Theorem i1faddlem 23460

Description: Decompose the preimage of a sum. (Contributed by Mario Carneiro, 19-Jun-2014.)

Hypotheses

Ref	Expression
i1fadd.1	|- ( ph -> F e. dom S.1 )
i1fadd.2	|- ( ph -> G e. dom S.1 )

Assertion

Ref	Expression
i1faddlem	|- ( ( ph /\ A e. CC ) -> ( `' ( F oF + G ) " { A } ) = U_ y e. ran G ( ( `' F " { ( A - y ) } ) i^i ( `' G " { y } ) ) )

Distinct variable groups:   y, A   y, F   y, G   ph, y

Proof of Theorem i1faddlem

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		i1fadd.1	. . . . . . . . 9 |- ( ph -> F e. dom S.1 )
2		i1ff 23443	. . . . . . . . 9 |- ( F e. dom S.1 -> F : RR --> RR )
3	1, 2	syl 17	. . . . . . . 8 |- ( ph -> F : RR --> RR )
4		ffn 6045	. . . . . . . 8 |- ( F : RR --> RR -> F Fn RR )
5	3, 4	syl 17	. . . . . . 7 |- ( ph -> F Fn RR )
6		i1fadd.2	. . . . . . . . 9 |- ( ph -> G e. dom S.1 )
7		i1ff 23443	. . . . . . . . 9 |- ( G e. dom S.1 -> G : RR --> RR )
8	6, 7	syl 17	. . . . . . . 8 |- ( ph -> G : RR --> RR )
9		ffn 6045	. . . . . . . 8 |- ( G : RR --> RR -> G Fn RR )
10	8, 9	syl 17	. . . . . . 7 |- ( ph -> G Fn RR )
11		reex 10027	. . . . . . . 8 |- RR e. _V
12	11	a1i 11	. . . . . . 7 |- ( ph -> RR e. _V )
13		inidm 3822	. . . . . . 7 |- ( RR i^i RR ) = RR
14	5, 10, 12, 12, 13	offn 6908	. . . . . 6 |- ( ph -> ( F oF + G ) Fn RR )
15	14	adantr 481	. . . . 5 |- ( ( ph /\ A e. CC ) -> ( F oF + G ) Fn RR )
16		fniniseg 6338	. . . . 5 |- ( ( F oF + G ) Fn RR -> ( z e. ( `' ( F oF + G ) " { A } ) <-> ( z e. RR /\ ( ( F oF + G ) ` z ) = A ) ) )
17	15, 16	syl 17	. . . 4 |- ( ( ph /\ A e. CC ) -> ( z e. ( `' ( F oF + G ) " { A } ) <-> ( z e. RR /\ ( ( F oF + G ) ` z ) = A ) ) )
18	10	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F oF + G ) ` z ) = A ) ) -> G Fn RR )
19		simprl 794	. . . . . . . 8 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F oF + G ) ` z ) = A ) ) -> z e. RR )
20		fnfvelrn 6356	. . . . . . . 8 |- ( ( G Fn RR /\ z e. RR ) -> ( G ` z ) e. ran G )
21	18, 19, 20	syl2anc 693	. . . . . . 7 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F oF + G ) ` z ) = A ) ) -> ( G ` z ) e. ran G )
22		simprr 796	. . . . . . . . . . . 12 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F oF + G ) ` z ) = A ) ) -> ( ( F oF + G ) ` z ) = A )
23		eqidd 2623	. . . . . . . . . . . . . 14 |- ( ( ph /\ z e. RR ) -> ( F ` z ) = ( F ` z ) )
24		eqidd 2623	. . . . . . . . . . . . . 14 |- ( ( ph /\ z e. RR ) -> ( G ` z ) = ( G ` z ) )
25	5, 10, 12, 12, 13, 23, 24	ofval 6906	. . . . . . . . . . . . 13 |- ( ( ph /\ z e. RR ) -> ( ( F oF + G ) ` z ) = ( ( F ` z ) + ( G ` z ) ) )
26	25	ad2ant2r 783	. . . . . . . . . . . 12 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F oF + G ) ` z ) = A ) ) -> ( ( F oF + G ) ` z ) = ( ( F ` z ) + ( G ` z ) ) )
27	22, 26	eqtr3d 2658	. . . . . . . . . . 11 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F oF + G ) ` z ) = A ) ) -> A = ( ( F ` z ) + ( G ` z ) ) )
28	27	oveq1d 6665	. . . . . . . . . 10 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F oF + G ) ` z ) = A ) ) -> ( A - ( G ` z ) ) = ( ( ( F ` z ) + ( G ` z ) ) - ( G ` z ) ) )
29		ax-resscn 9993	. . . . . . . . . . . . . 14 |- RR C_ CC
