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Theorem negdii 8218
Description: Distribution of negative over addition. (Contributed by NM, 28-Jul-1999.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
Hypotheses
Ref Expression
negidi.1 𝐴 ∈ ℂ
pncan3i.2 𝐵 ∈ ℂ
Assertion
Ref Expression
negdii -(𝐴 + 𝐵) = (-𝐴 + -𝐵)

Proof of Theorem negdii
StepHypRef Expression
1 negidi.1 . . . . 5 𝐴 ∈ ℂ
2 pncan3i.2 . . . . 5 𝐵 ∈ ℂ
31, 2addcli 7939 . . . 4 (𝐴 + 𝐵) ∈ ℂ
43negidi 8203 . . 3 ((𝐴 + 𝐵) + -(𝐴 + 𝐵)) = 0
51negidi 8203 . . . . 5 (𝐴 + -𝐴) = 0
62negidi 8203 . . . . 5 (𝐵 + -𝐵) = 0
75, 6oveq12i 5880 . . . 4 ((𝐴 + -𝐴) + (𝐵 + -𝐵)) = (0 + 0)
8 00id 8075 . . . 4 (0 + 0) = 0
97, 8eqtri 2198 . . 3 ((𝐴 + -𝐴) + (𝐵 + -𝐵)) = 0
101negcli 8202 . . . 4 -𝐴 ∈ ℂ
112negcli 8202 . . . 4 -𝐵 ∈ ℂ
121, 10, 2, 11add4i 8099 . . 3 ((𝐴 + -𝐴) + (𝐵 + -𝐵)) = ((𝐴 + 𝐵) + (-𝐴 + -𝐵))
134, 9, 123eqtr2i 2204 . 2 ((𝐴 + 𝐵) + -(𝐴 + 𝐵)) = ((𝐴 + 𝐵) + (-𝐴 + -𝐵))
143negcli 8202 . . 3 -(𝐴 + 𝐵) ∈ ℂ
1510, 11addcli 7939 . . 3 (-𝐴 + -𝐵) ∈ ℂ
163, 14, 15addcani 8116 . 2 (((𝐴 + 𝐵) + -(𝐴 + 𝐵)) = ((𝐴 + 𝐵) + (-𝐴 + -𝐵)) ↔ -(𝐴 + 𝐵) = (-𝐴 + -𝐵))
1713, 16mpbi 145 1 -(𝐴 + 𝐵) = (-𝐴 + -𝐵)
Colors of variables: wff set class
Syntax hints:   = wceq 1353  wcel 2148  (class class class)co 5868  cc 7787  0cc0 7789   + caddc 7792  -cneg 8106
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 4118  ax-pow 4171  ax-pr 4205  ax-setind 4532  ax-resscn 7881  ax-1cn 7882  ax-icn 7884  ax-addcl 7885  ax-addrcl 7886  ax-mulcl 7887  ax-addcom 7889  ax-addass 7891  ax-distr 7893  ax-i2m1 7894  ax-0id 7897  ax-rnegex 7898  ax-cnre 7900
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 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-iota 5173  df-fun 5213  df-fv 5219  df-riota 5824  df-ov 5871  df-oprab 5872  df-mpo 5873  df-sub 8107  df-neg 8108
This theorem is referenced by:  negsubdii  8219
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