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Theorem sgrp2nmndlem4 17730
Description: Lemma 4 for sgrp2nmnd 17732: 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 13434 . . 3 ((♯‘𝑆) = 2 → (𝐴𝑆𝐵𝑆𝐴𝐵))
3 3simpa 1179 . . 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 17725 . . 3 ((𝐴𝑆𝐵𝑆) → 𝑀 ∈ Mgm)
72, 3, 63syl 18 . 2 ((♯‘𝑆) = 2 → 𝑀 ∈ Mgm)
8 eqid 2800 . . . . . . . . . . 11 (+g𝑀) = (+g𝑀)
91, 4, 5, 8sgrp2nmndlem2 17726 . . . . . . . . . 10 ((𝐴𝑆𝐴𝑆) → (𝐴(+g𝑀)𝐴) = 𝐴)
109oveq1d 6894 . . . . . . . . 9 ((𝐴𝑆𝐴𝑆) → ((𝐴(+g𝑀)𝐴)(+g𝑀)𝐴) = (𝐴(+g𝑀)𝐴))
119oveq2d 6895 . . . . . . . . 9 ((𝐴𝑆𝐴𝑆) → (𝐴(+g𝑀)(𝐴(+g𝑀)𝐴)) = (𝐴(+g𝑀)𝐴))
1210, 11eqtr4d 2837 . . . . . . . 8 ((𝐴𝑆𝐴𝑆) → ((𝐴(+g𝑀)𝐴)(+g𝑀)𝐴) = (𝐴(+g𝑀)(𝐴(+g𝑀)𝐴)))
1312anidms 563 . . . . . . 7 (𝐴𝑆 → ((𝐴(+g𝑀)𝐴)(+g𝑀)𝐴) = (𝐴(+g𝑀)(𝐴(+g𝑀)𝐴)))
14133ad2ant1 1164 . . . . . 6 ((𝐴𝑆𝐵𝑆𝐴𝐵) → ((𝐴(+g𝑀)𝐴)(+g𝑀)𝐴) = (𝐴(+g𝑀)(𝐴(+g𝑀)𝐴)))
159anidms 563 . . . . . . . . . 10 (𝐴𝑆 → (𝐴(+g𝑀)𝐴) = 𝐴)
1615adantr 473 . . . . . . . . 9 ((𝐴𝑆𝐵𝑆) → (𝐴(+g𝑀)𝐴) = 𝐴)
1716oveq1d 6894 . . . . . . . 8 ((𝐴𝑆𝐵𝑆) → ((𝐴(+g𝑀)𝐴)(+g𝑀)𝐵) = (𝐴(+g𝑀)𝐵))
181, 4, 5, 8sgrp2nmndlem2 17726 . . . . . . . . . 10 ((𝐴𝑆𝐵𝑆) → (𝐴(+g𝑀)𝐵) = 𝐴)
1918oveq2d 6895 . . . . . . . . 9 ((𝐴𝑆𝐵𝑆) → (𝐴(+g𝑀)(𝐴(+g𝑀)𝐵)) = (𝐴(+g𝑀)𝐴))
2016, 19, 183eqtr4rd 2845 . . . . . . . 8 ((𝐴𝑆𝐵𝑆) → (𝐴(+g𝑀)𝐵) = (𝐴(+g𝑀)(𝐴(+g𝑀)𝐵)))
2117, 20eqtrd 2834 . . . . . . 7 ((𝐴𝑆𝐵𝑆) → ((𝐴(+g𝑀)𝐴)(+g𝑀)𝐵) = (𝐴(+g𝑀)(𝐴(+g𝑀)𝐵)))
22213adant3 1163 . . . . . 6 ((𝐴𝑆𝐵𝑆𝐴𝐵) → ((𝐴(+g𝑀)𝐴)(+g𝑀)𝐵) = (𝐴(+g𝑀)(𝐴(+g𝑀)𝐵)))
2314, 22jca 508 . . . . 5 ((𝐴𝑆𝐵𝑆𝐴𝐵) → (((𝐴(+g𝑀)𝐴)(+g𝑀)𝐴) = (𝐴(+g𝑀)(𝐴(+g𝑀)𝐴)) ∧ ((𝐴(+g𝑀)𝐴)(+g𝑀)𝐵) = (𝐴(+g𝑀)(𝐴(+g𝑀)𝐵))))
24183adant3 1163 . . . . . 6 ((𝐴𝑆𝐵𝑆𝐴𝐵) → (𝐴(+g𝑀)𝐵) = 𝐴)
251, 4, 5, 8sgrp2nmndlem3 17727 . . . . . . 7 ((𝐴𝑆𝐵𝑆𝐴𝐵) → (𝐵(+g𝑀)𝐴) = 𝐵)
2625oveq2d 6895 . . . . . 6 ((𝐴𝑆𝐵𝑆𝐴𝐵) → (𝐴(+g𝑀)(𝐵(+g𝑀)𝐴)) = (𝐴(+g𝑀)𝐵))
