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Theorem grpoid 30490

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 30484	. . . . 5 ⊢ (𝐺 ∈ GrpOp → 𝑈 ∈ 𝑋)
4	1	grporcan 30488	. . . . . 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 30486	. . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝑈𝐺𝐴) = 𝐴)
10	9	eqeq2d 2741	. 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 1541 ∈ wcel 2110 ran crn 5615 ‘cfv 6477 (class class class)co 7341 GrpOpcgr 30459 GIdcgi 30460

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 1911 ax-6 1968 ax-7 2009 ax-8 2112 ax-9 2120 ax-10 2143 ax-11 2159 ax-12 2179 ax-ext 2702 ax-sep 5232 ax-nul 5242 ax-pr 5368 ax-un 7663

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-ral 3046 df-rex 3055 df-reu 3345 df-rab 3394 df-v 3436 df-sbc 3740 df-csb 3849 df-dif 3903 df-un 3905 df-in 3907 df-ss 3917 df-nul 4282 df-if 4474 df-sn 4575 df-pr 4577 df-op 4581 df-uni 4858 df-iun 4941 df-br 5090 df-opab 5152 df-mpt 5171 df-id 5509 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-iota 6433 df-fun 6479 df-fn 6480 df-f 6481 df-fo 6483 df-fv 6485 df-riota 7298 df-ov 7344 df-grpo 30463 df-gid 30464

This theorem is referenced by: hhssnv 31234 ghomidOLD 37908

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