Theorem vitalilem2 23378

Description: Lemma for vitali 23382. (Contributed by Mario Carneiro, 16-Jun-2014.)

Hypotheses

Ref	Expression
vitali.1	|- .~ = { <. x , y >. | ( ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] 1 ) ) /\ ( x - y ) e. QQ ) }
vitali.2	|- S = ( ( 0 [,] 1 ) /. .~ )
vitali.3	|- ( ph -> F Fn S )
vitali.4	|- ( ph -> A. z e. S ( z =/= (/) -> ( F ` z ) e. z ) )
vitali.5	|- ( ph -> G : NN -1-1-onto-> ( QQ i^i ( -u 1 [,] 1 ) ) )
vitali.6	|- T = ( n e. NN |-> { s e. RR | ( s - ( G ` n ) ) e. ran F } )
vitali.7	|- ( ph -> -. ran F e. ( ~P RR \ dom vol ) )

Assertion

Ref	Expression
vitalilem2	|- ( ph -> ( ran F C_ ( 0 [,] 1 ) /\ ( 0 [,] 1 ) C_ U_ m e. NN ( T ` m ) /\ U_ m e. NN ( T ` m ) C_ ( -u 1 [,] 2 ) ) )

Distinct variable groups:   m, n, s, x, y, z, G   ph, m, n, x, z   z, S   T, m, x   m, F, n, s, x, y, z   .~ , m, n, s, x, y, z

Allowed substitution hints:   ph( y, s)   S( x, y, m, n, s)   T( y, z, n, s)

Proof of Theorem vitalilem2

Dummy variables v w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vitali.3	. . . 4 |- ( ph -> F Fn S )
2		vitali.4	. . . . 5 |- ( ph -> A. z e. S ( z =/= (/) -> ( F ` z ) e. z ) )
3		vitali.2	. . . . . . . . 9 |- S = ( ( 0 [,] 1 ) /. .~ )
4		neeq1 2856	. . . . . . . . 9 |- ( [ v ] .~ = z -> ( [ v ] .~ =/= (/) <-> z =/= (/) ) )
5		vitali.1	. . . . . . . . . . . . . 14 |- .~ = { <. x , y >. | ( ( x e. ( 0 [,] 1 ) /\ y e. ( 0 [,] 1 ) ) /\ ( x - y ) e. QQ ) }
6	5	vitalilem1 23376	. . . . . . . . . . . . 13 |- .~ Er ( 0 [,] 1 )
7		erdm 7752	. . . . . . . . . . . . 13 |- ( .~ Er ( 0 [,] 1 ) -> dom .~ = ( 0 [,] 1 ) )
8	6, 7	ax-mp 5	. . . . . . . . . . . 12 |- dom .~ = ( 0 [,] 1 )
9	8	eleq2i 2693	. . . . . . . . . . 11 |- ( v e. dom .~ <-> v e. ( 0 [,] 1 ) )
10		ecdmn0 7789	. . . . . . . . . . 11 |- ( v e. dom .~ <-> [ v ] .~ =/= (/) )
11	9, 10	bitr3i 266	. . . . . . . . . 10 |- ( v e. ( 0 [,] 1 ) <-> [ v ] .~ =/= (/) )
12	11	biimpi 206	. . . . . . . . 9 |- ( v e. ( 0 [,] 1 ) -> [ v ] .~ =/= (/) )
13	3, 4, 12	ectocl 7815	. . . . . . . 8 |- ( z e. S -> z =/= (/) )
14	13	adantl 482	. . . . . . 7 |- ( ( ph /\ z e. S ) -> z =/= (/) )
15		sseq1 3626	. . . . . . . . . 10 |- ( [ w ] .~ = z -> ( [ w ] .~ C_ ( 0 [,] 1 ) <-> z C_ ( 0 [,] 1 ) ) )
16	6	a1i 11	. . . . . . . . . . 11 |- ( w e. ( 0 [,] 1 ) -> .~ Er ( 0 [,] 1 ) )
17	16	ecss 7788	. . . . . . . . . 10 |- ( w e. ( 0 [,] 1 ) -> [ w ] .~ C_ ( 0 [,] 1 ) )
18	3, 15, 17	ectocl 7815	. . . . . . . . 9 |- ( z e. S -> z C_ ( 0 [,] 1 ) )
19	18	adantl 482	. . . . . . . 8 |- ( ( ph /\ z e. S ) -> z C_ ( 0 [,] 1 ) )
20	19	sseld 3602	. . . . . . 7 |- ( ( ph /\ z e. S ) -> ( ( F ` z ) e. z -> ( F ` z ) e. ( 0 [,] 1 ) ) )
21	14, 20	embantd 59	. . . . . 6 |- ( ( ph /\ z e. S ) -> ( ( z =/= (/) -> ( F ` z ) e. z ) -> ( F ` z ) e. ( 0 [,] 1 ) ) )
