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Theorem mul02lem1 9545
Description: Lemma for mul02 9547. 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 9386 . . . . 5  |-  0  e.  RR
2 remulcl 9367 . . . . 5  |-  ( ( 0  e.  RR  /\  A  e.  RR )  ->  ( 0  x.  A
)  e.  RR )
31, 2mpan 670 . . . 4  |-  ( A  e.  RR  ->  (
0  x.  A )  e.  RR )
4 ax-rrecex 9354 . . . 4  |-  ( ( ( 0  x.  A
)  e.  RR  /\  ( 0  x.  A
)  =/=  0 )  ->  E. y  e.  RR  ( ( 0  x.  A )  x.  y
)  =  1 )
53, 4sylan 471 . . 3  |-  ( ( A  e.  RR  /\  ( 0  x.  A
)  =/=  0 )  ->  E. y  e.  RR  ( ( 0  x.  A )  x.  y
)  =  1 )
65adantr 465 . 2  |-  ( ( ( A  e.  RR  /\  ( 0  x.  A
)  =/=  0 )  /\  B  e.  CC )  ->  E. y  e.  RR  ( ( 0  x.  A )  x.  y
)  =  1 )
7 00id 9544 . . . . 5  |-  ( 0  +  0 )  =  0
87oveq2i 6102 . . . 4  |-  ( ( ( y  x.  A
)  x.  B )  x.  ( 0  +  0 ) )  =  ( ( ( y  x.  A )  x.  B )  x.  0 )
98eqcomi 2447 . . 3  |-  ( ( ( y  x.  A
)  x.  B )  x.  0 )  =  ( ( ( y  x.  A )  x.  B )  x.  (
0  +  0 ) )
10 simprl 755 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  ( 0  x.  A )  =/=  0 )  /\  B  e.  CC )  /\  (
y  e.  RR  /\  ( ( 0  x.  A )  x.  y
)  =  1 ) )  ->  y  e.  RR )
1110recnd 9412 . . . . . 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 757 . . . . . . 7  |-  ( ( ( ( A  e.  RR  /\  ( 0  x.  A )  =/=  0 )  /\  B  e.  CC )  /\  (
y  e.  RR  /\  ( ( 0  x.  A )  x.  y
)  =  1 ) )  ->  A  e.  RR )
1312recnd 9412 . . . . . 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 9406 . . . . 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 754 . . . . 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 9378 . . . . . 6  |-  0  e.  CC
17 mul32 9536 . . . . . 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 1303 . . . . 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 661 . . . 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 9537 . . . . . . . . 9  |-  ( ( y  e.  CC  /\  A  e.  CC  /\  0  e.  CC )  ->  (
( y  x.  A
)  x.  0 )  =  ( ( 0  x.  A )  x.  y ) )
2116, 20mp3an3 1303 . . . . . . . 8  |-  ( ( y  e.  CC  /\  A  e.  CC )  ->  ( ( y  x.  A )  x.  0 )  =  ( ( 0  x.  A )  x.  y ) )
2211, 13, 21syl2anc 661 . . . . . . 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 756 . . . . . . 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 2475 . . . . . 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 6106 . . . . 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 9384 . . . . . 6  |-  ( B  e.  CC  ->  (
1  x.  B )  =  B )
2726ad2antlr 726 . . . . 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 2475 . . . 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 2475 . . 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 9406 . . . . 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 9371 . . . . . 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 1306 . . . . 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 6109 . . . 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 2475 . . 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 2499 . 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 2845 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 369    = wceq 1369    e. wcel 1756    =/= wne 2606   E.wrex 2716  (class class class)co 6091   CCcc 9280   RRcr 9281   0cc0 9282   1c1 9283    + caddc 9285    x. cmul 9287
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-po 4641  df-so 4642  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-ov 6094  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-pnf 9420  df-mnf 9421  df-ltxr 9423
This theorem is referenced by:  mul02lem2  9546
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