Theorem sgrp2nmndlem4 18868

Description: Lemma 4 for sgrp2nmnd 18870: 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	⊢ (+g‘𝑀) = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ 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.

Step	Hyp	Ref	Expression
1		mgm2nsgrp.s	. . . 4 ⊢ 𝑆 = {𝐴, 𝐵}
2	1	hashprdifel 14333	. . 3 ⊢ ((♯‘𝑆) = 2 → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵))
3		3simpa 1149	. . 3 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆))
4		mgm2nsgrp.b	. . . 4 ⊢ (Base‘𝑀) = 𝑆
5		sgrp2nmnd.o	. . . 4 ⊢ (+g‘𝑀) = (𝑥 ∈ 𝑆, 𝑦 ∈ 𝑆 ↦ if(𝑥 = 𝐴, 𝐴, 𝐵))
6	1, 4, 5	sgrp2nmndlem1 18863	. . 3 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → 𝑀 ∈ Mgm)
7	2, 3, 6	3syl 18	. 2 ⊢ ((♯‘𝑆) = 2 → 𝑀 ∈ Mgm)
8		eqid 2737	. . . . . . . . . . 11 ⊢ (+g‘𝑀) = (+g‘𝑀)
9	1, 4, 5, 8	sgrp2nmndlem2 18864	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → (𝐴(+g‘𝑀)𝐴) = 𝐴)
10	9	oveq1d 7383	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐴) = (𝐴(+g‘𝑀)𝐴))
11	9	oveq2d 7384	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐴)) = (𝐴(+g‘𝑀)𝐴))
12	10, 11	eqtr4d 2775	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐴) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐴)))
13	12	anidms 566	. . . . . . 7 ⊢ (𝐴 ∈ 𝑆 → ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐴) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐴)))
14	13	3ad2ant1 1134	. . . . . 6 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐴) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐴)))
15	9	anidms 566	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑆 → (𝐴(+g‘𝑀)𝐴) = 𝐴)
16	15	adantr 480	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴(+g‘𝑀)𝐴) = 𝐴)
17	16	oveq1d 7383	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐵) = (𝐴(+g‘𝑀)𝐵))
18	1, 4, 5, 8	sgrp2nmndlem2 18864	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴(+g‘𝑀)𝐵) = 𝐴)
19	18	oveq2d 7384	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐵)) = (𝐴(+g‘𝑀)𝐴))
20	16, 19, 18	3eqtr4rd 2783	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴(+g‘𝑀)𝐵) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐵)))
21	17, 20	eqtrd 2772	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐵) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐵)))
22	21	3adant3 1133	. . . . . 6 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐵) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐵)))
23	14, 22	jca 511	. . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → (((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐴) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐴)) ∧ ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐵) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐵))))
24	18	3adant3 1133	. . . . . 6 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → (𝐴(+g‘𝑀)𝐵) = 𝐴)
25	1, 4, 5, 8	sgrp2nmndlem3 18865	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → (𝐵(+g‘𝑀)𝐴) = 𝐵)
26	25	oveq2d 7384	. . . . . 6 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝐴)) = (𝐴(+g‘𝑀)𝐵))
27	24	oveq1d 7383	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝐴) = (𝐴(+g‘𝑀)𝐴))
28	15	3ad2ant1 1134	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → (𝐴(+g‘𝑀)𝐴) = 𝐴)
29	27, 28	eqtrd 2772	. . . . . 6 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝐴) = 𝐴)
30	24, 26, 29	3eqtr4rd 2783	. . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝐴) = (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝐴)))
31		simp2 1138	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → 𝐵 ∈ 𝑆)
32	1, 4, 5, 8	sgrp2nmndlem3 18865	. . . . . . . 8 ⊢ ((𝐵 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → (𝐵(+g‘𝑀)𝐵) = 𝐵)
33	31, 32	syld3an1 1413	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → (𝐵(+g‘𝑀)𝐵) = 𝐵)
34	33	oveq2d 7384	. . . . . 6 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝐵)) = (𝐴(+g‘𝑀)𝐵))
35	18	oveq1d 7383	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝐵) = (𝐴(+g‘𝑀)𝐵))
36	35, 18	eqtrd 2772	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝐵) = 𝐴)
37	36	3adant3 1133	. . . . . 6 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝐵) = 𝐴)
38	24, 34, 37	3eqtr4rd 2783	. . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝐵) = (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝐵)))
