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Mirrors > Home > MPE Home > Th. List > negsubdi2i | Structured version Visualization version GIF version |
Description: Distribution of negative over subtraction. (Contributed by NM, 1-Oct-1999.) |
Ref | Expression |
---|---|
negidi.1 | ⊢ 𝐴 ∈ ℂ |
pncan3i.2 | ⊢ 𝐵 ∈ ℂ |
Ref | Expression |
---|---|
negsubdi2i | ⊢ -(𝐴 − 𝐵) = (𝐵 − 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | negidi.1 | . . 3 ⊢ 𝐴 ∈ ℂ | |
2 | pncan3i.2 | . . 3 ⊢ 𝐵 ∈ ℂ | |
3 | 1, 2 | negsubdii 11236 | . 2 ⊢ -(𝐴 − 𝐵) = (-𝐴 + 𝐵) |
4 | 1 | negcli 11219 | . . 3 ⊢ -𝐴 ∈ ℂ |
5 | 2, 1 | negsubi 11229 | . . 3 ⊢ (𝐵 + -𝐴) = (𝐵 − 𝐴) |
6 | 2, 4, 5 | addcomli 11097 | . 2 ⊢ (-𝐴 + 𝐵) = (𝐵 − 𝐴) |
7 | 3, 6 | eqtri 2766 | 1 ⊢ -(𝐴 − 𝐵) = (𝐵 − 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1539 ∈ wcel 2108 (class class class)co 7255 ℂcc 10800 + caddc 10805 − cmin 11135 -cneg 11136 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4837 df-br 5071 df-opab 5133 df-mpt 5154 df-id 5480 df-po 5494 df-so 5495 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-er 8456 df-en 8692 df-dom 8693 df-sdom 8694 df-pnf 10942 df-mnf 10943 df-ltxr 10945 df-sub 11137 df-neg 11138 |
This theorem is referenced by: zeo 12336 geo2sum2 15514 bpoly2 15695 bpoly3 15696 cos2bnd 15825 3dvds 15968 ppiub 26257 ax5seglem7 27206 ipasslem10 29102 norm3adifii 29411 lnophmlem2 30280 ballotlem2 32355 lhe4.4ex1a 41836 stoweidlem26 43457 dirkertrigeqlem3 43531 |
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