Theorem grpidssd 13278

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 2196	. . . . . . 7 ⊢ (0g‘𝑆) = (0g‘𝑆)
4	2, 3	grpidcl 13231	. . . . . 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 5932	. . . . . . 7 ⊢ (𝑥 = (0g‘𝑆) → (𝑥(+g‘𝑀)𝑦) = ((0g‘𝑆)(+g‘𝑀)𝑦))
8		oveq1 5932	. . . . . . 7 ⊢ (𝑥 = (0g‘𝑆) → (𝑥(+g‘𝑆)𝑦) = ((0g‘𝑆)(+g‘𝑆)𝑦))
9	7, 8	eqeq12d 2211	. . . . . 6 ⊢ (𝑥 = (0g‘𝑆) → ((𝑥(+g‘𝑀)𝑦) = (𝑥(+g‘𝑆)𝑦) ↔ ((0g‘𝑆)(+g‘𝑀)𝑦) = ((0g‘𝑆)(+g‘𝑆)𝑦)))
10		oveq2 5933	. . . . . . 7 ⊢ (𝑦 = (0g‘𝑆) → ((0g‘𝑆)(+g‘𝑀)𝑦) = ((0g‘𝑆)(+g‘𝑀)(0g‘𝑆)))
11		oveq2 5933	. . . . . . 7 ⊢ (𝑦 = (0g‘𝑆) → ((0g‘𝑆)(+g‘𝑆)𝑦) = ((0g‘𝑆)(+g‘𝑆)(0g‘𝑆)))
12	10, 11	eqeq12d 2211	. . . . . 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 2882	. . . . 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 2196	. . . . . 6 ⊢ (+g‘𝑆) = (+g‘𝑆)
16	2, 15, 3	grplid 13233	. . . . 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 2229	. . 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 3185	. . . 4 ⊢ (𝜑 → (0g‘𝑆) ∈ (Base‘𝑀))
22		eqid 2196	. . . . 5 ⊢ (Base‘𝑀) = (Base‘𝑀)
23		eqid 2196	. . . . 5 ⊢ (+g‘𝑀) = (+g‘𝑀)
24		eqid 2196	. . . . 5 ⊢ (0g‘𝑀) = (0g‘𝑀)
25	22, 23, 24	grpidlcan 13268	. . . 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 2202	1 ⊢ (𝜑 → (0g‘𝑀) = (0g‘𝑆))

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1364   ∈ wcel 2167  ∀wral 2475   ⊆ wss 3157  ‘cfv 5259  (class class class)co 5925  Basecbs 12703  +_gcplusg 12780  0_gc0g 12958  Grpcgrp 13202

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-cnex 7987  ax-resscn 7988  ax-1re 7990  ax-addrcl 7993

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-iota 5220  df-fun 5261  df-fn 5262  df-fv 5267  df-riota 5880  df-ov 5928  df-inn 9008  df-2 9066  df-ndx 12706  df-slot 12707  df-base 12709  df-plusg 12793  df-0g 12960  df-mgm 13058  df-sgrp 13104  df-mnd 13119  df-grp 13205

This theorem is referenced by:  grpinvssd  13279