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Theorem lsmsubm 18000
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 17278 . . . 4 (𝑇 ∈ (SubMnd‘𝐺) → 𝐺 ∈ Mnd)
213ad2ant1 1080 . . 3 ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → 𝐺 ∈ Mnd)
3 eqid 2621 . . . . 5 (Base‘𝐺) = (Base‘𝐺)
43submss 17282 . . . 4 (𝑇 ∈ (SubMnd‘𝐺) → 𝑇 ⊆ (Base‘𝐺))
543ad2ant1 1080 . . 3 ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → 𝑇 ⊆ (Base‘𝐺))
63submss 17282 . . . 4 (𝑈 ∈ (SubMnd‘𝐺) → 𝑈 ⊆ (Base‘𝐺))
763ad2ant2 1081 . . 3 ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → 𝑈 ⊆ (Base‘𝐺))
8 lsmsubg.p . . . 4 = (LSSum‘𝐺)
93, 8lsmssv 17990 . . 3 ((𝐺 ∈ Mnd ∧ 𝑇 ⊆ (Base‘𝐺) ∧ 𝑈 ⊆ (Base‘𝐺)) → (𝑇 𝑈) ⊆ (Base‘𝐺))
102, 5, 7, 9syl3anc 1323 . 2 ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → (𝑇 𝑈) ⊆ (Base‘𝐺))
11 simp2 1060 . . . 4 ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → 𝑈 ∈ (SubMnd‘𝐺))
123, 8lsmub1x 17993 . . . 4 ((𝑇 ⊆ (Base‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺)) → 𝑇 ⊆ (𝑇 𝑈))
135, 11, 12syl2anc 692 . . 3 ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → 𝑇 ⊆ (𝑇 𝑈))
14 eqid 2621 . . . . 5 (0g𝐺) = (0g𝐺)
1514subm0cl 17284 . . . 4 (𝑇 ∈ (SubMnd‘𝐺) → (0g𝐺) ∈ 𝑇)
16153ad2ant1 1080 . . 3 ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → (0g𝐺) ∈ 𝑇)
1713, 16sseldd 3588 . 2 ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → (0g𝐺) ∈ (𝑇 𝑈))
18 eqid 2621 . . . . . . 7 (+g𝐺) = (+g𝐺)
193, 18, 8lsmelvalx 17987 . . . . . 6 ((𝐺 ∈ Mnd ∧ 𝑇 ⊆ (Base‘𝐺) ∧ 𝑈 ⊆ (Base‘𝐺)) → (𝑥 ∈ (𝑇 𝑈) ↔ ∃𝑎𝑇𝑐𝑈 𝑥 = (𝑎(+g𝐺)𝑐)))
202, 5, 7, 19syl3anc 1323 . . . . 5 ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → (𝑥 ∈ (𝑇 𝑈) ↔ ∃𝑎𝑇𝑐𝑈 𝑥 = (𝑎(+g𝐺)𝑐)))
213, 18, 8lsmelvalx 17987 . . . . . 6 ((𝐺 ∈ Mnd ∧ 𝑇 ⊆ (Base‘𝐺) ∧ 𝑈 ⊆ (Base‘𝐺)) → (𝑦 ∈ (𝑇 𝑈) ↔ ∃𝑏𝑇𝑑𝑈 𝑦 = (𝑏(+g𝐺)𝑑)))
222, 5, 7, 21syl3anc 1323 . . . . 5 ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → (𝑦 ∈ (𝑇 𝑈) ↔ ∃𝑏𝑇𝑑𝑈 𝑦 = (𝑏(+g𝐺)𝑑)))
2320, 22anbi12d 746 . . . 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 3100 . . . . 5 (∃𝑎𝑇𝑏𝑇 (∃𝑐𝑈 𝑥 = (𝑎(+g𝐺)𝑐) ∧ ∃𝑑𝑈 𝑦 = (𝑏(+g𝐺)𝑑)) ↔ (∃𝑎𝑇𝑐𝑈 𝑥 = (𝑎(+g𝐺)𝑐) ∧ ∃𝑏𝑇𝑑𝑈 𝑦 = (𝑏(+g𝐺)𝑑)))
25 reeanv 3100 . . . . . . 7 (∃𝑐𝑈𝑑𝑈 (𝑥 = (𝑎(+g𝐺)𝑐) ∧ 𝑦 = (𝑏(+g𝐺)𝑑)) ↔ (∃𝑐𝑈 𝑥 = (𝑎(+g𝐺)𝑐) ∧ ∃𝑑𝑈 𝑦 = (𝑏(+g𝐺)𝑑)))
262adantr 481 . . . . . . . . . . . 12 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → 𝐺 ∈ Mnd)
275adantr 481 . . . . . . . . . . . . 13 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → 𝑇 ⊆ (Base‘𝐺))
28 simprll 801 . . . . . . . . . . . . 13 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → 𝑎𝑇)
2927, 28sseldd 3588 . . . . . . . . . . . 12 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → 𝑎 ∈ (Base‘𝐺))
30 simprlr 802 . . . . . . . . . . . . 13 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → 𝑏𝑇)
3127, 30sseldd 3588 . . . . . . . . . . . 12 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → 𝑏 ∈ (Base‘𝐺))
327adantr 481 . . . . . . . . . . . . 13 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → 𝑈 ⊆ (Base‘𝐺))
33 simprrl 803 . . . . . . . . . . . . 13 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → 𝑐𝑈)
3432, 33sseldd 3588 . . . . . . . . . . . 12 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → 𝑐 ∈ (Base‘𝐺))
35 simprrr 804 . . . . . . . . . . . . 13 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → 𝑑𝑈)
3632, 35sseldd 3588 . . . . . . . . . . . 12 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → 𝑑 ∈ (Base‘𝐺))
