Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > grpoid | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
grpoinveu.1 | ⊢ 𝑋 = ran 𝐺 |
grpoinveu.2 | ⊢ 𝑈 = (GId‘𝐺) |
Ref | Expression |
---|---|
grpoid | ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝐴 = 𝑈 ↔ (𝐴𝐺𝐴) = 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grpoinveu.1 | . . . . . 6 ⊢ 𝑋 = ran 𝐺 | |
2 | grpoinveu.2 | . . . . . 6 ⊢ 𝑈 = (GId‘𝐺) | |
3 | 1, 2 | grpoidcl 28291 | . . . . 5 ⊢ (𝐺 ∈ GrpOp → 𝑈 ∈ 𝑋) |
4 | 1 | grporcan 28295 | . . . . . 6 ⊢ ((𝐺 ∈ GrpOp ∧ (𝐴 ∈ 𝑋 ∧ 𝑈 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → ((𝐴𝐺𝐴) = (𝑈𝐺𝐴) ↔ 𝐴 = 𝑈)) |
5 | 4 | 3exp2 1350 | . . . . 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 409 | . 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → ((𝐴𝐺𝐴) = (𝑈𝐺𝐴) ↔ 𝐴 = 𝑈)) |
9 | 1, 2 | grpolid 28293 | . . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝑈𝐺𝐴) = 𝐴) |
10 | 9 | eqeq2d 2832 | . 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → ((𝐴𝐺𝐴) = (𝑈𝐺𝐴) ↔ (𝐴𝐺𝐴) = 𝐴)) |
11 | 8, 10 | bitr3d 283 | 1 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋) → (𝐴 = 𝑈 ↔ (𝐴𝐺𝐴) = 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ran crn 5556 ‘cfv 6355 (class class class)co 7156 GrpOpcgr 28266 GIdcgi 28267 |
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 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-fo 6361 df-fv 6363 df-riota 7114 df-ov 7159 df-grpo 28270 df-gid 28271 |
This theorem is referenced by: hhssnv 29041 ghomidOLD 35182 |
Copyright terms: Public domain | W3C validator |