Theorem mndpropd 18852

Description: If two structures have the same base set, and the values of their group (addition) operations are equal for all pairs of elements of the base set, one is a monoid iff the other one is. (Contributed by Mario Carneiro, 6-Jan-2015.)

Hypotheses

Ref	Expression
mndpropd.1	⊢ (𝜑 → 𝐵 = (Base‘𝐾))
mndpropd.2	⊢ (𝜑 → 𝐵 = (Base‘𝐿))
mndpropd.3	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))

Assertion

Ref	Expression
mndpropd	⊢ (𝜑 → (𝐾 ∈ Mnd ↔ 𝐿 ∈ Mnd))

Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐾,𝑦   𝜑,𝑥,𝑦   𝑥,𝐿,𝑦

Proof of Theorem mndpropd

Dummy variables 𝑢 𝑠 𝑣 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simplr 781	. . . . . 6 ⊢ (((𝜑 ∧ 𝐾 ∈ Mnd) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐾 ∈ Mnd)
2		simprl 783	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐾 ∈ Mnd) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ 𝐵)
3		mndpropd.1	. . . . . . . 8 ⊢ (𝜑 → 𝐵 = (Base‘𝐾))
4	3	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐾 ∈ Mnd) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐵 = (Base‘𝐾))
5	2, 4	eleqtrd 2867	. . . . . 6 ⊢ (((𝜑 ∧ 𝐾 ∈ Mnd) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ (Base‘𝐾))
6		simprr 785	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐾 ∈ Mnd) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ 𝐵)
7	6, 4	eleqtrd 2867	. . . . . 6 ⊢ (((𝜑 ∧ 𝐾 ∈ Mnd) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ (Base‘𝐾))
8		eqid 2765	. . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾)
9		eqid 2765	. . . . . . 7 ⊢ (+g‘𝐾) = (+g‘𝐾)
10	8, 9	mndcl 18832	. . . . . 6 ⊢ ((𝐾 ∈ Mnd ∧ 𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) → (𝑥(+g‘𝐾)𝑦) ∈ (Base‘𝐾))
11	1, 5, 7, 10	syl3anc 1398	. . . . 5 ⊢ (((𝜑 ∧ 𝐾 ∈ Mnd) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) ∈ (Base‘𝐾))
12	11, 4	eleqtrrd 2868	. . . 4 ⊢ (((𝜑 ∧ 𝐾 ∈ Mnd) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) ∈ 𝐵)
13	12	ralrimivva 3210	. . 3 ⊢ ((𝜑 ∧ 𝐾 ∈ Mnd) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵)
14	13	ex 418	. 2 ⊢ (𝜑 → (𝐾 ∈ Mnd → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵))
