Theorem submcl 18826

Description: Submonoids are closed under the monoid operation. (Contributed by Mario Carneiro, 10-Mar-2015.)

Hypothesis

Ref	Expression
submcl.p	⊢ + = (+g‘𝑀)

Assertion

Ref	Expression
submcl	⊢ ((𝑆 ∈ (SubMnd‘𝑀) ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (𝑋 + 𝑌) ∈ 𝑆)

Proof of Theorem submcl

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

Step	Hyp	Ref	Expression
1		submrcl 18816	. . . . . . 7 ⊢ (𝑆 ∈ (SubMnd‘𝑀) → 𝑀 ∈ Mnd)
2		eqid 2736	. . . . . . . 8 ⊢ (Base‘𝑀) = (Base‘𝑀)
3		eqid 2736	. . . . . . . 8 ⊢ (0g‘𝑀) = (0g‘𝑀)
4		submcl.p	. . . . . . . 8 ⊢ + = (+g‘𝑀)
5	2, 3, 4	issubm 18817	. . . . . . 7 ⊢ (𝑀 ∈ Mnd → (𝑆 ∈ (SubMnd‘𝑀) ↔ (𝑆 ⊆ (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆)))
6	1, 5	syl 17	. . . . . 6 ⊢ (𝑆 ∈ (SubMnd‘𝑀) → (𝑆 ∈ (SubMnd‘𝑀) ↔ (𝑆 ⊆ (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆)))
7	6	ibi 267	. . . . 5 ⊢ (𝑆 ∈ (SubMnd‘𝑀) → (𝑆 ⊆ (Base‘𝑀) ∧ (0g‘𝑀) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆))
8	7	simp3d 1144	. . . 4 ⊢ (𝑆 ∈ (SubMnd‘𝑀) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆)
9		ovrspc2v 7458	. . . 4 ⊢ (((𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆) → (𝑋 + 𝑌) ∈ 𝑆)
10	8, 9	sylan2 593	. . 3 ⊢ (((𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ∧ 𝑆 ∈ (SubMnd‘𝑀)) → (𝑋 + 𝑌) ∈ 𝑆)
11	10	ancoms 458	. 2 ⊢ ((𝑆 ∈ (SubMnd‘𝑀) ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆)) → (𝑋 + 𝑌) ∈ 𝑆)
12	11	3impb 1114	1 ⊢ ((𝑆 ∈ (SubMnd‘𝑀) ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (𝑋 + 𝑌) ∈ 𝑆)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1539   ∈ wcel 2107  ∀wral 3060   ⊆ wss 3950  ‘cfv 6560  (class class class)co 7432  Basecbs 17248  +_gcplusg 17298  0_gc0g 17485  Mndcmnd 18748  SubMndcsubmnd 18796

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fv 6568  df-ov 7435  df-submnd 18798

This theorem is referenced by:  resmhm  18834  mhmima  18839  gsumwsubmcl  18851  submmulgcl  19136  symggen  19489  lsmsubm  19672  smndlsmidm  19675  gsumzadd  19941  gsumzoppg  19963  submcld  33041  erler  33270  rlocaddval  33273  rlocmulval  33274