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Mirrors > Home > MPE Home > Th. List > negsubdi | Structured version Visualization version GIF version |
Description: Distribution of negative over subtraction. (Contributed by NM, 15-Nov-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
negsubdi | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → -(𝐴 − 𝐵) = (-𝐴 + 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0cn 10368 | . . 3 ⊢ 0 ∈ ℂ | |
2 | subsub 10653 | . . 3 ⊢ ((0 ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (0 − (𝐴 − 𝐵)) = ((0 − 𝐴) + 𝐵)) | |
3 | 1, 2 | mp3an1 1521 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (0 − (𝐴 − 𝐵)) = ((0 − 𝐴) + 𝐵)) |
4 | df-neg 10609 | . 2 ⊢ -(𝐴 − 𝐵) = (0 − (𝐴 − 𝐵)) | |
5 | df-neg 10609 | . . 3 ⊢ -𝐴 = (0 − 𝐴) | |
6 | 5 | oveq1i 6932 | . 2 ⊢ (-𝐴 + 𝐵) = ((0 − 𝐴) + 𝐵) |
7 | 3, 4, 6 | 3eqtr4g 2839 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → -(𝐴 − 𝐵) = (-𝐴 + 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2107 (class class class)co 6922 ℂcc 10270 0cc0 10272 + caddc 10275 − cmin 10606 -cneg 10607 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4672 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-po 5274 df-so 5275 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-pnf 10413 df-mnf 10414 df-ltxr 10416 df-sub 10608 df-neg 10609 |
This theorem is referenced by: negdi 10680 negsubdi2 10682 neg2sub 10683 negsubdid 10749 rebtwnz 12094 odd2np1 15469 sin2pim 24675 cos2pim 24676 dya2ub 30930 jm2.24 38493 sigarms 41976 onego 42588 |
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