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| Mirrors > Home > MPE Home > Th. List > Mathboxes > dmnnzd | Structured version Visualization version GIF version | ||
| Description: A domain has no zero-divisors (besides zero). (Contributed by Jeff Madsen, 19-Jun-2010.) |
| Ref | Expression |
|---|---|
| dmnnzd.1 | ⊢ 𝐺 = (1st ‘𝑅) |
| dmnnzd.2 | ⊢ 𝐻 = (2nd ‘𝑅) |
| dmnnzd.3 | ⊢ 𝑋 = ran 𝐺 |
| dmnnzd.4 | ⊢ 𝑍 = (GId‘𝐺) |
| Ref | Expression |
|---|---|
| dmnnzd | ⊢ ((𝑅 ∈ Dmn ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ (𝐴𝐻𝐵) = 𝑍)) → (𝐴 = 𝑍 ∨ 𝐵 = 𝑍)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dmnnzd.1 | . . . . . 6 ⊢ 𝐺 = (1st ‘𝑅) | |
| 2 | dmnnzd.2 | . . . . . 6 ⊢ 𝐻 = (2nd ‘𝑅) | |
| 3 | dmnnzd.3 | . . . . . 6 ⊢ 𝑋 = ran 𝐺 | |
| 4 | dmnnzd.4 | . . . . . 6 ⊢ 𝑍 = (GId‘𝐺) | |
| 5 | eqid 2737 | . . . . . 6 ⊢ (GId‘𝐻) = (GId‘𝐻) | |
| 6 | 1, 2, 3, 4, 5 | isdmn3 38081 | . . . . 5 ⊢ (𝑅 ∈ Dmn ↔ (𝑅 ∈ CRingOps ∧ (GId‘𝐻) ≠ 𝑍 ∧ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍 ∨ 𝑏 = 𝑍)))) |
| 7 | 6 | simp3bi 1148 | . . . 4 ⊢ (𝑅 ∈ Dmn → ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍 ∨ 𝑏 = 𝑍))) |
| 8 | oveq1 7438 | . . . . . . 7 ⊢ (𝑎 = 𝐴 → (𝑎𝐻𝑏) = (𝐴𝐻𝑏)) | |
| 9 | 8 | eqeq1d 2739 | . . . . . 6 ⊢ (𝑎 = 𝐴 → ((𝑎𝐻𝑏) = 𝑍 ↔ (𝐴𝐻𝑏) = 𝑍)) |
| 10 | eqeq1 2741 | . . . . . . 7 ⊢ (𝑎 = 𝐴 → (𝑎 = 𝑍 ↔ 𝐴 = 𝑍)) | |
| 11 | 10 | orbi1d 917 | . . . . . 6 ⊢ (𝑎 = 𝐴 → ((𝑎 = 𝑍 ∨ 𝑏 = 𝑍) ↔ (𝐴 = 𝑍 ∨ 𝑏 = 𝑍))) |
| 12 | 9, 11 | imbi12d 344 | . . . . 5 ⊢ (𝑎 = 𝐴 → (((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍 ∨ 𝑏 = 𝑍)) ↔ ((𝐴𝐻𝑏) = 𝑍 → (𝐴 = 𝑍 ∨ 𝑏 = 𝑍)))) |
| 13 | oveq2 7439 | . . . . . . 7 ⊢ (𝑏 = 𝐵 → (𝐴𝐻𝑏) = (𝐴𝐻𝐵)) | |
| 14 | 13 | eqeq1d 2739 | . . . . . 6 ⊢ (𝑏 = 𝐵 → ((𝐴𝐻𝑏) = 𝑍 ↔ (𝐴𝐻𝐵) = 𝑍)) |
| 15 | eqeq1 2741 | . . . . . . 7 ⊢ (𝑏 = 𝐵 → (𝑏 = 𝑍 ↔ 𝐵 = 𝑍)) | |
| 16 | 15 | orbi2d 916 | . . . . . 6 ⊢ (𝑏 = 𝐵 → ((𝐴 = 𝑍 ∨ 𝑏 = 𝑍) ↔ (𝐴 = 𝑍 ∨ 𝐵 = 𝑍))) |
| 17 | 14, 16 | imbi12d 344 | . . . . 5 ⊢ (𝑏 = 𝐵 → (((𝐴𝐻𝑏) = 𝑍 → (𝐴 = 𝑍 ∨ 𝑏 = 𝑍)) ↔ ((𝐴𝐻𝐵) = 𝑍 → (𝐴 = 𝑍 ∨ 𝐵 = 𝑍)))) |
| 18 | 12, 17 | rspc2v 3633 | . . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍 ∨ 𝑏 = 𝑍)) → ((𝐴𝐻𝐵) = 𝑍 → (𝐴 = 𝑍 ∨ 𝐵 = 𝑍)))) |
| 19 | 7, 18 | syl5com 31 | . . 3 ⊢ (𝑅 ∈ Dmn → ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝐻𝐵) = 𝑍 → (𝐴 = 𝑍 ∨ 𝐵 = 𝑍)))) |
| 20 | 19 | expd 415 | . 2 ⊢ (𝑅 ∈ Dmn → (𝐴 ∈ 𝑋 → (𝐵 ∈ 𝑋 → ((𝐴𝐻𝐵) = 𝑍 → (𝐴 = 𝑍 ∨ 𝐵 = 𝑍))))) |
| 21 | 20 | 3imp2 1350 | 1 ⊢ ((𝑅 ∈ Dmn ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ (𝐴𝐻𝐵) = 𝑍)) → (𝐴 = 𝑍 ∨ 𝐵 = 𝑍)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∨ wo 848 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 ran crn 5686 ‘cfv 6561 (class class class)co 7431 1st c1st 8012 2nd c2nd 8013 GIdcgi 30509 CRingOpsccring 38000 Dmncdmn 38054 |
| 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-rmo 3380 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-int 4947 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-suc 6390 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-1o 8506 df-en 8986 df-grpo 30512 df-gid 30513 df-ginv 30514 df-ablo 30564 df-ass 37850 df-exid 37852 df-mgmOLD 37856 df-sgrOLD 37868 df-mndo 37874 df-rngo 37902 df-com2 37997 df-crngo 38001 df-idl 38017 df-pridl 38018 df-prrngo 38055 df-dmn 38056 df-igen 38067 |
| This theorem is referenced by: dmncan1 38083 |
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