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Theorem lsmsubm 18189
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 17468 . . . 4 (𝑇 ∈ (SubMnd‘𝐺) → 𝐺 ∈ Mnd)
213ad2ant1 1125 . . 3 ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → 𝐺 ∈ Mnd)
3 eqid 2724 . . . . 5 (Base‘𝐺) = (Base‘𝐺)
43submss 17472 . . . 4 (𝑇 ∈ (SubMnd‘𝐺) → 𝑇 ⊆ (Base‘𝐺))
543ad2ant1 1125 . . 3 ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → 𝑇 ⊆ (Base‘𝐺))
63submss 17472 . . . 4 (𝑈 ∈ (SubMnd‘𝐺) → 𝑈 ⊆ (Base‘𝐺))
763ad2ant2 1126 . . 3 ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → 𝑈 ⊆ (Base‘𝐺))
8 lsmsubg.p . . . 4 = (LSSum‘𝐺)
93, 8lsmssv 18179 . . 3 ((𝐺 ∈ Mnd ∧ 𝑇 ⊆ (Base‘𝐺) ∧ 𝑈 ⊆ (Base‘𝐺)) → (𝑇 𝑈) ⊆ (Base‘𝐺))
102, 5, 7, 9syl3anc 1439 . 2 ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → (𝑇 𝑈) ⊆ (Base‘𝐺))
11 simp2 1129 . . . 4 ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → 𝑈 ∈ (SubMnd‘𝐺))
123, 8lsmub1x 18182 . . . 4 ((𝑇 ⊆ (Base‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺)) → 𝑇 ⊆ (𝑇 𝑈))
135, 11, 12syl2anc 696 . . 3 ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → 𝑇 ⊆ (𝑇 𝑈))
14 eqid 2724 . . . . 5 (0g𝐺) = (0g𝐺)
1514subm0cl 17474 . . . 4 (𝑇 ∈ (SubMnd‘𝐺) → (0g𝐺) ∈ 𝑇)
16153ad2ant1 1125 . . 3 ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → (0g𝐺) ∈ 𝑇)
1713, 16sseldd 3710 . 2 ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → (0g𝐺) ∈ (𝑇 𝑈))
18 eqid 2724 . . . . . . 7 (+g𝐺) = (+g𝐺)
193, 18, 8lsmelvalx 18176 . . . . . 6 ((𝐺 ∈ Mnd ∧ 𝑇 ⊆ (Base‘𝐺) ∧ 𝑈 ⊆ (Base‘𝐺)) → (𝑥 ∈ (𝑇 𝑈) ↔ ∃𝑎𝑇𝑐𝑈 𝑥 = (𝑎(+g𝐺)𝑐)))
202, 5, 7, 19syl3anc 1439 . . . . 5 ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → (𝑥 ∈ (𝑇 𝑈) ↔ ∃𝑎𝑇𝑐𝑈 𝑥 = (𝑎(+g𝐺)𝑐)))
213, 18, 8lsmelvalx 18176 . . . . . 6 ((𝐺 ∈ Mnd ∧ 𝑇 ⊆ (Base‘𝐺) ∧ 𝑈 ⊆ (Base‘𝐺)) → (𝑦 ∈ (𝑇 𝑈) ↔ ∃𝑏𝑇𝑑𝑈 𝑦 = (𝑏(+g𝐺)𝑑)))
222, 5, 7, 21syl3anc 1439 . . . . 5 ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → (𝑦 ∈ (𝑇 𝑈) ↔ ∃𝑏𝑇𝑑𝑈 𝑦 = (𝑏(+g𝐺)𝑑)))
2320, 22anbi12d 749 . . . 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 3209 . . . . 5 (∃𝑎𝑇𝑏𝑇 (∃𝑐𝑈 𝑥 = (𝑎(+g𝐺)𝑐) ∧ ∃𝑑𝑈 𝑦 = (𝑏(+g𝐺)𝑑)) ↔ (∃𝑎𝑇𝑐𝑈 𝑥 = (𝑎(+g𝐺)𝑐) ∧ ∃𝑏𝑇𝑑𝑈 𝑦 = (𝑏(+g𝐺)𝑑)))
25 reeanv 3209 . . . . . . 7 (∃𝑐𝑈𝑑𝑈 (𝑥 = (𝑎(+g𝐺)𝑐) ∧ 𝑦 = (𝑏(+g𝐺)𝑑)) ↔ (∃𝑐𝑈 𝑥 = (𝑎(+g𝐺)𝑐) ∧ ∃𝑑𝑈 𝑦 = (𝑏(+g𝐺)𝑑)))
262adantr 472 . . . . . . . . . . . 12 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → 𝐺 ∈ Mnd)
275adantr 472 . . . . . . . . . . . . 13 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → 𝑇 ⊆ (Base‘𝐺))
28 simprll 821 . . . . . . . . . . . . 13 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → 𝑎𝑇)
2927, 28sseldd 3710 . . . . . . . . . . . 12 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → 𝑎 ∈ (Base‘𝐺))
30 simprlr 822 . . . . . . . . . . . . 13 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → 𝑏𝑇)
3127, 30sseldd 3710 . . . . . . . . . . . 12 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → 𝑏 ∈ (Base‘𝐺))
327adantr 472 . . . . . . . . . . . . 13 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → 𝑈 ⊆ (Base‘𝐺))
33 simprrl 823 . . . . . . . . . . . . 13 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → 𝑐𝑈)
3432, 33sseldd 3710 . . . . . . . . . . . 12 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → 𝑐 ∈ (Base‘𝐺))
35 simprrr 824 . . . . . . . . . . . . 13 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → 𝑑𝑈)
3632, 35sseldd 3710 . . . . . . . . . . . 12 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → 𝑑 ∈ (Base‘𝐺))
37 simpl3 1208 . . . . . . . . . . . . . 14 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → 𝑇 ⊆ (𝑍𝑈))
3837, 30sseldd 3710 . . . . . . . . . . . . 13 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → 𝑏 ∈ (𝑍𝑈))
39 lsmsubg.z . . . . . . . . . . . . . 14 𝑍 = (Cntz‘𝐺)
4018, 39cntzi 17883 . . . . . . . . . . . . 13 ((𝑏 ∈ (𝑍𝑈) ∧ 𝑐𝑈) → (𝑏(+g𝐺)𝑐) = (𝑐(+g𝐺)𝑏))
4138, 33, 40syl2anc 696 . . . . . . . . . . . 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 17429 . . . . . . . . . . 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 1204 . . . . . . . . . . . . 13 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → 𝑇 ∈ (SubMnd‘𝐺))
