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Mirrors > Home > MPE Home > Th. List > subnegd | Structured version Visualization version GIF version |
Description: Relationship between subtraction and negative. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
negidd.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
pncand.2 | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
Ref | Expression |
---|---|
subnegd | ⊢ (𝜑 → (𝐴 − -𝐵) = (𝐴 + 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | negidd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
2 | pncand.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
3 | subneg 11546 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 − -𝐵) = (𝐴 + 𝐵)) | |
4 | 1, 2, 3 | syl2anc 582 | 1 ⊢ (𝜑 → (𝐴 − -𝐵) = (𝐴 + 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 (class class class)co 7419 ℂcc 11143 + caddc 11148 − cmin 11481 -cneg 11482 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-resscn 11202 ax-1cn 11203 ax-icn 11204 ax-addcl 11205 ax-addrcl 11206 ax-mulcl 11207 ax-mulrcl 11208 ax-mulcom 11209 ax-addass 11210 ax-mulass 11211 ax-distr 11212 ax-i2m1 11213 ax-1ne0 11214 ax-1rid 11215 ax-rnegex 11216 ax-rrecex 11217 ax-cnre 11218 ax-pre-lttri 11219 ax-pre-lttrn 11220 ax-pre-ltadd 11221 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-po 5590 df-so 5591 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11287 df-mnf 11288 df-ltxr 11290 df-sub 11483 df-neg 11484 |
This theorem is referenced by: possumd 11876 dfceil2 13845 addmodlteq 13952 ipcnval 15131 fallfacfwd 16021 cossub 16154 znunit 21519 cphsqrtcl2 25163 ulmshft 26376 ptolemy 26481 efeq1 26512 quad2 26821 dcubic2 26826 dcubic 26828 mcubic 26829 dquartlem1 26833 quart 26843 asinlem 26850 asinlem2 26851 sinasin 26871 asinsin 26874 atandmtan 26902 atantan 26905 lgamgulmlem2 27012 lgambdd 27019 lgamucov 27020 lgseisenlem2 27359 rpvmasum2 27495 chpdifbndlem1 27536 pntrsumo1 27548 pntrlog2bndlem4 27563 nvabs 30559 breprexplemc 34397 logdivsqrle 34415 irrdiff 36938 poimirlem29 37255 areacirc 37319 posbezout 41705 acongrep 42545 acongeq 42548 jm2.25 42564 jm2.26lem3 42566 sqrtcvallem4 43213 sqrtcval 43215 radcnvrat 43895 dvradcnv2 43928 binomcxplemnotnn0 43937 fperiodmul 44826 itgsincmulx 45502 fourierdlem103 45737 fourierdlem109 45743 fourierdlem111 45745 sqwvfoura 45756 etransclem46 45808 hoicvrrex 46084 sigarms 46384 fmtnorec3 47027 2pwp1prm 47068 eenglngeehlnmlem1 47998 itsclc0yqsol 48025 |
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