Theorem grpidssd 19034

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	⊢ (𝜑 → 𝑀 ∈ Grp)
grpidssd.s	⊢ (𝜑 → 𝑆 ∈ Grp)
grpidssd.b	⊢ 𝐵 = (Base‘𝑆)
grpidssd.c	⊢ (𝜑 → 𝐵 ⊆ (Base‘𝑀))
grpidssd.o	⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝑀)𝑦) = (𝑥(+g‘𝑆)𝑦))

Assertion

Ref	Expression
grpidssd	⊢ (𝜑 → (0g‘𝑀) = (0g‘𝑆))

Distinct variable groups:   𝑥,𝐵,𝑦   𝑥,𝑀,𝑦   𝑥,𝑆,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem grpidssd

Step	Hyp	Ref	Expression
1		grpidssd.s	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ Grp)
2		grpidssd.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑆)
3		eqid 2737	. . . . . . 7 ⊢ (0g‘𝑆) = (0g‘𝑆)
4	2, 3	grpidcl 18983	. . . . . 6 ⊢ (𝑆 ∈ Grp → (0g‘𝑆) ∈ 𝐵)
5	1, 4	syl 17	. . . . 5 ⊢ (𝜑 → (0g‘𝑆) ∈ 𝐵)
6		grpidssd.o	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝑀)𝑦) = (𝑥(+g‘𝑆)𝑦))
7		oveq1 7438	. . . . . . 7 ⊢ (𝑥 = (0g‘𝑆) → (𝑥(+g‘𝑀)𝑦) = ((0g‘𝑆)(+g‘𝑀)𝑦))
8		oveq1 7438	. . . . . . 7 ⊢ (𝑥 = (0g‘𝑆) → (𝑥(+g‘𝑆)𝑦) = ((0g‘𝑆)(+g‘𝑆)𝑦))
9	7, 8	eqeq12d 2753	. . . . . 6 ⊢ (𝑥 = (0g‘𝑆) → ((𝑥(+g‘𝑀)𝑦) = (𝑥(+g‘𝑆)𝑦) ↔ ((0g‘𝑆)(+g‘𝑀)𝑦) = ((0g‘𝑆)(+g‘𝑆)𝑦)))
10		oveq2 7439	. . . . . . 7 ⊢ (𝑦 = (0g‘𝑆) → ((0g‘𝑆)(+g‘𝑀)𝑦) = ((0g‘𝑆)(+g‘𝑀)(0g‘𝑆)))
11		oveq2 7439	. . . . . . 7 ⊢ (𝑦 = (0g‘𝑆) → ((0g‘𝑆)(+g‘𝑆)𝑦) = ((0g‘𝑆)(+g‘𝑆)(0g‘𝑆)))
12	10, 11	eqeq12d 2753	. . . . . 6 ⊢ (𝑦 = (0g‘𝑆) → (((0g‘𝑆)(+g‘𝑀)𝑦) = ((0g‘𝑆)(+g‘𝑆)𝑦) ↔ ((0g‘𝑆)(+g‘𝑀)(0g‘𝑆)) = ((0g‘𝑆)(+g‘𝑆)(0g‘𝑆))))
13	9, 12	rspc2va 3634	. . . . 5 ⊢ ((((0g‘𝑆) ∈ 𝐵 ∧ (0g‘𝑆) ∈ 𝐵) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝑀)𝑦) = (𝑥(+g‘𝑆)𝑦)) → ((0g‘𝑆)(+g‘𝑀)(0g‘𝑆)) = ((0g‘𝑆)(+g‘𝑆)(0g‘𝑆)))
14	5, 5, 6, 13	syl21anc 838	. . . 4 ⊢ (𝜑 → ((0g‘𝑆)(+g‘𝑀)(0g‘𝑆)) = ((0g‘𝑆)(+g‘𝑆)(0g‘𝑆)))
15		eqid 2737	. . . . . 6 ⊢ (+g‘𝑆) = (+g‘𝑆)
16	2, 15, 3	grplid 18985	. . . . 5 ⊢ ((𝑆 ∈ Grp ∧ (0g‘𝑆) ∈ 𝐵) → ((0g‘𝑆)(+g‘𝑆)(0g‘𝑆)) = (0g‘𝑆))
17	1, 4, 16	syl2anc2 585	. . . 4 ⊢ (𝜑 → ((0g‘𝑆)(+g‘𝑆)(0g‘𝑆)) = (0g‘𝑆))
18	14, 17	eqtrd 2777	. . 3 ⊢ (𝜑 → ((0g‘𝑆)(+g‘𝑀)(0g‘𝑆)) = (0g‘𝑆))
19		grpidssd.m	. . . 4 ⊢ (𝜑 → 𝑀 ∈ Grp)
20		grpidssd.c	. . . . 5 ⊢ (𝜑 → 𝐵 ⊆ (Base‘𝑀))
21	20, 5	sseldd 3984	. . . 4 ⊢ (𝜑 → (0g‘𝑆) ∈ (Base‘𝑀))
22		eqid 2737	. . . . 5 ⊢ (Base‘𝑀) = (Base‘𝑀)
23		eqid 2737	. . . . 5 ⊢ (+g‘𝑀) = (+g‘𝑀)
24		eqid 2737	. . . . 5 ⊢ (0g‘𝑀) = (0g‘𝑀)
25	22, 23, 24	grpidlcan 19022	. . . 4 ⊢ ((𝑀 ∈ Grp ∧ (0g‘𝑆) ∈ (Base‘𝑀) ∧ (0g‘𝑆) ∈ (Base‘𝑀)) → (((0g‘𝑆)(+g‘𝑀)(0g‘𝑆)) = (0g‘𝑆) ↔ (0g‘𝑆) = (0g‘𝑀)))
26	19, 21, 21, 25	syl3anc 1373	. . 3 ⊢ (𝜑 → (((0g‘𝑆)(+g‘𝑀)(0g‘𝑆)) = (0g‘𝑆) ↔ (0g‘𝑆) = (0g‘𝑀)))
27	18, 26	mpbid 232	. 2 ⊢ (𝜑 → (0g‘𝑆) = (0g‘𝑀))
28	27	eqcomd 2743	1 ⊢ (𝜑 → (0g‘𝑀) = (0g‘𝑆))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1540   ∈ wcel 2108  ∀wral 3061   ⊆ wss 3951  ‘cfv 6561  (class class class)co 7431  Basecbs 17247  +_gcplusg 17297  0_gc0g 17484  Grpcgrp 18951

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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-riota 7388  df-ov 7434  df-0g 17486  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-grp 18954

This theorem is referenced by:  grpinvssd  19035