Theorem insubm 18629

Description: The intersection of two submonoids is a submonoid. (Contributed by AV, 25-Feb-2024.)

Assertion

Ref	Expression
insubm	⊢ ((𝐴 ∈ (SubMnd‘𝑀) ∧ 𝐵 ∈ (SubMnd‘𝑀)) → (𝐴 ∩ 𝐵) ∈ (SubMnd‘𝑀))

Proof of Theorem insubm

Dummy variables 𝑎 𝑏 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		submrcl 18613	. . 3 ⊢ (𝐴 ∈ (SubMnd‘𝑀) → 𝑀 ∈ Mnd)
2		ssinss1 4197	. . . . . . . . 9 ⊢ (𝐴 ⊆ (Base‘𝑀) → (𝐴 ∩ 𝐵) ⊆ (Base‘𝑀))
3	2	3ad2ant1 1133	. . . . . . . 8 ⊢ ((𝐴 ⊆ (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝐴 ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 (𝑎(+g‘𝑀)𝑏) ∈ 𝐴) → (𝐴 ∩ 𝐵) ⊆ (Base‘𝑀))
4	3	ad2antrl 726	. . . . . . 7 ⊢ ((𝑀 ∈ Mnd ∧ ((𝐴 ⊆ (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝐴 ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 (𝑎(+g‘𝑀)𝑏) ∈ 𝐴) ∧ (𝐵 ⊆ (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝐵 ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝑎(+g‘𝑀)𝑏) ∈ 𝐵))) → (𝐴 ∩ 𝐵) ⊆ (Base‘𝑀))
5		elin 3926	. . . . . . . . . . . . 13 ⊢ ((0g‘𝑀) ∈ (𝐴 ∩ 𝐵) ↔ ((0g‘𝑀) ∈ 𝐴 ∧ (0g‘𝑀) ∈ 𝐵))
6	5	simplbi2com 503	. . . . . . . . . . . 12 ⊢ ((0g‘𝑀) ∈ 𝐵 → ((0g‘𝑀) ∈ 𝐴 → (0g‘𝑀) ∈ (𝐴 ∩ 𝐵)))
7	6	3ad2ant2 1134	. . . . . . . . . . 11 ⊢ ((𝐵 ⊆ (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝐵 ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝑎(+g‘𝑀)𝑏) ∈ 𝐵) → ((0g‘𝑀) ∈ 𝐴 → (0g‘𝑀) ∈ (𝐴 ∩ 𝐵)))
8	7	com12 32	. . . . . . . . . 10 ⊢ ((0g‘𝑀) ∈ 𝐴 → ((𝐵 ⊆ (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝐵 ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝑎(+g‘𝑀)𝑏) ∈ 𝐵) → (0g‘𝑀) ∈ (𝐴 ∩ 𝐵)))
9	8	3ad2ant2 1134	. . . . . . . . 9 ⊢ ((𝐴 ⊆ (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝐴 ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 (𝑎(+g‘𝑀)𝑏) ∈ 𝐴) → ((𝐵 ⊆ (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝐵 ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝑎(+g‘𝑀)𝑏) ∈ 𝐵) → (0g‘𝑀) ∈ (𝐴 ∩ 𝐵)))
10	9	imp 407	. . . . . . . 8 ⊢ (((𝐴 ⊆ (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝐴 ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 (𝑎(+g‘𝑀)𝑏) ∈ 𝐴) ∧ (𝐵 ⊆ (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝐵 ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝑎(+g‘𝑀)𝑏) ∈ 𝐵)) → (0g‘𝑀) ∈ (𝐴 ∩ 𝐵))
11	10	adantl 482	. . . . . . 7 ⊢ ((𝑀 ∈ Mnd ∧ ((𝐴 ⊆ (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝐴 ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 (𝑎(+g‘𝑀)𝑏) ∈ 𝐴) ∧ (𝐵 ⊆ (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝐵 ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝑎(+g‘𝑀)𝑏) ∈ 𝐵))) → (0g‘𝑀) ∈ (𝐴 ∩ 𝐵))
12		elin 3926	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
13		elin 3926	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝐴 ∩ 𝐵) ↔ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
14	12, 13	anbi12i 627	. . . . . . . . 9 ⊢ ((𝑥 ∈ (𝐴 ∩ 𝐵) ∧ 𝑦 ∈ (𝐴 ∩ 𝐵)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
