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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 11520 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) − 𝐶) = ((𝐴 − 𝐶) + 𝐵)) | |
| 5 | 1, 2, 3, 4 | syl3anc 1372 | 1 ⊢ (𝜑 → ((𝐴 + 𝐵) − 𝐶) = ((𝐴 − 𝐶) + 𝐵)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 (class class class)co 7432 ℂcc 11154 + caddc 11159 − cmin 11493 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-po 5591 df-so 5592 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-pnf 11298 df-mnf 11299 df-ltxr 11301 df-sub 11495 | 
| This theorem is referenced by: lesub2 11759 fzoshftral 13824 modadd1 13949 discr 14280 bcp1n 14356 bcpasc 14361 revccat 14805 crre 15154 isercoll2 15706 binomlem 15866 climcndslem1 15886 binomfallfaclem2 16077 pythagtriplem14 16867 vdwlem6 17025 gsumsgrpccat 18854 srgbinomlem3 20226 itgcnlem 25826 dvcvx 26060 dvfsumlem1 26067 dvfsumlem2 26068 dvfsumlem2OLD 26069 plymullem1 26254 aaliou3lem2 26386 abelthlem2 26477 tangtx 26548 loglesqrt 26805 dcubic1 26889 quart1lem 26899 quartlem1 26901 basellem3 27127 basellem5 27129 chtub 27257 logfaclbnd 27267 bcp1ctr 27324 lgsquad2lem1 27429 2lgslem3b 27442 selberglem1 27590 selberg3 27604 selbergr 27613 selberg3r 27614 pntlemf 27650 pntlemo 27652 brbtwn2 28921 colinearalglem1 28922 colinearalglem2 28923 crctcsh 29845 clwwlkccatlem 30009 clwwlkel 30066 clwwlkwwlksb 30074 clwwlknonex2lem1 30127 ltesubnnd 32825 constrrtlc1 33774 constrrtcclem 33776 ballotlemfp1 34495 swrdwlk 35133 subfacp1lem6 35191 fwddifnp1 36167 poimirlem25 37653 poimirlem26 37654 2np3bcnp1 42146 sticksstones12a 42159 jm2.24nn 42976 jm2.18 43005 jm2.25 43016 dvnmul 45963 fourierdlem4 46131 fourierdlem26 46153 fourierdlem42 46169 vonicclem1 46703 cnambpcma 47311 cnapbmcpd 47312 fmtnorec4 47541 ltsubaddb 48436 ltsubadd2b 48438 2itscplem3 48706 | 
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