15		simplr 781	. . . . . 6 ⊢ (((𝜑 ∧ 𝐿 ∈ Mnd) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐿 ∈ Mnd)
16		simprl 783	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐿 ∈ Mnd) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ 𝐵)
17		mndpropd.2	. . . . . . . 8 ⊢ (𝜑 → 𝐵 = (Base‘𝐿))
18	17	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐿 ∈ Mnd) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐵 = (Base‘𝐿))
19	16, 18	eleqtrd 2867	. . . . . 6 ⊢ (((𝜑 ∧ 𝐿 ∈ Mnd) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ (Base‘𝐿))
20		simprr 785	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐿 ∈ Mnd) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ 𝐵)
21	20, 18	eleqtrd 2867	. . . . . 6 ⊢ (((𝜑 ∧ 𝐿 ∈ Mnd) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ (Base‘𝐿))
22		eqid 2765	. . . . . . 7 ⊢ (Base‘𝐿) = (Base‘𝐿)
23		eqid 2765	. . . . . . 7 ⊢ (+g‘𝐿) = (+g‘𝐿)
24	22, 23	mndcl 18832	. . . . . 6 ⊢ ((𝐿 ∈ Mnd ∧ 𝑥 ∈ (Base‘𝐿) ∧ 𝑦 ∈ (Base‘𝐿)) → (𝑥(+g‘𝐿)𝑦) ∈ (Base‘𝐿))
25	15, 19, 21, 24	syl3anc 1398	. . . . 5 ⊢ (((𝜑 ∧ 𝐿 ∈ Mnd) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐿)𝑦) ∈ (Base‘𝐿))
26		mndpropd.3	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))
27	26	adantlr 728	. . . . 5 ⊢ (((𝜑 ∧ 𝐿 ∈ Mnd) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))
28	25, 27, 18	3eltr4d 2880	. . . 4 ⊢ (((𝜑 ∧ 𝐿 ∈ Mnd) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) ∈ 𝐵)
29	28	ralrimivva 3210	. . 3 ⊢ ((𝜑 ∧ 𝐿 ∈ Mnd) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵)
30	29	ex 418	. 2 ⊢ (𝜑 → (𝐿 ∈ Mnd → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵))
31	26	oveqrspc2v 7446	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → (𝑢(+g‘𝐾)𝑣) = (𝑢(+g‘𝐿)𝑣))
32	31	adantlr 728	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → (𝑢(+g‘𝐾)𝑣) = (𝑢(+g‘𝐿)𝑣))
33	32	eleq1d 2850	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → ((𝑢(+g‘𝐾)𝑣) ∈ 𝐵 ↔ (𝑢(+g‘𝐿)𝑣) ∈ 𝐵))
34		simplll 787	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑤 ∈ 𝐵) → 𝜑)
35		simplrl 789	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑤 ∈ 𝐵) → 𝑢 ∈ 𝐵)
