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Theorem ofmulrt 20191
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 958 . . . . . . . 8  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  F : A --> CC )
2 ffn 5583 . . . . . . . 8  |-  ( F : A --> CC  ->  F  Fn  A )
31, 2syl 16 . . . . . . 7  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  F  Fn  A
)
4 simp3 959 . . . . . . . 8  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  G : A --> CC )
5 ffn 5583 . . . . . . . 8  |-  ( G : A --> CC  ->  G  Fn  A )
64, 5syl 16 . . . . . . 7  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  G  Fn  A
)
7 simp1 957 . . . . . . 7  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  A  e.  V
)
8 inidm 3542 . . . . . . 7  |-  ( A  i^i  A )  =  A
9 eqidd 2436 . . . . . . 7  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  /\  x  e.  A
)  ->  ( F `  x )  =  ( F `  x ) )
10 eqidd 2436 . . . . . . 7  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  /\  x  e.  A
)  ->  ( G `  x )  =  ( G `  x ) )
113, 6, 7, 7, 8, 9, 10ofval 6306 . . . . . 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 2443 . . . . 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 ) )
131ffvelrnda 5862 . . . . . 6  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  /\  x  e.  A
)  ->  ( F `  x )  e.  CC )
144ffvelrnda 5862 . . . . . 6  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  /\  x  e.  A
)  ->  ( G `  x )  e.  CC )
1513, 14mul0ord 9664 . . . . 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 ) ) )
1612, 15bitrd 245 . . . 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 ) ) )
1716pm5.32da 623 . . 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 ) ) ) )
183, 6, 7, 7, 8offn 6308 . . . 4  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  ( F  o F  x.  G )  Fn  A )
19 fniniseg 5843 . . . 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 ) ) )
2018, 19syl 16 . . 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 ) ) )
21 fniniseg 5843 . . . . . 6  |-  ( F  Fn  A  ->  (
x  e.  ( `' F " { 0 } )  <->  ( x  e.  A  /\  ( F `  x )  =  0 ) ) )
223, 21syl 16 . . . . 5  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  ( x  e.  ( `' F " { 0 } )  <-> 
( x  e.  A  /\  ( F `  x
)  =  0 ) ) )
23 fniniseg 5843 . . . . . 6  |-  ( G  Fn  A  ->  (
x  e.  ( `' G " { 0 } )  <->  ( x  e.  A  /\  ( G `  x )  =  0 ) ) )
246, 23syl 16 . . . . 5  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  ( x  e.  ( `' G " { 0 } )  <-> 
( x  e.  A  /\  ( G `  x
)  =  0 ) ) )
2522, 24orbi12d 691 . . . 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 ) ) ) )
26 elun 3480 . . . 4  |-  ( x  e.  ( ( `' F " { 0 } )  u.  ( `' G " { 0 } ) )  <->  ( x  e.  ( `' F " { 0 } )  \/  x  e.  ( `' G " { 0 } ) ) )
27 andi 838 . . . 4  |-  ( ( x  e.  A  /\  ( ( F `  x )  =  0  \/  ( G `  x )  =  0 ) )  <->  ( (
x  e.  A  /\  ( F `  x )  =  0 )  \/  ( x  e.  A  /\  ( G `  x
)  =  0 ) ) )
2825, 26, 273bitr4g 280 . . 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 ) ) ) )
2917, 20, 283bitr4d 277 . 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 } ) ) ) )
3029eqrdv 2433 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 177    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    u. cun 3310   {csn 3806   `'ccnv 4869   "cima 4873    Fn wfn 5441   -->wf 5442   ` cfv 5446  (class class class)co 6073    o Fcof 6295   CCcc 8980   0cc0 8982    x. cmul 8987
This theorem is referenced by:  plyrem  20214  fta1lem  20216  vieta1lem2  20220
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-po 4495  df-so 4496  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-riota 6541  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286
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