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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 6905 | . . 3 ⊢ 𝑍 ∈ V |
3 | 2 | zrdivrng 36816 | . 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 36814 | . . . . . 6 ⊢ (𝑅 ∈ DivRingOps → (𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp)) |
8 | 7 | simpld 495 | . . . . 5 ⊢ (𝑅 ∈ DivRingOps → 𝑅 ∈ RingOps) |
9 | dvrunz.5 | . . . . . 6 ⊢ 𝑈 = (GId‘𝐻) | |
10 | 4, 5, 1, 9, 6 | rngoueqz 36803 | . . . . 5 ⊢ (𝑅 ∈ RingOps → (𝑋 ≈ 1o ↔ 𝑈 = 𝑍)) |
11 | 8, 10 | syl 17 | . . . 4 ⊢ (𝑅 ∈ DivRingOps → (𝑋 ≈ 1o ↔ 𝑈 = 𝑍)) |
12 | 4, 6, 1 | rngosn6 36789 | . . . . . . 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 2821 | . . . . . . 7 ⊢ (𝑅 = ⟨{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}, {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩ → (𝑅 ∈ DivRingOps ↔ ⟨{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}, {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩ ∈ DivRingOps)) | |
15 | 14 | biimpd 228 | . . . . . 6 ⊢ (𝑅 = ⟨{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}, {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩ → (𝑅 ∈ DivRingOps → ⟨{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}, {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩ ∈ DivRingOps)) |
16 | 13, 15 | syl6bi 252 | . . . . 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 259 | . . 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 205 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ∖ cdif 3945 {csn 4628 ⟨cop 4634 class class class wbr 5148 × cxp 5674 ran crn 5677 ↾ cres 5678 ‘cfv 6543 1st c1st 7972 2nd c2nd 7973 1oc1o 8458 ≈ cen 8935 GrpOpcgr 29737 GIdcgi 29738 RingOpscrngo 36757 DivRingOpscdrng 36811 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7364 df-ov 7411 df-1st 7974 df-2nd 7975 df-1o 8465 df-en 8939 df-grpo 29741 df-gid 29742 df-ablo 29793 df-ass 36706 df-exid 36708 df-mgmOLD 36712 df-sgrOLD 36724 df-mndo 36730 df-rngo 36758 df-drngo 36812 |
This theorem is referenced by: isdrngo2 36821 divrngpr 36916 isfldidl 36931 |
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