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

Theorem rexmul 10783
Description: The extended real multiplication when both arguments are real. (Contributed by Mario Carneiro, 20-Aug-2015.)
Assertion
Ref Expression
rexmul  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A x e B )  =  ( A  x.  B ) )

Proof of Theorem rexmul
StepHypRef Expression
1 renepnf 9066 . . . . . . . . . . 11  |-  ( A  e.  RR  ->  A  =/=  +oo )
21adantr 452 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  A  =/=  +oo )
32necon2bi 2597 . . . . . . . . 9  |-  ( A  =  +oo  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
43adantl 453 . . . . . . . 8  |-  ( ( 0  <  B  /\  A  =  +oo )  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
5 renemnf 9067 . . . . . . . . . . 11  |-  ( A  e.  RR  ->  A  =/=  -oo )
65adantr 452 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  A  =/=  -oo )
76necon2bi 2597 . . . . . . . . 9  |-  ( A  =  -oo  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
87adantl 453 . . . . . . . 8  |-  ( ( B  <  0  /\  A  =  -oo )  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
94, 8jaoi 369 . . . . . . 7  |-  ( ( ( 0  <  B  /\  A  =  +oo )  \/  ( B  <  0  /\  A  = 
-oo ) )  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
10 renepnf 9066 . . . . . . . . . . 11  |-  ( B  e.  RR  ->  B  =/=  +oo )
1110adantl 453 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  B  =/=  +oo )
1211necon2bi 2597 . . . . . . . . 9  |-  ( B  =  +oo  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
1312adantl 453 . . . . . . . 8  |-  ( ( 0  <  A  /\  B  =  +oo )  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
14 renemnf 9067 . . . . . . . . . . 11  |-  ( B  e.  RR  ->  B  =/=  -oo )
1514adantl 453 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  B  =/=  -oo )
1615necon2bi 2597 . . . . . . . . 9  |-  ( B  =  -oo  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
1716adantl 453 . . . . . . . 8  |-  ( ( A  <  0  /\  B  =  -oo )  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
1813, 17jaoi 369 . . . . . . 7  |-  ( ( ( 0  <  A  /\  B  =  +oo )  \/  ( A  <  0  /\  B  = 
-oo ) )  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
199, 18jaoi 369 . . . . . 6  |-  ( ( ( ( 0  < 
B  /\  A  =  +oo )  \/  ( B  <  0  /\  A  =  -oo ) )  \/  ( ( 0  < 
A  /\  B  =  +oo )  \/  ( A  <  0  /\  B  =  -oo ) ) )  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
2019con2i 114 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  -.  ( ( ( 0  <  B  /\  A  =  +oo )  \/  ( B  <  0  /\  A  =  -oo ) )  \/  (
( 0  <  A  /\  B  =  +oo )  \/  ( A  <  0  /\  B  = 
-oo ) ) ) )
21 iffalse 3690 . . . . 5  |-  ( -.  ( ( ( 0  <  B  /\  A  =  +oo )  \/  ( B  <  0  /\  A  =  -oo ) )  \/  ( ( 0  < 
A  /\  B  =  +oo )  \/  ( A  <  0  /\  B  =  -oo ) ) )  ->  if ( ( ( ( 0  < 
B  /\  A  =  +oo )  \/  ( B  <  0  /\  A  =  -oo ) )  \/  ( ( 0  < 
A  /\  B  =  +oo )  \/  ( A  <  0  /\  B  =  -oo ) ) ) ,  +oo ,  if ( ( ( ( 0  <  B  /\  A  =  -oo )  \/  ( B  <  0  /\  A  =  +oo ) )  \/  (
( 0  <  A  /\  B  =  -oo )  \/  ( A  <  0  /\  B  = 
+oo ) ) ) ,  -oo ,  ( A  x.  B ) ) )  =  if ( ( ( ( 0  <  B  /\  A  =  -oo )  \/  ( B  <  0  /\  A  =  +oo ) )  \/  (
( 0  <  A  /\  B  =  -oo )  \/  ( A  <  0  /\  B  = 
+oo ) ) ) ,  -oo ,  ( A  x.  B ) ) )
