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Theorem submss 12821
Description: Submonoids are subsets of the base set. (Contributed by Mario Carneiro, 7-Mar-2015.)
Hypothesis
Ref Expression
submss.b 𝐵 = (Base‘𝑀)
Assertion
Ref Expression
submss (𝑆 ∈ (SubMnd‘𝑀) → 𝑆𝐵)

Proof of Theorem submss
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 submrcl 12816 . . . 4 (𝑆 ∈ (SubMnd‘𝑀) → 𝑀 ∈ Mnd)
2 submss.b . . . . 5 𝐵 = (Base‘𝑀)
3 eqid 2177 . . . . 5 (0g𝑀) = (0g𝑀)
4 eqid 2177 . . . . 5 (+g𝑀) = (+g𝑀)
52, 3, 4issubm 12817 . . . 4 (𝑀 ∈ Mnd → (𝑆 ∈ (SubMnd‘𝑀) ↔ (𝑆𝐵 ∧ (0g𝑀) ∈ 𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥(+g𝑀)𝑦) ∈ 𝑆)))
61, 5syl 14 . . 3 (𝑆 ∈ (SubMnd‘𝑀) → (𝑆 ∈ (SubMnd‘𝑀) ↔ (𝑆𝐵 ∧ (0g𝑀) ∈ 𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥(+g𝑀)𝑦) ∈ 𝑆)))
76ibi 176 . 2 (𝑆 ∈ (SubMnd‘𝑀) → (𝑆𝐵 ∧ (0g𝑀) ∈ 𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥(+g𝑀)𝑦) ∈ 𝑆))
87simp1d 1009 1 (𝑆 ∈ (SubMnd‘𝑀) → 𝑆𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105  w3a 978   = wceq 1353  wcel 2148  wral 2455  wss 3129  cfv 5216  (class class class)co 5874  Basecbs 12456  +gcplusg 12530  0gc0g 12695  Mndcmnd 12771  SubMndcsubmnd 12804
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-cnex 7901  ax-resscn 7902  ax-1re 7904  ax-addrcl 7907
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-fv 5224  df-ov 5877  df-inn 8918  df-ndx 12459  df-slot 12460  df-base 12462  df-submnd 12806
This theorem is referenced by:  mhmima  12829  submmulgcl  12979
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