Metamath Proof Explorer


Theorem itg2addnclem2

Description: Lemma for itg2addnc . The function described is a simple function. (Contributed by Brendan Leahy, 29-Oct-2017)

Ref Expression
Hypotheses itg2addnc.f1 φ F MblFn
itg2addnc.f2 φ F : 0 +∞
Assertion itg2addnclem2 φ h dom 1 v + x if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x dom 1

Proof

Step Hyp Ref Expression
1 itg2addnc.f1 φ F MblFn
2 itg2addnc.f2 φ F : 0 +∞
3 rge0ssre 0 +∞
4 fss F : 0 +∞ 0 +∞ F :
5 2 3 4 sylancl φ F :
6 5 ad2antrr φ h dom 1 v + F :
7 6 ffvelrnda φ h dom 1 v + x F x
8 rpre v + v
9 3re 3
10 3ne0 3 0
11 9 10 pm3.2i 3 3 0
12 redivcl v 3 3 0 v 3
13 12 3expb v 3 3 0 v 3
14 8 11 13 sylancl v + v 3
15 14 ad2antlr φ h dom 1 v + x v 3
16 rpcnne0 v + v v 0
17 3cn 3
18 17 10 pm3.2i 3 3 0
19 divne0 v v 0 3 3 0 v 3 0
20 16 18 19 sylancl v + v 3 0
21 20 ad2antlr φ h dom 1 v + x v 3 0
22 7 15 21 redivcld φ h dom 1 v + x F x v 3
23 reflcl F x v 3 F x v 3
24 22 23 syl φ h dom 1 v + x F x v 3
25 peano2rem F x v 3 F x v 3 1
26 24 25 syl φ h dom 1 v + x F x v 3 1
27 26 15 remulcld φ h dom 1 v + x F x v 3 1 v 3
28 i1ff h dom 1 h :
29 28 ad2antlr φ h dom 1 v + h :
30 29 ffvelrnda φ h dom 1 v + x h x
31 27 30 ifcld φ h dom 1 v + x if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x
32 31 fmpttd φ h dom 1 v + x if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x :
33 fzfi 0 sup ran h < v 3 + 1 Fin
34 ovex t 1 v 3 V
35 eqid t 0 sup ran h < v 3 + 1 t 1 v 3 = t 0 sup ran h < v 3 + 1 t 1 v 3
36 34 35 fnmpti t 0 sup ran h < v 3 + 1 t 1 v 3 Fn 0 sup ran h < v 3 + 1
37 dffn4 t 0 sup ran h < v 3 + 1 t 1 v 3 Fn 0 sup ran h < v 3 + 1 t 0 sup ran h < v 3 + 1 t 1 v 3 : 0 sup ran h < v 3 + 1 onto ran t 0 sup ran h < v 3 + 1 t 1 v 3
38 36 37 mpbi t 0 sup ran h < v 3 + 1 t 1 v 3 : 0 sup ran h < v 3 + 1 onto ran t 0 sup ran h < v 3 + 1 t 1 v 3
39 fofi 0 sup ran h < v 3 + 1 Fin t 0 sup ran h < v 3 + 1 t 1 v 3 : 0 sup ran h < v 3 + 1 onto ran t 0 sup ran h < v 3 + 1 t 1 v 3 ran t 0 sup ran h < v 3 + 1 t 1 v 3 Fin
40 33 38 39 mp2an ran t 0 sup ran h < v 3 + 1 t 1 v 3 Fin
41 i1frn h dom 1 ran h Fin
42 41 ad2antlr φ h dom 1 v + ran h Fin
43 unfi ran t 0 sup ran h < v 3 + 1 t 1 v 3 Fin ran h Fin ran t 0 sup ran h < v 3 + 1 t 1 v 3 ran h Fin
44 40 42 43 sylancr φ h dom 1 v + ran t 0 sup ran h < v 3 + 1 t 1 v 3 ran h Fin
45 0zd φ h dom 1 v + x F x v 3 1 v 3 h x h x 0 0
46 28 frnd h dom 1 ran h
47 i1f0rn h dom 1 0 ran h
48 elex2 0 ran h x x ran h
49 47 48 syl h dom 1 x x ran h
50 n0 ran h x x ran h
51 49 50 sylibr h dom 1 ran h
52 fimaxre2 ran h ran h Fin x y ran h y x
53 46 41 52 syl2anc h dom 1 x y ran h y x
54 suprcl ran h ran h x y ran h y x sup ran h <
55 46 51 53 54 syl3anc h dom 1 sup ran h <
56 55 ad3antlr φ h dom 1 v + x sup ran h <
57 56 15 21 redivcld φ h dom 1 v + x sup ran h < v 3
58 peano2re sup ran h < v 3 sup ran h < v 3 + 1
59 57 58 syl φ h dom 1 v + x sup ran h < v 3 + 1
60 ceicl sup ran h < v 3 + 1 sup ran h < v 3 + 1
61 59 60 syl φ h dom 1 v + x sup ran h < v 3 + 1
62 61 adantr φ h dom 1 v + x F x v 3 1 v 3 h x h x 0 sup ran h < v 3 + 1
63 22 flcld φ h dom 1 v + x F x v 3
64 63 adantr φ h dom 1 v + x F x v 3 1 v 3 h x h x 0 F x v 3
65 3nn 3
66 nnrp 3 3 +
67 65 66 ax-mp 3 +
68 rpdivcl v + 3 + v 3 +
69 67 68 mpan2 v + v 3 +
70 69 ad2antlr φ h dom 1 v + x v 3 +
71 2 ad2antrr φ h dom 1 v + F : 0 +∞
72 71 ffvelrnda φ h dom 1 v + x F x 0 +∞
73 elrege0 F x 0 +∞ F x 0 F x
74 72 73 sylib φ h dom 1 v + x F x 0 F x
75 74 simprd φ h dom 1 v + x 0 F x
76 7 70 75 divge0d φ h dom 1 v + x 0 F x v 3
77 flge0nn0 F x v 3 0 F x v 3 F x v 3 0
78 22 76 77 syl2anc φ h dom 1 v + x F x v 3 0
79 78 nn0ge0d φ h dom 1 v + x 0 F x v 3
80 79 adantr φ h dom 1 v + x F x v 3 1 v 3 h x h x 0 0 F x v 3
81 46 51 53 3jca h dom 1 ran h ran h x y ran h y x
82 81 ad3antlr φ h dom 1 v + x ran h ran h x y ran h y x
83 ffn h : h Fn
84 28 83 syl h dom 1 h Fn
85 dffn3 h Fn h : ran h
86 84 85 sylib h dom 1 h : ran h
87 86 ad2antlr φ h dom 1 v + h : ran h
88 87 ffvelrnda φ h dom 1 v + x h x ran h
89 suprub ran h ran h x y ran h y x h x ran h h x sup ran h <
90 82 88 89 syl2anc φ h dom 1 v + x h x sup ran h <
91 letr F x v 3 1 v 3 h x sup ran h < F x v 3 1 v 3 h x h x sup ran h < F x v 3 1 v 3 sup ran h <
92 27 30 56 91 syl3anc φ h dom 1 v + x F x v 3 1 v 3 h x h x sup ran h < F x v 3 1 v 3 sup ran h <
93 26 56 70 lemuldivd φ h dom 1 v + x F x v 3 1 v 3 sup ran h < F x v 3 1 sup ran h < v 3
94 1red φ h dom 1 v + x 1
95 24 94 57 lesubaddd φ h dom 1 v + x F x v 3 1 sup ran h < v 3 F x v 3 sup ran h < v 3 + 1
96 93 95 bitrd φ h dom 1 v + x F x v 3 1 v 3 sup ran h < F x v 3 sup ran h < v 3 + 1
97 ceige sup ran h < v 3 + 1 sup ran h < v 3 + 1 sup ran h < v 3 + 1
98 59 97 syl φ h dom 1 v + x sup ran h < v 3 + 1 sup ran h < v 3 + 1
99 61 zred φ h dom 1 v + x sup ran h < v 3 + 1
100 letr F x v 3 sup ran h < v 3 + 1 sup ran h < v 3 + 1 F x v 3 sup ran h < v 3 + 1 sup ran h < v 3 + 1 sup ran h < v 3 + 1 F x v 3 sup ran h < v 3 + 1
101 24 59 99 100 syl3anc φ h dom 1 v + x F x v 3 sup ran h < v 3 + 1 sup ran h < v 3 + 1 sup ran h < v 3 + 1 F x v 3 sup ran h < v 3 + 1
