Step |
Hyp |
Ref |
Expression |
1 |
|
hofcl.m |
|
2 |
|
hofcl.o |
|
3 |
|
hofcl.d |
|
4 |
|
hofcl.c |
|
5 |
|
hofcl.u |
|
6 |
|
hofcl.h |
|
7 |
|
eqid |
|
8 |
|
eqid |
|
9 |
|
eqid |
|
10 |
1 4 7 8 9
|
hofval |
|
11 |
|
fvex |
|
12 |
|
fvex |
|
13 |
12 12
|
xpex |
|
14 |
13 13
|
mpoex |
|
15 |
11 14
|
op2ndd |
|
16 |
10 15
|
syl |
|
17 |
16
|
opeq2d |
|
18 |
10 17
|
eqtr4d |
|
19 |
|
eqid |
|
20 |
2 7
|
oppcbas |
|
21 |
19 20 7
|
xpcbas |
|
22 |
|
eqid |
|
23 |
|
eqid |
|
24 |
|
eqid |
|
25 |
|
eqid |
|
26 |
|
eqid |
|
27 |
|
eqid |
|
28 |
|
eqid |
|
29 |
2
|
oppccat |
|
30 |
4 29
|
syl |
|
31 |
19 30 4
|
xpccat |
|
32 |
3
|
setccat |
|
33 |
5 32
|
syl |
|
34 |
|
eqid |
|
35 |
34 7
|
homffn |
|
36 |
35
|
a1i |
|
37 |
|
df-f |
|
38 |
36 6 37
|
sylanbrc |
|
39 |
3 5
|
setcbas |
|
40 |
39
|
feq3d |
|
41 |
38 40
|
mpbid |
|
42 |
|
eqid |
|
43 |
|
ovex |
|
44 |
|
ovex |
|
45 |
43 44
|
mpoex |
|
46 |
42 45
|
fnmpoi |
|
47 |
16
|
fneq1d |
|
48 |
46 47
|
mpbiri |
|
49 |
4
|
ad3antrrr |
|
50 |
|
simplrr |
|
51 |
|
xp1st |
|
52 |
50 51
|
syl |
|
53 |
52
|
adantr |
|
54 |
|
simplrl |
|
55 |
|
xp1st |
|
56 |
54 55
|
syl |
|
57 |
56
|
adantr |
|
58 |
|
xp2nd |
|
59 |
50 58
|
syl |
|
60 |
59
|
adantr |
|
61 |
|
simplrl |
|
62 |
|
1st2nd2 |
|
63 |
54 62
|
syl |
|
64 |
63
|
adantr |
|
65 |
64
|
oveq1d |
|
66 |
65
|
oveqd |
|
67 |
|
xp2nd |
|
68 |
54 67
|
syl |
|
69 |
68
|
adantr |
|
70 |
63
|
fveq2d |
|
71 |
|
df-ov |
|
72 |
70 71
|
eqtr4di |
|
73 |
72
|
eleq2d |
|
74 |
73
|
biimpa |
|
75 |
|
simplrr |
|
76 |
7 8 9 49 57 69 60 74 75
|
catcocl |
|
77 |
66 76
|
eqeltrd |
|
78 |
7 8 9 49 53 57 60 61 77
|
catcocl |
|
79 |
|
1st2nd2 |
|
80 |
50 79
|
syl |
|
81 |
80
|
fveq2d |
|
82 |
|
df-ov |
|
83 |
81 82
|
eqtr4di |
|
84 |
83
|
adantr |
|
85 |
78 84
|
eleqtrrd |
|
86 |
85
|
fmpttd |
|
87 |
5
|
ad2antrr |
|
88 |
34 7 8 56 68
|
homfval |
|
89 |
63
|
fveq2d |
|
90 |
|
df-ov |
|
91 |
89 90
|
eqtr4di |
|
92 |
88 91 72
|
3eqtr4d |
|
93 |
38
|
ad2antrr |
|
94 |
93 54
|
ffvelrnd |
|
95 |
92 94
|
eqeltrrd |
|
96 |
34 7 8 52 59
|
homfval |
|
97 |
80
|
fveq2d |
|
98 |
|
df-ov |
|
99 |
97 98
|
eqtr4di |
|
100 |
96 99 83
|
3eqtr4d |
|
101 |
93 50
|
ffvelrnd |
|
102 |
100 101
|
eqeltrrd |
|
103 |
3 87 24 95 102
|
elsetchom |
|
104 |
86 103
|
mpbird |
|
105 |
92 100
|
oveq12d |
|
106 |
104 105
|
eleqtrrd |
|
107 |
106
|
ralrimivva |
|
108 |
|
eqid |
|
109 |
108
|
fmpo |
|
110 |
107 109
|
sylib |
|
111 |
16
|
oveqd |
|
112 |
42
|
ovmpt4g |
|
113 |
45 112
|
mp3an3 |
|
114 |
111 113
|
sylan9eq |
|
115 |
|
eqid |
|
116 |
|
simprl |
|
117 |
|
simprr |
|
118 |
19 21 115 8 23 116 117
|
xpchom |
|
119 |
8 2
|
oppchom |
|
120 |
119
|
xpeq1i |
|
121 |
118 120
|
eqtrdi |
|
122 |
114 121
|
feq12d |
|
123 |
110 122
|
mpbird |
|
124 |
|
eqid |
|
125 |
4
|
ad2antrr |
|
126 |
55
|
adantl |
|
127 |
126
|
adantr |
|
128 |
67
|
adantl |
|
129 |
128
|
adantr |
|
130 |
|
simpr |
|
131 |
7 8 124 125 127 9 129 130
|
catlid |
|
132 |
131
|
oveq1d |
|
133 |
7 8 124 125 127 9 129 130
|
catrid |
|
134 |
132 133
|
eqtrd |
|
135 |
134
|
