MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mul02lem2 Structured version   Unicode version

Theorem mul02lem2 9245
Description: Lemma for mul02 9246. Zero times a real is zero. (Contributed by Scott Fenton, 3-Jan-2013.)
Assertion
Ref Expression
mul02lem2  |-  ( A  e.  RR  ->  (
0  x.  A )  =  0 )

Proof of Theorem mul02lem2
StepHypRef Expression
1 ax-1ne0 9061 . 2  |-  1  =/=  0
2 ax-1cn 9050 . . . . . . . . 9  |-  1  e.  CC
3 mul02lem1 9244 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  ( 0  x.  A
)  =/=  0 )  /\  1  e.  CC )  ->  1  =  ( 1  +  1 ) )
42, 3mpan2 654 . . . . . . . 8  |-  ( ( A  e.  RR  /\  ( 0  x.  A
)  =/=  0 )  ->  1  =  ( 1  +  1 ) )
54eqcomd 2443 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( 0  x.  A
)  =/=  0 )  ->  ( 1  +  1 )  =  1 )
65oveq2d 6099 . . . . . 6  |-  ( ( A  e.  RR  /\  ( 0  x.  A
)  =/=  0 )  ->  ( ( _i  x.  _i )  +  ( 1  +  1 ) )  =  ( ( _i  x.  _i )  +  1 ) )
7 ax-icn 9051 . . . . . . . . 9  |-  _i  e.  CC
87, 7mulcli 9097 . . . . . . . 8  |-  ( _i  x.  _i )  e.  CC
98, 2, 2addassi 9100 . . . . . . 7  |-  ( ( ( _i  x.  _i )  +  1 )  +  1 )  =  ( ( _i  x.  _i )  +  (
1  +  1 ) )
10 ax-i2m1 9060 . . . . . . . 8  |-  ( ( _i  x.  _i )  +  1 )  =  0
1110oveq1i 6093 . . . . . . 7  |-  ( ( ( _i  x.  _i )  +  1 )  +  1 )  =  ( 0  +  1 )
129, 11eqtr3i 2460 . . . . . 6  |-  ( ( _i  x.  _i )  +  ( 1  +  1 ) )  =  ( 0  +  1 )
13 00id 9243 . . . . . . 7  |-  ( 0  +  0 )  =  0
1410, 13eqtr4i 2461 . . . . . 6  |-  ( ( _i  x.  _i )  +  1 )  =  ( 0  +  0 )
156, 12, 143eqtr3g 2493 . . . . 5  |-  ( ( A  e.  RR  /\  ( 0  x.  A
)  =/=  0 )  ->  ( 0  +  1 )  =  ( 0  +  0 ) )
16 1re 9092 . . . . . 6  |-  1  e.  RR
17 0re 9093 . . . . . 6  |-  0  e.  RR
18 readdcan 9242 . . . . . 6  |-  ( ( 1  e.  RR  /\  0  e.  RR  /\  0  e.  RR )  ->  (
( 0  +  1 )  =  ( 0  +  0 )  <->  1  = 
0 ) )
1916, 17, 17, 18mp3an 1280 . . . . 5  |-  ( ( 0  +  1 )  =  ( 0  +  0 )  <->  1  = 
0 )
2015, 19sylib 190 . . . 4  |-  ( ( A  e.  RR  /\  ( 0  x.  A
)  =/=  0 )  ->  1  =  0 )
2120ex 425 . . 3  |-  ( A  e.  RR  ->  (
( 0  x.  A
)  =/=  0  -> 
1  =  0 ) )
2221necon1d 2675 . 2  |-  ( A  e.  RR  ->  (
1  =/=  0  -> 
( 0  x.  A
)  =  0 ) )
231, 22mpi 17 1  |-  ( A  e.  RR  ->  (
0  x.  A )  =  0 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601  (class class class)co 6083   CCcc 8990   RRcr 8991   0cc0 8992   1c1 8993   _ici 8994    + caddc 8995    x. cmul 8997
This theorem is referenced by:  mul02  9246  rexmul  10852  mbfmulc2lem  19541  i1fmulc  19597  itg1mulc  19598  stoweidlem34  27761  cshweqrep  28296
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-po 4505  df-so 4506  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-pnf 9124  df-mnf 9125  df-ltxr 9127
  Copyright terms: Public domain W3C validator