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Theorem grpinvssd 12952
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 12926 . . . . . 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 5884 . . . . . . 7 (𝑥 = ((invg𝑆)‘𝑋) → (𝑥(+g𝑀)𝑦) = (((invg𝑆)‘𝑋)(+g𝑀)𝑦))
10 oveq1 5884 . . . . . . 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 5885 . . . . . . 7 (𝑦 = 𝑋 → (((invg𝑆)‘𝑋)(+g𝑀)𝑦) = (((invg𝑆)‘𝑋)(+g𝑀)𝑋))
13 oveq2 5885 . . . . . . 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 2857 . . . . 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 12927 . . . . 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 3157 . . . . . 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 12927 . . . . . 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 12951 . . . . . 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 3158 . . . 4 ((𝜑𝑋𝐵) → ((invg𝑆)‘𝑋) ∈ (Base‘𝑀))
3724, 27grpinvcl 12926 . . . . 5 ((𝑀 ∈ Grp ∧ 𝑋 ∈ (Base‘𝑀)) → ((invg𝑀)‘𝑋) ∈ (Base‘𝑀))
3821, 23, 37syl2an2r 595 . . . 4 ((𝜑𝑋𝐵) → ((invg𝑀)‘𝑋) ∈ (Base‘𝑀))
3924, 25grprcan 12915 . . . 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 3131  cfv 5218  (class class class)co 5877  Basecbs 12464  +gcplusg 12538  0gc0g 12710  Grpcgrp 12882  invgcminusg 12883
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 4120  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-cnex 7904  ax-resscn 7905  ax-1re 7907  ax-addrcl 7910
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 2741  df-sbc 2965  df-csb 3060  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-riota 5833  df-ov 5880  df-inn 8922  df-2 8980  df-ndx 12467  df-slot 12468  df-base 12470  df-plusg 12551  df-0g 12712  df-mgm 12780  df-sgrp 12813  df-mnd 12823  df-grp 12885  df-minusg 12886
This theorem is referenced by:  grpissubg  13059
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