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| Mirrors > Home > MPE Home > Th. List > Mathboxes > dvrunz | Structured version Visualization version GIF version | ||
| Description: In a division ring the ring unit is different from the zero. (Contributed by FL, 14-Feb-2010.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| dvrunz.1 | ⊢ 𝐺 = (1st ‘𝑅) |
| dvrunz.2 | ⊢ 𝐻 = (2nd ‘𝑅) |
| dvrunz.3 | ⊢ 𝑋 = ran 𝐺 |
| dvrunz.4 | ⊢ 𝑍 = (GId‘𝐺) |
| dvrunz.5 | ⊢ 𝑈 = (GId‘𝐻) |
| Ref | Expression |
|---|---|
| dvrunz | ⊢ (𝑅 ∈ DivRingOps → 𝑈 ≠ 𝑍) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dvrunz.4 | . . . 4 ⊢ 𝑍 = (GId‘𝐺) | |
| 2 | 1 | fvexi 6920 | . . 3 ⊢ 𝑍 ∈ V |
| 3 | 2 | zrdivrng 37960 | . 2 ⊢ ¬ 〈{〈〈𝑍, 𝑍〉, 𝑍〉}, {〈〈𝑍, 𝑍〉, 𝑍〉}〉 ∈ DivRingOps |
| 4 | dvrunz.1 | . . . . . . 7 ⊢ 𝐺 = (1st ‘𝑅) | |
| 5 | dvrunz.2 | . . . . . . 7 ⊢ 𝐻 = (2nd ‘𝑅) | |
| 6 | dvrunz.3 | . . . . . . 7 ⊢ 𝑋 = ran 𝐺 | |
| 7 | 4, 5, 6, 1 | drngoi 37958 | . . . . . 6 ⊢ (𝑅 ∈ DivRingOps → (𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp)) |
| 8 | 7 | simpld 494 | . . . . 5 ⊢ (𝑅 ∈ DivRingOps → 𝑅 ∈ RingOps) |
| 9 | dvrunz.5 | . . . . . 6 ⊢ 𝑈 = (GId‘𝐻) | |
| 10 | 4, 5, 1, 9, 6 | rngoueqz 37947 | . . . . 5 ⊢ (𝑅 ∈ RingOps → (𝑋 ≈ 1o ↔ 𝑈 = 𝑍)) |
| 11 | 8, 10 | syl 17 | . . . 4 ⊢ (𝑅 ∈ DivRingOps → (𝑋 ≈ 1o ↔ 𝑈 = 𝑍)) |
| 12 | 4, 6, 1 | rngosn6 37933 | . . . . . . 7 ⊢ (𝑅 ∈ RingOps → (𝑋 ≈ 1o ↔ 𝑅 = 〈{〈〈𝑍, 𝑍〉, 𝑍〉}, {〈〈𝑍, 𝑍〉, 𝑍〉}〉)) |
| 13 | 8, 12 | syl 17 | . . . . . 6 ⊢ (𝑅 ∈ DivRingOps → (𝑋 ≈ 1o ↔ 𝑅 = 〈{〈〈𝑍, 𝑍〉, 𝑍〉}, {〈〈𝑍, 𝑍〉, 𝑍〉}〉)) |
| 14 | eleq1 2829 | . . . . . . 7 ⊢ (𝑅 = 〈{〈〈𝑍, 𝑍〉, 𝑍〉}, {〈〈𝑍, 𝑍〉, 𝑍〉}〉 → (𝑅 ∈ DivRingOps ↔ 〈{〈〈𝑍, 𝑍〉, 𝑍〉}, {〈〈𝑍, 𝑍〉, 𝑍〉}〉 ∈ DivRingOps)) | |
| 15 | 14 | biimpd 229 | . . . . . 6 ⊢ (𝑅 = 〈{〈〈𝑍, 𝑍〉, 𝑍〉}, {〈〈𝑍, 𝑍〉, 𝑍〉}〉 → (𝑅 ∈ DivRingOps → 〈{〈〈𝑍, 𝑍〉, 𝑍〉}, {〈〈𝑍, 𝑍〉, 𝑍〉}〉 ∈ DivRingOps)) |
| 16 | 13, 15 | biimtrdi 253 | . . . . 5 ⊢ (𝑅 ∈ DivRingOps → (𝑋 ≈ 1o → (𝑅 ∈ DivRingOps → 〈{〈〈𝑍, 𝑍〉, 𝑍〉}, {〈〈𝑍, 𝑍〉, 𝑍〉}〉 ∈ DivRingOps))) |
| 17 | 16 | pm2.43a 54 | . . . 4 ⊢ (𝑅 ∈ DivRingOps → (𝑋 ≈ 1o → 〈{〈〈𝑍, 𝑍〉, 𝑍〉}, {〈〈𝑍, 𝑍〉, 𝑍〉}〉 ∈ DivRingOps)) |
| 18 | 11, 17 | sylbird 260 | . . 3 ⊢ (𝑅 ∈ DivRingOps → (𝑈 = 𝑍 → 〈{〈〈𝑍, 𝑍〉, 𝑍〉}, {〈〈𝑍, 𝑍〉, 𝑍〉}〉 ∈ DivRingOps)) |
| 19 | 18 | necon3bd 2954 | . 2 ⊢ (𝑅 ∈ DivRingOps → (¬ 〈{〈〈𝑍, 𝑍〉, 𝑍〉}, {〈〈𝑍, 𝑍〉, 𝑍〉}〉 ∈ DivRingOps → 𝑈 ≠ 𝑍)) |
| 20 | 3, 19 | mpi 20 | 1 ⊢ (𝑅 ∈ DivRingOps → 𝑈 ≠ 𝑍) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∖ cdif 3948 {csn 4626 〈cop 4632 class class class wbr 5143 × cxp 5683 ran crn 5686 ↾ cres 5687 ‘cfv 6561 1st c1st 8012 2nd c2nd 8013 1oc1o 8499 ≈ cen 8982 GrpOpcgr 30508 GIdcgi 30509 RingOpscrngo 37901 DivRingOpscdrng 37955 |
| 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-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-1st 8014 df-2nd 8015 df-1o 8506 df-en 8986 df-grpo 30512 df-gid 30513 df-ablo 30564 df-ass 37850 df-exid 37852 df-mgmOLD 37856 df-sgrOLD 37868 df-mndo 37874 df-rngo 37902 df-drngo 37956 |
| This theorem is referenced by: isdrngo2 37965 divrngpr 38060 isfldidl 38075 |
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