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Mirrors > Home > ILE Home > Th. List > negsub | Unicode version |
Description: Relationship between subtraction and negative. Theorem I.3 of [Apostol] p. 18. (Contributed by NM, 21-Jan-1997.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
negsub |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-neg 7282 | . . . 4 | |
2 | 1 | oveq2i 5543 | . . 3 |
3 | 2 | a1i 9 | . 2 |
4 | 0cn 7111 | . . 3 | |
5 | addsubass 7318 | . . 3 | |
6 | 4, 5 | mp3an2 1256 | . 2 |
7 | simpl 107 | . . . 4 | |
8 | 7 | addid1d 7257 | . . 3 |
9 | 8 | oveq1d 5547 | . 2 |
10 | 3, 6, 9 | 3eqtr2d 2119 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wceq 1284 wcel 1433 (class class class)co 5532 cc 6979 cc0 6981 caddc 6984 cmin 7279 cneg 7280 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-setind 4280 ax-resscn 7068 ax-1cn 7069 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-addcom 7076 ax-addass 7078 ax-distr 7080 ax-i2m1 7081 ax-0id 7084 ax-rnegex 7085 ax-cnre 7087 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-iota 4887 df-fun 4924 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-sub 7281 df-neg 7282 |
This theorem is referenced by: negdi2 7366 negsubdi2 7367 resubcli 7371 resubcl 7372 negsubi 7386 negsubd 7425 submul2 7503 mulsub 7505 divsubdirap 7796 zsubcl 8392 elz2 8419 qsubcl 8723 fzsubel 9078 expsubap 9524 binom2sub 9587 resub 9757 imsub 9765 cjsub 9779 cjreim 9790 absdiflt 9978 absdifle 9979 abs2dif2 9993 subcn2 10150 dvdssub 10240 modgcd 10382 |
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