Theorem fwddifval 32269

Description: Calculate the value of the forward difference operator at a point. (Contributed by Scott Fenton, 18-May-2020.)

Hypotheses

Ref	Expression
fwddifval.1	|- ( ph -> A C_ CC )
fwddifval.2	|- ( ph -> F : A --> CC )
fwddifval.3	|- ( ph -> X e. A )
fwddifval.4	|- ( ph -> ( X + 1 ) e. A )

Assertion

Ref	Expression
fwddifval	|- ( ph -> ( ( _/_\ ` F ) ` X ) = ( ( F ` ( X + 1 ) ) - ( F ` X ) ) )

Proof of Theorem fwddifval

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

Step	Hyp	Ref	Expression
1		df-fwddif 32266	. . . . 5 |- _/_\ = ( f e. ( CC ^pm CC ) |-> ( x e. { y e. dom f | ( y + 1 ) e. dom f } |-> ( ( f ` ( x + 1 ) ) - ( f ` x ) ) ) )
2	1	a1i 11	. . . 4 |- ( ph -> _/_\ = ( f e. ( CC ^pm CC ) |-> ( x e. { y e. dom f | ( y + 1 ) e. dom f } |-> ( ( f ` ( x + 1 ) ) - ( f ` x ) ) ) ) )
3		dmeq 5324	. . . . . . 7 |- ( f = F -> dom f = dom F )
4	3	eleq2d 2687	. . . . . . 7 |- ( f = F -> ( ( y + 1 ) e. dom f <-> ( y + 1 ) e. dom F ) )
5	3, 4	rabeqbidv 3195	. . . . . 6 |- ( f = F -> { y e. dom f | ( y + 1 ) e. dom f } = { y e. dom F | ( y + 1 ) e. dom F } )
6		fveq1 6190	. . . . . . 7 |- ( f = F -> ( f ` ( x + 1 ) ) = ( F ` ( x + 1 ) ) )
7		fveq1 6190	. . . . . . 7 |- ( f = F -> ( f ` x ) = ( F ` x ) )
8	6, 7	oveq12d 6668	. . . . . 6 |- ( f = F -> ( ( f ` ( x + 1 ) ) - ( f ` x ) ) = ( ( F ` ( x + 1 ) ) - ( F ` x ) ) )
9	5, 8	mpteq12dv 4733	. . . . 5 |- ( f = F -> ( x e. { y e. dom f | ( y + 1 ) e. dom f } |-> ( ( f ` ( x + 1 ) ) - ( f ` x ) ) ) = ( x e. { y e. dom F | ( y + 1 ) e. dom F } |-> ( ( F ` ( x + 1 ) ) - ( F ` x ) ) ) )
10	9	adantl 482	. . . 4 |- ( ( ph /\ f = F ) -> ( x e. { y e. dom f | ( y + 1 ) e. dom f } |-> ( ( f ` ( x + 1 ) ) - ( f ` x ) ) ) = ( x e. { y e. dom F | ( y + 1 ) e. dom F } |-> ( ( F ` ( x + 1 ) ) - ( F ` x ) ) ) )
11		fwddifval.2	. . . . 5 |- ( ph -> F : A --> CC )
12		fwddifval.1	. . . . 5 |- ( ph -> A C_ CC )
13		cnex 10017	. . . . . 6 |- CC e. _V
14		elpm2r 7875	. . . . . 6 |- ( ( ( CC e. _V /\ CC e. _V ) /\ ( F : A --> CC /\ A C_ CC ) ) -> F e. ( CC ^pm CC ) )
15	13, 13, 14	mpanl12 718	. . . . 5 |- ( ( F : A --> CC /\ A C_ CC ) -> F e. ( CC ^pm CC ) )
16	11, 12, 15	syl2anc 693	. . . 4 |- ( ph -> F e. ( CC ^pm CC ) )
17		fdm 6051	. . . . . . 7 |- ( F : A --> CC -> dom F = A )
18	11, 17	syl 17	. . . . . 6 |- ( ph -> dom F = A )
19	13	a1i 11	. . . . . . 7 |- ( ph -> CC e. _V )
20	19, 12	ssexd 4805	. . . . . 6 |- ( ph -> A e. _V )
21	18, 20	eqeltrd 2701	. . . . 5 |- ( ph -> dom F e. _V )
22		rabexg 4812	. . . . 5 |- ( dom F e. _V -> { y e. dom F | ( y + 1 ) e. dom F } e. _V )