36		simplrr 790	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑤 ∈ 𝐵) → 𝑣 ∈ 𝐵)
37		simpllr 788	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑤 ∈ 𝐵) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵)
38		ovrspc2v 7445	. . . . . . . . . . . . 13 ⊢ (((𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) → (𝑢(+g‘𝐾)𝑣) ∈ 𝐵)
39	35, 36, 37, 38	syl21anc 851	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑤 ∈ 𝐵) → (𝑢(+g‘𝐾)𝑣) ∈ 𝐵)
40		simpr 490	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑤 ∈ 𝐵) → 𝑤 ∈ 𝐵)
41	26	oveqrspc2v 7446	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑢(+g‘𝐾)𝑣) ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((𝑢(+g‘𝐾)𝑣)(+g‘𝐾)𝑤) = ((𝑢(+g‘𝐾)𝑣)(+g‘𝐿)𝑤))
42	34, 39, 40, 41	syl12anc 850	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑤 ∈ 𝐵) → ((𝑢(+g‘𝐾)𝑣)(+g‘𝐾)𝑤) = ((𝑢(+g‘𝐾)𝑣)(+g‘𝐿)𝑤))
43	34, 35, 36, 31	syl12anc 850	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑤 ∈ 𝐵) → (𝑢(+g‘𝐾)𝑣) = (𝑢(+g‘𝐿)𝑣))
44	43	oveq1d 7434	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑤 ∈ 𝐵) → ((𝑢(+g‘𝐾)𝑣)(+g‘𝐿)𝑤) = ((𝑢(+g‘𝐿)𝑣)(+g‘𝐿)𝑤))
45	42, 44	eqtrd 2800	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑤 ∈ 𝐵) → ((𝑢(+g‘𝐾)𝑣)(+g‘𝐾)𝑤) = ((𝑢(+g‘𝐿)𝑣)(+g‘𝐿)𝑤))
46		ovrspc2v 7445	. . . . . . . . . . . . 13 ⊢ (((𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) → (𝑣(+g‘𝐾)𝑤) ∈ 𝐵)
47	36, 40, 37, 46	syl21anc 851	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑤 ∈ 𝐵) → (𝑣(+g‘𝐾)𝑤) ∈ 𝐵)
48	26	oveqrspc2v 7446	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐵 ∧ (𝑣(+g‘𝐾)𝑤) ∈ 𝐵)) → (𝑢(+g‘𝐾)(𝑣(+g‘𝐾)𝑤)) = (𝑢(+g‘𝐿)(𝑣(+g‘𝐾)𝑤)))
49	34, 35, 47, 48	syl12anc 850	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑤 ∈ 𝐵) → (𝑢(+g‘𝐾)(𝑣(+g‘𝐾)𝑤)) = (𝑢(+g‘𝐿)(𝑣(+g‘𝐾)𝑤)))
50	26	oveqrspc2v 7446	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑣(+g‘𝐾)𝑤) = (𝑣(+g‘𝐿)𝑤))
51	34, 36, 40, 50	syl12anc 850	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑤 ∈ 𝐵) → (𝑣(+g‘𝐾)𝑤) = (𝑣(+g‘𝐿)𝑤))
52	51	oveq2d 7435	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑤 ∈ 𝐵) → (𝑢(+g‘𝐿)(𝑣(+g‘𝐾)𝑤)) = (𝑢(+g‘𝐿)(𝑣(+g‘𝐿)𝑤)))
