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Theorem submgmcl 46174
Description: Submagmas are closed under the monoid operation. (Contributed by AV, 26-Feb-2020.)
Hypothesis
Ref Expression
submgmcl.p + = (+g𝑀)
Assertion
Ref Expression
submgmcl ((𝑆 ∈ (SubMgm‘𝑀) ∧ 𝑋𝑆𝑌𝑆) → (𝑋 + 𝑌) ∈ 𝑆)

Proof of Theorem submgmcl
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 submgmrcl 46162 . . . . . . 7 (𝑆 ∈ (SubMgm‘𝑀) → 𝑀 ∈ Mgm)
2 eqid 2733 . . . . . . . 8 (Base‘𝑀) = (Base‘𝑀)
3 submgmcl.p . . . . . . . 8 + = (+g𝑀)
42, 3issubmgm 46169 . . . . . . 7 (𝑀 ∈ Mgm → (𝑆 ∈ (SubMgm‘𝑀) ↔ (𝑆 ⊆ (Base‘𝑀) ∧ ∀𝑥𝑆𝑦𝑆 (𝑥 + 𝑦) ∈ 𝑆)))
51, 4syl 17 . . . . . 6 (𝑆 ∈ (SubMgm‘𝑀) → (𝑆 ∈ (SubMgm‘𝑀) ↔ (𝑆 ⊆ (Base‘𝑀) ∧ ∀𝑥𝑆𝑦𝑆 (𝑥 + 𝑦) ∈ 𝑆)))
65ibi 267 . . . . 5 (𝑆 ∈ (SubMgm‘𝑀) → (𝑆 ⊆ (Base‘𝑀) ∧ ∀𝑥𝑆𝑦𝑆 (𝑥 + 𝑦) ∈ 𝑆))
76simprd 497 . . . 4 (𝑆 ∈ (SubMgm‘𝑀) → ∀𝑥𝑆𝑦𝑆 (𝑥 + 𝑦) ∈ 𝑆)
8 ovrspc2v 7384 . . . 4 (((𝑋𝑆𝑌𝑆) ∧ ∀𝑥𝑆𝑦𝑆 (𝑥 + 𝑦) ∈ 𝑆) → (𝑋 + 𝑌) ∈ 𝑆)
97, 8sylan2 594 . . 3 (((𝑋𝑆𝑌𝑆) ∧ 𝑆 ∈ (SubMgm‘𝑀)) → (𝑋 + 𝑌) ∈ 𝑆)
109ancoms 460 . 2 ((𝑆 ∈ (SubMgm‘𝑀) ∧ (𝑋𝑆𝑌𝑆)) → (𝑋 + 𝑌) ∈ 𝑆)
11103impb 1116 1 ((𝑆 ∈ (SubMgm‘𝑀) ∧ 𝑋𝑆𝑌𝑆) → (𝑋 + 𝑌) ∈ 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  wral 3061  wss 3911  cfv 6497  (class class class)co 7358  Basecbs 17088  +gcplusg 17138  Mgmcmgm 18500  SubMgmcsubmgm 46158
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fv 6505  df-ov 7361  df-submgm 46160
This theorem is referenced by:  resmgmhm  46178  mgmhmima  46182
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