39	23, 30, 38	jca32 515	. . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ((((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐴) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐴)) ∧ ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐵) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐵))) ∧ (((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝐴) = (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝐴)) ∧ ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝐵) = (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝐵)))))
40	25	oveq1d 7383	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝐴) = (𝐵(+g‘𝑀)𝐴))
41	28	oveq2d 7384	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝐴)) = (𝐵(+g‘𝑀)𝐴))
42	40, 41	eqtr4d 2775	. . . . . 6 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝐴) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝐴)))
43	24	oveq2d 7384	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝐵)) = (𝐵(+g‘𝑀)𝐴))
44	25	oveq1d 7383	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝐵) = (𝐵(+g‘𝑀)𝐵))
45	44, 33	eqtrd 2772	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝐵) = 𝐵)
46	25, 43, 45	3eqtr4rd 2783	. . . . . 6 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝐵) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝐵)))
47	42, 46	jca 511	. . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → (((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝐴) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝐴)) ∧ ((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝐵) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝐵))))
48	25	oveq2d 7384	. . . . . 6 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝐴)) = (𝐵(+g‘𝑀)𝐵))
49	33	oveq1d 7383	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝐴) = (𝐵(+g‘𝑀)𝐴))
50	49, 25	eqtrd 2772	. . . . . 6 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝐴) = 𝐵)
51	33, 48, 50	3eqtr4rd 2783	. . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝐴) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝐴)))
52	32	oveq1d 7383	. . . . . . 7 ⊢ ((𝐵 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝐵) = (𝐵(+g‘𝑀)𝐵))
53	32	oveq2d 7384	. . . . . . 7 ⊢ ((𝐵 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝐵)) = (𝐵(+g‘𝑀)𝐵))
54	52, 53	eqtr4d 2775	. . . . . 6 ⊢ ((𝐵 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝐵) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝐵)))
55	31, 54	syld3an1 1413	. . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝐵) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝐵)))
56	47, 51, 55	jca32 515	. . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ((((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝐴) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝐴)) ∧ ((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝐵) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝐵))) ∧ (((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝐴) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝐴)) ∧ ((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝐵) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝐵)))))
57		oveq1 7375	. . . . . . . . . 10 ⊢ (𝑎 = 𝐴 → (𝑎(+g‘𝑀)𝑏) = (𝐴(+g‘𝑀)𝑏))
58	57	oveq1d 7383	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → ((𝑎(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = ((𝐴(+g‘𝑀)𝑏)(+g‘𝑀)𝑐))
59		oveq1 7375	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → (𝑎(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)) = (𝐴(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)))
60	58, 59	eqeq12d 2753	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → (((𝑎(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝑎(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)) ↔ ((𝐴(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)(𝑏(+g‘𝑀)𝑐))))
61	60	2ralbidv 3202	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (∀𝑏 ∈ {𝐴, 𝐵}∀𝑐 ∈ {𝐴, 𝐵} ((𝑎(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝑎(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)) ↔ ∀𝑏 ∈ {𝐴, 𝐵}∀𝑐 ∈ {𝐴, 𝐵} ((𝐴(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)(𝑏(+g‘𝑀)𝑐))))
62		oveq1 7375	. . . . . . . . . 10 ⊢ (𝑎 = 𝐵 → (𝑎(+g‘𝑀)𝑏) = (𝐵(+g‘𝑀)𝑏))
63	62	oveq1d 7383	. . . . . . . . 9 ⊢ (𝑎 = 𝐵 → ((𝑎(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = ((𝐵(+g‘𝑀)𝑏)(+g‘𝑀)𝑐))
64		oveq1 7375	. . . . . . . . 9 ⊢ (𝑎 = 𝐵 → (𝑎(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)) = (𝐵(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)))
65	63, 64	eqeq12d 2753	. . . . . . . 8 ⊢ (𝑎 = 𝐵 → (((𝑎(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝑎(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)) ↔ ((𝐵(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)(𝑏(+g‘𝑀)𝑐))))
