Theorem lsmsubm 19435

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 18613	. . . 4 ⊢ (𝑇 ∈ (SubMnd‘𝐺) → 𝐺 ∈ Mnd)
2	1	3ad2ant1 1133	. . 3 ⊢ ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → 𝐺 ∈ Mnd)
3		eqid 2736	. . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺)
4	3	submss 18620	. . . 4 ⊢ (𝑇 ∈ (SubMnd‘𝐺) → 𝑇 ⊆ (Base‘𝐺))
5	4	3ad2ant1 1133	. . 3 ⊢ ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → 𝑇 ⊆ (Base‘𝐺))
6	3	submss 18620	. . . 4 ⊢ (𝑈 ∈ (SubMnd‘𝐺) → 𝑈 ⊆ (Base‘𝐺))
7	6	3ad2ant2 1134	. . 3 ⊢ ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → 𝑈 ⊆ (Base‘𝐺))
8		lsmsubg.p	. . . 4 ⊢ ⊕ = (LSSum‘𝐺)
9	3, 8	lsmssv 19425	. . 3 ⊢ ((𝐺 ∈ Mnd ∧ 𝑇 ⊆ (Base‘𝐺) ∧ 𝑈 ⊆ (Base‘𝐺)) → (𝑇 ⊕ 𝑈) ⊆ (Base‘𝐺))
10	2, 5, 7, 9	syl3anc 1371	. 2 ⊢ ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → (𝑇 ⊕ 𝑈) ⊆ (Base‘𝐺))
11		simp2 1137	. . . 4 ⊢ ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → 𝑈 ∈ (SubMnd‘𝐺))
12	3, 8	lsmub1x 19428	. . . 4 ⊢ ((𝑇 ⊆ (Base‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺)) → 𝑇 ⊆ (𝑇 ⊕ 𝑈))
13	5, 11, 12	syl2anc 584	. . 3 ⊢ ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → 𝑇 ⊆ (𝑇 ⊕ 𝑈))
14		eqid 2736	. . . . 5 ⊢ (0g‘𝐺) = (0g‘𝐺)
15	14	subm0cl 18622	. . . 4 ⊢ (𝑇 ∈ (SubMnd‘𝐺) → (0g‘𝐺) ∈ 𝑇)
16	15	3ad2ant1 1133	. . 3 ⊢ ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → (0g‘𝐺) ∈ 𝑇)
17	13, 16	sseldd 3945	. 2 ⊢ ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → (0g‘𝐺) ∈ (𝑇 ⊕ 𝑈))
18		eqid 2736	. . . . . . 7 ⊢ (+g‘𝐺) = (+g‘𝐺)
19	3, 18, 8	lsmelvalx 19422	. . . . . 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 1371	. . . . 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 19422	. . . . . 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 1371	. . . . 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 631	. . . 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 3217	. . . . 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 3217	. . . . . . 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 481	. . . . . . . . . . . 12 ⊢ (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ ((𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑇) ∧ (𝑐 ∈ 𝑈 ∧ 𝑑 ∈ 𝑈))) → 𝐺 ∈ Mnd)
27	5	adantr 481	. . . . . . . . . . . . 13 ⊢ (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ ((𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑇) ∧ (𝑐 ∈ 𝑈 ∧ 𝑑 ∈ 𝑈))) → 𝑇 ⊆ (Base‘𝐺))
28		simprll 777	. . . . . . . . . . . . 13 ⊢ (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ ((𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑇) ∧ (𝑐 ∈ 𝑈 ∧ 𝑑 ∈ 𝑈))) → 𝑎 ∈ 𝑇)
29	27, 28	sseldd 3945	. . . . . . . . . . . 12 ⊢ (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ ((𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑇) ∧ (𝑐 ∈ 𝑈 ∧ 𝑑 ∈ 𝑈))) → 𝑎 ∈ (Base‘𝐺))
30		simprlr 778	. . . . . . . . . . . . 13 ⊢ (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ ((𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑇) ∧ (𝑐 ∈ 𝑈 ∧ 𝑑 ∈ 𝑈))) → 𝑏 ∈ 𝑇)
31	27, 30	sseldd 3945	. . . . . . . . . . . 12 ⊢ (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ ((𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑇) ∧ (𝑐 ∈ 𝑈 ∧ 𝑑 ∈ 𝑈))) → 𝑏 ∈ (Base‘𝐺))
32	7	adantr 481	. . . . . . . . . . . . 13 ⊢ (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ ((𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑇) ∧ (𝑐 ∈ 𝑈 ∧ 𝑑 ∈ 𝑈))) → 𝑈 ⊆ (Base‘𝐺))
33		simprrl 779	. . . . . . . . . . . . 13 ⊢ (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ ((𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑇) ∧ (𝑐 ∈ 𝑈 ∧ 𝑑 ∈ 𝑈))) → 𝑐 ∈ 𝑈)
34	32, 33	sseldd 3945	. . . . . . . . . . . 12 ⊢ (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ ((𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑇) ∧ (𝑐 ∈ 𝑈 ∧ 𝑑 ∈ 𝑈))) → 𝑐 ∈ (Base‘𝐺))
