Step |
Hyp |
Ref |
Expression |
1 |
|
dvdsmulf1o.1 |
|
2 |
|
dvdsmulf1o.2 |
|
3 |
|
dvdsmulf1o.3 |
|
4 |
|
dvdsmulf1o.x |
|
5 |
|
dvdsmulf1o.y |
|
6 |
|
dvdsmulf1o.z |
|
7 |
|
ax-mulf |
|
8 |
|
ffn |
|
9 |
7 8
|
ax-mp |
|
10 |
4
|
ssrab3 |
|
11 |
|
nnsscn |
|
12 |
10 11
|
sstri |
|
13 |
5
|
ssrab3 |
|
14 |
13 11
|
sstri |
|
15 |
|
xpss12 |
|
16 |
12 14 15
|
mp2an |
|
17 |
|
fnssres |
|
18 |
9 16 17
|
mp2an |
|
19 |
18
|
a1i |
|
20 |
|
ovres |
|
21 |
20
|
adantl |
|
22 |
|
breq1 |
|
23 |
22 4
|
elrab2 |
|
24 |
23
|
simplbi |
|
25 |
24
|
ad2antrl |
|
26 |
|
breq1 |
|
27 |
26 5
|
elrab2 |
|
28 |
27
|
simplbi |
|
29 |
28
|
ad2antll |
|
30 |
25 29
|
nnmulcld |
|
31 |
27
|
simprbi |
|
32 |
31
|
ad2antll |
|
33 |
23
|
simprbi |
|
34 |
33
|
ad2antrl |
|
35 |
29
|
nnzd |
|
36 |
2
|
adantr |
|
37 |
36
|
nnzd |
|
38 |
25
|
nnzd |
|
39 |
|
dvdscmul |
|
40 |
35 37 38 39
|
syl3anc |
|
41 |
1
|
adantr |
|
42 |
41
|
nnzd |
|
43 |
|
dvdsmulc |
|
44 |
38 42 37 43
|
syl3anc |
|
45 |
30
|
nnzd |
|
46 |
38 37
|
zmulcld |
|
47 |
42 37
|
zmulcld |
|
48 |
|
dvdstr |
|
49 |
45 46 47 48
|
syl3anc |
|
50 |
40 44 49
|
syl2and |
|
51 |
32 34 50
|
mp2and |
|
52 |
|
breq1 |
|
53 |
52 6
|
elrab2 |
|
54 |
30 51 53
|
sylanbrc |
|
55 |
21 54
|
eqeltrd |
|
56 |
55
|
ralrimivva |
|
57 |
|
ffnov |
|
58 |
19 56 57
|
sylanbrc |
|
59 |
25
|
adantr |
|
60 |
59
|
nnnn0d |
|
61 |
|
simprll |
|
62 |
|
breq1 |
|
63 |
62 4
|
elrab2 |
|
64 |
63
|
simplbi |
|
65 |
61 64
|
syl |
|
66 |
65
|
nnnn0d |
|
67 |
59
|
nnzd |
|
68 |
29
|
adantr |
|
69 |
68
|
nnzd |
|
70 |
|
dvdsmul1 |
|
71 |
67 69 70
|
syl2anc |
|
72 |
|
simprr |
|
73 |
12 61
|
sseldi |
|
74 |
|
simprlr |
|
75 |
|
breq1 |
|
76 |
75 5
|
elrab2 |
|
77 |
76
|
simplbi |
|
78 |
74 77
|
syl |
|
79 |
78
|
nncnd |
|
80 |
73 79
|
mulcomd |
|
81 |
72 80
|
eqtrd |
|
82 |
71 81
|
breqtrd |
|
83 |
78
|
nnzd |
|
84 |
37
|
adantr |
|
85 |
67 84
|
gcdcomd |
|
86 |
42
|
adantr |
|
87 |
2
|
nnzd |
|
88 |
1
|
nnzd |
|
89 |
87 88
|
gcdcomd |
|
90 |
89 3
|
eqtrd |
|
91 |
90
|
ad2antrr |
|
92 |
34
|
adantr |
|
93 |
|
rpdvds |
|
94 |
84 67 86 91 92 93
|
syl32anc |
|
95 |
85 94
|
eqtrd |
|
96 |
76
|
simprbi |
|
97 |
74 96
|
syl |
|
98 |
|
rpdvds |
|
99 |
67 83 84 95 97 98
|
syl32anc |
|
100 |
65
|
nnzd |
|
101 |
|
coprmdvds |
|
102 |
67 83 100 101
|
syl3anc |
|
103 |
82 99 102
|
mp2and |
|
104 |
|
dvdsmul1 |
|
105 |
100 83 104
|
syl2anc |
|
106 |
59
|
nncnd |
|
107 |
68
|
nncnd |
|
108 |
106 107
|
mulcomd |
|
109 |
72 108
|
eqtr3d |
|
110 |
105 109
|
breqtrd |
|
111 |
100 84
|
gcdcomd |
|
112 |