22	21	ralimdva 2962	. . . . 5 |- ( ph -> ( A. z e. S ( z =/= (/) -> ( F ` z ) e. z ) -> A. z e. S ( F ` z ) e. ( 0 [,] 1 ) ) )
23	2, 22	mpd 15	. . . 4 |- ( ph -> A. z e. S ( F ` z ) e. ( 0 [,] 1 ) )
24		ffnfv 6388	. . . 4 |- ( F : S --> ( 0 [,] 1 ) <-> ( F Fn S /\ A. z e. S ( F ` z ) e. ( 0 [,] 1 ) ) )
25	1, 23, 24	sylanbrc 698	. . 3 |- ( ph -> F : S --> ( 0 [,] 1 ) )
26		frn 6053	. . 3 |- ( F : S --> ( 0 [,] 1 ) -> ran F C_ ( 0 [,] 1 ) )
27	25, 26	syl 17	. 2 |- ( ph -> ran F C_ ( 0 [,] 1 ) )
28		vitali.5	. . . . . . . . 9 |- ( ph -> G : NN -1-1-onto-> ( QQ i^i ( -u 1 [,] 1 ) ) )
29	28	adantr 481	. . . . . . . 8 |- ( ( ph /\ v e. ( 0 [,] 1 ) ) -> G : NN -1-1-onto-> ( QQ i^i ( -u 1 [,] 1 ) ) )
30		f1ocnv 6149	. . . . . . . 8 |- ( G : NN -1-1-onto-> ( QQ i^i ( -u 1 [,] 1 ) ) -> `' G : ( QQ i^i ( -u 1 [,] 1 ) ) -1-1-onto-> NN )
31		f1of 6137	. . . . . . . 8 |- ( `' G : ( QQ i^i ( -u 1 [,] 1 ) ) -1-1-onto-> NN -> `' G : ( QQ i^i ( -u 1 [,] 1 ) ) --> NN )
32	29, 30, 31	3syl 18	. . . . . . 7 |- ( ( ph /\ v e. ( 0 [,] 1 ) ) -> `' G : ( QQ i^i ( -u 1 [,] 1 ) ) --> NN )
33		ovex 6678	. . . . . . . . . . . . . . 15 |- ( 0 [,] 1 ) e. _V
34		erex 7766	. . . . . . . . . . . . . . 15 |- ( .~ Er ( 0 [,] 1 ) -> ( ( 0 [,] 1 ) e. _V -> .~ e. _V ) )
35	6, 33, 34	mp2 9	. . . . . . . . . . . . . 14 |- .~ e. _V
36	35	ecelqsi 7803	. . . . . . . . . . . . 13 |- ( v e. ( 0 [,] 1 ) -> [ v ] .~ e. ( ( 0 [,] 1 ) /. .~ ) )
37	36	adantl 482	. . . . . . . . . . . 12 |- ( ( ph /\ v e. ( 0 [,] 1 ) ) -> [ v ] .~ e. ( ( 0 [,] 1 ) /. .~ ) )
38	37, 3	syl6eleqr 2712	. . . . . . . . . . 11 |- ( ( ph /\ v e. ( 0 [,] 1 ) ) -> [ v ] .~ e. S )
39	2	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ v e. ( 0 [,] 1 ) ) -> A. z e. S ( z =/= (/) -> ( F ` z ) e. z ) )
40		simpr 477	. . . . . . . . . . . 12 |- ( ( ph /\ v e. ( 0 [,] 1 ) ) -> v e. ( 0 [,] 1 ) )
41	40, 11	sylib 208	. . . . . . . . . . 11 |- ( ( ph /\ v e. ( 0 [,] 1 ) ) -> [ v ] .~ =/= (/) )
42		neeq1 2856	. . . . . . . . . . . . 13 |- ( z = [ v ] .~ -> ( z =/= (/) <-> [ v ] .~ =/= (/) ) )
43		fveq2 6191	. . . . . . . . . . . . . 14 |- ( z = [ v ] .~ -> ( F ` z ) = ( F ` [ v ] .~ ) )
44		id 22	. . . . . . . . . . . . . 14 |- ( z = [ v ] .~ -> z = [ v ] .~ )
45	43, 44	eleq12d 2695	. . . . . . . . . . . . 13 |- ( z = [ v ] .~ -> ( ( F ` z ) e. z <-> ( F ` [ v ] .~ ) e. [ v ] .~ ) )
46	42, 45	imbi12d 334	. . . . . . . . . . . 12 |- ( z = [ v ] .~ -> ( ( z =/= (/) -> ( F ` z ) e. z ) <-> ( [ v ] .~ =/= (/) -> ( F ` [ v ] .~ ) e. [ v ] .~ ) ) )
47	46	rspcv 3305	. . . . . . . . . . 11 |- ( [ v ] .~ e. S -> ( A. z e. S ( z =/= (/) -> ( F ` z ) e. z ) -> ( [ v ] .~ =/= (/) -> ( F ` [ v ] .~ ) e. [ v ] .~ ) ) )
48	38, 39, 41, 47	syl3c 66	. . . . . . . . . 10 |- ( ( ph /\ v e. ( 0 [,] 1 ) ) -> ( F ` [ v ] .~ ) e. [ v ] .~ )
