Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > mulneg1 | Unicode version |
Description: Product with negative is negative of product. Theorem I.12 of [Apostol] p. 18. (Contributed by NM, 14-May-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
mulneg1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0cn 7751 | . . . 4 | |
2 | subdir 8141 | . . . 4 | |
3 | 1, 2 | mp3an1 1302 | . . 3 |
4 | simpr 109 | . . . . 5 | |
5 | 4 | mul02d 8147 | . . . 4 |
6 | 5 | oveq1d 5782 | . . 3 |
7 | 3, 6 | eqtrd 2170 | . 2 |
8 | df-neg 7929 | . . 3 | |
9 | 8 | oveq1i 5777 | . 2 |
10 | df-neg 7929 | . 2 | |
11 | 7, 9, 10 | 3eqtr4g 2195 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 (class class class)co 5767 cc 7611 cc0 7613 cmul 7618 cmin 7926 cneg 7927 |
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-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-setind 4447 ax-resscn 7705 ax-1cn 7706 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-addcom 7713 ax-mulcom 7714 ax-addass 7715 ax-distr 7717 ax-i2m1 7718 ax-0id 7721 ax-rnegex 7722 ax-cnre 7724 |
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-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-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-sub 7928 df-neg 7929 |
This theorem is referenced by: mulneg2 8151 mulneg12 8152 mulm1 8155 mulneg1i 8159 mulneg1d 8166 divnegap 8459 zmulcl 9100 cjreim 10668 tanval3ap 11410 dvdsnegb 11499 odd2np1 11559 modgcd 11668 sinperlem 12878 |
Copyright terms: Public domain | W3C validator |