Theorem grpidssd 19041

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 2761	. . . . . . 7 ⊢ (0g‘𝑆) = (0g‘𝑆)
4	2, 3	grpidcl 18990	. . . . . 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 7399	. . . . . . 7 ⊢ (𝑥 = (0g‘𝑆) → (𝑥(+g‘𝑀)𝑦) = ((0g‘𝑆)(+g‘𝑀)𝑦))
8		oveq1 7399	. . . . . . 7 ⊢ (𝑥 = (0g‘𝑆) → (𝑥(+g‘𝑆)𝑦) = ((0g‘𝑆)(+g‘𝑆)𝑦))
9	7, 8	eqeq12d 2777	. . . . . 6 ⊢ (𝑥 = (0g‘𝑆) → ((𝑥(+g‘𝑀)𝑦) = (𝑥(+g‘𝑆)𝑦) ↔ ((0g‘𝑆)(+g‘𝑀)𝑦) = ((0g‘𝑆)(+g‘𝑆)𝑦)))
10		oveq2 7400	. . . . . . 7 ⊢ (𝑦 = (0g‘𝑆) → ((0g‘𝑆)(+g‘𝑀)𝑦) = ((0g‘𝑆)(+g‘𝑀)(0g‘𝑆)))
11		oveq2 7400	. . . . . . 7 ⊢ (𝑦 = (0g‘𝑆) → ((0g‘𝑆)(+g‘𝑆)𝑦) = ((0g‘𝑆)(+g‘𝑆)(0g‘𝑆)))
12	10, 11	eqeq12d 2777	. . . . . 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 3593	. . . . 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 848	. . . 4 ⊢ (𝜑 → ((0g‘𝑆)(+g‘𝑀)(0g‘𝑆)) = ((0g‘𝑆)(+g‘𝑆)(0g‘𝑆)))
15		eqid 2761	. . . . . 6 ⊢ (+g‘𝑆) = (+g‘𝑆)
16	2, 15, 3	grplid 18992	. . . . 5 ⊢ ((𝑆 ∈ Grp ∧ (0g‘𝑆) ∈ 𝐵) → ((0g‘𝑆)(+g‘𝑆)(0g‘𝑆)) = (0g‘𝑆))
17	1, 4, 16	syl2anc2 594	. . . 4 ⊢ (𝜑 → ((0g‘𝑆)(+g‘𝑆)(0g‘𝑆)) = (0g‘𝑆))
18	14, 17	eqtrd 2796	. . 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 3937	. . . 4 ⊢ (𝜑 → (0g‘𝑆) ∈ (Base‘𝑀))
22		eqid 2761	. . . . 5 ⊢ (Base‘𝑀) = (Base‘𝑀)
23		eqid 2761	. . . . 5 ⊢ (+g‘𝑀) = (+g‘𝑀)
24		eqid 2761	. . . . 5 ⊢ (0g‘𝑀) = (0g‘𝑀)
25	22, 23, 24	grpidlcan 19029	. . . 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 1389	. . 3 ⊢ (𝜑 → (((0g‘𝑆)(+g‘𝑀)(0g‘𝑆)) = (0g‘𝑆) ↔ (0g‘𝑆) = (0g‘𝑀)))
27	18, 26	mpbid 234	. 2 ⊢ (𝜑 → (0g‘𝑆) = (0g‘𝑀))
28	27	eqcomd 2767	1 ⊢ (𝜑 → (0g‘𝑀) = (0g‘𝑆))

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1559   ∈ wcel 2141  ∀wral 3075   ⊆ wss 3904  ‘cfv 6517  (class class class)co 7392  Basecbs 17228  +_gcplusg 17269  0_gc0g 17451  Grpcgrp 18958

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-iota 6473  df-fun 6519  df-fv 6525  df-riota 7349  df-ov 7395  df-0g 17453  df-mgm 18657  df-sgrp 18736  df-mnd 18752  df-grp 18961

This theorem is referenced by:  grpinvssd  19042