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Theorem mndissubm 18820
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 19164. (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 1195 . . 3 (((𝐺 ∈ Mnd ∧ 𝐻 ∈ Mnd) ∧ (𝑆𝐵0𝑆 ∧ (+g𝐻) = ((+g𝐺) ↾ (𝑆 × 𝑆)))) → 𝑆𝐵)
2 simpr2 1196 . . 3 (((𝐺 ∈ Mnd ∧ 𝐻 ∈ Mnd) ∧ (𝑆𝐵0𝑆 ∧ (+g𝐻) = ((+g𝐺) ↾ (𝑆 × 𝑆)))) → 0𝑆)
3 mndmgm 18754 . . . . . . 7 (𝐺 ∈ Mnd → 𝐺 ∈ Mgm)
4 mndmgm 18754 . . . . . . 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 1150 . . . . . 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 18658 . . . . 5 (((𝐺 ∈ Mgm ∧ 𝐻 ∈ Mgm) ∧ (𝑆𝐵 ∧ (+g𝐻) = ((+g𝐺) ↾ (𝑆 × 𝑆))) ∧ (𝑎𝑆𝑏𝑆)) → (𝑎(+g𝐺)𝑏) ∈ 𝑆)
136, 8, 9, 12syl3anc 1373 . . . 4 ((((𝐺 ∈ Mnd ∧ 𝐻 ∈ Mnd) ∧ (𝑆𝐵0𝑆 ∧ (+g𝐻) = ((+g𝐺) ↾ (𝑆 × 𝑆)))) ∧ (𝑎𝑆𝑏𝑆)) → (𝑎(+g𝐺)𝑏) ∈ 𝑆)
1413ralrimivva 3202 . . 3 (((𝐺 ∈ Mnd ∧ 𝐻 ∈ Mnd) ∧ (𝑆𝐵0𝑆 ∧ (+g𝐻) = ((+g𝐺) ↾ (𝑆 × 𝑆)))) → ∀𝑎𝑆𝑏𝑆 (𝑎(+g𝐺)𝑏) ∈ 𝑆)
15 mndissubm.z . . . . 5 0 = (0g𝐺)
16 eqid 2737 . . . . 5 (+g𝐺) = (+g𝐺)
1710, 15, 16issubm 18816 . . . 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 1343 . 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 1087   = wceq 1540  wcel 2108  wral 3061  wss 3951   × cxp 5683  cres 5687  cfv 6561  (class class class)co 7431  Basecbs 17247  +gcplusg 17297  0gc0g 17484  Mgmcmgm 18651  Mndcmnd 18747  SubMndcsubmnd 18795
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-res 5697  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-submnd 18797
This theorem is referenced by:  resmndismnd  18821  submefmnd  18908
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