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Theorem mul02lem1 9827
Description: Lemma for mul02 9829. 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 9661 . . . . 5  |-  0  e.  RR
2 remulcl 9642 . . . . 5  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( 0  x.  A
)  e.  RR )
31, 2mpan 684 . . . 4  |-  ( A  e.  RR  ->  (
0  x.  A )  e.  RR )
4 ax-rrecex 9629 . . . 4  |-  ( ( ( 0  x.  A
)  e.  RR  /\  ( 0  x.  A
)  =/=  0 )  ->  E. y  e.  RR  ( ( 0  x.  A )  x.  y
)  =  1 )
53, 4sylan 479 . . 3  |-  ( ( A  e.  RR  /\  ( 0  x.  A
)  =/=  0 )  ->  E. y  e.  RR  ( ( 0  x.  A )  x.  y
)  =  1 )
65adantr 472 . 2  |-  ( ( ( A  e.  RR  /\  ( 0  x.  A
)  =/=  0 )  /\  B  e.  CC )  ->  E. y  e.  RR  ( ( 0  x.  A )  x.  y
)  =  1 )
7 00id 9826 . . . . 5  |-  ( 0  +  0 )  =  0
87oveq2i 6319 . . . 4  |-  ( ( ( y  x.  A
)  x.  B )  x.  ( 0  +  0 ) )  =  ( ( ( y  x.  A )  x.  B )  x.  0 )
98eqcomi 2480 . . 3  |-  ( ( ( y  x.  A
)  x.  B )  x.  0 )  =  ( ( ( y  x.  A )  x.  B )  x.  (
0  +  0 ) )
10 simprl 772 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  ( 0  x.  A )  =/=  0 )  /\  B  e.  CC )  /\  (
y  e.  RR  /\  ( ( 0  x.  A )  x.  y
)  =  1 ) )  ->  y  e.  RR )
1110recnd 9687 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  ( 0  x.  A )  =/=  0 )  /\  B  e.  CC )  /\  (
y  e.  RR  /\  ( ( 0  x.  A )  x.  y
)  =  1 ) )  ->  y  e.  CC )
12 simplll 776 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  ( 0  x.  A )  =/=  0 )  /\  B  e.  CC )  /\  (
y  e.  RR  /\  ( ( 0  x.  A )  x.  y
)  =  1 ) )  ->  A  e.  RR )
1312recnd 9687 . . . . . 6  |-  ( ( ( ( A  e.  RR  /\  ( 0  x.  A )  =/=  0 )  /\  B  e.  CC )  /\  (
y  e.  RR  /\  ( ( 0  x.  A )  x.  y
)  =  1 ) )  ->  A  e.  CC )
1411, 13mulcld 9681 . . . . 5  |-  ( ( ( ( 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 770 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  ( 0  x.  A )  =/=  0 )  /\  B  e.  CC )  /\  (
y  e.  RR  /\  ( ( 0  x.  A )  x.  y
)  =  1 ) )  ->  B  e.  CC )
16 0cn 9653 . . . . . 6  |-  0  e.  CC
17 mul32 9818 . . . . . 6  |-  ( ( ( 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 1379 . . . . 5  |-  ( ( ( 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 673 . . . 4  |-  ( ( ( ( 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 9819 . . . . . . . . 9  |-  ( ( y  e.  CC  /\  A  e.  CC  /\  0  e.  CC )  ->  (
( y  x.  A
)  x.  0 )  =  ( ( 0  x.  A )  x.  y ) )
2116, 20mp3an3 1379 . . . . . . . 8  |-  ( ( y  e.  CC  /\  A  e.  CC )  ->  ( ( y  x.  A )  x.  0 )  =  ( ( 0  x.  A )  x.  y ) )
2211, 13, 21syl2anc 673 . . . . . . 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 )  =  ( ( 0  x.  A )  x.  y
) )
23 simprr 774 . . . . . . 7  |-  ( ( ( ( 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 2505 . . . . . 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 )  =  1 )
2524oveq1d 6323 . . . . 5  |-  ( ( ( ( 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 9659 . . . . . 6  |-  ( B  e.  CC  ->  (
1  x.  B )  =  B )
2726ad2antlr 741 . . . . 5  |-  ( ( ( ( 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 2505 . . . 4  |-  ( ( ( ( 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 2505 . . 3  |-  ( ( ( ( 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 9681 . . . . 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 )  e.  CC )
31 adddi 9646 . . . . . 6  |-  ( ( ( ( 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 1382 . . . . 5  |-  ( ( ( 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 17 . . . 4  |-  ( ( ( ( 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 6326 . . . 4  |-  ( ( ( ( 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 2505 . . 3  |-  ( ( ( ( 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 2529 . 2  |-  ( ( ( ( A  e.  RR  /\  ( 0  x.  A )  =/=  0 )  /\  B  e.  CC )  /\  (
y  e.  RR  /\  ( ( 0  x.  A )  x.  y
)  =  1 ) )  ->  B  =  ( B  +  B
) )
376, 36rexlimddv 2875 1  |-  ( ( ( A  e.  RR  /\  ( 0  x.  A
)  =/=  0 )  /\  B  e.  CC )  ->  B  =  ( B  +  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376    = wceq 1452    e. wcel 1904    =/= wne 2641   E.wrex 2757  (class class class)co 6308   CCcc 9555   RRcr 9556   0cc0 9557   1c1 9558    + caddc 9560    x. cmul 9562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-po 4760  df-so 4761  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-ov 6311  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-pnf 9695  df-mnf 9696  df-ltxr 9698
This theorem is referenced by:  mul02lem2  9828
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