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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 37939 | . 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 37937 | . . . . . 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 37926 | . . . . 5 ⊢ (𝑅 ∈ RingOps → (𝑋 ≈ 1o ↔ 𝑈 = 𝑍)) |
11 | 8, 10 | syl 17 | . . . 4 ⊢ (𝑅 ∈ DivRingOps → (𝑋 ≈ 1o ↔ 𝑈 = 𝑍)) |
12 | 4, 6, 1 | rngosn6 37912 | . . . . . . 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 2826 | . . . . . . 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 2951 | . 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 1536 ∈ wcel 2105 ≠ wne 2937 ∖ cdif 3959 {csn 4630 〈cop 4636 class class class wbr 5147 × cxp 5686 ran crn 5689 ↾ cres 5690 ‘cfv 6562 1st c1st 8010 2nd c2nd 8011 1oc1o 8497 ≈ cen 8980 GrpOpcgr 30517 GIdcgi 30518 RingOpscrngo 37880 DivRingOpscdrng 37934 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-1st 8012 df-2nd 8013 df-1o 8504 df-en 8984 df-grpo 30521 df-gid 30522 df-ablo 30573 df-ass 37829 df-exid 37831 df-mgmOLD 37835 df-sgrOLD 37847 df-mndo 37853 df-rngo 37881 df-drngo 37935 |
This theorem is referenced by: isdrngo2 37944 divrngpr 38039 isfldidl 38054 |
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