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Theorem submgmcl 18640
Description: Submagmas are closed under the magma 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 18628 . . . . . . 7 (𝑆 ∈ (SubMgm‘𝑀) → 𝑀 ∈ Mgm)
2 eqid 2726 . . . . . . . 8 (Base‘𝑀) = (Base‘𝑀)
3 submgmcl.p . . . . . . . 8 + = (+g𝑀)
42, 3issubmgm 18635 . . . . . . 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 495 . . . 4 (𝑆 ∈ (SubMgm‘𝑀) → ∀𝑥𝑆𝑦𝑆 (𝑥 + 𝑦) ∈ 𝑆)
8 ovrspc2v 7431 . . . 4 (((𝑋𝑆𝑌𝑆) ∧ ∀𝑥𝑆𝑦𝑆 (𝑥 + 𝑦) ∈ 𝑆) → (𝑋 + 𝑌) ∈ 𝑆)
97, 8sylan2 592 . . 3 (((𝑋𝑆𝑌𝑆) ∧ 𝑆 ∈ (SubMgm‘𝑀)) → (𝑋 + 𝑌) ∈ 𝑆)
109ancoms 458 . 2 ((𝑆 ∈ (SubMgm‘𝑀) ∧ (𝑋𝑆𝑌𝑆)) → (𝑋 + 𝑌) ∈ 𝑆)
11103impb 1112 1 ((𝑆 ∈ (SubMgm‘𝑀) ∧ 𝑋𝑆𝑌𝑆) → (𝑋 + 𝑌) ∈ 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1084   = wceq 1533  wcel 2098  wral 3055  wss 3943  cfv 6537  (class class class)co 7405  Basecbs 17153  +gcplusg 17206  Mgmcmgm 18571  SubMgmcsubmgm 18624
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fv 6545  df-ov 7408  df-submgm 18626
This theorem is referenced by:  resmgmhm  18644  mgmhmima  18648
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