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Theorem mul02lem1 9243
Description: Lemma for mul02 9245. 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 9092 . . . . 5  |-  0  e.  RR
2 remulcl 9076 . . . . 5  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( 0  x.  A
)  e.  RR )
31, 2mpan 653 . . . 4  |-  ( A  e.  RR  ->  (
0  x.  A )  e.  RR )
4 ax-rrecex 9063 . . . 4  |-  ( ( ( 0  x.  A
)  e.  RR  /\  ( 0  x.  A
)  =/=  0 )  ->  E. y  e.  RR  ( ( 0  x.  A )  x.  y
)  =  1 )
53, 4sylan 459 . . 3  |-  ( ( A  e.  RR  /\  ( 0  x.  A
)  =/=  0 )  ->  E. y  e.  RR  ( ( 0  x.  A )  x.  y
)  =  1 )
65adantr 453 . 2  |-  ( ( ( A  e.  RR  /\  ( 0  x.  A
)  =/=  0 )  /\  B  e.  CC )  ->  E. y  e.  RR  ( ( 0  x.  A )  x.  y
)  =  1 )
7 00id 9242 . . . . 5  |-  ( 0  +  0 )  =  0
87oveq2i 6093 . . . 4  |-  ( ( ( y  x.  A
)  x.  B )  x.  ( 0  +  0 ) )  =  ( ( ( y  x.  A )  x.  B )  x.  0 )
98eqcomi 2441 . . 3  |-  ( ( ( y  x.  A
)  x.  B )  x.  0 )  =  ( ( ( y  x.  A )  x.  B )  x.  (
0  +  0 ) )
10 simprl 734 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  ( 0  x.  A )  =/=  0 )  /\  B  e.  CC )  /\  (
y  e.  RR  /\  ( ( 0  x.  A )  x.  y
)  =  1 ) )  ->  y  e.  RR )
1110recnd 9115 . . . . . 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 736 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  ( 0  x.  A )  =/=  0 )  /\  B  e.  CC )  /\  (
y  e.  RR  /\  ( ( 0  x.  A )  x.  y
)  =  1 ) )  ->  A  e.  RR )
1312recnd 9115 . . . . . 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 9109 . . . . 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 733 . . . . 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 9085 . . . . . 6  |-  0  e.  CC
17 mul32 9234 . . . . . 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 1269 . . . . 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 644 . . . 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 9235 . . . . . . . . 9  |-  ( ( y  e.  CC  /\  A  e.  CC  /\  0  e.  CC )  ->  (
( y  x.  A
)  x.  0 )  =  ( ( 0  x.  A )  x.  y ) )
2116, 20mp3an3 1269 . . . . . . . 8  |-  ( ( y  e.  CC  /\  A  e.  CC )  ->  ( ( y  x.  A )  x.  0 )  =  ( ( 0  x.  A )  x.  y ) )
2211, 13, 21syl2anc 644 . . . . . . 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 735 . . . . . . 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 2469 . . . . . 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 6097 . . . . 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 9090 . . . . . 6  |-  ( B  e.  CC  ->  (
1  x.  B )  =  B )
2726ad2antlr 709 . . . . 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 2469 . . . 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 2469 . . 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 9109 . . . . 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 9080 . . . . . 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 1272 . . . . 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 16 . . . 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 6100 . . . 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 2469 . . 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 2493 . 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 2835 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 360    = wceq 1653    e. wcel 1726    =/= wne 2600   E.wrex 2707  (class class class)co 6082   CCcc 8989   RRcr 8990   0cc0 8991   1c1 8992    + caddc 8994    x. cmul 8996
This theorem is referenced by:  mul02lem2  9244
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-resscn 9048  ax-1cn 9049  ax-icn 9050  ax-addcl 9051  ax-addrcl 9052  ax-mulcl 9053  ax-mulrcl 9054  ax-mulcom 9055  ax-addass 9056  ax-mulass 9057  ax-distr 9058  ax-i2m1 9059  ax-1ne0 9060  ax-1rid 9061  ax-rnegex 9062  ax-rrecex 9063  ax-cnre 9064  ax-pre-lttri 9065  ax-pre-lttrn 9066  ax-pre-ltadd 9067
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-po 4504  df-so 4505  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-er 6906  df-en 7111  df-dom 7112  df-sdom 7113  df-pnf 9123  df-mnf 9124  df-ltxr 9126
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