Step |
Hyp |
Ref |
Expression |
1 |
|
prmrec.1 |
|
2 |
|
prmrec.2 |
|
3 |
|
prmrec.3 |
|
4 |
|
prmrec.4 |
|
5 |
|
prmreclem2.5 |
|
6 |
|
ovex |
|
7 |
|
fveqeq2 |
|
8 |
7
|
elrab |
|
9 |
4
|
ssrab3 |
|
10 |
|
simprl |
|
11 |
10
|
ad2antrr |
|
12 |
9 11
|
sseldi |
|
13 |
|
elfznn |
|
14 |
12 13
|
syl |
|
15 |
|
simpr |
|
16 |
|
prmuz2 |
|
17 |
15 16
|
syl |
|
18 |
5
|
prmreclem1 |
|
19 |
18
|
simp3d |
|
20 |
14 17 19
|
sylc |
|
21 |
|
simprr |
|
22 |
21
|
ad2antrr |
|
23 |
22
|
oveq1d |
|
24 |
|
sq1 |
|
25 |
23 24
|
eqtrdi |
|
26 |
25
|
oveq2d |
|
27 |
14
|
nncnd |
|
28 |
27
|
div1d |
|
29 |
26 28
|
eqtrd |
|
30 |
29
|
breq2d |
|
31 |
14
|
nnzd |
|
32 |
|
2nn0 |
|
33 |
32
|
a1i |
|
34 |
|
pcdvdsb |
|
35 |
15 31 33 34
|
syl3anc |
|
36 |
30 35
|
bitr4d |
|
37 |
20 36
|
mtbid |
|
38 |
15 14
|
pccld |
|
39 |
38
|
nn0red |
|
40 |
|
2re |
|
41 |
|
ltnle |
|
42 |
39 40 41
|
sylancl |
|
43 |
37 42
|
mpbird |
|
44 |
|
df-2 |
|
45 |
43 44
|
breqtrdi |
|
46 |
38
|
nn0zd |
|
47 |
|
1z |
|
48 |
|
zleltp1 |
|
49 |
46 47 48
|
sylancl |
|
50 |
45 49
|
mpbird |
|
51 |
|
nn0uz |
|
52 |
38 51
|
eleqtrdi |
|
53 |
|
elfz5 |
|
54 |
52 47 53
|
sylancl |
|
55 |
50 54
|
mpbird |
|
56 |
|
0z |
|
57 |
|
fzpr |
|
58 |
56 57
|
ax-mp |
|
59 |
|
1e0p1 |
|
60 |
59
|
oveq2i |
|
61 |
59
|
preq2i |
|
62 |
58 60 61
|
3eqtr4i |
|
63 |
55 62
|
eleqtrdi |
|
64 |
|
c0ex |
|
65 |
64
|
prid1 |
|
66 |
65
|
a1i |
|
67 |
63 66
|
ifclda |
|
68 |
67
|
fmpttd |
|
69 |
|
prex |
|
70 |
|
ovex |
|
71 |
69 70
|
elmap |
|
72 |
68 71
|
sylibr |
|
73 |
72
|
ex |
|
74 |
8 73
|
syl5bi |
|
75 |
|
fveqeq2 |
|
76 |
75
|
elrab |
|
77 |
8 76
|
anbi12i |
|
78 |
|
ovex |
|
79 |
78 64
|
ifex |
|
80 |
|
eqid |
|
81 |
79 80
|
fnmpti |
|
82 |
|
ovex |
|
83 |
82 64
|
ifex |
|
84 |
|
eqid |
|
85 |
83 84
|
fnmpti |
|
86 |
|
eqfnfv |
|
87 |
81 85 86
|
mp2an |
|
88 |
|
eleq1w |
|
89 |
|
oveq1 |
|
90 |
88 89
|
ifbieq1d |
|
91 |
|
ovex |
|
92 |
91 64
|
ifex |
|
93 |
90 80 92
|
fvmpt |
|
94 |
|
oveq1 |
|
95 |
88 94
|
ifbieq1d |
|
96 |
|
ovex |
|
97 |
96 64
|
ifex |
|
98 |
95 84 97
|
fvmpt |
|
99 |
93 98
|
eqeq12d |
|
100 |
99
|
ralbiia |
|
101 |
87 100
|
bitri |
|
102 |
|
simprll |
|
103 |
