Theorem mndpropd 12705

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 528	. . . . . 6 ⊢ (((𝜑 ∧ 𝐾 ∈ Mnd) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐾 ∈ Mnd)
2		simprl 529	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐾 ∈ Mnd) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ 𝐵)
3		mndpropd.1	. . . . . . . 8 ⊢ (𝜑 → 𝐵 = (Base‘𝐾))
4	3	ad2antrr 488	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐾 ∈ Mnd) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐵 = (Base‘𝐾))
5	2, 4	eleqtrd 2254	. . . . . 6 ⊢ (((𝜑 ∧ 𝐾 ∈ Mnd) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ (Base‘𝐾))
6		simprr 531	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐾 ∈ Mnd) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ 𝐵)
7	6, 4	eleqtrd 2254	. . . . . 6 ⊢ (((𝜑 ∧ 𝐾 ∈ Mnd) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ (Base‘𝐾))
8		eqid 2175	. . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾)
9		eqid 2175	. . . . . . 7 ⊢ (+g‘𝐾) = (+g‘𝐾)
10	8, 9	mndcl 12688	. . . . . 6 ⊢ ((𝐾 ∈ Mnd ∧ 𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾)) → (𝑥(+g‘𝐾)𝑦) ∈ (Base‘𝐾))
11	1, 5, 7, 10	syl3anc 1238	. . . . 5 ⊢ (((𝜑 ∧ 𝐾 ∈ Mnd) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) ∈ (Base‘𝐾))
12	11, 4	eleqtrrd 2255	. . . 4 ⊢ (((𝜑 ∧ 𝐾 ∈ Mnd) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) ∈ 𝐵)
13	12	ralrimivva 2557	. . 3 ⊢ ((𝜑 ∧ 𝐾 ∈ Mnd) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵)
14	13	ex 115	. 2 ⊢ (𝜑 → (𝐾 ∈ Mnd → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵))
15		simplr 528	. . . . . 6 ⊢ (((𝜑 ∧ 𝐿 ∈ Mnd) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐿 ∈ Mnd)
16		simprl 529	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐿 ∈ Mnd) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ 𝐵)
17		mndpropd.2	. . . . . . . 8 ⊢ (𝜑 → 𝐵 = (Base‘𝐿))
18	17	ad2antrr 488	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐿 ∈ Mnd) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐵 = (Base‘𝐿))
19	16, 18	eleqtrd 2254	. . . . . 6 ⊢ (((𝜑 ∧ 𝐿 ∈ Mnd) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ (Base‘𝐿))
20		simprr 531	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐿 ∈ Mnd) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ 𝐵)
21	20, 18	eleqtrd 2254	. . . . . 6 ⊢ (((𝜑 ∧ 𝐿 ∈ Mnd) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ (Base‘𝐿))
22		eqid 2175	. . . . . . 7 ⊢ (Base‘𝐿) = (Base‘𝐿)
23		eqid 2175	. . . . . . 7 ⊢ (+g‘𝐿) = (+g‘𝐿)
24	22, 23	mndcl 12688	. . . . . 6 ⊢ ((𝐿 ∈ Mnd ∧ 𝑥 ∈ (Base‘𝐿) ∧ 𝑦 ∈ (Base‘𝐿)) → (𝑥(+g‘𝐿)𝑦) ∈ (Base‘𝐿))
25	15, 19, 21, 24	syl3anc 1238	. . . . 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 477	. . . . 5 ⊢ (((𝜑 ∧ 𝐿 ∈ Mnd) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))
28	25, 27, 18	3eltr4d 2259	. . . 4 ⊢ (((𝜑 ∧ 𝐿 ∈ Mnd) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) ∈ 𝐵)
29	28	ralrimivva 2557	. . 3 ⊢ ((𝜑 ∧ 𝐿 ∈ Mnd) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵)
30	29	ex 115	. 2 ⊢ (𝜑 → (𝐿 ∈ Mnd → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵))
31	26	oveqrspc2v 5892	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) → (𝑢(+g‘𝐾)𝑣) = (𝑢(+g‘𝐿)𝑣))
32	31	adantlr 477	. . . . . . . . 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 2244	. . . . . . . 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 533	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑤 ∈ 𝐵) → 𝜑)
35		simplrl 535	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑤 ∈ 𝐵) → 𝑢 ∈ 𝐵)
36		simplrr 536	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑤 ∈ 𝐵) → 𝑣 ∈ 𝐵)
37		simpllr 534	. . . . . . . . . . . . 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 5891	. . . . . . . . . . . . 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 1237	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑤 ∈ 𝐵) → (𝑢(+g‘𝐾)𝑣) ∈ 𝐵)
40		simpr 110	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵)) ∧ 𝑤 ∈ 𝐵) → 𝑤 ∈ 𝐵)
41	26	oveqrspc2v 5892	. . . . . . . . . . . 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 1236	. . . . . . . . . . 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 1236	. . . . . . . . . . . 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 5880	. . . . . . . . . . 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 2208	. . . . . . . . . 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 5891	. . . . . . . . . . . . 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 1237	. . . . . . . . . . . 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 5892	. . . . . . . . . . . 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 1236	. . . . . . . . . . 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 5892	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑣(+g‘𝐾)𝑤) = (𝑣(+g‘𝐿)𝑤))
