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Theorem rexmul 11582
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 xe B )  =  ( A  x.  B ) )

Proof of Theorem rexmul
StepHypRef Expression
1 renepnf 9706 . . . . . . . . . . 11  |-  ( A  e.  RR  ->  A  =/= +oo )
21adantr 472 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  A  =/= +oo )
32necon2bi 2673 . . . . . . . . 9  |-  ( A  = +oo  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
43adantl 473 . . . . . . . 8  |-  ( ( 0  <  B  /\  A  = +oo )  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
5 renemnf 9707 . . . . . . . . . . 11  |-  ( A  e.  RR  ->  A  =/= -oo )
65adantr 472 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  A  =/= -oo )
76necon2bi 2673 . . . . . . . . 9  |-  ( A  = -oo  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
87adantl 473 . . . . . . . 8  |-  ( ( B  <  0  /\  A  = -oo )  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
94, 8jaoi 386 . . . . . . 7  |-  ( ( ( 0  <  B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) )  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
10 renepnf 9706 . . . . . . . . . . 11  |-  ( B  e.  RR  ->  B  =/= +oo )
1110adantl 473 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  B  =/= +oo )
1211necon2bi 2673 . . . . . . . . 9  |-  ( B  = +oo  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
1312adantl 473 . . . . . . . 8  |-  ( ( 0  <  A  /\  B  = +oo )  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
14 renemnf 9707 . . . . . . . . . . 11  |-  ( B  e.  RR  ->  B  =/= -oo )
1514adantl 473 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  B  =/= -oo )
1615necon2bi 2673 . . . . . . . . 9  |-  ( B  = -oo  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
1716adantl 473 . . . . . . . 8  |-  ( ( A  <  0  /\  B  = -oo )  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
1813, 17jaoi 386 . . . . . . 7  |-  ( ( ( 0  <  A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) )  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
199, 18jaoi 386 . . . . . 6  |-  ( ( ( ( 0  < 
B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) )  \/  ( ( 0  < 
A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) ) )  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
2019con2i 124 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  -.  ( ( ( 0  <  B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) )  \/  (
( 0  <  A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) ) ) )
2120iffalsed 3883 . . . 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 ) ) )
227adantl 473 . . . . . . . 8  |-  ( ( 0  <  B  /\  A  = -oo )  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
233adantl 473 . . . . . . . 8  |-  ( ( B  <  0  /\  A  = +oo )  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
2422, 23jaoi 386 . . . . . . 7  |-  ( ( ( 0  <  B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
2516adantl 473 . . . . . . . 8  |-  ( ( 0  <  A  /\  B  = -oo )  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
2612adantl 473 . . . . . . . 8  |-  ( ( A  <  0  /\  B  = +oo )  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
2725, 26jaoi 386 . . . . . . 7  |-  ( ( ( 0  <  A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) )  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
2824, 27jaoi 386 . . . . . 6  |-  ( ( ( ( 0  < 
B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  \/  ( ( 0  < 
A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) )  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
2928con2i 124 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  -.  ( ( ( 0  <  B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  \/  (
( 0  <  A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) ) )
3029iffalsed 3883 . . . 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 ) )
3121, 30eqtrd 2505 . . 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 ) )
3231ifeq2d 3891 . 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 ) ) )
33 rexr 9704 . . 3  |-  ( A  e.  RR  ->  A  e.  RR* )
34 rexr 9704 . . 3  |-  ( B  e.  RR  ->  B  e.  RR* )
35 xmulval 11541 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A xe 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 ) ) ) ) )
3633, 34, 35syl2an 485 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A xe 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 ) ) ) ) )
37 ifid 3909 . . 3  |-  if ( ( A  =  0  \/  B  =  0 ) ,  ( A  x.  B ) ,  ( A  x.  B
) )  =  ( A  x.  B )
38 oveq1 6315 . . . . . 6  |-  ( A  =  0  ->  ( A  x.  B )  =  ( 0  x.  B ) )
39 mul02lem2 9828 . . . . . . 7  |-  ( B  e.  RR  ->  (
0  x.  B )  =  0 )
4039adantl 473 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 0  x.  B
)  =  0 )
4138, 40sylan9eqr 2527 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  =  0 )  ->  ( A  x.  B )  =  0 )
42 oveq2 6316 . . . . . 6  |-  ( B  =  0  ->  ( A  x.  B )  =  ( A  x.  0 ) )
43 recn 9647 . . . . . . . 8  |-  ( A  e.  RR  ->  A  e.  CC )
4443mul01d 9850 . . . . . . 7  |-  ( A  e.  RR  ->  ( A  x.  0 )  =  0 )
4544adantr 472 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  x.  0 )  =  0 )
4642, 45sylan9eqr 2527 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  =  0 )  ->  ( A  x.  B )  =  0 )
4741, 46jaodan 802 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  =  0  \/  B  =  0 ) )  -> 
( A  x.  B
)  =  0 )
4847ifeq1da 3902 . . 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 ) ) )
4937, 48syl5eqr 2519 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  x.  B
)  =  if ( ( A  =  0  \/  B  =  0 ) ,  0 ,  ( A  x.  B
) ) )
5032, 36, 493eqtr4d 2515 1  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A xe B )  =  ( A  x.  B ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 375    /\ wa 376    = wceq 1452    e. wcel 1904    =/= wne 2641   ifcif 3872   class class class wbr 4395  (class class class)co 6308   RRcr 9556   0cc0 9557    x. cmul 9562   +oocpnf 9690   -oocmnf 9691   RR*cxr 9692    < clt 9693   xecxmu 11431
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-po 4760  df-so 4761  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-xmul 11434
This theorem is referenced by:  xmulid1  11590  xmulgt0  11594  xmulasslem3  11597  xlemul1a  11599  xlemul1  11601  xadddilem  11605  nmoix  21812  nmoi2  21813  nmoixOLD  21828  nmoi2OLD  21829  metnrmlem3  21956  metnrmlem3OLD  21971  nmoleub2lem  22206  xrecex  28464  rexdiv  28470  pnfinf  28574  xrge0slmod  28681  esumcst  28958  omssubadd  29201  omssubaddOLD  29205
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