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Mirrors > Home > ILE Home > Th. List > negsubdi2 | GIF version |
Description: Distribution of negative over subtraction. (Contributed by NM, 4-Oct-1999.) |
Ref | Expression |
---|---|
negsubdi2 | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → -(𝐴 − 𝐵) = (𝐵 − 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | negsubdi 7986 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → -(𝐴 − 𝐵) = (-𝐴 + 𝐵)) | |
2 | negcl 7930 | . . 3 ⊢ (𝐴 ∈ ℂ → -𝐴 ∈ ℂ) | |
3 | addcom 7867 | . . 3 ⊢ ((-𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (-𝐴 + 𝐵) = (𝐵 + -𝐴)) | |
4 | 2, 3 | sylan 281 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (-𝐴 + 𝐵) = (𝐵 + -𝐴)) |
5 | negsub 7978 | . . 3 ⊢ ((𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (𝐵 + -𝐴) = (𝐵 − 𝐴)) | |
6 | 5 | ancoms 266 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐵 + -𝐴) = (𝐵 − 𝐴)) |
7 | 1, 4, 6 | 3eqtrd 2154 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → -(𝐴 − 𝐵) = (𝐵 − 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1316 ∈ wcel 1465 (class class class)co 5742 ℂcc 7586 + caddc 7591 − cmin 7901 -cneg 7902 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-setind 4422 ax-resscn 7680 ax-1cn 7681 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-addcom 7688 ax-addass 7690 ax-distr 7692 ax-i2m1 7693 ax-0id 7696 ax-rnegex 7697 ax-cnre 7699 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-iota 5058 df-fun 5095 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-sub 7903 df-neg 7904 |
This theorem is referenced by: neg2sub 7990 negsubdi2d 8057 subeqrev 8106 mulsub2 8132 div2subap 8564 elz2 9090 fzshftral 9856 sqsubswap 10321 abssub 10841 abs2difabs 10848 sin2pim 12821 cos2pim 12822 ptolemy 12832 |
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