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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 7929 | . . . 4 | |
2 | 1 | oveq2i 5778 | . . 3 |
3 | 2 | a1i 9 | . 2 |
4 | 0cn 7751 | . . 3 | |
5 | addsubass 7965 | . . 3 | |
6 | 4, 5 | mp3an2 1303 | . 2 |
7 | simpl 108 | . . . 4 | |
8 | 7 | addid1d 7904 | . . 3 |
9 | 8 | oveq1d 5782 | . 2 |
10 | 3, 6, 9 | 3eqtr2d 2176 | 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 caddc 7616 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-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: negdi2 8013 negsubdi2 8014 resubcli 8018 resubcl 8019 negsubi 8033 negsubd 8072 submul2 8154 mulsub 8156 subap0 8398 divsubdirap 8461 zsubcl 9088 elz2 9115 qsubcl 9423 rexsub 9629 fzsubel 9833 expsubap 10334 binom2sub 10398 resub 10635 imsub 10643 cjsub 10657 cjreim 10668 absdiflt 10857 absdifle 10858 abs2dif2 10872 subcn2 11073 efsub 11376 efi4p 11413 sinsub 11436 cossub 11437 demoivreALT 11469 dvdssub 11527 modgcd 11668 |
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