Theorem lsmsubm 19590

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 18736	. . . 4 ⊢ (𝑇 ∈ (SubMnd‘𝐺) → 𝐺 ∈ Mnd)
2	1	3ad2ant1 1133	. . 3 ⊢ ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → 𝐺 ∈ Mnd)
3		eqid 2730	. . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺)
4	3	submss 18743	. . . 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 18743	. . . 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 19580	. . 3 ⊢ ((𝐺 ∈ Mnd ∧ 𝑇 ⊆ (Base‘𝐺) ∧ 𝑈 ⊆ (Base‘𝐺)) → (𝑇 ⊕ 𝑈) ⊆ (Base‘𝐺))
10	2, 5, 7, 9	syl3anc 1373	. 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 19583	. . . 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 2730	. . . . 5 ⊢ (0g‘𝐺) = (0g‘𝐺)
15	14	subm0cl 18745	. . . 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 3950	. 2 ⊢ ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → (0g‘𝐺) ∈ (𝑇 ⊕ 𝑈))
18		eqid 2730	. . . . . . 7 ⊢ (+g‘𝐺) = (+g‘𝐺)
19	3, 18, 8	lsmelvalx 19577	. . . . . 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 1373	. . . . 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 19577	. . . . . 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 1373	. . . . 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 632	. . . 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 3210	. . . . 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 3210	. . . . . . 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 480	. . . . . . . . . . . 12 ⊢ (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ ((𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑇) ∧ (𝑐 ∈ 𝑈 ∧ 𝑑 ∈ 𝑈))) → 𝐺 ∈ Mnd)
27	5	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ ((𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑇) ∧ (𝑐 ∈ 𝑈 ∧ 𝑑 ∈ 𝑈))) → 𝑇 ⊆ (Base‘𝐺))
28		simprll 778	. . . . . . . . . . . . 13 ⊢ (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ ((𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑇) ∧ (𝑐 ∈ 𝑈 ∧ 𝑑 ∈ 𝑈))) → 𝑎 ∈ 𝑇)
29	27, 28	sseldd 3950	. . . . . . . . . . . 12 ⊢ (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ ((𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑇) ∧ (𝑐 ∈ 𝑈 ∧ 𝑑 ∈ 𝑈))) → 𝑎 ∈ (Base‘𝐺))
30		simprlr 779	. . . . . . . . . . . . 13 ⊢ (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ ((𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑇) ∧ (𝑐 ∈ 𝑈 ∧ 𝑑 ∈ 𝑈))) → 𝑏 ∈ 𝑇)
31	27, 30	sseldd 3950	. . . . . . . . . . . 12 ⊢ (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ ((𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑇) ∧ (𝑐 ∈ 𝑈 ∧ 𝑑 ∈ 𝑈))) → 𝑏 ∈ (Base‘𝐺))
32	7	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ ((𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑇) ∧ (𝑐 ∈ 𝑈 ∧ 𝑑 ∈ 𝑈))) → 𝑈 ⊆ (Base‘𝐺))
33		simprrl 780	. . . . . . . . . . . . 13 ⊢ (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ ((𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑇) ∧ (𝑐 ∈ 𝑈 ∧ 𝑑 ∈ 𝑈))) → 𝑐 ∈ 𝑈)
34	32, 33	sseldd 3950	. . . . . . . . . . . 12 ⊢ (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ ((𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑇) ∧ (𝑐 ∈ 𝑈 ∧ 𝑑 ∈ 𝑈))) → 𝑐 ∈ (Base‘𝐺))
35		simprrr 781	. . . . . . . . . . . . 13 ⊢ (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ ((𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑇) ∧ (𝑐 ∈ 𝑈 ∧ 𝑑 ∈ 𝑈))) → 𝑑 ∈ 𝑈)
36	32, 35	sseldd 3950	. . . . . . . . . . . 12 ⊢ (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ ((𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑇) ∧ (𝑐 ∈ 𝑈 ∧ 𝑑 ∈ 𝑈))) → 𝑑 ∈ (Base‘𝐺))
37		simpl3 1194	. . . . . . . . . . . . . 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 3950	. . . . . . . . . . . . 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 19268	. . . . . . . . . . . . 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 18682	. . . . . . . . . . 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 1192	. . . . . . . . . . . . 13 ⊢ (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ ((𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑇) ∧ (𝑐 ∈ 𝑈 ∧ 𝑑 ∈ 𝑈))) → 𝑇 ∈ (SubMnd‘𝐺))
44	18	submcl 18746	. . . . . . . . . . . . 13 ⊢ ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑇) → (𝑎(+g‘𝐺)𝑏) ∈ 𝑇)
45	43, 28, 30, 44	syl3anc 1373	. . . . . . . . . . . 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 1193	. . . . . . . . . . . . 13 ⊢ (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ ((𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑇) ∧ (𝑐 ∈ 𝑈 ∧ 𝑑 ∈ 𝑈))) → 𝑈 ∈ (SubMnd‘𝐺))
47	18	submcl 18746	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑐 ∈ 𝑈 ∧ 𝑑 ∈ 𝑈) → (𝑐(+g‘𝐺)𝑑) ∈ 𝑈)
48	46, 33, 35, 47	syl3anc 1373	. . . . . . . . . . . 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 19578	. . . . . . . . . . . 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 1380	. . . . . . . . . . 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 2830	. . . . . . . . . 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 7399	. . . . . . . . . . 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 2814	. . . . . . . . . 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 247	. . . . . . . . 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 467	. . . . . . . 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 3195	. . . . . . 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 243	. . . . . 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 3195	. . . . 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 243	. . . 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 240	. . 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 3179	. 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 18737	. . 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 1343	1 ⊢ ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → (𝑇 ⊕ 𝑈) ∈ (SubMnd‘𝐺))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3045  ∃wrex 3054   ⊆ wss 3917  ‘cfv 6514  (class class class)co 7390  Basecbs 17186  +_gcplusg 17227  0_gc0g 17409  Mndcmnd 18668  SubMndcsubmnd 18716  Cntzccntz 19254  LSSumclsm 19571

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-er 8674  df-en 8922  df-dom 8923  df-sdom 8924  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-nn 12194  df-2 12256  df-sets 17141  df-slot 17159  df-ndx 17171  df-base 17187  df-ress 17208  df-plusg 17240  df-0g 17411  df-mgm 18574  df-sgrp 18653  df-mnd 18669  df-submnd 18718  df-cntz 19256  df-lsm 19573

This theorem is referenced by:  lsmsubg  19591