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Theorem grpinvssd 18829
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 2733 . . . . . . 7 (invg𝑆) = (invg𝑆)
42, 3grpinvcl 18803 . . . . . 6 ((𝑆 ∈ Grp ∧ 𝑋𝐵) → ((invg𝑆)‘𝑋) ∈ 𝐵)
51, 4sylan 581 . . . . 5 ((𝜑𝑋𝐵) → ((invg𝑆)‘𝑋) ∈ 𝐵)
6 simpr 486 . . . . 5 ((𝜑𝑋𝐵) → 𝑋𝐵)
7 grpidssd.o . . . . . 6 (𝜑 → ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝑀)𝑦) = (𝑥(+g𝑆)𝑦))
87adantr 482 . . . . 5 ((𝜑𝑋𝐵) → ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝑀)𝑦) = (𝑥(+g𝑆)𝑦))
9 oveq1 7365 . . . . . . 7 (𝑥 = ((invg𝑆)‘𝑋) → (𝑥(+g𝑀)𝑦) = (((invg𝑆)‘𝑋)(+g𝑀)𝑦))
10 oveq1 7365 . . . . . . 7 (𝑥 = ((invg𝑆)‘𝑋) → (𝑥(+g𝑆)𝑦) = (((invg𝑆)‘𝑋)(+g𝑆)𝑦))
119, 10eqeq12d 2749 . . . . . 6 (𝑥 = ((invg𝑆)‘𝑋) → ((𝑥(+g𝑀)𝑦) = (𝑥(+g𝑆)𝑦) ↔ (((invg𝑆)‘𝑋)(+g𝑀)𝑦) = (((invg𝑆)‘𝑋)(+g𝑆)𝑦)))
12 oveq2 7366 . . . . . . 7 (𝑦 = 𝑋 → (((invg𝑆)‘𝑋)(+g𝑀)𝑦) = (((invg𝑆)‘𝑋)(+g𝑀)𝑋))
13 oveq2 7366 . . . . . . 7 (𝑦 = 𝑋 → (((invg𝑆)‘𝑋)(+g𝑆)𝑦) = (((invg𝑆)‘𝑋)(+g𝑆)𝑋))
1412, 13eqeq12d 2749 . . . . . 6 (𝑦 = 𝑋 → ((((invg𝑆)‘𝑋)(+g𝑀)𝑦) = (((invg𝑆)‘𝑋)(+g𝑆)𝑦) ↔ (((invg𝑆)‘𝑋)(+g𝑀)𝑋) = (((invg𝑆)‘𝑋)(+g𝑆)𝑋)))
1511, 14rspc2va 3590 . . . . 5 (((((invg𝑆)‘𝑋) ∈ 𝐵𝑋𝐵) ∧ ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝑀)𝑦) = (𝑥(+g𝑆)𝑦)) → (((invg𝑆)‘𝑋)(+g𝑀)𝑋) = (((invg𝑆)‘𝑋)(+g𝑆)𝑋))
165, 6, 8, 15syl21anc 837 . . . 4 ((𝜑𝑋𝐵) → (((invg𝑆)‘𝑋)(+g𝑀)𝑋) = (((invg𝑆)‘𝑋)(+g𝑆)𝑋))
17 eqid 2733 . . . . . 6 (+g𝑆) = (+g𝑆)
18 eqid 2733 . . . . . 6 (0g𝑆) = (0g𝑆)
192, 17, 18, 3grplinv 18805 . . . . 5 ((𝑆 ∈ Grp ∧ 𝑋𝐵) → (((invg𝑆)‘𝑋)(+g𝑆)𝑋) = (0g𝑆))
201, 19sylan 581 . . . 4 ((𝜑𝑋𝐵) → (((invg𝑆)‘𝑋)(+g𝑆)𝑋) = (0g𝑆))
21 grpidssd.m . . . . . 6 (𝜑𝑀 ∈ Grp)
22 grpidssd.c . . . . . . 7 (𝜑𝐵 ⊆ (Base‘𝑀))
2322sselda 3945 . . . . . 6 ((𝜑𝑋𝐵) → 𝑋 ∈ (Base‘𝑀))
24 eqid 2733 . . . . . . 7 (Base‘𝑀) = (Base‘𝑀)
25 eqid 2733 . . . . . . 7 (+g𝑀) = (+g𝑀)
26 eqid 2733 . . . . . . 7 (0g𝑀) = (0g𝑀)
27 eqid 2733 . . . . . . 7 (invg𝑀) = (invg𝑀)
2824, 25, 26, 27grplinv 18805 . . . . . 6 ((𝑀 ∈ Grp ∧ 𝑋 ∈ (Base‘𝑀)) → (((invg𝑀)‘𝑋)(+g𝑀)𝑋) = (0g𝑀))
2921, 23, 28syl2an2r 684 . . . . 5 ((𝜑𝑋𝐵) → (((invg𝑀)‘𝑋)(+g𝑀)𝑋) = (0g𝑀))
3021, 1, 2, 22, 7grpidssd 18828 . . . . . 6 (𝜑 → (0g𝑀) = (0g𝑆))
3130adantr 482 . . . . 5 ((𝜑𝑋𝐵) → (0g𝑀) = (0g𝑆))
3229, 31eqtr2d 2774 . . . 4 ((𝜑𝑋𝐵) → (0g𝑆) = (((invg𝑀)‘𝑋)(+g𝑀)𝑋))
3316, 20, 323eqtrd 2777 . . 3 ((𝜑𝑋𝐵) → (((invg𝑆)‘𝑋)(+g𝑀)𝑋) = (((invg𝑀)‘𝑋)(+g𝑀)𝑋))
3421adantr 482 . . . 4 ((𝜑𝑋𝐵) → 𝑀 ∈ Grp)
3522adantr 482 . . . . 5 ((𝜑𝑋𝐵) → 𝐵 ⊆ (Base‘𝑀))
3635, 5sseldd 3946 . . . 4 ((𝜑𝑋𝐵) → ((invg𝑆)‘𝑋) ∈ (Base‘𝑀))
3724, 27grpinvcl 18803 . . . . 5 ((𝑀 ∈ Grp ∧ 𝑋 ∈ (Base‘𝑀)) → ((invg𝑀)‘𝑋) ∈ (Base‘𝑀))
3821, 23, 37syl2an2r 684 . . . 4 ((𝜑𝑋𝐵) → ((invg𝑀)‘𝑋) ∈ (Base‘𝑀))
3924, 25grprcan 18789 . . . 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)))
4034, 36, 38, 23, 39syl13anc 1373 . . 3 ((𝜑𝑋𝐵) → ((((invg𝑆)‘𝑋)(+g𝑀)𝑋) = (((invg𝑀)‘𝑋)(+g𝑀)𝑋) ↔ ((invg𝑆)‘𝑋) = ((invg𝑀)‘𝑋)))
4133, 40mpbid 231 . 2 ((𝜑𝑋𝐵) → ((invg𝑆)‘𝑋) = ((invg𝑀)‘𝑋))
4241ex 414 1 (𝜑 → (𝑋𝐵 → ((invg𝑆)‘𝑋) = ((invg𝑀)‘𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wral 3061  wss 3911  cfv 6497  (class class class)co 7358  Basecbs 17088  +gcplusg 17138  0gc0g 17326  Grpcgrp 18753  invgcminusg 18754
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fv 6505  df-riota 7314  df-ov 7361  df-0g 17328  df-mgm 18502  df-sgrp 18551  df-mnd 18562  df-grp 18756  df-minusg 18757
This theorem is referenced by:  grpissubg  18953
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