Theorem submcl 18728

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 18718	. . . . . . 7 ⊢ (𝑆 ∈ (SubMnd‘𝑀) → 𝑀 ∈ Mnd)
2		eqid 2733	. . . . . . . 8 ⊢ (Base‘𝑀) = (Base‘𝑀)
3		eqid 2733	. . . . . . . 8 ⊢ (0g‘𝑀) = (0g‘𝑀)
4		submcl.p	. . . . . . . 8 ⊢ + = (+g‘𝑀)
5	2, 3, 4	issubm 18719	. . . . . . 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 7381	. . . 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 1541   ∈ wcel 2113  ∀wral 3048   ⊆ wss 3898  ‘cfv 6489  (class class class)co 7355  Basecbs 17127  +_gcplusg 17168  0_gc0g 17350  Mndcmnd 18650  SubMndcsubmnd 18698

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 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374

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 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fv 6497  df-ov 7358  df-submnd 18700

This theorem is referenced by:  resmhm  18736  mhmima  18741  gsumwsubmcl  18753  submmulgcl  19038  symggen  19390  lsmsubm  19573  smndlsmidm  19576  gsumzadd  19842  gsumzoppg  19864  submcld  33045  erler  33275  rlocaddval  33278  rlocmulval  33279