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Theorem grpidssd 18935
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 2730 . . . . . . 7 (0g𝑆) = (0g𝑆)
42, 3grpidcl 18886 . . . . . 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 7418 . . . . . . 7 (𝑥 = (0g𝑆) → (𝑥(+g𝑀)𝑦) = ((0g𝑆)(+g𝑀)𝑦))
8 oveq1 7418 . . . . . . 7 (𝑥 = (0g𝑆) → (𝑥(+g𝑆)𝑦) = ((0g𝑆)(+g𝑆)𝑦))
97, 8eqeq12d 2746 . . . . . 6 (𝑥 = (0g𝑆) → ((𝑥(+g𝑀)𝑦) = (𝑥(+g𝑆)𝑦) ↔ ((0g𝑆)(+g𝑀)𝑦) = ((0g𝑆)(+g𝑆)𝑦)))
10 oveq2 7419 . . . . . . 7 (𝑦 = (0g𝑆) → ((0g𝑆)(+g𝑀)𝑦) = ((0g𝑆)(+g𝑀)(0g𝑆)))
11 oveq2 7419 . . . . . . 7 (𝑦 = (0g𝑆) → ((0g𝑆)(+g𝑆)𝑦) = ((0g𝑆)(+g𝑆)(0g𝑆)))
1210, 11eqeq12d 2746 . . . . . 6 (𝑦 = (0g𝑆) → (((0g𝑆)(+g𝑀)𝑦) = ((0g𝑆)(+g𝑆)𝑦) ↔ ((0g𝑆)(+g𝑀)(0g𝑆)) = ((0g𝑆)(+g𝑆)(0g𝑆))))
139, 12rspc2va 3622 . . . . 5 ((((0g𝑆) ∈ 𝐵 ∧ (0g𝑆) ∈ 𝐵) ∧ ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝑀)𝑦) = (𝑥(+g𝑆)𝑦)) → ((0g𝑆)(+g𝑀)(0g𝑆)) = ((0g𝑆)(+g𝑆)(0g𝑆)))
145, 5, 6, 13syl21anc 834 . . . 4 (𝜑 → ((0g𝑆)(+g𝑀)(0g𝑆)) = ((0g𝑆)(+g𝑆)(0g𝑆)))
15 eqid 2730 . . . . . 6 (+g𝑆) = (+g𝑆)
162, 15, 3grplid 18888 . . . . 5 ((𝑆 ∈ Grp ∧ (0g𝑆) ∈ 𝐵) → ((0g𝑆)(+g𝑆)(0g𝑆)) = (0g𝑆))
171, 4, 16syl2anc2 583 . . . 4 (𝜑 → ((0g𝑆)(+g𝑆)(0g𝑆)) = (0g𝑆))
1814, 17eqtrd 2770 . . 3 (𝜑 → ((0g𝑆)(+g𝑀)(0g𝑆)) = (0g𝑆))
19 grpidssd.m . . . 4 (𝜑𝑀 ∈ Grp)
20 grpidssd.c . . . . 5 (𝜑𝐵 ⊆ (Base‘𝑀))
2120, 5sseldd 3982 . . . 4 (𝜑 → (0g𝑆) ∈ (Base‘𝑀))
22 eqid 2730 . . . . 5 (Base‘𝑀) = (Base‘𝑀)
23 eqid 2730 . . . . 5 (+g𝑀) = (+g𝑀)
24 eqid 2730 . . . . 5 (0g𝑀) = (0g𝑀)
2522, 23, 24grpidlcan 18925 . . . 4 ((𝑀 ∈ Grp ∧ (0g𝑆) ∈ (Base‘𝑀) ∧ (0g𝑆) ∈ (Base‘𝑀)) → (((0g𝑆)(+g𝑀)(0g𝑆)) = (0g𝑆) ↔ (0g𝑆) = (0g𝑀)))
2619, 21, 21, 25syl3anc 1369 . . 3 (𝜑 → (((0g𝑆)(+g𝑀)(0g𝑆)) = (0g𝑆) ↔ (0g𝑆) = (0g𝑀)))
2718, 26mpbid 231 . 2 (𝜑 → (0g𝑆) = (0g𝑀))
2827eqcomd 2736 1 (𝜑 → (0g𝑀) = (0g𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1539  wcel 2104  wral 3059  wss 3947  cfv 6542  (class class class)co 7411  Basecbs 17148  +gcplusg 17201  0gc0g 17389  Grpcgrp 18855
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-iota 6494  df-fun 6544  df-fv 6550  df-riota 7367  df-ov 7414  df-0g 17391  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-grp 18858
This theorem is referenced by:  grpinvssd  18936
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