2724oveq1d 6894 . . . . . . 7 ((𝐴𝑆𝐵𝑆𝐴𝐵) → ((𝐴(+g𝑀)𝐵)(+g𝑀)𝐴) = (𝐴(+g𝑀)𝐴))
28153ad2ant1 1164 . . . . . . 7 ((𝐴𝑆𝐵𝑆𝐴𝐵) → (𝐴(+g𝑀)𝐴) = 𝐴)
2927, 28eqtrd 2834 . . . . . 6 ((𝐴𝑆𝐵𝑆𝐴𝐵) → ((𝐴(+g𝑀)𝐵)(+g𝑀)𝐴) = 𝐴)
3024, 26, 293eqtr4rd 2845 . . . . 5 ((𝐴𝑆𝐵𝑆𝐴𝐵) → ((𝐴(+g𝑀)𝐵)(+g𝑀)𝐴) = (𝐴(+g𝑀)(𝐵(+g𝑀)𝐴)))
31 simp2 1168 . . . . . . . 8 ((𝐴𝑆𝐵𝑆𝐴𝐵) → 𝐵𝑆)
321, 4, 5, 8sgrp2nmndlem3 17727 . . . . . . . 8 ((𝐵𝑆𝐵𝑆𝐴𝐵) → (𝐵(+g𝑀)𝐵) = 𝐵)
3331, 32syld3an1 1530 . . . . . . 7 ((𝐴𝑆𝐵𝑆𝐴𝐵) → (𝐵(+g𝑀)𝐵) = 𝐵)
3433oveq2d 6895 . . . . . 6 ((𝐴𝑆𝐵𝑆𝐴𝐵) → (𝐴(+g𝑀)(𝐵(+g𝑀)𝐵)) = (𝐴(+g𝑀)𝐵))
3518oveq1d 6894 . . . . . . . 8 ((𝐴𝑆𝐵𝑆) → ((𝐴(+g𝑀)𝐵)(+g𝑀)𝐵) = (𝐴(+g𝑀)𝐵))
3635, 18eqtrd 2834 . . . . . . 7 ((𝐴𝑆𝐵𝑆) → ((𝐴(+g𝑀)𝐵)(+g𝑀)𝐵) = 𝐴)
37363adant3 1163 . . . . . 6 ((𝐴𝑆𝐵𝑆𝐴𝐵) → ((𝐴(+g𝑀)𝐵)(+g𝑀)𝐵) = 𝐴)
3824, 34, 373eqtr4rd 2845 . . . . 5 ((𝐴𝑆𝐵𝑆𝐴𝐵) → ((𝐴(+g𝑀)𝐵)(+g𝑀)𝐵) = (𝐴(+g𝑀)(𝐵(+g𝑀)𝐵)))
3923, 30, 38jca32 512 . . . 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 6894 . . . . . . 7 ((𝐴𝑆𝐵𝑆𝐴𝐵) → ((𝐵(+g𝑀)𝐴)(+g𝑀)𝐴) = (𝐵(+g𝑀)𝐴))
4128oveq2d 6895 . . . . . . 7 ((𝐴𝑆𝐵𝑆𝐴𝐵) → (𝐵(+g𝑀)(𝐴(+g𝑀)𝐴)) = (𝐵(+g𝑀)𝐴))
4240, 41eqtr4d 2837 . . . . . 6 ((𝐴𝑆𝐵𝑆𝐴𝐵) → ((𝐵(+g𝑀)𝐴)(+g𝑀)𝐴) = (𝐵(+g𝑀)(𝐴(+g𝑀)𝐴)))
4324oveq2d 6895 . . . . . . 7 ((𝐴𝑆𝐵𝑆𝐴𝐵) → (𝐵(+g𝑀)(𝐴(+g𝑀)𝐵)) = (𝐵(+g𝑀)𝐴))
4425oveq1d 6894 . . . . . . . 8 ((𝐴𝑆𝐵𝑆𝐴𝐵) → ((𝐵(+g𝑀)𝐴)(+g𝑀)𝐵) = (𝐵(+g𝑀)𝐵))
4544, 33eqtrd 2834 . . . . . . 7 ((𝐴𝑆𝐵𝑆𝐴𝐵) → ((𝐵(+g𝑀)𝐴)(+g𝑀)𝐵) = 𝐵)
4625, 43, 453eqtr4rd 2845 . . . . . 6 ((𝐴𝑆𝐵𝑆𝐴𝐵) → ((𝐵(+g𝑀)𝐴)(+g𝑀)𝐵) = (𝐵(+g𝑀)(𝐴(+g𝑀)𝐵)))
4742, 46jca 508 . . . . 5 ((𝐴𝑆𝐵𝑆𝐴𝐵) → (((𝐵(+g𝑀)𝐴)(+g𝑀)𝐴) = (𝐵(+g𝑀)(𝐴(+g𝑀)𝐴)) ∧ ((𝐵(+g𝑀)𝐴)(+g𝑀)𝐵) = (𝐵(+g𝑀)(𝐴(+g𝑀)𝐵))))
4825oveq2d 6895 . . . . . 6 ((𝐴𝑆𝐵𝑆𝐴𝐵) → (𝐵(+g𝑀)(𝐵(+g𝑀)𝐴)) = (𝐵(+g𝑀)𝐵))
4933oveq1d 6894 . . . . . . 7 ((𝐴𝑆𝐵𝑆𝐴𝐵) → ((𝐵(+g𝑀)𝐵)(+g𝑀)𝐴) = (𝐵(+g𝑀)𝐴))
5049, 25eqtrd 2834 . . . . . 6 ((𝐴𝑆𝐵𝑆𝐴𝐵) → ((𝐵(+g𝑀)𝐵)(+g𝑀)𝐴) = 𝐵)
