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Theorem rexmul 10842
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 9124 . . . . . . . . . . 11  |-  ( A  e.  RR  ->  A  =/=  +oo )
21adantr 452 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  A  =/=  +oo )
32necon2bi 2644 . . . . . . . . 9  |-  ( A  =  +oo  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
43adantl 453 . . . . . . . 8  |-  ( ( 0  <  B  /\  A  =  +oo )  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
5 renemnf 9125 . . . . . . . . . . 11  |-  ( A  e.  RR  ->  A  =/=  -oo )
65adantr 452 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  A  =/=  -oo )
76necon2bi 2644 . . . . . . . . 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 9124 . . . . . . . . . . 11  |-  ( B  e.  RR  ->  B  =/=  +oo )
1110adantl 453 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  B  =/=  +oo )
1211necon2bi 2644 . . . . . . . . 9  |-  ( B  =  +oo  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
1312adantl 453 . . . . . . . 8  |-  ( ( 0  <  A  /\  B  =  +oo )  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
14 renemnf 9125 . . . . . . . . . . 11  |-  ( B  e.  RR  ->  B  =/=  -oo )
1514adantl 453 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  B  =/=  -oo )
1615necon2bi 2644 . . . . . . . . 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 3738 . . . . 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 3738 . . . . 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 2467 . . 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 3746 . 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 9122 . . 3  |-  ( A  e.  RR  ->  A  e.  RR* )
36 rexr 9122 . . 3  |-  ( B  e.  RR  ->  B  e.  RR* )
37 xmulval 10803 . . 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 3763 . . 3  |-  if ( ( A  =  0  \/  B  =  0 ) ,  ( A  x.  B ) ,  ( A  x.  B
) )  =  ( A  x.  B )
40 oveq1 6080 . . . . . 6  |-  ( A  =  0  ->  ( A  x.  B )  =  ( 0  x.  B ) )
41 mul02lem2 9235 . . . . . . 7  |-  ( B  e.  RR  ->  (
0  x.  B )  =  0 )
4241adantl 453 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 0  x.  B
)  =  0 )
4340, 42sylan9eqr 2489 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  =  0 )  ->  ( A  x.  B )  =  0 )
44 oveq2 6081 . . . . . 6  |-  ( B  =  0  ->  ( A  x.  B )  =  ( A  x.  0 ) )
45 recn 9072 . . . . . . . 8  |-  ( A  e.  RR  ->  A  e.  CC )
4645mul01d 9257 . . . . . . 7  |-  ( A  e.  RR  ->  ( A  x.  0 )  =  0 )
4746adantr 452 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  x.  0 )  =  0 )
4844, 47sylan9eqr 2489 . . . . 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 3756 . . 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 2481 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  x.  B
)  =  if ( ( A  =  0  \/  B  =  0 ) ,  0 ,  ( A  x.  B
) ) )
5234, 38, 513eqtr4d 2477 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 1652    e. wcel 1725    =/= wne 2598   ifcif 3731   class class class wbr 4204  (class class class)co 6073   RRcr 8981   0cc0 8982    x. cmul 8987    +oocpnf 9109    -oocmnf 9110   RR*cxr 9111    < clt 9112   x ecxmu 10701
This theorem is referenced by:  xmulid1  10850  xmulgt0  10854  xmulasslem3  10857  xlemul1a  10859  xlemul1  10861  xadddilem  10865  nmoix  18755  nmoi2  18756  metnrmlem3  18883  nmoleub2lem  19114  xrecex  24158  rexdiv  24164  pnfinf  24245  esumcst  24447
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-po 4495  df-so 4496  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-xmul 10704
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