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Theorem grpinvssd 17700
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 2771 . . . . . . 7 (invg𝑆) = (invg𝑆)
42, 3grpinvcl 17675 . . . . . 6 ((𝑆 ∈ Grp ∧ 𝑋𝐵) → ((invg𝑆)‘𝑋) ∈ 𝐵)
51, 4sylan 569 . . . . 5 ((𝜑𝑋𝐵) → ((invg𝑆)‘𝑋) ∈ 𝐵)
6 simpr 471 . . . . 5 ((𝜑𝑋𝐵) → 𝑋𝐵)
7 grpidssd.o . . . . . 6 (𝜑 → ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝑀)𝑦) = (𝑥(+g𝑆)𝑦))
87adantr 466 . . . . 5 ((𝜑𝑋𝐵) → ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝑀)𝑦) = (𝑥(+g𝑆)𝑦))
9 oveq1 6803 . . . . . . 7 (𝑥 = ((invg𝑆)‘𝑋) → (𝑥(+g𝑀)𝑦) = (((invg𝑆)‘𝑋)(+g𝑀)𝑦))
10 oveq1 6803 . . . . . . 7 (𝑥 = ((invg𝑆)‘𝑋) → (𝑥(+g𝑆)𝑦) = (((invg𝑆)‘𝑋)(+g𝑆)𝑦))
119, 10eqeq12d 2786 . . . . . 6 (𝑥 = ((invg𝑆)‘𝑋) → ((𝑥(+g𝑀)𝑦) = (𝑥(+g𝑆)𝑦) ↔ (((invg𝑆)‘𝑋)(+g𝑀)𝑦) = (((invg𝑆)‘𝑋)(+g𝑆)𝑦)))
12 oveq2 6804 . . . . . . 7 (𝑦 = 𝑋 → (((invg𝑆)‘𝑋)(+g𝑀)𝑦) = (((invg𝑆)‘𝑋)(+g𝑀)𝑋))
13 oveq2 6804 . . . . . . 7 (𝑦 = 𝑋 → (((invg𝑆)‘𝑋)(+g𝑆)𝑦) = (((invg𝑆)‘𝑋)(+g𝑆)𝑋))
1412, 13eqeq12d 2786 . . . . . 6 (𝑦 = 𝑋 → ((((invg𝑆)‘𝑋)(+g𝑀)𝑦) = (((invg𝑆)‘𝑋)(+g𝑆)𝑦) ↔ (((invg𝑆)‘𝑋)(+g𝑀)𝑋) = (((invg𝑆)‘𝑋)(+g𝑆)𝑋)))
1511, 14rspc2va 3473 . . . . 5 (((((invg𝑆)‘𝑋) ∈ 𝐵𝑋𝐵) ∧ ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝑀)𝑦) = (𝑥(+g𝑆)𝑦)) → (((invg𝑆)‘𝑋)(+g𝑀)𝑋) = (((invg𝑆)‘𝑋)(+g𝑆)𝑋))
165, 6, 8, 15syl21anc 1475 . . . 4 ((𝜑𝑋𝐵) → (((invg𝑆)‘𝑋)(+g𝑀)𝑋) = (((invg𝑆)‘𝑋)(+g𝑆)𝑋))
17 eqid 2771 . . . . . 6 (+g𝑆) = (+g𝑆)
18 eqid 2771 . . . . . 6 (0g𝑆) = (0g𝑆)
192, 17, 18, 3grplinv 17676 . . . . 5 ((𝑆 ∈ Grp ∧ 𝑋𝐵) → (((invg𝑆)‘𝑋)(+g𝑆)𝑋) = (0g𝑆))
201, 19sylan 569 . . . 4 ((𝜑𝑋𝐵) → (((invg𝑆)‘𝑋)(+g𝑆)𝑋) = (0g𝑆))
21 grpidssd.m . . . . . . 7 (𝜑𝑀 ∈ Grp)
2221adantr 466 . . . . . 6 ((𝜑𝑋𝐵) → 𝑀 ∈ Grp)
23 grpidssd.c . . . . . . 7 (𝜑𝐵 ⊆ (Base‘𝑀))
2423sselda 3752 . . . . . 6 ((𝜑𝑋𝐵) → 𝑋 ∈ (Base‘𝑀))
25 eqid 2771 . . . . . . 7 (Base‘𝑀) = (Base‘𝑀)
26 eqid 2771 . . . . . . 7 (+g𝑀) = (+g𝑀)
27 eqid 2771 . . . . . . 7 (0g𝑀) = (0g𝑀)
28 eqid 2771 . . . . . . 7 (invg𝑀) = (invg𝑀)
2925, 26, 27, 28grplinv 17676 . . . . . 6 ((𝑀 ∈ Grp ∧ 𝑋 ∈ (Base‘𝑀)) → (((invg𝑀)‘𝑋)(+g𝑀)𝑋) = (0g𝑀))
3022, 24, 29syl2anc 573 . . . . 5 ((𝜑𝑋𝐵) → (((invg𝑀)‘𝑋)(+g𝑀)𝑋) = (0g𝑀))
3121, 1, 2, 23, 7grpidssd 17699 . . . . . 6 (𝜑 → (0g𝑀) = (0g𝑆))
3231adantr 466 . . . . 5 ((𝜑𝑋𝐵) → (0g𝑀) = (0g𝑆))
3330, 32eqtr2d 2806 . . . 4 ((𝜑𝑋𝐵) → (0g𝑆) = (((invg𝑀)‘𝑋)(+g𝑀)𝑋))
3416, 20, 333eqtrd 2809 . . 3 ((𝜑𝑋𝐵) → (((invg𝑆)‘𝑋)(+g𝑀)𝑋) = (((invg𝑀)‘𝑋)(+g𝑀)𝑋))
3523adantr 466 . . . . 5 ((𝜑𝑋𝐵) → 𝐵 ⊆ (Base‘𝑀))
3635, 5sseldd 3753 . . . 4 ((𝜑𝑋𝐵) → ((invg𝑆)‘𝑋) ∈ (Base‘𝑀))
3725, 28grpinvcl 17675 . . . . 5 ((𝑀 ∈ Grp ∧ 𝑋 ∈ (Base‘𝑀)) → ((invg𝑀)‘𝑋) ∈ (Base‘𝑀))
3822, 24, 37syl2anc 573 . . . 4 ((𝜑𝑋𝐵) → ((invg𝑀)‘𝑋) ∈ (Base‘𝑀))
3925, 26grprcan 17663 . . . 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 1478 . . 3 ((𝜑𝑋𝐵) → ((((invg𝑆)‘𝑋)(+g𝑀)𝑋) = (((invg𝑀)‘𝑋)(+g𝑀)𝑋) ↔ ((invg𝑆)‘𝑋) = ((invg𝑀)‘𝑋)))
4134, 40mpbid 222 . 2 ((𝜑𝑋𝐵) → ((invg𝑆)‘𝑋) = ((invg𝑀)‘𝑋))
4241ex 397 1 (𝜑 → (𝑋𝐵 → ((invg𝑆)‘𝑋) = ((invg𝑀)‘𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  wral 3061  wss 3723  cfv 6030  (class class class)co 6796  Basecbs 16064  +gcplusg 16149  0gc0g 16308  Grpcgrp 17630  invgcminusg 17631
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-riota 6757  df-ov 6799  df-0g 16310  df-mgm 17450  df-sgrp 17492  df-mnd 17503  df-grp 17633  df-minusg 17634
This theorem is referenced by:  grpissubg  17822
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