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Theorem submcl 18628
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 18618 . . . . . . 7 (𝑆 ∈ (SubMndβ€˜π‘€) β†’ 𝑀 ∈ Mnd)
2 eqid 2733 . . . . . . . 8 (Baseβ€˜π‘€) = (Baseβ€˜π‘€)
3 eqid 2733 . . . . . . . 8 (0gβ€˜π‘€) = (0gβ€˜π‘€)
4 submcl.p . . . . . . . 8 + = (+gβ€˜π‘€)
52, 3, 4issubm 18619 . . . . . . 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 1145 . . . 4 (𝑆 ∈ (SubMndβ€˜π‘€) β†’ βˆ€π‘₯ ∈ 𝑆 βˆ€π‘¦ ∈ 𝑆 (π‘₯ + 𝑦) ∈ 𝑆)
9 ovrspc2v 7384 . . . 4 (((𝑋 ∈ 𝑆 ∧ π‘Œ ∈ 𝑆) ∧ βˆ€π‘₯ ∈ 𝑆 βˆ€π‘¦ ∈ 𝑆 (π‘₯ + 𝑦) ∈ 𝑆) β†’ (𝑋 + π‘Œ) ∈ 𝑆)
108, 9sylan2 594 . . 3 (((𝑋 ∈ 𝑆 ∧ π‘Œ ∈ 𝑆) ∧ 𝑆 ∈ (SubMndβ€˜π‘€)) β†’ (𝑋 + π‘Œ) ∈ 𝑆)
1110ancoms 460 . 2 ((𝑆 ∈ (SubMndβ€˜π‘€) ∧ (𝑋 ∈ 𝑆 ∧ π‘Œ ∈ 𝑆)) β†’ (𝑋 + π‘Œ) ∈ 𝑆)
12113impb 1116 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061   βŠ† wss 3911  β€˜cfv 6497  (class class class)co 7358  Basecbs 17088  +gcplusg 17138  0gc0g 17326  Mndcmnd 18561  SubMndcsubmnd 18605
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fv 6505  df-ov 7361  df-submnd 18607
This theorem is referenced by:  resmhm  18636  mhmima  18640  gsumwsubmcl  18652  submmulgcl  18924  symggen  19257  lsmsubm  19440  smndlsmidm  19443  gsumzadd  19704  gsumzoppg  19726
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