Theorem i1faddlem 21863

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 21846	. . . . . . . . 9 |- ( F e. dom S.1 -> F : RR --> RR )
3	1, 2	syl 16	. . . . . . . 8 |- ( ph -> F : RR --> RR )
4		ffn 5731	. . . . . . . 8 |- ( F : RR --> RR -> F Fn RR )
5	3, 4	syl 16	. . . . . . 7 |- ( ph -> F Fn RR )
6		i1fadd.2	. . . . . . . . 9 |- ( ph -> G e. dom S.1 )
7		i1ff 21846	. . . . . . . . 9 |- ( G e. dom S.1 -> G : RR --> RR )
8	6, 7	syl 16	. . . . . . . 8 |- ( ph -> G : RR --> RR )
9		ffn 5731	. . . . . . . 8 |- ( G : RR --> RR -> G Fn RR )
10	8, 9	syl 16	. . . . . . 7 |- ( ph -> G Fn RR )
11		reex 9583	. . . . . . . 8 |- RR e. _V
12	11	a1i 11	. . . . . . 7 |- ( ph -> RR e. _V )
13		inidm 3707	. . . . . . 7 |- ( RR i^i RR ) = RR
14	5, 10, 12, 12, 13	offn 6535	. . . . . 6 |- ( ph -> ( F oF + G ) Fn RR )
15	14	adantr 465	. . . . 5 |- ( ( ph /\ A e. CC ) -> ( F oF + G ) Fn RR )
16		fniniseg 6002	. . . . 5 |- ( ( F oF + G ) Fn RR -> ( z e. ( `' ( F oF + G ) " { A } ) <-> ( z e. RR /\ ( ( F oF + G ) ` z ) = A ) ) )
17	15, 16	syl 16	. . . 4 |- ( ( ph /\ A e. CC ) -> ( z e. ( `' ( F oF + G ) " { A } ) <-> ( z e. RR /\ ( ( F oF + G ) ` z ) = A ) ) )
18	10	ad2antrr 725	. . . . . . . 8 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F oF + G ) ` z ) = A ) ) -> G Fn RR )
19		simprl 755	. . . . . . . 8 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F oF + G ) ` z ) = A ) ) -> z e. RR )
20		fnfvelrn 6018	. . . . . . . 8 |- ( ( G Fn RR /\ z e. RR ) -> ( G ` z ) e. ran G )
21	18, 19, 20	syl2anc 661	. . . . . . 7 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F oF + G ) ` z ) = A ) ) -> ( G ` z ) e. ran G )
22		simprr 756	. . . . . . . . . . . 12 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F oF + G ) ` z ) = A ) ) -> ( ( F oF + G ) ` z ) = A )
23		eqidd 2468	. . . . . . . . . . . . . 14 |- ( ( ph /\ z e. RR ) -> ( F ` z ) = ( F ` z ) )
24		eqidd 2468	. . . . . . . . . . . . . 14 |- ( ( ph /\ z e. RR ) -> ( G ` z ) = ( G ` z ) )
25	5, 10, 12, 12, 13, 23, 24	ofval 6533	. . . . . . . . . . . . 13 |- ( ( ph /\ z e. RR ) -> ( ( F oF + G ) ` z ) = ( ( F ` z ) + ( G ` z ) ) )
26	25	ad2ant2r 746	. . . . . . . . . . . 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 2510	. . . . . . . . . . 11 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F oF + G ) ` z ) = A ) ) -> A = ( ( F ` z ) + ( G ` z ) ) )
28	27	oveq1d 6299	. . . . . . . . . 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 9549	. . . . . . . . . . . . . 14 |- RR C_ CC
30		fss 5739	. . . . . . . . . . . . . 14 |- ( ( F : RR --> RR /\ RR C_ CC ) -> F : RR --> CC )
31	3, 29, 30	sylancl 662	. . . . . . . . . . . . 13 |- ( ph -> F : RR --> CC )
32	31	ad2antrr 725	. . . . . . . . . . . 12 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F oF + G ) ` z ) = A ) ) -> F : RR --> CC )
33	32, 19	ffvelrnd 6022	. . . . . . . . . . 11 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F oF + G ) ` z ) = A ) ) -> ( F ` z ) e. CC )
34		fss 5739	. . . . . . . . . . . . . 14 |- ( ( G : RR --> RR /\ RR C_ CC ) -> G : RR --> CC )
35	8, 29, 34	sylancl 662	. . . . . . . . . . . . 13 |- ( ph -> G : RR --> CC )
36	35	ad2antrr 725	. . . . . . . . . . . 12 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F oF + G ) ` z ) = A ) ) -> G : RR --> CC )