63
|
simprbi |
|
113 |
61 112
|
syl |
|
114 |
|
rpdvds |
|
115 |
84 100 86 91 113 114
|
syl32anc |
|
116 |
111 115
|
eqtrd |
|
117 |
32
|
adantr |
|
118 |
|
rpdvds |
|
119 |
100 69 84 116 117 118
|
syl32anc |
|
120 |
|
coprmdvds |
|
121 |
100 69 67 120
|
syl3anc |
|
122 |
110 119 121
|
mp2and |
|
123 |
|
dvdseq |
|
124 |
60 66 103 122 123
|
syl22anc |
|
125 |
59
|
nnne0d |
|
126 |
124
|
oveq1d |
|
127 |
72 126
|
eqtr4d |
|
128 |
107 79 106 125 127
|
mulcanad |
|
129 |
124 128
|
opeq12d |
|
130 |
129
|
expr |
|
131 |
130
|
ralrimivva |
|
132 |
131
|
ralrimivva |
|
133 |
|
fvres |
|
134 |
|
fvres |
|
135 |
133 134
|
eqeqan12d |
|
136 |
135
|
imbi1d |
|
137 |
136
|
ralbidva |
|
138 |
137
|
ralbiia |
|
139 |
|
fveq2 |
|
140 |
|
df-ov |
|
141 |
139 140
|
eqtr4di |
|
142 |
141
|
eqeq2d |
|
143 |
|
eqeq2 |
|
144 |
142 143
|
imbi12d |
|
145 |
144
|
ralxp |
|
146 |
|
fveq2 |
|
147 |
|
df-ov |
|
148 |
146 147
|
eqtr4di |
|
149 |
148
|
eqeq1d |
|
150 |
|
eqeq1 |
|
151 |
149 150
|
imbi12d |
|
152 |
151
|
2ralbidv |
|
153 |
145 152
|
syl5bb |
|
154 |
153
|
ralxp |
|
155 |
138 154
|
bitri |
|
156 |
132 155
|
sylibr |
|
157 |
|
dff13 |
|
158 |
58 156 157
|
sylanbrc |
|
159 |
|
breq1 |
|
160 |
159 6
|
elrab2 |
|
161 |
160
|
simplbi |
|
162 |
161
|
adantl |
|
163 |
162
|
nnzd |
|
164 |
1
|
adantr |
|
165 |
164
|
nnzd |
|
166 |
164
|
nnne0d |
|
167 |
|
simpr |
|
168 |
167
|
necon3ai |
|
169 |
166 168
|
syl |
|
170 |
|
gcdn0cl |
|
171 |
163 165 169 170
|
syl21anc |
|
172 |
|
gcddvds |
|
173 |
163 165 172
|
syl2anc |
|
174 |
173
|
simprd |
|
175 |
|
breq1 |
|
176 |
175 4
|
elrab2 |
|
177 |
171 174 176
|
sylanbrc |
|
178 |
2
|
adantr |
|
179 |
178
|
nnzd |
|
180 |
178
|
nnne0d |
|
181 |
|
simpr |
|
182 |
181
|
necon3ai |
|
183 |
180 182
|
syl |
|
184 |
|
gcdn0cl |
|
185 |
163 179 183 184
|
syl21anc |
|
186 |
|
gcddvds |
|
187 |
163 179 186
|
syl2anc |
|
188 |
187
|
simprd |
|
189 |
|
breq1 |
|
190 |
189 5
|
elrab2 |
|
191 |
185 188 190
|
sylanbrc |
|
192 |
177 191
|
opelxpd |
|
193 |
192
|
fvresd |
|
194 |
3
|
adantr |
|
195 |
|
rpmulgcd2 |
|
196 |
163 165 179 194 195
|
syl31anc |
|
197 |
|
df-ov |
|
198 |
196 197
|
eqtrdi |
|
199 |
160
|
simprbi |
|
200 |
199
|
adantl |
|
201 |
1 2
|
nnmulcld |
|
202 |
|
gcdeq |
|
203 |
161 201 202
|
syl2anr |
|
204 |
200 203
|
mpbird |
|
205 |
193 198 204
|
3eqtr2rd |
|
206 |
|
fveq2 |
|
207 |
206
|
rspceeqv |
|
208 |
192 205 207
|
syl2anc |
|
209 |
208
|
ralrimiva |
|
210 |
|
dffo3 |
|
211 |
58 209 210
|
sylanbrc |
|
212 |
|
df-f1o |
|
213 |
158 211 212
|
sylanbrc |
|