102 98 101 mpan2d φ h dom 1 v + x F x v 3 sup ran h < v 3 + 1 F x v 3 sup ran h < v 3 + 1
103 96 102 sylbid φ h dom 1 v + x F x v 3 1 v 3 sup ran h < F x v 3 sup ran h < v 3 + 1
104 92 103 syld φ h dom 1 v + x F x v 3 1 v 3 h x h x sup ran h < F x v 3 sup ran h < v 3 + 1
105 90 104 mpan2d φ h dom 1 v + x F x v 3 1 v 3 h x F x v 3 sup ran h < v 3 + 1
106 105 adantrd φ h dom 1 v + x F x v 3 1 v 3 h x h x 0 F x v 3 sup ran h < v 3 + 1
107 106 imp φ h dom 1 v + x F x v 3 1 v 3 h x h x 0 F x v 3 sup ran h < v 3 + 1
108 45 62 64 80 107 elfzd φ h dom 1 v + x F x v 3 1 v 3 h x h x 0 F x v 3 0 sup ran h < v 3 + 1
109 eqid F x v 3 1 v 3 = F x v 3 1 v 3
110 oveq1 t = F x v 3 t 1 = F x v 3 1
111 110 oveq1d t = F x v 3 t 1 v 3 = F x v 3 1 v 3
112 111 rspceeqv F x v 3 0 sup ran h < v 3 + 1 F x v 3 1 v 3 = F x v 3 1 v 3 t 0 sup ran h < v 3 + 1 F x v 3 1 v 3 = t 1 v 3
113 108 109 112 sylancl φ h dom 1 v + x F x v 3 1 v 3 h x h x 0 t 0 sup ran h < v 3 + 1 F x v 3 1 v 3 = t 1 v 3
114 ovex F x v 3 1 v 3 V
115 35 elrnmpt F x v 3 1 v 3 V F x v 3 1 v 3 ran t 0 sup ran h < v 3 + 1 t 1 v 3 t 0 sup ran h < v 3 + 1 F x v 3 1 v 3 = t 1 v 3
116 114 115 ax-mp F x v 3 1 v 3 ran t 0 sup ran h < v 3 + 1 t 1 v 3 t 0 sup ran h < v 3 + 1 F x v 3 1 v 3 = t 1 v 3
117 113 116 sylibr φ h dom 1 v + x F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 ran t 0 sup ran h < v 3 + 1 t 1 v 3
118 elun1 F x v 3 1 v 3 ran t 0 sup ran h < v 3 + 1 t 1 v 3 F x v 3 1 v 3 ran t 0 sup ran h < v 3 + 1 t 1 v 3 ran h
119 117 118 syl φ h dom 1 v + x F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 ran t 0 sup ran h < v 3 + 1 t 1 v 3 ran h
120 elun2 h x ran h h x ran t 0 sup ran h < v 3 + 1 t 1 v 3 ran h
121 88 120 syl φ h dom 1 v + x h x ran t 0 sup ran h < v 3 + 1 t 1 v 3 ran h
122 121 adantr φ h dom 1 v + x ¬ F x v 3 1 v 3 h x h x 0 h x ran t 0 sup ran h < v 3 + 1 t 1 v 3 ran h
123 119 122 ifclda φ h dom 1 v + x if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x ran t 0 sup ran h < v 3 + 1 t 1 v 3 ran h
124 123 fmpttd φ h dom 1 v + x if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x : ran t 0 sup ran h < v 3 + 1 t 1 v 3 ran h
125 124 frnd φ h dom 1 v + ran x if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x ran t 0 sup ran h < v 3 + 1 t 1 v 3 ran h
126 ssfi ran t 0 sup ran h < v 3 + 1 t 1 v 3 ran h Fin ran x if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x ran t 0 sup ran h < v 3 + 1 t 1 v 3 ran h ran x if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x Fin
127 44 125 126 syl2anc φ h dom 1 v + ran x if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x Fin
128 eqid x if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x = x if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x
129 128 mptpreima x if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x -1 t = x | if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x t
130 unrab x | F x v 3 1 v 3 h x h x 0 t = F x v 3 1 v 3 x | ¬ F x v 3 1 v 3 h x h x = 0 t = h x = x | F x v 3 1 v 3 h x h x 0 t = F x v 3 1 v 3 ¬ F x v 3 1 v 3 h x h x = 0 t = h x
131 inrab x | F x v 3 1 v 3 h x x | h x 0 = x | F x v 3 1 v 3 h x h x 0
132 131 ineq1i x | F x v 3 1 v 3 h x x | h x 0 x | t = F x v 3 1 v 3 = x | F x v 3 1 v 3 h x h x 0 x | t = F x v 3 1 v 3
133 inrab x | F x v 3 1 v 3 h x h x 0 x | t = F x v 3 1 v 3 = x | F x v 3 1 v 3 h x h x 0 t = F x v 3 1 v 3
134 132 133 eqtri x | F x v 3 1 v 3 h x x | h x 0 x | t = F x v 3 1 v 3 = x | F x v 3 1 v 3 h x h x 0 t = F x v 3 1 v 3
135 unrab x | ¬ F x v 3 1 v 3 h x x | h x = 0 = x | ¬ F x v 3 1 v 3 h x h x = 0
136 135 ineq1i x | ¬ F x v 3 1 v 3 h x x | h x = 0 x | t = h x = x | ¬ F x v 3 1 v 3 h x h x = 0 x | t = h x
137 inrab x | ¬ F x v 3 1 v 3 h x h x = 0 x | t = h x = x | ¬ F x v 3 1 v 3 h x h x = 0 t = h x
138 136 137 eqtri x | ¬ F x v 3 1 v 3 h x x | h x = 0 x | t = h x = x | ¬ F x v 3 1 v 3 h x h x = 0 t = h x
139 134 138 uneq12i x | F x v 3 1 v 3 h x x | h x 0 x | t = F x v 3 1 v 3 x | ¬ F x v 3 1 v 3 h x x | h x = 0 x | t = h x = x | F x v 3 1 v 3 h x h x 0 t = F x v 3 1 v 3 x | ¬ F x v 3 1 v 3 h x h x = 0 t = h x
140 eqcom if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x = t t = if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x
141 fvex h x V
142 114 141 ifex if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x V
143 142 elsn if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x t if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x = t
144 ianor ¬ F x v 3 1 v 3 h x h x 0 ¬ F x v 3 1 v 3 h x ¬ h x 0
145 nne ¬ h x 0 h x = 0
146 145 orbi2i ¬ F x v 3 1 v 3 h x ¬ h x 0 ¬ F x v 3 1 v 3 h x h x = 0
147 144 146 bitr2i ¬ F x v 3 1 v 3 h x h x = 0 ¬ F x v 3 1 v 3 h x h x 0
148 147 anbi1i ¬ F x v 3 1 v 3 h x h x = 0 t = h x ¬ F x v 3 1 v 3 h x h x 0 t = h x
149 148 orbi2i F x v 3 1 v 3 h x h x 0 t = F x v 3 1 v 3 ¬ F x v 3 1 v 3 h x h x = 0 t = h x F x v 3 1 v 3 h x h x 0 t = F x v 3 1 v 3 ¬ F x v 3 1 v 3 h x h x 0 t = h x
150 eqif t = if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x F x v 3 1 v 3 h x h x 0 t = F x v 3 1 v 3 ¬ F x v 3 1 v 3 h x h x 0 t = h x
