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Theorem mul02lem1 9033
Description: Lemma for mul02 9035. 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 8883 . . . . 5  |-  0  e.  RR
2 remulcl 8867 . . . . 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 8854 . . . 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 9032 . . . . . . 7  |-  ( 0  +  0 )  =  0
87oveq2i 5911 . . . . . 6  |-  ( ( ( y  x.  A
)  x.  B )  x.  ( 0  +  0 ) )  =  ( ( ( y  x.  A )  x.  B )  x.  0 )
98eqcomi 2320 . . . . 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 8906 . . . . . . . 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 8906 . . . . . . . 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 8900 . . . . . . 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 8876 . . . . . . . 8  |-  0  e.  CC
17 mul32 9024 . . . . . . . 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 9025 . . . . . . . . . . 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 2348 . . . . . . . 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 5915 . . . . . . 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 8881 . . . . . . . 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 2348 . . . . . 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 2348 . . . . 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 8900 . . . . . . 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 8871 . . . . . . . 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 5918 . . . . . 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 2348 . . . . 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 2372 . . . 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 2700 . 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 1633    e. wcel 1701    =/= wne 2479   E.wrex 2578  (class class class)co 5900   CCcc 8780   RRcr 8781   0cc0 8782   1c1 8783    + caddc 8785    x. cmul 8787
This theorem is referenced by:  mul02lem2  9034
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858
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 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-po 4351  df-so 4352  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-er 6702  df-en 6907  df-dom 6908  df-sdom 6909  df-pnf 8914  df-mnf 8915  df-ltxr 8917
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