|
breq2 |
|
104 |
103
|
notbid |
|
105 |
104
|
ralbidv |
|
106 |
105 4
|
elrab2 |
|
107 |
106
|
simprbi |
|
108 |
102 107
|
syl |
|
109 |
|
simprrl |
|
110 |
|
breq2 |
|
111 |
110
|
notbid |
|
112 |
111
|
ralbidv |
|
113 |
112 4
|
elrab2 |
|
114 |
113
|
simprbi |
|
115 |
109 114
|
syl |
|
116 |
|
r19.26 |
|
117 |
|
eldifi |
|
118 |
|
fz1ssnn |
|
119 |
9 118
|
sstri |
|
120 |
119 102
|
sseldi |
|
121 |
|
pceq0 |
|
122 |
117 120 121
|
syl2anr |
|
123 |
119 109
|
sseldi |
|
124 |
|
pceq0 |
|
125 |
117 123 124
|
syl2anr |
|
126 |
122 125
|
anbi12d |
|
127 |
|
eqtr3 |
|
128 |
126 127
|
syl6bir |
|
129 |
128
|
ralimdva |
|
130 |
116 129
|
syl5bir |
|
131 |
108 115 130
|
mp2and |
|
132 |
131
|
biantrud |
|
133 |
|
incom |
|
134 |
133
|
uneq1i |
|
135 |
|
inundif |
|
136 |
134 135
|
eqtri |
|
137 |
136
|
raleqi |
|
138 |
|
ralunb |
|
139 |
137 138
|
bitr3i |
|
140 |
|
eldifn |
|
141 |
|
iffalse |
|
142 |
|
iffalse |
|
143 |
141 142
|
eqtr4d |
|
144 |
140 143
|
syl |
|
145 |
144
|
rgen |
|
146 |
145
|
biantru |
|
147 |
|
elinel1 |
|
148 |
|
iftrue |
|
149 |
|
iftrue |
|
150 |
148 149
|
eqeq12d |
|
151 |
147 150
|
syl |
|
152 |
151
|
ralbiia |
|
153 |
146 152
|
bitr3i |
|
154 |
139 153
|
bitri |
|
155 |
|
inundif |
|
156 |
155
|
raleqi |
|
157 |
|
ralunb |
|
158 |
156 157
|
bitr3i |
|
159 |
132 154 158
|
3bitr4g |
|
160 |
120
|
nnnn0d |
|
161 |
123
|
nnnn0d |
|
162 |
|
pc11 |
|
163 |
160 161 162
|
syl2anc |
|
164 |
159 163
|
bitr4d |
|
165 |
101 164
|
syl5bb |
|
166 |
165
|
ex |
|
167 |
77 166
|
syl5bi |
|
168 |
74 167
|
dom2d |
|
169 |
6 168
|
mpi |
|
170 |
|
fzfi |
|
171 |
|
ssrab2 |
|
172 |
|
ssfi |
|
173 |
170 171 172
|
mp2an |
|
174 |
4 173
|
eqeltri |
|
175 |
|
ssrab2 |
|
176 |
|
ssfi |
|
177 |
174 175 176
|
mp2an |
|
178 |
|
prfi |
|
179 |
|
fzfid |
|
180 |
|
mapfi |
|
181 |
178 179 180
|
sylancr |
|
182 |
|
hashdom |
|
183 |
177 181 182
|
sylancr |
|
184 |
169 183
|
mpbird |
|
185 |
|
hashmap |
|
186 |
178 179 185
|
sylancr |
|
187 |
|
prhash2ex |
|
188 |
187
|
a1i |
|
189 |
2
|
nnnn0d |
|
190 |
|
hashfz1 |
|
191 |
189 190
|
syl |
|
192 |
188 191
|
oveq12d |
|
193 |
186 192
|
eqtrd |
|
194 |
184 193
|
breqtrd |
|