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Theorem submcl 17973
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 17963 . . . . . . 7 (𝑆 ∈ (SubMnd‘𝑀) → 𝑀 ∈ Mnd)
2 eqid 2801 . . . . . . . 8 (Base‘𝑀) = (Base‘𝑀)
3 eqid 2801 . . . . . . . 8 (0g𝑀) = (0g𝑀)
4 submcl.p . . . . . . . 8 + = (+g𝑀)
52, 3, 4issubm 17964 . . . . . . 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 270 . . . . 5 (𝑆 ∈ (SubMnd‘𝑀) → (𝑆 ⊆ (Base‘𝑀) ∧ (0g𝑀) ∈ 𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥 + 𝑦) ∈ 𝑆))
87simp3d 1141 . . . 4 (𝑆 ∈ (SubMnd‘𝑀) → ∀𝑥𝑆𝑦𝑆 (𝑥 + 𝑦) ∈ 𝑆)
9 ovrspc2v 7165 . . . 4 (((𝑋𝑆𝑌𝑆) ∧ ∀𝑥𝑆𝑦𝑆 (𝑥 + 𝑦) ∈ 𝑆) → (𝑋 + 𝑌) ∈ 𝑆)
108, 9sylan2 595 . . 3 (((𝑋𝑆𝑌𝑆) ∧ 𝑆 ∈ (SubMnd‘𝑀)) → (𝑋 + 𝑌) ∈ 𝑆)
1110ancoms 462 . 2 ((𝑆 ∈ (SubMnd‘𝑀) ∧ (𝑋𝑆𝑌𝑆)) → (𝑋 + 𝑌) ∈ 𝑆)
12113impb 1112 1 ((𝑆 ∈ (SubMnd‘𝑀) ∧ 𝑋𝑆𝑌𝑆) → (𝑋 + 𝑌) ∈ 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2112  wral 3109  wss 3884  cfv 6328  (class class class)co 7139  Basecbs 16479  +gcplusg 16561  0gc0g 16709  Mndcmnd 17907  SubMndcsubmnd 17951
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 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fv 6336  df-ov 7142  df-submnd 17953
This theorem is referenced by:  resmhm  17981  mhmima  17985  gsumwsubmcl  17997  submmulgcl  18266  symggen  18594  lsmsubm  18774  smndlsmidm  18777  gsumzadd  19039  gsumzoppg  19061
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