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Theorem mndissubm 18446
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 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 18775. (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 18392 . . . . . . 7 (𝐺 ∈ Mnd → 𝐺 ∈ Mgm)
4 mndmgm 18392 . . . . . . 7 (𝐻 ∈ Mnd → 𝐻 ∈ Mgm)
53, 4anim12i 613 . . . . . 6 ((𝐺 ∈ Mnd ∧ 𝐻 ∈ Mnd) → (𝐺 ∈ Mgm ∧ 𝐻 ∈ Mgm))
65ad2antrr 723 . . . . 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 724 . . . . 5 ((((𝐺 ∈ Mnd ∧ 𝐻 ∈ Mnd) ∧ (𝑆𝐵0𝑆 ∧ (+g𝐻) = ((+g𝐺) ↾ (𝑆 × 𝑆)))) ∧ (𝑎𝑆𝑏𝑆)) → (𝑆𝐵 ∧ (+g𝐻) = ((+g𝐺) ↾ (𝑆 × 𝑆))))
9 simpr 485 . . . . 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 18331 . . . . 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 3123 . . 3 (((𝐺 ∈ Mnd ∧ 𝐻 ∈ Mnd) ∧ (𝑆𝐵0𝑆 ∧ (+g𝐻) = ((+g𝐺) ↾ (𝑆 × 𝑆)))) → ∀𝑎𝑆𝑏𝑆 (𝑎(+g𝐺)𝑏) ∈ 𝑆)
15 mndissubm.z . . . . 5 0 = (0g𝐺)
16 eqid 2738 . . . . 5 (+g𝐺) = (+g𝐺)
1710, 15, 16issubm 18442 . . . 4 (𝐺 ∈ Mnd → (𝑆 ∈ (SubMnd‘𝐺) ↔ (𝑆𝐵0𝑆 ∧ ∀𝑎𝑆𝑏𝑆 (𝑎(+g𝐺)𝑏) ∈ 𝑆)))
1817ad2antrr 723 . . 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 413 1 ((𝐺 ∈ Mnd ∧ 𝐻 ∈ Mnd) → ((𝑆𝐵0𝑆 ∧ (+g𝐻) = ((+g𝐺) ↾ (𝑆 × 𝑆))) → 𝑆 ∈ (SubMnd‘𝐺)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1086   = wceq 1539  wcel 2106  wral 3064  wss 3887   × cxp 5587  cres 5591  cfv 6433  (class class class)co 7275  Basecbs 16912  +gcplusg 16962  0gc0g 17150  Mgmcmgm 18324  Mndcmnd 18385  SubMndcsubmnd 18429
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-res 5601  df-iota 6391  df-fun 6435  df-fv 6441  df-ov 7278  df-mgm 18326  df-sgrp 18375  df-mnd 18386  df-submnd 18431
This theorem is referenced by:  resmndismnd  18447  submefmnd  18534
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