Theorem fdivmptfv 42339

Description: The function value of a quotient of two functions into the complex numbers. (Contributed by AV, 19-May-2020.)

Assertion

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

Proof of Theorem fdivmptfv

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fdivmpt 42334	. . 3 |- ( ( F : A --> CC /\ G : A --> CC /\ A e. V ) -> ( F /_f G ) = ( x e. ( G supp 0 ) |-> ( ( F ` x ) / ( G ` x ) ) ) )
2	1	adantr 481	. 2 |- ( ( ( F : A --> CC /\ G : A --> CC /\ A e. V ) /\ X e. ( G supp 0 ) ) -> ( F /_f G ) = ( x e. ( G supp 0 ) |-> ( ( F ` x ) / ( G ` x ) ) ) )
3		fveq2 6191	. . . 4 |- ( x = X -> ( F ` x ) = ( F ` X ) )
4		fveq2 6191	. . . 4 |- ( x = X -> ( G ` x ) = ( G ` X ) )
5	3, 4	oveq12d 6668	. . 3 |- ( x = X -> ( ( F ` x ) / ( G ` x ) ) = ( ( F ` X ) / ( G ` X ) ) )
6	5	adantl 482	. 2 |- ( ( ( ( F : A --> CC /\ G : A --> CC /\ A e. V ) /\ X e. ( G supp 0 ) ) /\ x = X ) -> ( ( F ` x ) / ( G ` x ) ) = ( ( F ` X ) / ( G ` X ) ) )
7		simpr 477	. 2 |- ( ( ( F : A --> CC /\ G : A --> CC /\ A e. V ) /\ X e. ( G supp 0 ) ) -> X e. ( G supp 0 ) )
8		ovexd 6680	. 2 |- ( ( ( F : A --> CC /\ G : A --> CC /\ A e. V ) /\ X e. ( G supp 0 ) ) -> ( ( F ` X ) / ( G ` X ) ) e. _V )
9	2, 6, 7, 8	fvmptd 6288	1 |- ( ( ( F : A --> CC /\ G : A --> CC /\ A e. V ) /\ X e. ( G supp 0 ) ) -> ( ( F /_f G ) ` X ) = ( ( F ` X ) / ( G ` X ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   |-> cmpt 4729   -->wf 5884   ` cfv 5888  (class class class)co 6650   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: (None)