Theorem i1faddlem 22226

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 22209	. . . . . . . . 9 |- ( F e. dom S.1 -> F : RR --> RR )
3	1, 2	syl 16	. . . . . . . 8 |- ( ph -> F : RR --> RR )
4		ffn 5737	. . . . . . . 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 22209	. . . . . . . . 9 |- ( G e. dom S.1 -> G : RR --> RR )
8	6, 7	syl 16	. . . . . . . 8 |- ( ph -> G : RR --> RR )
9		ffn 5737	. . . . . . . 8 |- ( G : RR --> RR -> G Fn RR )
10	8, 9	syl 16	. . . . . . 7 |- ( ph -> G Fn RR )
11		reex 9600	. . . . . . . 8 |- RR e. _V
12	11	a1i 11	. . . . . . 7 |- ( ph -> RR e. _V )
13		inidm 3703	. . . . . . 7 |- ( RR i^i RR ) = RR
14	5, 10, 12, 12, 13	offn 6550	. . . . . 6 |- ( ph -> ( F oF + G ) Fn RR )
15	14	adantr 465	. . . . 5 |- ( ( ph /\ A e. CC ) -> ( F oF + G ) Fn RR )
16		fniniseg 6009	. . . . 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 756	. . . . . . . 8 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F oF + G ) ` z ) = A ) ) -> z e. RR )
20		fnfvelrn 6029	. . . . . . . 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 757	. . . . . . . . . . . 12 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F oF + G ) ` z ) = A ) ) -> ( ( F oF + G ) ` z ) = A )
23		eqidd 2458	. . . . . . . . . . . . . 14 |- ( ( ph /\ z e. RR ) -> ( F ` z ) = ( F ` z ) )
24		eqidd 2458	. . . . . . . . . . . . . 14 |- ( ( ph /\ z e. RR ) -> ( G ` z ) = ( G ` z ) )
25	5, 10, 12, 12, 13, 23, 24	ofval 6548	. . . . . . . . . . . . 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 2500	. . . . . . . . . . 11 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F oF + G ) ` z ) = A ) ) -> A = ( ( F ` z ) + ( G ` z ) ) )
28	27	oveq1d 6311	. . . . . . . . . 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 9566	. . . . . . . . . . . . . 14 |- RR C_ CC
30		fss 5745	. . . . . . . . . . . . . 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 6033	. . . . . . . . . . 11 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F oF + G ) ` z ) = A ) ) -> ( F ` z ) e. CC )
34		fss 5745	. . . . . . . . . . . . . 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 6033	. . . . . . . . . . 11 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F oF + G ) ` z ) = A ) ) -> ( G ` z ) e. CC )
38	33, 37	pncand 9951	. . . . . . . . . 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 2499	. . . . . . . . 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 6009	. . . . . . . . . 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 922	. . . . . . . 8 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F oF + G ) ` z ) = A ) ) -> z e. ( `' F " { ( A - ( G ` z ) ) } ) )
44		eqidd 2458	. . . . . . . . 9 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F oF + G ) ` z ) = A ) ) -> ( G ` z ) = ( G ` z ) )
45		fniniseg 6009	. . . . . . . . . 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 922	. . . . . . . 8 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F oF + G ) ` z ) = A ) ) -> z e. ( `' G " { ( G ` z ) } ) )
48	43, 47	elind 3684	. . . . . . 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 6304	. . . . . . . . . . . 12 |- ( y = ( G ` z ) -> ( A - y ) = ( A - ( G ` z ) ) )
50	49	sneqd 4044	. . . . . . . . . . 11 |- ( y = ( G ` z ) -> { ( A - y ) } = { ( A - ( G ` z ) ) } )
51	50	imaeq2d 5347	. . . . . . . . . 10 |- ( y = ( G ` z ) -> ( `' F " { ( A - y ) } ) = ( `' F " { ( A - ( G ` z ) ) } ) )
52		sneq 4042	. . . . . . . . . . 11 |- ( y = ( G ` z ) -> { y } = { ( G ` z ) } )
53	52	imaeq2d 5347	. . . . . . . . . 10 |- ( y = ( G ` z ) -> ( `' G " { y } ) = ( `' G " { ( G ` z ) } ) )
54	51, 53	ineq12d 3697	. . . . . . . . 9 |- ( y = ( G ` z ) -> ( ( `' F " { ( A - y ) } ) i^i ( `' G " { y } ) ) = ( ( `' F " { ( A - ( G ` z ) ) } ) i^i ( `' G " { ( G ` z ) } ) ) )
55	54	eleq2d 2527	. . . . . . . 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 3210	. . . . . . 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 3683	. . . . . . 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 6009	. . . . . . . . . 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 6009	. . . . . . . . . 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 828	. . . . . . . . 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 756	. . . . . . . . . . 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 765	. . . . . . . . . . . . 13 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F ` z ) = ( A - y ) /\ ( G ` z ) = y ) ) ) -> ( F ` z ) = ( A - y ) )
71		simprrr 766	. . . . . . . . . . . . 13 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F ` z ) = ( A - y ) /\ ( G ` z ) = y ) ) ) -> ( G ` z ) = y )
72	70, 71	oveq12d 6314	. . . . . . . . . . . 12 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F ` z ) = ( A - y ) /\ ( G ` z ) = y ) ) ) -> ( ( F ` z ) + ( G ` z ) ) = ( ( A - y ) + y ) )
73		simplr 755	. . . . . . . . . . . . 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 6033	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F ` z ) = ( A - y ) /\ ( G ` z ) = y ) ) ) -> ( G ` z ) e. CC )
76	71, 75	eqeltrrd 2546	. . . . . . . . . . . . 13 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F ` z ) = ( A - y ) /\ ( G ` z ) = y ) ) ) -> y e. CC )
77	73, 76	npcand 9954	. . . . . . . . . . . 12 |- ( ( ( ph /\ A e. CC ) /\ ( z e. RR /\ ( ( F ` z ) = ( A - y ) /\ ( G ` z ) = y ) ) ) -> ( ( A - y ) + y ) = A )
78	69, 72, 77	3eqtrd 2502	. . . . . . . . . . 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 2952	. . . . 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 4337	. . 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 2454	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 1395   e. wcel 1819   E.wrex 2808   _Vcvv 3109   i^i cin 3470   C_ wss 3471   {csn 4032   U_ciun 4332   `'ccnv 5007   dom cdm 5008   ran crn 5009   "cima 5011   Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6296   oFcof 6537   CCcc 9507   RRcr 9508   + caddc 9512   - cmin 9824   S.1citg1 22150

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-po 4809  df-so 4810  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-ltxr 9650  df-sub 9826  df-sum 13521  df-itg1 22155

This theorem is referenced by:  i1fadd  22228  itg1addlem4  22232