Theorem i1faddlem 22651

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 22634	. . . . . . . . 9 |- ( F e. dom S.1 -> F : RR --> RR )
3	1, 2	syl 17	. . . . . . . 8 |- ( ph -> F : RR --> RR )
4		ffn 5728	. . . . . . . 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 22634	. . . . . . . . 9 |- ( G e. dom S.1 -> G : RR --> RR )
8	6, 7	syl 17	. . . . . . . 8 |- ( ph -> G : RR --> RR )
9		ffn 5728	. . . . . . . 8 |- ( G : RR --> RR -> G Fn RR )
10	8, 9	syl 17	. . . . . . 7 |- ( ph -> G Fn RR )
11		reex 9630	. . . . . . . 8 |- RR e. _V
12	11	a1i 11	. . . . . . 7 |- ( ph -> RR e. _V )
13		inidm 3641	. . . . . . 7 |- ( RR i^i RR ) = RR
14	5, 10, 12, 12, 13	offn 6542	. . . . . 6 |- ( ph -> ( F oF + G ) Fn RR )
15	14	adantr 467	. . . . 5 |- ( ( ph /\ A e. CC ) -> ( F oF + G ) Fn RR )
16		fniniseg 6003	. . . . 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 732	. . . . . . . 8 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F oF + G ) ` z ) = A ) ) -> G Fn RR )
19		simprl 764	. . . . . . . 8 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F oF + G ) ` z ) = A ) ) -> z e. RR )
20		fnfvelrn 6019	. . . . . . . 8 |- ( ( G Fn RR /\ z e. RR ) -> ( G ` z ) e. ran G )
21	18, 19, 20	syl2anc 667	. . . . . . 7 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F oF + G ) ` z ) = A ) ) -> ( G ` z ) e. ran G )
22		simprr 766	. . . . . . . . . . . 12 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F oF + G ) ` z ) = A ) ) -> ( ( F oF + G ) ` z ) = A )
23		eqidd 2452	. . . . . . . . . . . . . 14 |- ( ( ph /\ z e. RR ) -> ( F ` z ) = ( F ` z ) )
24		eqidd 2452	. . . . . . . . . . . . . 14 |- ( ( ph /\ z e. RR ) -> ( G ` z ) = ( G ` z ) )
25	5, 10, 12, 12, 13, 23, 24	ofval 6540	. . . . . . . . . . . . 13 |- ( ( ph /\ z e. RR ) -> ( ( F oF + G ) ` z ) = ( ( F ` z ) + ( G ` z ) ) )
26	25	ad2ant2r 753	. . . . . . . . . . . 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 2487	. . . . . . . . . . 11 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F oF + G ) ` z ) = A ) ) -> A = ( ( F ` z ) + ( G ` z ) ) )
28	27	oveq1d 6305	. . . . . . . . . 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 9596	. . . . . . . . . . . . . 14 |- RR C_ CC
30		fss 5737	. . . . . . . . . . . . . 14 |- ( ( F : RR --> RR /\ RR C_ CC ) -> F : RR --> CC )
31	3, 29, 30	sylancl 668	. . . . . . . . . . . . 13 |- ( ph -> F : RR --> CC )
32	31	ad2antrr 732	. . . . . . . . . . . 12 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F oF + G ) ` z ) = A ) ) -> F : RR --> CC )
33	32, 19	ffvelrnd 6023	. . . . . . . . . . 11 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F oF + G ) ` z ) = A ) ) -> ( F ` z ) e. CC )
34		fss 5737	. . . . . . . . . . . . . 14 |- ( ( G : RR --> RR /\ RR C_ CC ) -> G : RR --> CC )
35	8, 29, 34	sylancl 668	. . . . . . . . . . . . 13 |- ( ph -> G : RR --> CC )
36	35	ad2antrr 732	. . . . . . . . . . . 12 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F oF + G ) ` z ) = A ) ) -> G : RR --> CC )
37	36, 19	ffvelrnd 6023	. . . . . . . . . . 11 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F oF + G ) ` z ) = A ) ) -> ( G ` z ) e. CC )
38	33, 37	pncand 9987	. . . . . . . . . 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 2486	. . . . . . . . 9 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F oF + G ) ` z ) = A ) ) -> ( F ` z ) = ( A - ( G ` z ) ) )
40	5	ad2antrr 732	. . . . . . . . . 10 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F oF + G ) ` z ) = A ) ) -> F Fn RR )
41		fniniseg 6003	. . . . . . . . . 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 933	. . . . . . . 8 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F oF + G ) ` z ) = A ) ) -> z e. ( `' F " { ( A - ( G ` z ) ) } ) )
44		eqidd 2452	. . . . . . . . 9 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F oF + G ) ` z ) = A ) ) -> ( G ` z ) = ( G ` z ) )
45		fniniseg 6003	. . . . . . . . . 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 933	. . . . . . . 8 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F oF + G ) ` z ) = A ) ) -> z e. ( `' G " { ( G ` z ) } ) )
48	43, 47	elind 3618	. . . . . . 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 6298	. . . . . . . . . . . 12 |- ( y = ( G ` z ) -> ( A - y ) = ( A - ( G ` z ) ) )
50	49	sneqd 3980	. . . . . . . . . . 11 |- ( y = ( G ` z ) -> { ( A - y ) } = { ( A - ( G ` z ) ) } )
51	50	imaeq2d 5168	. . . . . . . . . 10 |- ( y = ( G ` z ) -> ( `' F " { ( A - y ) } ) = ( `' F " { ( A - ( G ` z ) ) } ) )
