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Mirrors > Home > MPE Home > Th. List > Mathboxes > dmncan2 | Structured version Visualization version Unicode version |
Description: Cancellation law for domains. (Contributed by Jeff Madsen, 6-Jan-2011.) |
Ref | Expression |
---|---|
dmncan.1 | |
dmncan.2 | |
dmncan.3 | |
dmncan.4 | GId |
Ref | Expression |
---|---|
dmncan2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmncrng 33855 | . . . 4 CRingOps | |
2 | dmncan.1 | . . . . . . 7 | |
3 | dmncan.2 | . . . . . . 7 | |
4 | dmncan.3 | . . . . . . 7 | |
5 | 2, 3, 4 | crngocom 33800 | . . . . . 6 CRingOps |
6 | 5 | 3adant3r2 1275 | . . . . 5 CRingOps |
7 | 2, 3, 4 | crngocom 33800 | . . . . . 6 CRingOps |
8 | 7 | 3adant3r1 1274 | . . . . 5 CRingOps |
9 | 6, 8 | eqeq12d 2637 | . . . 4 CRingOps |
10 | 1, 9 | sylan 488 | . . 3 |
11 | 10 | adantr 481 | . 2 |
12 | 3anrot 1043 | . . . 4 | |
13 | 12 | biimpri 218 | . . 3 |
14 | dmncan.4 | . . . 4 GId | |
15 | 2, 3, 4, 14 | dmncan1 33875 | . . 3 |
16 | 13, 15 | sylanl2 683 | . 2 |
17 | 11, 16 | sylbid 230 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 w3a 1037 wceq 1483 wcel 1990 wne 2794 crn 5115 cfv 5888 (class class class)co 6650 c1st 7166 c2nd 7167 GIdcgi 27344 CRingOpsccring 33792 cdmn 33846 |
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-int 4476 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-oprab 6654 df-mpt2 6655 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-ginv 27349 df-gdiv 27350 df-ablo 27399 df-ass 33642 df-exid 33644 df-mgmOLD 33648 df-sgrOLD 33660 df-mndo 33666 df-rngo 33694 df-com2 33789 df-crngo 33793 df-idl 33809 df-pridl 33810 df-prrngo 33847 df-dmn 33848 df-igen 33859 |
This theorem is referenced by: (None) |
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