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Theorem lsmsubm 18769
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 17958 . . . 4 (𝑇 ∈ (SubMnd‘𝐺) → 𝐺 ∈ Mnd)
213ad2ant1 1130 . . 3 ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → 𝐺 ∈ Mnd)
3 eqid 2822 . . . . 5 (Base‘𝐺) = (Base‘𝐺)
43submss 17965 . . . 4 (𝑇 ∈ (SubMnd‘𝐺) → 𝑇 ⊆ (Base‘𝐺))
543ad2ant1 1130 . . 3 ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → 𝑇 ⊆ (Base‘𝐺))
63submss 17965 . . . 4 (𝑈 ∈ (SubMnd‘𝐺) → 𝑈 ⊆ (Base‘𝐺))
763ad2ant2 1131 . . 3 ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → 𝑈 ⊆ (Base‘𝐺))
8 lsmsubg.p . . . 4 = (LSSum‘𝐺)
93, 8lsmssv 18759 . . 3 ((𝐺 ∈ Mnd ∧ 𝑇 ⊆ (Base‘𝐺) ∧ 𝑈 ⊆ (Base‘𝐺)) → (𝑇 𝑈) ⊆ (Base‘𝐺))
102, 5, 7, 9syl3anc 1368 . 2 ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → (𝑇 𝑈) ⊆ (Base‘𝐺))
11 simp2 1134 . . . 4 ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → 𝑈 ∈ (SubMnd‘𝐺))
123, 8lsmub1x 18762 . . . 4 ((𝑇 ⊆ (Base‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺)) → 𝑇 ⊆ (𝑇 𝑈))
135, 11, 12syl2anc 587 . . 3 ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → 𝑇 ⊆ (𝑇 𝑈))
14 eqid 2822 . . . . 5 (0g𝐺) = (0g𝐺)
1514subm0cl 17967 . . . 4 (𝑇 ∈ (SubMnd‘𝐺) → (0g𝐺) ∈ 𝑇)
16153ad2ant1 1130 . . 3 ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → (0g𝐺) ∈ 𝑇)
1713, 16sseldd 3943 . 2 ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → (0g𝐺) ∈ (𝑇 𝑈))
18 eqid 2822 . . . . . . 7 (+g𝐺) = (+g𝐺)
193, 18, 8lsmelvalx 18756 . . . . . 6 ((𝐺 ∈ Mnd ∧ 𝑇 ⊆ (Base‘𝐺) ∧ 𝑈 ⊆ (Base‘𝐺)) → (𝑥 ∈ (𝑇 𝑈) ↔ ∃𝑎𝑇𝑐𝑈 𝑥 = (𝑎(+g𝐺)𝑐)))
202, 5, 7, 19syl3anc 1368 . . . . 5 ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → (𝑥 ∈ (𝑇 𝑈) ↔ ∃𝑎𝑇𝑐𝑈 𝑥 = (𝑎(+g𝐺)𝑐)))
213, 18, 8lsmelvalx 18756 . . . . . 6 ((𝐺 ∈ Mnd ∧ 𝑇 ⊆ (Base‘𝐺) ∧ 𝑈 ⊆ (Base‘𝐺)) → (𝑦 ∈ (𝑇 𝑈) ↔ ∃𝑏𝑇𝑑𝑈 𝑦 = (𝑏(+g𝐺)𝑑)))
222, 5, 7, 21syl3anc 1368 . . . . 5 ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → (𝑦 ∈ (𝑇 𝑈) ↔ ∃𝑏𝑇𝑑𝑈 𝑦 = (𝑏(+g𝐺)𝑑)))
2320, 22anbi12d 633 . . . 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 3348 . . . . 5 (∃𝑎𝑇𝑏𝑇 (∃𝑐𝑈 𝑥 = (𝑎(+g𝐺)𝑐) ∧ ∃𝑑𝑈 𝑦 = (𝑏(+g𝐺)𝑑)) ↔ (∃𝑎𝑇𝑐𝑈 𝑥 = (𝑎(+g𝐺)𝑐) ∧ ∃𝑏𝑇𝑑𝑈 𝑦 = (𝑏(+g𝐺)𝑑)))
25 reeanv 3348 . . . . . . 7 (∃𝑐𝑈𝑑𝑈 (𝑥 = (𝑎(+g𝐺)𝑐) ∧ 𝑦 = (𝑏(+g𝐺)𝑑)) ↔ (∃𝑐𝑈 𝑥 = (𝑎(+g𝐺)𝑐) ∧ ∃𝑑𝑈 𝑦 = (𝑏(+g𝐺)𝑑)))
262adantr 484 . . . . . . . . . . . 12 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → 𝐺 ∈ Mnd)
275adantr 484 . . . . . . . . . . . . 13 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → 𝑇 ⊆ (Base‘𝐺))
28 simprll 778 . . . . . . . . . . . . 13 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → 𝑎𝑇)
2927, 28sseldd 3943 . . . . . . . . . . . 12 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → 𝑎 ∈ (Base‘𝐺))
30 simprlr 779 . . . . . . . . . . . . 13 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → 𝑏𝑇)
3127, 30sseldd 3943 . . . . . . . . . . . 12 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → 𝑏 ∈ (Base‘𝐺))
327adantr 484 . . . . . . . . . . . . 13 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → 𝑈 ⊆ (Base‘𝐺))
33 simprrl 780 . . . . . . . . . . . . 13 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → 𝑐𝑈)
3432, 33sseldd 3943 . . . . . . . . . . . 12 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → 𝑐 ∈ (Base‘𝐺))
35 simprrr 781 . . . . . . . . . . . . 13 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → 𝑑𝑈)
3632, 35sseldd 3943 . . . . . . . . . . . 12 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → 𝑑 ∈ (Base‘𝐺))
