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Theorem grpidssd 18113
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 both groups have the same identity element. (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
grpidssd (𝜑 → (0g𝑀) = (0g𝑆))
Distinct variable groups:   𝑥,𝐵,𝑦   𝑥,𝑀,𝑦   𝑥,𝑆,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem grpidssd
StepHypRef Expression
1 grpidssd.s . . . . . 6 (𝜑𝑆 ∈ Grp)
2 grpidssd.b . . . . . . 7 𝐵 = (Base‘𝑆)
3 eqid 2818 . . . . . . 7 (0g𝑆) = (0g𝑆)
42, 3grpidcl 18069 . . . . . 6 (𝑆 ∈ Grp → (0g𝑆) ∈ 𝐵)
51, 4syl 17 . . . . 5 (𝜑 → (0g𝑆) ∈ 𝐵)
6 grpidssd.o . . . . 5 (𝜑 → ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝑀)𝑦) = (𝑥(+g𝑆)𝑦))
7 oveq1 7152 . . . . . . 7 (𝑥 = (0g𝑆) → (𝑥(+g𝑀)𝑦) = ((0g𝑆)(+g𝑀)𝑦))
8 oveq1 7152 . . . . . . 7 (𝑥 = (0g𝑆) → (𝑥(+g𝑆)𝑦) = ((0g𝑆)(+g𝑆)𝑦))
97, 8eqeq12d 2834 . . . . . 6 (𝑥 = (0g𝑆) → ((𝑥(+g𝑀)𝑦) = (𝑥(+g𝑆)𝑦) ↔ ((0g𝑆)(+g𝑀)𝑦) = ((0g𝑆)(+g𝑆)𝑦)))
10 oveq2 7153 . . . . . . 7 (𝑦 = (0g𝑆) → ((0g𝑆)(+g𝑀)𝑦) = ((0g𝑆)(+g𝑀)(0g𝑆)))
11 oveq2 7153 . . . . . . 7 (𝑦 = (0g𝑆) → ((0g𝑆)(+g𝑆)𝑦) = ((0g𝑆)(+g𝑆)(0g𝑆)))
1210, 11eqeq12d 2834 . . . . . 6 (𝑦 = (0g𝑆) → (((0g𝑆)(+g𝑀)𝑦) = ((0g𝑆)(+g𝑆)𝑦) ↔ ((0g𝑆)(+g𝑀)(0g𝑆)) = ((0g𝑆)(+g𝑆)(0g𝑆))))
139, 12rspc2va 3631 . . . . 5 ((((0g𝑆) ∈ 𝐵 ∧ (0g𝑆) ∈ 𝐵) ∧ ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝑀)𝑦) = (𝑥(+g𝑆)𝑦)) → ((0g𝑆)(+g𝑀)(0g𝑆)) = ((0g𝑆)(+g𝑆)(0g𝑆)))
145, 5, 6, 13syl21anc 833 . . . 4 (𝜑 → ((0g𝑆)(+g𝑀)(0g𝑆)) = ((0g𝑆)(+g𝑆)(0g𝑆)))
15 eqid 2818 . . . . . 6 (+g𝑆) = (+g𝑆)
162, 15, 3grplid 18071 . . . . 5 ((𝑆 ∈ Grp ∧ (0g𝑆) ∈ 𝐵) → ((0g𝑆)(+g𝑆)(0g𝑆)) = (0g𝑆))
171, 4, 16syl2anc2 585 . . . 4 (𝜑 → ((0g𝑆)(+g𝑆)(0g𝑆)) = (0g𝑆))
1814, 17eqtrd 2853 . . 3 (𝜑 → ((0g𝑆)(+g𝑀)(0g𝑆)) = (0g𝑆))
19 grpidssd.m . . . 4 (𝜑𝑀 ∈ Grp)
20 grpidssd.c . . . . 5 (𝜑𝐵 ⊆ (Base‘𝑀))
2120, 5sseldd 3965 . . . 4 (𝜑 → (0g𝑆) ∈ (Base‘𝑀))
22 eqid 2818 . . . . 5 (Base‘𝑀) = (Base‘𝑀)
23 eqid 2818 . . . . 5 (+g𝑀) = (+g𝑀)
24 eqid 2818 . . . . 5 (0g𝑀) = (0g𝑀)
2522, 23, 24grpidlcan 18103 . . . 4 ((𝑀 ∈ Grp ∧ (0g𝑆) ∈ (Base‘𝑀) ∧ (0g𝑆) ∈ (Base‘𝑀)) → (((0g𝑆)(+g𝑀)(0g𝑆)) = (0g𝑆) ↔ (0g𝑆) = (0g𝑀)))
2619, 21, 21, 25syl3anc 1363 . . 3 (𝜑 → (((0g𝑆)(+g𝑀)(0g𝑆)) = (0g𝑆) ↔ (0g𝑆) = (0g𝑀)))
2718, 26mpbid 233 . 2 (𝜑 → (0g𝑆) = (0g𝑀))
2827eqcomd 2824 1 (𝜑 → (0g𝑀) = (0g𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207   = wceq 1528  wcel 2105  wral 3135  wss 3933  cfv 6348  (class class class)co 7145  Basecbs 16471  +gcplusg 16553  0gc0g 16701  Grpcgrp 18041
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-riota 7103  df-ov 7148  df-0g 16703  df-mgm 17840  df-sgrp 17889  df-mnd 17900  df-grp 18044
This theorem is referenced by:  grpinvssd  18114
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