2220, 21syl 16 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  if ( ( ( ( 0  <  B  /\  A  =  +oo )  \/  ( B  <  0  /\  A  = 
-oo ) )  \/  ( ( 0  < 
A  /\  B  =  +oo )  \/  ( A  <  0  /\  B  =  -oo ) ) ) ,  +oo ,  if ( ( ( ( 0  <  B  /\  A  =  -oo )  \/  ( B  <  0  /\  A  =  +oo ) )  \/  (
( 0  <  A  /\  B  =  -oo )  \/  ( A  <  0  /\  B  = 
+oo ) ) ) ,  -oo ,  ( A  x.  B ) ) )  =  if ( ( ( ( 0  <  B  /\  A  =  -oo )  \/  ( B  <  0  /\  A  =  +oo ) )  \/  (
( 0  <  A  /\  B  =  -oo )  \/  ( A  <  0  /\  B  = 
+oo ) ) ) ,  -oo ,  ( A  x.  B ) ) )
237adantl 453 . . . . . . . 8  |-  ( ( 0  <  B  /\  A  =  -oo )  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
243adantl 453 . . . . . . . 8  |-  ( ( B  <  0  /\  A  =  +oo )  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
2523, 24jaoi 369 . . . . . . 7  |-  ( ( ( 0  <  B  /\  A  =  -oo )  \/  ( B  <  0  /\  A  = 
+oo ) )  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
2616adantl 453 . . . . . . . 8  |-  ( ( 0  <  A  /\  B  =  -oo )  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
2712adantl 453 . . . . . . . 8  |-  ( ( A  <  0  /\  B  =  +oo )  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
2826, 27jaoi 369 . . . . . . 7  |-  ( ( ( 0  <  A  /\  B  =  -oo )  \/  ( A  <  0  /\  B  = 
+oo ) )  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
2925, 28jaoi 369 . . . . . 6  |-  ( ( ( ( 0  < 
B  /\  A  =  -oo )  \/  ( B  <  0  /\  A  =  +oo ) )  \/  ( ( 0  < 
A  /\  B  =  -oo )  \/  ( A  <  0  /\  B  =  +oo ) ) )  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
3029con2i 114 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  -.  ( ( ( 0  <  B  /\  A  =  -oo )  \/  ( B  <  0  /\  A  =  +oo ) )  \/  (
( 0  <  A  /\  B  =  -oo )  \/  ( A  <  0  /\  B  = 
+oo ) ) ) )
31 iffalse 3690 . . . . 5  |-  ( -.  ( ( ( 0  <  B  /\  A  =  -oo )  \/  ( B  <  0  /\  A  =  +oo ) )  \/  ( ( 0  < 
A  /\  B  =  -oo )  \/  ( A  <  0  /\  B  =  +oo ) ) )  ->  if ( ( ( ( 0  < 
B  /\  A  =  -oo )  \/  ( B  <  0  /\  A  =  +oo ) )  \/  ( ( 0  < 
A  /\  B  =  -oo )  \/  ( A  <  0  /\  B  =  +oo ) ) ) ,  -oo ,  ( A  x.  B ) )  =  ( A  x.  B ) )
3230, 31syl 16 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  if ( ( ( ( 0  <  B  /\  A  =  -oo )  \/  ( B  <  0  /\  A  = 
+oo ) )  \/  ( ( 0  < 
A  /\  B  =  -oo )  \/  ( A  <  0  /\  B  =  +oo ) ) ) ,  -oo ,  ( A  x.  B ) )  =  ( A  x.  B ) )
3322, 32eqtrd 2420 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  if ( ( ( ( 0  <  B  /\  A  =  +oo )  \/  ( B  <  0  /\  A  = 
-oo ) )  \/  ( ( 0  < 
A  /\  B  =  +oo )  \/  ( A  <  0  /\  B  =  -oo ) ) ) ,  +oo ,  if ( ( ( ( 0  <  B  /\  A  =  -oo )  \/  ( B  <  0  /\  A  =  +oo ) )  \/  (
( 0  <  A  /\  B  =  -oo )  \/  ( A  <  0  /\  B  = 
+oo ) ) ) ,  -oo ,  ( A  x.  B ) ) )  =  ( A  x.  B ) )
3433ifeq2d 3698 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  if ( ( A  =  0  \/  B  =  0 ) ,  0 ,  if ( ( ( ( 0  <  B  /\  A  =  +oo )  \/  ( B  <  0  /\  A  =  -oo ) )  \/  ( ( 0  < 
A  /\  B  =  +oo )  \/  ( A  <  0  /\  B  =  -oo ) ) ) ,  +oo ,  if ( ( ( ( 0  <  B  /\  A  =  -oo )  \/  ( B  <  0  /\  A  =  +oo ) )  \/  (
( 0  <  A  /\  B  =  -oo )  \/  ( A  <  0  /\  B  = 
+oo ) ) ) ,  -oo ,  ( A  x.  B ) ) ) )  =  if ( ( A  =  0  \/  B  =  0 ) ,  0 ,  ( A  x.  B ) ) )
