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Theorem rexmul 10591
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 8879 . . . . . . . . . . 11  |-  ( A  e.  RR  ->  A  =/=  +oo )
21adantr 451 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  A  =/=  +oo )
32necon2bi 2492 . . . . . . . . 9  |-  ( A  =  +oo  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
43adantl 452 . . . . . . . 8  |-  ( ( 0  <  B  /\  A  =  +oo )  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
5 renemnf 8880 . . . . . . . . . . 11  |-  ( A  e.  RR  ->  A  =/=  -oo )
65adantr 451 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  A  =/=  -oo )
76necon2bi 2492 . . . . . . . . 9  |-  ( A  =  -oo  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
87adantl 452 . . . . . . . 8  |-  ( ( B  <  0  /\  A  =  -oo )  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
94, 8jaoi 368 . . . . . . 7  |-  ( ( ( 0  <  B  /\  A  =  +oo )  \/  ( B  <  0  /\  A  = 
-oo ) )  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
10 renepnf 8879 . . . . . . . . . . 11  |-  ( B  e.  RR  ->  B  =/=  +oo )
1110adantl 452 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  B  =/=  +oo )
1211necon2bi 2492 . . . . . . . . 9  |-  ( B  =  +oo  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
1312adantl 452 . . . . . . . 8  |-  ( ( 0  <  A  /\  B  =  +oo )  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
14 renemnf 8880 . . . . . . . . . . 11  |-  ( B  e.  RR  ->  B  =/=  -oo )
1514adantl 452 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  B  =/=  -oo )
1615necon2bi 2492 . . . . . . . . 9  |-  ( B  =  -oo  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
1716adantl 452 . . . . . . . 8  |-  ( ( A  <  0  /\  B  =  -oo )  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
1813, 17jaoi 368 . . . . . . 7  |-  ( ( ( 0  <  A  /\  B  =  +oo )  \/  ( A  <  0  /\  B  = 
-oo ) )  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
199, 18jaoi 368 . . . . . 6  |-  ( ( ( ( 0  < 
B  /\  A  =  +oo )  \/  ( B  <  0  /\  A  =  -oo ) )  \/  ( ( 0  < 
A  /\  B  =  +oo )  \/  ( A  <  0  /\  B  =  -oo ) ) )  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
2019con2i 112 . . . . 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 3572 . . . . 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 15 . . . 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 452 . . . . . . . 8  |-  ( ( 0  <  B  /\  A  =  -oo )  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
243adantl 452 . . . . . . . 8  |-  ( ( B  <  0  /\  A  =  +oo )  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
2523, 24jaoi 368 . . . . . . 7  |-  ( ( ( 0  <  B  /\  A  =  -oo )  \/  ( B  <  0  /\  A  = 
+oo ) )  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
2616adantl 452 . . . . . . . 8  |-  ( ( 0  <  A  /\  B  =  -oo )  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
2712adantl 452 . . . . . . . 8  |-  ( ( A  <  0  /\  B  =  +oo )  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
2826, 27jaoi 368 . . . . . . 7  |-  ( ( ( 0  <  A  /\  B  =  -oo )  \/  ( A  <  0  /\  B  = 
+oo ) )  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
2925, 28jaoi 368 . . . . . 6  |-  ( ( ( ( 0  < 
B  /\  A  =  -oo )  \/  ( B  <  0  /\  A  =  +oo ) )  \/  ( ( 0  < 
A  /\  B  =  -oo )  \/  ( A  <  0  /\  B  =  +oo ) ) )  ->  -.  ( A  e.  RR  /\  B  e.  RR ) )
3029con2i 112 . . . . 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 3572 . . . . 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 15 . . . 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 2315 . . 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 3580 . 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 8877 . . 3  |-  ( A  e.  RR  ->  A  e.  RR* )
36 rexr 8877 . . 3  |-  ( B  e.  RR  ->  B  e.  RR* )
37 xmulval 10552 . . 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 463 . 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 3597 . . 3  |-  if ( ( A  =  0  \/  B  =  0 ) ,  ( A  x.  B ) ,  ( A  x.  B
) )  =  ( A  x.  B )
40 oveq1 5865 . . . . . 6  |-  ( A  =  0  ->  ( A  x.  B )  =  ( 0  x.  B ) )
41 mul02lem2 8989 . . . . . . 7  |-  ( B  e.  RR  ->  (
0  x.  B )  =  0 )
4241adantl 452 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 0  x.  B
)  =  0 )
4340, 42sylan9eqr 2337 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  =  0 )  ->  ( A  x.  B )  =  0 )
44 oveq2 5866 . . . . . 6  |-  ( B  =  0  ->  ( A  x.  B )  =  ( A  x.  0 ) )
45 recn 8827 . . . . . . . 8  |-  ( A  e.  RR  ->  A  e.  CC )
4645mul01d 9011 . . . . . . 7  |-  ( A  e.  RR  ->  ( A  x.  0 )  =  0 )
4746adantr 451 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  x.  0 )  =  0 )
4844, 47sylan9eqr 2337 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  =  0 )  ->  ( A  x.  B )  =  0 )
4943, 48jaodan 760 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  =  0  \/  B  =  0 ) )  -> 
( A  x.  B
)  =  0 )
5049ifeq1da 3590 . . 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 2329 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  x.  B
)  =  if ( ( A  =  0  \/  B  =  0 ) ,  0 ,  ( A  x.  B
) ) )
5234, 38, 513eqtr4d 2325 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 357    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   ifcif 3565   class class class wbr 4023  (class class class)co 5858   RRcr 8736   0cc0 8737    x. cmul 8742    +oocpnf 8864    -oocmnf 8865   RR*cxr 8866    < clt 8867   x ecxmu 10451
This theorem is referenced by:  xmulid1  10599  xmulgt0  10603  xmulasslem3  10606  xlemul1a  10608  xlemul1  10610  xadddilem  10614  nmoix  18238  nmoi2  18239  metnrmlem3  18365  nmoleub2lem  18595  xrecex  23103  rexdiv  23109  esumcst  23436
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-xmul 10454
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