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Theorem grpoid 28783

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 28777	. . . . 5 ⊢ (𝐺 ∈ GrpOp → 𝑈 ∈ 𝑋)
4	1	grporcan 28781	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝑈 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → ((𝐴𝐺𝐴) = (𝑈𝐺𝐴) ↔ 𝐴 = 𝑈))
5	4	3exp2 1352	. . . . 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 28779	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝑈𝐺𝐴) = 𝐴)
10	9	eqeq2d 2749	. 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → ((𝐴𝐺𝐴) = (𝑈𝐺𝐴) ↔ (𝐴𝐺𝐴) = 𝐴))
11	8, 10	bitr3d 280	1 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝐴 = 𝑈 ↔ (𝐴𝐺𝐴) = 𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1539 ∈ wcel 2108 ran crn 5581 ‘cfv 6418 (class class class)co 7255 GrpOpcgr 28752 GIdcgi 28753

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pr 5347 ax-un 7566

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ral 3068 df-rex 3069 df-reu 3070 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4457 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-id 5480 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-fo 6424 df-fv 6426 df-riota 7212 df-ov 7258 df-grpo 28756 df-gid 28757

This theorem is referenced by: hhssnv 29527 ghomidOLD 35974

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