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Mirrors > Home > MPE Home > Th. List > negdi | Structured version Visualization version GIF version |
Description: Distribution of negative over addition. (Contributed by NM, 10-May-2004.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
negdi | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → -(𝐴 + 𝐵) = (-𝐴 + -𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subneg 10789 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 − -𝐵) = (𝐴 + 𝐵)) | |
2 | 1 | negeqd 10733 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → -(𝐴 − -𝐵) = -(𝐴 + 𝐵)) |
3 | negcl 10739 | . . 3 ⊢ (𝐵 ∈ ℂ → -𝐵 ∈ ℂ) | |
4 | negsubdi 10796 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ -𝐵 ∈ ℂ) → -(𝐴 − -𝐵) = (-𝐴 + -𝐵)) | |
5 | 3, 4 | sylan2 592 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → -(𝐴 − -𝐵) = (-𝐴 + -𝐵)) |
6 | 2, 5 | eqtr3d 2835 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → -(𝐴 + 𝐵) = (-𝐴 + -𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1525 ∈ wcel 2083 (class class class)co 7023 ℂcc 10388 + caddc 10393 − cmin 10723 -cneg 10724 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326 ax-resscn 10447 ax-1cn 10448 ax-icn 10449 ax-addcl 10450 ax-addrcl 10451 ax-mulcl 10452 ax-mulrcl 10453 ax-mulcom 10454 ax-addass 10455 ax-mulass 10456 ax-distr 10457 ax-i2m1 10458 ax-1ne0 10459 ax-1rid 10460 ax-rnegex 10461 ax-rrecex 10462 ax-cnre 10463 ax-pre-lttri 10464 ax-pre-lttrn 10465 ax-pre-ltadd 10466 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-nel 3093 df-ral 3112 df-rex 3113 df-reu 3114 df-rab 3116 df-v 3442 df-sbc 3712 df-csb 3818 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-nul 4218 df-if 4388 df-pw 4461 df-sn 4479 df-pr 4481 df-op 4485 df-uni 4752 df-br 4969 df-opab 5031 df-mpt 5048 df-id 5355 df-po 5369 df-so 5370 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-iota 6196 df-fun 6234 df-fn 6235 df-f 6236 df-f1 6237 df-fo 6238 df-f1o 6239 df-fv 6240 df-riota 6984 df-ov 7026 df-oprab 7027 df-mpo 7028 df-er 8146 df-en 8365 df-dom 8366 df-sdom 8367 df-pnf 10530 df-mnf 10531 df-ltxr 10533 df-sub 10725 df-neg 10726 |
This theorem is referenced by: negdi2 10798 negdii 10824 negdid 10864 mulsub 10937 zeo 11922 xnegdi 12495 ceim1l 13069 binomrisefac 15233 mulgneg2 18019 archirngz 30452 |
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