Theorem lsmsubm 19867

Description: The sum of two commuting submonoids is a submonoid. (Contributed by Mario Carneiro, 19-Apr-2016.)

Hypotheses

Ref	Expression
lsmsubg.p	⊢ ⊕ = (LSSum‘𝐺)
lsmsubg.z	⊢ 𝑍 = (Cntz‘𝐺)

Assertion

Ref	Expression
lsmsubm	⊢ ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → (𝑇 ⊕ 𝑈) ∈ (SubMnd‘𝐺))

Proof of Theorem lsmsubm

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

Step	Hyp	Ref	Expression
1		submrcl 18997	. . . 4 ⊢ (𝑇 ∈ (SubMnd‘𝐺) → 𝐺 ∈ Mnd)
2	1	3ad2ant1 1151	. . 3 ⊢ ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → 𝐺 ∈ Mnd)
3		eqid 2761	. . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺)
4	3	submss 19004	. . . 4 ⊢ (𝑇 ∈ (SubMnd‘𝐺) → 𝑇 ⊆ (Base‘𝐺))
5	4	3ad2ant1 1151	. . 3 ⊢ ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → 𝑇 ⊆ (Base‘𝐺))
6	3	submss 19004	. . . 4 ⊢ (𝑈 ∈ (SubMnd‘𝐺) → 𝑈 ⊆ (Base‘𝐺))
7	6	3ad2ant2 1152	. . 3 ⊢ ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → 𝑈 ⊆ (Base‘𝐺))
8		lsmsubg.p	. . . 4 ⊢ ⊕ = (LSSum‘𝐺)
9	3, 8	lsmssv 19857	. . 3 ⊢ ((𝐺 ∈ Mnd ∧ 𝑇 ⊆ (Base‘𝐺) ∧ 𝑈 ⊆ (Base‘𝐺)) → (𝑇 ⊕ 𝑈) ⊆ (Base‘𝐺))
10	2, 5, 7, 9	syl3anc 1398	. 2 ⊢ ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → (𝑇 ⊕ 𝑈) ⊆ (Base‘𝐺))
11		simp2 1155	. . . 4 ⊢ ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → 𝑈 ∈ (SubMnd‘𝐺))
12	3, 8	lsmub1x 19860	. . . 4 ⊢ ((𝑇 ⊆ (Base‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺)) → 𝑇 ⊆ (𝑇 ⊕ 𝑈))
13	5, 11, 12	syl2anc 596	. . 3 ⊢ ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → 𝑇 ⊆ (𝑇 ⊕ 𝑈))
14		eqid 2761	. . . . 5 ⊢ (0g‘𝐺) = (0g‘𝐺)
15	14	subm0cl 19006	. . . 4 ⊢ (𝑇 ∈ (SubMnd‘𝐺) → (0g‘𝐺) ∈ 𝑇)
16	15	3ad2ant1 1151	. . 3 ⊢ ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → (0g‘𝐺) ∈ 𝑇)
17	13, 16	sseldd 3932	. 2 ⊢ ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → (0g‘𝐺) ∈ (𝑇 ⊕ 𝑈))
18		eqid 2761	. . . . . . 7 ⊢ (+g‘𝐺) = (+g‘𝐺)
19	3, 18, 8	lsmelvalx 19854	. . . . . 6 ⊢ ((𝐺 ∈ Mnd ∧ 𝑇 ⊆ (Base‘𝐺) ∧ 𝑈 ⊆ (Base‘𝐺)) → (𝑥 ∈ (𝑇 ⊕ 𝑈) ↔ ∃𝑎 ∈ 𝑇 ∃𝑐 ∈ 𝑈 𝑥 = (𝑎(+g‘𝐺)𝑐)))
20	2, 5, 7, 19	syl3anc 1398	. . . . 5 ⊢ ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → (𝑥 ∈ (𝑇 ⊕ 𝑈) ↔ ∃𝑎 ∈ 𝑇 ∃𝑐 ∈ 𝑈 𝑥 = (𝑎(+g‘𝐺)𝑐)))
