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Theorem grpoid 30578

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 30572	. . . . 5 ⊢ (𝐺 ∈ GrpOp → 𝑈 ∈ 𝑋)
4	1	grporcan 30576	. . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝑈 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → ((𝐴𝐺𝐴) = (𝑈𝐺𝐴) ↔ 𝐴 = 𝑈))
5	4	3exp2 1356	. . . . 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 30574	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝑈𝐺𝐴) = 𝐴)
10	9	eqeq2d 2748	. 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 1542 ∈ wcel 2114 ran crn 5626 ‘cfv 6493 (class class class)co 7360 GrpOpcgr 30547 GIdcgi 30548

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5242 ax-nul 5252 ax-pr 5378 ax-un 7682

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3062 df-reu 3352 df-rab 3401 df-v 3443 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-nul 4287 df-if 4481 df-sn 4582 df-pr 4584 df-op 4588 df-uni 4865 df-iun 4949 df-br 5100 df-opab 5162 df-mpt 5181 df-id 5520 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-fo 6499 df-fv 6501 df-riota 7317 df-ov 7363 df-grpo 30551 df-gid 30552

This theorem is referenced by: hhssnv 31322 ghomidOLD 38061

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