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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 10620 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) − 𝐶) = ((𝐴 − 𝐶) + 𝐵)) | |
5 | 1, 2, 3, 4 | syl3anc 1494 | 1 ⊢ (𝜑 → ((𝐴 + 𝐵) − 𝐶) = ((𝐴 − 𝐶) + 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1656 ∈ wcel 2164 (class class class)co 6910 ℂcc 10257 + caddc 10262 − cmin 10592 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-br 4876 df-opab 4938 df-mpt 4955 df-id 5252 df-po 5265 df-so 5266 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-er 8014 df-en 8229 df-dom 8230 df-sdom 8231 df-pnf 10400 df-mnf 10401 df-ltxr 10403 df-sub 10594 |
This theorem is referenced by: lesub2 10854 fzoshftral 12887 modadd1 13009 discr 13302 bcp1n 13403 bcpasc 13408 revccat 13889 crre 14238 isercoll2 14783 binomlem 14942 climcndslem1 14962 binomfallfaclem2 15150 pythagtriplem14 15911 vdwlem6 16068 gsumccat 17738 srgbinomlem3 18903 itgcnlem 23962 dvcvx 24189 dvfsumlem1 24195 dvfsumlem2 24196 plymullem1 24376 aaliou3lem2 24504 abelthlem2 24592 tangtx 24664 loglesqrt 24908 dcubic1 24992 quart1lem 25002 quartlem1 25004 basellem3 25229 basellem5 25231 chtub 25357 logfaclbnd 25367 bcp1ctr 25424 lgsquad2lem1 25529 2lgslem3b 25542 selberglem1 25654 selberg3 25668 selbergr 25677 selberg3r 25678 pntlemf 25714 pntlemo 25716 brbtwn2 26211 colinearalglem1 26212 colinearalglem2 26213 crctcsh 27130 clwwlkccatlem 27325 clwwlkel 27392 clwwlkwwlksb 27406 clwwlknonex2lem1 27478 ltesubnnd 30111 ballotlemfp1 31095 subfacp1lem6 31709 fwddifnp1 32806 poimirlem25 33977 poimirlem26 33978 jm2.24nn 38368 jm2.18 38397 jm2.25 38408 dvnmul 40951 fourierdlem4 41120 fourierdlem26 41142 fourierdlem42 41158 vonicclem1 41689 cnambpcma 42196 cnapbmcpd 42197 fmtnorec4 42309 ltsubaddb 43169 ltsubadd2b 43171 |
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