![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > negsubdi | 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 7948 | . . 3 ⊢ 0 ∈ ℂ | |
2 | subsub 8185 | . . 3 ⊢ ((0 ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (0 − (𝐴 − 𝐵)) = ((0 − 𝐴) + 𝐵)) | |
3 | 1, 2 | mp3an1 1324 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (0 − (𝐴 − 𝐵)) = ((0 − 𝐴) + 𝐵)) |
4 | df-neg 8129 | . 2 ⊢ -(𝐴 − 𝐵) = (0 − (𝐴 − 𝐵)) | |
5 | df-neg 8129 | . . 3 ⊢ -𝐴 = (0 − 𝐴) | |
6 | 5 | oveq1i 5884 | . 2 ⊢ (-𝐴 + 𝐵) = ((0 − 𝐴) + 𝐵) |
7 | 3, 4, 6 | 3eqtr4g 2235 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → -(𝐴 − 𝐵) = (-𝐴 + 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1353 ∈ wcel 2148 (class class class)co 5874 ℂcc 7808 0cc0 7810 + caddc 7813 − cmin 8126 -cneg 8127 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-setind 4536 ax-resscn 7902 ax-1cn 7903 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-addcom 7910 ax-addass 7912 ax-distr 7914 ax-i2m1 7915 ax-0id 7918 ax-rnegex 7919 ax-cnre 7921 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4004 df-opab 4065 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-iota 5178 df-fun 5218 df-fv 5224 df-riota 5830 df-ov 5877 df-oprab 5878 df-mpo 5879 df-sub 8128 df-neg 8129 |
This theorem is referenced by: negdi 8212 negsubdi2 8214 neg2sub 8215 negsubdid 8281 odd2np1 11872 sin2pim 14165 cos2pim 14166 |
Copyright terms: Public domain | W3C validator |