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Mirrors > Home > MPE Home > Th. List > Mathboxes > dvrunz | Structured version Visualization version Unicode 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 | |
dvrunz.2 | |
dvrunz.3 | |
dvrunz.4 | GId |
dvrunz.5 | GId |
Ref | Expression |
---|---|
dvrunz |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvrunz.4 | . . . 4 GId | |
2 | fvex 6201 | . . . 4 GId | |
3 | 1, 2 | eqeltri 2697 | . . 3 |
4 | 3 | zrdivrng 33752 | . 2 |
5 | dvrunz.1 | . . . . . . 7 | |
6 | dvrunz.2 | . . . . . . 7 | |
7 | dvrunz.3 | . . . . . . 7 | |
8 | 5, 6, 7, 1 | drngoi 33750 | . . . . . 6 |
9 | 8 | simpld 475 | . . . . 5 |
10 | dvrunz.5 | . . . . . 6 GId | |
11 | 5, 6, 1, 10, 7 | rngoueqz 33739 | . . . . 5 |
12 | 9, 11 | syl 17 | . . . 4 |
13 | 5, 7, 1 | rngosn6 33725 | . . . . . . 7 |
14 | 9, 13 | syl 17 | . . . . . 6 |
15 | eleq1 2689 | . . . . . . 7 | |
16 | 15 | biimpd 219 | . . . . . 6 |
17 | 14, 16 | syl6bi 243 | . . . . 5 |
18 | 17 | pm2.43a 54 | . . . 4 |
19 | 12, 18 | sylbird 250 | . . 3 |
20 | 19 | necon3bd 2808 | . 2 |
21 | 4, 20 | mpi 20 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wb 196 wceq 1483 wcel 1990 wne 2794 cvv 3200 cdif 3571 csn 4177 cop 4183 class class class wbr 4653 cxp 5112 crn 5115 cres 5116 cfv 5888 c1st 7166 c2nd 7167 c1o 7553 cen 7952 cgr 27343 GIdcgi 27344 crngo 33693 cdrng 33747 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-om 7066 df-1st 7168 df-2nd 7169 df-1o 7560 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-grpo 27347 df-gid 27348 df-ablo 27399 df-ass 33642 df-exid 33644 df-mgmOLD 33648 df-sgrOLD 33660 df-mndo 33666 df-rngo 33694 df-drngo 33748 |
This theorem is referenced by: isdrngo2 33757 divrngpr 33852 isfldidl 33867 |
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