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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 11447 | . . . 4 ⊢ -𝐵 = (0 − 𝐵) | |
2 | 1 | oveq2i 7420 | . . 3 ⊢ (𝐴 + -𝐵) = (𝐴 + (0 − 𝐵)) |
3 | 2 | a1i 11 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + -𝐵) = (𝐴 + (0 − 𝐵))) |
4 | 0cn 11206 | . . 3 ⊢ 0 ∈ ℂ | |
5 | addsubass 11470 | . . 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 11414 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 0) = 𝐴) |
9 | 8 | oveq1d 7424 | . 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 7409 ℂcc 11108 0cc0 11110 + caddc 11113 − cmin 11444 -cneg 11445 |
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 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 |
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 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-po 5589 df-so 5590 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-pnf 11250 df-mnf 11251 df-ltxr 11253 df-sub 11446 df-neg 11447 |
This theorem is referenced by: negdi2 11518 negsubdi2 11519 resubcli 11522 resubcl 11524 negsubi 11538 negsubd 11577 submul2 11654 addneg1mul 11656 mulsub 11657 divsubdir 11908 difgtsumgt 12525 elz2 12576 zsubcl 12604 qsubcl 12952 rexsub 13212 fzsubel 13537 ceim1l 13812 modcyc2 13872 negmod 13881 modsumfzodifsn 13909 expsub 14076 binom2sub 14183 seqshft 15032 resub 15074 imsub 15082 cjsub 15096 cjreim 15107 absdiflt 15264 absdifle 15265 abs2dif2 15280 subcn2 15539 bpoly2 16001 bpoly3 16002 efsub 16043 efi4p 16080 sinsub 16111 cossub 16112 demoivreALT 16144 dvdssub 16247 modgcd 16474 gzsubcl 16873 psgnunilem2 19363 cnfldsub 20973 itg1sub 25227 plyremlem 25817 sineq0 26033 logneg2 26123 ang180lem2 26315 asinsin 26397 atanneg 26412 atancj 26415 atanlogadd 26419 atanlogsublem 26420 atanlogsub 26421 2efiatan 26423 tanatan 26424 cosatan 26426 atans2 26436 dvatan 26440 zetacvg 26519 wilthlem1 26572 wilthlem2 26573 basellem8 26592 lgsvalmod 26819 cnnvm 29935 cncph 30072 hvsubdistr2 30303 lnfnsubi 31299 subfacval2 34178 itg2addnclem3 36541 lcmineqlem1 40894 2xp3dxp2ge1d 41022 pellexlem6 41572 pell14qrdich 41607 rmxm1 41673 rmym1 41674 addsubeq0 46004 omoeALTV 46353 omeoALTV 46354 emoo 46372 emee 46374 zlmodzxzequap 47180 flsubz 47203 |
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