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Theorem grpinvssd 12901
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 2177 . . . . . . 7 (invg𝑆) = (invg𝑆)
42, 3grpinvcl 12875 . . . . . 6 ((𝑆 ∈ Grp ∧ 𝑋𝐵) → ((invg𝑆)‘𝑋) ∈ 𝐵)
51, 4sylan 283 . . . . 5 ((𝜑𝑋𝐵) → ((invg𝑆)‘𝑋) ∈ 𝐵)
6 simpr 110 . . . . 5 ((𝜑𝑋𝐵) → 𝑋𝐵)
7 grpidssd.o . . . . . 6 (𝜑 → ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝑀)𝑦) = (𝑥(+g𝑆)𝑦))
87adantr 276 . . . . 5 ((𝜑𝑋𝐵) → ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝑀)𝑦) = (𝑥(+g𝑆)𝑦))
9 oveq1 5881 . . . . . . 7 (𝑥 = ((invg𝑆)‘𝑋) → (𝑥(+g𝑀)𝑦) = (((invg𝑆)‘𝑋)(+g𝑀)𝑦))
10 oveq1 5881 . . . . . . 7 (𝑥 = ((invg𝑆)‘𝑋) → (𝑥(+g𝑆)𝑦) = (((invg𝑆)‘𝑋)(+g𝑆)𝑦))
119, 10eqeq12d 2192 . . . . . 6 (𝑥 = ((invg𝑆)‘𝑋) → ((𝑥(+g𝑀)𝑦) = (𝑥(+g𝑆)𝑦) ↔ (((invg𝑆)‘𝑋)(+g𝑀)𝑦) = (((invg𝑆)‘𝑋)(+g𝑆)𝑦)))
12 oveq2 5882 . . . . . . 7 (𝑦 = 𝑋 → (((invg𝑆)‘𝑋)(+g𝑀)𝑦) = (((invg𝑆)‘𝑋)(+g𝑀)𝑋))
13 oveq2 5882 . . . . . . 7 (𝑦 = 𝑋 → (((invg𝑆)‘𝑋)(+g𝑆)𝑦) = (((invg𝑆)‘𝑋)(+g𝑆)𝑋))
1412, 13eqeq12d 2192 . . . . . 6 (𝑦 = 𝑋 → ((((invg𝑆)‘𝑋)(+g𝑀)𝑦) = (((invg𝑆)‘𝑋)(+g𝑆)𝑦) ↔ (((invg𝑆)‘𝑋)(+g𝑀)𝑋) = (((invg𝑆)‘𝑋)(+g𝑆)𝑋)))
1511, 14rspc2va 2855 . . . . 5 (((((invg𝑆)‘𝑋) ∈ 𝐵𝑋𝐵) ∧ ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝑀)𝑦) = (𝑥(+g𝑆)𝑦)) → (((invg𝑆)‘𝑋)(+g𝑀)𝑋) = (((invg𝑆)‘𝑋)(+g𝑆)𝑋))
165, 6, 8, 15syl21anc 1237 . . . 4 ((𝜑𝑋𝐵) → (((invg𝑆)‘𝑋)(+g𝑀)𝑋) = (((invg𝑆)‘𝑋)(+g𝑆)𝑋))
17 eqid 2177 . . . . . 6 (+g𝑆) = (+g𝑆)
18 eqid 2177 . . . . . 6 (0g𝑆) = (0g𝑆)
192, 17, 18, 3grplinv 12876 . . . . 5 ((𝑆 ∈ Grp ∧ 𝑋𝐵) → (((invg𝑆)‘𝑋)(+g𝑆)𝑋) = (0g𝑆))
201, 19sylan 283 . . . 4 ((𝜑𝑋𝐵) → (((invg𝑆)‘𝑋)(+g𝑆)𝑋) = (0g𝑆))
21 grpidssd.m . . . . . 6 (𝜑𝑀 ∈ Grp)
22 grpidssd.c . . . . . . 7 (𝜑𝐵 ⊆ (Base‘𝑀))
2322sselda 3155 . . . . . 6 ((𝜑𝑋𝐵) → 𝑋 ∈ (Base‘𝑀))
24 eqid 2177 . . . . . . 7 (Base‘𝑀) = (Base‘𝑀)
25 eqid 2177 . . . . . . 7 (+g𝑀) = (+g𝑀)
