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Theorem grpidssd 18899
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 2733 . . . . . . 7 (0g𝑆) = (0g𝑆)
42, 3grpidcl 18850 . . . . . 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 7416 . . . . . . 7 (𝑥 = (0g𝑆) → (𝑥(+g𝑀)𝑦) = ((0g𝑆)(+g𝑀)𝑦))
8 oveq1 7416 . . . . . . 7 (𝑥 = (0g𝑆) → (𝑥(+g𝑆)𝑦) = ((0g𝑆)(+g𝑆)𝑦))
97, 8eqeq12d 2749 . . . . . 6 (𝑥 = (0g𝑆) → ((𝑥(+g𝑀)𝑦) = (𝑥(+g𝑆)𝑦) ↔ ((0g𝑆)(+g𝑀)𝑦) = ((0g𝑆)(+g𝑆)𝑦)))
10 oveq2 7417 . . . . . . 7 (𝑦 = (0g𝑆) → ((0g𝑆)(+g𝑀)𝑦) = ((0g𝑆)(+g𝑀)(0g𝑆)))
11 oveq2 7417 . . . . . . 7 (𝑦 = (0g𝑆) → ((0g𝑆)(+g𝑆)𝑦) = ((0g𝑆)(+g𝑆)(0g𝑆)))
1210, 11eqeq12d 2749 . . . . . 6 (𝑦 = (0g𝑆) → (((0g𝑆)(+g𝑀)𝑦) = ((0g𝑆)(+g𝑆)𝑦) ↔ ((0g𝑆)(+g𝑀)(0g𝑆)) = ((0g𝑆)(+g𝑆)(0g𝑆))))
139, 12rspc2va 3624 . . . . 5 ((((0g𝑆) ∈ 𝐵 ∧ (0g𝑆) ∈ 𝐵) ∧ ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝑀)𝑦) = (𝑥(+g𝑆)𝑦)) → ((0g𝑆)(+g𝑀)(0g𝑆)) = ((0g𝑆)(+g𝑆)(0g𝑆)))
145, 5, 6, 13syl21anc 837 . . . 4 (𝜑 → ((0g𝑆)(+g𝑀)(0g𝑆)) = ((0g𝑆)(+g𝑆)(0g𝑆)))
15 eqid 2733 . . . . . 6 (+g𝑆) = (+g𝑆)
162, 15, 3grplid 18852 . . . . 5 ((𝑆 ∈ Grp ∧ (0g𝑆) ∈ 𝐵) → ((0g𝑆)(+g𝑆)(0g𝑆)) = (0g𝑆))
171, 4, 16syl2anc2 586 . . . 4 (𝜑 → ((0g𝑆)(+g𝑆)(0g𝑆)) = (0g𝑆))
1814, 17eqtrd 2773 . . 3 (𝜑 → ((0g𝑆)(+g𝑀)(0g𝑆)) = (0g𝑆))
19 grpidssd.m . . . 4 (𝜑𝑀 ∈ Grp)
20 grpidssd.c . . . . 5 (𝜑𝐵 ⊆ (Base‘𝑀))
2120, 5sseldd 3984 . . . 4 (𝜑 → (0g𝑆) ∈ (Base‘𝑀))
22 eqid 2733 . . . . 5 (Base‘𝑀) = (Base‘𝑀)
23 eqid 2733 . . . . 5 (+g𝑀) = (+g𝑀)
24 eqid 2733 . . . . 5 (0g𝑀) = (0g𝑀)
2522, 23, 24grpidlcan 18889 . . . 4 ((𝑀 ∈ Grp ∧ (0g𝑆) ∈ (Base‘𝑀) ∧ (0g𝑆) ∈ (Base‘𝑀)) → (((0g𝑆)(+g𝑀)(0g𝑆)) = (0g𝑆) ↔ (0g𝑆) = (0g𝑀)))
2619, 21, 21, 25syl3anc 1372 . . 3 (𝜑 → (((0g𝑆)(+g𝑀)(0g𝑆)) = (0g𝑆) ↔ (0g𝑆) = (0g𝑀)))
2718, 26mpbid 231 . 2 (𝜑 → (0g𝑆) = (0g𝑀))
2827eqcomd 2739 1 (𝜑 → (0g𝑀) = (0g𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1542  wcel 2107  wral 3062  wss 3949  cfv 6544  (class class class)co 7409  Basecbs 17144  +gcplusg 17197  0gc0g 17385  Grpcgrp 18819
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 5300  ax-nul 5307  ax-pr 5428
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 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6496  df-fun 6546  df-fv 6552  df-riota 7365  df-ov 7412  df-0g 17387  df-mgm 18561  df-sgrp 18610  df-mnd 18626  df-grp 18822
This theorem is referenced by:  grpinvssd  18900
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