21	3, 18, 8	lsmelvalx 19854	. . . . . 6 ⊢ ((𝐺 ∈ Mnd ∧ 𝑇 ⊆ (Base‘𝐺) ∧ 𝑈 ⊆ (Base‘𝐺)) → (𝑦 ∈ (𝑇 ⊕ 𝑈) ↔ ∃𝑏 ∈ 𝑇 ∃𝑑 ∈ 𝑈 𝑦 = (𝑏(+g‘𝐺)𝑑)))
22	2, 5, 7, 21	syl3anc 1398	. . . . 5 ⊢ ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → (𝑦 ∈ (𝑇 ⊕ 𝑈) ↔ ∃𝑏 ∈ 𝑇 ∃𝑑 ∈ 𝑈 𝑦 = (𝑏(+g‘𝐺)𝑑)))
23	20, 22	anbi12d 644	. . . 4 ⊢ ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → ((𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈)) ↔ (∃𝑎 ∈ 𝑇 ∃𝑐 ∈ 𝑈 𝑥 = (𝑎(+g‘𝐺)𝑐) ∧ ∃𝑏 ∈ 𝑇 ∃𝑑 ∈ 𝑈 𝑦 = (𝑏(+g‘𝐺)𝑑))))
24		reeanv 3235	. . . . 5 ⊢ (∃𝑎 ∈ 𝑇 ∃𝑏 ∈ 𝑇 (∃𝑐 ∈ 𝑈 𝑥 = (𝑎(+g‘𝐺)𝑐) ∧ ∃𝑑 ∈ 𝑈 𝑦 = (𝑏(+g‘𝐺)𝑑)) ↔ (∃𝑎 ∈ 𝑇 ∃𝑐 ∈ 𝑈 𝑥 = (𝑎(+g‘𝐺)𝑐) ∧ ∃𝑏 ∈ 𝑇 ∃𝑑 ∈ 𝑈 𝑦 = (𝑏(+g‘𝐺)𝑑)))
25		reeanv 3235	. . . . . . 7 ⊢ (∃𝑐 ∈ 𝑈 ∃𝑑 ∈ 𝑈 (𝑥 = (𝑎(+g‘𝐺)𝑐) ∧ 𝑦 = (𝑏(+g‘𝐺)𝑑)) ↔ (∃𝑐 ∈ 𝑈 𝑥 = (𝑎(+g‘𝐺)𝑐) ∧ ∃𝑑 ∈ 𝑈 𝑦 = (𝑏(+g‘𝐺)𝑑)))
26	2	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ ((𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑇) ∧ (𝑐 ∈ 𝑈 ∧ 𝑑 ∈ 𝑈))) → 𝐺 ∈ Mnd)
27	5	adantr 486	. . . . . . . . . . . . 13 ⊢ (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ ((𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑇) ∧ (𝑐 ∈ 𝑈 ∧ 𝑑 ∈ 𝑈))) → 𝑇 ⊆ (Base‘𝐺))
28		simprll 791	. . . . . . . . . . . . 13 ⊢ (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ ((𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑇) ∧ (𝑐 ∈ 𝑈 ∧ 𝑑 ∈ 𝑈))) → 𝑎 ∈ 𝑇)
29	27, 28	sseldd 3932	. . . . . . . . . . . 12 ⊢ (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ ((𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑇) ∧ (𝑐 ∈ 𝑈 ∧ 𝑑 ∈ 𝑈))) → 𝑎 ∈ (Base‘𝐺))
30		simprlr 792	. . . . . . . . . . . . 13 ⊢ (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ ((𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑇) ∧ (𝑐 ∈ 𝑈 ∧ 𝑑 ∈ 𝑈))) → 𝑏 ∈ 𝑇)
31	27, 30	sseldd 3932	. . . . . . . . . . . 12 ⊢ (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ ((𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑇) ∧ (𝑐 ∈ 𝑈 ∧ 𝑑 ∈ 𝑈))) → 𝑏 ∈ (Base‘𝐺))
32	7	adantr 486	. . . . . . . . . . . . 13 ⊢ (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ ((𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑇) ∧ (𝑐 ∈ 𝑈 ∧ 𝑑 ∈ 𝑈))) → 𝑈 ⊆ (Base‘𝐺))