37 simpl3 1190 . . . . . . . . . . . . . 14 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → 𝑇 ⊆ (𝑍𝑈))
3837, 30sseldd 3943 . . . . . . . . . . . . 13 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → 𝑏 ∈ (𝑍𝑈))
39 lsmsubg.z . . . . . . . . . . . . . 14 𝑍 = (Cntz‘𝐺)
4018, 39cntzi 18450 . . . . . . . . . . . . 13 ((𝑏 ∈ (𝑍𝑈) ∧ 𝑐𝑈) → (𝑏(+g𝐺)𝑐) = (𝑐(+g𝐺)𝑏))
4138, 33, 40syl2anc 587 . . . . . . . . . . . 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 17916 . . . . . . . . . . 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 1188 . . . . . . . . . . . . 13 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → 𝑇 ∈ (SubMnd‘𝐺))
4418submcl 17968 . . . . . . . . . . . . 13 ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑎𝑇𝑏𝑇) → (𝑎(+g𝐺)𝑏) ∈ 𝑇)
4543, 28, 30, 44syl3anc 1368 . . . . . . . . . . . 12 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → (𝑎(+g𝐺)𝑏) ∈ 𝑇)
46 simpl2 1189 . . . . . . . . . . . . 13 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → 𝑈 ∈ (SubMnd‘𝐺))
4718submcl 17968 . . . . . . . . . . . . 13 ((𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑐𝑈𝑑𝑈) → (𝑐(+g𝐺)𝑑) ∈ 𝑈)
4846, 33, 35, 47syl3anc 1368 . . . . . . . . . . . 12 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → (𝑐(+g𝐺)𝑑) ∈ 𝑈)
493, 18, 8lsmelvalix 18757 . . . . . . . . . . . 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 1375 . . . . . . . . . . 11 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → ((𝑎(+g𝐺)𝑏)(+g𝐺)(𝑐(+g𝐺)𝑑)) ∈ (𝑇 𝑈))
5142, 50eqeltrrd 2915 . . . . . . . . . 10 (((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) ∧ ((𝑎𝑇𝑏𝑇) ∧ (𝑐𝑈𝑑𝑈))) → ((𝑎(+g𝐺)𝑐)(+g𝐺)(𝑏(+g𝐺)𝑑)) ∈ (𝑇 𝑈))
52 oveq12 7149 . . . . . . . . . . 11 ((𝑥 = (𝑎(+g𝐺)𝑐) ∧ 𝑦 = (𝑏(+g𝐺)𝑑)) → (𝑥(+g𝐺)𝑦) = ((𝑎(+g𝐺)𝑐)(+g𝐺)(𝑏(+g𝐺)𝑑)))
5352eleq1d 2898 . . . . . . . . . 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 250 . . . . . . . . 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 471 . . . . . . . 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 3280 . . . . . . 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 246 . . . . . 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 3280 . . . . 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 246 . . . 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 243 . . 3 ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → ((𝑥 ∈ (𝑇 𝑈) ∧ 𝑦 ∈ (𝑇 𝑈)) → (𝑥(+g𝐺)𝑦) ∈ (𝑇 𝑈)))
6160ralrimivv 3180 . 2 ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → ∀𝑥 ∈ (𝑇 𝑈)∀𝑦 ∈ (𝑇 𝑈)(𝑥(+g𝐺)𝑦) ∈ (𝑇 𝑈))
623, 14, 18issubm 17959 . . 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 1339 1 ((𝑇 ∈ (SubMnd‘𝐺) ∧ 𝑈 ∈ (SubMnd‘𝐺) ∧ 𝑇 ⊆ (𝑍𝑈)) → (𝑇 𝑈) ∈ (SubMnd‘𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2114  wral 3130  wrex 3131  wss 3908  cfv 6334  (class class class)co 7140  Basecbs 16474  +gcplusg 16556  0gc0g 16704  Mndcmnd 17902  SubMndcsubmnd 17946  Cntzccntz 18436  LSSumclsm 18750
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-nel 3116  df-ral 3135  df-rex 3136  df-reu 3137  df-rmo 3138  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145  df-om 7566  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11626  df-2 11688  df-ndx 16477  df-slot 16478  df-base 16480  df-sets 16481  df-ress 16482  df-plusg 16569  df-0g 16706  df-mgm 17843  df-sgrp 17892  df-mnd 17903  df-submnd 17948  df-cntz 18438  df-lsm 18752
This theorem is referenced by:  lsmsubg  18770
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