Theorem mndissubm 18741

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 19085. (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 1195	. . 3 ⊢ (((𝐺 ∈ Mnd ∧ 𝐻 ∈ Mnd) ∧ (𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆)))) → 𝑆 ⊆ 𝐵)
2		simpr2 1196	. . 3 ⊢ (((𝐺 ∈ Mnd ∧ 𝐻 ∈ Mnd) ∧ (𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆)))) → 0 ∈ 𝑆)
3		mndmgm 18675	. . . . . . 7 ⊢ (𝐺 ∈ Mnd → 𝐺 ∈ Mgm)
4		mndmgm 18675	. . . . . . 7 ⊢ (𝐻 ∈ Mnd → 𝐻 ∈ Mgm)
5	3, 4	anim12i 613	. . . . . 6 ⊢ ((𝐺 ∈ Mnd ∧ 𝐻 ∈ Mnd) → (𝐺 ∈ Mgm ∧ 𝐻 ∈ Mgm))
6	5	ad2antrr 726	. . . . 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 1149	. . . . . 6 ⊢ ((𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆))) → (𝑆 ⊆ 𝐵 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆))))
8	7	ad2antlr 727	. . . . 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 484	. . . . 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 18579	. . . . 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 1373	. . . 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 3181	. . 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 2730	. . . . 5 ⊢ (+g‘𝐺) = (+g‘𝐺)
17	10, 15, 16	issubm 18737	. . . 4 ⊢ (𝐺 ∈ Mnd → (𝑆 ∈ (SubMnd‘𝐺) ↔ (𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ∧ ∀𝑎 ∈ 𝑆 ∀𝑏 ∈ 𝑆 (𝑎(+g‘𝐺)𝑏) ∈ 𝑆)))
18	17	ad2antrr 726	. . 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 1343	. 2 ⊢ (((𝐺 ∈ Mnd ∧ 𝐻 ∈ Mnd) ∧ (𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆)))) → 𝑆 ∈ (SubMnd‘𝐺))
20	19	ex 412	1 ⊢ ((𝐺 ∈ Mnd ∧ 𝐻 ∈ Mnd) → ((𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆))) → 𝑆 ∈ (SubMnd‘𝐺)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3045   ⊆ wss 3917   × cxp 5639   ↾ cres 5643  ‘cfv 6514  (class class class)co 7390  Basecbs 17186  +_gcplusg 17227  0_gc0g 17409  Mgmcmgm 18572  Mndcmnd 18668  SubMndcsubmnd 18716

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-res 5653  df-iota 6467  df-fun 6516  df-fv 6522  df-ov 7393  df-mgm 18574  df-sgrp 18653  df-mnd 18669  df-submnd 18718

This theorem is referenced by:  resmndismnd  18742  submefmnd  18829