![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > nncan | Structured version Visualization version GIF version |
Description: Cancellation law for subtraction. (Contributed by NM, 21-Jun-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) |
Ref | Expression |
---|---|
nncan | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 − (𝐴 − 𝐵)) = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subsub2 10713 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 − (𝐴 − 𝐵)) = (𝐴 + (𝐵 − 𝐴))) | |
2 | 1 | 3anidm12 1400 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 − (𝐴 − 𝐵)) = (𝐴 + (𝐵 − 𝐴))) |
3 | pncan3 10692 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + (𝐵 − 𝐴)) = 𝐵) | |
4 | 2, 3 | eqtrd 2807 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 − (𝐴 − 𝐵)) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1508 ∈ wcel 2051 (class class class)co 6974 ℂcc 10331 + caddc 10336 − cmin 10668 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 ax-resscn 10390 ax-1cn 10391 ax-icn 10392 ax-addcl 10393 ax-addrcl 10394 ax-mulcl 10395 ax-mulrcl 10396 ax-mulcom 10397 ax-addass 10398 ax-mulass 10399 ax-distr 10400 ax-i2m1 10401 ax-1ne0 10402 ax-1rid 10403 ax-rnegex 10404 ax-rrecex 10405 ax-cnre 10406 ax-pre-lttri 10407 ax-pre-lttrn 10408 ax-pre-ltadd 10409 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-nel 3067 df-ral 3086 df-rex 3087 df-reu 3088 df-rab 3090 df-v 3410 df-sbc 3675 df-csb 3780 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4709 df-br 4926 df-opab 4988 df-mpt 5005 df-id 5308 df-po 5322 df-so 5323 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-riota 6935 df-ov 6977 df-oprab 6978 df-mpo 6979 df-er 8087 df-en 8305 df-dom 8306 df-sdom 8307 df-pnf 10474 df-mnf 10475 df-ltxr 10477 df-sub 10670 |
This theorem is referenced by: nnncan1 10721 nncand 10801 elz2 11809 fzrev2 12785 fzrevral 12806 fzrevral2 12807 bccmpl 13482 revrev 13984 fsumrev 14992 geolim2 15085 dvdssub2 15509 efgredleme 18640 psrcom 19915 psropprmul 20124 icccvx 23272 lebnumii 23288 pcorevlem 23348 pcorev2 23350 pi1xfrcnv 23379 efcvx 24755 cosne0 24830 logtayl 24959 logtayl2 24961 logccv 24962 acoscos 25187 sinacos 25199 cvxcl 25279 scvxcvx 25280 basellem5 25379 logfacbnd3 25516 bposlem1 25577 gausslemma2dlem1a 25658 lgsquadlem2 25674 chtppilimlem2 25767 rplogsumlem1 25777 rpvmasumlem 25780 brbtwn2 26409 ax5seglem1 26432 resconn 32115 dvasin 34456 fouriersw 41979 subsubelfzo0 42964 |
Copyright terms: Public domain | W3C validator |