151 149 150 bitr4i F x v 3 1 v 3 h x h x 0 t = F x v 3 1 v 3 ¬ F x v 3 1 v 3 h x h x = 0 t = h x t = if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x
152 140 143 151 3bitr4i if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x t F x v 3 1 v 3 h x h x 0 t = F x v 3 1 v 3 ¬ F x v 3 1 v 3 h x h x = 0 t = h x
153 152 rabbii x | if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x t = x | F x v 3 1 v 3 h x h x 0 t = F x v 3 1 v 3 ¬ F x v 3 1 v 3 h x h x = 0 t = h x
154 130 139 153 3eqtr4ri x | if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x t = x | F x v 3 1 v 3 h x x | h x 0 x | t = F x v 3 1 v 3 x | ¬ F x v 3 1 v 3 h x x | h x = 0 x | t = h x
155 129 154 eqtri x if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x -1 t = x | F x v 3 1 v 3 h x x | h x 0 x | t = F x v 3 1 v 3 x | ¬ F x v 3 1 v 3 h x x | h x = 0 x | t = h x
156 eldifi t ran x if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x 0 t ran x if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x
157 32 frnd φ h dom 1 v + ran x if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x
158 157 sseld φ h dom 1 v + t ran x if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x t
159 156 158 syl5 φ h dom 1 v + t ran x if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x 0 t
160 159 imdistani φ h dom 1 v + t ran x if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x 0 φ h dom 1 v + t
161 rabiun x t ran h h -1 t | F x v 3 1 v 3 h x = t ran h x h -1 t | F x v 3 1 v 3 h x
162 cnvimarndm h -1 ran h = dom h
163 iunid t ran h t = ran h
164 163 imaeq2i h -1 t ran h t = h -1 ran h
165 imaiun h -1 t ran h t = t ran h h -1 t
166 164 165 eqtr3i h -1 ran h = t ran h h -1 t
167 162 166 eqtr3i dom h = t ran h h -1 t
168 28 fdmd h dom 1 dom h =
169 167 168 eqtr3id h dom 1 t ran h h -1 t =
170 169 ad2antlr φ h dom 1 v + t ran h h -1 t =
171 rabeq t ran h h -1 t = x t ran h h -1 t | F x v 3 1 v 3 h x = x | F x v 3 1 v 3 h x
172 170 171 syl φ h dom 1 v + x t ran h h -1 t | F x v 3 1 v 3 h x = x | F x v 3 1 v 3 h x
173 161 172 eqtr3id φ h dom 1 v + t ran h x h -1 t | F x v 3 1 v 3 h x = x | F x v 3 1 v 3 h x
174 fniniseg h Fn x h -1 t x h x = t
175 28 83 174 3syl h dom 1 x h -1 t x h x = t
176 175 simplbda h dom 1 x h -1 t h x = t
177 176 breq2d h dom 1 x h -1 t F x v 3 1 v 3 h x F x v 3 1 v 3 t
178 177 rabbidva h dom 1 x h -1 t | F x v 3 1 v 3 h x = x h -1 t | F x v 3 1 v 3 t
179 inrab2 x | F x v 3 1 v 3 t h -1 t = x h -1 t | F x v 3 1 v 3 t
180 imassrn h -1 t ran h -1
181 dfdm4 dom h = ran h -1
182 181 168 eqtr3id h dom 1 ran h -1 =
183 180 182 sseqtrid h dom 1 h -1 t
184 sseqin2 h -1 t h -1 t = h -1 t
185 183 184 sylib h dom 1 h -1 t = h -1 t
186 rabeq h -1 t = h -1 t x h -1 t | F x v 3 1 v 3 t = x h -1 t | F x v 3 1 v 3 t
187 185 186 syl h dom 1 x h -1 t | F x v 3 1 v 3 t = x h -1 t | F x v 3 1 v 3 t
188 179 187 syl5eq h dom 1 x | F x v 3 1 v 3 t h -1 t = x h -1 t | F x v 3 1 v 3 t
189 178 188 eqtr4d h dom 1 x h -1 t | F x v 3 1 v 3 h x = x | F x v 3 1 v 3 t h -1 t
190 189 ad3antlr φ h dom 1 v + t ran h x h -1 t | F x v 3 1 v 3 h x = x | F x v 3 1 v 3 t h -1 t
191 26 adantlr φ h dom 1 v + t ran h x F x v 3 1
192 46 ad2antlr φ h dom 1 v + ran h
193 192 sselda φ h dom 1 v + t ran h t
194 193 adantr φ h dom 1 v + t ran h x t
195 69 ad3antlr φ h dom 1 v + t ran h x v 3 +
196 191 194 195 lemuldivd φ h dom 1 v + t ran h x F x v 3 1 v 3 t F x v 3 1 t v 3
197 24 adantlr φ h dom 1 v + t ran h x F x v 3
198 1red φ h dom 1 v + t ran h x 1
199 14 ad3antlr φ h dom 1 v + t ran h x v 3
200 20 ad3antlr φ h dom 1 v + t ran h x v 3 0
201 194 199 200 redivcld φ h dom 1 v + t ran h x t v 3
202 197 198 201 lesubaddd φ h dom 1 v + t ran h x F x v 3 1 t v 3 F x v 3 t v 3 + 1
203 7 adantlr φ h dom 1 v + t ran h x F x
204 peano2re t v 3 t v 3 + 1
205 201 204 syl φ h dom 1 v + t ran h x t v 3 + 1
206 reflcl t v 3 + 1 t v 3 + 1
207 205 206 syl φ h dom 1 v + t ran h x t v 3 + 1
208 peano2re t v 3 + 1 t v 3 + 1 + 1
209 207 208 syl φ h dom 1 v + t ran h x t v 3 + 1 + 1
210 203 209 195 ltdivmuld φ h dom 1 v + t ran h x F x v 3 < t v 3 + 1 + 1 F x < v 3 t v 3 + 1 + 1
211 22 adantlr φ h dom 1 v + t ran h x F x v 3
212 flflp1 F x v 3 t v 3 + 1 F x v 3 t v 3 + 1 F x v 3 < t v 3 + 1 + 1
213 211 205 212 syl2anc φ h dom 1 v + t ran h x F x v 3 t v 3 + 1 F x v 3 < t v 3 + 1 + 1
214 199 209 remulcld φ h dom 1 v + t ran h x v 3 t v 3 + 1 + 1
215 214 rexrd φ h dom 1 v + t ran h x v 3 t v 3 + 1 + 1 *
216 elioomnf v 3 t v 3 + 1 + 1 * F x −∞ v 3 t v 3 + 1 + 1 F x F x < v 3 t v 3 + 1 + 1
217 215 216 syl φ h dom 1 v + t ran h x F x −∞ v 3 t v 3 + 1 + 1 F x F x < v 3 t v 3 + 1 + 1
218 203 biantrurd φ h dom 1 v + t ran h x F x < v 3 t v 3 + 1 + 1 F x F x < v 3 t v 3 + 1 + 1
219 217 218 bitr4d φ h dom 1 v + t ran h x F x −∞ v 3 t v 3 + 1 + 1 F x < v 3 t v 3 + 1 + 1
220 210 213 219 3bitr4d φ h dom 1 v + t ran h x F x v 3 t v 3 + 1 F x −∞ v 3 t v 3 + 1 + 1
221 196 202 220 3bitrd φ h dom 1 v + t ran h x F x v 3 1 v 3 t F x −∞ v 3 t v 3 + 1 + 1
222 221 rabbidva φ h dom 1 v + t ran h x | F x v 3 1 v 3 t = x | F x −∞ v 3 t v 3 + 1 + 1
223 2 feqmptd φ F = x F x
224 223 cnveqd φ F -1 = x F x -1
225 224 imaeq1d φ F -1 −∞ v 3 t v 3 + 1 + 1 = x F x -1 −∞ v 3 t v 3 + 1 + 1
