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Mirrors > Home > MPE Home > Th. List > addsubd | Structured version Visualization version GIF version |
Description: Law for subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
negidd.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
pncand.2 | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
subaddd.3 | ⊢ (𝜑 → 𝐶 ∈ ℂ) |
Ref | Expression |
---|---|
addsubd | ⊢ (𝜑 → ((𝐴 + 𝐵) − 𝐶) = ((𝐴 − 𝐶) + 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | negidd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
2 | pncand.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
3 | subaddd.3 | . 2 ⊢ (𝜑 → 𝐶 ∈ ℂ) | |
4 | addsub 10584 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) − 𝐶) = ((𝐴 − 𝐶) + 𝐵)) | |
5 | 1, 2, 3, 4 | syl3anc 1491 | 1 ⊢ (𝜑 → ((𝐴 + 𝐵) − 𝐶) = ((𝐴 − 𝐶) + 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1653 ∈ wcel 2157 (class class class)co 6878 ℂcc 10222 + caddc 10227 − cmin 10556 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-op 4375 df-uni 4629 df-br 4844 df-opab 4906 df-mpt 4923 df-id 5220 df-po 5233 df-so 5234 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-er 7982 df-en 8196 df-dom 8197 df-sdom 8198 df-pnf 10365 df-mnf 10366 df-ltxr 10368 df-sub 10558 |
This theorem is referenced by: lesub2 10815 fzoshftral 12840 modadd1 12962 discr 13255 bcp1n 13356 bcpasc 13361 revccat 13846 crre 14195 isercoll2 14740 binomlem 14899 climcndslem1 14919 binomfallfaclem2 15107 pythagtriplem14 15866 vdwlem6 16023 gsumccat 17693 srgbinomlem3 18858 itgcnlem 23897 dvcvx 24124 dvfsumlem1 24130 dvfsumlem2 24131 plymullem1 24311 aaliou3lem2 24439 abelthlem2 24527 tangtx 24599 loglesqrt 24843 dcubic1 24924 quart1lem 24934 quartlem1 24936 basellem3 25161 basellem5 25163 chtub 25289 logfaclbnd 25299 bcp1ctr 25356 lgsquad2lem1 25461 2lgslem3b 25474 selberglem1 25586 selberg3 25600 selbergr 25609 selberg3r 25610 pntlemf 25646 pntlemo 25648 brbtwn2 26142 colinearalglem1 26143 colinearalglem2 26144 crctcsh 27075 clwwlkccatlem 27282 clwwlkel 27355 clwwlkwwlksb 27370 clwwlknonex2lem1 27447 ltesubnnd 30086 ballotlemfp1 31070 subfacp1lem6 31684 fwddifnp1 32785 poimirlem25 33923 poimirlem26 33924 jm2.24nn 38311 jm2.18 38340 jm2.25 38351 dvnmul 40902 fourierdlem4 41071 fourierdlem26 41093 fourierdlem42 41109 vonicclem1 41643 cnambpcma 42150 cnapbmcpd 42151 fmtnorec4 42243 ltsubaddb 43103 ltsubadd2b 43105 |
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