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Theorem lsmsubm 18452
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 17732 . . . 4 (𝑇 ∈ (SubMnd‘𝐺) → 𝐺 ∈ Mnd)
213ad2ant1 1124 . . 3 ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → 𝐺 ∈ Mnd)
3 eqid 2778 . . . . 5 (Base‘𝐺) = (Base‘𝐺)
43submss 17736 . . . 4 (𝑇 ∈ (SubMnd‘𝐺) → 𝑇 ⊆ (Base‘𝐺))
543ad2ant1 1124 . . 3 ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → 𝑇 ⊆ (Base‘𝐺))
63submss 17736 . . . 4 (𝑈 ∈ (SubMnd‘𝐺) → 𝑈 ⊆ (Base‘𝐺))
763ad2ant2 1125 . . 3 ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → 𝑈 ⊆ (Base‘𝐺))
8 lsmsubg.p . . . 4 = (LSSum‘𝐺)
93, 8lsmssv 18442 . . 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 1128 . . . 4 ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → 𝑈 ∈ (SubMnd‘𝐺))
123, 8lsmub1x 18445 . . . 4 ((𝑇 ⊆ (Base‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺)) → 𝑇 ⊆ (𝑇 𝑈))
135, 11, 12syl2anc 579 . . 3 ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → 𝑇 ⊆ (𝑇 𝑈))
14 eqid 2778 . . . . 5 (0g𝐺) = (0g𝐺)
1514subm0cl 17738 . . . 4 (𝑇 ∈ (SubMnd‘𝐺) → (0g𝐺) ∈ 𝑇)
16153ad2ant1 1124 . . 3 ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → (0g𝐺) ∈ 𝑇)
1713, 16sseldd 3822 . 2 ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → (0g𝐺) ∈ (𝑇 𝑈))
18 eqid 2778 . . . . . . 7 (+g𝐺) = (+g𝐺)
193, 18, 8lsmelvalx 18439 . . . . . 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 18439 . . . . . 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 624 . . . 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 3293 . . . . 5 (∃𝑎𝑇𝑏𝑇 (∃𝑐𝑈 𝑥 = (𝑎(+g𝐺)𝑐) ∧ ∃𝑑𝑈 𝑦 = (𝑏(+g𝐺)𝑑)) ↔ (∃𝑎𝑇𝑐𝑈 𝑥 = (𝑎(+g𝐺)𝑐) ∧ ∃𝑏𝑇𝑑𝑈 𝑦 = (𝑏(+g𝐺)𝑑)))
25 reeanv 3293 . . . . . . 7 (∃𝑐𝑈𝑑𝑈 (𝑥 = (𝑎(+g𝐺)𝑐) ∧ 𝑦 = (𝑏(+g𝐺)𝑑)) ↔ (∃𝑐𝑈 𝑥 = (𝑎(+g𝐺)𝑐) ∧ ∃𝑑𝑈 𝑦 = (𝑏(+g𝐺)𝑑)))
262adantr 474 . . . . . . . . . . . 12 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → 𝐺 ∈ Mnd)
275adantr 474 . . . . . . . . . . . . 13 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → 𝑇 ⊆ (Base‘𝐺))
28 simprll 769 . . . . . . . . . . . . 13 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → 𝑎𝑇)
2927, 28sseldd 3822 . . . . . . . . . . . 12 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → 𝑎 ∈ (Base‘𝐺))
30 simprlr 770 . . . . . . . . . . . . 13 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → 𝑏𝑇)
3127, 30sseldd 3822 . . . . . . . . . . . 12 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → 𝑏 ∈ (Base‘𝐺))
327adantr 474 . . . . . . . . . . . . 13 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → 𝑈 ⊆ (Base‘𝐺))
33 simprrl 771 . . . . . . . . . . . . 13 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → 𝑐𝑈)
3432, 33sseldd 3822 . . . . . . . . . . . 12 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → 𝑐 ∈ (Base‘𝐺))
35 simprrr 772 . . . . . . . . . . . . 13 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → 𝑑𝑈)
3632, 35sseldd 3822 . . . . . . . . . . . 12 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → 𝑑 ∈ (Base‘𝐺))
37 simpl3 1203 . . . . . . . . . . . . . 14 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → 𝑇 ⊆ (𝑍𝑈))
3837, 30sseldd 3822 . . . . . . . . . . . . 13 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → 𝑏 ∈ (𝑍𝑈))
39 lsmsubg.z . . . . . . . . . . . . . 14 𝑍 = (Cntz‘𝐺)
4018, 39cntzi 18145 . . . . . . . . . . . . 13 ((𝑏 ∈ (𝑍𝑈) ∧ 𝑐𝑈) → (𝑏(+g𝐺)𝑐) = (𝑐(+g𝐺)𝑏))