5133, 48, 503eqtr4rd 2845 . . . . 5 ((𝐴𝑆𝐵𝑆𝐴𝐵) → ((𝐵(+g𝑀)𝐵)(+g𝑀)𝐴) = (𝐵(+g𝑀)(𝐵(+g𝑀)𝐴)))
5232oveq1d 6894 . . . . . . 7 ((𝐵𝑆𝐵𝑆𝐴𝐵) → ((𝐵(+g𝑀)𝐵)(+g𝑀)𝐵) = (𝐵(+g𝑀)𝐵))
5332oveq2d 6895 . . . . . . 7 ((𝐵𝑆𝐵𝑆𝐴𝐵) → (𝐵(+g𝑀)(𝐵(+g𝑀)𝐵)) = (𝐵(+g𝑀)𝐵))
5452, 53eqtr4d 2837 . . . . . 6 ((𝐵𝑆𝐵𝑆𝐴𝐵) → ((𝐵(+g𝑀)𝐵)(+g𝑀)𝐵) = (𝐵(+g𝑀)(𝐵(+g𝑀)𝐵)))
5531, 54syld3an1 1530 . . . . 5 ((𝐴𝑆𝐵𝑆𝐴𝐵) → ((𝐵(+g𝑀)𝐵)(+g𝑀)𝐵) = (𝐵(+g𝑀)(𝐵(+g𝑀)𝐵)))
5647, 51, 55jca32 512 . . . 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 6886 . . . . . . . . . 10 (𝑎 = 𝐴 → (𝑎(+g𝑀)𝑏) = (𝐴(+g𝑀)𝑏))
5857oveq1d 6894 . . . . . . . . 9 (𝑎 = 𝐴 → ((𝑎(+g𝑀)𝑏)(+g𝑀)𝑐) = ((𝐴(+g𝑀)𝑏)(+g𝑀)𝑐))
59 oveq1 6886 . . . . . . . . 9 (𝑎 = 𝐴 → (𝑎(+g𝑀)(𝑏(+g𝑀)𝑐)) = (𝐴(+g𝑀)(𝑏(+g𝑀)𝑐)))
6058, 59eqeq12d 2815 . . . . . . . 8 (𝑎 = 𝐴 → (((𝑎(+g𝑀)𝑏)(+g𝑀)𝑐) = (𝑎(+g𝑀)(𝑏(+g𝑀)𝑐)) ↔ ((𝐴(+g𝑀)𝑏)(+g𝑀)𝑐) = (𝐴(+g𝑀)(𝑏(+g𝑀)𝑐))))
61602ralbidv 3171 . . . . . . 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 6886 . . . . . . . . . 10 (𝑎 = 𝐵 → (𝑎(+g𝑀)𝑏) = (𝐵(+g𝑀)𝑏))
6362oveq1d 6894 . . . . . . . . 9 (𝑎 = 𝐵 → ((𝑎(+g𝑀)𝑏)(+g𝑀)𝑐) = ((𝐵(+g𝑀)𝑏)(+g𝑀)𝑐))
64 oveq1 6886 . . . . . . . . 9 (𝑎 = 𝐵 → (𝑎(+g𝑀)(𝑏(+g𝑀)𝑐)) = (𝐵(+g𝑀)(𝑏(+g𝑀)𝑐)))
6563, 64eqeq12d 2815 . . . . . . . 8 (𝑎 = 𝐵 → (((𝑎(+g𝑀)𝑏)(+g𝑀)𝑐) = (𝑎(+g𝑀)(𝑏(+g𝑀)𝑐)) ↔ ((𝐵(+g𝑀)𝑏)(+g𝑀)𝑐) = (𝐵(+g𝑀)(𝑏(+g𝑀)𝑐))))
66652ralbidv 3171 . . . . . . 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 4425 . . . . . 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 6887 . . . . . . . . . . 11 (𝑏 = 𝐴 → (𝐴(+g𝑀)𝑏) = (𝐴(+g𝑀)𝐴))
6968oveq1d 6894 . . . . . . . . . 10 (𝑏 = 𝐴 → ((𝐴(+g𝑀)𝑏)(+g𝑀)𝑐) = ((𝐴(+g𝑀)𝐴)(+g𝑀)𝑐))
70 oveq1 6886 . . . . . . . . . . 11 (𝑏 = 𝐴 → (𝑏(+g𝑀)𝑐) = (𝐴(+g𝑀)𝑐))
7170oveq2d 6895 . . . . . . . . . 10 (𝑏 = 𝐴 → (𝐴(+g𝑀)(𝑏(+g𝑀)𝑐)) = (𝐴(+g𝑀)(𝐴(+g𝑀)𝑐)))