15		oveq1 7364	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = 𝑥 → (𝑎(+g‘𝑀)𝑏) = (𝑥(+g‘𝑀)𝑏))
16	15	eleq1d 2822	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝑥 → ((𝑎(+g‘𝑀)𝑏) ∈ 𝐴 ↔ (𝑥(+g‘𝑀)𝑏) ∈ 𝐴))
17		oveq2 7365	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 𝑦 → (𝑥(+g‘𝑀)𝑏) = (𝑥(+g‘𝑀)𝑦))
18	17	eleq1d 2822	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = 𝑦 → ((𝑥(+g‘𝑀)𝑏) ∈ 𝐴 ↔ (𝑥(+g‘𝑀)𝑦) ∈ 𝐴))
19		simpl 483	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐴)
20	19	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ 𝐴)
21		eqidd 2737	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑎 = 𝑥) → 𝐴 = 𝐴)
22		simpl 483	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐴)
23	22	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ 𝐴)
24	16, 18, 20, 21, 23	rspc2vd 3906	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 (𝑎(+g‘𝑀)𝑏) ∈ 𝐴 → (𝑥(+g‘𝑀)𝑦) ∈ 𝐴))
25	24	com12 32	. . . . . . . . . . . . . 14 ⊢ (∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 (𝑎(+g‘𝑀)𝑏) ∈ 𝐴 → (((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝑀)𝑦) ∈ 𝐴))
26	25	3ad2ant3 1135	. . . . . . . . . . . . 13 ⊢ ((𝐴 ⊆ (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝐴 ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 (𝑎(+g‘𝑀)𝑏) ∈ 𝐴) → (((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝑀)𝑦) ∈ 𝐴))
27	26	ad2antrl 726	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ Mnd ∧ ((𝐴 ⊆ (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝐴 ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 (𝑎(+g‘𝑀)𝑏) ∈ 𝐴) ∧ (𝐵 ⊆ (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝐵 ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝑎(+g‘𝑀)𝑏) ∈ 𝐵))) → (((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝑀)𝑦) ∈ 𝐴))
28	27	imp 407	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ Mnd ∧ ((𝐴 ⊆ (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝐴 ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 (𝑎(+g‘𝑀)𝑏) ∈ 𝐴) ∧ (𝐵 ⊆ (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝐵 ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝑎(+g‘𝑀)𝑏) ∈ 𝐵))) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))) → (𝑥(+g‘𝑀)𝑦) ∈ 𝐴)
29	15	eleq1d 2822	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = 𝑥 → ((𝑎(+g‘𝑀)𝑏) ∈ 𝐵 ↔ (𝑥(+g‘𝑀)𝑏) ∈ 𝐵))
30	17	eleq1d 2822	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 𝑦 → ((𝑥(+g‘𝑀)𝑏) ∈ 𝐵 ↔ (𝑥(+g‘𝑀)𝑦) ∈ 𝐵))
31		simpr 485	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
32	31	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ 𝐵)
33		eqidd 2737	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑎 = 𝑥) → 𝐵 = 𝐵)
34		simpr 485	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵)
35	34	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ 𝐵)
36	29, 30, 32, 33, 35	rspc2vd 3906	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝑎(+g‘𝑀)𝑏) ∈ 𝐵 → (𝑥(+g‘𝑀)𝑦) ∈ 𝐵))
