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Theorem rexmul 11564
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 9695 . . . . . . . . . . 11  |-  ( A  e.  RR  ->  A  =/= +oo )
21adantr 466 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  A  =/= +oo )
32necon2bi 2657 . . . . . . . . 9  |-  ( A  = +oo  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
43adantl 467 . . . . . . . 8  |-  ( ( 0  <  B  /\  A  = +oo )  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
5 renemnf 9696 . . . . . . . . . . 11  |-  ( A  e.  RR  ->  A  =/= -oo )
65adantr 466 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  A  =/= -oo )
76necon2bi 2657 . . . . . . . . 9  |-  ( A  = -oo  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
87adantl 467 . . . . . . . 8  |-  ( ( B  <  0  /\  A  = -oo )  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
94, 8jaoi 380 . . . . . . 7  |-  ( ( ( 0  <  B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) )  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
10 renepnf 9695 . . . . . . . . . . 11  |-  ( B  e.  RR  ->  B  =/= +oo )
1110adantl 467 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  B  =/= +oo )
1211necon2bi 2657 . . . . . . . . 9  |-  ( B  = +oo  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
1312adantl 467 . . . . . . . 8  |-  ( ( 0  <  A  /\  B  = +oo )  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
14 renemnf 9696 . . . . . . . . . . 11  |-  ( B  e.  RR  ->  B  =/= -oo )
1514adantl 467 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  B  =/= -oo )
1615necon2bi 2657 . . . . . . . . 9  |-  ( B  = -oo  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
1716adantl 467 . . . . . . . 8  |-  ( ( A  <  0  /\  B  = -oo )  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
1813, 17jaoi 380 . . . . . . 7  |-  ( ( ( 0  <  A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) )  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
199, 18jaoi 380 . . . . . 6  |-  ( ( ( ( 0  < 
B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) )  \/  ( ( 0  < 
A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) ) )  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
2019con2i 123 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  -.  ( ( ( 0  <  B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) )  \/  (
( 0  <  A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) ) ) )
2120iffalsed 3922 . . . 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 467 . . . . . . . 8  |-  ( ( 0  <  B  /\  A  = -oo )  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
233adantl 467 . . . . . . . 8  |-  ( ( B  <  0  /\  A  = +oo )  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
2422, 23jaoi 380 . . . . . . 7  |-  ( ( ( 0  <  B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
2516adantl 467 . . . . . . . 8  |-  ( ( 0  <  A  /\  B  = -oo )  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
2612adantl 467 . . . . . . . 8  |-  ( ( A  <  0  /\  B  = +oo )  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
2725, 26jaoi 380 . . . . . . 7  |-  ( ( ( 0  <  A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) )  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
2824, 27jaoi 380 . . . . . 6  |-  ( ( ( ( 0  < 
B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  \/  ( ( 0  < 
A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) )  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
2928con2i 123 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  -.  ( ( ( 0  <  B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  \/  (
( 0  <  A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) ) )
3029iffalsed 3922 . . . 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 2463 . . 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 3930 . 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 9693 . . 3  |-  ( A  e.  RR  ->  A  e.  RR* )
34 rexr 9693 . . 3  |-  ( B  e.  RR  ->  B  e.  RR* )
35 xmulval 11525 . . 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 479 . 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 3948 . . 3  |-  if ( ( A  =  0  \/  B  =  0 ) ,  ( A  x.  B ) ,  ( A  x.  B
) )  =  ( A  x.  B )
38 oveq1 6312 . . . . . 6  |-  ( A  =  0  ->  ( A  x.  B )  =  ( 0  x.  B ) )
39 mul02lem2 9817 . . . . . . 7  |-  ( B  e.  RR  ->  (
0  x.  B )  =  0 )
4039adantl 467 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 0  x.  B
)  =  0 )
4138, 40sylan9eqr 2485 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  =  0 )  ->  ( A  x.  B )  =  0 )
42 oveq2 6313 . . . . . 6  |-  ( B  =  0  ->  ( A  x.  B )  =  ( A  x.  0 ) )
43 recn 9636 . . . . . . . 8  |-  ( A  e.  RR  ->  A  e.  CC )
4443mul01d 9839 . . . . . . 7  |-  ( A  e.  RR  ->  ( A  x.  0 )  =  0 )
4544adantr 466 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  x.  0 )  =  0 )
4642, 45sylan9eqr 2485 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  =  0 )  ->  ( A  x.  B )  =  0 )
4741, 46jaodan 792 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  =  0  \/  B  =  0 ) )  -> 
( A  x.  B
)  =  0 )
4847ifeq1da 3941 . . 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 2477 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  x.  B
)  =  if ( ( A  =  0  \/  B  =  0 ) ,  0 ,  ( A  x.  B
) ) )
5032, 36, 493eqtr4d 2473 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 369    /\ wa 370    = wceq 1437    e. wcel 1872    =/= wne 2614   ifcif 3911   class class class wbr 4423  (class class class)co 6305   RRcr 9545   0cc0 9546    x. cmul 9551   +oocpnf 9679   -oocmnf 9680   RR*cxr 9681    < clt 9682   xecxmu 11415
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597  ax-cnex 9602  ax-resscn 9603  ax-1cn 9604  ax-icn 9605  ax-addcl 9606  ax-addrcl 9607  ax-mulcl 9608  ax-mulrcl 9609  ax-mulcom 9610  ax-addass 9611  ax-mulass 9612  ax-distr 9613  ax-i2m1 9614  ax-1ne0 9615  ax-1rid 9616  ax-rnegex 9617  ax-rrecex 9618  ax-cnre 9619  ax-pre-lttri 9620  ax-pre-lttrn 9621  ax-pre-ltadd 9622
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-po 4774  df-so 4775  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-er 7374  df-en 7581  df-dom 7582  df-sdom 7583  df-pnf 9684  df-mnf 9685  df-xr 9686  df-ltxr 9687  df-xmul 11418
This theorem is referenced by:  xmulid1  11572  xmulgt0  11576  xmulasslem3  11579  xlemul1a  11581  xlemul1  11583  xadddilem  11587  nmoix  21732  nmoi2  21733  nmoixOLD  21748  nmoi2OLD  21749  metnrmlem3  21876  metnrmlem3OLD  21891  nmoleub2lem  22126  xrecex  28396  rexdiv  28402  pnfinf  28507  xrge0slmod  28615  esumcst  28892  omssubadd  29136  omssubaddOLD  29140
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