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Theorem ofmulrt 19678
Description: The set of roots of a product is the union of the roots of the terms. (Contributed by Mario Carneiro, 28-Jul-2014.)
Assertion
Ref Expression
ofmulrt  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  ( `' ( F  o F  x.  G ) " {
0 } )  =  ( ( `' F " { 0 } )  u.  ( `' G " { 0 } ) ) )

Proof of Theorem ofmulrt
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 simp2 956 . . . . . . . 8  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  F : A --> CC )
2 ffn 5405 . . . . . . . 8  |-  ( F : A --> CC  ->  F  Fn  A )
31, 2syl 15 . . . . . . 7  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  F  Fn  A
)
4 simp3 957 . . . . . . . 8  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  G : A --> CC )
5 ffn 5405 . . . . . . . 8  |-  ( G : A --> CC  ->  G  Fn  A )
64, 5syl 15 . . . . . . 7  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  G  Fn  A
)
7 simp1 955 . . . . . . 7  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  A  e.  V
)
8 inidm 3391 . . . . . . 7  |-  ( A  i^i  A )  =  A
9 eqidd 2297 . . . . . . 7  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  /\  x  e.  A
)  ->  ( F `  x )  =  ( F `  x ) )
10 eqidd 2297 . . . . . . 7  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  /\  x  e.  A
)  ->  ( G `  x )  =  ( G `  x ) )
113, 6, 7, 7, 8, 9, 10ofval 6103 . . . . . 6  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  /\  x  e.  A
)  ->  ( ( F  o F  x.  G
) `  x )  =  ( ( F `
 x )  x.  ( G `  x
) ) )
1211eqeq1d 2304 . . . . 5  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  /\  x  e.  A
)  ->  ( (
( F  o F  x.  G ) `  x )  =  0  <-> 
( ( F `  x )  x.  ( G `  x )
)  =  0 ) )
13 ffvelrn 5679 . . . . . . 7  |-  ( ( F : A --> CC  /\  x  e.  A )  ->  ( F `  x
)  e.  CC )
141, 13sylan 457 . . . . . 6  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  /\  x  e.  A
)  ->  ( F `  x )  e.  CC )
15 ffvelrn 5679 . . . . . . 7  |-  ( ( G : A --> CC  /\  x  e.  A )  ->  ( G `  x
)  e.  CC )
164, 15sylan 457 . . . . . 6  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  /\  x  e.  A
)  ->  ( G `  x )  e.  CC )
1714, 16mul0ord 9434 . . . . 5  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  /\  x  e.  A
)  ->  ( (
( F `  x
)  x.  ( G `
 x ) )  =  0  <->  ( ( F `  x )  =  0  \/  ( G `  x )  =  0 ) ) )
1812, 17bitrd 244 . . . 4  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  /\  x  e.  A
)  ->  ( (
( F  o F  x.  G ) `  x )  =  0  <-> 
( ( F `  x )  =  0  \/  ( G `  x )  =  0 ) ) )
1918pm5.32da 622 . . 3  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  ( ( x  e.  A  /\  (
( F  o F  x.  G ) `  x )  =  0 )  <->  ( x  e.  A  /\  ( ( F `  x )  =  0  \/  ( G `  x )  =  0 ) ) ) )
203, 6, 7, 7, 8offn 6105 . . . 4  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  ( F  o F  x.  G )  Fn  A )
21 fniniseg 5662 . . . 4  |-  ( ( F  o F  x.  G )  Fn  A  ->  ( x  e.  ( `' ( F  o F  x.  G ) " { 0 } )  <-> 
( x  e.  A  /\  ( ( F  o F  x.  G ) `  x )  =  0 ) ) )
2220, 21syl 15 . . 3  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  ( x  e.  ( `' ( F  o F  x.  G
) " { 0 } )  <->  ( x  e.  A  /\  (
( F  o F  x.  G ) `  x )  =  0 ) ) )
23 fniniseg 5662 . . . . . 6  |-  ( F  Fn  A  ->  (
x  e.  ( `' F " { 0 } )  <->  ( x  e.  A  /\  ( F `  x )  =  0 ) ) )
243, 23syl 15 . . . . 5  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  ( x  e.  ( `' F " { 0 } )  <-> 
( x  e.  A  /\  ( F `  x
)  =  0 ) ) )
25 fniniseg 5662 . . . . . 6  |-  ( G  Fn  A  ->  (
x  e.  ( `' G " { 0 } )  <->  ( x  e.  A  /\  ( G `  x )  =  0 ) ) )
266, 25syl 15 . . . . 5  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  ( x  e.  ( `' G " { 0 } )  <-> 
( x  e.  A  /\  ( G `  x
)  =  0 ) ) )
2724, 26orbi12d 690 . . . 4  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  ( ( x  e.  ( `' F " { 0 } )  \/  x  e.  ( `' G " { 0 } ) )  <->  ( (
x  e.  A  /\  ( F `  x )  =  0 )  \/  ( x  e.  A  /\  ( G `  x
)  =  0 ) ) ) )
28 elun 3329 . . . 4  |-  ( x  e.  ( ( `' F " { 0 } )  u.  ( `' G " { 0 } ) )  <->  ( x  e.  ( `' F " { 0 } )  \/  x  e.  ( `' G " { 0 } ) ) )
29 andi 837 . . . 4  |-  ( ( x  e.  A  /\  ( ( F `  x )  =  0  \/  ( G `  x )  =  0 ) )  <->  ( (
x  e.  A  /\  ( F `  x )  =  0 )  \/  ( x  e.  A  /\  ( G `  x
)  =  0 ) ) )
3027, 28, 293bitr4g 279 . . 3  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  ( x  e.  ( ( `' F " { 0 } )  u.  ( `' G " { 0 } ) )  <->  ( x  e.  A  /\  ( ( F `  x )  =  0  \/  ( G `  x )  =  0 ) ) ) )
3119, 22, 303bitr4d 276 . 2  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  ( x  e.  ( `' ( F  o F  x.  G
) " { 0 } )  <->  x  e.  ( ( `' F " { 0 } )  u.  ( `' G " { 0 } ) ) ) )
3231eqrdv 2294 1  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  ( `' ( F  o F  x.  G ) " {
0 } )  =  ( ( `' F " { 0 } )  u.  ( `' G " { 0 } ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    u. cun 3163   {csn 3653   `'ccnv 4704   "cima 4708    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874    o Fcof 6092   CCcc 8751   0cc0 8753    x. cmul 8758
This theorem is referenced by:  plyrem  19701  fta1lem  19703  vieta1lem2  19707
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-riota 6320  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056
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