4418submcl 17475 . . . . . . . . . . . . 13 ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑎𝑇𝑏𝑇) → (𝑎(+g𝐺)𝑏) ∈ 𝑇)
4543, 28, 30, 44syl3anc 1439 . . . . . . . . . . . 12 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → (𝑎(+g𝐺)𝑏) ∈ 𝑇)
46 simpl2 1206 . . . . . . . . . . . . 13 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → 𝑈 ∈ (SubMnd‘𝐺))
4718submcl 17475 . . . . . . . . . . . . 13 ((𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑐𝑈𝑑𝑈) → (𝑐(+g𝐺)𝑑) ∈ 𝑈)
4846, 33, 35, 47syl3anc 1439 . . . . . . . . . . . 12 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → (𝑐(+g𝐺)𝑑) ∈ 𝑈)
493, 18, 8lsmelvalix 18177 . . . . . . . . . . . 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 1447 . . . . . . . . . . 11 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → ((𝑎(+g𝐺)𝑏)(+g𝐺)(𝑐(+g𝐺)𝑑)) ∈ (𝑇 𝑈))
5142, 50eqeltrrd 2804 . . . . . . . . . 10 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → ((𝑎(+g𝐺)𝑐)(+g𝐺)(𝑏(+g𝐺)𝑑)) ∈ (𝑇 𝑈))
52 oveq12 6774 . . . . . . . . . . 11 ((𝑥 = (𝑎(+g𝐺)𝑐) ∧ 𝑦 = (𝑏(+g𝐺)𝑑)) → (𝑥(+g𝐺)𝑦) = ((𝑎(+g𝐺)𝑐)(+g𝐺)(𝑏(+g𝐺)𝑑)))
5352eleq1d 2788 . . . . . . . . . 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 237 . . . . . . . . 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 683 . . . . . . . 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 3140 . . . . . . 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 233 . . . . . 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 3140 . . . . 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 233 . . . 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 230 . . 3 ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → ((𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈)) → (𝑥(+g𝐺)𝑦) ∈ (𝑇 𝑈)))
6160ralrimivv 3072 . 2 ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → ∀𝑥 ∈ (𝑇 𝑈)∀𝑦 ∈ (𝑇 𝑈)(𝑥(+g𝐺)𝑦) ∈ (𝑇 𝑈))
623, 14, 18issubm 17469 . . 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 1382 1 ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → (𝑇 𝑈) ∈ (SubMnd‘𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1596  wcel 2103  wral 3014  wrex 3015  wss 3680  cfv 6001  (class class class)co 6765  Basecbs 15980  +gcplusg 16064  0gc0g 16223  Mndcmnd 17416  SubMndcsubmnd 17456  Cntzccntz 17869  LSSumclsm 18170
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066  ax-cnex 10105  ax-resscn 10106  ax-1cn 10107  ax-icn 10108  ax-addcl 10109  ax-addrcl 10110  ax-mulcl 10111  ax-mulrcl 10112  ax-mulcom 10113  ax-addass 10114  ax-mulass 10115  ax-distr 10116  ax-i2m1 10117  ax-1ne0 10118  ax-1rid 10119  ax-rnegex 10120  ax-rrecex 10121  ax-cnre 10122  ax-pre-lttri 10123  ax-pre-lttrn 10124  ax-pre-ltadd 10125  ax-pre-mulgt0 10126
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-nel 3000  df-ral 3019  df-rex 3020  df-reu 3021  df-rmo 3022  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-pss 3696  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-tp 4290  df-op 4292  df-uni 4545  df-iun 4630  df-br 4761  df-opab 4821  df-mpt 4838  df-tr 4861  df-id 5128  df-eprel 5133  df-po 5139  df-so 5140  df-fr 5177  df-we 5179  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-pred 5793  df-ord 5839  df-on 5840  df-lim 5841  df-suc 5842  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-riota 6726  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-om 7183  df-1st 7285  df-2nd 7286  df-wrecs 7527  df-recs 7588  df-rdg 7626  df-er 7862  df-en 8073  df-dom 8074  df-sdom 8075  df-pnf 10189  df-mnf 10190  df-xr 10191  df-ltxr 10192  df-le 10193  df-sub 10381  df-neg 10382  df-nn 11134  df-2 11192  df-ndx 15983  df-slot 15984  df-base 15986  df-sets 15987  df-ress 15988  df-plusg 16077  df-0g 16225  df-mgm 17364  df-sgrp 17406  df-mnd 17417  df-submnd 17458  df-cntz 17871  df-lsm 18172
This theorem is referenced by:  lsmsubg  18190
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