| 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 11495 | . . . 4 ⊢ -𝐵 = (0 − 𝐵) | |
| 2 | 1 | oveq2i 7442 | . . 3 ⊢ (𝐴 + -𝐵) = (𝐴 + (0 − 𝐵)) |
| 3 | 2 | a1i 11 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + -𝐵) = (𝐴 + (0 − 𝐵))) |
| 4 | 0cn 11253 | . . 3 ⊢ 0 ∈ ℂ | |
| 5 | addsubass 11518 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 0 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 0) − 𝐵) = (𝐴 + (0 − 𝐵))) | |
| 6 | 4, 5 | mp3an2 1451 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 0) − 𝐵) = (𝐴 + (0 − 𝐵))) |
| 7 | simpl 482 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 𝐴 ∈ ℂ) | |
| 8 | 7 | addridd 11461 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 0) = 𝐴) |
| 9 | 8 | oveq1d 7446 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 0) − 𝐵) = (𝐴 − 𝐵)) |
| 10 | 3, 6, 9 | 3eqtr2d 2783 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + -𝐵) = (𝐴 − 𝐵)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 (class class class)co 7431 ℂcc 11153 0cc0 11155 + caddc 11158 − cmin 11492 -cneg 11493 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-ltxr 11300 df-sub 11494 df-neg 11495 |
| This theorem is referenced by: negdi2 11567 negsubdi2 11568 resubcli 11571 resubcl 11573 negsubi 11587 negsubd 11626 submul2 11703 addneg1mul 11705 mulsub 11706 divsubdir 11961 difgtsumgt 12579 elz2 12631 zsubcl 12659 qsubcl 13010 rexsub 13275 fzsubel 13600 ceim1l 13887 modcyc2 13947 negmod 13957 modsumfzodifsn 13985 expsub 14151 binom2sub 14259 seqshft 15124 resub 15166 imsub 15174 cjsub 15188 cjreim 15199 absdiflt 15356 absdifle 15357 abs2dif2 15372 subcn2 15631 bpoly2 16093 bpoly3 16094 efsub 16136 efi4p 16173 sinsub 16204 cossub 16205 demoivreALT 16237 dvdssub 16341 modgcd 16569 gzsubcl 16978 psgnunilem2 19513 cnfldsub 21410 itg1sub 25744 plyremlem 26346 sineq0 26566 logneg2 26657 ang180lem2 26853 asinsin 26935 atanneg 26950 atancj 26953 atanlogadd 26957 atanlogsublem 26958 atanlogsub 26959 2efiatan 26961 tanatan 26962 cosatan 26964 atans2 26974 dvatan 26978 zetacvg 27058 wilthlem1 27111 wilthlem2 27112 basellem8 27131 lgsvalmod 27360 cnnvm 30701 cncph 30838 hvsubdistr2 31069 lnfnsubi 32065 subfacval2 35192 itg2addnclem3 37680 lcmineqlem1 42030 2xp3dxp2ge1d 42242 pellexlem6 42845 pell14qrdich 42880 rmxm1 42946 rmym1 42947 addsubeq0 47308 omoeALTV 47672 omeoALTV 47673 emoo 47691 emee 47693 zlmodzxzequap 48416 flsubz 48439 |
| Copyright terms: Public domain | W3C validator |