30		fss 6056	. . . . . . . . . . . . . 14 |- ( ( F : RR --> RR /\ RR C_ CC ) -> F : RR --> CC )
31	3, 29, 30	sylancl 694	. . . . . . . . . . . . 13 |- ( ph -> F : RR --> CC )
32	31	ad2antrr 762	. . . . . . . . . . . 12 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F oF + G ) ` z ) = A ) ) -> F : RR --> CC )
33	32, 19	ffvelrnd 6360	. . . . . . . . . . 11 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F oF + G ) ` z ) = A ) ) -> ( F ` z ) e. CC )
34		fss 6056	. . . . . . . . . . . . . 14 |- ( ( G : RR --> RR /\ RR C_ CC ) -> G : RR --> CC )
35	8, 29, 34	sylancl 694	. . . . . . . . . . . . 13 |- ( ph -> G : RR --> CC )
36	35	ad2antrr 762	. . . . . . . . . . . 12 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F oF + G ) ` z ) = A ) ) -> G : RR --> CC )
37	36, 19	ffvelrnd 6360	. . . . . . . . . . 11 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F oF + G ) ` z ) = A ) ) -> ( G ` z ) e. CC )
38	33, 37	pncand 10393	. . . . . . . . . 10 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F oF + G ) ` z ) = A ) ) -> ( ( ( F ` z ) + ( G ` z ) ) - ( G ` z ) ) = ( F ` z ) )
39	28, 38	eqtr2d 2657	. . . . . . . . 9 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F oF + G ) ` z ) = A ) ) -> ( F ` z ) = ( A - ( G ` z ) ) )
40	5	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F oF + G ) ` z ) = A ) ) -> F Fn RR )
41		fniniseg 6338	. . . . . . . . . 10 |- ( F Fn RR -> ( z e. ( `' F " { ( A - ( G ` z ) ) } ) <-> ( z e. RR /\ ( F ` z ) = ( A - ( G ` z ) ) ) ) )
42	40, 41	syl 17	. . . . . . . . 9 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F oF + G ) ` z ) = A ) ) -> ( z e. ( `' F " { ( A - ( G ` z ) ) } ) <-> ( z e. RR /\ ( F ` z ) = ( A - ( G ` z ) ) ) ) )
43	19, 39, 42	mpbir2and 957	. . . . . . . 8 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F oF + G ) ` z ) = A ) ) -> z e. ( `' F " { ( A - ( G ` z ) ) } ) )
44		eqidd 2623	. . . . . . . . 9 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F oF + G ) ` z ) = A ) ) -> ( G ` z ) = ( G ` z ) )
45		fniniseg 6338	. . . . . . . . . 10 |- ( G Fn RR -> ( z e. ( `' G " { ( G ` z ) } ) <-> ( z e. RR /\ ( G ` z ) = ( G ` z ) ) ) )
46	18, 45	syl 17	. . . . . . . . 9 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F oF + G ) ` z ) = A ) ) -> ( z e. ( `' G " { ( G ` z ) } ) <-> ( z e. RR /\ ( G ` z ) = ( G ` z ) ) ) )
47	19, 44, 46	mpbir2and 957	. . . . . . . 8 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F oF + G ) ` z ) = A ) ) -> z e. ( `' G " { ( G ` z ) } ) )
48	43, 47	elind 3798	. . . . . . 7 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F oF + G ) ` z ) = A ) ) -> z e. ( ( `' F " { ( A - ( G ` z ) ) } ) i^i ( `' G " { ( G ` z ) } ) ) )
49		oveq2 6658	. . . . . . . . . . . 12 |- ( y = ( G ` z ) -> ( A - y ) = ( A - ( G ` z ) ) )
50	49	sneqd 4189	. . . . . . . . . . 11 |- ( y = ( G ` z ) -> { ( A - y ) } = { ( A - ( G ` z ) ) } )
51	50	imaeq2d 5466	. . . . . . . . . 10 |- ( y = ( G ` z ) -> ( `' F " { ( A - y ) } ) = ( `' F " { ( A - ( G ` z ) ) } ) )
52		sneq 4187	. . . . . . . . . . 11 |- ( y = ( G ` z ) -> { y } = { ( G ` z ) } )
53	52	imaeq2d 5466	. . . . . . . . . 10 |- ( y = ( G ` z ) -> ( `' G " { y } ) = ( `' G " { ( G ` z ) } ) )
