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Mirrors > Home > MPE Home > Th. List > Mathboxes > dvrunz | Structured version Visualization version GIF version |
Description: In a division ring the 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 | fvex 6113 | . . . 4 ⊢ (GId‘𝐺) ∈ V | |
3 | 1, 2 | eqeltri 2684 | . . 3 ⊢ 𝑍 ∈ V |
4 | 3 | zrdivrng 32922 | . 2 ⊢ ¬ 〈{〈〈𝑍, 𝑍〉, 𝑍〉}, {〈〈𝑍, 𝑍〉, 𝑍〉}〉 ∈ DivRingOps |
5 | dvrunz.1 | . . . . . . 7 ⊢ 𝐺 = (1st ‘𝑅) | |
6 | dvrunz.2 | . . . . . . 7 ⊢ 𝐻 = (2nd ‘𝑅) | |
7 | dvrunz.3 | . . . . . . 7 ⊢ 𝑋 = ran 𝐺 | |
8 | 5, 6, 7, 1 | drngoi 32920 | . . . . . 6 ⊢ (𝑅 ∈ DivRingOps → (𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp)) |
9 | 8 | simpld 474 | . . . . 5 ⊢ (𝑅 ∈ DivRingOps → 𝑅 ∈ RingOps) |
10 | dvrunz.5 | . . . . . 6 ⊢ 𝑈 = (GId‘𝐻) | |
11 | 5, 6, 1, 10, 7 | rngoueqz 32909 | . . . . 5 ⊢ (𝑅 ∈ RingOps → (𝑋 ≈ 1𝑜 ↔ 𝑈 = 𝑍)) |
12 | 9, 11 | syl 17 | . . . 4 ⊢ (𝑅 ∈ DivRingOps → (𝑋 ≈ 1𝑜 ↔ 𝑈 = 𝑍)) |
13 | 5, 7, 1 | rngosn6 32895 | . . . . . . 7 ⊢ (𝑅 ∈ RingOps → (𝑋 ≈ 1𝑜 ↔ 𝑅 = 〈{〈〈𝑍, 𝑍〉, 𝑍〉}, {〈〈𝑍, 𝑍〉, 𝑍〉}〉)) |
14 | 9, 13 | syl 17 | . . . . . 6 ⊢ (𝑅 ∈ DivRingOps → (𝑋 ≈ 1𝑜 ↔ 𝑅 = 〈{〈〈𝑍, 𝑍〉, 𝑍〉}, {〈〈𝑍, 𝑍〉, 𝑍〉}〉)) |
15 | eleq1 2676 | . . . . . . 7 ⊢ (𝑅 = 〈{〈〈𝑍, 𝑍〉, 𝑍〉}, {〈〈𝑍, 𝑍〉, 𝑍〉}〉 → (𝑅 ∈ DivRingOps ↔ 〈{〈〈𝑍, 𝑍〉, 𝑍〉}, {〈〈𝑍, 𝑍〉, 𝑍〉}〉 ∈ DivRingOps)) | |
16 | 15 | biimpd 218 | . . . . . 6 ⊢ (𝑅 = 〈{〈〈𝑍, 𝑍〉, 𝑍〉}, {〈〈𝑍, 𝑍〉, 𝑍〉}〉 → (𝑅 ∈ DivRingOps → 〈{〈〈𝑍, 𝑍〉, 𝑍〉}, {〈〈𝑍, 𝑍〉, 𝑍〉}〉 ∈ DivRingOps)) |
17 | 14, 16 | syl6bi 242 | . . . . 5 ⊢ (𝑅 ∈ DivRingOps → (𝑋 ≈ 1𝑜 → (𝑅 ∈ DivRingOps → 〈{〈〈𝑍, 𝑍〉, 𝑍〉}, {〈〈𝑍, 𝑍〉, 𝑍〉}〉 ∈ DivRingOps))) |
18 | 17 | pm2.43a 52 | . . . 4 ⊢ (𝑅 ∈ DivRingOps → (𝑋 ≈ 1𝑜 → 〈{〈〈𝑍, 𝑍〉, 𝑍〉}, {〈〈𝑍, 𝑍〉, 𝑍〉}〉 ∈ DivRingOps)) |
19 | 12, 18 | sylbird 249 | . . 3 ⊢ (𝑅 ∈ DivRingOps → (𝑈 = 𝑍 → 〈{〈〈𝑍, 𝑍〉, 𝑍〉}, {〈〈𝑍, 𝑍〉, 𝑍〉}〉 ∈ DivRingOps)) |
20 | 19 | necon3bd 2796 | . 2 ⊢ (𝑅 ∈ DivRingOps → (¬ 〈{〈〈𝑍, 𝑍〉, 𝑍〉}, {〈〈𝑍, 𝑍〉, 𝑍〉}〉 ∈ DivRingOps → 𝑈 ≠ 𝑍)) |
21 | 4, 20 | mpi 20 | 1 ⊢ (𝑅 ∈ DivRingOps → 𝑈 ≠ 𝑍) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 Vcvv 3173 ∖ cdif 3537 {csn 4125 〈cop 4131 class class class wbr 4583 × cxp 5036 ran crn 5039 ↾ cres 5040 ‘cfv 5804 1st c1st 7057 2nd c2nd 7058 1𝑜c1o 7440 ≈ cen 7838 GrpOpcgr 26727 GIdcgi 26728 RingOpscrngo 32863 DivRingOpscdrng 32917 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-om 6958 df-1st 7059 df-2nd 7060 df-1o 7447 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-grpo 26731 df-gid 26732 df-ablo 26783 df-ass 32812 df-exid 32814 df-mgmOLD 32818 df-sgrOLD 32830 df-mndo 32836 df-rngo 32864 df-drngo 32918 |
This theorem is referenced by: isdrngo2 32927 divrngpr 33022 isfldidl 33037 |
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