226 eqid x F x = x F x
227 226 mptpreima x F x -1 −∞ v 3 t v 3 + 1 + 1 = x | F x −∞ v 3 t v 3 + 1 + 1
228 225 227 eqtrdi φ F -1 −∞ v 3 t v 3 + 1 + 1 = x | F x −∞ v 3 t v 3 + 1 + 1
229 228 ad3antrrr φ h dom 1 v + t ran h F -1 −∞ v 3 t v 3 + 1 + 1 = x | F x −∞ v 3 t v 3 + 1 + 1
230 222 229 eqtr4d φ h dom 1 v + t ran h x | F x v 3 1 v 3 t = F -1 −∞ v 3 t v 3 + 1 + 1
231 mbfima F MblFn F : F -1 −∞ v 3 t v 3 + 1 + 1 dom vol
232 1 5 231 syl2anc φ F -1 −∞ v 3 t v 3 + 1 + 1 dom vol
233 232 ad3antrrr φ h dom 1 v + t ran h F -1 −∞ v 3 t v 3 + 1 + 1 dom vol
234 230 233 eqeltrd φ h dom 1 v + t ran h x | F x v 3 1 v 3 t dom vol
235 46 sseld h dom 1 t ran h t
236 235 ad2antlr φ h dom 1 v + t ran h t
237 236 imdistani φ h dom 1 v + t ran h φ h dom 1 v + t
238 i1fmbf h dom 1 h MblFn
239 238 28 jca h dom 1 h MblFn h :
240 239 ad2antlr φ h dom 1 v + h MblFn h :
241 mbfimasn h MblFn h : t h -1 t dom vol
242 241 3expa h MblFn h : t h -1 t dom vol
243 240 242 sylan φ h dom 1 v + t h -1 t dom vol
244 237 243 syl φ h dom 1 v + t ran h h -1 t dom vol
245 inmbl x | F x v 3 1 v 3 t dom vol h -1 t dom vol x | F x v 3 1 v 3 t h -1 t dom vol
246 234 244 245 syl2anc φ h dom 1 v + t ran h x | F x v 3 1 v 3 t h -1 t dom vol
247 190 246 eqeltrd φ h dom 1 v + t ran h x h -1 t | F x v 3 1 v 3 h x dom vol
248 247 ralrimiva φ h dom 1 v + t ran h x h -1 t | F x v 3 1 v 3 h x dom vol
249 finiunmbl ran h Fin t ran h x h -1 t | F x v 3 1 v 3 h x dom vol t ran h x h -1 t | F x v 3 1 v 3 h x dom vol
250 42 248 249 syl2anc φ h dom 1 v + t ran h x h -1 t | F x v 3 1 v 3 h x dom vol
251 173 250 eqeltrrd φ h dom 1 v + x | F x v 3 1 v 3 h x dom vol
252 unrab x | h x −∞ 0 x | h x 0 +∞ = x | h x −∞ 0 h x 0 +∞
253 28 feqmptd h dom 1 h = x h x
254 253 cnveqd h dom 1 h -1 = x h x -1
255 254 imaeq1d h dom 1 h -1 −∞ 0 = x h x -1 −∞ 0
256 eqid x h x = x h x
257 256 mptpreima x h x -1 −∞ 0 = x | h x −∞ 0
258 255 257 eqtrdi h dom 1 h -1 −∞ 0 = x | h x −∞ 0
259 254 imaeq1d h dom 1 h -1 0 +∞ = x h x -1 0 +∞
260 256 mptpreima x h x -1 0 +∞ = x | h x 0 +∞
261 259 260 eqtrdi h dom 1 h -1 0 +∞ = x | h x 0 +∞
262 258 261 uneq12d h dom 1 h -1 −∞ 0 h -1 0 +∞ = x | h x −∞ 0 x | h x 0 +∞
263 28 ffvelrnda h dom 1 x h x
264 0re 0
265 lttri2 h x 0 h x 0 h x < 0 0 < h x
266 264 265 mpan2 h x h x 0 h x < 0 0 < h x
267 ibar h x h x < 0 0 < h x h x h x < 0 0 < h x
268 andi h x h x < 0 0 < h x h x h x < 0 h x 0 < h x
269 0xr 0 *
270 elioomnf 0 * h x −∞ 0 h x h x < 0
271 elioopnf 0 * h x 0 +∞ h x 0 < h x
272 270 271 orbi12d 0 * h x −∞ 0 h x 0 +∞ h x h x < 0 h x 0 < h x
273 269 272 ax-mp h x −∞ 0 h x 0 +∞ h x h x < 0 h x 0 < h x
274 268 273 bitr4i h x h x < 0 0 < h x h x −∞ 0 h x 0 +∞
275 267 274 bitrdi h x h x < 0 0 < h x h x −∞ 0 h x 0 +∞
276 266 275 bitrd h x h x 0 h x −∞ 0 h x 0 +∞
277 263 276 syl h dom 1 x h x 0 h x −∞ 0 h x 0 +∞
278 277 rabbidva h dom 1 x | h x 0 = x | h x −∞ 0 h x 0 +∞
279 252 262 278 3eqtr4a h dom 1 h -1 −∞ 0 h -1 0 +∞ = x | h x 0
280 i1fima h dom 1 h -1 −∞ 0 dom vol
281 i1fima h dom 1 h -1 0 +∞ dom vol
282 unmbl h -1 −∞ 0 dom vol h -1 0 +∞ dom vol h -1 −∞ 0 h -1 0 +∞ dom vol
283 280 281 282 syl2anc h dom 1 h -1 −∞ 0 h -1 0 +∞ dom vol
284 279 283 eqeltrrd h dom 1 x | h x 0 dom vol
285 284 ad2antlr φ h dom 1 v + x | h x 0 dom vol
286 inmbl x | F x v 3 1 v 3 h x dom vol x | h x 0 dom vol x | F x v 3 1 v 3 h x x | h x 0 dom vol
287 251 285 286 syl2anc φ h dom 1 v + x | F x v 3 1 v 3 h x x | h x 0 dom vol
288 287 adantr φ h dom 1 v + t x | F x v 3 1 v 3 h x x | h x 0 dom vol
289 24 recnd φ h dom 1 v + x F x v 3
290 289 adantlr φ h dom 1 v + t x F x v 3
291 1cnd φ h dom 1 v + t x 1
292 simplr φ h dom 1 v + t x t
293 14 ad3antlr φ h dom 1 v + t x v 3
294 20 ad3antlr φ h dom 1 v + t x v 3 0
295 292 293 294 redivcld φ h dom 1 v + t x t v 3
296 295 recnd φ h dom 1 v + t x t v 3
297 290 291 296 subadd2d φ h dom 1 v + t x F x v 3 1 = t v 3 t v 3 + 1 = F x v 3
298 eqcom F x v 3 1 = t v 3 t v 3 = F x v 3 1
299 recn t t
300 299 ad2antlr φ h dom 1 v + t x t
301 26 recnd φ h dom 1 v + x F x v 3 1
302 301 adantlr φ h dom 1 v + t x F x v 3 1
303 14 recnd v + v 3
304 303 ad3antlr φ h dom 1 v + t x v 3
305 300 302 304 294 divmul3d φ h dom 1 v + t x t v 3 = F x v 3 1 t = F x v 3 1 v 3
306 298 305 syl5bb φ h dom 1 v + t x F x v 3 1 = t v 3 t = F x v 3 1 v 3
307 297 306 bitr3d φ h dom 1 v + t x t v 3 + 1 = F x v 3 t = F x v 3 1 v 3
308 307 rabbidva φ h dom 1 v + t x | t v 3 + 1 = F x v 3 = x | t = F x v 3 1 v 3
309 imaundi F -1 t v 3 + 1 v 3 t v 3 + 1 v 3 v 3 t v 3 + 1 + 1 = F -1 t v 3 + 1 v 3 F -1 t v 3 + 1 v 3 v 3 t v 3 + 1 + 1
310 224 ad4antr φ h dom 1 v + t t v 3 + 1 F -1 = x F x -1
311 zre t v 3 + 1 t v 3 + 1
312 311 adantl φ h dom 1 v + t t v 3 + 1 t v 3 + 1
313 14 ad3antlr φ h dom 1 v + t t v 3 + 1 v 3
314 312 313 remulcld φ h dom 1 v + t t v 3 + 1 t v 3 + 1 v 3
315 314 rexrd φ h dom 1 v + t t v 3 + 1 t v 3 + 1 v 3 *
316 peano2z t v 3 + 1 t v 3 + 1 + 1
317 316 zred t v 3 + 1 t v 3 + 1 + 1
318 317 adantl φ h dom 1 v + t t v 3 + 1 t v 3 + 1 + 1
319 313 318 remulcld φ h dom 1 v + t t v 3 + 1 v 3 t v 3 + 1 + 1
320 319 rexrd φ h dom 1 v + t t v 3 + 1 v 3 t v 3 + 1 + 1 *