53	49, 52	eqtrd 2800	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑤 ∈ 𝐵) → (𝑢(+g‘𝐾)(𝑣(+g‘𝐾)𝑤)) = (𝑢(+g‘𝐿)(𝑣(+g‘𝐿)𝑤)))
54	45, 53	eqeq12d 2781	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑤 ∈ 𝐵) → (((𝑢(+g‘𝐾)𝑣)(+g‘𝐾)𝑤) = (𝑢(+g‘𝐾)(𝑣(+g‘𝐾)𝑤)) ↔ ((𝑢(+g‘𝐿)𝑣)(+g‘𝐿)𝑤) = (𝑢(+g‘𝐿)(𝑣(+g‘𝐿)𝑤))))
55	54	ralbidva 3188	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → (∀𝑤 ∈ 𝐵 ((𝑢(+g‘𝐾)𝑣)(+g‘𝐾)𝑤) = (𝑢(+g‘𝐾)(𝑣(+g‘𝐾)𝑤)) ↔ ∀𝑤 ∈ 𝐵 ((𝑢(+g‘𝐿)𝑣)(+g‘𝐿)𝑤) = (𝑢(+g‘𝐿)(𝑣(+g‘𝐿)𝑤))))
56	33, 55	anbi12d 644	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → (((𝑢(+g‘𝐾)𝑣) ∈ 𝐵 ∧ ∀𝑤 ∈ 𝐵 ((𝑢(+g‘𝐾)𝑣)(+g‘𝐾)𝑤) = (𝑢(+g‘𝐾)(𝑣(+g‘𝐾)𝑤))) ↔ ((𝑢(+g‘𝐿)𝑣) ∈ 𝐵 ∧ ∀𝑤 ∈ 𝐵 ((𝑢(+g‘𝐿)𝑣)(+g‘𝐿)𝑤) = (𝑢(+g‘𝐿)(𝑣(+g‘𝐿)𝑤)))))
57	56	2ralbidva 3229	. . . . . 6 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) → (∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 ((𝑢(+g‘𝐾)𝑣) ∈ 𝐵 ∧ ∀𝑤 ∈ 𝐵 ((𝑢(+g‘𝐾)𝑣)(+g‘𝐾)𝑤) = (𝑢(+g‘𝐾)(𝑣(+g‘𝐾)𝑤))) ↔ ∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 ((𝑢(+g‘𝐿)𝑣) ∈ 𝐵 ∧ ∀𝑤 ∈ 𝐵 ((𝑢(+g‘𝐿)𝑣)(+g‘𝐿)𝑤) = (𝑢(+g‘𝐿)(𝑣(+g‘𝐿)𝑤)))))
58	3	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) → 𝐵 = (Base‘𝐾))
59	58	eleq2d 2851	. . . . . . . . 9 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) → ((𝑢(+g‘𝐾)𝑣) ∈ 𝐵 ↔ (𝑢(+g‘𝐾)𝑣) ∈ (Base‘𝐾)))
60	58	raleqdv 3325	. . . . . . . . 9 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) → (∀𝑤 ∈ 𝐵 ((𝑢(+g‘𝐾)𝑣)(+g‘𝐾)𝑤) = (𝑢(+g‘𝐾)(𝑣(+g‘𝐾)𝑤)) ↔ ∀𝑤 ∈ (Base‘𝐾)((𝑢(+g‘𝐾)𝑣)(+g‘𝐾)𝑤) = (𝑢(+g‘𝐾)(𝑣(+g‘𝐾)𝑤))))
61	59, 60	anbi12d 644	. . . . . . . 8 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) → (((𝑢(+g‘𝐾)𝑣) ∈ 𝐵 ∧ ∀𝑤 ∈ 𝐵 ((𝑢(+g‘𝐾)𝑣)(+g‘𝐾)𝑤) = (𝑢(+g‘𝐾)(𝑣(+g‘𝐾)𝑤))) ↔ ((𝑢(+g‘𝐾)𝑣) ∈ (Base‘𝐾) ∧ ∀𝑤 ∈ (Base‘𝐾)((𝑢(+g‘𝐾)𝑣)(+g‘𝐾)𝑤) = (𝑢(+g‘𝐾)(𝑣(+g‘𝐾)𝑤)))))
62	58, 61	raleqbidv 3340	. . . . . . 7 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) → (∀𝑣 ∈ 𝐵 ((𝑢(+g‘𝐾)𝑣) ∈ 𝐵 ∧ ∀𝑤 ∈ 𝐵 ((𝑢(+g‘𝐾)𝑣)(+g‘𝐾)𝑤) = (𝑢(+g‘𝐾)(𝑣(+g‘𝐾)𝑤))) ↔ ∀𝑣 ∈ (Base‘𝐾)((𝑢(+g‘𝐾)𝑣) ∈ (Base‘𝐾) ∧ ∀𝑤 ∈ (Base‘𝐾)((𝑢(+g‘𝐾)𝑣)(+g‘𝐾)𝑤) = (𝑢(+g‘𝐾)(𝑣(+g‘𝐾)𝑤)))))