49		fvex 6201	. . . . . . . . . . . 12 |- ( F ` [ v ] .~ ) e. _V
50		vex 3203	. . . . . . . . . . . 12 |- v e. _V
51	49, 50	elec 7786	. . . . . . . . . . 11 |- ( ( F ` [ v ] .~ ) e. [ v ] .~ <-> v .~ ( F ` [ v ] .~ ) )
52		oveq12 6659	. . . . . . . . . . . . 13 |- ( ( x = v /\ y = ( F ` [ v ] .~ ) ) -> ( x - y ) = ( v - ( F ` [ v ] .~ ) ) )
53	52	eleq1d 2686	. . . . . . . . . . . 12 |- ( ( x = v /\ y = ( F ` [ v ] .~ ) ) -> ( ( x - y ) e. QQ <-> ( v - ( F ` [ v ] .~ ) ) e. QQ ) )
54	53, 5	brab2a 5194	. . . . . . . . . . 11 |- ( v .~ ( F ` [ v ] .~ ) <-> ( ( v e. ( 0 [,] 1 ) /\ ( F ` [ v ] .~ ) e. ( 0 [,] 1 ) ) /\ ( v - ( F ` [ v ] .~ ) ) e. QQ ) )
55	51, 54	bitri 264	. . . . . . . . . 10 |- ( ( F ` [ v ] .~ ) e. [ v ] .~ <-> ( ( v e. ( 0 [,] 1 ) /\ ( F ` [ v ] .~ ) e. ( 0 [,] 1 ) ) /\ ( v - ( F ` [ v ] .~ ) ) e. QQ ) )
56	48, 55	sylib 208	. . . . . . . . 9 |- ( ( ph /\ v e. ( 0 [,] 1 ) ) -> ( ( v e. ( 0 [,] 1 ) /\ ( F ` [ v ] .~ ) e. ( 0 [,] 1 ) ) /\ ( v - ( F ` [ v ] .~ ) ) e. QQ ) )
57	56	simprd 479	. . . . . . . 8 |- ( ( ph /\ v e. ( 0 [,] 1 ) ) -> ( v - ( F ` [ v ] .~ ) ) e. QQ )
58		0re 10040	. . . . . . . . . . . . 13 |- 0 e. RR
59		1re 10039	. . . . . . . . . . . . 13 |- 1 e. RR
60	58, 59	elicc2i 12239	. . . . . . . . . . . 12 |- ( v e. ( 0 [,] 1 ) <-> ( v e. RR /\ 0 <_ v /\ v <_ 1 ) )
61	40, 60	sylib 208	. . . . . . . . . . 11 |- ( ( ph /\ v e. ( 0 [,] 1 ) ) -> ( v e. RR /\ 0 <_ v /\ v <_ 1 ) )
62	61	simp1d 1073	. . . . . . . . . 10 |- ( ( ph /\ v e. ( 0 [,] 1 ) ) -> v e. RR )
63	56	simpld 475	. . . . . . . . . . . . 13 |- ( ( ph /\ v e. ( 0 [,] 1 ) ) -> ( v e. ( 0 [,] 1 ) /\ ( F ` [ v ] .~ ) e. ( 0 [,] 1 ) ) )
64	63	simprd 479	. . . . . . . . . . . 12 |- ( ( ph /\ v e. ( 0 [,] 1 ) ) -> ( F ` [ v ] .~ ) e. ( 0 [,] 1 ) )
65	58, 59	elicc2i 12239	. . . . . . . . . . . 12 |- ( ( F ` [ v ] .~ ) e. ( 0 [,] 1 ) <-> ( ( F ` [ v ] .~ ) e. RR /\ 0 <_ ( F ` [ v ] .~ ) /\ ( F ` [ v ] .~ ) <_ 1 ) )
66	64, 65	sylib 208	. . . . . . . . . . 11 |- ( ( ph /\ v e. ( 0 [,] 1 ) ) -> ( ( F ` [ v ] .~ ) e. RR /\ 0 <_ ( F ` [ v ] .~ ) /\ ( F ` [ v ] .~ ) <_ 1 ) )
67	66	simp1d 1073	. . . . . . . . . 10 |- ( ( ph /\ v e. ( 0 [,] 1 ) ) -> ( F ` [ v ] .~ ) e. RR )
68	62, 67	resubcld 10458	. . . . . . . . 9 |- ( ( ph /\ v e. ( 0 [,] 1 ) ) -> ( v - ( F ` [ v ] .~ ) ) e. RR )
69	67, 62	resubcld 10458	. . . . . . . . . . . 12 |- ( ( ph /\ v e. ( 0 [,] 1 ) ) -> ( ( F ` [ v ] .~ ) - v ) e. RR )
70		1red 10055	. . . . . . . . . . . 12 |- ( ( ph /\ v e. ( 0 [,] 1 ) ) -> 1 e. RR )
71	61	simp2d 1074	. . . . . . . . . . . . 13 |- ( ( ph /\ v e. ( 0 [,] 1 ) ) -> 0 <_ v )
72	67, 62	subge02d 10619	. . . . . . . . . . . . 13 |- ( ( ph /\ v e. ( 0 [,] 1 ) ) -> ( 0 <_ v <-> ( ( F ` [ v ] .~ ) - v ) <_ ( F ` [ v ] .~ ) ) )
