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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 11443 | . . . 4 ⊢ -𝐵 = (0 − 𝐵) | |
2 | 1 | oveq2i 7415 | . . 3 ⊢ (𝐴 + -𝐵) = (𝐴 + (0 − 𝐵)) |
3 | 2 | a1i 11 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + -𝐵) = (𝐴 + (0 − 𝐵))) |
4 | 0cn 11202 | . . 3 ⊢ 0 ∈ ℂ | |
5 | addsubass 11466 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 0 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 0) − 𝐵) = (𝐴 + (0 − 𝐵))) | |
6 | 4, 5 | mp3an2 1450 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 0) − 𝐵) = (𝐴 + (0 − 𝐵))) |
7 | simpl 484 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 𝐴 ∈ ℂ) | |
8 | 7 | addridd 11410 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 0) = 𝐴) |
9 | 8 | oveq1d 7419 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 0) − 𝐵) = (𝐴 − 𝐵)) |
10 | 3, 6, 9 | 3eqtr2d 2779 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + -𝐵) = (𝐴 − 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 (class class class)co 7404 ℂcc 11104 0cc0 11106 + caddc 11109 − cmin 11440 -cneg 11441 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-po 5587 df-so 5588 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-ltxr 11249 df-sub 11442 df-neg 11443 |
This theorem is referenced by: negdi2 11514 negsubdi2 11515 resubcli 11518 resubcl 11520 negsubi 11534 negsubd 11573 submul2 11650 addneg1mul 11652 mulsub 11653 divsubdir 11904 difgtsumgt 12521 elz2 12572 zsubcl 12600 qsubcl 12948 rexsub 13208 fzsubel 13533 ceim1l 13808 modcyc2 13868 negmod 13877 modsumfzodifsn 13905 expsub 14072 binom2sub 14179 seqshft 15028 resub 15070 imsub 15078 cjsub 15092 cjreim 15103 absdiflt 15260 absdifle 15261 abs2dif2 15276 subcn2 15535 bpoly2 15997 bpoly3 15998 efsub 16039 efi4p 16076 sinsub 16107 cossub 16108 demoivreALT 16140 dvdssub 16243 modgcd 16470 gzsubcl 16869 psgnunilem2 19356 cnfldsub 20958 itg1sub 25209 plyremlem 25799 sineq0 26015 logneg2 26105 ang180lem2 26295 asinsin 26377 atanneg 26392 atancj 26395 atanlogadd 26399 atanlogsublem 26400 atanlogsub 26401 2efiatan 26403 tanatan 26404 cosatan 26406 atans2 26416 dvatan 26420 zetacvg 26499 wilthlem1 26552 wilthlem2 26553 basellem8 26572 lgsvalmod 26799 cnnvm 29913 cncph 30050 hvsubdistr2 30281 lnfnsubi 31277 subfacval2 34116 itg2addnclem3 36479 lcmineqlem1 40832 2xp3dxp2ge1d 40960 pellexlem6 41505 pell14qrdich 41540 rmxm1 41606 rmym1 41607 addsubeq0 45939 omoeALTV 46288 omeoALTV 46289 emoo 46307 emee 46309 zlmodzxzequap 47082 flsubz 47105 |
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