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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 11556 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 − -𝐵) = (𝐴 + 𝐵)) | |
4 | 1, 2, 3 | syl2anc 584 | 1 ⊢ (𝜑 → (𝐴 − -𝐵) = (𝐴 + 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 (class class class)co 7431 ℂcc 11151 + caddc 11156 − cmin 11490 -cneg 11491 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-ltxr 11298 df-sub 11492 df-neg 11493 |
This theorem is referenced by: possumd 11886 dfceil2 13876 addmodlteq 13984 ipcnval 15179 fallfacfwd 16069 cossub 16202 znunit 21600 cphsqrtcl2 25234 ulmshft 26448 ptolemy 26553 efeq1 26585 quad2 26897 dcubic2 26902 dcubic 26904 mcubic 26905 dquartlem1 26909 quart 26919 asinlem 26926 asinlem2 26927 sinasin 26947 asinsin 26950 atandmtan 26978 atantan 26981 lgamgulmlem2 27088 lgambdd 27095 lgamucov 27096 lgseisenlem2 27435 rpvmasum2 27571 chpdifbndlem1 27612 pntrsumo1 27624 pntrlog2bndlem4 27639 nvabs 30701 breprexplemc 34626 logdivsqrle 34644 irrdiff 37309 poimirlem29 37636 areacirc 37700 posbezout 42082 acongrep 42969 acongeq 42972 jm2.25 42988 jm2.26lem3 42990 sqrtcvallem4 43629 sqrtcval 43631 radcnvrat 44310 dvradcnv2 44343 binomcxplemnotnn0 44352 fperiodmul 45255 itgsincmulx 45930 fourierdlem103 46165 fourierdlem109 46171 fourierdlem111 46173 sqwvfoura 46184 etransclem46 46236 hoicvrrex 46512 sigarms 46812 fmtnorec3 47473 2pwp1prm 47514 eenglngeehlnmlem1 48587 itsclc0yqsol 48614 |
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