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Theorem submcl 17560
Description: Submonoids are closed under the monoid operation. (Contributed by Mario Carneiro, 10-Mar-2015.)
Hypothesis
Ref Expression
submcl.p + = (+g𝑀)
Assertion
Ref Expression
submcl ((𝑆 ∈ (SubMnd‘𝑀) ∧ 𝑋𝑆𝑌𝑆) → (𝑋 + 𝑌) ∈ 𝑆)

Proof of Theorem submcl
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 submrcl 17553 . . . . . . 7 (𝑆 ∈ (SubMnd‘𝑀) → 𝑀 ∈ Mnd)
2 eqid 2770 . . . . . . . 8 (Base‘𝑀) = (Base‘𝑀)
3 eqid 2770 . . . . . . . 8 (0g𝑀) = (0g𝑀)
4 submcl.p . . . . . . . 8 + = (+g𝑀)
52, 3, 4issubm 17554 . . . . . . 7 (𝑀 ∈ Mnd → (𝑆 ∈ (SubMnd‘𝑀) ↔ (𝑆 ⊆ (Base‘𝑀) ∧ (0g𝑀) ∈ 𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥 + 𝑦) ∈ 𝑆)))
61, 5syl 17 . . . . . 6 (𝑆 ∈ (SubMnd‘𝑀) → (𝑆 ∈ (SubMnd‘𝑀) ↔ (𝑆 ⊆ (Base‘𝑀) ∧ (0g𝑀) ∈ 𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥 + 𝑦) ∈ 𝑆)))
76ibi 256 . . . . 5 (𝑆 ∈ (SubMnd‘𝑀) → (𝑆 ⊆ (Base‘𝑀) ∧ (0g𝑀) ∈ 𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥 + 𝑦) ∈ 𝑆))
87simp3d 1137 . . . 4 (𝑆 ∈ (SubMnd‘𝑀) → ∀𝑥𝑆𝑦𝑆 (𝑥 + 𝑦) ∈ 𝑆)
9 ovrspc2v 6816 . . . 4 (((𝑋𝑆𝑌𝑆) ∧ ∀𝑥𝑆𝑦𝑆 (𝑥 + 𝑦) ∈ 𝑆) → (𝑋 + 𝑌) ∈ 𝑆)
108, 9sylan2 572 . . 3 (((𝑋𝑆𝑌𝑆) ∧ 𝑆 ∈ (SubMnd‘𝑀)) → (𝑋 + 𝑌) ∈ 𝑆)
1110ancoms 455 . 2 ((𝑆 ∈ (SubMnd‘𝑀) ∧ (𝑋𝑆𝑌𝑆)) → (𝑋 + 𝑌) ∈ 𝑆)
12113impb 1106 1 ((𝑆 ∈ (SubMnd‘𝑀) ∧ 𝑋𝑆𝑌𝑆) → (𝑋 + 𝑌) ∈ 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  w3a 1070   = wceq 1630  wcel 2144  wral 3060  wss 3721  cfv 6031  (class class class)co 6792  Basecbs 16063  +gcplusg 16148  0gc0g 16307  Mndcmnd 17501  SubMndcsubmnd 17541
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fv 6039  df-ov 6795  df-submnd 17543
This theorem is referenced by:  resmhm  17566  mhmima  17570  gsumwsubmcl  17582  submmulgcl  17792  symggen  18096  lsmsubm  18274  gsumzadd  18528  gsumzoppg  18550
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