63	58, 62	raleqbidv 3340	. . . . . 6 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) → (∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 ((𝑢(+g‘𝐾)𝑣) ∈ 𝐵 ∧ ∀𝑤 ∈ 𝐵 ((𝑢(+g‘𝐾)𝑣)(+g‘𝐾)𝑤) = (𝑢(+g‘𝐾)(𝑣(+g‘𝐾)𝑤))) ↔ ∀𝑢 ∈ (Base‘𝐾)∀𝑣 ∈ (Base‘𝐾)((𝑢(+g‘𝐾)𝑣) ∈ (Base‘𝐾) ∧ ∀𝑤 ∈ (Base‘𝐾)((𝑢(+g‘𝐾)𝑣)(+g‘𝐾)𝑤) = (𝑢(+g‘𝐾)(𝑣(+g‘𝐾)𝑤)))))
64	17	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) → 𝐵 = (Base‘𝐿))
65	64	eleq2d 2851	. . . . . . . . 9 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) → ((𝑢(+g‘𝐿)𝑣) ∈ 𝐵 ↔ (𝑢(+g‘𝐿)𝑣) ∈ (Base‘𝐿)))
66	64	raleqdv 3325	. . . . . . . . 9 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) → (∀𝑤 ∈ 𝐵 ((𝑢(+g‘𝐿)𝑣)(+g‘𝐿)𝑤) = (𝑢(+g‘𝐿)(𝑣(+g‘𝐿)𝑤)) ↔ ∀𝑤 ∈ (Base‘𝐿)((𝑢(+g‘𝐿)𝑣)(+g‘𝐿)𝑤) = (𝑢(+g‘𝐿)(𝑣(+g‘𝐿)𝑤))))
67	65, 66	anbi12d 644	. . . . . . . 8 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) → (((𝑢(+g‘𝐿)𝑣) ∈ 𝐵 ∧ ∀𝑤 ∈ 𝐵 ((𝑢(+g‘𝐿)𝑣)(+g‘𝐿)𝑤) = (𝑢(+g‘𝐿)(𝑣(+g‘𝐿)𝑤))) ↔ ((𝑢(+g‘𝐿)𝑣) ∈ (Base‘𝐿) ∧ ∀𝑤 ∈ (Base‘𝐿)((𝑢(+g‘𝐿)𝑣)(+g‘𝐿)𝑤) = (𝑢(+g‘𝐿)(𝑣(+g‘𝐿)𝑤)))))
68	64, 67	raleqbidv 3340	. . . . . . 7 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) → (∀𝑣 ∈ 𝐵 ((𝑢(+g‘𝐿)𝑣) ∈ 𝐵 ∧ ∀𝑤 ∈ 𝐵 ((𝑢(+g‘𝐿)𝑣)(+g‘𝐿)𝑤) = (𝑢(+g‘𝐿)(𝑣(+g‘𝐿)𝑤))) ↔ ∀𝑣 ∈ (Base‘𝐿)((𝑢(+g‘𝐿)𝑣) ∈ (Base‘𝐿) ∧ ∀𝑤 ∈ (Base‘𝐿)((𝑢(+g‘𝐿)𝑣)(+g‘𝐿)𝑤) = (𝑢(+g‘𝐿)(𝑣(+g‘𝐿)𝑤)))))
69	64, 68	raleqbidv 3340	. . . . . 6 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) → (∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 ((𝑢(+g‘𝐿)𝑣) ∈ 𝐵 ∧ ∀𝑤 ∈ 𝐵 ((𝑢(+g‘𝐿)𝑣)(+g‘𝐿)𝑤) = (𝑢(+g‘𝐿)(𝑣(+g‘𝐿)𝑤))) ↔ ∀𝑢 ∈ (Base‘𝐿)∀𝑣 ∈ (Base‘𝐿)((𝑢(+g‘𝐿)𝑣) ∈ (Base‘𝐿) ∧ ∀𝑤 ∈ (Base‘𝐿)((𝑢(+g‘𝐿)𝑣)(+g‘𝐿)𝑤) = (𝑢(+g‘𝐿)(𝑣(+g‘𝐿)𝑤)))))
70	57, 63, 69	3bitr3d 312	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) → (∀𝑢 ∈ (Base‘𝐾)∀𝑣 ∈ (Base‘𝐾)((𝑢(+g‘𝐾)𝑣) ∈ (Base‘𝐾) ∧ ∀𝑤 ∈ (Base‘𝐾)((𝑢(+g‘𝐾)𝑣)(+g‘𝐾)𝑤) = (𝑢(+g‘𝐾)(𝑣(+g‘𝐾)𝑤))) ↔ ∀𝑢 ∈ (Base‘𝐿)∀𝑣 ∈ (Base‘𝐿)((𝑢(+g‘𝐿)𝑣) ∈ (Base‘𝐿) ∧ ∀𝑤 ∈ (Base‘𝐿)((𝑢(+g‘𝐿)𝑣)(+g‘𝐿)𝑤) = (𝑢(+g‘𝐿)(𝑣(+g‘𝐿)𝑤)))))
71		simplll 787	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) ∧ 𝑠 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) → 𝜑)
72		simplr 781	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) ∧ 𝑠 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) → 𝑠 ∈ 𝐵)
73		simpr 490	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) ∧ 𝑠 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) → 𝑢 ∈ 𝐵)
