Theorem lsmsubm 18772

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 17961	. . . 4 ⊢ (𝑇 ∈ (SubMnd‘𝐺) → 𝐺 ∈ Mnd)
2	1	3ad2ant1 1129	. . 3 ⊢ ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → 𝐺 ∈ Mnd)
3		eqid 2821	. . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺)
4	3	submss 17968	. . . 4 ⊢ (𝑇 ∈ (SubMnd‘𝐺) → 𝑇 ⊆ (Base‘𝐺))
5	4	3ad2ant1 1129	. . 3 ⊢ ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → 𝑇 ⊆ (Base‘𝐺))
6	3	submss 17968	. . . 4 ⊢ (𝑈 ∈ (SubMnd‘𝐺) → 𝑈 ⊆ (Base‘𝐺))
7	6	3ad2ant2 1130	. . 3 ⊢ ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → 𝑈 ⊆ (Base‘𝐺))
8		lsmsubg.p	. . . 4 ⊢ ⊕ = (LSSum‘𝐺)
9	3, 8	lsmssv 18762	. . 3 ⊢ ((𝐺 ∈ Mnd ∧ 𝑇 ⊆ (Base‘𝐺) ∧ 𝑈 ⊆ (Base‘𝐺)) → (𝑇 ⊕ 𝑈) ⊆ (Base‘𝐺))
10	2, 5, 7, 9	syl3anc 1367	. 2 ⊢ ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → (𝑇 ⊕ 𝑈) ⊆ (Base‘𝐺))
11		simp2 1133	. . . 4 ⊢ ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → 𝑈 ∈ (SubMnd‘𝐺))
12	3, 8	lsmub1x 18765	. . . 4 ⊢ ((𝑇 ⊆ (Base‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺)) → 𝑇 ⊆ (𝑇 ⊕ 𝑈))
13	5, 11, 12	syl2anc 586	. . 3 ⊢ ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → 𝑇 ⊆ (𝑇 ⊕ 𝑈))
14		eqid 2821	. . . . 5 ⊢ (0g‘𝐺) = (0g‘𝐺)
15	14	subm0cl 17970	. . . 4 ⊢ (𝑇 ∈ (SubMnd‘𝐺) → (0g‘𝐺) ∈ 𝑇)
16	15	3ad2ant1 1129	. . 3 ⊢ ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → (0g‘𝐺) ∈ 𝑇)
17	13, 16	sseldd 3967	. 2 ⊢ ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → (0g‘𝐺) ∈ (𝑇 ⊕ 𝑈))
18		eqid 2821	. . . . . . 7 ⊢ (+g‘𝐺) = (+g‘𝐺)
19	3, 18, 8	lsmelvalx 18759	. . . . . 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 1367	. . . . 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 18759	. . . . . 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 1367	. . . . 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 3367	. . . . 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 3367	. . . . . . 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 483	. . . . . . . . . . . 12 ⊢ (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ ((𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑇) ∧ (𝑐 ∈ 𝑈 ∧ 𝑑 ∈ 𝑈))) → 𝐺 ∈ Mnd)
27	5	adantr 483	. . . . . . . . . . . . 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 3967	. . . . . . . . . . . 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 3967	. . . . . . . . . . . 12 ⊢ (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ ((𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑇) ∧ (𝑐 ∈ 𝑈 ∧ 𝑑 ∈ 𝑈))) → 𝑏 ∈ (Base‘𝐺))
32	7	adantr 483	. . . . . . . . . . . . 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 3967	. . . . . . . . . . . 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 3967	. . . . . . . . . . . 12 ⊢ (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ ((𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑇) ∧ (𝑐 ∈ 𝑈 ∧ 𝑑 ∈ 𝑈))) → 𝑑 ∈ (Base‘𝐺))
37		simpl3 1189	. . . . . . . . . . . . . 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 3967	. . . . . . . . . . . . 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 18453	. . . . . . . . . . . . 13 ⊢ ((𝑏 ∈ (𝑍‘𝑈) ∧ 𝑐 ∈ 𝑈) → (𝑏(+g‘𝐺)𝑐) = (𝑐(+g‘𝐺)𝑏))
41	38, 33, 40	syl2anc 586	. . . . . . . . . . . 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 17919	. . . . . . . . . . 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 1187	. . . . . . . . . . . . 13 ⊢ (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ ((𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑇) ∧ (𝑐 ∈ 𝑈 ∧ 𝑑 ∈ 𝑈))) → 𝑇 ∈ (SubMnd‘𝐺))
44	18	submcl 17971	. . . . . . . . . . . . 13 ⊢ ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑇) → (𝑎(+g‘𝐺)𝑏) ∈ 𝑇)
45	43, 28, 30, 44	syl3anc 1367	. . . . . . . . . . . 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 1188	. . . . . . . . . . . . 13 ⊢ (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) ∧ ((𝑎 ∈ 𝑇 ∧ 𝑏 ∈ 𝑇) ∧ (𝑐 ∈ 𝑈 ∧ 𝑑 ∈ 𝑈))) → 𝑈 ∈ (SubMnd‘𝐺))
47	18	submcl 17971	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑐 ∈ 𝑈 ∧ 𝑑 ∈ 𝑈) → (𝑐(+g‘𝐺)𝑑) ∈ 𝑈)
48	46, 33, 35, 47	syl3anc 1367	. . . . . . . . . . . 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 18760	. . . . . . . . . . . 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 1374	. . . . . . . . . . 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 2914	. . . . . . . . . 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 7159	. . . . . . . . . . 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 2897	. . . . . . . . . 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 249	. . . . . . . . 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 470	. . . . . . . 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 3294	. . . . . . 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	syl5bir 245	. . . . . 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 3294	. . . . 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	syl5bir 245	. . . 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 242	. . 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 3190	. 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 17962	. . 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 1338	1 ⊢ ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍‘𝑈)) → (𝑇 ⊕ 𝑈) ∈ (SubMnd‘𝐺))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110  ∀wral 3138  ∃wrex 3139   ⊆ wss 3935  ‘cfv 6349  (class class class)co 7150  Basecbs 16477  +_gcplusg 16559  0_gc0g 16707  Mndcmnd 17905  SubMndcsubmnd 17949  Cntzccntz 18439  LSSumclsm 18753

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-uni 4832  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7575  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-er 8283  df-en 8504  df-dom 8505  df-sdom 8506  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-nn 11633  df-2 11694  df-ndx 16480  df-slot 16481  df-base 16483  df-sets 16484  df-ress 16485  df-plusg 16572  df-0g 16709  df-mgm 17846  df-sgrp 17895  df-mnd 17906  df-submnd 17951  df-cntz 18441  df-lsm 18755

This theorem is referenced by:  lsmsubg  18773