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 Description: Addition of complex numbers in terms of signed reals. (Contributed by NM, 28-May-1995.) (New usage is discouraged.)
Assertion
Ref Expression

Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opex 4253 . 2
2 oveq1 5881 . . . 4
3 oveq1 5881 . . . 4
4 opeq12 3814 . . . 4
52, 3, 4syl2an 463 . . 3
6 oveq2 5882 . . . 4
7 oveq2 5882 . . . 4
8 opeq12 3814 . . . 4
96, 7, 8syl2an 463 . . 3
105, 9sylan9eq 2348 . 2
11 df-add 8764 . . 3
12 df-c 8759 . . . . . . 7
1312eleq2i 2360 . . . . . 6
1412eleq2i 2360 . . . . . 6
1513, 14anbi12i 678 . . . . 5
1615anbi1i 676 . . . 4
1716oprabbii 5919 . . 3
1811, 17eqtri 2316 . 2
191, 10, 18ov3 6000 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 358  wex 1531   wceq 1632   wcel 1696  cop 3656   cxp 4703  (class class class)co 5874  coprab 5875  cnr 8505   cplr 8509  cc 8751   caddc 8756 This theorem is referenced by:  addresr  8776  addcnsrec  8781  axaddf  8783  axcnre  8802 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-oprab 5878  df-c 8759  df-add 8764
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