Theorem submgmcl 44081

Description: Submagmas are closed under the monoid operation. (Contributed by AV, 26-Feb-2020.)

Hypothesis

Ref	Expression
submgmcl.p	⊢ + = (+g‘𝑀)

Assertion

Ref	Expression
submgmcl	⊢ ((𝑆 ∈ (SubMgm‘𝑀) ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (𝑋 + 𝑌) ∈ 𝑆)

Proof of Theorem submgmcl

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		submgmrcl 44069	. . . . . . 7 ⊢ (𝑆 ∈ (SubMgm‘𝑀) → 𝑀 ∈ Mgm)
2		eqid 2821	. . . . . . . 8 ⊢ (Base‘𝑀) = (Base‘𝑀)
3		submgmcl.p	. . . . . . . 8 ⊢ + = (+g‘𝑀)
4	2, 3	issubmgm 44076	. . . . . . 7 ⊢ (𝑀 ∈ Mgm → (𝑆 ∈ (SubMgm‘𝑀) ↔ (𝑆 ⊆ (Base‘𝑀) ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆)))
5	1, 4	syl 17	. . . . . 6 ⊢ (𝑆 ∈ (SubMgm‘𝑀) → (𝑆 ∈ (SubMgm‘𝑀) ↔ (𝑆 ⊆ (Base‘𝑀) ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆)))
6	5	ibi 269	. . . . 5 ⊢ (𝑆 ∈ (SubMgm‘𝑀) → (𝑆 ⊆ (Base‘𝑀) ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆))
7	6	simprd 498	. . . 4 ⊢ (𝑆 ∈ (SubMgm‘𝑀) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆)
8		ovrspc2v 7182	. . . 4 ⊢ (((𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆) → (𝑋 + 𝑌) ∈ 𝑆)
9	7, 8	sylan2 594	. . 3 ⊢ (((𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ∧ 𝑆 ∈ (SubMgm‘𝑀)) → (𝑋 + 𝑌) ∈ 𝑆)
10	9	ancoms 461	. 2 ⊢ ((𝑆 ∈ (SubMgm‘𝑀) ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆)) → (𝑋 + 𝑌) ∈ 𝑆)
11	10	3impb 1111	1 ⊢ ((𝑆 ∈ (SubMgm‘𝑀) ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (𝑋 + 𝑌) ∈ 𝑆)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  ∀wral 3138   ⊆ wss 3936  ‘cfv 6355  (class class class)co 7156  Basecbs 16483  +_gcplusg 16565  Mgmcmgm 17850  SubMgmcsubmgm 44065

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-iota 6314  df-fun 6357  df-fv 6363  df-ov 7159  df-submgm 44067

This theorem is referenced by:  resmgmhm  44085  mgmhmima  44089