mpteq2dva |
|
136 |
|
df-ov |
|
137 |
4
|
adantr |
|
138 |
7 8 124 137 126
|
catidcl |
|
139 |
7 8 124 137 128
|
catidcl |
|
140 |
1 137 7 8 126 128 126 128 9 138 139
|
hof2val |
|
141 |
136 140
|
syl5eqr |
|
142 |
62
|
adantl |
|
143 |
142
|
fveq2d |
|
144 |
143 90
|
eqtr4di |
|
145 |
34 7 8 126 128
|
homfval |
|
146 |
144 145
|
eqtrd |
|
147 |
146
|
reseq2d |
|
148 |
|
mptresid |
|
149 |
147 148
|
eqtrdi |
|
150 |
135 141 149
|
3eqtr4d |
|
151 |
142 142
|
oveq12d |
|
152 |
142
|
fveq2d |
|
153 |
30
|
adantr |
|
154 |
|
eqid |
|
155 |
19 153 137 20 7 154 124 25 126 128
|
xpcid |
|
156 |
2 124
|
oppcid |
|
157 |
137 156
|
syl |
|
158 |
157
|
fveq1d |
|
159 |
158
|
opeq1d |
|
160 |
152 155 159
|
3eqtrd |
|
161 |
151 160
|
fveq12d |
|
162 |
5
|
adantr |
|
163 |
38
|
ffvelrnda |
|
164 |
3 26 162 163
|
setcid |
|
165 |
150 161 164
|
3eqtr4d |
|
166 |
4
|
3ad2ant1 |
|
167 |
5
|
3ad2ant1 |
|
168 |
6
|
3ad2ant1 |
|
169 |
|
simp21 |
|
170 |
169 55
|
syl |
|
171 |
169 67
|
syl |
|
172 |
|
simp22 |
|
173 |
172 51
|
syl |
|
174 |
172 58
|
syl |
|
175 |
|
simp23 |
|
176 |
|
xp1st |
|
177 |
175 176
|
syl |
|
178 |
|
xp2nd |
|
179 |
175 178
|
syl |
|
180 |
|
simp3l |
|
181 |
19 21 115 8 23 169 172
|
xpchom |
|
182 |
180 181
|
eleqtrd |
|
183 |
|
xp1st |
|
184 |
182 183
|
syl |
|
185 |
184 119
|
eleqtrdi |
|
186 |
|
xp2nd |
|
187 |
182 186
|
syl |
|
188 |
|
simp3r |
|
189 |
19 21 115 8 23 172 175
|
xpchom |
|
190 |
188 189
|
eleqtrd |
|
191 |
|
xp1st |
|
192 |
190 191
|
syl |
|
193 |
8 2
|
oppchom |
|
194 |
192 193
|
eleqtrdi |
|
195 |
|
xp2nd |
|
196 |
190 195
|
syl |
|
197 |
1 2 3 166 167 168 7 8 170 171 173 174 177 179 185 187 194 196
|
hofcllem |
|
198 |
169 62
|
syl |
|
199 |
|
1st2nd2 |
|
200 |
175 199
|
syl |
|
201 |
198 200
|
oveq12d |
|
202 |
172 79
|
syl |
|
203 |
198 202
|
opeq12d |
|
204 |
203 200
|
oveq12d |
|
205 |
|
1st2nd2 |
|
206 |
190 205
|
syl |
|
207 |
|
1st2nd2 |
|
208 |
182 207
|
syl |
|
209 |
204 206 208
|
oveq123d |
|
210 |
|
eqid |
|
211 |
19 20 7 115 8 170 171 173 174 210 9 27 177 179 184 187 192 196
|
xpcco2 |
|
212 |
7 9 2 170 173 177
|
oppcco |
|
213 |
212
|
opeq1d |
|
214 |
209 211 213
|
3eqtrd |
|
215 |
201 214
|
fveq12d |
|
216 |
|
df-ov |
|
217 |
215 216
|
eqtr4di |
|
218 |
198
|
fveq2d |
|
219 |
218 90
|
eqtr4di |
|
220 |
34 7 8 170 171
|
homfval |
|
221 |
219 220
|
eqtrd |
|
222 |
202
|
fveq2d |
|
223 |
222 98
|
eqtr4di |
|
224 |
34 7 8 173 174
|
homfval |
|
225 |
223 224
|
eqtrd |
|
226 |
221 225
|
opeq12d |
|
227 |
200
|
fveq2d |
|
228 |
|
df-ov |
|
229 |
227 228
|
eqtr4di |
|
230 |
34 7 8 177 179
|
homfval |
|
231 |
229 230
|
eqtrd |
|
232 |
226 231
|
oveq12d |
|
233 |
202 200
|
oveq12d |
|
234 |
233 206
|
fveq12d |
|
235 |
|
df-ov |
|
236 |
234 235
|
eqtr4di |
|
237 |
198 202
|
oveq12d |
|
238 |
237 208
|
fveq12d |
|
239 |
|
df-ov |
|
240 |
238 239
|
eqtr4di |
|
241 |
232 236 240
|
oveq123d |
|
242 |
197 217 241
|
3eqtr4d |
|
243 |
21 22 23 24 25 26 27 28 31 33 41 48 123 165 242
|
isfuncd |
|
244 |
|
df-br |
|
245 |
243 244
|
sylib |
|
246 |
18 245
|
eqeltrd |
|