74	26	oveqrspc2v 7446	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑢 ∈ 𝐵)) → (𝑠(+g‘𝐾)𝑢) = (𝑠(+g‘𝐿)𝑢))
75	71, 72, 73, 74	syl12anc 850	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) ∧ 𝑠 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) → (𝑠(+g‘𝐾)𝑢) = (𝑠(+g‘𝐿)𝑢))
76	75	eqeq1d 2767	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) ∧ 𝑠 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) → ((𝑠(+g‘𝐾)𝑢) = 𝑢 ↔ (𝑠(+g‘𝐿)𝑢) = 𝑢))
77	26	oveqrspc2v 7446	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐵 ∧ 𝑠 ∈ 𝐵)) → (𝑢(+g‘𝐾)𝑠) = (𝑢(+g‘𝐿)𝑠))
78	71, 73, 72, 77	syl12anc 850	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) ∧ 𝑠 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) → (𝑢(+g‘𝐾)𝑠) = (𝑢(+g‘𝐿)𝑠))
79	78	eqeq1d 2767	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) ∧ 𝑠 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) → ((𝑢(+g‘𝐾)𝑠) = 𝑢 ↔ (𝑢(+g‘𝐿)𝑠) = 𝑢))
80	76, 79	anbi12d 644	. . . . . . . 8 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) ∧ 𝑠 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) → (((𝑠(+g‘𝐾)𝑢) = 𝑢 ∧ (𝑢(+g‘𝐾)𝑠) = 𝑢) ↔ ((𝑠(+g‘𝐿)𝑢) = 𝑢 ∧ (𝑢(+g‘𝐿)𝑠) = 𝑢)))
81	80	ralbidva 3188	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) ∧ 𝑠 ∈ 𝐵) → (∀𝑢 ∈ 𝐵 ((𝑠(+g‘𝐾)𝑢) = 𝑢 ∧ (𝑢(+g‘𝐾)𝑠) = 𝑢) ↔ ∀𝑢 ∈ 𝐵 ((𝑠(+g‘𝐿)𝑢) = 𝑢 ∧ (𝑢(+g‘𝐿)𝑠) = 𝑢)))
82	81	rexbidva 3189	. . . . . 6 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) → (∃𝑠 ∈ 𝐵 ∀𝑢 ∈ 𝐵 ((𝑠(+g‘𝐾)𝑢) = 𝑢 ∧ (𝑢(+g‘𝐾)𝑠) = 𝑢) ↔ ∃𝑠 ∈ 𝐵 ∀𝑢 ∈ 𝐵 ((𝑠(+g‘𝐿)𝑢) = 𝑢 ∧ (𝑢(+g‘𝐿)𝑠) = 𝑢)))
83	58	raleqdv 3325	. . . . . . 7 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) → (∀𝑢 ∈ 𝐵 ((𝑠(+g‘𝐾)𝑢) = 𝑢 ∧ (𝑢(+g‘𝐾)𝑠) = 𝑢) ↔ ∀𝑢 ∈ (Base‘𝐾)((𝑠(+g‘𝐾)𝑢) = 𝑢 ∧ (𝑢(+g‘𝐾)𝑠) = 𝑢)))
84	58, 83	rexeqbidv 3341	. . . . . 6 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) → (∃𝑠 ∈ 𝐵 ∀𝑢 ∈ 𝐵 ((𝑠(+g‘𝐾)𝑢) = 𝑢 ∧ (𝑢(+g‘𝐾)𝑠) = 𝑢) ↔ ∃𝑠 ∈ (Base‘𝐾)∀𝑢 ∈ (Base‘𝐾)((𝑠(+g‘𝐾)𝑢) = 𝑢 ∧ (𝑢(+g‘𝐾)𝑠) = 𝑢)))
85	64	raleqdv 3325	. . . . . . 7 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) → (∀𝑢 ∈ 𝐵 ((𝑠(+g‘𝐿)𝑢) = 𝑢 ∧ (𝑢(+g‘𝐿)𝑠) = 𝑢) ↔ ∀𝑢 ∈ (Base‘𝐿)((𝑠(+g‘𝐿)𝑢) = 𝑢 ∧ (𝑢(+g‘𝐿)𝑠) = 𝑢)))