37	36	com12 32	. . . . . . . . . . . . . . 15 ⊢ (∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝑎(+g‘𝑀)𝑏) ∈ 𝐵 → (((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝑀)𝑦) ∈ 𝐵))
38	37	3ad2ant3 1135	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ⊆ (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝐵 ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝑎(+g‘𝑀)𝑏) ∈ 𝐵) → (((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝑀)𝑦) ∈ 𝐵))
39	38	adantl 482	. . . . . . . . . . . . 13 ⊢ (((𝐴 ⊆ (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝐴 ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 (𝑎(+g‘𝑀)𝑏) ∈ 𝐴) ∧ (𝐵 ⊆ (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝐵 ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝑎(+g‘𝑀)𝑏) ∈ 𝐵)) → (((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝑀)𝑦) ∈ 𝐵))
40	39	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ Mnd ∧ ((𝐴 ⊆ (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝐴 ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 (𝑎(+g‘𝑀)𝑏) ∈ 𝐴) ∧ (𝐵 ⊆ (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝐵 ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝑎(+g‘𝑀)𝑏) ∈ 𝐵))) → (((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝑀)𝑦) ∈ 𝐵))
41	40	imp 407	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ Mnd ∧ ((𝐴 ⊆ (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝐴 ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 (𝑎(+g‘𝑀)𝑏) ∈ 𝐴) ∧ (𝐵 ⊆ (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝐵 ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝑎(+g‘𝑀)𝑏) ∈ 𝐵))) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))) → (𝑥(+g‘𝑀)𝑦) ∈ 𝐵)
42	28, 41	elind 4154	. . . . . . . . . 10 ⊢ (((𝑀 ∈ Mnd ∧ ((𝐴 ⊆ (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝐴 ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 (𝑎(+g‘𝑀)𝑏) ∈ 𝐴) ∧ (𝐵 ⊆ (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝐵 ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝑎(+g‘𝑀)𝑏) ∈ 𝐵))) ∧ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))) → (𝑥(+g‘𝑀)𝑦) ∈ (𝐴 ∩ 𝐵))
43	42	ex 413	. . . . . . . . 9 ⊢ ((𝑀 ∈ Mnd ∧ ((𝐴 ⊆ (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝐴 ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 (𝑎(+g‘𝑀)𝑏) ∈ 𝐴) ∧ (𝐵 ⊆ (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝐵 ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝑎(+g‘𝑀)𝑏) ∈ 𝐵))) → (((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝑀)𝑦) ∈ (𝐴 ∩ 𝐵)))
44	14, 43	biimtrid 241	. . . . . . . 8 ⊢ ((𝑀 ∈ Mnd ∧ ((𝐴 ⊆ (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝐴 ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 (𝑎(+g‘𝑀)𝑏) ∈ 𝐴) ∧ (𝐵 ⊆ (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝐵 ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝑎(+g‘𝑀)𝑏) ∈ 𝐵))) → ((𝑥 ∈ (𝐴 ∩ 𝐵) ∧ 𝑦 ∈ (𝐴 ∩ 𝐵)) → (𝑥(+g‘𝑀)𝑦) ∈ (𝐴 ∩ 𝐵)))
