Theorem xmul01 12097

Description: Extended real version of mul01 10215. (Contributed by Mario Carneiro, 20-Aug-2015.)

Assertion

Ref	Expression
xmul01	|- ( A e. RR* -> ( A xe 0 ) = 0 )

Proof of Theorem xmul01

Step	Hyp	Ref	Expression
1		0xr 10086	. . 3 |- 0 e. RR*
2		xmulval 12056	. . 3 |- ( ( A e. RR* /\ 0 e. RR* ) -> ( A xe 0 ) = if ( ( A = 0 \/ 0 = 0 ) , 0 , if ( ( ( ( 0 < 0 /\ A = +oo ) \/ ( 0 < 0 /\ A = -oo ) ) \/ ( ( 0 < A /\ 0 = +oo ) \/ ( A < 0 /\ 0 = -oo ) ) ) , +oo , if ( ( ( ( 0 < 0 /\ A = -oo ) \/ ( 0 < 0 /\ A = +oo ) ) \/ ( ( 0 < A /\ 0 = -oo ) \/ ( A < 0 /\ 0 = +oo ) ) ) , -oo , ( A x. 0 ) ) ) ) )
3	1, 2	mpan2 707	. 2 |- ( A e. RR* -> ( A xe 0 ) = if ( ( A = 0 \/ 0 = 0 ) , 0 , if ( ( ( ( 0 < 0 /\ A = +oo ) \/ ( 0 < 0 /\ A = -oo ) ) \/ ( ( 0 < A /\ 0 = +oo ) \/ ( A < 0 /\ 0 = -oo ) ) ) , +oo , if ( ( ( ( 0 < 0 /\ A = -oo ) \/ ( 0 < 0 /\ A = +oo ) ) \/ ( ( 0 < A /\ 0 = -oo ) \/ ( A < 0 /\ 0 = +oo ) ) ) , -oo , ( A x. 0 ) ) ) ) )
4		eqid 2622	. . . 4 |- 0 = 0
5	4	olci 406	. . 3 |- ( A = 0 \/ 0 = 0 )
6	5	iftruei 4093	. 2 |- if ( ( A = 0 \/ 0 = 0 ) , 0 , if ( ( ( ( 0 < 0 /\ A = +oo ) \/ ( 0 < 0 /\ A = -oo ) ) \/ ( ( 0 < A /\ 0 = +oo ) \/ ( A < 0 /\ 0 = -oo ) ) ) , +oo , if ( ( ( ( 0 < 0 /\ A = -oo ) \/ ( 0 < 0 /\ A = +oo ) ) \/ ( ( 0 < A /\ 0 = -oo ) \/ ( A < 0 /\ 0 = +oo ) ) ) , -oo , ( A x. 0 ) ) ) ) = 0
7	3, 6	syl6eq 2672	1 |- ( A e. RR* -> ( A xe 0 ) = 0 )

Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   ifcif 4086   class class class wbr 4653  (class class class)co 6650   0cc0 9936   x. cmul 9941   +oocpnf 10071   -oocmnf 10072   RR*cxr 10073   < clt 10074   xecxmu 11945

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-i2m1 10004  ax-1ne0 10005  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-pnf 10076  df-mnf 10077  df-xr 10078  df-xmul 11948

This theorem is referenced by:  xmul02  12098  xmulge0  12114  xmulass  12117  xlemul1a  12118  xadddilem  12124  xadddi2  12127  psmetge0  22117  xmetge0  22149  nmoix  22533  xrge0mulc1cn  29987  esumcst  30125