MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  resub Unicode version

Theorem resub 11860
Description: Real part distributes over subtraction. (Contributed by NM, 17-Mar-2005.)
Assertion
Ref Expression
resub  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( Re `  ( A  -  B )
)  =  ( ( Re `  A )  -  ( Re `  B ) ) )

Proof of Theorem resub
StepHypRef Expression
1 negcl 9239 . . . 4  |-  ( B  e.  CC  ->  -u B  e.  CC )
2 readd 11859 . . . 4  |-  ( ( A  e.  CC  /\  -u B  e.  CC )  ->  ( Re `  ( A  +  -u B
) )  =  ( ( Re `  A
)  +  ( Re
`  -u B ) ) )
31, 2sylan2 461 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( Re `  ( A  +  -u B ) )  =  ( ( Re `  A )  +  ( Re `  -u B ) ) )
4 reneg 11858 . . . . 5  |-  ( B  e.  CC  ->  (
Re `  -u B )  =  -u ( Re `  B ) )
54adantl 453 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( Re `  -u B
)  =  -u (
Re `  B )
)
65oveq2d 6037 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( Re `  A )  +  ( Re `  -u B
) )  =  ( ( Re `  A
)  +  -u (
Re `  B )
) )
73, 6eqtrd 2420 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( Re `  ( A  +  -u B ) )  =  ( ( Re `  A )  +  -u ( Re `  B ) ) )
8 negsub 9282 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  -u B )  =  ( A  -  B ) )
98fveq2d 5673 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( Re `  ( A  +  -u B ) )  =  ( Re
`  ( A  -  B ) ) )
10 recl 11843 . . . 4  |-  ( A  e.  CC  ->  (
Re `  A )  e.  RR )
1110recnd 9048 . . 3  |-  ( A  e.  CC  ->  (
Re `  A )  e.  CC )
12 recl 11843 . . . 4  |-  ( B  e.  CC  ->  (
Re `  B )  e.  RR )
1312recnd 9048 . . 3  |-  ( B  e.  CC  ->  (
Re `  B )  e.  CC )
14 negsub 9282 . . 3  |-  ( ( ( Re `  A
)  e.  CC  /\  ( Re `  B )  e.  CC )  -> 
( ( Re `  A )  +  -u ( Re `  B ) )  =  ( ( Re `  A )  -  ( Re `  B ) ) )
1511, 13, 14syl2an 464 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( Re `  A )  +  -u ( Re `  B ) )  =  ( ( Re `  A )  -  ( Re `  B ) ) )
167, 9, 153eqtr3d 2428 1  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( Re `  ( A  -  B )
)  =  ( ( Re `  A )  -  ( Re `  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   ` cfv 5395  (class class class)co 6021   CCcc 8922    + caddc 8927    - cmin 9224   -ucneg 9225   Recre 11830
This theorem is referenced by:  resubd  11949  recn2  12322  caucvgr  12397  tanregt0  20309  logcnlem4  20404  isosctrlem1  20530  acoscos  20601  acosbnd  20608  atanlogsublem  20623
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-po 4445  df-so 4446  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-riota 6486  df-er 6842  df-en 7047  df-dom 7048  df-sdom 7049  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-div 9611  df-2 9991  df-cj 11832  df-re 11833  df-im 11834
  Copyright terms: Public domain W3C validator