Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 7894 | . . . 4 ⊢ (𝐴 + 𝐵) ∈ ℂ |
4 | 3 | negidi 8158 | . . 3 ⊢ ((𝐴 + 𝐵) + -(𝐴 + 𝐵)) = 0 |
5 | 1 | negidi 8158 | . . . . 5 ⊢ (𝐴 + -𝐴) = 0 |
6 | 2 | negidi 8158 | . . . . 5 ⊢ (𝐵 + -𝐵) = 0 |
7 | 5, 6 | oveq12i 5848 | . . . 4 ⊢ ((𝐴 + -𝐴) + (𝐵 + -𝐵)) = (0 + 0) |
8 | 00id 8030 | . . . 4 ⊢ (0 + 0) = 0 | |
9 | 7, 8 | eqtri 2185 | . . 3 ⊢ ((𝐴 + -𝐴) + (𝐵 + -𝐵)) = 0 |
10 | 1 | negcli 8157 | . . . 4 ⊢ -𝐴 ∈ ℂ |
11 | 2 | negcli 8157 | . . . 4 ⊢ -𝐵 ∈ ℂ |
12 | 1, 10, 2, 11 | add4i 8054 | . . 3 ⊢ ((𝐴 + -𝐴) + (𝐵 + -𝐵)) = ((𝐴 + 𝐵) + (-𝐴 + -𝐵)) |
13 | 4, 9, 12 | 3eqtr2i 2191 | . 2 ⊢ ((𝐴 + 𝐵) + -(𝐴 + 𝐵)) = ((𝐴 + 𝐵) + (-𝐴 + -𝐵)) |
14 | 3 | negcli 8157 | . . 3 ⊢ -(𝐴 + 𝐵) ∈ ℂ |
15 | 10, 11 | addcli 7894 | . . 3 ⊢ (-𝐴 + -𝐵) ∈ ℂ |
16 | 3, 14, 15 | addcani 8071 | . 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 1342 ∈ wcel 2135 (class class class)co 5836 ℂcc 7742 0cc0 7744 + caddc 7747 -cneg 8061 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-setind 4508 ax-resscn 7836 ax-1cn 7837 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-addcom 7844 ax-addass 7846 ax-distr 7848 ax-i2m1 7849 ax-0id 7852 ax-rnegex 7853 ax-cnre 7855 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-iota 5147 df-fun 5184 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-sub 8062 df-neg 8063 |
This theorem is referenced by: negsubdii 8174 |
Copyright terms: Public domain | W3C validator |