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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 10886 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) − 𝐶) = ((𝐴 − 𝐶) + 𝐵)) | |
5 | 1, 2, 3, 4 | syl3anc 1368 | 1 ⊢ (𝜑 → ((𝐴 + 𝐵) − 𝐶) = ((𝐴 − 𝐶) + 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 (class class class)co 7135 ℂcc 10524 + caddc 10529 − cmin 10859 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-ltxr 10669 df-sub 10861 |
This theorem is referenced by: lesub2 11124 fzoshftral 13149 modadd1 13271 discr 13597 bcp1n 13672 bcpasc 13677 revccat 14119 crre 14465 isercoll2 15017 binomlem 15176 climcndslem1 15196 binomfallfaclem2 15386 pythagtriplem14 16155 vdwlem6 16312 gsumsgrpccat 17996 gsumccatOLD 17997 srgbinomlem3 19285 itgcnlem 24393 dvcvx 24623 dvfsumlem1 24629 dvfsumlem2 24630 plymullem1 24811 aaliou3lem2 24939 abelthlem2 25027 tangtx 25098 loglesqrt 25347 dcubic1 25431 quart1lem 25441 quartlem1 25443 basellem3 25668 basellem5 25670 chtub 25796 logfaclbnd 25806 bcp1ctr 25863 lgsquad2lem1 25968 2lgslem3b 25981 selberglem1 26129 selberg3 26143 selbergr 26152 selberg3r 26153 pntlemf 26189 pntlemo 26191 brbtwn2 26699 colinearalglem1 26700 colinearalglem2 26701 crctcsh 27610 clwwlkccatlem 27774 clwwlkel 27831 clwwlkwwlksb 27839 clwwlknonex2lem1 27892 ltesubnnd 30564 ballotlemfp1 31859 swrdwlk 32486 subfacp1lem6 32545 fwddifnp1 33739 poimirlem25 35082 poimirlem26 35083 2np3bcnp1 39348 jm2.24nn 39900 jm2.18 39929 jm2.25 39940 dvnmul 42585 fourierdlem4 42753 fourierdlem26 42775 fourierdlem42 42791 vonicclem1 43322 cnambpcma 43851 cnapbmcpd 43852 fmtnorec4 44066 ltsubaddb 44923 ltsubadd2b 44925 2itscplem3 45194 |
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