Theorem mgmsscl 18685

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 19188. (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 7618	. . 3 ⊢ ((𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (𝑋((+g‘𝐺) ↾ (𝑆 × 𝑆))𝑌) = (𝑋(+g‘𝐺)𝑌))
2	1	3ad2ant3 1135	. 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 1198	. . . . 5 ⊢ (((𝐺 ∈ Mgm ∧ 𝐻 ∈ Mgm) ∧ (𝑆 ⊆ 𝐵 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆))) ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆)) → 𝐻 ∈ Mgm)
4		simp3 1138	. . . . 5 ⊢ (((𝐺 ∈ Mgm ∧ 𝐻 ∈ Mgm) ∧ (𝑆 ⊆ 𝐵 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆))) ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆)) → (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆))
5		3anass 1095	. . . . 5 ⊢ ((𝐻 ∈ Mgm ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ↔ (𝐻 ∈ Mgm ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆)))
6	3, 4, 5	sylanbrc 582	. . . 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 2740	. . . . 5 ⊢ (+g‘𝐻) = (+g‘𝐻)
9	7, 8	mgmcl 18683	. . . 4 ⊢ ((𝐻 ∈ Mgm ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (𝑋(+g‘𝐻)𝑌) ∈ 𝑆)
10	6, 9	syl 17	. . 3 ⊢ (((𝐺 ∈ Mgm ∧ 𝐻 ∈ Mgm) ∧ (𝑆 ⊆ 𝐵 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆))) ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆)) → (𝑋(+g‘𝐻)𝑌) ∈ 𝑆)
11		oveq 7456	. . . . . . 7 ⊢ (((+g‘𝐺) ↾ (𝑆 × 𝑆)) = (+g‘𝐻) → (𝑋((+g‘𝐺) ↾ (𝑆 × 𝑆))𝑌) = (𝑋(+g‘𝐻)𝑌))
12	11	eleq1d 2829	. . . . . 6 ⊢ (((+g‘𝐺) ↾ (𝑆 × 𝑆)) = (+g‘𝐻) → ((𝑋((+g‘𝐺) ↾ (𝑆 × 𝑆))𝑌) ∈ 𝑆 ↔ (𝑋(+g‘𝐻)𝑌) ∈ 𝑆))
13	12	eqcoms 2748	. . . . 5 ⊢ ((+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆)) → ((𝑋((+g‘𝐺) ↾ (𝑆 × 𝑆))𝑌) ∈ 𝑆 ↔ (𝑋(+g‘𝐻)𝑌) ∈ 𝑆))
14	13	adantl 481	. . . 4 ⊢ ((𝑆 ⊆ 𝐵 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆))) → ((𝑋((+g‘𝐺) ↾ (𝑆 × 𝑆))𝑌) ∈ 𝑆 ↔ (𝑋(+g‘𝐻)𝑌) ∈ 𝑆))
15	14	3ad2ant2 1134	. . 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 257	. 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 2845	1 ⊢ (((𝐺 ∈ Mgm ∧ 𝐻 ∈ Mgm) ∧ (𝑆 ⊆ 𝐵 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆))) ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆)) → (𝑋(+g‘𝐺)𝑌) ∈ 𝑆)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108   ⊆ wss 3976   × cxp 5698   ↾ cres 5702  ‘cfv 6575  (class class class)co 7450  Basecbs 17260  +_gcplusg 17313  Mgmcmgm 18678

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-xp 5706  df-res 5712  df-iota 6527  df-fv 6583  df-ov 7453  df-mgm 18680

This theorem is referenced by:  mndissubm  18844  grpissubg  19188