Step |
Hyp |
Ref |
Expression |
1 |
|
dprdsplit.2 |
|
2 |
|
dprdsplit.i |
|
3 |
|
dprdsplit.u |
|
4 |
|
dmdprdsplit.z |
|
5 |
|
dmdprdsplit.0 |
|
6 |
|
dmdprdsplit2.1 |
|
7 |
|
dmdprdsplit2.2 |
|
8 |
|
dmdprdsplit2.3 |
|
9 |
|
dmdprdsplit2.4 |
|
10 |
|
dmdprdsplit2lem.k |
|
11 |
3
|
adantr |
|
12 |
11
|
eleq2d |
|
13 |
|
elun |
|
14 |
12 13
|
bitrdi |
|
15 |
6
|
ad2antrr |
|
16 |
|
ssun1 |
|
17 |
16 3
|
sseqtrrid |
|
18 |
1 17
|
fssresd |
|
19 |
18
|
fdmd |
|
20 |
19
|
ad2antrr |
|
21 |
|
simplr |
|
22 |
|
simprl |
|
23 |
|
simprr |
|
24 |
15 20 21 22 23 4
|
dprdcntz |
|
25 |
|
fvres |
|
26 |
25
|
ad2antlr |
|
27 |
|
fvres |
|
28 |
27
|
ad2antrl |
|
29 |
28
|
fveq2d |
|
30 |
24 26 29
|
3sstr3d |
|
31 |
30
|
exp32 |
|
32 |
25
|
ad2antlr |
|
33 |
6
|
ad2antrr |
|
34 |
19
|
ad2antrr |
|
35 |
|
simplr |
|
36 |
33 34 35
|
dprdub |
|
37 |
32 36
|
eqsstrrd |
|
38 |
8
|
ad2antrr |
|
39 |
|
eqid |
|
40 |
39
|
dprdssv |
|
41 |
|
fvres |
|
42 |
41
|
ad2antrl |
|
43 |
7
|
ad2antrr |
|
44 |
|
ssun2 |
|
45 |
44 3
|
sseqtrrid |
|
46 |
1 45
|
fssresd |
|
47 |
46
|
fdmd |
|
48 |
47
|
ad2antrr |
|
49 |
|
simprl |
|
50 |
43 48 49
|
dprdub |
|
51 |
42 50
|
eqsstrrd |
|
52 |
39 4
|
cntz2ss |
|
53 |
40 51 52
|
sylancr |
|
54 |
38 53
|
sstrd |
|
55 |
37 54
|
sstrd |
|
56 |
55
|
exp32 |
|
57 |
31 56
|
jaod |
|
58 |
14 57
|
sylbid |
|
59 |
|
dprdgrp |
|
60 |
6 59
|
syl |
|
61 |
60
|
adantr |
|
62 |
39
|
subgacs |
|
63 |
|
acsmre |
|
64 |
61 62 63
|
3syl |
|
65 |
|
difundir |
|
66 |
11
|
difeq1d |
|
67 |
|
simpr |
|
68 |
67
|
snssd |
|
69 |
|
sslin |
|
70 |
68 69
|
syl |
|
71 |
|
incom |
|
72 |
2
|
adantr |
|
73 |
71 72
|
syl5eqr |
|
74 |
|
sseq0 |
|
75 |
70 73 74
|
syl2anc |
|
76 |
|
disj3 |
|
77 |
75 76
|
sylib |
|
78 |
77
|
uneq2d |
|
79 |
65 66 78
|
3eqtr4a |
|
80 |
79
|
imaeq2d |
|
81 |
|
imaundi |
|
82 |
80 81
|
eqtrdi |
|
83 |
82
|
unieqd |
|
84 |
|
uniun |
|
85 |
83 84
|
eqtrdi |
|
86 |
|
difss |
|
87 |
|
imass2 |
|
88 |
|
uniss |
|
89 |
86 87 88
|
mp2b |
|
90 |
|
imassrn |
|
91 |
1
|
frnd |
|
92 |
91
|
adantr |
|
93 |
|
mresspw |
|
94 |
64 93
|
syl |
|
95 |
92 94
|
sstrd |
|
96 |
90 95
|
sstrid |
|
97 |
|
sspwuni |
|
98 |
96 97
|
sylib |
|
99 |
89 98
|
sstrid |
|
100 |
64 10 99
|
mrcssidd |
|
101 |
|
imassrn |
|
102 |
101 95
|
sstrid |
|
103 |
|
sspwuni |
|
104 |
102 103
|
sylib |
|
105 |
64 10 104
|
mrcssidd |
|
106 |
10
|
dprdspan |
|
107 |
7 106
|
syl |
|
108 |
|
df-ima |
|
109 |
108
|
unieqi |
|
110 |
109
|
fveq2i |
|
111 |