66	65	2ralbidv 3202	. . . . . . 7 ⊢ (𝑎 = 𝐵 → (∀𝑏 ∈ {𝐴, 𝐵}∀𝑐 ∈ {𝐴, 𝐵} ((𝑎(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝑎(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)) ↔ ∀𝑏 ∈ {𝐴, 𝐵}∀𝑐 ∈ {𝐴, 𝐵} ((𝐵(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)(𝑏(+g‘𝑀)𝑐))))
67	61, 66	ralprg 4655	. . . . . 6 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (∀𝑎 ∈ {𝐴, 𝐵}∀𝑏 ∈ {𝐴, 𝐵}∀𝑐 ∈ {𝐴, 𝐵} ((𝑎(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝑎(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)) ↔ (∀𝑏 ∈ {𝐴, 𝐵}∀𝑐 ∈ {𝐴, 𝐵} ((𝐴(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)) ∧ ∀𝑏 ∈ {𝐴, 𝐵}∀𝑐 ∈ {𝐴, 𝐵} ((𝐵(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)))))
68		oveq2 7376	. . . . . . . . . . 11 ⊢ (𝑏 = 𝐴 → (𝐴(+g‘𝑀)𝑏) = (𝐴(+g‘𝑀)𝐴))
69	68	oveq1d 7383	. . . . . . . . . 10 ⊢ (𝑏 = 𝐴 → ((𝐴(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝑐))
70		oveq1 7375	. . . . . . . . . . 11 ⊢ (𝑏 = 𝐴 → (𝑏(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)𝑐))
71	70	oveq2d 7384	. . . . . . . . . 10 ⊢ (𝑏 = 𝐴 → (𝐴(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝑐)))
72	69, 71	eqeq12d 2753	. . . . . . . . 9 ⊢ (𝑏 = 𝐴 → (((𝐴(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)) ↔ ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝑐))))
73	72	ralbidv 3161	. . . . . . . 8 ⊢ (𝑏 = 𝐴 → (∀𝑐 ∈ {𝐴, 𝐵} ((𝐴(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)) ↔ ∀𝑐 ∈ {𝐴, 𝐵} ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝑐))))
74		oveq2 7376	. . . . . . . . . . 11 ⊢ (𝑏 = 𝐵 → (𝐴(+g‘𝑀)𝑏) = (𝐴(+g‘𝑀)𝐵))
75	74	oveq1d 7383	. . . . . . . . . 10 ⊢ (𝑏 = 𝐵 → ((𝐴(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝑐))
76		oveq1 7375	. . . . . . . . . . 11 ⊢ (𝑏 = 𝐵 → (𝑏(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)𝑐))
77	76	oveq2d 7384	. . . . . . . . . 10 ⊢ (𝑏 = 𝐵 → (𝐴(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)) = (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝑐)))
78	75, 77	eqeq12d 2753	. . . . . . . . 9 ⊢ (𝑏 = 𝐵 → (((𝐴(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)) ↔ ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝑐))))
79	78	ralbidv 3161	. . . . . . . 8 ⊢ (𝑏 = 𝐵 → (∀𝑐 ∈ {𝐴, 𝐵} ((𝐴(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)) ↔ ∀𝑐 ∈ {𝐴, 𝐵} ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝑐))))
80	73, 79	ralprg 4655	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (∀𝑏 ∈ {𝐴, 𝐵}∀𝑐 ∈ {𝐴, 𝐵} ((𝐴(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)) ↔ (∀𝑐 ∈ {𝐴, 𝐵} ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝑐)) ∧ ∀𝑐 ∈ {𝐴, 𝐵} ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝑐)))))
81		oveq2 7376	. . . . . . . . . . 11 ⊢ (𝑏 = 𝐴 → (𝐵(+g‘𝑀)𝑏) = (𝐵(+g‘𝑀)𝐴))
82	81	oveq1d 7383	. . . . . . . . . 10 ⊢ (𝑏 = 𝐴 → ((𝐵(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = ((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝑐))
83	70	oveq2d 7384	. . . . . . . . . 10 ⊢ (𝑏 = 𝐴 → (𝐵(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝑐)))
84	82, 83	eqeq12d 2753	. . . . . . . . 9 ⊢ (𝑏 = 𝐴 → (((𝐵(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)) ↔ ((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝑐))))
85	84	ralbidv 3161	. . . . . . . 8 ⊢ (𝑏 = 𝐴 → (∀𝑐 ∈ {𝐴, 𝐵} ((𝐵(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)) ↔ ∀𝑐 ∈ {𝐴, 𝐵} ((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝑐))))
86		oveq2 7376	. . . . . . . . . . 11 ⊢ (𝑏 = 𝐵 → (𝐵(+g‘𝑀)𝑏) = (𝐵(+g‘𝑀)𝐵))
87	86	oveq1d 7383	. . . . . . . . . 10 ⊢ (𝑏 = 𝐵 → ((𝐵(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = ((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝑐))
88	76	oveq2d 7384	. . . . . . . . . 10 ⊢ (𝑏 = 𝐵 → (𝐵(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝑐)))
89	87, 88	eqeq12d 2753	. . . . . . . . 9 ⊢ (𝑏 = 𝐵 → (((𝐵(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)) ↔ ((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝑐))))
90	89	ralbidv 3161	. . . . . . . 8 ⊢ (𝑏 = 𝐵 → (∀𝑐 ∈ {𝐴, 𝐵} ((𝐵(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)) ↔ ∀𝑐 ∈ {𝐴, 𝐵} ((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝑐))))
91	85, 90	ralprg 4655	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (∀𝑏 ∈ {𝐴, 𝐵}∀𝑐 ∈ {𝐴, 𝐵} ((𝐵(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)) ↔ (∀𝑐 ∈ {𝐴, 𝐵} ((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝑐)) ∧ ∀𝑐 ∈ {𝐴, 𝐵} ((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝑐)))))