52		sneq 3978	. . . . . . . . . . 11 |- ( y = ( G ` z ) -> { y } = { ( G ` z ) } )
53	52	imaeq2d 5168	. . . . . . . . . 10 |- ( y = ( G ` z ) -> ( `' G " { y } ) = ( `' G " { ( G ` z ) } ) )
54	51, 53	ineq12d 3635	. . . . . . . . 9 |- ( y = ( G ` z ) -> ( ( `' F " { ( A - y ) } ) i^i ( `' G " { y } ) ) = ( ( `' F " { ( A - ( G ` z ) ) } ) i^i ( `' G " { ( G ` z ) } ) ) )
55	54	eleq2d 2514	. . . . . . . 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 3150	. . . . . . 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 667	. . . . . 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 436	. . . . 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 3617	. . . . . . 7 |- ( z e. ( ( `' F " { ( A - y ) } ) i^i ( `' G " { y } ) ) <-> ( z e. ( `' F " { ( A - y ) } ) /\ z e. ( `' G " { y } ) ) )
60	5	adantr 467	. . . . . . . . . 10 |- ( ( ph /\ A e. CC ) -> F Fn RR )
61		fniniseg 6003	. . . . . . . . . 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 467	. . . . . . . . . 10 |- ( ( ph /\ A e. CC ) -> G Fn RR )
64		fniniseg 6003	. . . . . . . . . 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 717	. . . . . . . 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 837	. . . . . . . . 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 764	. . . . . . . . . . 11 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F ` z ) = ( A - y ) /\ ( G ` z ) = y ) ) ) -> z e. RR )
69	25	ad2ant2r 753	. . . . . . . . . . . 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 774	. . . . . . . . . . . . 13 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F ` z ) = ( A - y ) /\ ( G ` z ) = y ) ) ) -> ( F ` z ) = ( A - y ) )
71		simprrr 775	. . . . . . . . . . . . 13 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F ` z ) = ( A - y ) /\ ( G ` z ) = y ) ) ) -> ( G ` z ) = y )
72	70, 71	oveq12d 6308	. . . . . . . . . . . 12 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F ` z ) = ( A - y ) /\ ( G ` z ) = y ) ) ) -> ( ( F ` z ) + ( G ` z ) ) = ( ( A - y ) + y ) )
73		simplr 762	. . . . . . . . . . . . 13 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F ` z ) = ( A - y ) /\ ( G ` z ) = y ) ) ) -> A e. CC )
74	35	ad2antrr 732	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F ` z ) = ( A - y ) /\ ( G ` z ) = y ) ) ) -> G : RR --> CC )
75	74, 68	ffvelrnd 6023	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F ` z ) = ( A - y ) /\ ( G ` z ) = y ) ) ) -> ( G ` z ) e. CC )
76	71, 75	eqeltrrd 2530	. . . . . . . . . . . . 13 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F ` z ) = ( A - y ) /\ ( G ` z ) = y ) ) ) -> y e. CC )
77	73, 76	npcand 9990	. . . . . . . . . . . 12 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F ` z ) = ( A - y ) /\ ( G ` z ) = y ) ) ) -> ( ( A - y ) + y ) = A )
78	69, 72, 77	3eqtrd 2489	. . . . . . . . . . 11 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F ` z ) = ( A - y ) /\ ( G ` z ) = y ) ) ) -> ( ( F oF + G ) ` z ) = A )
79	68, 78	jca 535	. . . . . . . . . 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 436	. . . . . . . . 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 222	. . . . . . . 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 219	. . . . . . 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 221	. . . . . 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 2882	. . . . 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 194	. . . 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 257	. . 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 4283	. . 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 267	. 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 2449	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 188   /\ wa 371   = wceq 1444   e. wcel 1887   E.wrex 2738   _Vcvv 3045   i^i cin 3403   C_ wss 3404   {csn 3968   U_ciun 4278   `'ccnv 4833   dom cdm 4834   ran crn 4835   "cima 4837   Fn wfn 5577   -->wf 5578   ` cfv 5582  (class class class)co 6290   oFcof 6529   CCcc 9537   RRcr 9538   + caddc 9542   - cmin 9860   S.1citg1 22573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-po 4755  df-so 4756  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-of 6531  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-pnf 9677  df-mnf 9678  df-ltxr 9680  df-sub 9862  df-sum 13753  df-itg1 22578

This theorem is referenced by:  i1fadd  22653  itg1addlem4  22657