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Theorem mul02lem1 8988
Description: Lemma for mul02 8990. If any real does not produce  0 when multiplied by  0, then any complex is equal to double itself. (Contributed by Scott Fenton, 3-Jan-2013.)
Assertion
Ref Expression
mul02lem1  |-  ( ( ( A  e.  RR  /\  ( 0  x.  A
)  =/=  0 )  /\  B  e.  CC )  ->  B  =  ( B  +  B ) )

Proof of Theorem mul02lem1
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 0re 8838 . . . . 5  |-  0  e.  RR
2 remulcl 8822 . . . . 5  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( 0  x.  A
)  e.  RR )
31, 2mpan 651 . . . 4  |-  ( A  e.  RR  ->  (
0  x.  A )  e.  RR )
4 ax-rrecex 8809 . . . 4  |-  ( ( ( 0  x.  A
)  e.  RR  /\  ( 0  x.  A
)  =/=  0 )  ->  E. y  e.  RR  ( ( 0  x.  A )  x.  y
)  =  1 )
53, 4sylan 457 . . 3  |-  ( ( A  e.  RR  /\  ( 0  x.  A
)  =/=  0 )  ->  E. y  e.  RR  ( ( 0  x.  A )  x.  y
)  =  1 )
65adantr 451 . 2  |-  ( ( ( A  e.  RR  /\  ( 0  x.  A
)  =/=  0 )  /\  B  e.  CC )  ->  E. y  e.  RR  ( ( 0  x.  A )  x.  y
)  =  1 )
7 00id 8987 . . . . . . 7  |-  ( 0  +  0 )  =  0
87oveq2i 5869 . . . . . 6  |-  ( ( ( y  x.  A
)  x.  B )  x.  ( 0  +  0 ) )  =  ( ( ( y  x.  A )  x.  B )  x.  0 )
98eqcomi 2287 . . . . 5  |-  ( ( ( y  x.  A
)  x.  B )  x.  0 )  =  ( ( ( y  x.  A )  x.  B )  x.  (
0  +  0 ) )
10 simprl 732 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR  /\  ( 0  x.  A )  =/=  0 )  /\  B  e.  CC )  /\  (
y  e.  RR  /\  ( ( 0  x.  A )  x.  y
)  =  1 ) )  ->  y  e.  RR )
1110recnd 8861 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  ( 0  x.  A )  =/=  0 )  /\  B  e.  CC )  /\  (
y  e.  RR  /\  ( ( 0  x.  A )  x.  y
)  =  1 ) )  ->  y  e.  CC )
12 simplll 734 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR  /\  ( 0  x.  A )  =/=  0 )  /\  B  e.  CC )  /\  (
y  e.  RR  /\  ( ( 0  x.  A )  x.  y
)  =  1 ) )  ->  A  e.  RR )
1312recnd 8861 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  ( 0  x.  A )  =/=  0 )  /\  B  e.  CC )  /\  (
y  e.  RR  /\  ( ( 0  x.  A )  x.  y
)  =  1 ) )  ->  A  e.  CC )
1411, 13mulcld 8855 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  ( 0  x.  A )  =/=  0 )  /\  B  e.  CC )  /\  (
y  e.  RR  /\  ( ( 0  x.  A )  x.  y
)  =  1 ) )  ->  ( y  x.  A )  e.  CC )
15 simplr 731 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  ( 0  x.  A )  =/=  0 )  /\  B  e.  CC )  /\  (
y  e.  RR  /\  ( ( 0  x.  A )  x.  y
)  =  1 ) )  ->  B  e.  CC )
16 0cn 8831 . . . . . . . 8  |-  0  e.  CC
17 mul32 8979 . . . . . . . 8  |-  ( ( ( y  x.  A
)  e.  CC  /\  B  e.  CC  /\  0  e.  CC )  ->  (
( ( y  x.  A )  x.  B
)  x.  0 )  =  ( ( ( y  x.  A )  x.  0 )  x.  B ) )
1816, 17mp3an3 1266 . . . . . . 7  |-  ( ( ( y  x.  A
)  e.  CC  /\  B  e.  CC )  ->  ( ( ( y  x.  A )  x.  B )  x.  0 )  =  ( ( ( y  x.  A
)  x.  0 )  x.  B ) )
1914, 15, 18syl2anc 642 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  ( 0  x.  A )  =/=  0 )  /\  B  e.  CC )  /\  (
y  e.  RR  /\  ( ( 0  x.  A )  x.  y
)  =  1 ) )  ->  ( (
( y  x.  A
)  x.  B )  x.  0 )  =  ( ( ( y  x.  A )  x.  0 )  x.  B
) )
20 mul31 8980 . . . . . . . . . . 11  |-  ( ( y  e.  CC  /\  A  e.  CC  /\  0  e.  CC )  ->  (
( y  x.  A
)  x.  0 )  =  ( ( 0  x.  A )  x.  y ) )
2116, 20mp3an3 1266 . . . . . . . . . 10  |-  ( ( y  e.  CC  /\  A  e.  CC )  ->  ( ( y  x.  A )  x.  0 )  =  ( ( 0  x.  A )  x.  y ) )
2211, 13, 21syl2anc 642 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR  /\  ( 0  x.  A )  =/=  0 )  /\  B  e.  CC )  /\  (
y  e.  RR  /\  ( ( 0  x.  A )  x.  y
)  =  1 ) )  ->  ( (
y  x.  A )  x.  0 )  =  ( ( 0  x.  A )  x.  y
) )
