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Theorem submgmcl 44051
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 44039 . . . . . . 7 (𝑆 ∈ (SubMgm‘𝑀) → 𝑀 ∈ Mgm)
2 eqid 2819 . . . . . . . 8 (Base‘𝑀) = (Base‘𝑀)
3 submgmcl.p . . . . . . . 8 + = (+g𝑀)
42, 3issubmgm 44046 . . . . . . 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 269 . . . . 5 (𝑆 ∈ (SubMgm‘𝑀) → (𝑆 ⊆ (Base‘𝑀) ∧ ∀𝑥𝑆𝑦𝑆 (𝑥 + 𝑦) ∈ 𝑆))
76simprd 498 . . . 4 (𝑆 ∈ (SubMgm‘𝑀) → ∀𝑥𝑆𝑦𝑆 (𝑥 + 𝑦) ∈ 𝑆)
8 ovrspc2v 7174 . . . 4 (((𝑋𝑆𝑌𝑆) ∧ ∀𝑥𝑆𝑦𝑆 (𝑥 + 𝑦) ∈ 𝑆) → (𝑋 + 𝑌) ∈ 𝑆)
97, 8sylan2 594 . . 3 (((𝑋𝑆𝑌𝑆) ∧ 𝑆 ∈ (SubMgm‘𝑀)) → (𝑋 + 𝑌) ∈ 𝑆)
109ancoms 461 . 2 ((𝑆 ∈ (SubMgm‘𝑀) ∧ (𝑋𝑆𝑌𝑆)) → (𝑋 + 𝑌) ∈ 𝑆)
11103impb 1110 1 ((𝑆 ∈ (SubMgm‘𝑀) ∧ 𝑋𝑆𝑌𝑆) → (𝑋 + 𝑌) ∈ 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  w3a 1082   = wceq 1531  wcel 2108  wral 3136  wss 3934  cfv 6348  (class class class)co 7148  Basecbs 16475  +gcplusg 16557  Mgmcmgm 17842  SubMgmcsubmgm 44035
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7151  df-submgm 44037
This theorem is referenced by:  resmgmhm  44055  mgmhmima  44059
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