37	36, 19	ffvelrnd 6022	. . . . . . . . . . 11 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F oF + G ) ` z ) = A ) ) -> ( G ` z ) e. CC )
38	33, 37	pncand 9931	. . . . . . . . . 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 2509	. . . . . . . . 9 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F oF + G ) ` z ) = A ) ) -> ( F ` z ) = ( A - ( G ` z ) ) )
40	5	ad2antrr 725	. . . . . . . . . 10 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F oF + G ) ` z ) = A ) ) -> F Fn RR )
41		fniniseg 6002	. . . . . . . . . 10 |- ( F Fn RR -> ( z e. ( `' F " { ( A - ( G ` z ) ) } ) <-> ( z e. RR /\ ( F ` z ) = ( A - ( G ` z ) ) ) ) )
42	40, 41	syl 16	. . . . . . . . 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 920	. . . . . . . 8 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F oF + G ) ` z ) = A ) ) -> z e. ( `' F " { ( A - ( G ` z ) ) } ) )
44		eqidd 2468	. . . . . . . . 9 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F oF + G ) ` z ) = A ) ) -> ( G ` z ) = ( G ` z ) )
45		fniniseg 6002	. . . . . . . . . 10 |- ( G Fn RR -> ( z e. ( `' G " { ( G ` z ) } ) <-> ( z e. RR /\ ( G ` z ) = ( G ` z ) ) ) )
46	18, 45	syl 16	. . . . . . . . 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 920	. . . . . . . 8 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F oF + G ) ` z ) = A ) ) -> z e. ( `' G " { ( G ` z ) } ) )
48	43, 47	elind 3688	. . . . . . 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 6292	. . . . . . . . . . . 12 |- ( y = ( G ` z ) -> ( A - y ) = ( A - ( G ` z ) ) )
50	49	sneqd 4039	. . . . . . . . . . 11 |- ( y = ( G ` z ) -> { ( A - y ) } = { ( A - ( G ` z ) ) } )
51	50	imaeq2d 5337	. . . . . . . . . 10 |- ( y = ( G ` z ) -> ( `' F " { ( A - y ) } ) = ( `' F " { ( A - ( G ` z ) ) } ) )
52		sneq 4037	. . . . . . . . . . 11 |- ( y = ( G ` z ) -> { y } = { ( G ` z ) } )
53	52	imaeq2d 5337	. . . . . . . . . 10 |- ( y = ( G ` z ) -> ( `' G " { y } ) = ( `' G " { ( G ` z ) } ) )
54	51, 53	ineq12d 3701	. . . . . . . . 9 |- ( y = ( G ` z ) -> ( ( `' F " { ( A - y ) } ) i^i ( `' G " { y } ) ) = ( ( `' F " { ( A - ( G ` z ) ) } ) i^i ( `' G " { ( G ` z ) } ) ) )
55	54	eleq2d 2537	. . . . . . . 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 3214	. . . . . . 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 661	. . . . . 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 434	. . . . 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 3687	. . . . . . 7 |- ( z e. ( ( `' F " { ( A - y ) } ) i^i ( `' G " { y } ) ) <-> ( z e. ( `' F " { ( A - y ) } ) /\ z e. ( `' G " { y } ) ) )
60	5	adantr 465	. . . . . . . . . 10 |- ( ( ph /\ A e. CC ) -> F Fn RR )
61		fniniseg 6002	. . . . . . . . . 10 |- ( F Fn RR -> ( z e. ( `' F " { ( A - y ) } ) <-> ( z e. RR /\ ( F ` z ) = ( A - y ) ) ) )
62	60, 61	syl 16	. . . . . . . . 9 |- ( ( ph /\ A e. CC ) -> ( z e. ( `' F " { ( A - y ) } ) <-> ( z e. RR /\ ( F ` z ) = ( A - y ) ) ) )
63	10	adantr 465	. . . . . . . . . 10 |- ( ( ph /\ A e. CC ) -> G Fn RR )
64		fniniseg 6002	. . . . . . . . . 10 |- ( G Fn RR -> ( z e. ( `' G " { y } ) <-> ( z e. RR /\ ( G ` z ) = y ) ) )
65	63, 64	syl 16	. . . . . . . . 9 |- ( ( ph /\ A e. CC ) -> ( z e. ( `' G " { y } ) <-> ( z e. RR /\ ( G ` z ) = y ) ) )
