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Theorem mul02lem2 9647
Description: Lemma for mul02 9648. 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 9452 . 2  |-  1  =/=  0
2 ax-1cn 9441 . . . . . . . . 9  |-  1  e.  CC
3 mul02lem1 9646 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  ( 0  x.  A
)  =/=  0 )  /\  1  e.  CC )  ->  1  =  ( 1  +  1 ) )
42, 3mpan2 671 . . . . . . . 8  |-  ( ( A  e.  RR  /\  ( 0  x.  A
)  =/=  0 )  ->  1  =  ( 1  +  1 ) )
54eqcomd 2459 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( 0  x.  A
)  =/=  0 )  ->  ( 1  +  1 )  =  1 )
65oveq2d 6206 . . . . . 6  |-  ( ( A  e.  RR  /\  ( 0  x.  A
)  =/=  0 )  ->  ( ( _i  x.  _i )  +  ( 1  +  1 ) )  =  ( ( _i  x.  _i )  +  1 ) )
7 ax-icn 9442 . . . . . . . . 9  |-  _i  e.  CC
87, 7mulcli 9492 . . . . . . . 8  |-  ( _i  x.  _i )  e.  CC
98, 2, 2addassi 9495 . . . . . . 7  |-  ( ( ( _i  x.  _i )  +  1 )  +  1 )  =  ( ( _i  x.  _i )  +  (
1  +  1 ) )
10 ax-i2m1 9451 . . . . . . . 8  |-  ( ( _i  x.  _i )  +  1 )  =  0
1110oveq1i 6200 . . . . . . 7  |-  ( ( ( _i  x.  _i )  +  1 )  +  1 )  =  ( 0  +  1 )
129, 11eqtr3i 2482 . . . . . 6  |-  ( ( _i  x.  _i )  +  ( 1  +  1 ) )  =  ( 0  +  1 )
13 00id 9645 . . . . . . 7  |-  ( 0  +  0 )  =  0
1410, 13eqtr4i 2483 . . . . . 6  |-  ( ( _i  x.  _i )  +  1 )  =  ( 0  +  0 )
156, 12, 143eqtr3g 2515 . . . . 5  |-  ( ( A  e.  RR  /\  ( 0  x.  A
)  =/=  0 )  ->  ( 0  +  1 )  =  ( 0  +  0 ) )
16 1re 9486 . . . . . 6  |-  1  e.  RR
17 0re 9487 . . . . . 6  |-  0  e.  RR
18 readdcan 9644 . . . . . 6  |-  ( ( 1  e.  RR  /\  0  e.  RR  /\  0  e.  RR )  ->  (
( 0  +  1 )  =  ( 0  +  0 )  <->  1  = 
0 ) )
1916, 17, 17, 18mp3an 1315 . . . . 5  |-  ( ( 0  +  1 )  =  ( 0  +  0 )  <->  1  = 
0 )
2015, 19sylib 196 . . . 4  |-  ( ( A  e.  RR  /\  ( 0  x.  A
)  =/=  0 )  ->  1  =  0 )
2120ex 434 . . 3  |-  ( A  e.  RR  ->  (
( 0  x.  A
)  =/=  0  -> 
1  =  0 ) )
2221necon1d 2673 . 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2644  (class class class)co 6190   CCcc 9381   RRcr 9382   0cc0 9383   1c1 9384   _ici 9385    + caddc 9386    x. cmul 9388
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-op 3982  df-uni 4190  df-br 4391  df-opab 4449  df-mpt 4450  df-id 4734  df-po 4739  df-so 4740  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-ov 6193  df-er 7201  df-en 7411  df-dom 7412  df-sdom 7413  df-pnf 9521  df-mnf 9522  df-ltxr 9524
This theorem is referenced by:  mul02  9648  rexmul  11335  mbfmulc2lem  21241  i1fmulc  21297  itg1mulc  21298  stoweidlem34  29967  ztprmneprm  30877
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