Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > negsub | Structured version Visualization version GIF 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 | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + -𝐵) = (𝐴 − 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-neg 10269 | . . . 4 ⊢ -𝐵 = (0 − 𝐵) | |
2 | 1 | oveq2i 6661 | . . 3 ⊢ (𝐴 + -𝐵) = (𝐴 + (0 − 𝐵)) |
3 | 2 | a1i 11 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + -𝐵) = (𝐴 + (0 − 𝐵))) |
4 | 0cn 10032 | . . 3 ⊢ 0 ∈ ℂ | |
5 | addsubass 10291 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 0 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 0) − 𝐵) = (𝐴 + (0 − 𝐵))) | |
6 | 4, 5 | mp3an2 1412 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 0) − 𝐵) = (𝐴 + (0 − 𝐵))) |
7 | simpl 473 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 𝐴 ∈ ℂ) | |
8 | 7 | addid1d 10236 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 0) = 𝐴) |
9 | 8 | oveq1d 6665 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 0) − 𝐵) = (𝐴 − 𝐵)) |
10 | 3, 6, 9 | 3eqtr2d 2662 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + -𝐵) = (𝐴 − 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 (class class class)co 6650 ℂcc 9934 0cc0 9936 + caddc 9939 − cmin 10266 -cneg 10267 |
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 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 |
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-nel 2898 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-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 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-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-ltxr 10079 df-sub 10268 df-neg 10269 |
This theorem is referenced by: negdi2 10339 negsubdi2 10340 resubcli 10343 resubcl 10345 negsubi 10359 negsubd 10398 submul2 10470 addneg1mul 10472 mulsub 10473 divsubdir 10721 difgtsumgt 11346 elz2 11394 zsubcl 11419 qsubcl 11807 rexsub 12064 fzsubel 12377 ceim1l 12646 modcyc2 12706 negmod 12715 modsumfzodifsn 12743 expsub 12908 binom2sub 12981 seqshft 13825 resub 13867 imsub 13875 cjsub 13889 cjreim 13900 absdiflt 14057 absdifle 14058 abs2dif2 14073 subcn2 14325 bpoly2 14788 bpoly3 14789 efsub 14830 efi4p 14867 sinsub 14898 cossub 14899 demoivreALT 14931 dvdssub 15026 modgcd 15253 gzsubcl 15644 psgnunilem2 17915 cnfldsub 19774 itg1sub 23476 plyremlem 24059 sineq0 24273 logneg2 24361 ang180lem2 24540 asinsin 24619 atanneg 24634 atancj 24637 atanlogadd 24641 atanlogsublem 24642 atanlogsub 24643 2efiatan 24645 tanatan 24646 cosatan 24648 atans2 24658 dvatan 24662 zetacvg 24741 wilthlem1 24794 wilthlem2 24795 basellem8 24814 lgsvalmod 25041 cnnvm 27537 cncph 27674 hvsubdistr2 27907 lnfnsubi 28905 subfacval2 31169 itg2addnclem3 33463 pellexlem6 37398 pell14qrdich 37433 rmxm1 37499 rmym1 37500 omoeALTV 41596 omeoALTV 41597 emoo 41613 emee 41615 zlmodzxzequap 42288 flsubz 42312 |
Copyright terms: Public domain | W3C validator |