66	62, 65	anbi12d 710	. . . . . . . 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 826	. . . . . . . . 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 755	. . . . . . . . . . 11 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F ` z ) = ( A - y ) /\ ( G ` z ) = y ) ) ) -> z e. RR )
69	25	ad2ant2r 746	. . . . . . . . . . . 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 763	. . . . . . . . . . . . 13 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F ` z ) = ( A - y ) /\ ( G ` z ) = y ) ) ) -> ( F ` z ) = ( A - y ) )
71		simprrr 764	. . . . . . . . . . . . 13 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F ` z ) = ( A - y ) /\ ( G ` z ) = y ) ) ) -> ( G ` z ) = y )
72	70, 71	oveq12d 6302	. . . . . . . . . . . 12 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F ` z ) = ( A - y ) /\ ( G ` z ) = y ) ) ) -> ( ( F ` z ) + ( G ` z ) ) = ( ( A - y ) + y ) )
73		simplr 754	. . . . . . . . . . . . 13 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F ` z ) = ( A - y ) /\ ( G ` z ) = y ) ) ) -> A e. CC )
74	35	ad2antrr 725	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F ` z ) = ( A - y ) /\ ( G ` z ) = y ) ) ) -> G : RR --> CC )
75	74, 68	ffvelrnd 6022	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F ` z ) = ( A - y ) /\ ( G ` z ) = y ) ) ) -> ( G ` z ) e. CC )
76	71, 75	eqeltrrd 2556	. . . . . . . . . . . . 13 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F ` z ) = ( A - y ) /\ ( G ` z ) = y ) ) ) -> y e. CC )
77	73, 76	npcand 9934	. . . . . . . . . . . 12 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F ` z ) = ( A - y ) /\ ( G ` z ) = y ) ) ) -> ( ( A - y ) + y ) = A )
78	69, 72, 77	3eqtrd 2512	. . . . . . . . . . 11 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F ` z ) = ( A - y ) /\ ( G ` z ) = y ) ) ) -> ( ( F oF + G ) ` z ) = A )
79	68, 78	jca 532	. . . . . . . . . 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 434	. . . . . . . . 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 218	. . . . . . . 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 215	. . . . . . 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 217	. . . . . 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 2958	. . . . 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 191	. . . 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 253	. . 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 4330	. . 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 263	. 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 2464	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 184   /\ wa 369   = wceq 1379   e. wcel 1767   E.wrex 2815   _Vcvv 3113   i^i cin 3475   C_ wss 3476   {csn 4027   U_ciun 4325   `'ccnv 4998   dom cdm 4999   ran crn 5000   "cima 5002   Fn wfn 5583   -->wf 5584   ` cfv 5588  (class class class)co 6284   oFcof 6522   CCcc 9490   RRcr 9491   + caddc 9495   - cmin 9805   S.1citg1 21787

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-of 6524  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-pnf 9630  df-mnf 9631  df-ltxr 9633  df-sub 9807  df-sum 13472  df-itg1 21792

This theorem is referenced by:  i1fadd  21865  itg1addlem4  21869