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Theorem grpinvssd 17264
Description: If the base set of a group is contained in the base set of another group, and the group operation of the group is the restriction of the group operation of the other group to its base set, then the elements of the first group have the same inverses in both groups. (Contributed by AV, 15-Mar-2019.)
Hypotheses
Ref Expression
grpidssd.m (𝜑𝑀 ∈ Grp)
grpidssd.s (𝜑𝑆 ∈ Grp)
grpidssd.b 𝐵 = (Base‘𝑆)
grpidssd.c (𝜑𝐵 ⊆ (Base‘𝑀))
grpidssd.o (𝜑 → ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝑀)𝑦) = (𝑥(+g𝑆)𝑦))
Assertion
Ref Expression
grpinvssd (𝜑 → (𝑋𝐵 → ((invg𝑆)‘𝑋) = ((invg𝑀)‘𝑋)))
Distinct variable groups:   𝑥,𝐵,𝑦   𝑥,𝑀,𝑦   𝑥,𝑆,𝑦   𝑥,𝑋,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem grpinvssd
StepHypRef Expression
1 grpidssd.s . . . . . 6 (𝜑𝑆 ∈ Grp)
2 grpidssd.b . . . . . . 7 𝐵 = (Base‘𝑆)
3 eqid 2610 . . . . . . 7 (invg𝑆) = (invg𝑆)
42, 3grpinvcl 17239 . . . . . 6 ((𝑆 ∈ Grp ∧ 𝑋𝐵) → ((invg𝑆)‘𝑋) ∈ 𝐵)
51, 4sylan 487 . . . . 5 ((𝜑𝑋𝐵) → ((invg𝑆)‘𝑋) ∈ 𝐵)
6 simpr 476 . . . . 5 ((𝜑𝑋𝐵) → 𝑋𝐵)
7 grpidssd.o . . . . . 6 (𝜑 → ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝑀)𝑦) = (𝑥(+g𝑆)𝑦))
87adantr 480 . . . . 5 ((𝜑𝑋𝐵) → ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝑀)𝑦) = (𝑥(+g𝑆)𝑦))
9 oveq1 6534 . . . . . . 7 (𝑥 = ((invg𝑆)‘𝑋) → (𝑥(+g𝑀)𝑦) = (((invg𝑆)‘𝑋)(+g𝑀)𝑦))
10 oveq1 6534 . . . . . . 7 (𝑥 = ((invg𝑆)‘𝑋) → (𝑥(+g𝑆)𝑦) = (((invg𝑆)‘𝑋)(+g𝑆)𝑦))
119, 10eqeq12d 2625 . . . . . 6 (𝑥 = ((invg𝑆)‘𝑋) → ((𝑥(+g𝑀)𝑦) = (𝑥(+g𝑆)𝑦) ↔ (((invg𝑆)‘𝑋)(+g𝑀)𝑦) = (((invg𝑆)‘𝑋)(+g𝑆)𝑦)))
12 oveq2 6535 . . . . . . 7 (𝑦 = 𝑋 → (((invg𝑆)‘𝑋)(+g𝑀)𝑦) = (((invg𝑆)‘𝑋)(+g𝑀)𝑋))
13 oveq2 6535 . . . . . . 7 (𝑦 = 𝑋 → (((invg𝑆)‘𝑋)(+g𝑆)𝑦) = (((invg𝑆)‘𝑋)(+g𝑆)𝑋))
1412, 13eqeq12d 2625 . . . . . 6 (𝑦 = 𝑋 → ((((invg𝑆)‘𝑋)(+g𝑀)𝑦) = (((invg𝑆)‘𝑋)(+g𝑆)𝑦) ↔ (((invg𝑆)‘𝑋)(+g𝑀)𝑋) = (((invg𝑆)‘𝑋)(+g𝑆)𝑋)))
1511, 14rspc2va 3294 . . . . 5 (((((invg𝑆)‘𝑋) ∈ 𝐵𝑋𝐵) ∧ ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝑀)𝑦) = (𝑥(+g𝑆)𝑦)) → (((invg𝑆)‘𝑋)(+g𝑀)𝑋) = (((invg𝑆)‘𝑋)(+g𝑆)𝑋))
165, 6, 8, 15syl21anc 1317 . . . 4 ((𝜑𝑋𝐵) → (((invg𝑆)‘𝑋)(+g𝑀)𝑋) = (((invg𝑆)‘𝑋)(+g𝑆)𝑋))
17 eqid 2610 . . . . . 6 (+g𝑆) = (+g𝑆)
18 eqid 2610 . . . . . 6 (0g𝑆) = (0g𝑆)
