Step |
Hyp |
Ref |
Expression |
1 |
|
smat.s |
|
2 |
|
smat.m |
|
3 |
|
smat.n |
|
4 |
|
smat.k |
|
5 |
|
smat.l |
|
6 |
|
smat.a |
|
7 |
|
elmapi |
|
8 |
|
ffun |
|
9 |
6 7 8
|
3syl |
|
10 |
|
eqid |
|
11 |
10
|
mpofun |
|
12 |
11
|
a1i |
|
13 |
|
funco |
|
14 |
9 12 13
|
syl2anc |
|
15 |
|
fz1ssnn |
|
16 |
15 4
|
sseldi |
|
17 |
|
fz1ssnn |
|
18 |
17 5
|
sseldi |
|
19 |
|
smatfval |
|
20 |
16 18 6 19
|
syl3anc |
|
21 |
1 20
|
syl5eq |
|
22 |
21
|
funeqd |
|
23 |
14 22
|
mpbird |
|
24 |
|
fdmrn |
|
25 |
23 24
|
sylib |
|
26 |
21
|
dmeqd |
|
27 |
|
dmco |
|
28 |
|
fdm |
|
29 |
6 7 28
|
3syl |
|
30 |
29
|
imaeq2d |
|
31 |
30
|
eleq2d |
|
32 |
|
opex |
|
33 |
10 32
|
fnmpoi |
|
34 |
|
elpreima |
|
35 |
33 34
|
ax-mp |
|
36 |
35
|
a1i |
|
37 |
|
simplr |
|
38 |
37
|
fveq2d |
|
39 |
|
df-ov |
|
40 |
38 39
|
eqtr4di |
|
41 |
|
breq1 |
|
42 |
|
id |
|
43 |
|
oveq1 |
|
44 |
41 42 43
|
ifbieq12d |
|
45 |
44
|
opeq1d |
|
46 |
|
breq1 |
|
47 |
|
id |
|
48 |
|
oveq1 |
|
49 |
46 47 48
|
ifbieq12d |
|
50 |
49
|
opeq2d |
|
51 |
|
opex |
|
52 |
45 50 10 51
|
ovmpo |
|
53 |
52
|
adantl |
|
54 |
40 53
|
eqtrd |
|
55 |
54
|
eleq1d |
|
56 |
|
opelxp |
|
57 |
55 56
|
syl6bb |
|
58 |
|
ifel |
|
59 |
|
simplrl |
|
60 |
59
|
nnred |
|
61 |
16
|
nnred |
|
62 |
61
|
ad3antrrr |
|
63 |
2
|
nnred |
|
64 |
63
|
ad3antrrr |
|
65 |
|
simpr |
|
66 |
60 62 65
|
ltled |
|
67 |
|
elfzle2 |
|
68 |
4 67
|
syl |
|
69 |
68
|
ad3antrrr |
|
70 |
60 62 64 66 69
|
letrd |
|
71 |
59 70
|
jca |
|
72 |
2
|
nnzd |
|
73 |
|
fznn |
|
74 |
72 73
|
syl |
|
75 |
74
|
ad3antrrr |
|
76 |
71 75
|
mpbird |
|
77 |
60 62 64 65 69
|
ltletrd |
|
78 |
2
|
ad3antrrr |
|
79 |
|
nnltlem1 |
|
80 |
59 78 79
|
syl2anc |
|
81 |
77 80
|
mpbid |
|
82 |
76 81
|
2thd |
|
83 |
82
|
pm5.32da |
|
84 |
|
fznn |
|
85 |
72 84
|
syl |
|
86 |
85
|
ad2antrr |
|
87 |
|
simprl |
|
88 |
87
|
peano2nnd |
|
89 |
88
|
biantrurd |
|
90 |
87
|
nnzd |
|
91 |
72
|
ad2antrr |
|
92 |
|
zltp1le |
|
93 |
|
zltlem1 |
|
94 |
92 93
|
bitr3d |
|
95 |
90 91 94
|
syl2anc |
|
96 |
86 89 95
|
3bitr2d |
|
97 |
96
|
anbi2d |
|
98 |
83 97
|
orbi12d |
|
99 |
|
pm4.42 |
|
100 |
|
ancom |
|
101 |
|
ancom |
|
102 |
100 101
|
orbi12i |
|
103 |
99 102
|
bitri |
|
104 |
98 103
|
bitr4di |
|
105 |
58 104
|
syl5bb |
