Mathbox for Jeff Madsen |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > 0rngo | Structured version Visualization version Unicode version |
Description: In a ring, iff the ring contains only . (Contributed by Jeff Madsen, 6-Jan-2011.) |
Ref | Expression |
---|---|
0ring.1 | |
0ring.2 | |
0ring.3 | |
0ring.4 | GId |
0ring.5 | GId |
Ref | Expression |
---|---|
0rngo |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ring.4 | . . . . . . 7 GId | |
2 | fvex 6201 | . . . . . . 7 GId | |
3 | 1, 2 | eqeltri 2697 | . . . . . 6 |
4 | 3 | snid 4208 | . . . . 5 |
5 | eleq1 2689 | . . . . 5 | |
6 | 4, 5 | mpbii 223 | . . . 4 |
7 | 0ring.1 | . . . . . 6 | |
8 | 7, 1 | 0idl 33824 | . . . . 5 |
9 | 0ring.2 | . . . . . 6 | |
10 | 0ring.3 | . . . . . 6 | |
11 | 0ring.5 | . . . . . 6 GId | |
12 | 7, 9, 10, 11 | 1idl 33825 | . . . . 5 |
13 | 8, 12 | mpdan 702 | . . . 4 |
14 | 6, 13 | syl5ib 234 | . . 3 |
15 | eqcom 2629 | . . 3 | |
16 | 14, 15 | syl6ib 241 | . 2 |
17 | 7 | rneqi 5352 | . . . . 5 |
18 | 10, 17 | eqtri 2644 | . . . 4 |
19 | 18, 9, 11 | rngo1cl 33738 | . . 3 |
20 | eleq2 2690 | . . . 4 | |
21 | elsni 4194 | . . . . 5 | |
22 | 21 | eqcomd 2628 | . . . 4 |
23 | 20, 22 | syl6bi 243 | . . 3 |
24 | 19, 23 | syl5com 31 | . 2 |
25 | 16, 24 | impbid 202 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wceq 1483 wcel 1990 cvv 3200 csn 4177 crn 5115 cfv 5888 c1st 7166 c2nd 7167 GIdcgi 27344 crngo 33693 cidl 33806 |
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-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-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 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-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-1st 7168 df-2nd 7169 df-grpo 27347 df-gid 27348 df-ginv 27349 df-ablo 27399 df-ass 33642 df-exid 33644 df-mgmOLD 33648 df-sgrOLD 33660 df-mndo 33666 df-rngo 33694 df-idl 33809 |
This theorem is referenced by: smprngopr 33851 isfldidl2 33868 |
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