4138, 33, 40syl2anc 579 . . . . . . . . . . . 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 17693 . . . . . . . . . . 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 1199 . . . . . . . . . . . . 13 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → 𝑇 ∈ (SubMnd‘𝐺))
4418submcl 17739 . . . . . . . . . . . . 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 1201 . . . . . . . . . . . . 13 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → 𝑈 ∈ (SubMnd‘𝐺))
4718submcl 17739 . . . . . . . . . . . . 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 18440 . . . . . . . . . . . 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 1446 . . . . . . . . . . 11 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → ((𝑎(+g𝐺)𝑏)(+g𝐺)(𝑐(+g𝐺)𝑑)) ∈ (𝑇 𝑈))
5142, 50eqeltrrd 2860 . . . . . . . . . 10 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → ((𝑎(+g𝐺)𝑐)(+g𝐺)(𝑏(+g𝐺)𝑑)) ∈ (𝑇 𝑈))
52 oveq12 6931 . . . . . . . . . . 11 ((𝑥 = (𝑎(+g𝐺)𝑐) ∧ 𝑦 = (𝑏(+g𝐺)𝑑)) → (𝑥(+g𝐺)𝑦) = ((𝑎(+g𝐺)𝑐)(+g𝐺)(𝑏(+g𝐺)𝑑)))
5352eleq1d 2844 . . . . . . . . . 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 239 . . . . . . . . 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 461 . . . . . . . 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 3221 . . . . . . 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 235 . . . . . 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 3221 . . . . 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 235 . . . 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 232 . . 3 ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → ((𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈)) → (𝑥(+g𝐺)𝑦) ∈ (𝑇 𝑈)))
6160ralrimivv 3152 . 2 ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → ∀𝑥 ∈ (𝑇 𝑈)∀𝑦 ∈ (𝑇 𝑈)(𝑥(+g𝐺)𝑦) ∈ (𝑇 𝑈))
623, 14, 18issubm 17733 . . 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 1399 1 ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → (𝑇 𝑈) ∈ (SubMnd‘𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386  w3a 1071   = wceq 1601  wcel 2107  wral 3090  wrex 3091  wss 3792  cfv 6135  (class class class)co 6922  Basecbs 16255  +gcplusg 16338  0gc0g 16486  Mndcmnd 17680  SubMndcsubmnd 17720  Cntzccntz 18131  LSSumclsm 18433
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-cnex 10328  ax-resscn 10329  ax-1cn 10330  ax-icn 10331  ax-addcl 10332  ax-addrcl 10333  ax-mulcl 10334  ax-mulrcl 10335  ax-mulcom 10336  ax-addass 10337  ax-mulass 10338  ax-distr 10339  ax-i2m1 10340  ax-1ne0 10341  ax-1rid 10342  ax-rnegex 10343  ax-rrecex 10344  ax-cnre 10345  ax-pre-lttri 10346  ax-pre-lttrn 10347  ax-pre-ltadd 10348  ax-pre-mulgt0 10349
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-om 7344  df-1st 7445  df-2nd 7446  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-er 8026  df-en 8242  df-dom 8243  df-sdom 8244  df-pnf 10413  df-mnf 10414  df-xr 10415  df-ltxr 10416  df-le 10417  df-sub 10608  df-neg 10609  df-nn 11375  df-2 11438  df-ndx 16258  df-slot 16259  df-base 16261  df-sets 16262  df-ress 16263  df-plusg 16351  df-0g 16488  df-mgm 17628  df-sgrp 17670  df-mnd 17681  df-submnd 17722  df-cntz 18133  df-lsm 18435
This theorem is referenced by:  lsmsubg  18453
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