈ Smgrp ↔ (𝑀 ∈ Mgm ∧ ∀𝑎 ∈ {𝐴, 𝐵}∀𝑏 ∈ {𝐴, 𝐵}∀𝑐 ∈ {𝐴, 𝐵} ((𝑎(+g𝑀)𝑏)(+g𝑀)𝑐) = (𝑎(+g𝑀)(𝑏(+g𝑀)𝑐))))
1347, 131, 133sylanbrc 583 1 ((♯‘𝑆) = 2 → 𝑀 ∈ Smgrp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1081   = wceq 1530  wcel 2106  wne 3020  wral 3142  ifcif 4469  {cpr 4565  cfv 6351  (class class class)co 7151  cmpo 7153  2c2 11684  chash 13683  Basecbs 16475  +gcplusg 16557  Mgmcmgm 17842  Smgrpcsgrp 17891
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2615  df-eu 2649  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ne 3021  df-nel 3128  df-ral 3147  df-rex 3148  df-reu 3149  df-rmo 3150  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-int 4874  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-pred 6145  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7572  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-oadd 8100  df-er 8282  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-dju 9322  df-card 9360  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11631  df-2 11692  df-n0 11890  df-z 11974  df-uz 12236  df-fz 12886  df-hash 13684  df-mgm 17844  df-sgrp 17892
This theorem is referenced by:  sgrp2nmnd  18030  sgrpnmndex  18032
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