MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  lsmsubm Structured version   Visualization version   GIF version

Theorem lsmsubm 19354
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 ⊆ (𝑍𝑈)) → (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.
StepHypRef Expression
1 submrcl 18538 . . . 4 (𝑇 ∈ (SubMnd‘𝐺) → 𝐺 ∈ Mnd)
213ad2ant1 1133 . . 3 ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → 𝐺 ∈ Mnd)
3 eqid 2737 . . . . 5 (Base‘𝐺) = (Base‘𝐺)
43submss 18545 . . . 4 (𝑇 ∈ (SubMnd‘𝐺) → 𝑇 ⊆ (Base‘𝐺))
543ad2ant1 1133 . . 3 ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → 𝑇 ⊆ (Base‘𝐺))
63submss 18545 . . . 4 (𝑈 ∈ (SubMnd‘𝐺) → 𝑈 ⊆ (Base‘𝐺))
763ad2ant2 1134 . . 3 ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → 𝑈 ⊆ (Base‘𝐺))
8 lsmsubg.p . . . 4 = (LSSum‘𝐺)
93, 8lsmssv 19344 . . 3 ((𝐺 ∈ Mnd ∧ 𝑇 ⊆ (Base‘𝐺) ∧ 𝑈 ⊆ (Base‘𝐺)) → (𝑇 𝑈) ⊆ (Base‘𝐺))
102, 5, 7, 9syl3anc 1371 . 2 ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → (𝑇 𝑈) ⊆ (Base‘𝐺))
11 simp2 1137 . . . 4 ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → 𝑈 ∈ (SubMnd‘𝐺))
123, 8lsmub1x 19347 . . . 4 ((𝑇 ⊆ (Base‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺)) → 𝑇 ⊆ (𝑇 𝑈))
135, 11, 12syl2anc 585 . . 3 ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → 𝑇 ⊆ (𝑇 𝑈))
14 eqid 2737 . . . . 5 (0g𝐺) = (0g𝐺)
1514subm0cl 18547 . . . 4 (𝑇 ∈ (SubMnd‘𝐺) → (0g𝐺) ∈ 𝑇)
16153ad2ant1 1133 . . 3 ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → (0g𝐺) ∈ 𝑇)
1713, 16sseldd 3936 . 2 ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → (0g𝐺) ∈ (𝑇 𝑈))
18 eqid 2737 . . . . . . 7 (+g𝐺) = (+g𝐺)
193, 18, 8lsmelvalx 19341 . . . . . 6 ((𝐺 ∈ Mnd ∧ 𝑇 ⊆ (Base‘𝐺) ∧ 𝑈 ⊆ (Base‘𝐺)) → (𝑥 ∈ (𝑇 𝑈) ↔ ∃𝑎𝑇𝑐𝑈 𝑥 = (𝑎(+g𝐺)𝑐)))
202, 5, 7, 19syl3anc 1371 . . . . 5 ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → (𝑥 ∈ (𝑇 𝑈) ↔ ∃𝑎𝑇𝑐𝑈 𝑥 = (𝑎(+g𝐺)𝑐)))
213, 18, 8lsmelvalx 19341 . . . . . 6 ((𝐺 ∈ Mnd ∧ 𝑇 ⊆ (Base‘𝐺) ∧ 𝑈 ⊆ (Base‘𝐺)) → (𝑦 ∈ (𝑇 𝑈) ↔ ∃𝑏𝑇𝑑𝑈 𝑦 = (𝑏(+g𝐺)𝑑)))
222, 5, 7, 21syl3anc 1371 . . . . 5 ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → (𝑦 ∈ (𝑇 𝑈) ↔ ∃𝑏𝑇𝑑𝑈 𝑦 = (𝑏(+g𝐺)𝑑)))
2320, 22anbi12d 632 . . . 4 ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → ((𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈)) ↔ (∃𝑎𝑇𝑐𝑈 𝑥 = (𝑎(+g𝐺)𝑐) ∧ ∃𝑏𝑇𝑑𝑈 𝑦 = (𝑏(+g𝐺)𝑑))))
24 reeanv 3214 . . . . 5 (∃𝑎𝑇𝑏𝑇 (∃𝑐𝑈 𝑥 = (𝑎(+g𝐺)𝑐) ∧ ∃𝑑𝑈 𝑦 = (𝑏(+g𝐺)𝑑)) ↔ (∃𝑎𝑇𝑐𝑈 𝑥 = (𝑎(+g𝐺)𝑐) ∧ ∃𝑏𝑇𝑑𝑈 𝑦 = (𝑏(+g𝐺)𝑑)))