26 eqid 2177 . . . . . . 7 (0g𝑀) = (0g𝑀)
27 eqid 2177 . . . . . . 7 (invg𝑀) = (invg𝑀)
2824, 25, 26, 27grplinv 12876 . . . . . 6 ((𝑀 ∈ Grp ∧ 𝑋 ∈ (Base‘𝑀)) → (((invg𝑀)‘𝑋)(+g𝑀)𝑋) = (0g𝑀))
2921, 23, 28syl2an2r 595 . . . . 5 ((𝜑𝑋𝐵) → (((invg𝑀)‘𝑋)(+g𝑀)𝑋) = (0g𝑀))
3021, 1, 2, 22, 7grpidssd 12900 . . . . . 6 (𝜑 → (0g𝑀) = (0g𝑆))
3130adantr 276 . . . . 5 ((𝜑𝑋𝐵) → (0g𝑀) = (0g𝑆))
3229, 31eqtr2d 2211 . . . 4 ((𝜑𝑋𝐵) → (0g𝑆) = (((invg𝑀)‘𝑋)(+g𝑀)𝑋))
3316, 20, 323eqtrd 2214 . . 3 ((𝜑𝑋𝐵) → (((invg𝑆)‘𝑋)(+g𝑀)𝑋) = (((invg𝑀)‘𝑋)(+g𝑀)𝑋))
3421adantr 276 . . . 4 ((𝜑𝑋𝐵) → 𝑀 ∈ Grp)
3522adantr 276 . . . . 5 ((𝜑𝑋𝐵) → 𝐵 ⊆ (Base‘𝑀))
3635, 5sseldd 3156 . . . 4 ((𝜑𝑋𝐵) → ((invg𝑆)‘𝑋) ∈ (Base‘𝑀))
3724, 27grpinvcl 12875 . . . . 5 ((𝑀 ∈ Grp ∧ 𝑋 ∈ (Base‘𝑀)) → ((invg𝑀)‘𝑋) ∈ (Base‘𝑀))
3821, 23, 37syl2an2r 595 . . . 4 ((𝜑𝑋𝐵) → ((invg𝑀)‘𝑋) ∈ (Base‘𝑀))
3924, 25grprcan 12864 . . . 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 1240 . . 3 ((𝜑𝑋𝐵) → ((((invg𝑆)‘𝑋)(+g𝑀)𝑋) = (((invg𝑀)‘𝑋)(+g𝑀)𝑋) ↔ ((invg𝑆)‘𝑋) = ((invg𝑀)‘𝑋)))
4133, 40mpbid 147 . 2 ((𝜑𝑋𝐵) → ((invg𝑆)‘𝑋) = ((invg𝑀)‘𝑋))
4241ex 115 1 (𝜑 → (𝑋𝐵 → ((invg𝑆)‘𝑋) = ((invg𝑀)‘𝑋)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1353  wcel 2148  wral 2455  wss 3129  cfv 5216  (class class class)co 5874  Basecbs 12456  +gcplusg 12530  0gc0g 12695  Grpcgrp 12831  invgcminusg 12832
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-cnex 7901  ax-resscn 7902  ax-1re 7904  ax-addrcl 7907
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-riota 5830  df-ov 5877  df-inn 8918  df-2 8976  df-ndx 12459  df-slot 12460  df-base 12462  df-plusg 12543  df-0g 12697  df-mgm 12729  df-sgrp 12762  df-mnd 12772  df-grp 12834  df-minusg 12835
This theorem is referenced by:  grpissubg  13007
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