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Mirrors > Home > ILE Home > Th. List > mulcanapd | Unicode version |
Description: Cancellation law for multiplication. (Contributed by Jim Kingdon, 21-Feb-2020.) |
Ref | Expression |
---|---|
mulcand.1 | |
mulcand.2 | |
mulcand.3 | |
mulcand.4 | # |
Ref | Expression |
---|---|
mulcanapd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mulcand.3 | . . . 4 | |
2 | mulcand.4 | . . . 4 # | |
3 | recexap 8407 | . . . 4 # | |
4 | 1, 2, 3 | syl2anc 408 | . . 3 |
5 | oveq2 5775 | . . . 4 | |
6 | simprl 520 | . . . . . . . . 9 | |
7 | 1 | adantr 274 | . . . . . . . . 9 |
8 | 6, 7 | mulcomd 7780 | . . . . . . . 8 |
9 | simprr 521 | . . . . . . . 8 | |
10 | 8, 9 | eqtrd 2170 | . . . . . . 7 |
11 | 10 | oveq1d 5782 | . . . . . 6 |
12 | mulcand.1 | . . . . . . . 8 | |
13 | 12 | adantr 274 | . . . . . . 7 |
14 | 6, 7, 13 | mulassd 7782 | . . . . . 6 |
15 | 13 | mulid2d 7777 | . . . . . 6 |
16 | 11, 14, 15 | 3eqtr3d 2178 | . . . . 5 |
17 | 10 | oveq1d 5782 | . . . . . 6 |
18 | mulcand.2 | . . . . . . . 8 | |
19 | 18 | adantr 274 | . . . . . . 7 |
20 | 6, 7, 19 | mulassd 7782 | . . . . . 6 |
21 | 19 | mulid2d 7777 | . . . . . 6 |
22 | 17, 20, 21 | 3eqtr3d 2178 | . . . . 5 |
23 | 16, 22 | eqeq12d 2152 | . . . 4 |
24 | 5, 23 | syl5ib 153 | . . 3 |
25 | 4, 24 | rexlimddv 2552 | . 2 |
26 | oveq2 5775 | . 2 | |
27 | 25, 26 | impbid1 141 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wcel 1480 wrex 2415 class class class wbr 3924 (class class class)co 5767 cc 7611 cc0 7613 c1 7614 cmul 7618 # cap 8336 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-mulrcl 7712 ax-addcom 7713 ax-mulcom 7714 ax-addass 7715 ax-mulass 7716 ax-distr 7717 ax-i2m1 7718 ax-0lt1 7719 ax-1rid 7720 ax-0id 7721 ax-rnegex 7722 ax-precex 7723 ax-cnre 7724 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 ax-pre-apti 7728 ax-pre-ltadd 7729 ax-pre-mulgt0 7730 ax-pre-mulext 7731 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-id 4210 df-po 4213 df-iso 4214 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-iota 5083 df-fun 5120 df-fv 5126 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-sub 7928 df-neg 7929 df-reap 8330 df-ap 8337 |
This theorem is referenced by: mulcanap2d 8416 mulcanapad 8417 mulcanap 8419 div11ap 8453 eqneg 8485 dvdscmulr 11511 qredeq 11766 cncongr1 11773 |
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