92	80, 91	anbi12d 633	. . . . . 6 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((∀𝑏 ∈ {𝐴, 𝐵}∀𝑐 ∈ {𝐴, 𝐵} ((𝐴(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)) ∧ ∀𝑏 ∈ {𝐴, 𝐵}∀𝑐 ∈ {𝐴, 𝐵} ((𝐵(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)(𝑏(+g‘𝑀)𝑐))) ↔ ((∀𝑐 ∈ {𝐴, 𝐵} ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝑐)) ∧ ∀𝑐 ∈ {𝐴, 𝐵} ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝑐))) ∧ (∀𝑐 ∈ {𝐴, 𝐵} ((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝑐)) ∧ ∀𝑐 ∈ {𝐴, 𝐵} ((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝑐))))))
93		oveq2 7376	. . . . . . . . . 10 ⊢ (𝑐 = 𝐴 → ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝑐) = ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐴))
94		oveq2 7376	. . . . . . . . . . 11 ⊢ (𝑐 = 𝐴 → (𝐴(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)𝐴))
95	94	oveq2d 7384	. . . . . . . . . 10 ⊢ (𝑐 = 𝐴 → (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝑐)) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐴)))
96	93, 95	eqeq12d 2753	. . . . . . . . 9 ⊢ (𝑐 = 𝐴 → (((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝑐)) ↔ ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐴) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐴))))
97		oveq2 7376	. . . . . . . . . 10 ⊢ (𝑐 = 𝐵 → ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝑐) = ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐵))
98		oveq2 7376	. . . . . . . . . . 11 ⊢ (𝑐 = 𝐵 → (𝐴(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)𝐵))
99	98	oveq2d 7384	. . . . . . . . . 10 ⊢ (𝑐 = 𝐵 → (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝑐)) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐵)))
100	97, 99	eqeq12d 2753	. . . . . . . . 9 ⊢ (𝑐 = 𝐵 → (((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝑐)) ↔ ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐵) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐵))))
101	96, 100	ralprg 4655	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (∀𝑐 ∈ {𝐴, 𝐵} ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝑐)) ↔ (((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐴) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐴)) ∧ ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐵) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐵)))))
102		oveq2 7376	. . . . . . . . . 10 ⊢ (𝑐 = 𝐴 → ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝑐) = ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝐴))
103		oveq2 7376	. . . . . . . . . . 11 ⊢ (𝑐 = 𝐴 → (𝐵(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)𝐴))
104	103	oveq2d 7384	. . . . . . . . . 10 ⊢ (𝑐 = 𝐴 → (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝑐)) = (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝐴)))
105	102, 104	eqeq12d 2753	. . . . . . . . 9 ⊢ (𝑐 = 𝐴 → (((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝑐)) ↔ ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝐴) = (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝐴))))
106		oveq2 7376	. . . . . . . . . 10 ⊢ (𝑐 = 𝐵 → ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝑐) = ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝐵))
107		oveq2 7376	. . . . . . . . . . 11 ⊢ (𝑐 = 𝐵 → (𝐵(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)𝐵))
108	107	oveq2d 7384	. . . . . . . . . 10 ⊢ (𝑐 = 𝐵 → (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝑐)) = (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝐵)))
109	106, 108	eqeq12d 2753	. . . . . . . . 9 ⊢ (𝑐 = 𝐵 → (((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝑐)) ↔ ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝐵) = (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝐵))))
110	105, 109	ralprg 4655	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (∀𝑐 ∈ {𝐴, 𝐵} ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝑐)) ↔ (((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝐴) = (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝐴)) ∧ ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝐵) = (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝐵)))))
111	101, 110	anbi12d 633	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((∀𝑐 ∈ {𝐴, 𝐵} ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝑐)) ∧ ∀𝑐 ∈ {𝐴, 𝐵} ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝑐))) ↔ ((((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐴) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐴)) ∧ ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐵) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐵))) ∧ (((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝐴) = (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝐴)) ∧ ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝐵) = (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝐵))))))