321 zcn t v 3 + 1 t v 3 + 1
322 321 adantl φ h dom 1 v + t t v 3 + 1 t v 3 + 1
323 303 ad3antlr φ h dom 1 v + t t v 3 + 1 v 3
324 322 323 mulcomd φ h dom 1 v + t t v 3 + 1 t v 3 + 1 v 3 = v 3 t v 3 + 1
325 69 ad3antlr φ h dom 1 v + t t v 3 + 1 v 3 +
326 311 ltp1d t v 3 + 1 t v 3 + 1 < t v 3 + 1 + 1
327 326 adantl φ h dom 1 v + t t v 3 + 1 t v 3 + 1 < t v 3 + 1 + 1
328 312 318 325 327 ltmul2dd φ h dom 1 v + t t v 3 + 1 v 3 t v 3 + 1 < v 3 t v 3 + 1 + 1
329 324 328 eqbrtrd φ h dom 1 v + t t v 3 + 1 t v 3 + 1 v 3 < v 3 t v 3 + 1 + 1
330 snunioo t v 3 + 1 v 3 * v 3 t v 3 + 1 + 1 * t v 3 + 1 v 3 < v 3 t v 3 + 1 + 1 t v 3 + 1 v 3 t v 3 + 1 v 3 v 3 t v 3 + 1 + 1 = t v 3 + 1 v 3 v 3 t v 3 + 1 + 1
331 315 320 329 330 syl3anc φ h dom 1 v + t t v 3 + 1 t v 3 + 1 v 3 t v 3 + 1 v 3 v 3 t v 3 + 1 + 1 = t v 3 + 1 v 3 v 3 t v 3 + 1 + 1
332 310 331 imaeq12d φ h dom 1 v + t t v 3 + 1 F -1 t v 3 + 1 v 3 t v 3 + 1 v 3 v 3 t v 3 + 1 + 1 = x F x -1 t v 3 + 1 v 3 v 3 t v 3 + 1 + 1
333 309 332 eqtr3id φ h dom 1 v + t t v 3 + 1 F -1 t v 3 + 1 v 3 F -1 t v 3 + 1 v 3 v 3 t v 3 + 1 + 1 = x F x -1 t v 3 + 1 v 3 v 3 t v 3 + 1 + 1
334 226 mptpreima x F x -1 t v 3 + 1 v 3 v 3 t v 3 + 1 + 1 = x | F x t v 3 + 1 v 3 v 3 t v 3 + 1 + 1
335 5 ad3antrrr φ h dom 1 v + t F :
336 335 ffvelrnda φ h dom 1 v + t x F x
337 336 3biant1d φ h dom 1 v + t x t v 3 + 1 v 3 F x F x < v 3 t v 3 + 1 + 1 F x t v 3 + 1 v 3 F x F x < v 3 t v 3 + 1 + 1
338 337 adantr φ h dom 1 v + t x t v 3 + 1 t v 3 + 1 v 3 F x F x < v 3 t v 3 + 1 + 1 F x t v 3 + 1 v 3 F x F x < v 3 t v 3 + 1 + 1
339 311 adantl φ h dom 1 v + t x t v 3 + 1 t v 3 + 1
340 336 adantr φ h dom 1 v + t x t v 3 + 1 F x
341 69 ad4antlr φ h dom 1 v + t x t v 3 + 1 v 3 +
342 339 340 341 lemuldivd φ h dom 1 v + t x t v 3 + 1 t v 3 + 1 v 3 F x t v 3 + 1 F x v 3
343 317 adantl φ h dom 1 v + t x t v 3 + 1 t v 3 + 1 + 1
344 340 343 341 ltdivmuld φ h dom 1 v + t x t v 3 + 1 F x v 3 < t v 3 + 1 + 1 F x < v 3 t v 3 + 1 + 1
345 344 bicomd φ h dom 1 v + t x t v 3 + 1 F x < v 3 t v 3 + 1 + 1 F x v 3 < t v 3 + 1 + 1
346 342 345 anbi12d φ h dom 1 v + t x t v 3 + 1 t v 3 + 1 v 3 F x F x < v 3 t v 3 + 1 + 1 t v 3 + 1 F x v 3 F x v 3 < t v 3 + 1 + 1
347 338 346 bitr3d φ h dom 1 v + t x t v 3 + 1 F x t v 3 + 1 v 3 F x F x < v 3 t v 3 + 1 + 1 t v 3 + 1 F x v 3 F x v 3 < t v 3 + 1 + 1
348 elico2 t v 3 + 1 v 3 v 3 t v 3 + 1 + 1 * F x t v 3 + 1 v 3 v 3 t v 3 + 1 + 1 F x t v 3 + 1 v 3 F x F x < v 3 t v 3 + 1 + 1
349 314 320 348 syl2anc φ h dom 1 v + t t v 3 + 1 F x t v 3 + 1 v 3 v 3 t v 3 + 1 + 1 F x t v 3 + 1 v 3 F x F x < v 3 t v 3 + 1 + 1
350 349 adantlr φ h dom 1 v + t x t v 3 + 1 F x t v 3 + 1 v 3 v 3 t v 3 + 1 + 1 F x t v 3 + 1 v 3 F x F x < v 3 t v 3 + 1 + 1
351 eqcom t v 3 + 1 = F x v 3 F x v 3 = t v 3 + 1
352 22 adantlr φ h dom 1 v + t x F x v 3
353 flbi F x v 3 t v 3 + 1 F x v 3 = t v 3 + 1 t v 3 + 1 F x v 3 F x v 3 < t v 3 + 1 + 1
354 352 353 sylan φ h dom 1 v + t x t v 3 + 1 F x v 3 = t v 3 + 1 t v 3 + 1 F x v 3 F x v 3 < t v 3 + 1 + 1
355 351 354 syl5bb φ h dom 1 v + t x t v 3 + 1 t v 3 + 1 = F x v 3 t v 3 + 1 F x v 3 F x v 3 < t v 3 + 1 + 1
356 347 350 355 3bitr4d φ h dom 1 v + t x t v 3 + 1 F x t v 3 + 1 v 3 v 3 t v 3 + 1 + 1 t v 3 + 1 = F x v 3
357 356 an32s φ h dom 1 v + t t v 3 + 1 x F x t v 3 + 1 v 3 v 3 t v 3 + 1 + 1 t v 3 + 1 = F x v 3
358 357 rabbidva φ h dom 1 v + t t v 3 + 1 x | F x t v 3 + 1 v 3 v 3 t v 3 + 1 + 1 = x | t v 3 + 1 = F x v 3
359 334 358 syl5eq φ h dom 1 v + t t v 3 + 1 x F x -1 t v 3 + 1 v 3 v 3 t v 3 + 1 + 1 = x | t v 3 + 1 = F x v 3
360 333 359 eqtrd φ h dom 1 v + t t v 3 + 1 F -1 t v 3 + 1 v 3 F -1 t v 3 + 1 v 3 v 3 t v 3 + 1 + 1 = x | t v 3 + 1 = F x v 3
361 1 ad4antr φ h dom 1 v + t t v 3 + 1 F MblFn
362 5 ad4antr φ h dom 1 v + t t v 3 + 1 F :
363 mbfimasn F MblFn F : t v 3 + 1 v 3 F -1 t v 3 + 1 v 3 dom vol
364 361 362 314 363 syl3anc φ h dom 1 v + t t v 3 + 1 F -1 t v 3 + 1 v 3 dom vol
365 mbfima F MblFn F : F -1 t v 3 + 1 v 3 v 3 t v 3 + 1 + 1 dom vol
366 1 5 365 syl2anc φ F -1 t v 3 + 1 v 3 v 3 t v 3 + 1 + 1 dom vol
367 366 ad4antr φ h dom 1 v + t t v 3 + 1 F -1 t v 3 + 1 v 3 v 3 t v 3 + 1 + 1 dom vol
368 unmbl F -1 t v 3 + 1 v 3 dom vol F -1 t v 3 + 1 v 3 v 3 t v 3 + 1 + 1 dom vol F -1 t v 3 + 1 v 3 F -1 t v 3 + 1 v 3 v 3 t v 3 + 1 + 1 dom vol
369 364 367 368 syl2anc φ h dom 1 v + t t v 3 + 1 F -1 t v 3 + 1 v 3 F -1 t v 3 + 1 v 3 v 3 t v 3 + 1 + 1 dom vol
370 360 369 eqeltrrd φ h dom 1 v + t t v 3 + 1 x | t v 3 + 1 = F x v 3 dom vol
371 simpr φ h dom 1 v + t x t v 3 + 1 = F x v 3 t v 3 + 1 = F x v 3
372 352 flcld φ h dom 1 v + t x F x v 3
373 372 adantr φ h dom 1 v + t x t v 3 + 1 = F x v 3 F x v 3
374 371 373 eqeltrd φ h dom 1 v + t x t v 3 + 1 = F x v 3 t v 3 + 1
375 374 stoic1a φ h dom 1 v + t x ¬ t v 3 + 1 ¬ t v 3 + 1 = F x v 3
376 375 an32s φ h dom 1 v + t ¬ t v 3 + 1 x ¬ t v 3 + 1 = F x v 3
377 376 ralrimiva φ h dom 1 v + t ¬ t v 3 + 1 x ¬ t v 3 + 1 = F x v 3
378 rabeq0 x | t v 3 + 1 = F x v 3 = x ¬ t v 3 + 1 = F x v 3
379 377 378 sylibr φ h dom 1 v + t ¬ t v 3 + 1 x | t v 3 + 1 = F x v 3 =
380 0mbl dom vol
