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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 | 1 | fvexi 6687 | . . 3 ⊢ 𝑍 ∈ V |
3 | 2 | zrdivrng 35235 | . 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 35233 | . . . . . 6 ⊢ (𝑅 ∈ DivRingOps → (𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp)) |
8 | 7 | simpld 497 | . . . . 5 ⊢ (𝑅 ∈ DivRingOps → 𝑅 ∈ RingOps) |
9 | dvrunz.5 | . . . . . 6 ⊢ 𝑈 = (GId‘𝐻) | |
10 | 4, 5, 1, 9, 6 | rngoueqz 35222 | . . . . 5 ⊢ (𝑅 ∈ RingOps → (𝑋 ≈ 1o ↔ 𝑈 = 𝑍)) |
11 | 8, 10 | syl 17 | . . . 4 ⊢ (𝑅 ∈ DivRingOps → (𝑋 ≈ 1o ↔ 𝑈 = 𝑍)) |
12 | 4, 6, 1 | rngosn6 35208 | . . . . . . 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 2903 | . . . . . . 7 ⊢ (𝑅 = 〈{〈〈𝑍, 𝑍〉, 𝑍〉}, {〈〈𝑍, 𝑍〉, 𝑍〉}〉 → (𝑅 ∈ DivRingOps ↔ 〈{〈〈𝑍, 𝑍〉, 𝑍〉}, {〈〈𝑍, 𝑍〉, 𝑍〉}〉 ∈ DivRingOps)) | |
15 | 14 | biimpd 231 | . . . . . 6 ⊢ (𝑅 = 〈{〈〈𝑍, 𝑍〉, 𝑍〉}, {〈〈𝑍, 𝑍〉, 𝑍〉}〉 → (𝑅 ∈ DivRingOps → 〈{〈〈𝑍, 𝑍〉, 𝑍〉}, {〈〈𝑍, 𝑍〉, 𝑍〉}〉 ∈ DivRingOps)) |
16 | 13, 15 | syl6bi 255 | . . . . 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 262 | . . 3 ⊢ (𝑅 ∈ DivRingOps → (𝑈 = 𝑍 → 〈{〈〈𝑍, 𝑍〉, 𝑍〉}, {〈〈𝑍, 𝑍〉, 𝑍〉}〉 ∈ DivRingOps)) |
19 | 18 | necon3bd 3033 | . 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 208 = wceq 1536 ∈ wcel 2113 ≠ wne 3019 ∖ cdif 3936 {csn 4570 〈cop 4576 class class class wbr 5069 × cxp 5556 ran crn 5559 ↾ cres 5560 ‘cfv 6358 1st c1st 7690 2nd c2nd 7691 1oc1o 8098 ≈ cen 8509 GrpOpcgr 28269 GIdcgi 28270 RingOpscrngo 35176 DivRingOpscdrng 35230 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-om 7584 df-1st 7692 df-2nd 7693 df-1o 8105 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-grpo 28273 df-gid 28274 df-ablo 28325 df-ass 35125 df-exid 35127 df-mgmOLD 35131 df-sgrOLD 35143 df-mndo 35149 df-rngo 35177 df-drngo 35231 |
This theorem is referenced by: isdrngo2 35240 divrngpr 35335 isfldidl 35350 |
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