54	51, 53	ineq12d 3815	. . . . . . . . 9 |- ( y = ( G ` z ) -> ( ( `' F " { ( A - y ) } ) i^i ( `' G " { y } ) ) = ( ( `' F " { ( A - ( G ` z ) ) } ) i^i ( `' G " { ( G ` z ) } ) ) )
55	54	eleq2d 2687	. . . . . . . 8 |- ( y = ( G ` z ) -> ( z e. ( ( `' F " { ( A - y ) } ) i^i ( `' G " { y } ) ) <-> z e. ( ( `' F " { ( A - ( G ` z ) ) } ) i^i ( `' G " { ( G ` z ) } ) ) ) )
56	55	rspcev 3309	. . . . . . 7 |- ( ( ( G ` z ) e. ran G /\ z e. ( ( `' F " { ( A - ( G ` z ) ) } ) i^i ( `' G " { ( G ` z ) } ) ) ) -> E. y e. ran G z e. ( ( `' F " { ( A - y ) } ) i^i ( `' G " { y } ) ) )
57	21, 48, 56	syl2anc 693	. . . . . 6 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F oF + G ) ` z ) = A ) ) -> E. y e. ran G z e. ( ( `' F " { ( A - y ) } ) i^i ( `' G " { y } ) ) )
58	57	ex 450	. . . . 5 |- ( ( ph /\ A e. CC ) -> ( ( z e. RR /\ ( ( F oF + G ) ` z ) = A ) -> E. y e. ran G z e. ( ( `' F " { ( A - y ) } ) i^i ( `' G " { y } ) ) ) )
59		elin 3796	. . . . . . 7 |- ( z e. ( ( `' F " { ( A - y ) } ) i^i ( `' G " { y } ) ) <-> ( z e. ( `' F " { ( A - y ) } ) /\ z e. ( `' G " { y } ) ) )
60	5	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ A e. CC ) -> F Fn RR )
61		fniniseg 6338	. . . . . . . . . 10 |- ( F Fn RR -> ( z e. ( `' F " { ( A - y ) } ) <-> ( z e. RR /\ ( F ` z ) = ( A - y ) ) ) )
62	60, 61	syl 17	. . . . . . . . 9 |- ( ( ph /\ A e. CC ) -> ( z e. ( `' F " { ( A - y ) } ) <-> ( z e. RR /\ ( F ` z ) = ( A - y ) ) ) )
63	10	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ A e. CC ) -> G Fn RR )
64		fniniseg 6338	. . . . . . . . . 10 |- ( G Fn RR -> ( z e. ( `' G " { y } ) <-> ( z e. RR /\ ( G ` z ) = y ) ) )
65	63, 64	syl 17	. . . . . . . . 9 |- ( ( ph /\ A e. CC ) -> ( z e. ( `' G " { y } ) <-> ( z e. RR /\ ( G ` z ) = y ) ) )
66	62, 65	anbi12d 747	. . . . . . . 8 |- ( ( ph /\ A e. CC ) -> ( ( z e. ( `' F " { ( A - y ) } ) /\ z e. ( `' G " { y } ) ) <-> ( ( z e. RR /\ ( F ` z ) = ( A - y ) ) /\ ( z e. RR /\ ( G ` z ) = y ) ) ) )
67		anandi 871	. . . . . . . . 9 |- ( ( z e. RR /\ ( ( F ` z ) = ( A - y ) /\ ( G ` z ) = y ) ) <-> ( ( z e. RR /\ ( F ` z ) = ( A - y ) ) /\ ( z e. RR /\ ( G ` z ) = y ) ) )
68		simprl 794	. . . . . . . . . . 11 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F ` z ) = ( A - y ) /\ ( G ` z ) = y ) ) ) -> z e. RR )
69	25	ad2ant2r 783	. . . . . . . . . . . 12 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F ` z ) = ( A - y ) /\ ( G ` z ) = y ) ) ) -> ( ( F oF + G ) ` z ) = ( ( F ` z ) + ( G ` z ) ) )
70		simprrl 804	. . . . . . . . . . . . 13 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F ` z ) = ( A - y ) /\ ( G ` z ) = y ) ) ) -> ( F ` z ) = ( A - y ) )
71		simprrr 805	. . . . . . . . . . . . 13 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F ` z ) = ( A - y ) /\ ( G ` z ) = y ) ) ) -> ( G ` z ) = y )
72	70, 71	oveq12d 6668	. . . . . . . . . . . 12 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F ` z ) = ( A - y ) /\ ( G ` z ) = y ) ) ) -> ( ( F ` z ) + ( G ` z ) ) = ( ( A - y ) + y ) )
73		simplr 792	. . . . . . . . . . . . 13 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F ` z ) = ( A - y ) /\ ( G ` z ) = y ) ) ) -> A e. CC )
