Theorem grpidssd 13148

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 2193	. . . . . . 7 ⊢ (0g‘𝑆) = (0g‘𝑆)
4	2, 3	grpidcl 13101	. . . . . 6 ⊢ (𝑆 ∈ Grp → (0g‘𝑆) ∈ 𝐵)
5	1, 4	syl 14	. . . . 5 ⊢ (𝜑 → (0g‘𝑆) ∈ 𝐵)
6		grpidssd.o	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝑀)𝑦) = (𝑥(+g‘𝑆)𝑦))
7		oveq1 5925	. . . . . . 7 ⊢ (𝑥 = (0g‘𝑆) → (𝑥(+g‘𝑀)𝑦) = ((0g‘𝑆)(+g‘𝑀)𝑦))
8		oveq1 5925	. . . . . . 7 ⊢ (𝑥 = (0g‘𝑆) → (𝑥(+g‘𝑆)𝑦) = ((0g‘𝑆)(+g‘𝑆)𝑦))
9	7, 8	eqeq12d 2208	. . . . . 6 ⊢ (𝑥 = (0g‘𝑆) → ((𝑥(+g‘𝑀)𝑦) = (𝑥(+g‘𝑆)𝑦) ↔ ((0g‘𝑆)(+g‘𝑀)𝑦) = ((0g‘𝑆)(+g‘𝑆)𝑦)))
10		oveq2 5926	. . . . . . 7 ⊢ (𝑦 = (0g‘𝑆) → ((0g‘𝑆)(+g‘𝑀)𝑦) = ((0g‘𝑆)(+g‘𝑀)(0g‘𝑆)))
11		oveq2 5926	. . . . . . 7 ⊢ (𝑦 = (0g‘𝑆) → ((0g‘𝑆)(+g‘𝑆)𝑦) = ((0g‘𝑆)(+g‘𝑆)(0g‘𝑆)))
12	10, 11	eqeq12d 2208	. . . . . 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 2878	. . . . 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 1248	. . . 4 ⊢ (𝜑 → ((0g‘𝑆)(+g‘𝑀)(0g‘𝑆)) = ((0g‘𝑆)(+g‘𝑆)(0g‘𝑆)))
15		eqid 2193	. . . . . 6 ⊢ (+g‘𝑆) = (+g‘𝑆)
16	2, 15, 3	grplid 13103	. . . . 5 ⊢ ((𝑆 ∈ Grp ∧ (0g‘𝑆) ∈ 𝐵) → ((0g‘𝑆)(+g‘𝑆)(0g‘𝑆)) = (0g‘𝑆))
17	1, 4, 16	syl2anc2 412	. . . 4 ⊢ (𝜑 → ((0g‘𝑆)(+g‘𝑆)(0g‘𝑆)) = (0g‘𝑆))
18	14, 17	eqtrd 2226	. . 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 3180	. . . 4 ⊢ (𝜑 → (0g‘𝑆) ∈ (Base‘𝑀))
22		eqid 2193	. . . . 5 ⊢ (Base‘𝑀) = (Base‘𝑀)
23		eqid 2193	. . . . 5 ⊢ (+g‘𝑀) = (+g‘𝑀)
24		eqid 2193	. . . . 5 ⊢ (0g‘𝑀) = (0g‘𝑀)
25	22, 23, 24	grpidlcan 13138	. . . 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 1249	. . 3 ⊢ (𝜑 → (((0g‘𝑆)(+g‘𝑀)(0g‘𝑆)) = (0g‘𝑆) ↔ (0g‘𝑆) = (0g‘𝑀)))
27	18, 26	mpbid 147	. 2 ⊢ (𝜑 → (0g‘𝑆) = (0g‘𝑀))
28	27	eqcomd 2199	1 ⊢ (𝜑 → (0g‘𝑀) = (0g‘𝑆))

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1364   ∈ wcel 2164  ∀wral 2472   ⊆ wss 3153  ‘cfv 5254  (class class class)co 5918  Basecbs 12618  +_gcplusg 12695  0_gc0g 12867  Grpcgrp 13072

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-cnex 7963  ax-resscn 7964  ax-1re 7966  ax-addrcl 7969

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-iota 5215  df-fun 5256  df-fn 5257  df-fv 5262  df-riota 5873  df-ov 5921  df-inn 8983  df-2 9041  df-ndx 12621  df-slot 12622  df-base 12624  df-plusg 12708  df-0g 12869  df-mgm 12939  df-sgrp 12985  df-mnd 12998  df-grp 13075

This theorem is referenced by:  grpinvssd  13149