33		simprrl 793	. . . . . . . . . . . . 13 ⊢ (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ ((𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑇) ∧ (𝑐 ∈ 𝑈 ∧ 𝑑 ∈ 𝑈))) → 𝑐 ∈ 𝑈)
34	32, 33	sseldd 3932	. . . . . . . . . . . 12 ⊢ (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ ((𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑇) ∧ (𝑐 ∈ 𝑈 ∧ 𝑑 ∈ 𝑈))) → 𝑐 ∈ (Base‘𝐺))
35		simprrr 794	. . . . . . . . . . . . 13 ⊢ (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ ((𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑇) ∧ (𝑐 ∈ 𝑈 ∧ 𝑑 ∈ 𝑈))) → 𝑑 ∈ 𝑈)
36	32, 35	sseldd 3932	. . . . . . . . . . . 12 ⊢ (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ ((𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑇) ∧ (𝑐 ∈ 𝑈 ∧ 𝑑 ∈ 𝑈))) → 𝑑 ∈ (Base‘𝐺))
37		simpl3 1212	. . . . . . . . . . . . . 14 ⊢ (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ ((𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑇) ∧ (𝑐 ∈ 𝑈 ∧ 𝑑 ∈ 𝑈))) → 𝑇 ⊆ (𝑍‘𝑈))
38	37, 30	sseldd 3932	. . . . . . . . . . . . 13 ⊢ (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ ((𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑇) ∧ (𝑐 ∈ 𝑈 ∧ 𝑑 ∈ 𝑈))) → 𝑏 ∈ (𝑍‘𝑈))
39		lsmsubg.z	. . . . . . . . . . . . . 14 ⊢ 𝑍 = (Cntz‘𝐺)
40	18, 39	cntzi 19543	. . . . . . . . . . . . 13 ⊢ ((𝑏 ∈ (𝑍‘𝑈) ∧ 𝑐 ∈ 𝑈) → (𝑏(+g‘𝐺)𝑐) = (𝑐(+g‘𝐺)𝑏))
41	38, 33, 40	syl2anc 596	. . . . . . . . . . . 12 ⊢ (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ ((𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑇) ∧ (𝑐 ∈ 𝑈 ∧ 𝑑 ∈ 𝑈))) → (𝑏(+g‘𝐺)𝑐) = (𝑐(+g‘𝐺)𝑏))
42	3, 18, 26, 29, 31, 34, 36, 41	mnd4g 18938	. . . . . . . . . . 11 ⊢ (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ ((𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑇) ∧ (𝑐 ∈ 𝑈 ∧ 𝑑 ∈ 𝑈))) → ((𝑎(+g‘𝐺)𝑏)(+g‘𝐺)(𝑐(+g‘𝐺)𝑑)) = ((𝑎(+g‘𝐺)𝑐)(+g‘𝐺)(𝑏(+g‘𝐺)𝑑)))
43		simpl1 1210	. . . . . . . . . . . . 13 ⊢ (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ ((𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑇) ∧ (𝑐 ∈ 𝑈 ∧ 𝑑 ∈ 𝑈))) → 𝑇 ∈ (SubMnd‘𝐺))
44	18	submcl 19007	. . . . . . . . . . . . 13 ⊢ ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑇) → (𝑎(+g‘𝐺)𝑏) ∈ 𝑇)
