Theorem insubm 13117

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 13103	. . 3 ⊢ (𝐴 ∈ (SubMnd‘𝑀) → 𝑀 ∈ Mnd)
2		ssinss1 3392	. . . . . . . . 9 ⊢ (𝐴 ⊆ (Base‘𝑀) → (𝐴 ∩ 𝐵) ⊆ (Base‘𝑀))
3	2	3ad2ant1 1020	. . . . . . . 8 ⊢ ((𝐴 ⊆ (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝐴 ∧ ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 (𝑎(+g‘𝑀)𝑏) ∈ 𝐴) → (𝐴 ∩ 𝐵) ⊆ (Base‘𝑀))
4	3	ad2antrl 490	. . . . . . 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 3346	. . . . . . . . . . . . 13 ⊢ ((0g‘𝑀) ∈ (𝐴 ∩ 𝐵) ↔ ((0g‘𝑀) ∈ 𝐴 ∧ (0g‘𝑀) ∈ 𝐵))
6	5	simplbi2com 1455	. . . . . . . . . . . 12 ⊢ ((0g‘𝑀) ∈ 𝐵 → ((0g‘𝑀) ∈ 𝐴 → (0g‘𝑀) ∈ (𝐴 ∩ 𝐵)))
7	6	3ad2ant2 1021	. . . . . . . . . . 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 30	. . . . . . . . . 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 1021	. . . . . . . . 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 124	. . . . . . . 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 277	. . . . . . 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 3346	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
13		elin 3346	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝐴 ∩ 𝐵) ↔ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
14	12, 13	anbi12i 460	. . . . . . . . 9 ⊢ ((𝑥 ∈ (𝐴 ∩ 𝐵) ∧ 𝑦 ∈ (𝐴 ∩ 𝐵)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
15		oveq1 5929	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = 𝑥 → (𝑎(+g‘𝑀)𝑏) = (𝑥(+g‘𝑀)𝑏))
16	15	eleq1d 2265	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝑥 → ((𝑎(+g‘𝑀)𝑏) ∈ 𝐴 ↔ (𝑥(+g‘𝑀)𝑏) ∈ 𝐴))
17		oveq2 5930	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 𝑦 → (𝑥(+g‘𝑀)𝑏) = (𝑥(+g‘𝑀)𝑦))
18	17	eleq1d 2265	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = 𝑦 → ((𝑥(+g‘𝑀)𝑏) ∈ 𝐴 ↔ (𝑥(+g‘𝑀)𝑦) ∈ 𝐴))
19		simpl 109	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐴)
20	19	adantr 276	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ 𝐴)
21		eqidd 2197	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑎 = 𝑥) → 𝐴 = 𝐴)
22		simpl 109	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐴)
23	22	adantl 277	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ 𝐴)
24	16, 18, 20, 21, 23	rspc2vd 3153	. . . . . . . . . . . . . . 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 30	. . . . . . . . . . . . . 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 1022	. . . . . . . . . . . . 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 490	. . . . . . . . . . . 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 124	. . . . . . . . . . 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 2265	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = 𝑥 → ((𝑎(+g‘𝑀)𝑏) ∈ 𝐵 ↔ (𝑥(+g‘𝑀)𝑏) ∈ 𝐵))
30	17	eleq1d 2265	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 𝑦 → ((𝑥(+g‘𝑀)𝑏) ∈ 𝐵 ↔ (𝑥(+g‘𝑀)𝑦) ∈ 𝐵))
31		simpr 110	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
32	31	adantr 276	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ 𝐵)
33		eqidd 2197	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑎 = 𝑥) → 𝐵 = 𝐵)
34		simpr 110	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵)
35	34	adantl 277	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ∧ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ 𝐵)
36	29, 30, 32, 33, 35	rspc2vd 3153	. . . . . . . . . . . . . . . 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 30	. . . . . . . . . . . . . . 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 1022	. . . . . . . . . . . . . 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 277	. . . . . . . . . . . . 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 277	. . . . . . . . . . . 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 124	. . . . . . . . . . 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 3348	. . . . . . . . . 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 115	. . . . . . . . 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 152	. . . . . . . 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 2578	. . . . . . 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 1179	. . . . . 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 115	. . . . 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 2196	. . . . . . 7 ⊢ (Base‘𝑀) = (Base‘𝑀)
49		eqid 2196	. . . . . . 7 ⊢ (0g‘𝑀) = (0g‘𝑀)
50		eqid 2196	. . . . . . 7 ⊢ (+g‘𝑀) = (+g‘𝑀)
51	48, 49, 50	issubm 13104	. . . . . 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 13104	. . . . . 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 473	. . . . 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 13104	. . . . 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 203	. . . 4 ⊢ (𝑀 ∈ Mnd → ((𝐴 ∈ (SubMnd‘𝑀) ∧ 𝐵 ∈ (SubMnd‘𝑀)) → (𝐴 ∩ 𝐵) ∈ (SubMnd‘𝑀)))
56	55	expd 258	. . 3 ⊢ (𝑀 ∈ Mnd → (𝐴 ∈ (SubMnd‘𝑀) → (𝐵 ∈ (SubMnd‘𝑀) → (𝐴 ∩ 𝐵) ∈ (SubMnd‘𝑀))))
57	1, 56	mpcom 36	. 2 ⊢ (𝐴 ∈ (SubMnd‘𝑀) → (𝐵 ∈ (SubMnd‘𝑀) → (𝐴 ∩ 𝐵) ∈ (SubMnd‘𝑀)))
58	57	imp 124	1 ⊢ ((𝐴 ∈ (SubMnd‘𝑀) ∧ 𝐵 ∈ (SubMnd‘𝑀)) → (𝐴 ∩ 𝐵) ∈ (SubMnd‘𝑀))

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 980   ∈ wcel 2167  ∀wral 2475   ∩ cin 3156   ⊆ wss 3157  ‘cfv 5258  (class class class)co 5922  Basecbs 12678  +_gcplusg 12755  0_gc0g 12927  Mndcmnd 13057  SubMndcsubmnd 13090

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-cnex 7970  ax-resscn 7971  ax-1re 7973  ax-addrcl 7976

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-fv 5266  df-ov 5925  df-inn 8991  df-ndx 12681  df-slot 12682  df-base 12684  df-submnd 13092

This theorem is referenced by: (None)