7269, 71eqeq12d 2815 . . . . . . . . 9 (𝑏 = 𝐴 → (((𝐴(+g𝑀)𝑏)(+g𝑀)𝑐) = (𝐴(+g𝑀)(𝑏(+g𝑀)𝑐)) ↔ ((𝐴(+g𝑀)𝐴)(+g𝑀)𝑐) = (𝐴(+g𝑀)(𝐴(+g𝑀)𝑐))))
7372ralbidv 3168 . . . . . . . 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 6887 . . . . . . . . . . 11 (𝑏 = 𝐵 → (𝐴(+g𝑀)𝑏) = (𝐴(+g𝑀)𝐵))
7574oveq1d 6894 . . . . . . . . . 10 (𝑏 = 𝐵 → ((𝐴(+g𝑀)𝑏)(+g𝑀)𝑐) = ((𝐴(+g𝑀)𝐵)(+g𝑀)𝑐))
76 oveq1 6886 . . . . . . . . . . 11 (𝑏 = 𝐵 → (𝑏(+g𝑀)𝑐) = (𝐵(+g𝑀)𝑐))
7776oveq2d 6895 . . . . . . . . . 10 (𝑏 = 𝐵 → (𝐴(+g𝑀)(𝑏(+g𝑀)𝑐)) = (𝐴(+g𝑀)(𝐵(+g𝑀)𝑐)))
7875, 77eqeq12d 2815 . . . . . . . . 9 (𝑏 = 𝐵 → (((𝐴(+g𝑀)𝑏)(+g𝑀)𝑐) = (𝐴(+g𝑀)(𝑏(+g𝑀)𝑐)) ↔ ((𝐴(+g𝑀)𝐵)(+g𝑀)𝑐) = (𝐴(+g𝑀)(𝐵(+g𝑀)𝑐))))
7978ralbidv 3168 . . . . . . . 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 4425 . . . . . . 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 6887 . . . . . . . . . . 11 (𝑏 = 𝐴 → (𝐵(+g𝑀)𝑏) = (𝐵(+g𝑀)𝐴))
8281oveq1d 6894 . . . . . . . . . 10 (𝑏 = 𝐴 → ((𝐵(+g𝑀)𝑏)(+g𝑀)𝑐) = ((𝐵(+g𝑀)𝐴)(+g𝑀)𝑐))
8370oveq2d 6895 . . . . . . . . . 10 (𝑏 = 𝐴 → (𝐵(+g𝑀)(𝑏(+g𝑀)𝑐)) = (𝐵(+g𝑀)(𝐴(+g𝑀)𝑐)))
8482, 83eqeq12d 2815 . . . . . . . . 9 (𝑏 = 𝐴 → (((𝐵(+g𝑀)𝑏)(+g𝑀)𝑐) = (𝐵(+g𝑀)(𝑏(+g𝑀)𝑐)) ↔ ((𝐵(+g𝑀)𝐴)(+g𝑀)𝑐) = (𝐵(+g𝑀)(𝐴(+g𝑀)𝑐))))
8584ralbidv 3168 . . . . . . . 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 6887 . . . . . . . . . . 11 (𝑏 = 𝐵 → (𝐵(+g𝑀)𝑏) = (𝐵(+g𝑀)𝐵))
8786oveq1d 6894 . . . . . . . . . 10 (𝑏 = 𝐵 → ((𝐵(+g𝑀)𝑏)(+g𝑀)𝑐) = ((𝐵(+g𝑀)𝐵)(+g𝑀)𝑐))
8876oveq2d 6895 . . . . . . . . . 10 (𝑏 = 𝐵 → (𝐵(+g𝑀)(𝑏(+g𝑀)𝑐)) = (𝐵(+g𝑀)(𝐵(+g𝑀)𝑐)))
8987, 88eqeq12d 2815 . . . . . . . . 9 (𝑏 = 𝐵 → (((𝐵(+g𝑀)𝑏)(+g𝑀)𝑐) = (𝐵(+g𝑀)(𝑏(+g𝑀)𝑐)) ↔ ((𝐵(+g𝑀)𝐵)(+g𝑀)𝑐) = (𝐵(+g𝑀)(𝐵(+g𝑀)𝑐))))
9089ralbidv 3168 . . . . . . . 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 4425 . . . . . . 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 625 . . . . . 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 6887 . . . . . . . . . 10 (𝑐 = 𝐴 → ((𝐴(+g𝑀)𝐴)(+g𝑀)𝑐) = ((𝐴(+g𝑀)𝐴)(+g𝑀)𝐴))