23 simprr 733 . . . . . . . . 9  |-  ( ( ( ( A  e.  RR  /\  ( 0  x.  A )  =/=  0 )  /\  B  e.  CC )  /\  (
y  e.  RR  /\  ( ( 0  x.  A )  x.  y
)  =  1 ) )  ->  ( (
0  x.  A )  x.  y )  =  1 )
2422, 23eqtrd 2315 . . . . . . . 8  |-  ( ( ( ( A  e.  RR  /\  ( 0  x.  A )  =/=  0 )  /\  B  e.  CC )  /\  (
y  e.  RR  /\  ( ( 0  x.  A )  x.  y
)  =  1 ) )  ->  ( (
y  x.  A )  x.  0 )  =  1 )
2524oveq1d 5873 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  ( 0  x.  A )  =/=  0 )  /\  B  e.  CC )  /\  (
y  e.  RR  /\  ( ( 0  x.  A )  x.  y
)  =  1 ) )  ->  ( (
( y  x.  A
)  x.  0 )  x.  B )  =  ( 1  x.  B
) )
26 mulid2 8836 . . . . . . . 8  |-  ( B  e.  CC  ->  (
1  x.  B )  =  B )
2726ad2antlr 707 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  ( 0  x.  A )  =/=  0 )  /\  B  e.  CC )  /\  (
y  e.  RR  /\  ( ( 0  x.  A )  x.  y
)  =  1 ) )  ->  ( 1  x.  B )  =  B )
2825, 27eqtrd 2315 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  ( 0  x.  A )  =/=  0 )  /\  B  e.  CC )  /\  (
y  e.  RR  /\  ( ( 0  x.  A )  x.  y
)  =  1 ) )  ->  ( (
( y  x.  A
)  x.  0 )  x.  B )  =  B )
2919, 28eqtrd 2315 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  ( 0  x.  A )  =/=  0 )  /\  B  e.  CC )  /\  (
y  e.  RR  /\  ( ( 0  x.  A )  x.  y
)  =  1 ) )  ->  ( (
( y  x.  A
)  x.  B )  x.  0 )  =  B )
3014, 15mulcld 8855 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  ( 0  x.  A )  =/=  0 )  /\  B  e.  CC )  /\  (
y  e.  RR  /\  ( ( 0  x.  A )  x.  y
)  =  1 ) )  ->  ( (
y  x.  A )  x.  B )  e.  CC )
31 adddi 8826 . . . . . . . 8  |-  ( ( ( ( y  x.  A )  x.  B
)  e.  CC  /\  0  e.  CC  /\  0  e.  CC )  ->  (
( ( y  x.  A )  x.  B
)  x.  ( 0  +  0 ) )  =  ( ( ( ( y  x.  A
)  x.  B )  x.  0 )  +  ( ( ( y  x.  A )  x.  B )  x.  0 ) ) )
3216, 16, 31mp3an23 1269 . . . . . . 7  |-  ( ( ( y  x.  A
)  x.  B )  e.  CC  ->  (
( ( y  x.  A )  x.  B
)  x.  ( 0  +  0 ) )  =  ( ( ( ( y  x.  A
)  x.  B )  x.  0 )  +  ( ( ( y  x.  A )  x.  B )  x.  0 ) ) )
3330, 32syl 15 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  ( 0  x.  A )  =/=  0 )  /\  B  e.  CC )  /\  (
y  e.  RR  /\  ( ( 0  x.  A )  x.  y
)  =  1 ) )  ->  ( (
( y  x.  A
)  x.  B )  x.  ( 0  +  0 ) )  =  ( ( ( ( y  x.  A )  x.  B )  x.  0 )  +  ( ( ( y  x.  A )  x.  B
)  x.  0 ) ) )
3429, 29oveq12d 5876 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  ( 0  x.  A )  =/=  0 )  /\  B  e.  CC )  /\  (
y  e.  RR  /\  ( ( 0  x.  A )  x.  y
)  =  1 ) )  ->  ( (
( ( y  x.  A )  x.  B
)  x.  0 )  +  ( ( ( y  x.  A )  x.  B )  x.  0 ) )  =  ( B  +  B
) )
3533, 34eqtrd 2315 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  ( 0  x.  A )  =/=  0 )  /\  B  e.  CC )  /\  (
y  e.  RR  /\  ( ( 0  x.  A )  x.  y
)  =  1 ) )  ->  ( (
( y  x.  A
)  x.  B )  x.  ( 0  +  0 ) )  =  ( B  +  B
) )
369, 29, 353eqtr3a 2339 . . . 4  |-  ( ( ( ( A  e.  RR  /\  ( 0  x.  A )  =/=  0 )  /\  B  e.  CC )  /\  (
y  e.  RR  /\  ( ( 0  x.  A )  x.  y
)  =  1 ) )  ->  B  =  ( B  +  B
) )
3736exp32 588 . . 3  |-  ( ( ( A  e.  RR  /\  ( 0  x.  A
)  =/=  0 )  /\  B  e.  CC )  ->  ( y  e.  RR  ->  ( (
( 0  x.  A
)  x.  y )  =  1  ->  B  =  ( B  +  B ) ) ) )
3837rexlimdv 2666 . 2  |-  ( ( ( A  e.  RR  /\  ( 0  x.  A
)  =/=  0 )  /\  B  e.  CC )  ->  ( E. y  e.  RR  ( ( 0  x.  A )  x.  y )  =  1  ->  B  =  ( B  +  B ) ) )
396, 38mpd 14 1  |-  ( ( ( A  e.  RR  /\  ( 0  x.  A
)  =/=  0 )  /\  B  e.  CC )  ->  B  =  ( B  +  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742
This theorem is referenced by:  mul02lem2  8989
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-ltxr 8872
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