25 reeanv 3214 . . . . . . 7 (∃𝑐𝑈𝑑𝑈 (𝑥 = (𝑎(+g𝐺)𝑐) ∧ 𝑦 = (𝑏(+g𝐺)𝑑)) ↔ (∃𝑐𝑈 𝑥 = (𝑎(+g𝐺)𝑐) ∧ ∃𝑑𝑈 𝑦 = (𝑏(+g𝐺)𝑑)))
262adantr 482 . . . . . . . . . . . 12 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → 𝐺 ∈ Mnd)
275adantr 482 . . . . . . . . . . . . 13 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → 𝑇 ⊆ (Base‘𝐺))
28 simprll 777 . . . . . . . . . . . . 13 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → 𝑎𝑇)
2927, 28sseldd 3936 . . . . . . . . . . . 12 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → 𝑎 ∈ (Base‘𝐺))
30 simprlr 778 . . . . . . . . . . . . 13 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → 𝑏𝑇)
3127, 30sseldd 3936 . . . . . . . . . . . 12 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → 𝑏 ∈ (Base‘𝐺))
327adantr 482 . . . . . . . . . . . . 13 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → 𝑈 ⊆ (Base‘𝐺))
33 simprrl 779 . . . . . . . . . . . . 13 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → 𝑐𝑈)
3432, 33sseldd 3936 . . . . . . . . . . . 12 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → 𝑐 ∈ (Base‘𝐺))
35 simprrr 780 . . . . . . . . . . . . 13 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → 𝑑𝑈)
3632, 35sseldd 3936 . . . . . . . . . . . 12 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → 𝑑 ∈ (Base‘𝐺))
37 simpl3 1193 . . . . . . . . . . . . . 14 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → 𝑇 ⊆ (𝑍𝑈))
3837, 30sseldd 3936 . . . . . . . . . . . . 13 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → 𝑏 ∈ (𝑍𝑈))
39 lsmsubg.z . . . . . . . . . . . . . 14 𝑍 = (Cntz‘𝐺)
4018, 39cntzi 19031 . . . . . . . . . . . . 13 ((𝑏 ∈ (𝑍𝑈) ∧ 𝑐𝑈) → (𝑏(+g𝐺)𝑐) = (𝑐(+g𝐺)𝑏))
4138, 33, 40syl2anc 585 . . . . . . . . . . . 12 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → (𝑏(+g𝐺)𝑐) = (𝑐(+g𝐺)𝑏))
423, 18, 26, 29, 31, 34, 36, 41mnd4g 18496 . . . . . . . . . . 11 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → ((𝑎(+g𝐺)𝑏)(+g𝐺)(𝑐(+g𝐺)𝑑)) = ((𝑎(+g𝐺)𝑐)(+g𝐺)(𝑏(+g𝐺)𝑑)))
43 simpl1 1191 . . . . . . . . . . . . 13 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → 𝑇 ∈ (SubMnd‘𝐺))
4418submcl 18548 . . . . . . . . . . . . 13 ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑎𝑇𝑏𝑇) → (𝑎(+g𝐺)𝑏) ∈ 𝑇)
4543, 28, 30, 44syl3anc 1371 . . . . . . . . . . . 12 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → (𝑎(+g𝐺)𝑏) ∈ 𝑇)