94 oveq2 6887 . . . . . . . . . . 11 (𝑐 = 𝐴 → (𝐴(+g𝑀)𝑐) = (𝐴(+g𝑀)𝐴))
9594oveq2d 6895 . . . . . . . . . 10 (𝑐 = 𝐴 → (𝐴(+g𝑀)(𝐴(+g𝑀)𝑐)) = (𝐴(+g𝑀)(𝐴(+g𝑀)𝐴)))
9693, 95eqeq12d 2815 . . . . . . . . 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 6887 . . . . . . . . . 10 (𝑐 = 𝐵 → ((𝐴(+g𝑀)𝐴)(+g𝑀)𝑐) = ((𝐴(+g𝑀)𝐴)(+g𝑀)𝐵))
98 oveq2 6887 . . . . . . . . . . 11 (𝑐 = 𝐵 → (𝐴(+g𝑀)𝑐) = (𝐴(+g𝑀)𝐵))
9998oveq2d 6895 . . . . . . . . . 10 (𝑐 = 𝐵 → (𝐴(+g𝑀)(𝐴(+g𝑀)𝑐)) = (𝐴(+g𝑀)(𝐴(+g𝑀)𝐵)))
10097, 99eqeq12d 2815 . . . . . . . . 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 4425 . . . . . . . 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 6887 . . . . . . . . . 10 (𝑐 = 𝐴 → ((𝐴(+g𝑀)𝐵)(+g𝑀)𝑐) = ((𝐴(+g𝑀)𝐵)(+g𝑀)𝐴))
103 oveq2 6887 . . . . . . . . . . 11 (𝑐 = 𝐴 → (𝐵(+g𝑀)𝑐) = (𝐵(+g𝑀)𝐴))
104103oveq2d 6895 . . . . . . . . . 10 (𝑐 = 𝐴 → (𝐴(+g𝑀)(𝐵(+g𝑀)𝑐)) = (𝐴(+g𝑀)(𝐵(+g𝑀)𝐴)))
105102, 104eqeq12d 2815 . . . . . . . . 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 6887 . . . . . . . . . 10 (𝑐 = 𝐵 → ((𝐴(+g𝑀)𝐵)(+g𝑀)𝑐) = ((𝐴(+g𝑀)𝐵)(+g𝑀)𝐵))
107 oveq2 6887 . . . . . . . . . . 11 (𝑐 = 𝐵 → (𝐵(+g𝑀)𝑐) = (𝐵(+g𝑀)𝐵))
108107oveq2d 6895 . . . . . . . . . 10 (𝑐 = 𝐵 → (𝐴(+g𝑀)(𝐵(+g𝑀)𝑐)) = (𝐴(+g𝑀)(𝐵(+g𝑀)𝐵)))
109106, 108eqeq12d 2815 . . . . . . . . 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 4425 . . . . . . . 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 625 . . . . . . 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 6887 . . . . . . . . . 10 (𝑐 = 𝐴 → ((𝐵(+g𝑀)𝐴)(+g𝑀)𝑐) = ((𝐵(+g𝑀)𝐴)(+g𝑀)𝐴))
11394oveq2d 6895 . . . . . . . . . 10 (𝑐 = 𝐴 → (𝐵(+g𝑀)(𝐴(+g𝑀)𝑐)) = (𝐵(+g𝑀)(𝐴(+g𝑀)𝐴)))
114112, 113eqeq12d 2815 . . . . . . . . 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 6887 . . . . . . . . . 10 (𝑐 = 𝐵 → ((𝐵(+g𝑀)𝐴)(+g𝑀)𝑐) = ((𝐵(+g𝑀)𝐴)(+g𝑀)𝐵))
