Theorem fdivmpt 42334

Description: The quotient of two functions into the complex numbers as mapping. (Contributed by AV, 16-May-2020.)

Assertion

Ref	Expression
fdivmpt	|- ( ( F : A --> CC /\ G : A --> CC /\ A e. V ) -> ( F /_f G ) = ( x e. ( G supp 0 ) |-> ( ( F ` x ) / ( G ` x ) ) ) )

Distinct variable groups:   x, A   x, F   x, G   x, V

Proof of Theorem fdivmpt

Step	Hyp	Ref	Expression
1		fex 6490	. . . 4 |- ( ( F : A --> CC /\ A e. V ) -> F e. _V )
2	1	3adant2 1080	. . 3 |- ( ( F : A --> CC /\ G : A --> CC /\ A e. V ) -> F e. _V )
3		fex 6490	. . . 4 |- ( ( G : A --> CC /\ A e. V ) -> G e. _V )
4	3	3adant1 1079	. . 3 |- ( ( F : A --> CC /\ G : A --> CC /\ A e. V ) -> G e. _V )
5		fdivval 42333	. . . 4 |- ( ( F e. _V /\ G e. _V ) -> ( F /_f G ) = ( ( F oF / G ) |` ( G supp 0 ) ) )
6		offres 7163	. . . 4 |- ( ( F e. _V /\ G e. _V ) -> ( ( F oF / G ) |` ( G supp 0 ) ) = ( ( F |` ( G supp 0 ) ) oF / ( G |` ( G supp 0 ) ) ) )
7	5, 6	eqtrd 2656	. . 3 |- ( ( F e. _V /\ G e. _V ) -> ( F /_f G ) = ( ( F |` ( G supp 0 ) ) oF / ( G |` ( G supp 0 ) ) ) )
8	2, 4, 7	syl2anc 693	. 2 |- ( ( F : A --> CC /\ G : A --> CC /\ A e. V ) -> ( F /_f G ) = ( ( F |` ( G supp 0 ) ) oF / ( G |` ( G supp 0 ) ) ) )
9		ffn 6045	. . . . 5 |- ( F : A --> CC -> F Fn A )
10	9	3ad2ant1 1082	. . . 4 |- ( ( F : A --> CC /\ G : A --> CC /\ A e. V ) -> F Fn A )
11		suppssdm 7308	. . . . 5 |- ( G supp 0 ) C_ dom G
12		fdm 6051	. . . . . . 7 |- ( G : A --> CC -> dom G = A )
13	12	eqcomd 2628	. . . . . 6 |- ( G : A --> CC -> A = dom G )
14	13	3ad2ant2 1083	. . . . 5 |- ( ( F : A --> CC /\ G : A --> CC /\ A e. V ) -> A = dom G )
15	11, 14	syl5sseqr 3654	. . . 4 |- ( ( F : A --> CC /\ G : A --> CC /\ A e. V ) -> ( G supp 0 ) C_ A )
16		fnssres 6004	. . . 4 |- ( ( F Fn A /\ ( G supp 0 ) C_ A ) -> ( F |` ( G supp 0 ) ) Fn ( G supp 0 ) )
17	10, 15, 16	syl2anc 693	. . 3 |- ( ( F : A --> CC /\ G : A --> CC /\ A e. V ) -> ( F |` ( G supp 0 ) ) Fn ( G supp 0 ) )
18		ffn 6045	. . . . 5 |- ( G : A --> CC -> G Fn A )
19	18	3ad2ant2 1083	. . . 4 |- ( ( F : A --> CC /\ G : A --> CC /\ A e. V ) -> G Fn A )
20		fnssres 6004	. . . 4 |- ( ( G Fn A /\ ( G supp 0 ) C_ A ) -> ( G |` ( G supp 0 ) ) Fn ( G supp 0 ) )
21	19, 15, 20	syl2anc 693	. . 3 |- ( ( F : A --> CC /\ G : A --> CC /\ A e. V ) -> ( G |` ( G supp 0 ) ) Fn ( G supp 0 ) )
22		ovexd 6680	. . 3 |- ( ( F : A --> CC /\ G : A --> CC /\ A e. V ) -> ( G supp 0 ) e. _V )
23		inidm 3822	. . 3 |- ( ( G supp 0 ) i^i ( G supp 0 ) ) = ( G supp 0 )
24		fvres 6207	. . . 4 |- ( x e. ( G supp 0 ) -> ( ( F |` ( G supp 0 ) ) ` x ) = ( F ` x ) )
25	24	adantl 482	. . 3 |- ( ( ( F : A --> CC /\ G : A --> CC /\ A e. V ) /\ x e. ( G supp 0 ) ) -> ( ( F |` ( G supp 0 ) ) ` x ) = ( F ` x ) )
26		fvres 6207	. . . 4 |- ( x e. ( G supp 0 ) -> ( ( G |` ( G supp 0 ) ) ` x ) = ( G ` x ) )
27	26	adantl 482	. . 3 |- ( ( ( F : A --> CC /\ G : A --> CC /\ A e. V ) /\ x e. ( G supp 0 ) ) -> ( ( G |` ( G supp 0 ) ) ` x ) = ( G ` x ) )
28	17, 21, 22, 22, 23, 25, 27	offval 6904	. 2 |- ( ( F : A --> CC /\ G : A --> CC /\ A e. V ) -> ( ( F |` ( G supp 0 ) ) oF / ( G |` ( G supp 0 ) ) ) = ( x e. ( G supp 0 ) |-> ( ( F ` x ) / ( G ` x ) ) ) )
29	8, 28	eqtrd 2656	1 |- ( ( F : A --> CC /\ G : A --> CC /\ A e. V ) -> ( F /_f G ) = ( x e. ( G supp 0 ) |-> ( ( F ` x ) / ( G ` x ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   C_ wss 3574   |-> cmpt 4729   dom cdm 5114   |` cres 5116   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   oFcof 6895   supp csupp 7295   CCcc 9934   0cc0 9936   / cdiv 10684   /_f cfdiv 42331

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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-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-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-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-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-supp 7296  df-fdiv 42332

This theorem is referenced by:  fdivmptf  42335  refdivmptf  42336  fdivmptfv  42339  refdivmptfv  42340