Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > mulcanpi | Structured version Visualization version Unicode version |
Description: Multiplication cancellation law for positive integers. (Contributed by NM, 4-Feb-1996.) (Revised by Mario Carneiro, 10-May-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
mulcanpi |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mulclpi 9715 | . . . . . . . . . 10 | |
2 | eleq1 2689 | . . . . . . . . . 10 | |
3 | 1, 2 | syl5ib 234 | . . . . . . . . 9 |
4 | 3 | imp 445 | . . . . . . . 8 |
5 | dmmulpi 9713 | . . . . . . . . 9 | |
6 | 0npi 9704 | . . . . . . . . 9 | |
7 | 5, 6 | ndmovrcl 6820 | . . . . . . . 8 |
8 | simpr 477 | . . . . . . . 8 | |
9 | 4, 7, 8 | 3syl 18 | . . . . . . 7 |
10 | mulpiord 9707 | . . . . . . . . . 10 | |
11 | 10 | adantr 481 | . . . . . . . . 9 |
12 | mulpiord 9707 | . . . . . . . . . 10 | |
13 | 12 | adantlr 751 | . . . . . . . . 9 |
14 | 11, 13 | eqeq12d 2637 | . . . . . . . 8 |
15 | pinn 9700 | . . . . . . . . . . . . 13 | |
16 | pinn 9700 | . . . . . . . . . . . . 13 | |
17 | pinn 9700 | . . . . . . . . . . . . 13 | |
18 | elni2 9699 | . . . . . . . . . . . . . . . 16 | |
19 | 18 | simprbi 480 | . . . . . . . . . . . . . . 15 |
20 | nnmcan 7714 | . . . . . . . . . . . . . . . 16 | |
21 | 20 | biimpd 219 | . . . . . . . . . . . . . . 15 |
22 | 19, 21 | sylan2 491 | . . . . . . . . . . . . . 14 |
23 | 22 | ex 450 | . . . . . . . . . . . . 13 |
24 | 15, 16, 17, 23 | syl3an 1368 | . . . . . . . . . . . 12 |
25 | 24 | 3exp 1264 | . . . . . . . . . . 11 |
26 | 25 | com4r 94 | . . . . . . . . . 10 |
27 | 26 | pm2.43i 52 | . . . . . . . . 9 |
28 | 27 | imp31 448 | . . . . . . . 8 |
29 | 14, 28 | sylbid 230 | . . . . . . 7 |
30 | 9, 29 | sylan2 491 | . . . . . 6 |
31 | 30 | exp32 631 | . . . . 5 |
32 | 31 | imp4b 613 | . . . 4 |
33 | 32 | pm2.43i 52 | . . 3 |
34 | 33 | ex 450 | . 2 |
35 | oveq2 6658 | . 2 | |
36 | 34, 35 | impbid1 215 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 w3a 1037 wceq 1483 wcel 1990 c0 3915 (class class class)co 6650 com 7065 comu 7558 cnpi 9666 cmi 9668 |
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-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-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-pred 5680 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-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-oadd 7564 df-omul 7565 df-ni 9694 df-mi 9696 |
This theorem is referenced by: enqer 9743 nqereu 9751 adderpqlem 9776 mulerpqlem 9777 |
Copyright terms: Public domain | W3C validator |