86	64, 85	rexeqbidv 3341	. . . . . 6 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) → (∃𝑠 ∈ 𝐵 ∀𝑢 ∈ 𝐵 ((𝑠(+g‘𝐿)𝑢) = 𝑢 ∧ (𝑢(+g‘𝐿)𝑠) = 𝑢) ↔ ∃𝑠 ∈ (Base‘𝐿)∀𝑢 ∈ (Base‘𝐿)((𝑠(+g‘𝐿)𝑢) = 𝑢 ∧ (𝑢(+g‘𝐿)𝑠) = 𝑢)))
87	82, 84, 86	3bitr3d 312	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) → (∃𝑠 ∈ (Base‘𝐾)∀𝑢 ∈ (Base‘𝐾)((𝑠(+g‘𝐾)𝑢) = 𝑢 ∧ (𝑢(+g‘𝐾)𝑠) = 𝑢) ↔ ∃𝑠 ∈ (Base‘𝐿)∀𝑢 ∈ (Base‘𝐿)((𝑠(+g‘𝐿)𝑢) = 𝑢 ∧ (𝑢(+g‘𝐿)𝑠) = 𝑢)))
88	70, 87	anbi12d 644	. . . 4 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) → ((∀𝑢 ∈ (Base‘𝐾)∀𝑣 ∈ (Base‘𝐾)((𝑢(+g‘𝐾)𝑣) ∈ (Base‘𝐾) ∧ ∀𝑤 ∈ (Base‘𝐾)((𝑢(+g‘𝐾)𝑣)(+g‘𝐾)𝑤) = (𝑢(+g‘𝐾)(𝑣(+g‘𝐾)𝑤))) ∧ ∃𝑠 ∈ (Base‘𝐾)∀𝑢 ∈ (Base‘𝐾)((𝑠(+g‘𝐾)𝑢) = 𝑢 ∧ (𝑢(+g‘𝐾)𝑠) = 𝑢)) ↔ (∀𝑢 ∈ (Base‘𝐿)∀𝑣 ∈ (Base‘𝐿)((𝑢(+g‘𝐿)𝑣) ∈ (Base‘𝐿) ∧ ∀𝑤 ∈ (Base‘𝐿)((𝑢(+g‘𝐿)𝑣)(+g‘𝐿)𝑤) = (𝑢(+g‘𝐿)(𝑣(+g‘𝐿)𝑤))) ∧ ∃𝑠 ∈ (Base‘𝐿)∀𝑢 ∈ (Base‘𝐿)((𝑠(+g‘𝐿)𝑢) = 𝑢 ∧ (𝑢(+g‘𝐿)𝑠) = 𝑢))))
89	8, 9	ismnd 18827	. . . 4 ⊢ (𝐾 ∈ Mnd ↔ (∀𝑢 ∈ (Base‘𝐾)∀𝑣 ∈ (Base‘𝐾)((𝑢(+g‘𝐾)𝑣) ∈ (Base‘𝐾) ∧ ∀𝑤 ∈ (Base‘𝐾)((𝑢(+g‘𝐾)𝑣)(+g‘𝐾)𝑤) = (𝑢(+g‘𝐾)(𝑣(+g‘𝐾)𝑤))) ∧ ∃𝑠 ∈ (Base‘𝐾)∀𝑢 ∈ (Base‘𝐾)((𝑠(+g‘𝐾)𝑢) = 𝑢 ∧ (𝑢(+g‘𝐾)𝑠) = 𝑢)))
90	22, 23	ismnd 18827	. . . 4 ⊢ (𝐿 ∈ Mnd ↔ (∀𝑢 ∈ (Base‘𝐿)∀𝑣 ∈ (Base‘𝐿)((𝑢(+g‘𝐿)𝑣) ∈ (Base‘𝐿) ∧ ∀𝑤 ∈ (Base‘𝐿)((𝑢(+g‘𝐿)𝑣)(+g‘𝐿)𝑤) = (𝑢(+g‘𝐿)(𝑣(+g‘𝐿)𝑤))) ∧ ∃𝑠 ∈ (Base‘𝐿)∀𝑢 ∈ (Base‘𝐿)((𝑠(+g‘𝐿)𝑢) = 𝑢 ∧ (𝑢(+g‘𝐿)𝑠) = 𝑢)))
91	88, 89, 90	3bitr4g 317	. . 3 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) → (𝐾 ∈ Mnd ↔ 𝐿 ∈ Mnd))
92	91	ex 418	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵 → (𝐾 ∈ Mnd ↔ 𝐿 ∈ Mnd)))
93	14, 30, 92	pm5.21ndd 382	1 ⊢ (𝜑 → (𝐾 ∈ Mnd ↔ 𝐿 ∈ Mnd))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∀wral 3081  ∃wrex 3091  ‘cfv 6540  (class class class)co 7419  Basecbs 17291  +_gcplusg 17332  Mndcmnd 18824

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-nul 5271  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-iota 6496  df-fv 6548  df-ov 7422  df-mgm 18720  df-sgrp 18809  df-mnd 18825

This theorem is used by:  mndprop  18853  mhmpropd  18887  grppropd  19062  oppgmndb  19469  cmnpropd  19905  ringpropd  20417  prdsringd  20448  psrmnd  43369