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 10148 | . . . 4 ⊢ -𝐵 = (0 − 𝐵) | |
2 | 1 | oveq2i 6560 | . . 3 ⊢ (𝐴 + -𝐵) = (𝐴 + (0 − 𝐵)) |
3 | 2 | a1i 11 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + -𝐵) = (𝐴 + (0 − 𝐵))) |
4 | 0cn 9911 | . . 3 ⊢ 0 ∈ ℂ | |
5 | addsubass 10170 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 0 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 0) − 𝐵) = (𝐴 + (0 − 𝐵))) | |
6 | 4, 5 | mp3an2 1404 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 0) − 𝐵) = (𝐴 + (0 − 𝐵))) |
7 | simpl 472 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 𝐴 ∈ ℂ) | |
8 | 7 | addid1d 10115 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 0) = 𝐴) |
9 | 8 | oveq1d 6564 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 0) − 𝐵) = (𝐴 − 𝐵)) |
10 | 3, 6, 9 | 3eqtr2d 2650 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + -𝐵) = (𝐴 − 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 (class class class)co 6549 ℂcc 9813 0cc0 9815 + caddc 9818 − cmin 10145 -cneg 10146 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-ltxr 9958 df-sub 10147 df-neg 10148 |
This theorem is referenced by: negdi2 10218 negsubdi2 10219 resubcli 10222 resubcl 10224 negsubi 10238 negsubd 10277 submul2 10349 addneg1mul 10351 mulsub 10352 divsubdir 10600 difgtsumgt 11223 elz2 11271 zsubcl 11296 qsubcl 11683 rexsub 11938 fzsubel 12248 ceim1l 12508 modcyc2 12568 negmod 12577 modsumfzodifsn 12605 expsub 12770 binom2sub 12843 seqshft 13673 resub 13715 imsub 13723 cjsub 13737 cjreim 13748 absdiflt 13905 absdifle 13906 abs2dif2 13921 subcn2 14173 bpoly2 14627 bpoly3 14628 efsub 14669 efi4p 14706 sinsub 14737 cossub 14738 demoivreALT 14770 dvdssub 14864 modgcd 15091 gzsubcl 15482 psgnunilem2 17738 cnfldsub 19593 itg1sub 23282 plyremlem 23863 sineq0 24077 logneg2 24165 ang180lem2 24340 asinsin 24419 atanneg 24434 atancj 24437 atanlogadd 24441 atanlogsublem 24442 atanlogsub 24443 2efiatan 24445 tanatan 24446 cosatan 24448 atans2 24458 dvatan 24462 zetacvg 24541 wilthlem1 24594 wilthlem2 24595 basellem8 24614 lgsvalmod 24841 cnnvm 26921 cncph 27058 hvsubdistr2 27291 lnfnsubi 28289 subfacval2 30423 itg2addnclem3 32633 pellexlem6 36416 pell14qrdich 36451 rmxm1 36517 rmym1 36518 omoeALTV 40134 omeoALTV 40135 emoo 40151 emee 40153 zlmodzxzequap 42082 flsubz 42106 |
Copyright terms: Public domain | W3C validator |