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Theorem negsubi 8230
Description: Relationship between subtraction and negative. Theorem I.3 of [Apostol] p. 18. (Contributed by NM, 26-Nov-1994.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
Hypotheses
Ref Expression
negidi.1 𝐴 ∈ ℂ
pncan3i.2 𝐵 ∈ ℂ
Assertion
Ref Expression
negsubi (𝐴 + -𝐵) = (𝐴𝐵)

Proof of Theorem negsubi
StepHypRef Expression
1 negidi.1 . 2 𝐴 ∈ ℂ
2 pncan3i.2 . 2 𝐵 ∈ ℂ
3 negsub 8200 . 2 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + -𝐵) = (𝐴𝐵))
41, 2, 3mp2an 426 1 (𝐴 + -𝐵) = (𝐴𝐵)
Colors of variables: wff set class
Syntax hints:   = wceq 1353  wcel 2148  (class class class)co 5871  cc 7805   + caddc 7810  cmin 8123  -cneg 8124
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 4120  ax-pow 4173  ax-pr 4208  ax-setind 4535  ax-resscn 7899  ax-1cn 7900  ax-icn 7902  ax-addcl 7903  ax-addrcl 7904  ax-mulcl 7905  ax-addcom 7907  ax-addass 7909  ax-distr 7911  ax-i2m1 7912  ax-0id 7915  ax-rnegex 7916  ax-cnre 7918
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 4003  df-opab 4064  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-iota 5176  df-fun 5216  df-fv 5222  df-riota 5827  df-ov 5874  df-oprab 5875  df-mpo 5876  df-sub 8125  df-neg 8126
This theorem is referenced by:  negsubdii  8237  negsubdi2i  8238  cosq14gt0  14115  lgsdir2lem1  14291
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