MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mndissubm Structured version   Visualization version   GIF version

Theorem mndissubm 18088
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 18417. (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 18034 . . . . . . 7 (𝐺 ∈ Mnd → 𝐺 ∈ Mgm)
4 mndmgm 18034 . . . . . . 7 (𝐻 ∈ Mnd → 𝐻 ∈ Mgm)
53, 4anim12i 616 . . . . . 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 488 . . . . 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 17973 . . . . 5 (((𝐺 ∈ Mgm ∧ 𝐻 ∈ Mgm) ∧ (𝑆𝐵 ∧ (+g𝐻) = ((+g𝐺) ↾ (𝑆 × 𝑆))) ∧ (𝑎𝑆𝑏𝑆)) → (𝑎(+g𝐺)𝑏) ∈ 𝑆)
136, 8, 9, 12syl3anc 1372 . . . 4 ((((𝐺 ∈ Mnd ∧ 𝐻 ∈ Mnd) ∧ (𝑆𝐵0𝑆 ∧ (+g𝐻) = ((+g𝐺) ↾ (𝑆 × 𝑆)))) ∧ (𝑎𝑆𝑏𝑆)) → (𝑎(+g𝐺)𝑏) ∈ 𝑆)
1413ralrimivva 3103 . . 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 18084 . . . 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 416 1 ((𝐺 ∈ Mnd ∧ 𝐻 ∈ Mnd) → ((𝑆𝐵0𝑆 ∧ (+g𝐻) = ((+g𝐺) ↾ (𝑆 × 𝑆))) → 𝑆 ∈ (SubMnd‘𝐺)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1088   = wceq 1542  wcel 2114  wral 3053  wss 3843   × cxp 5523  cres 5527  cfv 6339  (class class class)co 7170  Basecbs 16586  +gcplusg 16668  0gc0g 16816  Mgmcmgm 17966  Mndcmnd 18027  SubMndcsubmnd 18071
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-sbc 3681  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-res 5537  df-iota 6297  df-fun 6341  df-fv 6347  df-ov 7173  df-mgm 17968  df-sgrp 18017  df-mnd 18028  df-submnd 18073
This theorem is referenced by:  resmndismnd  18089  submefmnd  18176
  Copyright terms: Public domain W3C validator