37 simpl3 1064 . . . . . . . . . . . . . 14 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → 𝑇 ⊆ (𝑍𝑈))
3837, 30sseldd 3588 . . . . . . . . . . . . 13 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → 𝑏 ∈ (𝑍𝑈))
39 lsmsubg.z . . . . . . . . . . . . . 14 𝑍 = (Cntz‘𝐺)
4018, 39cntzi 17694 . . . . . . . . . . . . 13 ((𝑏 ∈ (𝑍𝑈) ∧ 𝑐𝑈) → (𝑏(+g𝐺)𝑐) = (𝑐(+g𝐺)𝑏))
4138, 33, 40syl2anc 692 . . . . . . . . . . . 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 17239 . . . . . . . . . . 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 1062 . . . . . . . . . . . . 13 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → 𝑇 ∈ (SubMnd‘𝐺))
4418submcl 17285 . . . . . . . . . . . . 13 ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑎𝑇𝑏𝑇) → (𝑎(+g𝐺)𝑏) ∈ 𝑇)
4543, 28, 30, 44syl3anc 1323 . . . . . . . . . . . 12 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → (𝑎(+g𝐺)𝑏) ∈ 𝑇)
46 simpl2 1063 . . . . . . . . . . . . 13 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → 𝑈 ∈ (SubMnd‘𝐺))
4718submcl 17285 . . . . . . . . . . . . 13 ((𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑐𝑈𝑑𝑈) → (𝑐(+g𝐺)𝑑) ∈ 𝑈)
4846, 33, 35, 47syl3anc 1323 . . . . . . . . . . . 12 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → (𝑐(+g𝐺)𝑑) ∈ 𝑈)
493, 18, 8lsmelvalix 17988 . . . . . . . . . . . 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 1331 . . . . . . . . . . 11 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → ((𝑎(+g𝐺)𝑏)(+g𝐺)(𝑐(+g𝐺)𝑑)) ∈ (𝑇 𝑈))
5142, 50eqeltrrd 2699 . . . . . . . . . 10 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → ((𝑎(+g𝐺)𝑐)(+g𝐺)(𝑏(+g𝐺)𝑑)) ∈ (𝑇 𝑈))
52 oveq12 6619 . . . . . . . . . . 11 ((𝑥 = (𝑎(+g𝐺)𝑐) ∧ 𝑦 = (𝑏(+g𝐺)𝑑)) → (𝑥(+g𝐺)𝑦) = ((𝑎(+g𝐺)𝑐)(+g𝐺)(𝑏(+g𝐺)𝑑)))
5352eleq1d 2683 . . . . . . . . . 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 679 . . . . . . . 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 3032 . . . . . . 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 3032 . . . . 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 2965 . 2 ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → ∀𝑥 ∈ (𝑇 𝑈)∀𝑦 ∈ (𝑇 𝑈)(𝑥(+g𝐺)𝑦) ∈ (𝑇 𝑈))
623, 14, 18issubm 17279 . . 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 1243 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 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  wrex 2908  wss 3559  cfv 5852  (class class class)co 6610  Basecbs 15792  +gcplusg 15873  0gc0g 16032  Mndcmnd 17226  SubMndcsubmnd 17266  Cntzccntz 17680  LSSumclsm 17981
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9944  ax-resscn 9945  ax-1cn 9946  ax-icn 9947  ax-addcl 9948  ax-addrcl 9949  ax-mulcl 9950  ax-mulrcl 9951  ax-mulcom 9952  ax-addass 9953  ax-mulass 9954  ax-distr 9955  ax-i2m1 9956  ax-1ne0 9957  ax-1rid 9958  ax-rnegex 9959  ax-rrecex 9960  ax-cnre 9961  ax-pre-lttri 9962  ax-pre-lttrn 9963  ax-pre-ltadd 9964  ax-pre-mulgt0 9965
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-er 7694  df-en 7908  df-dom 7909  df-sdom 7910  df-pnf 10028  df-mnf 10029  df-xr 10030  df-ltxr 10031  df-le 10032  df-sub 10220  df-neg 10221  df-nn 10973  df-2 11031  df-ndx 15795  df-slot 15796  df-base 15797  df-sets 15798  df-ress 15799  df-plusg 15886  df-0g 16034  df-mgm 17174  df-sgrp 17216  df-mnd 17227  df-submnd 17268  df-cntz 17682  df-lsm 17983
This theorem is referenced by:  lsmsubg  18001
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