35 rexr 9064 . . 3  |-  ( A  e.  RR  ->  A  e.  RR* )
36 rexr 9064 . . 3  |-  ( B  e.  RR  ->  B  e.  RR* )
37 xmulval 10744 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A x e B )  =  if ( ( A  =  0  \/  B  =  0 ) ,  0 ,  if ( ( ( ( 0  <  B  /\  A  =  +oo )  \/  ( B  <  0  /\  A  =  -oo ) )  \/  (
( 0  <  A  /\  B  =  +oo )  \/  ( A  <  0  /\  B  = 
-oo ) ) ) ,  +oo ,  if ( ( ( ( 0  <  B  /\  A  =  -oo )  \/  ( B  <  0  /\  A  =  +oo ) )  \/  (
( 0  <  A  /\  B  =  -oo )  \/  ( A  <  0  /\  B  = 
+oo ) ) ) ,  -oo ,  ( A  x.  B ) ) ) ) )
3835, 36, 37syl2an 464 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A x e B )  =  if ( ( A  =  0  \/  B  =  0 ) ,  0 ,  if ( ( ( ( 0  < 
B  /\  A  =  +oo )  \/  ( B  <  0  /\  A  =  -oo ) )  \/  ( ( 0  < 
A  /\  B  =  +oo )  \/  ( A  <  0  /\  B  =  -oo ) ) ) ,  +oo ,  if ( ( ( ( 0  <  B  /\  A  =  -oo )  \/  ( B  <  0  /\  A  =  +oo ) )  \/  (
( 0  <  A  /\  B  =  -oo )  \/  ( A  <  0  /\  B  = 
+oo ) ) ) ,  -oo ,  ( A  x.  B ) ) ) ) )
39 ifid 3715 . . 3  |-  if ( ( A  =  0  \/  B  =  0 ) ,  ( A  x.  B ) ,  ( A  x.  B
) )  =  ( A  x.  B )
40 oveq1 6028 . . . . . 6  |-  ( A  =  0  ->  ( A  x.  B )  =  ( 0  x.  B ) )
41 mul02lem2 9176 . . . . . . 7  |-  ( B  e.  RR  ->  (
0  x.  B )  =  0 )
4241adantl 453 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 0  x.  B
)  =  0 )
4340, 42sylan9eqr 2442 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  =  0 )  ->  ( A  x.  B )  =  0 )
44 oveq2 6029 . . . . . 6  |-  ( B  =  0  ->  ( A  x.  B )  =  ( A  x.  0 ) )
45 recn 9014 . . . . . . . 8  |-  ( A  e.  RR  ->  A  e.  CC )
4645mul01d 9198 . . . . . . 7  |-  ( A  e.  RR  ->  ( A  x.  0 )  =  0 )
4746adantr 452 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  x.  0 )  =  0 )
4844, 47sylan9eqr 2442 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  =  0 )  ->  ( A  x.  B )  =  0 )
4943, 48jaodan 761 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  =  0  \/  B  =  0 ) )  -> 
( A  x.  B
)  =  0 )
5049ifeq1da 3708 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  if ( ( A  =  0  \/  B  =  0 ) ,  ( A  x.  B
) ,  ( A  x.  B ) )  =  if ( ( A  =  0  \/  B  =  0 ) ,  0 ,  ( A  x.  B ) ) )
5139, 50syl5eqr 2434 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  x.  B
)  =  if ( ( A  =  0  \/  B  =  0 ) ,  0 ,  ( A  x.  B
) ) )
5234, 38, 513eqtr4d 2430 1  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A x e B )  =  ( A  x.  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2551   ifcif 3683   class class class wbr 4154  (class class class)co 6021   RRcr 8923   0cc0 8924    x. cmul 8929    +oocpnf 9051    -oocmnf 9052   RR*cxr 9053    < clt 9054   x ecxmu 10642
This theorem is referenced by:  xmulid1  10791  xmulgt0  10795  xmulasslem3  10798  xlemul1a  10800  xlemul1  10802  xadddilem  10806  nmoix  18635  nmoi2  18636  metnrmlem3  18763  nmoleub2lem  18994  xrecex  24005  rexdiv  24011  esumcst  24252
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-cnex 8980  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
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-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-er 6842  df-en 7047  df-dom 7048  df-sdom 7049  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-xmul 10645
  Copyright terms: Public domain W3C validator