Theorem fdivval 42333

Description: The quotient of two functions into the complex numbers. (Contributed by AV, 15-May-2020.)

Assertion

Ref	Expression
fdivval	|- ( ( F e. V /\ G e. W ) -> ( F /_f G ) = ( ( F oF / G ) |` ( G supp 0 ) ) )

Proof of Theorem fdivval

Dummy variables x f g are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-fdiv 42332	. . 3 |- /_f = ( f e. _V , g e. _V |-> ( ( f oF / g ) |` ( g supp 0 ) ) )
2	1	a1i 11	. 2 |- ( ( F e. V /\ G e. W ) -> /_f = ( f e. _V , g e. _V |-> ( ( f oF / g ) |` ( g supp 0 ) ) ) )
3		oveq12 6659	. . . 4 |- ( ( f = F /\ g = G ) -> ( f oF / g ) = ( F oF / G ) )
4		oveq1 6657	. . . . 5 |- ( g = G -> ( g supp 0 ) = ( G supp 0 ) )
5	4	adantl 482	. . . 4 |- ( ( f = F /\ g = G ) -> ( g supp 0 ) = ( G supp 0 ) )
6	3, 5	reseq12d 5397	. . 3 |- ( ( f = F /\ g = G ) -> ( ( f oF / g ) |` ( g supp 0 ) ) = ( ( F oF / G ) |` ( G supp 0 ) ) )
7	6	adantl 482	. 2 |- ( ( ( F e. V /\ G e. W ) /\ ( f = F /\ g = G ) ) -> ( ( f oF / g ) |` ( g supp 0 ) ) = ( ( F oF / G ) |` ( G supp 0 ) ) )
8		elex 3212	. . 3 |- ( F e. V -> F e. _V )
9	8	adantr 481	. 2 |- ( ( F e. V /\ G e. W ) -> F e. _V )
10		elex 3212	. . 3 |- ( G e. W -> G e. _V )
11	10	adantl 482	. 2 |- ( ( F e. V /\ G e. W ) -> G e. _V )
12		funmpt 5926	. . . 4 |- Fun ( x e. ( dom F i^i dom G ) |-> ( ( F ` x ) / ( G ` x ) ) )
13		offval0 42299	. . . . 5 |- ( ( F e. V /\ G e. W ) -> ( F oF / G ) = ( x e. ( dom F i^i dom G ) |-> ( ( F ` x ) / ( G ` x ) ) ) )
14	13	funeqd 5910	. . . 4 |- ( ( F e. V /\ G e. W ) -> ( Fun ( F oF / G ) <-> Fun ( x e. ( dom F i^i dom G ) |-> ( ( F ` x ) / ( G ` x ) ) ) ) )
15	12, 14	mpbiri 248	. . 3 |- ( ( F e. V /\ G e. W ) -> Fun ( F oF / G ) )
16		ovex 6678	. . 3 |- ( G supp 0 ) e. _V
17		resfunexg 6479	. . 3 |- ( ( Fun ( F oF / G ) /\ ( G supp 0 ) e. _V ) -> ( ( F oF / G ) |` ( G supp 0 ) ) e. _V )
18	15, 16, 17	sylancl 694	. 2 |- ( ( F e. V /\ G e. W ) -> ( ( F oF / G ) |` ( G supp 0 ) ) e. _V )
19	2, 7, 9, 11, 18	ovmpt2d 6788	1 |- ( ( F e. V /\ G e. W ) -> ( F /_f G ) = ( ( F oF / G ) |` ( G supp 0 ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   i^i cin 3573   |-> cmpt 4729   dom cdm 5114   |` cres 5116   Fun wfun 5882   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   oFcof 6895   supp csupp 7295   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-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-fdiv 42332

This theorem is referenced by:  fdivmpt  42334