46 simpl2 1192 . . . . . . . . . . . . 13 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → 𝑈 ∈ (SubMnd‘𝐺))
4718submcl 18548 . . . . . . . . . . . . 13 ((𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑐𝑈𝑑𝑈) → (𝑐(+g𝐺)𝑑) ∈ 𝑈)
4846, 33, 35, 47syl3anc 1371 . . . . . . . . . . . 12 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → (𝑐(+g𝐺)𝑑) ∈ 𝑈)
493, 18, 8lsmelvalix 19342 . . . . . . . . . . . 12 (((𝐺 ∈ Mnd ∧ 𝑇 ⊆ (Base‘𝐺) ∧ 𝑈 ⊆ (Base‘𝐺)) ∧ ((𝑎(+g𝐺)𝑏) ∈ 𝑇 ∧ (𝑐(+g𝐺)𝑑) ∈ 𝑈)) → ((𝑎(+g𝐺)𝑏)(+g𝐺)(𝑐(+g𝐺)𝑑)) ∈ (𝑇 𝑈))
5026, 27, 32, 45, 48, 49syl32anc 1378 . . . . . . . . . . 11 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → ((𝑎(+g𝐺)𝑏)(+g𝐺)(𝑐(+g𝐺)𝑑)) ∈ (𝑇 𝑈))
5142, 50eqeltrrd 2839 . . . . . . . . . 10 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → ((𝑎(+g𝐺)𝑐)(+g𝐺)(𝑏(+g𝐺)𝑑)) ∈ (𝑇 𝑈))
52 oveq12 7350 . . . . . . . . . . 11 ((𝑥 = (𝑎(+g𝐺)𝑐) ∧ 𝑦 = (𝑏(+g𝐺)𝑑)) → (𝑥(+g𝐺)𝑦) = ((𝑎(+g𝐺)𝑐)(+g𝐺)(𝑏(+g𝐺)𝑑)))
5352eleq1d 2822 . . . . . . . . . 10 ((𝑥 = (𝑎(+g𝐺)𝑐) ∧ 𝑦 = (𝑏(+g𝐺)𝑑)) → ((𝑥(+g𝐺)𝑦) ∈ (𝑇 𝑈) ↔ ((𝑎(+g𝐺)𝑐)(+g𝐺)(𝑏(+g𝐺)𝑑)) ∈ (𝑇 𝑈)))
5451, 53syl5ibrcom 247 . . . . . . . . 9 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → ((𝑥 = (𝑎(+g𝐺)𝑐) ∧ 𝑦 = (𝑏(+g𝐺)𝑑)) → (𝑥(+g𝐺)𝑦) ∈ (𝑇 𝑈)))
5554anassrs 469 . . . . . . . 8 ((((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ (𝑎𝑇𝑏𝑇)) ∧ (𝑐𝑈𝑑𝑈)) → ((𝑥 = (𝑎(+g𝐺)𝑐) ∧ 𝑦 = (𝑏(+g𝐺)𝑑)) → (𝑥(+g𝐺)𝑦) ∈ (𝑇 𝑈)))
5655rexlimdvva 3202 . . . . . . 7 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ (𝑎𝑇𝑏𝑇)) → (∃𝑐𝑈𝑑𝑈 (𝑥 = (𝑎(+g𝐺)𝑐) ∧ 𝑦 = (𝑏(+g𝐺)𝑑)) → (𝑥(+g𝐺)𝑦) ∈ (𝑇 𝑈)))
5725, 56syl5bir 243 . . . . . 6 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ (𝑎𝑇𝑏𝑇)) → ((∃𝑐𝑈 𝑥 = (𝑎(+g𝐺)𝑐) ∧ ∃𝑑𝑈 𝑦 = (𝑏(+g𝐺)𝑑)) → (𝑥(+g𝐺)𝑦) ∈ (𝑇 𝑈)))
5857rexlimdvva 3202 . . . . 5 ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → (∃𝑎𝑇𝑏𝑇 (∃𝑐𝑈 𝑥 = (𝑎(+g𝐺)𝑐) ∧ ∃𝑑𝑈 𝑦 = (𝑏(+g𝐺)𝑑)) → (𝑥(+g𝐺)𝑦) ∈ (𝑇 𝑈)))
5924, 58syl5bir 243 . . . 4 ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → ((∃𝑎𝑇𝑐𝑈 𝑥 = (𝑎(+g𝐺)𝑐) ∧ ∃𝑏𝑇𝑑𝑈 𝑦 = (𝑏(+g𝐺)𝑑)) → (𝑥(+g𝐺)𝑦) ∈ (𝑇 𝑈)))
6023, 59sylbid 239 . . 3 ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → ((𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈)) → (𝑥(+g𝐺)𝑦) ∈ (𝑇 𝑈)))
6160ralrimivv 3192 . 2 ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → ∀𝑥 ∈ (𝑇 𝑈)∀𝑦 ∈ (𝑇 𝑈)(𝑥(+g𝐺)𝑦) ∈ (𝑇 𝑈))
623, 14, 18issubm 18539 . . 3 (𝐺 ∈ Mnd → ((𝑇 𝑈) ∈ (SubMnd‘𝐺) ↔ ((𝑇 𝑈) ⊆ (Base‘𝐺) ∧ (0g𝐺) ∈ (𝑇 𝑈) ∧ ∀𝑥 ∈ (𝑇 𝑈)∀𝑦 ∈ (𝑇 𝑈)(𝑥(+g𝐺)𝑦) ∈ (𝑇 𝑈))))