381 379 380 eqeltrdi φ h dom 1 v + t ¬ t v 3 + 1 x | t v 3 + 1 = F x v 3 dom vol
382 370 381 pm2.61dan φ h dom 1 v + t x | t v 3 + 1 = F x v 3 dom vol
383 308 382 eqeltrrd φ h dom 1 v + t x | t = F x v 3 1 v 3 dom vol
384 inmbl x | F x v 3 1 v 3 h x x | h x 0 dom vol x | t = F x v 3 1 v 3 dom vol x | F x v 3 1 v 3 h x x | h x 0 x | t = F x v 3 1 v 3 dom vol
385 288 383 384 syl2anc φ h dom 1 v + t x | F x v 3 1 v 3 h x x | h x 0 x | t = F x v 3 1 v 3 dom vol
386 rabiun x t ran h h -1 t | ¬ F x v 3 1 v 3 h x = t ran h x h -1 t | ¬ F x v 3 1 v 3 h x
387 rabeq t ran h h -1 t = x t ran h h -1 t | ¬ F x v 3 1 v 3 h x = x | ¬ F x v 3 1 v 3 h x
388 169 387 syl h dom 1 x t ran h h -1 t | ¬ F x v 3 1 v 3 h x = x | ¬ F x v 3 1 v 3 h x
389 386 388 eqtr3id h dom 1 t ran h x h -1 t | ¬ F x v 3 1 v 3 h x = x | ¬ F x v 3 1 v 3 h x
390 389 ad2antlr φ h dom 1 v + t ran h x h -1 t | ¬ F x v 3 1 v 3 h x = x | ¬ F x v 3 1 v 3 h x
391 177 notbid h dom 1 x h -1 t ¬ F x v 3 1 v 3 h x ¬ F x v 3 1 v 3 t
392 391 rabbidva h dom 1 x h -1 t | ¬ F x v 3 1 v 3 h x = x h -1 t | ¬ F x v 3 1 v 3 t
393 inrab2 x | ¬ F x v 3 1 v 3 t h -1 t = x h -1 t | ¬ F x v 3 1 v 3 t
394 rabeq h -1 t = h -1 t x h -1 t | ¬ F x v 3 1 v 3 t = x h -1 t | ¬ F x v 3 1 v 3 t
395 185 394 syl h dom 1 x h -1 t | ¬ F x v 3 1 v 3 t = x h -1 t | ¬ F x v 3 1 v 3 t
396 393 395 syl5eq h dom 1 x | ¬ F x v 3 1 v 3 t h -1 t = x h -1 t | ¬ F x v 3 1 v 3 t
397 392 396 eqtr4d h dom 1 x h -1 t | ¬ F x v 3 1 v 3 h x = x | ¬ F x v 3 1 v 3 t h -1 t
398 397 ad3antlr φ h dom 1 v + t ran h x h -1 t | ¬ F x v 3 1 v 3 h x = x | ¬ F x v 3 1 v 3 t h -1 t
399 imaundi F -1 t v 3 + 1 + 1 v 3 t v 3 + 1 + 1 v 3 +∞ = F -1 t v 3 + 1 + 1 v 3 F -1 t v 3 + 1 + 1 v 3 +∞
400 14 20 jca v + v 3 v 3 0
401 redivcl t v 3 v 3 0 t v 3
402 401 3expb t v 3 v 3 0 t v 3
403 400 402 sylan2 t v + t v 3
404 403 ancoms v + t t v 3
405 404 adantll φ h dom 1 v + t t v 3
406 405 204 syl φ h dom 1 v + t t v 3 + 1
407 peano2re t v 3 + 1 t v 3 + 1 + 1
408 reflcl t v 3 + 1 + 1 t v 3 + 1 + 1
409 406 407 408 3syl φ h dom 1 v + t t v 3 + 1 + 1
410 14 ad2antlr φ h dom 1 v + t v 3
411 409 410 remulcld φ h dom 1 v + t t v 3 + 1 + 1 v 3
412 411 rexrd φ h dom 1 v + t t v 3 + 1 + 1 v 3 *
413 pnfxr +∞ *
414 413 a1i φ h dom 1 v + t +∞ *
415 ltpnf t v 3 + 1 + 1 v 3 t v 3 + 1 + 1 v 3 < +∞
416 411 415 syl φ h dom 1 v + t t v 3 + 1 + 1 v 3 < +∞
417 snunioo t v 3 + 1 + 1 v 3 * +∞ * t v 3 + 1 + 1 v 3 < +∞ t v 3 + 1 + 1 v 3 t v 3 + 1 + 1 v 3 +∞ = t v 3 + 1 + 1 v 3 +∞
418 412 414 416 417 syl3anc φ h dom 1 v + t t v 3 + 1 + 1 v 3 t v 3 + 1 + 1 v 3 +∞ = t v 3 + 1 + 1 v 3 +∞
419 418 imaeq2d φ h dom 1 v + t F -1 t v 3 + 1 + 1 v 3 t v 3 + 1 + 1 v 3 +∞ = F -1 t v 3 + 1 + 1 v 3 +∞
420 399 419 eqtr3id φ h dom 1 v + t F -1 t v 3 + 1 + 1 v 3 F -1 t v 3 + 1 + 1 v 3 +∞ = F -1 t v 3 + 1 + 1 v 3 +∞
421 224 imaeq1d φ F -1 t v 3 + 1 + 1 v 3 +∞ = x F x -1 t v 3 + 1 + 1 v 3 +∞
422 226 mptpreima x F x -1 t v 3 + 1 + 1 v 3 +∞ = x | F x t v 3 + 1 + 1 v 3 +∞
423 421 422 eqtrdi φ F -1 t v 3 + 1 + 1 v 3 +∞ = x | F x t v 3 + 1 + 1 v 3 +∞
424 423 ad3antrrr φ h dom 1 v + t F -1 t v 3 + 1 + 1 v 3 +∞ = x | F x t v 3 + 1 + 1 v 3 +∞
425 406 407 syl φ h dom 1 v + t t v 3 + 1 + 1
426 425 adantr φ h dom 1 v + t x t v 3 + 1 + 1
427 flflp1 t v 3 + 1 + 1 F x v 3 t v 3 + 1 + 1 F x v 3 t v 3 + 1 + 1 < F x v 3 + 1
428 426 352 427 syl2anc φ h dom 1 v + t x t v 3 + 1 + 1 F x v 3 t v 3 + 1 + 1 < F x v 3 + 1
429 411 adantr φ h dom 1 v + t x t v 3 + 1 + 1 v 3
430 elicopnf t v 3 + 1 + 1 v 3 F x t v 3 + 1 + 1 v 3 +∞ F x t v 3 + 1 + 1 v 3 F x
431 429 430 syl φ h dom 1 v + t x F x t v 3 + 1 + 1 v 3 +∞ F x t v 3 + 1 + 1 v 3 F x
432 336 biantrurd φ h dom 1 v + t x t v 3 + 1 + 1 v 3 F x F x t v 3 + 1 + 1 v 3 F x
433 409 adantr φ h dom 1 v + t x t v 3 + 1 + 1
434 69 ad3antlr φ h dom 1 v + t x v 3 +
435 433 336 434 lemuldivd φ h dom 1 v + t x t v 3 + 1 + 1 v 3 F x t v 3 + 1 + 1 F x v 3
436 431 432 435 3bitr2d φ h dom 1 v + t x F x t v 3 + 1 + 1 v 3 +∞ t v 3 + 1 + 1 F x v 3
437 406 adantr φ h dom 1 v + t x t v 3 + 1
438 352 23 syl φ h dom 1 v + t x F x v 3
439 1red φ h dom 1 v + t x 1
440 437 438 439 ltadd1d φ h dom 1 v + t x t v 3 + 1 < F x v 3 t v 3 + 1 + 1 < F x v 3 + 1
441 428 436 440 3bitr4d φ h dom 1 v + t x F x t v 3 + 1 + 1 v 3 +∞ t v 3 + 1 < F x v 3
442 295 439 438 ltaddsubd φ h dom 1 v + t x t v 3 + 1 < F x v 3 t v 3 < F x v 3 1
443 441 442 bitrd φ h dom 1 v + t x F x t v 3 + 1 + 1 v 3 +∞ t v 3 < F x v 3 1
444 438 25 syl φ h dom 1 v + t x F x v 3 1
445 292 444 434 ltdivmul2d φ h dom 1 v + t x t v 3 < F x v 3 1 t < F x v 3 1 v 3
446 444 293 remulcld φ h dom 1 v + t x F x v 3 1 v 3
447 292 446 ltnled φ h dom 1 v + t x t < F x v 3 1 v 3 ¬ F x v 3 1 v 3 t
448 443 445 447 3bitrd φ h dom 1 v + t x F x t v 3 + 1 + 1 v 3 +∞ ¬ F x v 3 1 v 3 t
449 448 rabbidva φ h dom 1 v + t x | F x t v 3 + 1 + 1 v 3 +∞ = x | ¬ F x v 3 1 v 3 t
450 420 424 449 3eqtrd φ h dom 1 v + t F -1 t v 3 + 1 + 1 v 3 F -1 t v 3 + 1 + 1 v 3 +∞ = x | ¬ F x v 3 1 v 3 t
451 1 ad3antrrr φ h dom 1 v + t F MblFn
452 mbfimasn F MblFn F : t v 3 + 1 + 1 v 3 F -1 t v 3 + 1 + 1 v 3 dom vol
453 451 335 411 452 syl3anc φ h dom 1 v + t F -1 t v 3 + 1 + 1 v 3 dom vol
