Theorem grpidssd 18747

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 2736	. . . . . . 7 ⊢ (0g‘𝑆) = (0g‘𝑆)
4	2, 3	grpidcl 18703	. . . . . 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 7344	. . . . . . 7 ⊢ (𝑥 = (0g‘𝑆) → (𝑥(+g‘𝑀)𝑦) = ((0g‘𝑆)(+g‘𝑀)𝑦))
8		oveq1 7344	. . . . . . 7 ⊢ (𝑥 = (0g‘𝑆) → (𝑥(+g‘𝑆)𝑦) = ((0g‘𝑆)(+g‘𝑆)𝑦))
9	7, 8	eqeq12d 2752	. . . . . 6 ⊢ (𝑥 = (0g‘𝑆) → ((𝑥(+g‘𝑀)𝑦) = (𝑥(+g‘𝑆)𝑦) ↔ ((0g‘𝑆)(+g‘𝑀)𝑦) = ((0g‘𝑆)(+g‘𝑆)𝑦)))
10		oveq2 7345	. . . . . . 7 ⊢ (𝑦 = (0g‘𝑆) → ((0g‘𝑆)(+g‘𝑀)𝑦) = ((0g‘𝑆)(+g‘𝑀)(0g‘𝑆)))
11		oveq2 7345	. . . . . . 7 ⊢ (𝑦 = (0g‘𝑆) → ((0g‘𝑆)(+g‘𝑆)𝑦) = ((0g‘𝑆)(+g‘𝑆)(0g‘𝑆)))
12	10, 11	eqeq12d 2752	. . . . . 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 3580	. . . . 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 835	. . . 4 ⊢ (𝜑 → ((0g‘𝑆)(+g‘𝑀)(0g‘𝑆)) = ((0g‘𝑆)(+g‘𝑆)(0g‘𝑆)))
15		eqid 2736	. . . . . 6 ⊢ (+g‘𝑆) = (+g‘𝑆)
16	2, 15, 3	grplid 18705	. . . . 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 2776	. . 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 3933	. . . 4 ⊢ (𝜑 → (0g‘𝑆) ∈ (Base‘𝑀))
22		eqid 2736	. . . . 5 ⊢ (Base‘𝑀) = (Base‘𝑀)
23		eqid 2736	. . . . 5 ⊢ (+g‘𝑀) = (+g‘𝑀)
24		eqid 2736	. . . . 5 ⊢ (0g‘𝑀) = (0g‘𝑀)
25	22, 23, 24	grpidlcan 18737	. . . 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 1370	. . 3 ⊢ (𝜑 → (((0g‘𝑆)(+g‘𝑀)(0g‘𝑆)) = (0g‘𝑆) ↔ (0g‘𝑆) = (0g‘𝑀)))
27	18, 26	mpbid 231	. 2 ⊢ (𝜑 → (0g‘𝑆) = (0g‘𝑀))
28	27	eqcomd 2742	1 ⊢ (𝜑 → (0g‘𝑀) = (0g‘𝑆))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1540   ∈ wcel 2105  ∀wral 3061   ⊆ wss 3898  ‘cfv 6479  (class class class)co 7337  Basecbs 17009  +_gcplusg 17059  0_gc0g 17247  Grpcgrp 18673

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5243  ax-nul 5250  ax-pr 5372

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3349  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3728  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-br 5093  df-opab 5155  df-mpt 5176  df-id 5518  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-iota 6431  df-fun 6481  df-fv 6487  df-riota 7293  df-ov 7340  df-0g 17249  df-mgm 18423  df-sgrp 18472  df-mnd 18483  df-grp 18676

This theorem is referenced by:  grpinvssd  18748