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Theorem mul02lem2 9711
Description: Lemma for mul02 9712. 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 9511 . 2  |-  1  =/=  0
2 ax-1cn 9500 . . . . . . . . 9  |-  1  e.  CC
3 mul02lem1 9710 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  ( 0  x.  A
)  =/=  0 )  /\  1  e.  CC )  ->  1  =  ( 1  +  1 ) )
42, 3mpan2 669 . . . . . . . 8  |-  ( ( A  e.  RR  /\  ( 0  x.  A
)  =/=  0 )  ->  1  =  ( 1  +  1 ) )
54eqcomd 2410 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( 0  x.  A
)  =/=  0 )  ->  ( 1  +  1 )  =  1 )
65oveq2d 6250 . . . . . 6  |-  ( ( A  e.  RR  /\  ( 0  x.  A
)  =/=  0 )  ->  ( ( _i  x.  _i )  +  ( 1  +  1 ) )  =  ( ( _i  x.  _i )  +  1 ) )
7 ax-icn 9501 . . . . . . . . 9  |-  _i  e.  CC
87, 7mulcli 9551 . . . . . . . 8  |-  ( _i  x.  _i )  e.  CC
98, 2, 2addassi 9554 . . . . . . 7  |-  ( ( ( _i  x.  _i )  +  1 )  +  1 )  =  ( ( _i  x.  _i )  +  (
1  +  1 ) )
10 ax-i2m1 9510 . . . . . . . 8  |-  ( ( _i  x.  _i )  +  1 )  =  0
1110oveq1i 6244 . . . . . . 7  |-  ( ( ( _i  x.  _i )  +  1 )  +  1 )  =  ( 0  +  1 )
129, 11eqtr3i 2433 . . . . . 6  |-  ( ( _i  x.  _i )  +  ( 1  +  1 ) )  =  ( 0  +  1 )
13 00id 9709 . . . . . . 7  |-  ( 0  +  0 )  =  0
1410, 13eqtr4i 2434 . . . . . 6  |-  ( ( _i  x.  _i )  +  1 )  =  ( 0  +  0 )
156, 12, 143eqtr3g 2466 . . . . 5  |-  ( ( A  e.  RR  /\  ( 0  x.  A
)  =/=  0 )  ->  ( 0  +  1 )  =  ( 0  +  0 ) )
16 1re 9545 . . . . . 6  |-  1  e.  RR
17 0re 9546 . . . . . 6  |-  0  e.  RR
18 readdcan 9708 . . . . . 6  |-  ( ( 1  e.  RR  /\  0  e.  RR  /\  0  e.  RR )  ->  (
( 0  +  1 )  =  ( 0  +  0 )  <->  1  = 
0 ) )
1916, 17, 17, 18mp3an 1326 . . . . 5  |-  ( ( 0  +  1 )  =  ( 0  +  0 )  <->  1  = 
0 )
2015, 19sylib 196 . . . 4  |-  ( ( A  e.  RR  /\  ( 0  x.  A
)  =/=  0 )  ->  1  =  0 )
2120ex 432 . . 3  |-  ( A  e.  RR  ->  (
( 0  x.  A
)  =/=  0  -> 
1  =  0 ) )
2221necon1d 2628 . 2  |-  ( A  e.  RR  ->  (
1  =/=  0  -> 
( 0  x.  A
)  =  0 ) )
231, 22mpi 18 1  |-  ( A  e.  RR  ->  (
0  x.  A )  =  0 )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405    e. wcel 1842    =/= wne 2598  (class class class)co 6234   CCcc 9440   RRcr 9441   0cc0 9442   1c1 9443   _ici 9444    + caddc 9445    x. cmul 9447
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530  ax-resscn 9499  ax-1cn 9500  ax-icn 9501  ax-addcl 9502  ax-addrcl 9503  ax-mulcl 9504  ax-mulrcl 9505  ax-mulcom 9506  ax-addass 9507  ax-mulass 9508  ax-distr 9509  ax-i2m1 9510  ax-1ne0 9511  ax-1rid 9512  ax-rnegex 9513  ax-rrecex 9514  ax-cnre 9515  ax-pre-lttri 9516  ax-pre-lttrn 9517  ax-pre-ltadd 9518
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-po 4743  df-so 4744  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-ov 6237  df-er 7268  df-en 7475  df-dom 7476  df-sdom 7477  df-pnf 9580  df-mnf 9581  df-ltxr 9583
This theorem is referenced by:  mul02  9712  rexmul  11434  mbfmulc2lem  22238  i1fmulc  22294  itg1mulc  22295  stoweidlem34  37166  ztprmneprm  38427  nn0sumshdiglemA  38730  nn0sumshdiglem1  38732
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