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| Mirrors > Home > MPE Home > Th. List > Mathboxes > grpoeqdivid | Structured version Visualization version GIF version | ||
| Description: Two group elements are equal iff their quotient is the identity. (Contributed by Jeff Madsen, 6-Jan-2011.) | 
| Ref | Expression | 
|---|---|
| grpeqdivid.1 | ⊢ 𝑋 = ran 𝐺 | 
| grpeqdivid.2 | ⊢ 𝑈 = (GId‘𝐺) | 
| grpeqdivid.3 | ⊢ 𝐷 = ( /𝑔 ‘𝐺) | 
| Ref | Expression | 
|---|---|
| grpoeqdivid | ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 = 𝐵 ↔ (𝐴𝐷𝐵) = 𝑈)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | grpeqdivid.1 | . . . . 5 ⊢ 𝑋 = ran 𝐺 | |
| 2 | grpeqdivid.3 | . . . . 5 ⊢ 𝐷 = ( /𝑔 ‘𝐺) | |
| 3 | grpeqdivid.2 | . . . . 5 ⊢ 𝑈 = (GId‘𝐺) | |
| 4 | 1, 2, 3 | grpodivid 30561 | . . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐵 ∈ 𝑋) → (𝐵𝐷𝐵) = 𝑈) | 
| 5 | 4 | 3adant2 1132 | . . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐵𝐷𝐵) = 𝑈) | 
| 6 | oveq1 7438 | . . . 4 ⊢ (𝐴 = 𝐵 → (𝐴𝐷𝐵) = (𝐵𝐷𝐵)) | |
| 7 | 6 | eqeq1d 2739 | . . 3 ⊢ (𝐴 = 𝐵 → ((𝐴𝐷𝐵) = 𝑈 ↔ (𝐵𝐷𝐵) = 𝑈)) | 
| 8 | 5, 7 | syl5ibrcom 247 | . 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 = 𝐵 → (𝐴𝐷𝐵) = 𝑈)) | 
| 9 | oveq1 7438 | . . 3 ⊢ ((𝐴𝐷𝐵) = 𝑈 → ((𝐴𝐷𝐵)𝐺𝐵) = (𝑈𝐺𝐵)) | |
| 10 | 1, 2 | grponpcan 30562 | . . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝐷𝐵)𝐺𝐵) = 𝐴) | 
| 11 | 1, 3 | grpolid 30535 | . . . . 5 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐵 ∈ 𝑋) → (𝑈𝐺𝐵) = 𝐵) | 
| 12 | 11 | 3adant2 1132 | . . . 4 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑈𝐺𝐵) = 𝐵) | 
| 13 | 10, 12 | eqeq12d 2753 | . . 3 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (((𝐴𝐷𝐵)𝐺𝐵) = (𝑈𝐺𝐵) ↔ 𝐴 = 𝐵)) | 
| 14 | 9, 13 | imbitrid 244 | . 2 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝐷𝐵) = 𝑈 → 𝐴 = 𝐵)) | 
| 15 | 8, 14 | impbid 212 | 1 ⊢ ((𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 = 𝐵 ↔ (𝐴𝐷𝐵) = 𝑈)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ran crn 5686 ‘cfv 6561 (class class class)co 7431 GrpOpcgr 30508 GIdcgi 30509 /𝑔 cgs 30511 | 
| 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 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-grpo 30512 df-gid 30513 df-ginv 30514 df-gdiv 30515 | 
| This theorem is referenced by: grpokerinj 37900 dmncan1 38083 | 
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