73	71, 72	mpbid 222	. . . . . . . . . . . 12 |- ( ( ph /\ v e. ( 0 [,] 1 ) ) -> ( ( F ` [ v ] .~ ) - v ) <_ ( F ` [ v ] .~ ) )
74	66	simp3d 1075	. . . . . . . . . . . 12 |- ( ( ph /\ v e. ( 0 [,] 1 ) ) -> ( F ` [ v ] .~ ) <_ 1 )
75	69, 67, 70, 73, 74	letrd 10194	. . . . . . . . . . 11 |- ( ( ph /\ v e. ( 0 [,] 1 ) ) -> ( ( F ` [ v ] .~ ) - v ) <_ 1 )
76	69, 70	lenegd 10606	. . . . . . . . . . 11 |- ( ( ph /\ v e. ( 0 [,] 1 ) ) -> ( ( ( F ` [ v ] .~ ) - v ) <_ 1 <-> -u 1 <_ -u ( ( F ` [ v ] .~ ) - v ) ) )
77	75, 76	mpbid 222	. . . . . . . . . 10 |- ( ( ph /\ v e. ( 0 [,] 1 ) ) -> -u 1 <_ -u ( ( F ` [ v ] .~ ) - v ) )
78	67	recnd 10068	. . . . . . . . . . 11 |- ( ( ph /\ v e. ( 0 [,] 1 ) ) -> ( F ` [ v ] .~ ) e. CC )
79	62	recnd 10068	. . . . . . . . . . 11 |- ( ( ph /\ v e. ( 0 [,] 1 ) ) -> v e. CC )
80	78, 79	negsubdi2d 10408	. . . . . . . . . 10 |- ( ( ph /\ v e. ( 0 [,] 1 ) ) -> -u ( ( F ` [ v ] .~ ) - v ) = ( v - ( F ` [ v ] .~ ) ) )
81	77, 80	breqtrd 4679	. . . . . . . . 9 |- ( ( ph /\ v e. ( 0 [,] 1 ) ) -> -u 1 <_ ( v - ( F ` [ v ] .~ ) ) )
82	66	simp2d 1074	. . . . . . . . . . 11 |- ( ( ph /\ v e. ( 0 [,] 1 ) ) -> 0 <_ ( F ` [ v ] .~ ) )
83	62, 67	subge02d 10619	. . . . . . . . . . 11 |- ( ( ph /\ v e. ( 0 [,] 1 ) ) -> ( 0 <_ ( F ` [ v ] .~ ) <-> ( v - ( F ` [ v ] .~ ) ) <_ v ) )
84	82, 83	mpbid 222	. . . . . . . . . 10 |- ( ( ph /\ v e. ( 0 [,] 1 ) ) -> ( v - ( F ` [ v ] .~ ) ) <_ v )
85	61	simp3d 1075	. . . . . . . . . 10 |- ( ( ph /\ v e. ( 0 [,] 1 ) ) -> v <_ 1 )
86	68, 62, 70, 84, 85	letrd 10194	. . . . . . . . 9 |- ( ( ph /\ v e. ( 0 [,] 1 ) ) -> ( v - ( F ` [ v ] .~ ) ) <_ 1 )
87		neg1rr 11125	. . . . . . . . . 10 |- -u 1 e. RR
88	87, 59	elicc2i 12239	. . . . . . . . 9 |- ( ( v - ( F ` [ v ] .~ ) ) e. ( -u 1 [,] 1 ) <-> ( ( v - ( F ` [ v ] .~ ) ) e. RR /\ -u 1 <_ ( v - ( F ` [ v ] .~ ) ) /\ ( v - ( F ` [ v ] .~ ) ) <_ 1 ) )
89	68, 81, 86, 88	syl3anbrc 1246	. . . . . . . 8 |- ( ( ph /\ v e. ( 0 [,] 1 ) ) -> ( v - ( F ` [ v ] .~ ) ) e. ( -u 1 [,] 1 ) )
90	57, 89	elind 3798	. . . . . . 7 |- ( ( ph /\ v e. ( 0 [,] 1 ) ) -> ( v - ( F ` [ v ] .~ ) ) e. ( QQ i^i ( -u 1 [,] 1 ) ) )
91	32, 90	ffvelrnd 6360	. . . . . 6 |- ( ( ph /\ v e. ( 0 [,] 1 ) ) -> ( `' G ` ( v - ( F ` [ v ] .~ ) ) ) e. NN )
92		f1ocnvfv2 6533	. . . . . . . . . . . 12 |- ( ( G : NN -1-1-onto-> ( QQ i^i ( -u 1 [,] 1 ) ) /\ ( v - ( F ` [ v ] .~ ) ) e. ( QQ i^i ( -u 1 [,] 1 ) ) ) -> ( G ` ( `' G ` ( v - ( F ` [ v ] .~ ) ) ) ) = ( v - ( F ` [ v ] .~ ) ) )
93	29, 90, 92	syl2anc 693	. . . . . . . . . . 11 |- ( ( ph /\ v e. ( 0 [,] 1 ) ) -> ( G ` ( `' G ` ( v - ( F ` [ v ] .~ ) ) ) ) = ( v - ( F ` [ v ] .~ ) ) )
94	93	oveq2d 6666	. . . . . . . . . 10 |- ( ( ph /\ v e. ( 0 [,] 1 ) ) -> ( v - ( G ` ( `' G ` ( v - ( F ` [ v ] .~ ) ) ) ) ) = ( v - ( v - ( F ` [ v ] .~ ) ) ) )