|
106 |
|
ifel |
|
107 |
|
simplrr |
|
108 |
107
|
nnred |
|
109 |
18
|
nnred |
|
110 |
109
|
ad3antrrr |
|
111 |
3
|
nnred |
|
112 |
111
|
ad3antrrr |
|
113 |
|
simpr |
|
114 |
108 110 113
|
ltled |
|
115 |
|
elfzle2 |
|
116 |
5 115
|
syl |
|
117 |
116
|
ad3antrrr |
|
118 |
108 110 112 114 117
|
letrd |
|
119 |
107 118
|
jca |
|
120 |
3
|
nnzd |
|
121 |
|
fznn |
|
122 |
120 121
|
syl |
|
123 |
122
|
ad3antrrr |
|
124 |
119 123
|
mpbird |
|
125 |
108 110 112 113 117
|
ltletrd |
|
126 |
3
|
ad3antrrr |
|
127 |
|
nnltlem1 |
|
128 |
107 126 127
|
syl2anc |
|
129 |
125 128
|
mpbid |
|
130 |
124 129
|
2thd |
|
131 |
130
|
pm5.32da |
|
132 |
|
fznn |
|
133 |
120 132
|
syl |
|
134 |
133
|
ad2antrr |
|
135 |
|
simprr |
|
136 |
135
|
peano2nnd |
|
137 |
136
|
biantrurd |
|
138 |
135
|
nnzd |
|
139 |
120
|
ad2antrr |
|
140 |
|
zltp1le |
|
141 |
|
zltlem1 |
|
142 |
140 141
|
bitr3d |
|
143 |
138 139 142
|
syl2anc |
|
144 |
134 137 143
|
3bitr2d |
|
145 |
144
|
anbi2d |
|
146 |
131 145
|
orbi12d |
|
147 |
|
pm4.42 |
|
148 |
|
ancom |
|
149 |
|
ancom |
|
150 |
148 149
|
orbi12i |
|
151 |
147 150
|
bitri |
|
152 |
146 151
|
bitr4di |
|
153 |
106 152
|
syl5bb |
|
154 |
105 153
|
anbi12d |
|
155 |
57 154
|
bitrd |
|
156 |
155
|
pm5.32da |
|
157 |
|
1zzd |
|
158 |
72 157
|
zsubcld |
|
159 |
|
fznn |
|
160 |
158 159
|
syl |
|
161 |
120 157
|
zsubcld |
|
162 |
|
fznn |
|
163 |
161 162
|
syl |
|
164 |
160 163
|
anbi12d |
|
165 |
|
an4 |
|
166 |
164 165
|
syl6bb |
|
167 |
166
|
adantr |
|
168 |
156 167
|
bitr4d |
|
169 |
168
|
pm5.32da |
|
170 |
|
elxp6 |
|
171 |
170
|
anbi1i |
|
172 |
|
anass |
|
173 |
171 172
|
bitri |
|
174 |
|
elxp6 |
|
175 |
169 173 174
|
3bitr4g |
|
176 |
31 36 175
|
3bitrd |
|
177 |
176
|
eqrdv |
|
178 |
27 177
|
syl5eq |
|
179 |
26 178
|
eqtrd |
|
180 |
179
|
feq2d |
|
181 |
25 180
|
mpbid |
|
182 |
21
|
rneqd |
|
183 |
|
rncoss |
|
184 |
182 183
|
eqsstrdi |
|
185 |
|
frn |
|
186 |
6 7 185
|
3syl |
|
187 |
184 186
|
sstrd |
|
188 |
|
fss |
|
189 |
181 187 188
|
syl2anc |
|
190 |
|
reldmmap |
|
191 |
190
|
ovrcl |
|
192 |
6 191
|
syl |
|
193 |
192
|
simpld |
|
194 |
|
ovex |
|
195 |
|
ovex |
|
196 |
194 195
|
xpex |
|
197 |
|
elmapg |
|
198 |
193 196 197
|
sylancl |
|
199 |
189 198
|
mpbird |
|