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Theorem submcl 18764
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 18754 . . . . . . 7 (𝑆 ∈ (SubMndβ€˜π‘€) β†’ 𝑀 ∈ Mnd)
2 eqid 2728 . . . . . . . 8 (Baseβ€˜π‘€) = (Baseβ€˜π‘€)
3 eqid 2728 . . . . . . . 8 (0gβ€˜π‘€) = (0gβ€˜π‘€)
4 submcl.p . . . . . . . 8 + = (+gβ€˜π‘€)
52, 3, 4issubm 18755 . . . . . . 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 1142 . . . 4 (𝑆 ∈ (SubMndβ€˜π‘€) β†’ βˆ€π‘₯ ∈ 𝑆 βˆ€π‘¦ ∈ 𝑆 (π‘₯ + 𝑦) ∈ 𝑆)
9 ovrspc2v 7446 . . . 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 1113 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 1085   = wceq 1534   ∈ wcel 2099  βˆ€wral 3058   βŠ† wss 3947  β€˜cfv 6548  (class class class)co 7420  Basecbs 17180  +gcplusg 17233  0gc0g 17421  Mndcmnd 18694  SubMndcsubmnd 18739
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fv 6556  df-ov 7423  df-submnd 18741
This theorem is referenced by:  resmhm  18772  mhmima  18777  gsumwsubmcl  18789  submmulgcl  19072  symggen  19425  lsmsubm  19608  smndlsmidm  19611  gsumzadd  19877  gsumzoppg  19899  submcld  32768  erler  32992  rlocaddval  32995  rlocmulval  32996
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