45	43, 28, 30, 44	syl3anc 1398	. . . . . . . . . . . 12 ⊢ (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ ((𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑇) ∧ (𝑐 ∈ 𝑈 ∧ 𝑑 ∈ 𝑈))) → (𝑎(+g‘𝐺)𝑏) ∈ 𝑇)
46		simpl2 1211	. . . . . . . . . . . . 13 ⊢ (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ ((𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑇) ∧ (𝑐 ∈ 𝑈 ∧ 𝑑 ∈ 𝑈))) → 𝑈 ∈ (SubMnd‘𝐺))
47	18	submcl 19007	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑐 ∈ 𝑈 ∧ 𝑑 ∈ 𝑈) → (𝑐(+g‘𝐺)𝑑) ∈ 𝑈)
48	46, 33, 35, 47	syl3anc 1398	. . . . . . . . . . . 12 ⊢ (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ ((𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑇) ∧ (𝑐 ∈ 𝑈 ∧ 𝑑 ∈ 𝑈))) → (𝑐(+g‘𝐺)𝑑) ∈ 𝑈)
49	3, 18, 8	lsmelvalix 19855	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ Mnd ∧ 𝑇 ⊆ (Base‘𝐺) ∧ 𝑈 ⊆ (Base‘𝐺)) ∧ ((𝑎(+g‘𝐺)𝑏) ∈ 𝑇 ∧ (𝑐(+g‘𝐺)𝑑) ∈ 𝑈)) → ((𝑎(+g‘𝐺)𝑏)(+g‘𝐺)(𝑐(+g‘𝐺)𝑑)) ∈ (𝑇 ⊕ 𝑈))
50	26, 27, 32, 45, 48, 49	syl32anc 1405	. . . . . . . . . . 11 ⊢ (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ ((𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑇) ∧ (𝑐 ∈ 𝑈 ∧ 𝑑 ∈ 𝑈))) → ((𝑎(+g‘𝐺)𝑏)(+g‘𝐺)(𝑐(+g‘𝐺)𝑑)) ∈ (𝑇 ⊕ 𝑈))
51	42, 50	eqeltrrd 2862	. . . . . . . . . 10 ⊢ (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ ((𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑇) ∧ (𝑐 ∈ 𝑈 ∧ 𝑑 ∈ 𝑈))) → ((𝑎(+g‘𝐺)𝑐)(+g‘𝐺)(𝑏(+g‘𝐺)𝑑)) ∈ (𝑇 ⊕ 𝑈))
52		oveq12 7429	. . . . . . . . . . 11 ⊢ ((𝑥 = (𝑎(+g‘𝐺)𝑐) ∧ 𝑦 = (𝑏(+g‘𝐺)𝑑)) → (𝑥(+g‘𝐺)𝑦) = ((𝑎(+g‘𝐺)𝑐)(+g‘𝐺)(𝑏(+g‘𝐺)𝑑)))
53	52	eleq1d 2846	. . . . . . . . . 10 ⊢ ((𝑥 = (𝑎(+g‘𝐺)𝑐) ∧ 𝑦 = (𝑏(+g‘𝐺)𝑑)) → ((𝑥(+g‘𝐺)𝑦) ∈ (𝑇 ⊕ 𝑈) ↔ ((𝑎(+g‘𝐺)𝑐)(+g‘𝐺)(𝑏(+g‘𝐺)𝑑)) ∈ (𝑇 ⊕ 𝑈)))
54	51, 53	syl5ibrcom 250	. . . . . . . . 9 ⊢ (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ ((𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑇) ∧ (𝑐 ∈ 𝑈 ∧ 𝑑 ∈ 𝑈))) → ((𝑥 = (𝑎(+g‘𝐺)𝑐) ∧ 𝑦 = (𝑏(+g‘𝐺)𝑑)) → (𝑥(+g‘𝐺)𝑦) ∈ (𝑇 ⊕ 𝑈)))
55	54	anassrs 473	. . . . . . . 8 ⊢ ((((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ (𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑇)) ∧ (𝑐 ∈ 𝑈 ∧ 𝑑 ∈ 𝑈)) → ((𝑥 = (𝑎(+g‘𝐺)𝑐) ∧ 𝑦 = (𝑏(+g‘𝐺)𝑑)) → (𝑥(+g‘𝐺)𝑦) ∈ (𝑇 ⊕ 𝑈)))
