Theorem mgmsscl 18736

Description: If the base set of a magma is contained in the base set of another magma, and the group operation of the magma is the restriction of the group operation of the other magma to its base set, then the base set of the magma is closed under the group operation of the other magma. Formerly part of proof of grpissubg 19271. (Contributed by AV, 17-Feb-2024.)

Hypotheses

Ref	Expression
mgmsscl.b	⊢ 𝐵 = (Base‘𝐺)
mgmsscl.s	⊢ 𝑆 = (Base‘𝐻)

Assertion

Ref	Expression
mgmsscl	⊢ (((𝐺 ∈ Mgm ∧ 𝐻 ∈ Mgm) ∧ (𝑆 ⊆ 𝐵 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆))) ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆)) → (𝑋(+g‘𝐺)𝑌) ∈ 𝑆)

Proof of Theorem mgmsscl

Step	Hyp	Ref	Expression
1		ovres 7580	. . 3 ⊢ ((𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (𝑋((+g‘𝐺) ↾ (𝑆 × 𝑆))𝑌) = (𝑋(+g‘𝐺)𝑌))
2	1	3ad2ant3 1153	. 2 ⊢ (((𝐺 ∈ Mgm ∧ 𝐻 ∈ Mgm) ∧ (𝑆 ⊆ 𝐵 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆))) ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆)) → (𝑋((+g‘𝐺) ↾ (𝑆 × 𝑆))𝑌) = (𝑋(+g‘𝐺)𝑌))
3		simp1r 1217	. . . . 5 ⊢ (((𝐺 ∈ Mgm ∧ 𝐻 ∈ Mgm) ∧ (𝑆 ⊆ 𝐵 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆))) ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆)) → 𝐻 ∈ Mgm)
4		simp3 1156	. . . . 5 ⊢ (((𝐺 ∈ Mgm ∧ 𝐻 ∈ Mgm) ∧ (𝑆 ⊆ 𝐵 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆))) ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆)) → (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆))
5		3anass 1111	. . . . 5 ⊢ ((𝐻 ∈ Mgm ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ↔ (𝐻 ∈ Mgm ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆)))
6	3, 4, 5	sylanbrc 595	. . . 4 ⊢ (((𝐺 ∈ Mgm ∧ 𝐻 ∈ Mgm) ∧ (𝑆 ⊆ 𝐵 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆))) ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆)) → (𝐻 ∈ Mgm ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆))
7		mgmsscl.s	. . . . 5 ⊢ 𝑆 = (Base‘𝐻)
8		eqid 2760	. . . . 5 ⊢ (+g‘𝐻) = (+g‘𝐻)
9	7, 8	mgmcl 18734	. . . 4 ⊢ ((𝐻 ∈ Mgm ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (𝑋(+g‘𝐻)𝑌) ∈ 𝑆)
10	6, 9	syl 18	. . 3 ⊢ (((𝐺 ∈ Mgm ∧ 𝐻 ∈ Mgm) ∧ (𝑆 ⊆ 𝐵 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆))) ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆)) → (𝑋(+g‘𝐻)𝑌) ∈ 𝑆)
11		oveq 7420	. . . . . . 7 ⊢ (((+g‘𝐺) ↾ (𝑆 × 𝑆)) = (+g‘𝐻) → (𝑋((+g‘𝐺) ↾ (𝑆 × 𝑆))𝑌) = (𝑋(+g‘𝐻)𝑌))
12	11	eleq1d 2845	. . . . . 6 ⊢ (((+g‘𝐺) ↾ (𝑆 × 𝑆)) = (+g‘𝐻) → ((𝑋((+g‘𝐺) ↾ (𝑆 × 𝑆))𝑌) ∈ 𝑆 ↔ (𝑋(+g‘𝐻)𝑌) ∈ 𝑆))
13	12	eqcoms 2768	. . . . 5 ⊢ ((+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆)) → ((𝑋((+g‘𝐺) ↾ (𝑆 × 𝑆))𝑌) ∈ 𝑆 ↔ (𝑋(+g‘𝐻)𝑌) ∈ 𝑆))
14	13	adantl 487	. . . 4 ⊢ ((𝑆 ⊆ 𝐵 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆))) → ((𝑋((+g‘𝐺) ↾ (𝑆 × 𝑆))𝑌) ∈ 𝑆 ↔ (𝑋(+g‘𝐻)𝑌) ∈ 𝑆))
15	14	3ad2ant2 1152	. . 3 ⊢ (((𝐺 ∈ Mgm ∧ 𝐻 ∈ Mgm) ∧ (𝑆 ⊆ 𝐵 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆))) ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆)) → ((𝑋((+g‘𝐺) ↾ (𝑆 × 𝑆))𝑌) ∈ 𝑆 ↔ (𝑋(+g‘𝐻)𝑌) ∈ 𝑆))
16	10, 15	mpbird 260	. 2 ⊢ (((𝐺 ∈ Mgm ∧ 𝐻 ∈ Mgm) ∧ (𝑆 ⊆ 𝐵 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆))) ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆)) → (𝑋((+g‘𝐺) ↾ (𝑆 × 𝑆))𝑌) ∈ 𝑆)
17	2, 16	eqeltrrd 2861	1 ⊢ (((𝐺 ∈ Mgm ∧ 𝐻 ∈ Mgm) ∧ (𝑆 ⊆ 𝐵 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆))) ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆)) → (𝑋(+g‘𝐺)𝑌) ∈ 𝑆)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ⊆ wss 3899   × cxp 5653   ↾ cres 5657  ‘cfv 6533  (class class class)co 7414  Basecbs 17302  +_gcplusg 17343  Mgmcmgm 18729

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-xp 5661  df-res 5667  df-iota 6489  df-fv 6541  df-ov 7417  df-mgm 18731

This theorem is used by:  mndissubm  18916  grpissubg  19271