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Theorem ofmulrt 22803
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  oF  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 997 . . . . . . . 8  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  F : A --> CC )
2 ffn 5737 . . . . . . . 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 998 . . . . . . . 8  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  G : A --> CC )
5 ffn 5737 . . . . . . . 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 996 . . . . . . 7  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  A  e.  V
)
8 inidm 3703 . . . . . . 7  |-  ( A  i^i  A )  =  A
9 eqidd 2458 . . . . . . 7  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  /\  x  e.  A
)  ->  ( F `  x )  =  ( F `  x ) )
10 eqidd 2458 . . . . . . 7  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  /\  x  e.  A
)  ->  ( G `  x )  =  ( G `  x ) )
113, 6, 7, 7, 8, 9, 10ofval 6548 . . . . . 6  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  /\  x  e.  A
)  ->  ( ( F  oF  x.  G
) `  x )  =  ( ( F `
 x )  x.  ( G `  x
) ) )
1211eqeq1d 2459 . . . . 5  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  /\  x  e.  A
)  ->  ( (
( F  oF  x.  G ) `  x )  =  0  <-> 
( ( F `  x )  x.  ( G `  x )
)  =  0 ) )
131ffvelrnda 6032 . . . . . 6  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  /\  x  e.  A
)  ->  ( F `  x )  e.  CC )
144ffvelrnda 6032 . . . . . 6  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  /\  x  e.  A
)  ->  ( G `  x )  e.  CC )
1513, 14mul0ord 10220 . . . . 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 253 . . . 4  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  /\  x  e.  A
)  ->  ( (
( F  oF  x.  G ) `  x )  =  0  <-> 
( ( F `  x )  =  0  \/  ( G `  x )  =  0 ) ) )
1716pm5.32da 641 . . 3  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  ( ( x  e.  A  /\  (
( F  oF  x.  G ) `  x )  =  0 )  <->  ( x  e.  A  /\  ( ( F `  x )  =  0  \/  ( G `  x )  =  0 ) ) ) )
183, 6, 7, 7, 8offn 6550 . . . 4  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  ( F  oF  x.  G )  Fn  A )
19 fniniseg 6009 . . . 4  |-  ( ( F  oF  x.  G )  Fn  A  ->  ( x  e.  ( `' ( F  oF  x.  G ) " { 0 } )  <-> 
( x  e.  A  /\  ( ( F  oF  x.  G ) `  x )  =  0 ) ) )
2018, 19syl 16 . . 3  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  ( x  e.  ( `' ( F  oF  x.  G
) " { 0 } )  <->  ( x  e.  A  /\  (
( F  oF  x.  G ) `  x )  =  0 ) ) )
21 fniniseg 6009 . . . . . 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 6009 . . . . . 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 709 . . . 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 3641 . . . 4  |-  ( x  e.  ( ( `' F " { 0 } )  u.  ( `' G " { 0 } ) )  <->  ( x  e.  ( `' F " { 0 } )  \/  x  e.  ( `' G " { 0 } ) ) )
27 andi 867 . . . 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 288 . . 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 285 . 2  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  ( x  e.  ( `' ( F  oF  x.  G
) " { 0 } )  <->  x  e.  ( ( `' F " { 0 } )  u.  ( `' G " { 0 } ) ) ) )
3029eqrdv 2454 1  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> CC )  ->  ( `' ( F  oF  x.  G ) " {
0 } )  =  ( ( `' F " { 0 } )  u.  ( `' G " { 0 } ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819    u. cun 3469   {csn 4032   `'ccnv 5007   "cima 5011    Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6296    oFcof 6537   CCcc 9507   0cc0 9509    x. cmul 9514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-po 4809  df-so 4810  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827
This theorem is referenced by:  plyrem  22826  fta1lem  22828  vieta1lem2  22832
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