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Theorem mgmsscl 17849
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 18291. (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) ∧ (𝑆𝐵 ∧ (+g𝐻) = ((+g𝐺) ↾ (𝑆 × 𝑆))) ∧ (𝑋𝑆𝑌𝑆)) → (𝑋(+g𝐺)𝑌) ∈ 𝑆)

Proof of Theorem mgmsscl
StepHypRef Expression
1 ovres 7306 . . 3 ((𝑋𝑆𝑌𝑆) → (𝑋((+g𝐺) ↾ (𝑆 × 𝑆))𝑌) = (𝑋(+g𝐺)𝑌))
213ad2ant3 1130 . 2 (((𝐺 ∈ Mgm ∧ 𝐻 ∈ Mgm) ∧ (𝑆𝐵 ∧ (+g𝐻) = ((+g𝐺) ↾ (𝑆 × 𝑆))) ∧ (𝑋𝑆𝑌𝑆)) → (𝑋((+g𝐺) ↾ (𝑆 × 𝑆))𝑌) = (𝑋(+g𝐺)𝑌))
3 simp1r 1193 . . . . 5 (((𝐺 ∈ Mgm ∧ 𝐻 ∈ Mgm) ∧ (𝑆𝐵 ∧ (+g𝐻) = ((+g𝐺) ↾ (𝑆 × 𝑆))) ∧ (𝑋𝑆𝑌𝑆)) → 𝐻 ∈ Mgm)
4 simp3 1133 . . . . 5 (((𝐺 ∈ Mgm ∧ 𝐻 ∈ Mgm) ∧ (𝑆𝐵 ∧ (+g𝐻) = ((+g𝐺) ↾ (𝑆 × 𝑆))) ∧ (𝑋𝑆𝑌𝑆)) → (𝑋𝑆𝑌𝑆))
5 3anass 1090 . . . . 5 ((𝐻 ∈ Mgm ∧ 𝑋𝑆𝑌𝑆) ↔ (𝐻 ∈ Mgm ∧ (𝑋𝑆𝑌𝑆)))
63, 4, 5sylanbrc 585 . . . 4 (((𝐺 ∈ Mgm ∧ 𝐻 ∈ Mgm) ∧ (𝑆𝐵 ∧ (+g𝐻) = ((+g𝐺) ↾ (𝑆 × 𝑆))) ∧ (𝑋𝑆𝑌𝑆)) → (𝐻 ∈ Mgm ∧ 𝑋𝑆𝑌𝑆))
7 mgmsscl.s . . . . 5 𝑆 = (Base‘𝐻)
8 eqid 2819 . . . . 5 (+g𝐻) = (+g𝐻)
97, 8mgmcl 17847 . . . 4 ((𝐻 ∈ Mgm ∧ 𝑋𝑆𝑌𝑆) → (𝑋(+g𝐻)𝑌) ∈ 𝑆)
106, 9syl 17 . . 3 (((𝐺 ∈ Mgm ∧ 𝐻 ∈ Mgm) ∧ (𝑆𝐵 ∧ (+g𝐻) = ((+g𝐺) ↾ (𝑆 × 𝑆))) ∧ (𝑋𝑆𝑌𝑆)) → (𝑋(+g𝐻)𝑌) ∈ 𝑆)
11 oveq 7154 . . . . . . 7 (((+g𝐺) ↾ (𝑆 × 𝑆)) = (+g𝐻) → (𝑋((+g𝐺) ↾ (𝑆 × 𝑆))𝑌) = (𝑋(+g𝐻)𝑌))
1211eleq1d 2895 . . . . . 6 (((+g𝐺) ↾ (𝑆 × 𝑆)) = (+g𝐻) → ((𝑋((+g𝐺) ↾ (𝑆 × 𝑆))𝑌) ∈ 𝑆 ↔ (𝑋(+g𝐻)𝑌) ∈ 𝑆))
1312eqcoms 2827 . . . . 5 ((+g𝐻) = ((+g𝐺) ↾ (𝑆 × 𝑆)) → ((𝑋((+g𝐺) ↾ (𝑆 × 𝑆))𝑌) ∈ 𝑆 ↔ (𝑋(+g𝐻)𝑌) ∈ 𝑆))
1413adantl 484 . . . 4 ((𝑆𝐵 ∧ (+g𝐻) = ((+g𝐺) ↾ (𝑆 × 𝑆))) → ((𝑋((+g𝐺) ↾ (𝑆 × 𝑆))𝑌) ∈ 𝑆 ↔ (𝑋(+g𝐻)𝑌) ∈ 𝑆))
15143ad2ant2 1129 . . 3 (((𝐺 ∈ Mgm ∧ 𝐻 ∈ Mgm) ∧ (𝑆𝐵 ∧ (+g𝐻) = ((+g𝐺) ↾ (𝑆 × 𝑆))) ∧ (𝑋𝑆𝑌𝑆)) → ((𝑋((+g𝐺) ↾ (𝑆 × 𝑆))𝑌) ∈ 𝑆 ↔ (𝑋(+g𝐻)𝑌) ∈ 𝑆))
1610, 15mpbird 259 . 2 (((𝐺 ∈ Mgm ∧ 𝐻 ∈ Mgm) ∧ (𝑆𝐵 ∧ (+g𝐻) = ((+g𝐺) ↾ (𝑆 × 𝑆))) ∧ (𝑋𝑆𝑌𝑆)) → (𝑋((+g𝐺) ↾ (𝑆 × 𝑆))𝑌) ∈ 𝑆)
172, 16eqeltrrd 2912 1 (((𝐺 ∈ Mgm ∧ 𝐻 ∈ Mgm) ∧ (𝑆𝐵 ∧ (+g𝐻) = ((+g𝐺) ↾ (𝑆 × 𝑆))) ∧ (𝑋𝑆𝑌𝑆)) → (𝑋(+g𝐺)𝑌) ∈ 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  w3a 1082   = wceq 1531  wcel 2108  wss 3934   × cxp 5546  cres 5550  cfv 6348  (class class class)co 7148  Basecbs 16475  +gcplusg 16557  Mgmcmgm 17842
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-xp 5554  df-res 5560  df-iota 6307  df-fv 6356  df-ov 7151  df-mgm 17844
This theorem is referenced by:  mndissubm  17964  grpissubg  18291
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