23		mptexg 6484	. . . . 5 |- ( { y e. dom F | ( y + 1 ) e. dom F } e. _V -> ( x e. { y e. dom F | ( y + 1 ) e. dom F } |-> ( ( F ` ( x + 1 ) ) - ( F ` x ) ) ) e. _V )
24	21, 22, 23	3syl 18	. . . 4 |- ( ph -> ( x e. { y e. dom F | ( y + 1 ) e. dom F } |-> ( ( F ` ( x + 1 ) ) - ( F ` x ) ) ) e. _V )
25	2, 10, 16, 24	fvmptd 6288	. . 3 |- ( ph -> ( _/_\ ` F ) = ( x e. { y e. dom F | ( y + 1 ) e. dom F } |-> ( ( F ` ( x + 1 ) ) - ( F ` x ) ) ) )
26	18	eleq2d 2687	. . . . 5 |- ( ph -> ( ( y + 1 ) e. dom F <-> ( y + 1 ) e. A ) )
27	18, 26	rabeqbidv 3195	. . . 4 |- ( ph -> { y e. dom F | ( y + 1 ) e. dom F } = { y e. A | ( y + 1 ) e. A } )
28	27	mpteq1d 4738	. . 3 |- ( ph -> ( x e. { y e. dom F | ( y + 1 ) e. dom F } |-> ( ( F ` ( x + 1 ) ) - ( F ` x ) ) ) = ( x e. { y e. A | ( y + 1 ) e. A } |-> ( ( F ` ( x + 1 ) ) - ( F ` x ) ) ) )
29	25, 28	eqtrd 2656	. 2 |- ( ph -> ( _/_\ ` F ) = ( x e. { y e. A | ( y + 1 ) e. A } |-> ( ( F ` ( x + 1 ) ) - ( F ` x ) ) ) )
30		oveq1 6657	. . . . 5 |- ( x = X -> ( x + 1 ) = ( X + 1 ) )
31	30	fveq2d 6195	. . . 4 |- ( x = X -> ( F ` ( x + 1 ) ) = ( F ` ( X + 1 ) ) )
32		fveq2 6191	. . . 4 |- ( x = X -> ( F ` x ) = ( F ` X ) )
33	31, 32	oveq12d 6668	. . 3 |- ( x = X -> ( ( F ` ( x + 1 ) ) - ( F ` x ) ) = ( ( F ` ( X + 1 ) ) - ( F ` X ) ) )
34	33	adantl 482	. 2 |- ( ( ph /\ x = X ) -> ( ( F ` ( x + 1 ) ) - ( F ` x ) ) = ( ( F ` ( X + 1 ) ) - ( F ` X ) ) )
35		fwddifval.3	. . 3 |- ( ph -> X e. A )
36		fwddifval.4	. . 3 |- ( ph -> ( X + 1 ) e. A )
37		oveq1 6657	. . . . 5 |- ( y = X -> ( y + 1 ) = ( X + 1 ) )
38	37	eleq1d 2686	. . . 4 |- ( y = X -> ( ( y + 1 ) e. A <-> ( X + 1 ) e. A ) )
39	38	elrab 3363	. . 3 |- ( X e. { y e. A | ( y + 1 ) e. A } <-> ( X e. A /\ ( X + 1 ) e. A ) )
40	35, 36, 39	sylanbrc 698	. 2 |- ( ph -> X e. { y e. A | ( y + 1 ) e. A } )
41		ovexd 6680	. 2 |- ( ph -> ( ( F ` ( X + 1 ) ) - ( F ` X ) ) e. _V )
42	29, 34, 40, 41	fvmptd 6288	1 |- ( ph -> ( ( _/_\ ` F ) ` X ) = ( ( F ` ( X + 1 ) ) - ( F ` X ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   C_ wss 3574   |-> cmpt 4729   dom cdm 5114   -->wf 5884   ` cfv 5888  (class class class)co 6650   ^pm cpm 7858   CCcc 9934   1c1 9937   + caddc 9939   - cmin 10266   _/_\ cfwddif 32265

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-cnex 9992

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-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-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-pm 7860  df-fwddif 32266

This theorem is referenced by: (None)