Theorem insubm 18831

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 18815	. . 3 ⊢ (𝐴 ∈ (SubMnd‘𝑀) → 𝑀 ∈ Mnd)
2		ssinss1 4246	. . . . . . . . 9 ⊢ (𝐴 ⊆ (Base‘𝑀) → (𝐴 ∩ 𝐵) ⊆ (Base‘𝑀))
3	2	3ad2ant1 1134	. . . . . . . 8 ⊢ ((𝐴 ⊆ (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝐴 ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 (𝑎(+g‘𝑀)𝑏) ∈ 𝐴) → (𝐴 ∩ 𝐵) ⊆ (Base‘𝑀))
4	3	ad2antrl 728	. . . . . . 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 3967	. . . . . . . . . . . . 13 ⊢ ((0g‘𝑀) ∈ (𝐴 ∩ 𝐵) ↔ ((0g‘𝑀) ∈ 𝐴 ∧ (0g‘𝑀) ∈ 𝐵))
6	5	simplbi2com 502	. . . . . . . . . . . 12 ⊢ ((0g‘𝑀) ∈ 𝐵 → ((0g‘𝑀) ∈ 𝐴 → (0g‘𝑀) ∈ (𝐴 ∩ 𝐵)))
7	6	3ad2ant2 1135	. . . . . . . . . . 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 1135	. . . . . . . . 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 406	. . . . . . . 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 481	. . . . . . 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 3967	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
13		elin 3967	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝐴 ∩ 𝐵) ↔ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
14	12, 13	anbi12i 628	. . . . . . . . 9 ⊢ ((𝑥 ∈ (𝐴 ∩ 𝐵) ∧ 𝑦 ∈ (𝐴 ∩ 𝐵)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
15		oveq1 7438	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = 𝑥 → (𝑎(+g‘𝑀)𝑏) = (𝑥(+g‘𝑀)𝑏))
16	15	eleq1d 2826	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝑥 → ((𝑎(+g‘𝑀)𝑏) ∈ 𝐴 ↔ (𝑥(+g‘𝑀)𝑏) ∈ 𝐴))
17		oveq2 7439	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 𝑦 → (𝑥(+g‘𝑀)𝑏) = (𝑥(+g‘𝑀)𝑦))
18	17	eleq1d 2826	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = 𝑦 → ((𝑥(+g‘𝑀)𝑏) ∈ 𝐴 ↔ (𝑥(+g‘𝑀)𝑦) ∈ 𝐴))
19		simpl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐴)
20	19	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ 𝐴)
21		eqidd 2738	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑎 = 𝑥) → 𝐴 = 𝐴)
22		simpl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐴)
23	22	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ 𝐴)
24	16, 18, 20, 21, 23	rspc2vd 3947	. . . . . . . . . . . . . . 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 1136	. . . . . . . . . . . . 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 728	. . . . . . . . . . . 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 406	. . . . . . . . . . 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 2826	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = 𝑥 → ((𝑎(+g‘𝑀)𝑏) ∈ 𝐵 ↔ (𝑥(+g‘𝑀)𝑏) ∈ 𝐵))
30	17	eleq1d 2826	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 𝑦 → ((𝑥(+g‘𝑀)𝑏) ∈ 𝐵 ↔ (𝑥(+g‘𝑀)𝑦) ∈ 𝐵))
31		simpr 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
32	31	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ 𝐵)
33		eqidd 2738	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑎 = 𝑥) → 𝐵 = 𝐵)
34		simpr 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵)
35	34	adantl 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ 𝐵)
36	29, 30, 32, 33, 35	rspc2vd 3947	. . . . . . . . . . . . . . . 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 1136	. . . . . . . . . . . . . 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 481	. . . . . . . . . . . . 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 481	. . . . . . . . . . . 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 406	. . . . . . . . . . 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 4200	. . . . . . . . . 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 412	. . . . . . . . 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 242	. . . . . . . 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 3200	. . . . . . 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 1129	. . . . . 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 412	. . . . 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 2737	. . . . . . 7 ⊢ (Base‘𝑀) = (Base‘𝑀)
49		eqid 2737	. . . . . . 7 ⊢ (0g‘𝑀) = (0g‘𝑀)
50		eqid 2737	. . . . . . 7 ⊢ (+g‘𝑀) = (+g‘𝑀)
51	48, 49, 50	issubm 18816	. . . . . 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 18816	. . . . . 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 632	. . . . 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 18816	. . . . 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 294	. . . 4 ⊢ (𝑀 ∈ Mnd → ((𝐴 ∈ (SubMnd‘𝑀) ∧ 𝐵 ∈ (SubMnd‘𝑀)) → (𝐴 ∩ 𝐵) ∈ (SubMnd‘𝑀)))
56	55	expd 415	. . 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 406	1 ⊢ ((𝐴 ∈ (SubMnd‘𝑀) ∧ 𝐵 ∈ (SubMnd‘𝑀)) → (𝐴 ∩ 𝐵) ∈ (SubMnd‘𝑀))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   ∈ wcel 2108  ∀wral 3061   ∩ cin 3950   ⊆ wss 3951  ‘cfv 6561  (class class class)co 7431  Basecbs 17247  +_gcplusg 17297  0_gc0g 17484  Mndcmnd 18747  SubMndcsubmnd 18795

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-submnd 18797

This theorem is referenced by:  symgsubmefmnd  19416