11698oveq2d 6895 . . . . . . . . . 10 (𝑐 = 𝐵 → (𝐵(+g𝑀)(𝐴(+g𝑀)𝑐)) = (𝐵(+g𝑀)(𝐴(+g𝑀)𝐵)))
117115, 116eqeq12d 2815 . . . . . . . . 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 4425 . . . . . . . 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 6887 . . . . . . . . . 10 (𝑐 = 𝐴 → ((𝐵(+g𝑀)𝐵)(+g𝑀)𝑐) = ((𝐵(+g𝑀)𝐵)(+g𝑀)𝐴))
120103oveq2d 6895 . . . . . . . . . 10 (𝑐 = 𝐴 → (𝐵(+g𝑀)(𝐵(+g𝑀)𝑐)) = (𝐵(+g𝑀)(𝐵(+g𝑀)𝐴)))
121119, 120eqeq12d 2815 . . . . . . . . 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 6887 . . . . . . . . . 10 (𝑐 = 𝐵 → ((𝐵(+g𝑀)𝐵)(+g𝑀)𝑐) = ((𝐵(+g𝑀)𝐵)(+g𝑀)𝐵))
123107oveq2d 6895 . . . . . . . . . 10 (𝑐 = 𝐵 → (𝐵(+g𝑀)(𝐵(+g𝑀)𝑐)) = (𝐵(+g𝑀)(𝐵(+g𝑀)𝐵)))
124122, 123eqeq12d 2815 . . . . . . . . 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 4425 . . . . . . . 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 625 . . . . . . 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 625 . . . . . 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 297 . . . . 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 1163 . . . 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 705 . . 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 2823 . . 3 {𝐴, 𝐵} = (Base‘𝑀)
133132, 8issgrp 17599 . 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 579 1 ((♯‘𝑆) = 2 → 𝑀 ∈ SGrp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  w3a 1108   = wceq 1653  wcel 2157  wne 2972  wral 3090  ifcif 4278  {cpr 4371  cfv 6102  (class class class)co 6879  cmpt2 6881  2c2 11367  chash 13369  Basecbs 16183  +gcplusg 16266  Mgmcmgm 17554  SGrpcsgrp 17597