107 110
|
eqtr4di |
|
112 |
111
|
adantr |
|
113 |
105 112
|
sseqtrrd |
|
114 |
|
unss12 |
|
115 |
100 113 114
|
syl2anc |
|
116 |
10
|
mrccl |
|
117 |
64 99 116
|
syl2anc |
|
118 |
|
dprdsubg |
|
119 |
7 118
|
syl |
|
120 |
119
|
adantr |
|
121 |
|
eqid |
|
122 |
121
|
lsmunss |
|
123 |
117 120 122
|
syl2anc |
|
124 |
115 123
|
sstrd |
|
125 |
85 124
|
eqsstrd |
|
126 |
89
|
a1i |
|
127 |
64 10 126 98
|
mrcssd |
|
128 |
10
|
dprdspan |
|
129 |
6 128
|
syl |
|
130 |
|
df-ima |
|
131 |
130
|
unieqi |
|
132 |
131
|
fveq2i |
|
133 |
129 132
|
eqtr4di |
|
134 |
133
|
adantr |
|
135 |
127 134
|
sseqtrrd |
|
136 |
8
|
adantr |
|
137 |
135 136
|
sstrd |
|
138 |
121 4
|
lsmsubg |
|
139 |
117 120 137 138
|
syl3anc |
|
140 |
10
|
mrcsscl |
|
141 |
64 125 139 140
|
syl3anc |
|
142 |
|
sslin |
|
143 |
141 142
|
syl |
|
144 |
17
|
sselda |
|
145 |
1
|
ffvelrnda |
|
146 |
144 145
|
syldan |
|
147 |
25
|
adantl |
|
148 |
6
|
adantr |
|
149 |
19
|
adantr |
|
150 |
148 149 67
|
dprdub |
|
151 |
147 150
|
eqsstrrd |
|
152 |
|
dprdsubg |
|
153 |
6 152
|
syl |
|
154 |
153
|
adantr |
|
155 |
121
|
lsmlub |
|
156 |
146 117 154 155
|
syl3anc |
|
157 |
151 135 156
|
mpbi2and |
|
158 |
157
|
ssrind |
|
159 |
9
|
adantr |
|
160 |
158 159
|
sseqtrd |
|
161 |
121
|
lsmub1 |
|
162 |
146 117 161
|
syl2anc |
|
163 |
5
|
subg0cl |
|
164 |
146 163
|
syl |
|
165 |
162 164
|
sseldd |
|
166 |
5
|
subg0cl |
|
167 |
120 166
|
syl |
|
168 |
165 167
|
elind |
|
169 |
168
|
snssd |
|
170 |
160 169
|
eqssd |
|
171 |
|
resima2 |
|
172 |
86 171
|
mp1i |
|
173 |
172
|
unieqd |
|
174 |
173
|
fveq2d |
|
175 |
147 174
|
ineq12d |
|
176 |
148 149 67 5 10
|
dprddisj |
|
177 |
175 176
|
eqtr3d |
|
178 |
1
|
adantr |
|
179 |
|
ffun |
|
180 |
|
funiunfv |
|
181 |
178 179 180
|
3syl |
|
182 |
6
|
ad2antrr |
|
183 |
19
|
ad2antrr |
|
184 |
|
eldifi |
|
185 |
184
|
adantl |
|
186 |
|
simplr |
|
187 |
|
eldifsni |
|
188 |
187
|
adantl |
|
189 |
182 183 185 186 188 4
|
dprdcntz |
|
190 |
185
|
fvresd |
|
191 |
25
|
ad2antlr |
|
192 |
191
|
fveq2d |
|
193 |
189 190 192
|
3sstr3d |
|
194 |
193
|
ralrimiva |
|
195 |
|
iunss |
|
196 |
194 195
|
sylibr |
|
197 |
181 196
|
eqsstrrd |
|
198 |
39
|
subgss |
|
199 |
146 198
|
syl |
|
200 |
39 4
|
cntzsubg |
|
201 |
61 199 200
|
syl2anc |
|
202 |
10
|
mrcsscl |
|
203 |
64 197 201 202
|
syl3anc |
|
204 |
4 117 146 203
|
cntzrecd |
|
205 |
121 146 117 120 5 170 177 4 204
|
lsmdisj3 |
|
206 |
143 205
|
sseqtrd |
|
207 |
58 206
|
jca |
|