632, 62syl 17 . 2 ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → ((𝑇 𝑈) ∈ (SubMnd‘𝐺) ↔ ((𝑇 𝑈) ⊆ (Base‘𝐺) ∧ (0g𝐺) ∈ (𝑇 𝑈) ∧ ∀𝑥 ∈ (𝑇 𝑈)∀𝑦 ∈ (𝑇 𝑈)(𝑥(+g𝐺)𝑦) ∈ (𝑇 𝑈))))
6410, 17, 61, 63mpbir3and 1342 1 ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → (𝑇 𝑈) ∈ (SubMnd‘𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1087   = wceq 1541  wcel 2106  wral 3062  wrex 3071  wss 3901  cfv 6483  (class class class)co 7341  Basecbs 17009  +gcplusg 17059  0gc0g 17247  Mndcmnd 18482  SubMndcsubmnd 18526  Cntzccntz 19017  LSSumclsm 19335
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 2708  ax-rep 5233  ax-sep 5247  ax-nul 5254  ax-pow 5312  ax-pr 5376  ax-un 7654  ax-cnex 11032  ax-resscn 11033  ax-1cn 11034  ax-icn 11035  ax-addcl 11036  ax-addrcl 11037  ax-mulcl 11038  ax-mulrcl 11039  ax-mulcom 11040  ax-addass 11041  ax-mulass 11042  ax-distr 11043  ax-i2m1 11044  ax-1ne0 11045  ax-1rid 11046  ax-rnegex 11047  ax-rrecex 11048  ax-cnre 11049  ax-pre-lttri 11050  ax-pre-lttrn 11051  ax-pre-ltadd 11052  ax-pre-mulgt0 11053
This theorem depends on definitions:  df-bi 206  df-an 398  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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3350  df-reu 3351  df-rab 3405  df-v 3444  df-sbc 3731  df-csb 3847  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3920  df-nul 4274  df-if 4478  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4857  df-iun 4947  df-br 5097  df-opab 5159  df-mpt 5180  df-tr 5214  df-id 5522  df-eprel 5528  df-po 5536  df-so 5537  df-fr 5579  df-we 5581  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6242  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6435  df-fun 6485  df-fn 6486  df-f 6487  df-f1 6488  df-fo 6489  df-f1o 6490  df-fv 6491  df-riota 7297  df-ov 7344  df-oprab 7345  df-mpo 7346  df-om 7785  df-1st 7903  df-2nd 7904  df-frecs 8171  df-wrecs 8202  df-recs 8276  df-rdg 8315  df-er 8573  df-en 8809  df-dom 8810  df-sdom 8811  df-pnf 11116  df-mnf 11117  df-xr 11118  df-ltxr 11119  df-le 11120  df-sub 11312  df-neg 11313  df-nn 12079  df-2 12141  df-sets 16962  df-slot 16980  df-ndx 16992  df-base 17010  df-ress 17039  df-plusg 17072  df-0g 17249  df-mgm 18423  df-sgrp 18472  df-mnd 18483  df-submnd 18528  df-cntz 19019  df-lsm 19337
This theorem is referenced by:  lsmsubg  19355
  Copyright terms: Public domain W3C validator