MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ofmulrt Structured version   Visualization version   Unicode version
Theorem ofmulrt 24037
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 1062 . . . . . . . 8 |- ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) -> F : A --> CC )
2 ffn 6045 . . . . . . . 8 |- ( F : A --> CC -> F Fn A )
31, 2syl 17 . . . . . . 7 |- ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) -> F Fn A )
4 simp3 1063 . . . . . . . 8 |- ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) -> G : A --> CC )
5 ffn 6045 . . . . . . . 8 |- ( G : A --> CC -> G Fn A )
64, 5syl 17 . . . . . . 7 |- ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) -> G Fn A )
7 simp1 1061 . . . . . . 7 |- ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) -> A e. V )
8 inidm 3822 . . . . . . 7 |- ( A i^i A ) = A
9 eqidd 2623 . . . . . . 7 |- ( ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) /\ x e. A ) -> ( F ` x ) = ( F ` x ) )
10 eqidd 2623 . . . . . . 7 |- ( ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) /\ x e. A ) -> ( G ` x ) = ( G ` x ) )
113, 6, 7, 7, 8, 9, 10ofval 6906 . . . . . 6 |- ( ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) /\ x e. A ) -> ( ( F oF x. G ) ` x ) = ( ( F ` x ) x. ( G ` x ) ) )
1211eqeq1d 2624 . . . . 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 6359 . . . . . 6 |- ( ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) /\ x e. A ) -> ( F ` x ) e. CC )
144ffvelrnda 6359 . . . . . 6 |- ( ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) /\ x e. A ) -> ( G ` x ) e. CC )
1513, 14mul0ord 10677 . . . . 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 268 . . . 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 673 . . 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 6908 . . . 4 |- ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) -> ( F oF x. G ) Fn A )
19 fniniseg 6338 . . . 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 17 . . 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 6338 . . . . . 6 |- ( F Fn A -> ( x e. ( `' F " { 0 } ) <-> ( x e. A /\ ( F ` x ) = 0 ) ) )
223, 21syl 17 . . . . 5 |- ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) -> ( x e. ( `' F " { 0 } ) <-> ( x e. A /\ ( F ` x ) = 0 ) ) )
23 fniniseg 6338 . . . . . 6 |- ( G Fn A -> ( x e. ( `' G " { 0 } ) <-> ( x e. A /\ ( G ` x ) = 0 ) ) )
246, 23syl 17 . . . . 5 |- ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) -> ( x e. ( `' G " { 0 } ) <-> ( x e. A /\ ( G ` x ) = 0 ) ) )
2522, 24orbi12d 746 . . . 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 3753 . . . 4 |- ( x e. ( ( `' F " { 0 } ) u. ( `' G " { 0 } ) ) <-> ( x e. ( `' F " { 0 } ) \/ x e. ( `' G " { 0 } ) ) )
27 andi 911 . . . 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 303 . . 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 300 . 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 2620 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 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   u. cun 3572   {csn 4177   `'ccnv 5113   "cima 5117   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   oFcof 6895   CCcc 9934   0cc0 9936   x. cmul 9941
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269
This theorem is referenced by:  plyrem  24060  fta1lem  24062  vieta1lem2  24066
  Copyright terms: Public domain W3C validator