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Theorem mndissubm 18833
Description: If the base set of a monoid is contained in the base set of another monoid, and the group operation of the monoid is the restriction of the group operation of the other monoid to its base set, and the identity element of the other monoid is contained in the base set of the monoid, then the (base set of the) monoid is a submonoid of the other monoid. Analogous to grpissubg 19177. (Contributed by AV, 17-Feb-2024.)
Hypotheses
Ref Expression
mndissubm.b 𝐵 = (Base‘𝐺)
mndissubm.s 𝑆 = (Base‘𝐻)
mndissubm.z 0 = (0g𝐺)
Assertion
Ref Expression
mndissubm ((𝐺 ∈ Mnd ∧ 𝐻 ∈ Mnd) → ((𝑆𝐵0𝑆 ∧ (+g𝐻) = ((+g𝐺) ↾ (𝑆 × 𝑆))) → 𝑆 ∈ (SubMnd‘𝐺)))

Proof of Theorem mndissubm
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr1 1193 . . 3 (((𝐺 ∈ Mnd ∧ 𝐻 ∈ Mnd) ∧ (𝑆𝐵0𝑆 ∧ (+g𝐻) = ((+g𝐺) ↾ (𝑆 × 𝑆)))) → 𝑆𝐵)
2 simpr2 1194 . . 3 (((𝐺 ∈ Mnd ∧ 𝐻 ∈ Mnd) ∧ (𝑆𝐵0𝑆 ∧ (+g𝐻) = ((+g𝐺) ↾ (𝑆 × 𝑆)))) → 0𝑆)
3 mndmgm 18767 . . . . . . 7 (𝐺 ∈ Mnd → 𝐺 ∈ Mgm)
4 mndmgm 18767 . . . . . . 7 (𝐻 ∈ Mnd → 𝐻 ∈ Mgm)
53, 4anim12i 613 . . . . . 6 ((𝐺 ∈ Mnd ∧ 𝐻 ∈ Mnd) → (𝐺 ∈ Mgm ∧ 𝐻 ∈ Mgm))
65ad2antrr 726 . . . . 5 ((((𝐺 ∈ Mnd ∧ 𝐻 ∈ Mnd) ∧ (𝑆𝐵0𝑆 ∧ (+g𝐻) = ((+g𝐺) ↾ (𝑆 × 𝑆)))) ∧ (𝑎𝑆𝑏𝑆)) → (𝐺 ∈ Mgm ∧ 𝐻 ∈ Mgm))
7 3simpb 1148 . . . . . 6 ((𝑆𝐵0𝑆 ∧ (+g𝐻) = ((+g𝐺) ↾ (𝑆 × 𝑆))) → (𝑆𝐵 ∧ (+g𝐻) = ((+g𝐺) ↾ (𝑆 × 𝑆))))
87ad2antlr 727 . . . . 5 ((((𝐺 ∈ Mnd ∧ 𝐻 ∈ Mnd) ∧ (𝑆𝐵0𝑆 ∧ (+g𝐻) = ((+g𝐺) ↾ (𝑆 × 𝑆)))) ∧ (𝑎𝑆𝑏𝑆)) → (𝑆𝐵 ∧ (+g𝐻) = ((+g𝐺) ↾ (𝑆 × 𝑆))))
9 simpr 484 . . . . 5 ((((𝐺 ∈ Mnd ∧ 𝐻 ∈ Mnd) ∧ (𝑆𝐵0𝑆 ∧ (+g𝐻) = ((+g𝐺) ↾ (𝑆 × 𝑆)))) ∧ (𝑎𝑆𝑏𝑆)) → (𝑎𝑆𝑏𝑆))
10 mndissubm.b . . . . . 6 𝐵 = (Base‘𝐺)
11 mndissubm.s . . . . . 6 𝑆 = (Base‘𝐻)
1210, 11mgmsscl 18671 . . . . 5 (((𝐺 ∈ Mgm ∧ 𝐻 ∈ Mgm) ∧ (𝑆𝐵 ∧ (+g𝐻) = ((+g𝐺) ↾ (𝑆 × 𝑆))) ∧ (𝑎𝑆𝑏𝑆)) → (𝑎(+g𝐺)𝑏) ∈ 𝑆)
136, 8, 9, 12syl3anc 1370 . . . 4 ((((𝐺 ∈ Mnd ∧ 𝐻 ∈ Mnd) ∧ (𝑆𝐵0𝑆 ∧ (+g𝐻) = ((+g𝐺) ↾ (𝑆 × 𝑆)))) ∧ (𝑎𝑆𝑏𝑆)) → (𝑎(+g𝐺)𝑏) ∈ 𝑆)
1413ralrimivva 3200 . . 3 (((𝐺 ∈ Mnd ∧ 𝐻 ∈ Mnd) ∧ (𝑆𝐵0𝑆 ∧ (+g𝐻) = ((+g𝐺) ↾ (𝑆 × 𝑆)))) → ∀𝑎𝑆𝑏𝑆 (𝑎(+g𝐺)𝑏) ∈ 𝑆)
15 mndissubm.z . . . . 5 0 = (0g𝐺)
16 eqid 2735 . . . . 5 (+g𝐺) = (+g𝐺)
1710, 15, 16issubm 18829 . . . 4 (𝐺 ∈ Mnd → (𝑆 ∈ (SubMnd‘𝐺) ↔ (𝑆𝐵0𝑆 ∧ ∀𝑎𝑆𝑏𝑆 (𝑎(+g𝐺)𝑏) ∈ 𝑆)))
1817ad2antrr 726 . . 3 (((𝐺 ∈ Mnd ∧ 𝐻 ∈ Mnd) ∧ (𝑆𝐵0𝑆 ∧ (+g𝐻) = ((+g𝐺) ↾ (𝑆 × 𝑆)))) → (𝑆 ∈ (SubMnd‘𝐺) ↔ (𝑆𝐵0𝑆 ∧ ∀𝑎𝑆𝑏𝑆 (𝑎(+g𝐺)𝑏) ∈ 𝑆)))
191, 2, 14, 18mpbir3and 1341 . 2 (((𝐺 ∈ Mnd ∧ 𝐻 ∈ Mnd) ∧ (𝑆𝐵0𝑆 ∧ (+g𝐻) = ((+g𝐺) ↾ (𝑆 × 𝑆)))) → 𝑆 ∈ (SubMnd‘𝐺))
2019ex 412 1 ((𝐺 ∈ Mnd ∧ 𝐻 ∈ Mnd) → ((𝑆𝐵0𝑆 ∧ (+g𝐻) = ((+g𝐺) ↾ (𝑆 × 𝑆))) → 𝑆 ∈ (SubMnd‘𝐺)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wral 3059  wss 3963   × cxp 5687  cres 5691  cfv 6563  (class class class)co 7431  Basecbs 17245  +gcplusg 17298  0gc0g 17486  Mgmcmgm 18664  Mndcmnd 18760  SubMndcsubmnd 18808
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-res 5701  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-submnd 18810
This theorem is referenced by:  resmndismnd  18834  submefmnd  18921
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