| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > nncand | Structured version Visualization version GIF version | ||
| Description: Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.) |
| Ref | Expression |
|---|---|
| negidd.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
| pncand.2 | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
| Ref | Expression |
|---|---|
| nncand | ⊢ (𝜑 → (𝐴 − (𝐴 − 𝐵)) = 𝐵) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | negidd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
| 2 | pncand.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
| 3 | nncan 11538 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 − (𝐴 − 𝐵)) = 𝐵) | |
| 4 | 1, 2, 3 | syl2anc 584 | 1 ⊢ (𝜑 → (𝐴 − (𝐴 − 𝐵)) = 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 (class class class)co 7431 ℂcc 11153 − cmin 11492 |
| 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 |
| This theorem is referenced by: moddiffl 13922 flmod 13925 ccatswrd 14706 o1dif 15666 fprodser 15985 fprodrev 16013 fallfacval3 16048 efaddlem 16129 4sqlem5 16980 mul4sqlem 16991 4sqlem14 16996 znunit 21582 coe1tmmul2 22279 blssps 24434 blss 24435 metdstri 24873 ivthlem3 25488 ioorcl2 25607 vitalilem2 25644 dvexp3 26016 dvcvx 26059 iblulm 26450 chordthmlem4 26878 heron 26881 cubic 26892 dquartlem1 26894 birthdaylem2 26995 lgamgulmlem2 27073 lgamcvg2 27098 ftalem2 27117 basellem3 27126 gausslemma2dlem1a 27409 lgsquadlem1 27424 addsqrexnreu 27486 pntrlog2bndlem4 27624 axsegconlem1 28932 lt2addrd 32755 ballotlemsf1o 34516 revpfxsfxrev 35121 swrdrevpfx 35122 bcprod 35738 irrdiff 37327 sticksstones12a 42158 sticksstones12 42159 fltnltalem 42672 fltnlta 42673 lzenom 42781 rmspecfund 42920 fzmaxdif 42993 jm2.18 43000 jm2.19 43005 jm2.20nn 43009 supxrgere 45344 lptre2pt 45655 ioodvbdlimc2lem 45949 dvnprodlem1 45961 dvnprodlem2 45962 fourierdlem4 46126 fourierdlem26 46148 fourierdlem42 46164 fourierdlem48 46169 fourierdlem65 46186 fouriersw 46246 sge0gtfsumgt 46458 meaiininclem 46501 m1modne 47350 fmtnorec2lem 47529 goldbachthlem2 47533 pw2m1lepw2m1 48437 eenglngeehlnmlem2 48659 itsclquadb 48697 |
| Copyright terms: Public domain | W3C validator |