95	79, 78	nncand 10397	. . . . . . . . . 10 |- ( ( ph /\ v e. ( 0 [,] 1 ) ) -> ( v - ( v - ( F ` [ v ] .~ ) ) ) = ( F ` [ v ] .~ ) )
96	94, 95	eqtrd 2656	. . . . . . . . 9 |- ( ( ph /\ v e. ( 0 [,] 1 ) ) -> ( v - ( G ` ( `' G ` ( v - ( F ` [ v ] .~ ) ) ) ) ) = ( F ` [ v ] .~ ) )
97	1	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ v e. ( 0 [,] 1 ) ) -> F Fn S )
98		fnfvelrn 6356	. . . . . . . . . 10 |- ( ( F Fn S /\ [ v ] .~ e. S ) -> ( F ` [ v ] .~ ) e. ran F )
99	97, 38, 98	syl2anc 693	. . . . . . . . 9 |- ( ( ph /\ v e. ( 0 [,] 1 ) ) -> ( F ` [ v ] .~ ) e. ran F )
100	96, 99	eqeltrd 2701	. . . . . . . 8 |- ( ( ph /\ v e. ( 0 [,] 1 ) ) -> ( v - ( G ` ( `' G ` ( v - ( F ` [ v ] .~ ) ) ) ) ) e. ran F )
101		oveq1 6657	. . . . . . . . . 10 |- ( s = v -> ( s - ( G ` ( `' G ` ( v - ( F ` [ v ] .~ ) ) ) ) ) = ( v - ( G ` ( `' G ` ( v - ( F ` [ v ] .~ ) ) ) ) ) )
102	101	eleq1d 2686	. . . . . . . . 9 |- ( s = v -> ( ( s - ( G ` ( `' G ` ( v - ( F ` [ v ] .~ ) ) ) ) ) e. ran F <-> ( v - ( G ` ( `' G ` ( v - ( F ` [ v ] .~ ) ) ) ) ) e. ran F ) )
103	102	elrab 3363	. . . . . . . 8 |- ( v e. { s e. RR | ( s - ( G ` ( `' G ` ( v - ( F ` [ v ] .~ ) ) ) ) ) e. ran F } <-> ( v e. RR /\ ( v - ( G ` ( `' G ` ( v - ( F ` [ v ] .~ ) ) ) ) ) e. ran F ) )
104	62, 100, 103	sylanbrc 698	. . . . . . 7 |- ( ( ph /\ v e. ( 0 [,] 1 ) ) -> v e. { s e. RR | ( s - ( G ` ( `' G ` ( v - ( F ` [ v ] .~ ) ) ) ) ) e. ran F } )
105		fveq2 6191	. . . . . . . . . . . 12 |- ( n = ( `' G ` ( v - ( F ` [ v ] .~ ) ) ) -> ( G ` n ) = ( G ` ( `' G ` ( v - ( F ` [ v ] .~ ) ) ) ) )
106	105	oveq2d 6666	. . . . . . . . . . 11 |- ( n = ( `' G ` ( v - ( F ` [ v ] .~ ) ) ) -> ( s - ( G ` n ) ) = ( s - ( G ` ( `' G ` ( v - ( F ` [ v ] .~ ) ) ) ) ) )
107	106	eleq1d 2686	. . . . . . . . . 10 |- ( n = ( `' G ` ( v - ( F ` [ v ] .~ ) ) ) -> ( ( s - ( G ` n ) ) e. ran F <-> ( s - ( G ` ( `' G ` ( v - ( F ` [ v ] .~ ) ) ) ) ) e. ran F ) )
108	107	rabbidv 3189	. . . . . . . . 9 |- ( n = ( `' G ` ( v - ( F ` [ v ] .~ ) ) ) -> { s e. RR | ( s - ( G ` n ) ) e. ran F } = { s e. RR | ( s - ( G ` ( `' G ` ( v - ( F ` [ v ] .~ ) ) ) ) ) e. ran F } )
109		vitali.6	. . . . . . . . 9 |- T = ( n e. NN |-> { s e. RR | ( s - ( G ` n ) ) e. ran F } )
110		reex 10027	. . . . . . . . . 10 |- RR e. _V
111	110	rabex 4813	. . . . . . . . 9 |- { s e. RR | ( s - ( G ` ( `' G ` ( v - ( F ` [ v ] .~ ) ) ) ) ) e. ran F } e. _V
112	108, 109, 111	fvmpt 6282	. . . . . . . 8 |- ( ( `' G ` ( v - ( F ` [ v ] .~ ) ) ) e. NN -> ( T ` ( `' G ` ( v - ( F ` [ v ] .~ ) ) ) ) = { s e. RR | ( s - ( G ` ( `' G ` ( v - ( F ` [ v ] .~ ) ) ) ) ) e. ran F } )
113	91, 112	syl 17	. . . . . . 7 |- ( ( ph /\ v e. ( 0 [,] 1 ) ) -> ( T ` ( `' G ` ( v - ( F ` [ v ] .~ ) ) ) ) = { s e. RR | ( s - ( G ` ( `' G ` ( v - ( F ` [ v ] .~ ) ) ) ) ) e. ran F } )
