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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 11495 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 − (𝐴 − 𝐵)) = (𝐴 + (𝐵 − 𝐴))) | |
2 | 1 | 3anidm12 1418 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 − (𝐴 − 𝐵)) = (𝐴 + (𝐵 − 𝐴))) |
3 | pncan3 11475 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + (𝐵 − 𝐴)) = 𝐵) | |
4 | 2, 3 | eqtrd 2771 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 − (𝐴 − 𝐵)) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2105 (class class class)co 7412 ℂcc 11114 + caddc 11119 − cmin 11451 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-po 5588 df-so 5589 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-er 8709 df-en 8946 df-dom 8947 df-sdom 8948 df-pnf 11257 df-mnf 11258 df-ltxr 11260 df-sub 11453 |
This theorem is referenced by: nnncan1 11503 nncand 11583 elz2 12583 fzrev2 13572 fzrevral 13593 fzrevral2 13594 bccmpl 14276 revrev 14724 fsumrev 15732 geolim2 15824 dvdssub2 16251 efgredleme 19659 psrcom 21840 psropprmul 22080 icccvx 24795 lebnumii 24812 pcorevlem 24873 pcorev2 24875 pi1xfrcnv 24904 efcvx 26301 cosne0 26378 logtayl 26508 logtayl2 26510 logccv 26511 acoscos 26739 sinacos 26751 cvxcl 26830 scvxcvx 26831 basellem5 26930 logfacbnd3 27069 bposlem1 27130 gausslemma2dlem1a 27211 lgsquadlem2 27227 chtppilimlem2 27320 rplogsumlem1 27330 rpvmasumlem 27333 brbtwn2 28596 ax5seglem1 28619 resconn 34701 dvasin 37036 fouriersw 45406 subsubelfzo0 46493 |
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