56	55	rexlimdvva 3220	. . . . . . 7 ⊢ (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ (𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑇)) → (∃𝑐 ∈ 𝑈 ∃𝑑 ∈ 𝑈 (𝑥 = (𝑎(+g‘𝐺)𝑐) ∧ 𝑦 = (𝑏(+g‘𝐺)𝑑)) → (𝑥(+g‘𝐺)𝑦) ∈ (𝑇 ⊕ 𝑈)))
57	25, 56	biimtrrid 246	. . . . . 6 ⊢ (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ (𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑇)) → ((∃𝑐 ∈ 𝑈 𝑥 = (𝑎(+g‘𝐺)𝑐) ∧ ∃𝑑 ∈ 𝑈 𝑦 = (𝑏(+g‘𝐺)𝑑)) → (𝑥(+g‘𝐺)𝑦) ∈ (𝑇 ⊕ 𝑈)))
58	57	rexlimdvva 3220	. . . . 5 ⊢ ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → (∃𝑎 ∈ 𝑇 ∃𝑏 ∈ 𝑇 (∃𝑐 ∈ 𝑈 𝑥 = (𝑎(+g‘𝐺)𝑐) ∧ ∃𝑑 ∈ 𝑈 𝑦 = (𝑏(+g‘𝐺)𝑑)) → (𝑥(+g‘𝐺)𝑦) ∈ (𝑇 ⊕ 𝑈)))
59	24, 58	biimtrrid 246	. . . 4 ⊢ ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → ((∃𝑎 ∈ 𝑇 ∃𝑐 ∈ 𝑈 𝑥 = (𝑎(+g‘𝐺)𝑐) ∧ ∃𝑏 ∈ 𝑇 ∃𝑑 ∈ 𝑈 𝑦 = (𝑏(+g‘𝐺)𝑑)) → (𝑥(+g‘𝐺)𝑦) ∈ (𝑇 ⊕ 𝑈)))
60	23, 59	sylbid 243	. . 3 ⊢ ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → ((𝑥 ∈ (𝑇 ⊕ 𝑈) ∧ 𝑦 ∈ (𝑇 ⊕ 𝑈)) → (𝑥(+g‘𝐺)𝑦) ∈ (𝑇 ⊕ 𝑈)))
61	60	ralrimivv 3204	. 2 ⊢ ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → ∀𝑥 ∈ (𝑇 ⊕ 𝑈)∀𝑦 ∈ (𝑇 ⊕ 𝑈)(𝑥(+g‘𝐺)𝑦) ∈ (𝑇 ⊕ 𝑈))
62	3, 14, 18	issubm 18998	. . 3 ⊢ (𝐺 ∈ Mnd → ((𝑇 ⊕ 𝑈) ∈ (SubMnd‘𝐺) ↔ ((𝑇 ⊕ 𝑈) ⊆ (Base‘𝐺) ∧ (0g‘𝐺) ∈ (𝑇 ⊕ 𝑈) ∧ ∀𝑥 ∈ (𝑇 ⊕ 𝑈)∀𝑦 ∈ (𝑇 ⊕ 𝑈)(𝑥(+g‘𝐺)𝑦) ∈ (𝑇 ⊕ 𝑈))))
63	2, 62	syl 18	. 2 ⊢ ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → ((𝑇 ⊕ 𝑈) ∈ (SubMnd‘𝐺) ↔ ((𝑇 ⊕ 𝑈) ⊆ (Base‘𝐺) ∧ (0g‘𝐺) ∈ (𝑇 ⊕ 𝑈) ∧ ∀𝑥 ∈ (𝑇 ⊕ 𝑈)∀𝑦 ∈ (𝑇 ⊕ 𝑈)(𝑥(+g‘𝐺)𝑦) ∈ (𝑇 ⊕ 𝑈))))
64	10, 17, 61, 63	mpbir3and 1361	1 ⊢ ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → (𝑇 ⊕ 𝑈) ∈ (SubMnd‘𝐺))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3077  ∃wrex 3087   ⊆ wss 3899  ‘cfv 6538  (class class class)co 7420  Basecbs 17387  +_gcplusg 17428  0_gc0g 17610  Mndcmnd 18923  SubMndcsubmnd 18977  Cntzccntz 19529  LSSumclsm 19848

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-er 8717  df-en 8974  df-dom 8975  df-sdom 8976  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-nn 12336  df-2 12405  df-sets 17342  df-slot 17360  df-ndx 17372  df-base 17388  df-ress 17409  df-plusg 17441  df-0g 17612  df-mgm 18816  df-sgrp 18908  df-mnd 18924  df-submnd 18979  df-cntz 19531  df-lsm 19850

This theorem is used by:  lsmsubg  19868