Theorem submgmcl 18634

Description: Submagmas are closed under the magma 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 18622	. . . . . . 7 ⊢ (𝑆 ∈ (SubMgm‘𝑀) → 𝑀 ∈ Mgm)
2		eqid 2736	. . . . . . . 8 ⊢ (Base‘𝑀) = (Base‘𝑀)
3		submgmcl.p	. . . . . . . 8 ⊢ + = (+g‘𝑀)
4	2, 3	issubmgm 18629	. . . . . . 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 267	. . . . 5 ⊢ (𝑆 ∈ (SubMgm‘𝑀) → (𝑆 ⊆ (Base‘𝑀) ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆))
7	6	simprd 495	. . . 4 ⊢ (𝑆 ∈ (SubMgm‘𝑀) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆)
8		ovrspc2v 7384	. . . 4 ⊢ (((𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆) → (𝑋 + 𝑌) ∈ 𝑆)
9	7, 8	sylan2 593	. . 3 ⊢ (((𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ∧ 𝑆 ∈ (SubMgm‘𝑀)) → (𝑋 + 𝑌) ∈ 𝑆)
10	9	ancoms 458	. 2 ⊢ ((𝑆 ∈ (SubMgm‘𝑀) ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆)) → (𝑋 + 𝑌) ∈ 𝑆)
11	10	3impb 1114	1 ⊢ ((𝑆 ∈ (SubMgm‘𝑀) ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (𝑋 + 𝑌) ∈ 𝑆)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  ∀wral 3051   ⊆ wss 3901  ‘cfv 6492  (class class class)co 7358  Basecbs 17138  +_gcplusg 17179  Mgmcmgm 18565  SubMgmcsubmgm 18618

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fv 6500  df-ov 7361  df-submgm 18620

This theorem is referenced by:  resmgmhm  18638  mgmhmima  18642