112		oveq2 7376	. . . . . . . . . 10 ⊢ (𝑐 = 𝐴 → ((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝑐) = ((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝐴))
113	94	oveq2d 7384	. . . . . . . . . 10 ⊢ (𝑐 = 𝐴 → (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝑐)) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝐴)))
114	112, 113	eqeq12d 2753	. . . . . . . . 9 ⊢ (𝑐 = 𝐴 → (((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝑐)) ↔ ((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝐴) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝐴))))
115		oveq2 7376	. . . . . . . . . 10 ⊢ (𝑐 = 𝐵 → ((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝑐) = ((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝐵))
116	98	oveq2d 7384	. . . . . . . . . 10 ⊢ (𝑐 = 𝐵 → (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝑐)) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝐵)))
117	115, 116	eqeq12d 2753	. . . . . . . . 9 ⊢ (𝑐 = 𝐵 → (((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝑐)) ↔ ((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝐵) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝐵))))
118	114, 117	ralprg 4655	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (∀𝑐 ∈ {𝐴, 𝐵} ((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝑐)) ↔ (((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝐴) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝐴)) ∧ ((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝐵) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝐵)))))
119		oveq2 7376	. . . . . . . . . 10 ⊢ (𝑐 = 𝐴 → ((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝑐) = ((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝐴))
120	103	oveq2d 7384	. . . . . . . . . 10 ⊢ (𝑐 = 𝐴 → (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝑐)) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝐴)))
121	119, 120	eqeq12d 2753	. . . . . . . . 9 ⊢ (𝑐 = 𝐴 → (((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝑐)) ↔ ((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝐴) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝐴))))
122		oveq2 7376	. . . . . . . . . 10 ⊢ (𝑐 = 𝐵 → ((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝑐) = ((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝐵))
123	107	oveq2d 7384	. . . . . . . . . 10 ⊢ (𝑐 = 𝐵 → (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝑐)) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝐵)))
124	122, 123	eqeq12d 2753	. . . . . . . . 9 ⊢ (𝑐 = 𝐵 → (((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝑐)) ↔ ((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝐵) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝐵))))
125	121, 124	ralprg 4655	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (∀𝑐 ∈ {𝐴, 𝐵} ((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝑐)) ↔ (((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝐴) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝐴)) ∧ ((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝐵) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝐵)))))
126	118, 125	anbi12d 633	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((∀𝑐 ∈ {𝐴, 𝐵} ((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝑐)) ∧ ∀𝑐 ∈ {𝐴, 𝐵} ((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝑐))) ↔ ((((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝐴) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝐴)) ∧ ((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝐵) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝐵))) ∧ (((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝐴) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝐴)) ∧ ((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝐵) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝐵))))))
