Theorem mndissubm 18871

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 19219. (Contributed by AV, 17-Feb-2024.)

Hypotheses

Ref	Expression
mndissubm.b	⊢ 𝐵 = (Base‘𝐺)
mndissubm.s	⊢ 𝑆 = (Base‘𝐻)
mndissubm.z	⊢ 0 = (0g‘𝐺)

Assertion

Ref	Expression
mndissubm	⊢ ((𝐺 ∈ Mnd ∧ 𝐻 ∈ Mnd) → ((𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆))) → 𝑆 ∈ (SubMnd‘𝐺)))

Proof of Theorem mndissubm

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr1 1212	. . 3 ⊢ (((𝐺 ∈ Mnd ∧ 𝐻 ∈ Mnd) ∧ (𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆)))) → 𝑆 ⊆ 𝐵)
2		simpr2 1213	. . 3 ⊢ (((𝐺 ∈ Mnd ∧ 𝐻 ∈ Mnd) ∧ (𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆)))) → 0 ∈ 𝑆)
3		mndmgm 18805	. . . . . . 7 ⊢ (𝐺 ∈ Mnd → 𝐺 ∈ Mgm)
4		mndmgm 18805	. . . . . . 7 ⊢ (𝐻 ∈ Mnd → 𝐻 ∈ Mgm)
5	3, 4	anim12i 624	. . . . . 6 ⊢ ((𝐺 ∈ Mnd ∧ 𝐻 ∈ Mnd) → (𝐺 ∈ Mgm ∧ 𝐻 ∈ Mgm))
6	5	ad2antrr 738	. . . . 5 ⊢ ((((𝐺 ∈ Mnd ∧ 𝐻 ∈ Mnd) ∧ (𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆)))) ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆)) → (𝐺 ∈ Mgm ∧ 𝐻 ∈ Mgm))
7		3simpb 1166	. . . . . 6 ⊢ ((𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆))) → (𝑆 ⊆ 𝐵 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆))))
8	7	ad2antlr 739	. . . . 5 ⊢ ((((𝐺 ∈ Mnd ∧ 𝐻 ∈ Mnd) ∧ (𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆)))) ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆)) → (𝑆 ⊆ 𝐵 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆))))
9		simpr 489	. . . . 5 ⊢ ((((𝐺 ∈ Mnd ∧ 𝐻 ∈ Mnd) ∧ (𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆)))) ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆)) → (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆))
10		mndissubm.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐺)
11		mndissubm.s	. . . . . 6 ⊢ 𝑆 = (Base‘𝐻)
12	10, 11	mgmsscl 18709	. . . . 5 ⊢ (((𝐺 ∈ Mgm ∧ 𝐻 ∈ Mgm) ∧ (𝑆 ⊆ 𝐵 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆))) ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆)) → (𝑎(+g‘𝐺)𝑏) ∈ 𝑆)
13	6, 8, 9, 12	syl3anc 1397	. . . 4 ⊢ ((((𝐺 ∈ Mnd ∧ 𝐻 ∈ Mnd) ∧ (𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆)))) ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑆)) → (𝑎(+g‘𝐺)𝑏) ∈ 𝑆)
14	13	ralrimivva 3207	. . 3 ⊢ (((𝐺 ∈ Mnd ∧ 𝐻 ∈ Mnd) ∧ (𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆)))) → ∀𝑎 ∈ 𝑆 ∀𝑏 ∈ 𝑆 (𝑎(+g‘𝐺)𝑏) ∈ 𝑆)
15		mndissubm.z	. . . . 5 ⊢ 0 = (0g‘𝐺)
16		eqid 2762	. . . . 5 ⊢ (+g‘𝐺) = (+g‘𝐺)
17	10, 15, 16	issubm 18867	. . . 4 ⊢ (𝐺 ∈ Mnd → (𝑆 ∈ (SubMnd‘𝐺) ↔ (𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝑆 ∀𝑏 ∈ 𝑆 (𝑎(+g‘𝐺)𝑏) ∈ 𝑆)))
18	17	ad2antrr 738	. . 3 ⊢ (((𝐺 ∈ Mnd ∧ 𝐻 ∈ Mnd) ∧ (𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆)))) → (𝑆 ∈ (SubMnd‘𝐺) ↔ (𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝑆 ∀𝑏 ∈ 𝑆 (𝑎(+g‘𝐺)𝑏) ∈ 𝑆)))
19	1, 2, 14, 18	mpbir3and 1360	. 2 ⊢ (((𝐺 ∈ Mnd ∧ 𝐻 ∈ Mnd) ∧ (𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆)))) → 𝑆 ∈ (SubMnd‘𝐺))
20	19	ex 417	1 ⊢ ((𝐺 ∈ Mnd ∧ 𝐻 ∈ Mnd) → ((𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆))) → 𝑆 ∈ (SubMnd‘𝐺)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1102   = wceq 1569   ∈ wcel 2142  ∀wral 3078   ⊆ wss 3904   × cxp 5658   ↾ cres 5662  ‘cfv 6536  (class class class)co 7412  Basecbs 17275  +_gcplusg 17316  0_gc0g 17498  Mgmcmgm 18702  Mndcmnd 18798  SubMndcsubmnd 18846

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-sbc 3744  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-res 5672  df-iota 6492  df-fun 6538  df-fv 6544  df-ov 7415  df-mgm 18704  df-sgrp 18783  df-mnd 18799  df-submnd 18848

This theorem is used by:  resmndismnd  18872  submefmnd  18960