454 mbfima F MblFn F : F -1 t v 3 + 1 + 1 v 3 +∞ dom vol
455 1 5 454 syl2anc φ F -1 t v 3 + 1 + 1 v 3 +∞ dom vol
456 455 ad3antrrr φ h dom 1 v + t F -1 t v 3 + 1 + 1 v 3 +∞ dom vol
457 unmbl F -1 t v 3 + 1 + 1 v 3 dom vol F -1 t v 3 + 1 + 1 v 3 +∞ dom vol F -1 t v 3 + 1 + 1 v 3 F -1 t v 3 + 1 + 1 v 3 +∞ dom vol
458 453 456 457 syl2anc φ h dom 1 v + t F -1 t v 3 + 1 + 1 v 3 F -1 t v 3 + 1 + 1 v 3 +∞ dom vol
459 450 458 eqeltrrd φ h dom 1 v + t x | ¬ F x v 3 1 v 3 t dom vol
460 237 459 syl φ h dom 1 v + t ran h x | ¬ F x v 3 1 v 3 t dom vol
461 inmbl x | ¬ F x v 3 1 v 3 t dom vol h -1 t dom vol x | ¬ F x v 3 1 v 3 t h -1 t dom vol
462 460 244 461 syl2anc φ h dom 1 v + t ran h x | ¬ F x v 3 1 v 3 t h -1 t dom vol
463 398 462 eqeltrd φ h dom 1 v + t ran h x h -1 t | ¬ F x v 3 1 v 3 h x dom vol
464 463 ralrimiva φ h dom 1 v + t ran h x h -1 t | ¬ F x v 3 1 v 3 h x dom vol
465 finiunmbl ran h Fin t ran h x h -1 t | ¬ F x v 3 1 v 3 h x dom vol t ran h x h -1 t | ¬ F x v 3 1 v 3 h x dom vol
466 42 464 465 syl2anc φ h dom 1 v + t ran h x h -1 t | ¬ F x v 3 1 v 3 h x dom vol
467 390 466 eqeltrrd φ h dom 1 v + x | ¬ F x v 3 1 v 3 h x dom vol
468 254 imaeq1d h dom 1 h -1 0 = x h x -1 0
469 256 mptpreima x h x -1 0 = x | h x 0
470 141 elsn h x 0 h x = 0
471 470 rabbii x | h x 0 = x | h x = 0
472 469 471 eqtri x h x -1 0 = x | h x = 0
473 468 472 eqtrdi h dom 1 h -1 0 = x | h x = 0
474 i1fima h dom 1 h -1 0 dom vol
475 473 474 eqeltrrd h dom 1 x | h x = 0 dom vol
476 475 ad2antlr φ h dom 1 v + x | h x = 0 dom vol
477 unmbl x | ¬ F x v 3 1 v 3 h x dom vol x | h x = 0 dom vol x | ¬ F x v 3 1 v 3 h x x | h x = 0 dom vol
478 467 476 477 syl2anc φ h dom 1 v + x | ¬ F x v 3 1 v 3 h x x | h x = 0 dom vol
479 478 adantr φ h dom 1 v + t x | ¬ F x v 3 1 v 3 h x x | h x = 0 dom vol
480 254 imaeq1d h dom 1 h -1 t = x h x -1 t
481 256 mptpreima x h x -1 t = x | h x t
482 141 elsn h x t h x = t
483 eqcom h x = t t = h x
484 482 483 bitri h x t t = h x
485 484 rabbii x | h x t = x | t = h x
486 481 485 eqtri x h x -1 t = x | t = h x
487 480 486 eqtrdi h dom 1 h -1 t = x | t = h x
488 487 ad3antlr φ h dom 1 v + t h -1 t = x | t = h x
489 488 243 eqeltrrd φ h dom 1 v + t x | t = h x dom vol
490 inmbl x | ¬ F x v 3 1 v 3 h x x | h x = 0 dom vol x | t = h x dom vol x | ¬ F x v 3 1 v 3 h x x | h x = 0 x | t = h x dom vol
491 479 489 490 syl2anc φ h dom 1 v + t x | ¬ F x v 3 1 v 3 h x x | h x = 0 x | t = h x dom vol
492 unmbl x | F x v 3 1 v 3 h x x | h x 0 x | t = F x v 3 1 v 3 dom vol x | ¬ F x v 3 1 v 3 h x x | h x = 0 x | t = h x dom vol x | F x v 3 1 v 3 h x x | h x 0 x | t = F x v 3 1 v 3 x | ¬ F x v 3 1 v 3 h x x | h x = 0 x | t = h x dom vol
493 385 491 492 syl2anc φ h dom 1 v + t x | F x v 3 1 v 3 h x x | h x 0 x | t = F x v 3 1 v 3 x | ¬ F x v 3 1 v 3 h x x | h x = 0 x | t = h x dom vol
494 160 493 syl φ h dom 1 v + t ran x if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x 0 x | F x v 3 1 v 3 h x x | h x 0 x | t = F x v 3 1 v 3 x | ¬ F x v 3 1 v 3 h x x | h x = 0 x | t = h x dom vol
495 155 494 eqeltrid φ h dom 1 v + t ran x if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x 0 x if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x -1 t dom vol
496 mblvol x if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x -1 t dom vol vol x if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x -1 t = vol * x if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x -1 t
497 495 496 syl φ h dom 1 v + t ran x if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x 0 vol x if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x -1 t = vol * x if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x -1 t
498 eldifsn t ran x if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x 0 t ran x if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x t 0
499 158 anim1d φ h dom 1 v + t ran x if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x t 0 t t 0
500 498 499 syl5bi φ h dom 1 v + t ran x if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x 0 t t 0
501 500 imdistani φ h dom 1 v + t ran x if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x 0 φ h dom 1 v + t t 0
502 129 a1i h dom 1 x if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x -1 t = x | if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x t
503 468 469 eqtrdi h dom 1 h -1 0 = x | h x 0
504 502 503 ineq12d h dom 1 x if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x -1 t h -1 0 = x | if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x t x | h x 0
505 inrab x | if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x t x | h x 0 = x | if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x t h x 0
506 504 505 eqtrdi h dom 1 x if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x -1 t h -1 0 = x | if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x t h x 0
507 506 ad3antlr φ h dom 1 v + t t 0 x if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x -1 t h -1 0 = x | if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x t h x 0
508 145 biimpri h x = 0 ¬ h x 0
509 508 intnand h x = 0 ¬ F x v 3 1 v 3 h x h x 0