127	111, 126	anbi12d 633	. . . . . 6 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (((∀𝑐 ∈ {𝐴, 𝐵} ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝑐)) ∧ ∀𝑐 ∈ {𝐴, 𝐵} ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝑐) = (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝑐))) ∧ (∀𝑐 ∈ {𝐴, 𝐵} ((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝑐)) ∧ ∀𝑐 ∈ {𝐴, 𝐵} ((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝑐) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝑐)))) ↔ (((((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐴) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐴)) ∧ ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐵) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐵))) ∧ (((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝐴) = (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝐴)) ∧ ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝐵) = (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝐵)))) ∧ ((((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝐴) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝐴)) ∧ ((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝐵) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝐵))) ∧ (((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝐴) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝐴)) ∧ ((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝐵) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝐵)))))))
128	67, 92, 127	3bitrd 305	. . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (∀𝑎 ∈ {𝐴, 𝐵}∀𝑏 ∈ {𝐴, 𝐵}∀𝑐 ∈ {𝐴, 𝐵} ((𝑎(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝑎(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)) ↔ (((((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐴) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐴)) ∧ ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐵) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐵))) ∧ (((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝐴) = (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝐴)) ∧ ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝐵) = (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝐵)))) ∧ ((((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝐴) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝐴)) ∧ ((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝐵) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝐵))) ∧ (((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝐴) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝐴)) ∧ ((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝐵) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝐵)))))))
129	128	3adant3 1133	. . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → (∀𝑎 ∈ {𝐴, 𝐵}∀𝑏 ∈ {𝐴, 𝐵}∀𝑐 ∈ {𝐴, 𝐵} ((𝑎(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝑎(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)) ↔ (((((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐴) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐴)) ∧ ((𝐴(+g‘𝑀)𝐴)(+g‘𝑀)𝐵) = (𝐴(+g‘𝑀)(𝐴(+g‘𝑀)𝐵))) ∧ (((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝐴) = (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝐴)) ∧ ((𝐴(+g‘𝑀)𝐵)(+g‘𝑀)𝐵) = (𝐴(+g‘𝑀)(𝐵(+g‘𝑀)𝐵)))) ∧ ((((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝐴) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝐴)) ∧ ((𝐵(+g‘𝑀)𝐴)(+g‘𝑀)𝐵) = (𝐵(+g‘𝑀)(𝐴(+g‘𝑀)𝐵))) ∧ (((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝐴) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝐴)) ∧ ((𝐵(+g‘𝑀)𝐵)(+g‘𝑀)𝐵) = (𝐵(+g‘𝑀)(𝐵(+g‘𝑀)𝐵)))))))
130	39, 56, 129	mpbir2and 714	. . 3 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐴 ≠ 𝐵) → ∀𝑎 ∈ {𝐴, 𝐵}∀𝑏 ∈ {𝐴, 𝐵}∀𝑐 ∈ {𝐴, 𝐵} ((𝑎(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝑎(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)))
131	2, 130	syl 17	. 2 ⊢ ((♯‘𝑆) = 2 → ∀𝑎 ∈ {𝐴, 𝐵}∀𝑏 ∈ {𝐴, 𝐵}∀𝑐 ∈ {𝐴, 𝐵} ((𝑎(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝑎(+g‘𝑀)(𝑏(+g‘𝑀)𝑐)))
132	4, 1	eqtr2i 2761	. . 3 ⊢ {𝐴, 𝐵} = (Base‘𝑀)
133	132, 8	issgrp 18657	. 2 ⊢ (𝑀 ∈ Smgrp ↔ (𝑀 ∈ Mgm ∧ ∀𝑎 ∈ {𝐴, 𝐵}∀𝑏 ∈ {𝐴, 𝐵}∀𝑐 ∈ {𝐴, 𝐵} ((𝑎(+g‘𝑀)𝑏)(+g‘𝑀)𝑐) = (𝑎(+g‘𝑀)(𝑏(+g‘𝑀)𝑐))))
134	7, 131, 133	sylanbrc 584	1 ⊢ ((♯‘𝑆) = 2 → 𝑀 ∈ Smgrp)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  ifcif 4481  {cpr 4584  ‘cfv 6500  (class class class)co 7368   ∈ cmpo 7370  2c2 12212  ♯chash 14265  Basecbs 17148  +_gcplusg 17189  Mgmcmgm 18575  Smgrpcsgrp 18655

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-oadd 8411  df-er 8645  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-dju 9825  df-card 9863  df-pnf 11180  df-mnf 11181  df-xr 11182  df-ltxr 11183  df-le 11184  df-sub 11378  df-neg 11379  df-nn 12158  df-2 12220  df-n0 12414  df-z 12501  df-uz 12764  df-fz 13436  df-hash 14266  df-mgm 18577  df-sgrp 18656

This theorem is referenced by:  sgrp2nmnd  18870  sgrpnmndex  18872