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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 6665 | . . 3 ⊢ 𝑍 ∈ V |
3 | 2 | zrdivrng 35258 | . 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 35256 | . . . . . 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 35245 | . . . . 5 ⊢ (𝑅 ∈ RingOps → (𝑋 ≈ 1o ↔ 𝑈 = 𝑍)) |
11 | 8, 10 | syl 17 | . . . 4 ⊢ (𝑅 ∈ DivRingOps → (𝑋 ≈ 1o ↔ 𝑈 = 𝑍)) |
12 | 4, 6, 1 | rngosn6 35231 | . . . . . . 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 2898 | . . . . . . 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 3025 | . 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 1537 ∈ wcel 2114 ≠ wne 3011 ∖ cdif 3916 {csn 4548 〈cop 4554 class class class wbr 5047 × cxp 5534 ran crn 5537 ↾ cres 5538 ‘cfv 6336 1st c1st 7668 2nd c2nd 7669 1oc1o 8076 ≈ cen 8487 GrpOpcgr 28245 GIdcgi 28246 RingOpscrngo 35199 DivRingOpscdrng 35253 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2792 ax-rep 5171 ax-sep 5184 ax-nul 5191 ax-pow 5247 ax-pr 5311 ax-un 7442 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2891 df-nfc 2959 df-ne 3012 df-ral 3138 df-rex 3139 df-reu 3140 df-rmo 3141 df-rab 3142 df-v 3483 df-sbc 3759 df-csb 3867 df-dif 3922 df-un 3924 df-in 3926 df-ss 3935 df-pss 3937 df-nul 4275 df-if 4449 df-pw 4522 df-sn 4549 df-pr 4551 df-tp 4553 df-op 4555 df-uni 4820 df-iun 4902 df-br 5048 df-opab 5110 df-mpt 5128 df-tr 5154 df-id 5441 df-eprel 5446 df-po 5455 df-so 5456 df-fr 5495 df-we 5497 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6295 df-fun 6338 df-fn 6339 df-f 6340 df-f1 6341 df-fo 6342 df-f1o 6343 df-fv 6344 df-riota 7095 df-ov 7140 df-om 7562 df-1st 7670 df-2nd 7671 df-1o 8083 df-er 8270 df-en 8491 df-dom 8492 df-sdom 8493 df-fin 8494 df-grpo 28249 df-gid 28250 df-ablo 28301 df-ass 35148 df-exid 35150 df-mgmOLD 35154 df-sgrOLD 35166 df-mndo 35172 df-rngo 35200 df-drngo 35254 |
This theorem is referenced by: isdrngo2 35263 divrngpr 35358 isfldidl 35373 |
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