510 509 iffalsed h x = 0 if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x = h x
511 eqtr if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x = h x h x = 0 if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x = 0
512 510 511 mpancom h x = 0 if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x = 0
513 512 adantl t 0 x h x = 0 if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x = 0
514 simpll t 0 x h x = 0 t 0
515 514 necomd t 0 x h x = 0 0 t
516 513 515 eqnetrd t 0 x h x = 0 if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x t
517 516 ex t 0 x h x = 0 if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x t
518 orcom ¬ if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x t ¬ h x 0 ¬ h x 0 ¬ if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x t
519 ianor ¬ if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x t h x 0 ¬ if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x t ¬ h x 0
520 imor h x 0 ¬ if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x t ¬ h x 0 ¬ if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x t
521 518 519 520 3bitr4i ¬ if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x t h x 0 h x 0 ¬ if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x t
522 143 necon3bbii ¬ if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x t if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x t
523 470 522 imbi12i h x 0 ¬ if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x t h x = 0 if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x t
524 521 523 bitri ¬ if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x t h x 0 h x = 0 if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x t
525 517 524 sylibr t 0 x ¬ if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x t h x 0
526 525 ralrimiva t 0 x ¬ if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x t h x 0
527 rabeq0 x | if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x t h x 0 = x ¬ if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x t h x 0
528 526 527 sylibr t 0 x | if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x t h x 0 =
529 528 ad2antll φ h dom 1 v + t t 0 x | if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x t h x 0 =
530 507 529 eqtrd φ h dom 1 v + t t 0 x if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x -1 t h -1 0 =
531 imassrn x if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x -1 t ran x if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x -1
532 dfdm4 dom x if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x = ran x if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x -1
533 142 128 dmmpti dom x if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x =
534 532 533 eqtr3i ran x if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x -1 =
535 531 534 sseqtri x if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x -1 t
536 reldisj x if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x -1 t x if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x -1 t h -1 0 = x if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x -1 t h -1 0
537 535 536 ax-mp x if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x -1 t h -1 0 = x if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x -1 t h -1 0
538 530 537 sylib φ h dom 1 v + t t 0 x if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x -1 t h -1 0
539 ffun h : Fun h
540 difpreima Fun h h -1 ran h 0 = h -1 ran h h -1 0
541 539 540 syl h : h -1 ran h 0 = h -1 ran h h -1 0
542 fdm h : dom h =
543 162 542 syl5eq h : h -1 ran h =
544 543 difeq1d h : h -1 ran h h -1 0 = h -1 0
545 541 544 eqtrd h : h -1 ran h 0 = h -1 0
546 28 545 syl h dom 1 h -1 ran h 0 = h -1 0
547 546 ad3antlr φ h dom 1 v + t t 0 h -1 ran h 0 = h -1 0
548 538 547 sseqtrrd φ h dom 1 v + t t 0 x if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x -1 t h -1 ran h 0
549 imassrn h -1 ran h 0 ran h -1
550 549 182 sseqtrid h dom 1 h -1 ran h 0
551 550 ad3antlr φ h dom 1 v + t t 0 h -1 ran h 0
552 i1fima h dom 1 h -1 ran h 0 dom vol
553 mblvol h -1 ran h 0 dom vol vol h -1 ran h 0 = vol * h -1 ran h 0
554 552 553 syl h dom 1 vol h -1 ran h 0 = vol * h -1 ran h 0
555 neldifsn ¬ 0 ran h 0
556 i1fima2 h dom 1 ¬ 0 ran h 0 vol h -1 ran h 0
557 555 556 mpan2 h dom 1 vol h -1 ran h 0
558 554 557 eqeltrrd h dom 1 vol * h -1 ran h 0
559 558 ad3antlr φ h dom 1 v + t t 0 vol * h -1 ran h 0
560 ovolsscl x if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x -1 t h -1 ran h 0 h -1 ran h 0 vol * h -1 ran h 0 vol * x if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x -1 t
561 548 551 559 560 syl3anc φ h dom 1 v + t t 0 vol * x if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x -1 t
562 501 561 syl φ h dom 1 v + t ran x if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x 0 vol * x if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x -1 t
563 497 562 eqeltrd φ h dom 1 v + t ran x if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x 0 vol x if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x -1 t
564 32 127 495 563 i1fd φ h dom 1 v + x if F x v 3 1 v 3 h x h x 0 F x v 3 1 v 3 h x dom 1