114	104, 113	eleqtrrd 2704	. . . . . 6 |- ( ( ph /\ v e. ( 0 [,] 1 ) ) -> v e. ( T ` ( `' G ` ( v - ( F ` [ v ] .~ ) ) ) ) )
115	91, 114	jca 554	. . . . 5 |- ( ( ph /\ v e. ( 0 [,] 1 ) ) -> ( ( `' G ` ( v - ( F ` [ v ] .~ ) ) ) e. NN /\ v e. ( T ` ( `' G ` ( v - ( F ` [ v ] .~ ) ) ) ) ) )
116		fveq2 6191	. . . . . 6 |- ( m = ( `' G ` ( v - ( F ` [ v ] .~ ) ) ) -> ( T ` m ) = ( T ` ( `' G ` ( v - ( F ` [ v ] .~ ) ) ) ) )
117	116	eliuni 4526	. . . . 5 |- ( ( ( `' G ` ( v - ( F ` [ v ] .~ ) ) ) e. NN /\ v e. ( T ` ( `' G ` ( v - ( F ` [ v ] .~ ) ) ) ) ) -> v e. U_ m e. NN ( T ` m ) )
118	115, 117	syl 17	. . . 4 |- ( ( ph /\ v e. ( 0 [,] 1 ) ) -> v e. U_ m e. NN ( T ` m ) )
119	118	ex 450	. . 3 |- ( ph -> ( v e. ( 0 [,] 1 ) -> v e. U_ m e. NN ( T ` m ) ) )
120	119	ssrdv 3609	. 2 |- ( ph -> ( 0 [,] 1 ) C_ U_ m e. NN ( T ` m ) )
121		eliun 4524	. . . 4 |- ( x e. U_ m e. NN ( T ` m ) <-> E. m e. NN x e. ( T ` m ) )
122		fveq2 6191	. . . . . . . . . . . . . . . 16 |- ( n = m -> ( G ` n ) = ( G ` m ) )
123	122	oveq2d 6666	. . . . . . . . . . . . . . 15 |- ( n = m -> ( s - ( G ` n ) ) = ( s - ( G ` m ) ) )
124	123	eleq1d 2686	. . . . . . . . . . . . . 14 |- ( n = m -> ( ( s - ( G ` n ) ) e. ran F <-> ( s - ( G ` m ) ) e. ran F ) )
125	124	rabbidv 3189	. . . . . . . . . . . . 13 |- ( n = m -> { s e. RR | ( s - ( G ` n ) ) e. ran F } = { s e. RR | ( s - ( G ` m ) ) e. ran F } )
126	110	rabex 4813	. . . . . . . . . . . . 13 |- { s e. RR | ( s - ( G ` m ) ) e. ran F } e. _V
127	125, 109, 126	fvmpt 6282	. . . . . . . . . . . 12 |- ( m e. NN -> ( T ` m ) = { s e. RR | ( s - ( G ` m ) ) e. ran F } )
128	127	adantl 482	. . . . . . . . . . 11 |- ( ( ph /\ m e. NN ) -> ( T ` m ) = { s e. RR | ( s - ( G ` m ) ) e. ran F } )
129	128	eleq2d 2687	. . . . . . . . . 10 |- ( ( ph /\ m e. NN ) -> ( x e. ( T ` m ) <-> x e. { s e. RR | ( s - ( G ` m ) ) e. ran F } ) )
130	129	biimpa 501	. . . . . . . . 9 |- ( ( ( ph /\ m e. NN ) /\ x e. ( T ` m ) ) -> x e. { s e. RR | ( s - ( G ` m ) ) e. ran F } )
131		oveq1 6657	. . . . . . . . . . 11 |- ( s = x -> ( s - ( G ` m ) ) = ( x - ( G ` m ) ) )
132	131	eleq1d 2686	. . . . . . . . . 10 |- ( s = x -> ( ( s - ( G ` m ) ) e. ran F <-> ( x - ( G ` m ) ) e. ran F ) )
133	132	elrab 3363	. . . . . . . . 9 |- ( x e. { s e. RR | ( s - ( G ` m ) ) e. ran F } <-> ( x e. RR /\ ( x - ( G ` m ) ) e. ran F ) )
134	130, 133	sylib 208	. . . . . . . 8 |- ( ( ( ph /\ m e. NN ) /\ x e. ( T ` m ) ) -> ( x e. RR /\ ( x - ( G ` m ) ) e. ran F ) )
135	134	simpld 475	. . . . . . 7 |- ( ( ( ph /\ m e. NN ) /\ x e. ( T ` m ) ) -> x e. RR )
136	87	a1i 11	. . . . . . . 8 |- ( ( ( ph /\ m e. NN ) /\ x e. ( T ` m ) ) -> -u 1 e. RR )
137		iccssre 12255	. . . . . . . . . . 11 |- ( ( -u 1 e. RR /\ 1 e. RR ) -> ( -u 1 [,] 1 ) C_ RR )