35		simprrr 780	. . . . . . . . . . . . 13 ⊢ (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ ((𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑇) ∧ (𝑐 ∈ 𝑈 ∧ 𝑑 ∈ 𝑈))) → 𝑑 ∈ 𝑈)
36	32, 35	sseldd 3945	. . . . . . . . . . . 12 ⊢ (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ ((𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑇) ∧ (𝑐 ∈ 𝑈 ∧ 𝑑 ∈ 𝑈))) → 𝑑 ∈ (Base‘𝐺))
37		simpl3 1193	. . . . . . . . . . . . . 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 3945	. . . . . . . . . . . . 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 19109	. . . . . . . . . . . . 13 ⊢ ((𝑏 ∈ (𝑍‘𝑈) ∧ 𝑐 ∈ 𝑈) → (𝑏(+g‘𝐺)𝑐) = (𝑐(+g‘𝐺)𝑏))
41	38, 33, 40	syl2anc 584	. . . . . . . . . . . 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 18570	. . . . . . . . . . 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 1191	. . . . . . . . . . . . 13 ⊢ (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ ((𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑇) ∧ (𝑐 ∈ 𝑈 ∧ 𝑑 ∈ 𝑈))) → 𝑇 ∈ (SubMnd‘𝐺))
44	18	submcl 18623	. . . . . . . . . . . . 13 ⊢ ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑇) → (𝑎(+g‘𝐺)𝑏) ∈ 𝑇)
45	43, 28, 30, 44	syl3anc 1371	. . . . . . . . . . . 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 1192	. . . . . . . . . . . . 13 ⊢ (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ ((𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑇) ∧ (𝑐 ∈ 𝑈 ∧ 𝑑 ∈ 𝑈))) → 𝑈 ∈ (SubMnd‘𝐺))
47	18	submcl 18623	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑐 ∈ 𝑈 ∧ 𝑑 ∈ 𝑈) → (𝑐(+g‘𝐺)𝑑) ∈ 𝑈)
48	46, 33, 35, 47	syl3anc 1371	. . . . . . . . . . . 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 19423	. . . . . . . . . . . 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 1378	. . . . . . . . . . 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 2839	. . . . . . . . . 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 7366	. . . . . . . . . . 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 2822	. . . . . . . . . 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 246	. . . . . . . . 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 468	. . . . . . . 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 3205	. . . . . . 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 242	. . . . . 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 3205	. . . . 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 242	. . . 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 239	. . 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 3195	. 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 18614	. . 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 17	. 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 1342	1 ⊢ ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → (𝑇 ⊕ 𝑈) ∈ (SubMnd‘𝐺))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∀wral 3064  ∃wrex 3073   ⊆ wss 3910  ‘cfv 6496  (class class class)co 7357  Basecbs 17083  +_gcplusg 17133  0_gc0g 17321  Mndcmnd 18556  SubMndcsubmnd 18600  Cntzccntz 19095  LSSumclsm 19416

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-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  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-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  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-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  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-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-0g 17323  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-submnd 18602  df-cntz 19097  df-lsm 19418

This theorem is referenced by:  lsmsubg  19436