74	35	ad2antrr 762	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F ` z ) = ( A - y ) /\ ( G ` z ) = y ) ) ) -> G : RR --> CC )
75	74, 68	ffvelrnd 6360	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F ` z ) = ( A - y ) /\ ( G ` z ) = y ) ) ) -> ( G ` z ) e. CC )
76	71, 75	eqeltrrd 2702	. . . . . . . . . . . . 13 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F ` z ) = ( A - y ) /\ ( G ` z ) = y ) ) ) -> y e. CC )
77	73, 76	npcand 10396	. . . . . . . . . . . 12 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F ` z ) = ( A - y ) /\ ( G ` z ) = y ) ) ) -> ( ( A - y ) + y ) = A )
78	69, 72, 77	3eqtrd 2660	. . . . . . . . . . 11 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F ` z ) = ( A - y ) /\ ( G ` z ) = y ) ) ) -> ( ( F oF + G ) ` z ) = A )
79	68, 78	jca 554	. . . . . . . . . 10 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F ` z ) = ( A - y ) /\ ( G ` z ) = y ) ) ) -> ( z e. RR /\ ( ( F oF + G ) ` z ) = A ) )
80	79	ex 450	. . . . . . . . 9 |- ( ( ph /\ A e. CC ) -> ( ( z e. RR /\ ( ( F ` z ) = ( A - y ) /\ ( G ` z ) = y ) ) -> ( z e. RR /\ ( ( F oF + G ) ` z ) = A ) ) )
81	67, 80	syl5bir 233	. . . . . . . 8 |- ( ( ph /\ A e. CC ) -> ( ( ( z e. RR /\ ( F ` z ) = ( A - y ) ) /\ ( z e. RR /\ ( G ` z ) = y ) ) -> ( z e. RR /\ ( ( F oF + G ) ` z ) = A ) ) )
82	66, 81	sylbid 230	. . . . . . 7 |- ( ( ph /\ A e. CC ) -> ( ( z e. ( `' F " { ( A - y ) } ) /\ z e. ( `' G " { y } ) ) -> ( z e. RR /\ ( ( F oF + G ) ` z ) = A ) ) )
83	59, 82	syl5bi 232	. . . . . 6 |- ( ( ph /\ A e. CC ) -> ( z e. ( ( `' F " { ( A - y ) } ) i^i ( `' G " { y } ) ) -> ( z e. RR /\ ( ( F oF + G ) ` z ) = A ) ) )
84	83	rexlimdvw 3034	. . . . 5 |- ( ( ph /\ A e. CC ) -> ( E. y e. ran G z e. ( ( `' F " { ( A - y ) } ) i^i ( `' G " { y } ) ) -> ( z e. RR /\ ( ( F oF + G ) ` z ) = A ) ) )
85	58, 84	impbid 202	. . . 4 |- ( ( ph /\ A e. CC ) -> ( ( z e. RR /\ ( ( F oF + G ) ` z ) = A ) <-> E. y e. ran G z e. ( ( `' F " { ( A - y ) } ) i^i ( `' G " { y } ) ) ) )
86	17, 85	bitrd 268	. . 3 |- ( ( ph /\ A e. CC ) -> ( z e. ( `' ( F oF + G ) " { A } ) <-> E. y e. ran G z e. ( ( `' F " { ( A - y ) } ) i^i ( `' G " { y } ) ) ) )
87		eliun 4524	. . 3 |- ( z e. U_ y e. ran G ( ( `' F " { ( A - y ) } ) i^i ( `' G " { y } ) ) <-> E. y e. ran G z e. ( ( `' F " { ( A - y ) } ) i^i ( `' G " { y } ) ) )
88	86, 87	syl6bbr 278	. 2 |- ( ( ph /\ A e. CC ) -> ( z e. ( `' ( F oF + G ) " { A } ) <-> z e. U_ y e. ran G ( ( `' F " { ( A - y ) } ) i^i ( `' G " { y } ) ) ) )
89	88	eqrdv 2620	1 |- ( ( ph /\ A e. CC ) -> ( `' ( F oF + G ) " { A } ) = U_ y e. ran G ( ( `' F " { ( A - y ) } ) i^i ( `' G " { y } ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   _Vcvv 3200   i^i cin 3573   C_ wss 3574   {csn 4177   U_ciun 4520   `'ccnv 5113   dom cdm 5114   ran crn 5115   "cima 5117   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   oFcof 6895   CCcc 9934   RRcr 9935   + caddc 9939   - cmin 10266   S.1citg1 23384

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  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

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-ltxr 10079  df-sub 10268  df-sum 14417  df-itg1 23389

This theorem is referenced by:  i1fadd  23462  itg1addlem4  23466