138	87, 59, 137	mp2an 708	. . . . . . . . . 10 |- ( -u 1 [,] 1 ) C_ RR
139		inss2 3834	. . . . . . . . . . 11 |- ( QQ i^i ( -u 1 [,] 1 ) ) C_ ( -u 1 [,] 1 )
140		f1of 6137	. . . . . . . . . . . . 13 |- ( G : NN -1-1-onto-> ( QQ i^i ( -u 1 [,] 1 ) ) -> G : NN --> ( QQ i^i ( -u 1 [,] 1 ) ) )
141	28, 140	syl 17	. . . . . . . . . . . 12 |- ( ph -> G : NN --> ( QQ i^i ( -u 1 [,] 1 ) ) )
142	141	ffvelrnda 6359	. . . . . . . . . . 11 |- ( ( ph /\ m e. NN ) -> ( G ` m ) e. ( QQ i^i ( -u 1 [,] 1 ) ) )
143	139, 142	sseldi 3601	. . . . . . . . . 10 |- ( ( ph /\ m e. NN ) -> ( G ` m ) e. ( -u 1 [,] 1 ) )
144	138, 143	sseldi 3601	. . . . . . . . 9 |- ( ( ph /\ m e. NN ) -> ( G ` m ) e. RR )
145	144	adantr 481	. . . . . . . 8 |- ( ( ( ph /\ m e. NN ) /\ x e. ( T ` m ) ) -> ( G ` m ) e. RR )
146	143	adantr 481	. . . . . . . . . 10 |- ( ( ( ph /\ m e. NN ) /\ x e. ( T ` m ) ) -> ( G ` m ) e. ( -u 1 [,] 1 ) )
147	87, 59	elicc2i 12239	. . . . . . . . . 10 |- ( ( G ` m ) e. ( -u 1 [,] 1 ) <-> ( ( G ` m ) e. RR /\ -u 1 <_ ( G ` m ) /\ ( G ` m ) <_ 1 ) )
148	146, 147	sylib 208	. . . . . . . . 9 |- ( ( ( ph /\ m e. NN ) /\ x e. ( T ` m ) ) -> ( ( G ` m ) e. RR /\ -u 1 <_ ( G ` m ) /\ ( G ` m ) <_ 1 ) )
149	148	simp2d 1074	. . . . . . . 8 |- ( ( ( ph /\ m e. NN ) /\ x e. ( T ` m ) ) -> -u 1 <_ ( G ` m ) )
150	27	ad2antrr 762	. . . . . . . . . . . 12 |- ( ( ( ph /\ m e. NN ) /\ x e. ( T ` m ) ) -> ran F C_ ( 0 [,] 1 ) )
151	134	simprd 479	. . . . . . . . . . . 12 |- ( ( ( ph /\ m e. NN ) /\ x e. ( T ` m ) ) -> ( x - ( G ` m ) ) e. ran F )
152	150, 151	sseldd 3604	. . . . . . . . . . 11 |- ( ( ( ph /\ m e. NN ) /\ x e. ( T ` m ) ) -> ( x - ( G ` m ) ) e. ( 0 [,] 1 ) )
153	58, 59	elicc2i 12239	. . . . . . . . . . 11 |- ( ( x - ( G ` m ) ) e. ( 0 [,] 1 ) <-> ( ( x - ( G ` m ) ) e. RR /\ 0 <_ ( x - ( G ` m ) ) /\ ( x - ( G ` m ) ) <_ 1 ) )
154	152, 153	sylib 208	. . . . . . . . . 10 |- ( ( ( ph /\ m e. NN ) /\ x e. ( T ` m ) ) -> ( ( x - ( G ` m ) ) e. RR /\ 0 <_ ( x - ( G ` m ) ) /\ ( x - ( G ` m ) ) <_ 1 ) )
155	154	simp2d 1074	. . . . . . . . 9 |- ( ( ( ph /\ m e. NN ) /\ x e. ( T ` m ) ) -> 0 <_ ( x - ( G ` m ) ) )
156	135, 145	subge0d 10617	. . . . . . . . 9 |- ( ( ( ph /\ m e. NN ) /\ x e. ( T ` m ) ) -> ( 0 <_ ( x - ( G ` m ) ) <-> ( G ` m ) <_ x ) )
157	155, 156	mpbid 222	. . . . . . . 8 |- ( ( ( ph /\ m e. NN ) /\ x e. ( T ` m ) ) -> ( G ` m ) <_ x )
158	136, 145, 135, 149, 157	letrd 10194	. . . . . . 7 |- ( ( ( ph /\ m e. NN ) /\ x e. ( T ` m ) ) -> -u 1 <_ x )
159		peano2re 10209	. . . . . . . . 9 |- ( ( G ` m ) e. RR -> ( ( G ` m ) + 1 ) e. RR )
160	145, 159	syl 17	. . . . . . . 8 |- ( ( ( ph /\ m e. NN ) /\ x e. ( T ` m ) ) -> ( ( G ` m ) + 1 ) e. RR )
161		2re 11090	. . . . . . . . 9 |- 2 e. RR
162	161	a1i 11	. . . . . . . 8 |- ( ( ( ph /\ m e. NN ) /\ x e. ( T ` m ) ) -> 2 e. RR )
