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Theorem rexmul 11475
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 9653 . . . . . . . . . . 11  |-  ( A  e.  RR  ->  A  =/= +oo )
21adantr 465 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  A  =/= +oo )
32necon2bi 2704 . . . . . . . . 9  |-  ( A  = +oo  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
43adantl 466 . . . . . . . 8  |-  ( ( 0  <  B  /\  A  = +oo )  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
5 renemnf 9654 . . . . . . . . . . 11  |-  ( A  e.  RR  ->  A  =/= -oo )
65adantr 465 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  A  =/= -oo )
76necon2bi 2704 . . . . . . . . 9  |-  ( A  = -oo  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
87adantl 466 . . . . . . . 8  |-  ( ( B  <  0  /\  A  = -oo )  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
94, 8jaoi 379 . . . . . . 7  |-  ( ( ( 0  <  B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) )  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
10 renepnf 9653 . . . . . . . . . . 11  |-  ( B  e.  RR  ->  B  =/= +oo )
1110adantl 466 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  B  =/= +oo )
1211necon2bi 2704 . . . . . . . . 9  |-  ( B  = +oo  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
1312adantl 466 . . . . . . . 8  |-  ( ( 0  <  A  /\  B  = +oo )  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
14 renemnf 9654 . . . . . . . . . . 11  |-  ( B  e.  RR  ->  B  =/= -oo )
1514adantl 466 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  B  =/= -oo )
1615necon2bi 2704 . . . . . . . . 9  |-  ( B  = -oo  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
1716adantl 466 . . . . . . . 8  |-  ( ( A  <  0  /\  B  = -oo )  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
1813, 17jaoi 379 . . . . . . 7  |-  ( ( ( 0  <  A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) )  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
199, 18jaoi 379 . . . . . 6  |-  ( ( ( ( 0  < 
B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) )  \/  ( ( 0  < 
A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) ) )  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
2019con2i 120 . . . . 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 3954 . . . . 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 466 . . . . . . . 8  |-  ( ( 0  <  B  /\  A  = -oo )  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
243adantl 466 . . . . . . . 8  |-  ( ( B  <  0  /\  A  = +oo )  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
2523, 24jaoi 379 . . . . . . 7  |-  ( ( ( 0  <  B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
2616adantl 466 . . . . . . . 8  |-  ( ( 0  <  A  /\  B  = -oo )  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
2712adantl 466 . . . . . . . 8  |-  ( ( A  <  0  /\  B  = +oo )  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
2826, 27jaoi 379 . . . . . . 7  |-  ( ( ( 0  <  A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) )  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
2925, 28jaoi 379 . . . . . 6  |-  ( ( ( ( 0  < 
B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  \/  ( ( 0  < 
A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) )  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
3029con2i 120 . . . . 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 3954 . . . . 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 2508 . . 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 3964 . 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 9651 . . 3  |-  ( A  e.  RR  ->  A  e.  RR* )
36 rexr 9651 . . 3  |-  ( B  e.  RR  ->  B  e.  RR* )
37 xmulval 11436 . . 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 ) ) ) ) )
3835, 36, 37syl2an 477 . 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 ) ) ) ) )
39 ifid 3982 . . 3  |-  if ( ( A  =  0  \/  B  =  0 ) ,  ( A  x.  B ) ,  ( A  x.  B
) )  =  ( A  x.  B )
40 oveq1 6302 . . . . . 6  |-  ( A  =  0  ->  ( A  x.  B )  =  ( 0  x.  B ) )
41 mul02lem2 9768 . . . . . . 7  |-  ( B  e.  RR  ->  (
0  x.  B )  =  0 )
4241adantl 466 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 0  x.  B
)  =  0 )
4340, 42sylan9eqr 2530 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  =  0 )  ->  ( A  x.  B )  =  0 )
44 oveq2 6303 . . . . . 6  |-  ( B  =  0  ->  ( A  x.  B )  =  ( A  x.  0 ) )
45 recn 9594 . . . . . . . 8  |-  ( A  e.  RR  ->  A  e.  CC )
4645mul01d 9790 . . . . . . 7  |-  ( A  e.  RR  ->  ( A  x.  0 )  =  0 )
4746adantr 465 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  x.  0 )  =  0 )
4844, 47sylan9eqr 2530 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  =  0 )  ->  ( A  x.  B )  =  0 )
4943, 48jaodan 783 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  =  0  \/  B  =  0 ) )  -> 
( A  x.  B
)  =  0 )
5049ifeq1da 3975 . . 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 2522 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  x.  B
)  =  if ( ( A  =  0  \/  B  =  0 ) ,  0 ,  ( A  x.  B
) ) )
5234, 38, 513eqtr4d 2518 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 368    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   ifcif 3945   class class class wbr 4453  (class class class)co 6295   RRcr 9503   0cc0 9504    x. cmul 9509   +oocpnf 9637   -oocmnf 9638   RR*cxr 9639    < clt 9640   xecxmu 11329
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-po 4806  df-so 4807  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-xmul 11332
This theorem is referenced by:  xmulid1  11483  xmulgt0  11487  xmulasslem3  11490  xlemul1a  11492  xlemul1  11494  xadddilem  11498  nmoix  21104  nmoi2  21105  metnrmlem3  21233  nmoleub2lem  21465  xrecex  27440  rexdiv  27446  pnfinf  27551  xrge0slmod  27659  esumcst  27896
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