192, 17, 18, 3grplinv 17240 . . . . 5 ((𝑆 ∈ Grp ∧ 𝑋𝐵) → (((invg𝑆)‘𝑋)(+g𝑆)𝑋) = (0g𝑆))
201, 19sylan 487 . . . 4 ((𝜑𝑋𝐵) → (((invg𝑆)‘𝑋)(+g𝑆)𝑋) = (0g𝑆))
21 grpidssd.m . . . . . . 7 (𝜑𝑀 ∈ Grp)
2221adantr 480 . . . . . 6 ((𝜑𝑋𝐵) → 𝑀 ∈ Grp)
23 grpidssd.c . . . . . . 7 (𝜑𝐵 ⊆ (Base‘𝑀))
2423sselda 3568 . . . . . 6 ((𝜑𝑋𝐵) → 𝑋 ∈ (Base‘𝑀))
25 eqid 2610 . . . . . . 7 (Base‘𝑀) = (Base‘𝑀)
26 eqid 2610 . . . . . . 7 (+g𝑀) = (+g𝑀)
27 eqid 2610 . . . . . . 7 (0g𝑀) = (0g𝑀)
28 eqid 2610 . . . . . . 7 (invg𝑀) = (invg𝑀)
2925, 26, 27, 28grplinv 17240 . . . . . 6 ((𝑀 ∈ Grp ∧ 𝑋 ∈ (Base‘𝑀)) → (((invg𝑀)‘𝑋)(+g𝑀)𝑋) = (0g𝑀))
3022, 24, 29syl2anc 691 . . . . 5 ((𝜑𝑋𝐵) → (((invg𝑀)‘𝑋)(+g𝑀)𝑋) = (0g𝑀))
3121, 1, 2, 23, 7grpidssd 17263 . . . . . 6 (𝜑 → (0g𝑀) = (0g𝑆))
3231adantr 480 . . . . 5 ((𝜑𝑋𝐵) → (0g𝑀) = (0g𝑆))
3330, 32eqtr2d 2645 . . . 4 ((𝜑𝑋𝐵) → (0g𝑆) = (((invg𝑀)‘𝑋)(+g𝑀)𝑋))
3416, 20, 333eqtrd 2648 . . 3 ((𝜑𝑋𝐵) → (((invg𝑆)‘𝑋)(+g𝑀)𝑋) = (((invg𝑀)‘𝑋)(+g𝑀)𝑋))
3523adantr 480 . . . . 5 ((𝜑𝑋𝐵) → 𝐵 ⊆ (Base‘𝑀))
3635, 5sseldd 3569 . . . 4 ((𝜑𝑋𝐵) → ((invg𝑆)‘𝑋) ∈ (Base‘𝑀))
3725, 28grpinvcl 17239 . . . . 5 ((𝑀 ∈ Grp ∧ 𝑋 ∈ (Base‘𝑀)) → ((invg𝑀)‘𝑋) ∈ (Base‘𝑀))
3822, 24, 37syl2anc 691 . . . 4 ((𝜑𝑋𝐵) → ((invg𝑀)‘𝑋) ∈ (Base‘𝑀))
3925, 26grprcan 17227 . . . 4 ((𝑀 ∈ Grp ∧ (((invg𝑆)‘𝑋) ∈ (Base‘𝑀) ∧ ((invg𝑀)‘𝑋) ∈ (Base‘𝑀) ∧ 𝑋 ∈ (Base‘𝑀))) → ((((invg𝑆)‘𝑋)(+g𝑀)𝑋) = (((invg𝑀)‘𝑋)(+g𝑀)𝑋) ↔ ((invg𝑆)‘𝑋) = ((invg𝑀)‘𝑋)))
4022, 36, 38, 24, 39syl13anc 1320 . . 3 ((𝜑𝑋𝐵) → ((((invg𝑆)‘𝑋)(+g𝑀)𝑋) = (((invg𝑀)‘𝑋)(+g𝑀)𝑋) ↔ ((invg𝑆)‘𝑋) = ((invg𝑀)‘𝑋)))
4134, 40mpbid 221 . 2 ((𝜑𝑋𝐵) → ((invg𝑆)‘𝑋) = ((invg𝑀)‘𝑋))
4241ex 449 1 (𝜑 → (𝑋𝐵 → ((invg𝑆)‘𝑋) = ((invg𝑀)‘𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  wral 2896  wss 3540  cfv 5790  (class class class)co 6527  Basecbs 15644  +gcplusg 15717  0gc0g 15872  Grpcgrp 17194  invgcminusg 17195
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4694  ax-sep 4704  ax-nul 4712  ax-pow 4764  ax-pr 4828
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4368  df-iun 4452  df-br 4579  df-opab 4639  df-mpt 4640  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-0g 15874  df-mgm 17014  df-sgrp 17056  df-mnd 17067  df-grp 17197  df-minusg 17198
This theorem is referenced by:  grpissubg  17386
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