163	154	simp3d 1075	. . . . . . . . 9 |- ( ( ( ph /\ m e. NN ) /\ x e. ( T ` m ) ) -> ( x - ( G ` m ) ) <_ 1 )
164		1red 10055	. . . . . . . . . 10 |- ( ( ( ph /\ m e. NN ) /\ x e. ( T ` m ) ) -> 1 e. RR )
165	135, 145, 164	lesubadd2d 10626	. . . . . . . . 9 |- ( ( ( ph /\ m e. NN ) /\ x e. ( T ` m ) ) -> ( ( x - ( G ` m ) ) <_ 1 <-> x <_ ( ( G ` m ) + 1 ) ) )
166	163, 165	mpbid 222	. . . . . . . 8 |- ( ( ( ph /\ m e. NN ) /\ x e. ( T ` m ) ) -> x <_ ( ( G ` m ) + 1 ) )
167	148	simp3d 1075	. . . . . . . . . 10 |- ( ( ( ph /\ m e. NN ) /\ x e. ( T ` m ) ) -> ( G ` m ) <_ 1 )
168	145, 164, 164, 167	leadd1dd 10641	. . . . . . . . 9 |- ( ( ( ph /\ m e. NN ) /\ x e. ( T ` m ) ) -> ( ( G ` m ) + 1 ) <_ ( 1 + 1 ) )
169		df-2 11079	. . . . . . . . 9 |- 2 = ( 1 + 1 )
170	168, 169	syl6breqr 4695	. . . . . . . 8 |- ( ( ( ph /\ m e. NN ) /\ x e. ( T ` m ) ) -> ( ( G ` m ) + 1 ) <_ 2 )
171	135, 160, 162, 166, 170	letrd 10194	. . . . . . 7 |- ( ( ( ph /\ m e. NN ) /\ x e. ( T ` m ) ) -> x <_ 2 )
172	87, 161	elicc2i 12239	. . . . . . 7 |- ( x e. ( -u 1 [,] 2 ) <-> ( x e. RR /\ -u 1 <_ x /\ x <_ 2 ) )
173	135, 158, 171, 172	syl3anbrc 1246	. . . . . 6 |- ( ( ( ph /\ m e. NN ) /\ x e. ( T ` m ) ) -> x e. ( -u 1 [,] 2 ) )
174	173	ex 450	. . . . 5 |- ( ( ph /\ m e. NN ) -> ( x e. ( T ` m ) -> x e. ( -u 1 [,] 2 ) ) )
175	174	rexlimdva 3031	. . . 4 |- ( ph -> ( E. m e. NN x e. ( T ` m ) -> x e. ( -u 1 [,] 2 ) ) )
176	121, 175	syl5bi 232	. . 3 |- ( ph -> ( x e. U_ m e. NN ( T ` m ) -> x e. ( -u 1 [,] 2 ) ) )
177	176	ssrdv 3609	. 2 |- ( ph -> U_ m e. NN ( T ` m ) C_ ( -u 1 [,] 2 ) )
178	27, 120, 177	3jca 1242	1 |- ( ph -> ( ran F C_ ( 0 [,] 1 ) /\ ( 0 [,] 1 ) C_ U_ m e. NN ( T ` m ) /\ U_ m e. NN ( T ` m ) C_ ( -u 1 [,] 2 ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   {crab 2916   _Vcvv 3200   \ cdif 3571   i^i cin 3573   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   U_ciun 4520   class class class wbr 4653   {copab 4712   |-> cmpt 4729   `'ccnv 5113   dom cdm 5114   ran crn 5115   Fn wfn 5883   -->wf 5884   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   Er wer 7739   [cec 7740   /.cqs 7741   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   <_ cle 10075   - cmin 10266   -ucneg 10267   NNcn 11020   2c2 11070   QQcq 11788   [,]cicc 12178   volcvol 23232

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-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  ax-pre-mulgt0 10013

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-rmo 2920  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-ec 7744  df-qs 7748  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-n0 11293  df-z 11378  df-q 11789  df-icc 12182

This theorem is referenced by:  vitalilem3  23379  vitalilem4  23380  vitalilem5  23381