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Theorem grpoid 30447

Description: Two ways of saying that an element of a group is the identity element. (Contributed by Paul Chapman, 25-Feb-2008.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
grpoinveu.1	⊢ 𝑋 = ran 𝐺
grpoinveu.2	⊢ 𝑈 = (GId‘𝐺)

**Assertion**
Ref	Expression
grpoid	⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝐴 = 𝑈 ↔ (𝐴𝐺𝐴) = 𝐴))

**Proof of Theorem grpoid**
Step	Hyp	Ref	Expression
1		grpoinveu.1	. . . . . 6 ⊢ 𝑋 = ran 𝐺
2		grpoinveu.2	. . . . . 6 ⊢ 𝑈 = (GId‘𝐺)
3	1, 2	grpoidcl 30441	. . . . 5 ⊢ (𝐺 ∈ GrpOp → 𝑈 ∈ 𝑋)
4	1	grporcan 30445	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝑈 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → ((𝐴𝐺𝐴) = (𝑈𝐺𝐴) ↔ 𝐴 = 𝑈))
5	4	3exp2 1355	. . . . 5 ⊢ (𝐺 ∈ GrpOp → (𝐴 ∈ 𝑋 → (𝑈 ∈ 𝑋 → (𝐴 ∈ 𝑋 → ((𝐴𝐺𝐴) = (𝑈𝐺𝐴) ↔ 𝐴 = 𝑈)))))
6	3, 5	mpid 44	. . . 4 ⊢ (𝐺 ∈ GrpOp → (𝐴 ∈ 𝑋 → (𝐴 ∈ 𝑋 → ((𝐴𝐺𝐴) = (𝑈𝐺𝐴) ↔ 𝐴 = 𝑈))))
7	6	pm2.43d 53	. . 3 ⊢ (𝐺 ∈ GrpOp → (𝐴 ∈ 𝑋 → ((𝐴𝐺𝐴) = (𝑈𝐺𝐴) ↔ 𝐴 = 𝑈)))
8	7	imp 406	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → ((𝐴𝐺𝐴) = (𝑈𝐺𝐴) ↔ 𝐴 = 𝑈))
9	1, 2	grpolid 30443	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝑈𝐺𝐴) = 𝐴)
10	9	eqeq2d 2746	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → ((𝐴𝐺𝐴) = (𝑈𝐺𝐴) ↔ (𝐴𝐺𝐴) = 𝐴))
11	8, 10	bitr3d 281	1 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝐴 = 𝑈 ↔ (𝐴𝐺𝐴) = 𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ran crn 5655 ‘cfv 6530 (class class class)co 7403 GrpOpcgr 30416 GIdcgi 30417

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 2707 ax-sep 5266 ax-nul 5276 ax-pr 5402 ax-un 7727

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-ral 3052 df-rex 3061 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-nul 4309 df-if 4501 df-sn 4602 df-pr 4604 df-op 4608 df-uni 4884 df-iun 4969 df-br 5120 df-opab 5182 df-mpt 5202 df-id 5548 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-iota 6483 df-fun 6532 df-fn 6533 df-f 6534 df-fo 6536 df-fv 6538 df-riota 7360 df-ov 7406 df-grpo 30420 df-gid 30421

This theorem is referenced by: hhssnv 31191 ghomidOLD 37859

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