45	44	ralrimivv 3195	. . . . . . 7 ⊢ ((𝑀 ∈ Mnd ∧ ((𝐴 ⊆ (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝐴 ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 (𝑎(+g‘𝑀)𝑏) ∈ 𝐴) ∧ (𝐵 ⊆ (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝐵 ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝑎(+g‘𝑀)𝑏) ∈ 𝐵))) → ∀𝑥 ∈ (𝐴 ∩ 𝐵)∀𝑦 ∈ (𝐴 ∩ 𝐵)(𝑥(+g‘𝑀)𝑦) ∈ (𝐴 ∩ 𝐵))
46	4, 11, 45	3jca 1128	. . . . . 6 ⊢ ((𝑀 ∈ Mnd ∧ ((𝐴 ⊆ (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝐴 ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 (𝑎(+g‘𝑀)𝑏) ∈ 𝐴) ∧ (𝐵 ⊆ (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝐵 ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝑎(+g‘𝑀)𝑏) ∈ 𝐵))) → ((𝐴 ∩ 𝐵) ⊆ (Base‘𝑀) ∧ (0g‘𝑀) ∈ (𝐴 ∩ 𝐵) ∧ ∀𝑥 ∈ (𝐴 ∩ 𝐵)∀𝑦 ∈ (𝐴 ∩ 𝐵)(𝑥(+g‘𝑀)𝑦) ∈ (𝐴 ∩ 𝐵)))
47	46	ex 413	. . . . 5 ⊢ (𝑀 ∈ Mnd → (((𝐴 ⊆ (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝐴 ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 (𝑎(+g‘𝑀)𝑏) ∈ 𝐴) ∧ (𝐵 ⊆ (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝐵 ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝑎(+g‘𝑀)𝑏) ∈ 𝐵)) → ((𝐴 ∩ 𝐵) ⊆ (Base‘𝑀) ∧ (0g‘𝑀) ∈ (𝐴 ∩ 𝐵) ∧ ∀𝑥 ∈ (𝐴 ∩ 𝐵)∀𝑦 ∈ (𝐴 ∩ 𝐵)(𝑥(+g‘𝑀)𝑦) ∈ (𝐴 ∩ 𝐵))))
48		eqid 2736	. . . . . . 7 ⊢ (Base‘𝑀) = (Base‘𝑀)
49		eqid 2736	. . . . . . 7 ⊢ (0g‘𝑀) = (0g‘𝑀)
50		eqid 2736	. . . . . . 7 ⊢ (+g‘𝑀) = (+g‘𝑀)
51	48, 49, 50	issubm 18614	. . . . . 6 ⊢ (𝑀 ∈ Mnd → (𝐴 ∈ (SubMnd‘𝑀) ↔ (𝐴 ⊆ (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝐴 ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 (𝑎(+g‘𝑀)𝑏) ∈ 𝐴)))
52	48, 49, 50	issubm 18614	. . . . . 6 ⊢ (𝑀 ∈ Mnd → (𝐵 ∈ (SubMnd‘𝑀) ↔ (𝐵 ⊆ (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝐵 ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝑎(+g‘𝑀)𝑏) ∈ 𝐵)))
53	51, 52	anbi12d 631	. . . . 5 ⊢ (𝑀 ∈ Mnd → ((𝐴 ∈ (SubMnd‘𝑀) ∧ 𝐵 ∈ (SubMnd‘𝑀)) ↔ ((𝐴 ⊆ (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝐴 ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 (𝑎(+g‘𝑀)𝑏) ∈ 𝐴) ∧ (𝐵 ⊆ (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝐵 ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝑎(+g‘𝑀)𝑏) ∈ 𝐵))))
54	48, 49, 50	issubm 18614	. . . . 5 ⊢ (𝑀 ∈ Mnd → ((𝐴 ∩ 𝐵) ∈ (SubMnd‘𝑀) ↔ ((𝐴 ∩ 𝐵) ⊆ (Base‘𝑀) ∧ (0g‘𝑀) ∈ (𝐴 ∩ 𝐵) ∧ ∀𝑥 ∈ (𝐴 ∩ 𝐵)∀𝑦 ∈ (𝐴 ∩ 𝐵)(𝑥(+g‘𝑀)𝑦) ∈ (𝐴 ∩ 𝐵))))
55	47, 53, 54	3imtr4d 293	. . . 4 ⊢ (𝑀 ∈ Mnd → ((𝐴 ∈ (SubMnd‘𝑀) ∧ 𝐵 ∈ (SubMnd‘𝑀)) → (𝐴 ∩ 𝐵) ∈ (SubMnd‘𝑀)))
56	55	expd 416	. . 3 ⊢ (𝑀 ∈ Mnd → (𝐴 ∈ (SubMnd‘𝑀) → (𝐵 ∈ (SubMnd‘𝑀) → (𝐴 ∩ 𝐵) ∈ (SubMnd‘𝑀))))
57	1, 56	mpcom 38	. 2 ⊢ (𝐴 ∈ (SubMnd‘𝑀) → (𝐵 ∈ (SubMnd‘𝑀) → (𝐴 ∩ 𝐵) ∈ (SubMnd‘𝑀)))
58	57	imp 407	1 ⊢ ((𝐴 ∈ (SubMnd‘𝑀) ∧ 𝐵 ∈ (SubMnd‘𝑀)) → (𝐴 ∩ 𝐵) ∈ (SubMnd‘𝑀))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1087   ∈ wcel 2106  ∀wral 3064   ∩ cin 3909   ⊆ wss 3910  ‘cfv 6496  (class class class)co 7357  Basecbs 17083  +_gcplusg 17133  0_gc0g 17321  Mndcmnd 18556  SubMndcsubmnd 18600

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fv 6504  df-ov 7360  df-submnd 18602

This theorem is referenced by:  symgsubmefmnd  19180