51	34, 36, 40, 50	syl12anc 1236	. . . . . . . . . . . 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 5881	. . . . . . . . . . 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 2208	. . . . . . . . . 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 2190	. . . . . . . . 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 2471	. . . . . . . 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 473	. . . . . . 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 2497	. . . . . 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 276	. . . . . . 7 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) → 𝐵 = (Base‘𝐾))
59	58	eleq2d 2245	. . . . . . . . 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 2676	. . . . . . . . 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 473	. . . . . . . 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 2682	. . . . . . 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 2682	. . . . . 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 276	. . . . . . 7 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) → 𝐵 = (Base‘𝐿))
65	64	eleq2d 2245	. . . . . . . . 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 2676	. . . . . . . . 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 473	. . . . . . . 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 2682	. . . . . . 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 2682	. . . . . 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 218	. . . . 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 533	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) ∧ 𝑠 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) → 𝜑)
72		simplr 528	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) ∧ 𝑠 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) → 𝑠 ∈ 𝐵)
73		simpr 110	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) ∧ 𝑠 ∈ 𝐵) ∧ 𝑢 ∈ 𝐵) → 𝑢 ∈ 𝐵)
74	26	oveqrspc2v 5892	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑠 ∈ 𝐵 ∧ 𝑢 ∈ 𝐵)) → (𝑠(+g‘𝐾)𝑢) = (𝑠(+g‘𝐿)𝑢))
75	71, 72, 73, 74	syl12anc 1236	. . . . . . . . . 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 2184	. . . . . . . . 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 5892	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐵 ∧ 𝑠 ∈ 𝐵)) → (𝑢(+g‘𝐾)𝑠) = (𝑢(+g‘𝐿)𝑠))
78	71, 73, 72, 77	syl12anc 1236	. . . . . . . . . 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 2184	. . . . . . . . 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 473	. . . . . . . 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 2471	. . . . . . 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 2472	. . . . . 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 2676	. . . . . . 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 2683	. . . . . 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 2676	. . . . . . 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 2683	. . . . . 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 218	. . . . 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 473	. . . 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 12684	. . . 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 12684	. . . 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 223	. . 3 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵) → (𝐾 ∈ Mnd ↔ 𝐿 ∈ Mnd))
92	91	ex 115	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) ∈ 𝐵 → (𝐾 ∈ Mnd ↔ 𝐿 ∈ Mnd)))
93	14, 30, 92	pm5.21ndd 705	1 ⊢ (𝜑 → (𝐾 ∈ Mnd ↔ 𝐿 ∈ Mnd))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1353   ∈ wcel 2146  ∀wral 2453  ∃wrex 2454  ‘cfv 5208  (class class class)co 5865  Basecbs 12427  +_gcplusg 12491  Mndcmnd 12681

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-cnex 7877  ax-resscn 7878  ax-1re 7880  ax-addrcl 7883

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-rab 2462  df-v 2737  df-sbc 2961  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-br 3999  df-opab 4060  df-mpt 4061  df-id 4287  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-iota 5170  df-fun 5210  df-fn 5211  df-fv 5216  df-ov 5868  df-inn 8891  df-2 8949  df-ndx 12430  df-slot 12431  df-base 12433  df-plusg 12504  df-mgm 12639  df-sgrp 12672  df-mnd 12682

This theorem is referenced by:  mndprop  12706  mhmpropd  12718  grppropd  12753  cmnpropd  12894  ringpropd  13009  ring1  13028