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Theorem mul02lem2 9542
Description: Lemma for mul02 9543. 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 9347 . 2  |-  1  =/=  0
2 ax-1cn 9336 . . . . . . . . 9  |-  1  e.  CC
3 mul02lem1 9541 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  ( 0  x.  A
)  =/=  0 )  /\  1  e.  CC )  ->  1  =  ( 1  +  1 ) )
42, 3mpan2 666 . . . . . . . 8  |-  ( ( A  e.  RR  /\  ( 0  x.  A
)  =/=  0 )  ->  1  =  ( 1  +  1 ) )
54eqcomd 2446 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( 0  x.  A
)  =/=  0 )  ->  ( 1  +  1 )  =  1 )
65oveq2d 6106 . . . . . 6  |-  ( ( A  e.  RR  /\  ( 0  x.  A
)  =/=  0 )  ->  ( ( _i  x.  _i )  +  ( 1  +  1 ) )  =  ( ( _i  x.  _i )  +  1 ) )
7 ax-icn 9337 . . . . . . . . 9  |-  _i  e.  CC
87, 7mulcli 9387 . . . . . . . 8  |-  ( _i  x.  _i )  e.  CC
98, 2, 2addassi 9390 . . . . . . 7  |-  ( ( ( _i  x.  _i )  +  1 )  +  1 )  =  ( ( _i  x.  _i )  +  (
1  +  1 ) )
10 ax-i2m1 9346 . . . . . . . 8  |-  ( ( _i  x.  _i )  +  1 )  =  0
1110oveq1i 6100 . . . . . . 7  |-  ( ( ( _i  x.  _i )  +  1 )  +  1 )  =  ( 0  +  1 )
129, 11eqtr3i 2463 . . . . . 6  |-  ( ( _i  x.  _i )  +  ( 1  +  1 ) )  =  ( 0  +  1 )
13 00id 9540 . . . . . . 7  |-  ( 0  +  0 )  =  0
1410, 13eqtr4i 2464 . . . . . 6  |-  ( ( _i  x.  _i )  +  1 )  =  ( 0  +  0 )
156, 12, 143eqtr3g 2496 . . . . 5  |-  ( ( A  e.  RR  /\  ( 0  x.  A
)  =/=  0 )  ->  ( 0  +  1 )  =  ( 0  +  0 ) )
16 1re 9381 . . . . . 6  |-  1  e.  RR
17 0re 9382 . . . . . 6  |-  0  e.  RR
18 readdcan 9539 . . . . . 6  |-  ( ( 1  e.  RR  /\  0  e.  RR  /\  0  e.  RR )  ->  (
( 0  +  1 )  =  ( 0  +  0 )  <->  1  = 
0 ) )
1916, 17, 17, 18mp3an 1309 . . . . 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 2678 . 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 1364    e. wcel 1761    =/= wne 2604  (class class class)co 6090   CCcc 9276   RRcr 9277   0cc0 9278   1c1 9279   _ici 9280    + caddc 9281    x. cmul 9283
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-po 4637  df-so 4638  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-ov 6093  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-pnf 9416  df-mnf 9417  df-ltxr 9419
This theorem is referenced by:  mul02  9543  rexmul  11230  mbfmulc2lem  21084  i1fmulc  21140  itg1mulc  21141  stoweidlem34  29754  ztprmneprm  30663
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