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Theorem submcl 18735
Description: Submonoids are closed under the monoid operation. (Contributed by Mario Carneiro, 10-Mar-2015.)
Hypothesis
Ref Expression
submcl.p + = (+gβ€˜π‘€)
Assertion
Ref Expression
submcl ((𝑆 ∈ (SubMndβ€˜π‘€) ∧ 𝑋 ∈ 𝑆 ∧ π‘Œ ∈ 𝑆) β†’ (𝑋 + π‘Œ) ∈ 𝑆)

Proof of Theorem submcl
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 submrcl 18725 . . . . . . 7 (𝑆 ∈ (SubMndβ€˜π‘€) β†’ 𝑀 ∈ Mnd)
2 eqid 2726 . . . . . . . 8 (Baseβ€˜π‘€) = (Baseβ€˜π‘€)
3 eqid 2726 . . . . . . . 8 (0gβ€˜π‘€) = (0gβ€˜π‘€)
4 submcl.p . . . . . . . 8 + = (+gβ€˜π‘€)
52, 3, 4issubm 18726 . . . . . . 7 (𝑀 ∈ Mnd β†’ (𝑆 ∈ (SubMndβ€˜π‘€) ↔ (𝑆 βŠ† (Baseβ€˜π‘€) ∧ (0gβ€˜π‘€) ∈ 𝑆 ∧ βˆ€π‘₯ ∈ 𝑆 βˆ€π‘¦ ∈ 𝑆 (π‘₯ + 𝑦) ∈ 𝑆)))
61, 5syl 17 . . . . . 6 (𝑆 ∈ (SubMndβ€˜π‘€) β†’ (𝑆 ∈ (SubMndβ€˜π‘€) ↔ (𝑆 βŠ† (Baseβ€˜π‘€) ∧ (0gβ€˜π‘€) ∈ 𝑆 ∧ βˆ€π‘₯ ∈ 𝑆 βˆ€π‘¦ ∈ 𝑆 (π‘₯ + 𝑦) ∈ 𝑆)))
76ibi 267 . . . . 5 (𝑆 ∈ (SubMndβ€˜π‘€) β†’ (𝑆 βŠ† (Baseβ€˜π‘€) ∧ (0gβ€˜π‘€) ∈ 𝑆 ∧ βˆ€π‘₯ ∈ 𝑆 βˆ€π‘¦ ∈ 𝑆 (π‘₯ + 𝑦) ∈ 𝑆))
87simp3d 1141 . . . 4 (𝑆 ∈ (SubMndβ€˜π‘€) β†’ βˆ€π‘₯ ∈ 𝑆 βˆ€π‘¦ ∈ 𝑆 (π‘₯ + 𝑦) ∈ 𝑆)
9 ovrspc2v 7430 . . . 4 (((𝑋 ∈ 𝑆 ∧ π‘Œ ∈ 𝑆) ∧ βˆ€π‘₯ ∈ 𝑆 βˆ€π‘¦ ∈ 𝑆 (π‘₯ + 𝑦) ∈ 𝑆) β†’ (𝑋 + π‘Œ) ∈ 𝑆)
108, 9sylan2 592 . . 3 (((𝑋 ∈ 𝑆 ∧ π‘Œ ∈ 𝑆) ∧ 𝑆 ∈ (SubMndβ€˜π‘€)) β†’ (𝑋 + π‘Œ) ∈ 𝑆)
1110ancoms 458 . 2 ((𝑆 ∈ (SubMndβ€˜π‘€) ∧ (𝑋 ∈ 𝑆 ∧ π‘Œ ∈ 𝑆)) β†’ (𝑋 + π‘Œ) ∈ 𝑆)
12113impb 1112 1 ((𝑆 ∈ (SubMndβ€˜π‘€) ∧ 𝑋 ∈ 𝑆 ∧ π‘Œ ∈ 𝑆) β†’ (𝑋 + π‘Œ) ∈ 𝑆)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆ€wral 3055   βŠ† wss 3943  β€˜cfv 6536  (class class class)co 7404  Basecbs 17151  +gcplusg 17204  0gc0g 17392  Mndcmnd 18665  SubMndcsubmnd 18710
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 6488  df-fun 6538  df-fv 6544  df-ov 7407  df-submnd 18712
This theorem is referenced by:  resmhm  18743  mhmima  18748  gsumwsubmcl  18760  submmulgcl  19042  symggen  19388  lsmsubm  19571  smndlsmidm  19574  gsumzadd  19840  gsumzoppg  19862
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