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Mirrors > Home > ILE Home > Th. List > negdii | GIF version |
Description: Distribution of negative over addition. (Contributed by NM, 28-Jul-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) |
Ref | Expression |
---|---|
negidi.1 | ⊢ 𝐴 ∈ ℂ |
pncan3i.2 | ⊢ 𝐵 ∈ ℂ |
Ref | Expression |
---|---|
negdii | ⊢ -(𝐴 + 𝐵) = (-𝐴 + -𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | negidi.1 | . . . . 5 ⊢ 𝐴 ∈ ℂ | |
2 | pncan3i.2 | . . . . 5 ⊢ 𝐵 ∈ ℂ | |
3 | 1, 2 | addcli 7794 | . . . 4 ⊢ (𝐴 + 𝐵) ∈ ℂ |
4 | 3 | negidi 8055 | . . 3 ⊢ ((𝐴 + 𝐵) + -(𝐴 + 𝐵)) = 0 |
5 | 1 | negidi 8055 | . . . . 5 ⊢ (𝐴 + -𝐴) = 0 |
6 | 2 | negidi 8055 | . . . . 5 ⊢ (𝐵 + -𝐵) = 0 |
7 | 5, 6 | oveq12i 5794 | . . . 4 ⊢ ((𝐴 + -𝐴) + (𝐵 + -𝐵)) = (0 + 0) |
8 | 00id 7927 | . . . 4 ⊢ (0 + 0) = 0 | |
9 | 7, 8 | eqtri 2161 | . . 3 ⊢ ((𝐴 + -𝐴) + (𝐵 + -𝐵)) = 0 |
10 | 1 | negcli 8054 | . . . 4 ⊢ -𝐴 ∈ ℂ |
11 | 2 | negcli 8054 | . . . 4 ⊢ -𝐵 ∈ ℂ |
12 | 1, 10, 2, 11 | add4i 7951 | . . 3 ⊢ ((𝐴 + -𝐴) + (𝐵 + -𝐵)) = ((𝐴 + 𝐵) + (-𝐴 + -𝐵)) |
13 | 4, 9, 12 | 3eqtr2i 2167 | . 2 ⊢ ((𝐴 + 𝐵) + -(𝐴 + 𝐵)) = ((𝐴 + 𝐵) + (-𝐴 + -𝐵)) |
14 | 3 | negcli 8054 | . . 3 ⊢ -(𝐴 + 𝐵) ∈ ℂ |
15 | 10, 11 | addcli 7794 | . . 3 ⊢ (-𝐴 + -𝐵) ∈ ℂ |
16 | 3, 14, 15 | addcani 7968 | . 2 ⊢ (((𝐴 + 𝐵) + -(𝐴 + 𝐵)) = ((𝐴 + 𝐵) + (-𝐴 + -𝐵)) ↔ -(𝐴 + 𝐵) = (-𝐴 + -𝐵)) |
17 | 13, 16 | mpbi 144 | 1 ⊢ -(𝐴 + 𝐵) = (-𝐴 + -𝐵) |
Colors of variables: wff set class |
Syntax hints: = wceq 1332 ∈ wcel 1481 (class class class)co 5782 ℂcc 7642 0cc0 7644 + caddc 7647 -cneg 7958 |
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 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-setind 4460 ax-resscn 7736 ax-1cn 7737 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-addcom 7744 ax-addass 7746 ax-distr 7748 ax-i2m1 7749 ax-0id 7752 ax-rnegex 7753 ax-cnre 7755 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-iota 5096 df-fun 5133 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-sub 7959 df-neg 7960 |
This theorem is referenced by: negsubdii 8071 |
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