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-rep 4965  ax-sep 4976  ax-nul 4984  ax-pow 5036  ax-pr 5098  ax-un 7184  ax-cnex 10281  ax-resscn 10282  ax-1cn 10283  ax-icn 10284  ax-addcl 10285  ax-addrcl 10286  ax-mulcl 10287  ax-mulrcl 10288  ax-mulcom 10289  ax-addass 10290  ax-mulass 10291  ax-distr 10292  ax-i2m1 10293  ax-1ne0 10294  ax-1rid 10295  ax-rnegex 10296  ax-rrecex 10297  ax-cnre 10298  ax-pre-lttri 10299  ax-pre-lttrn 10300  ax-pre-ltadd 10301  ax-pre-mulgt0 10302
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2592  df-eu 2610  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ne 2973  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3388  df-sbc 3635  df-csb 3730  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-pss 3786  df-nul 4117  df-if 4279  df-pw 4352  df-sn 4370  df-pr 4372  df-tp 4374  df-op 4376  df-uni 4630  df-int 4669  df-iun 4713  df-br 4845  df-opab 4907  df-mpt 4924  df-tr 4947  df-id 5221  df-eprel 5226  df-po 5234  df-so 5235  df-fr 5272  df-we 5274  df-xp 5319  df-rel 5320  df-cnv 5321  df-co 5322  df-dm 5323  df-rn 5324  df-res 5325  df-ima 5326  df-pred 5899  df-ord 5945  df-on 5946  df-lim 5947  df-suc 5948  df-iota 6065  df-fun 6104  df-fn 6105  df-f 6106  df-f1 6107  df-fo 6108  df-f1o 6109  df-fv 6110  df-riota 6840  df-ov 6882  df-oprab 6883  df-mpt2 6884  df-om 7301  df-1st 7402  df-2nd 7403  df-wrecs 7646  df-recs 7708  df-rdg 7746  df-1o 7800  df-oadd 7804  df-er 7983  df-en 8197  df-dom 8198  df-sdom 8199  df-fin 8200  df-card 9052  df-cda 9279  df-pnf 10366  df-mnf 10367  df-xr 10368  df-ltxr 10369  df-le 10370  df-sub 10559  df-neg 10560  df-nn 11314  df-2 11375  df-n0 11580  df-z 11666  df-uz 11930  df-fz 12580  df-hash 13370  df-mgm 17556  df-sgrp 17598
This theorem is referenced by:  sgrp2nmnd  17732  sgrpnmndex  17734
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