Step |
Hyp |
Ref |
Expression |
1 |
|
satffunlem2lem2.s |
|
2 |
|
satffunlem2lem2.a |
|
3 |
|
satffunlem2lem2.b |
|
4 |
1
|
fveq1i |
|
5 |
4
|
dmeqi |
|
6 |
|
simprl |
|
7 |
|
simprr |
|
8 |
|
peano2 |
|
9 |
8
|
adantr |
|
10 |
6 7 9
|
3jca |
|
11 |
|
satfdmfmla |
|
12 |
10 11
|
syl |
|
13 |
12
|
adantr |
|
14 |
5 13
|
syl5eq |
|
15 |
|
ovex |
|
16 |
15
|
difexi |
|
17 |
2 16
|
eqeltri |
|
18 |
17
|
a1i |
|
19 |
18
|
ralrimiva |
|
20 |
3 15
|
rabex2 |
|
21 |
20
|
a1i |
|
22 |
21
|
ralrimiva |
|
23 |
19 22
|
jca |
|
24 |
23
|
ralrimiva |
|
25 |
|
simplr |
|
26 |
8
|
ancri |
|
27 |
26
|
ad2antrr |
|
28 |
25 27
|
jca |
|
29 |
|
sssucid |
|
30 |
1
|
satfsschain |
|
31 |
28 29 30
|
mpisyl |
|
32 |
|
dmopab3rexdif |
|
33 |
24 31 32
|
syl2anc |
|
34 |
|
simpr |
|
35 |
|
fveqeq2 |
|
36 |
35
|
adantl |
|
37 |
|
eqidd |
|
38 |
34 36 37
|
rspcedvd |
|
39 |
4
|
funeqi |
|
40 |
39
|
biimpi |
|
41 |
40
|
adantl |
|
42 |
1
|
fveq1i |
|
43 |
31 42 4
|
3sstr3g |
|
44 |
41 43
|
jca |
|
45 |
44
|
adantr |
|
46 |
|
funeldmdif |
|
47 |
45 46
|
syl |
|
48 |
38 47
|
mpbird |
|
49 |
48
|
ex |
|
50 |
4 42
|
difeq12i |
|
51 |
50
|
eleq2i |
|
52 |
51
|
a1i |
|
53 |
13
|
eqcomd |
|
54 |
|
simpl |
|
55 |
6 7 54
|
3jca |
|
56 |
|
satfdmfmla |
|
57 |
55 56
|
syl |
|
58 |
57
|
eqcomd |
|
59 |
58
|
adantr |
|
60 |
53 59
|
difeq12d |
|
61 |
60
|
eleq2d |
|
62 |
49 52 61
|
3imtr4d |
|
63 |
62
|
imp |
|
64 |
63
|
adantr |
|
65 |
|
oveq1 |
|
66 |
65
|
eqeq2d |
|
67 |
66
|
rexbidv |
|
68 |
|
eqidd |
|
69 |
|
id |
|
70 |
68 69
|
goaleq12d |
|
71 |
70
|
eqeq2d |
|
72 |
71
|
rexbidv |
|
73 |
67 72
|
orbi12d |
|
74 |
73
|
adantl |
|
75 |
6
|
adantr |
|
76 |
7
|
adantr |
|
77 |
8
|
ad2antrr |
|
78 |
75 76 77
|
3jca |
|
79 |
|
satfrel |
|
80 |
78 79
|
syl |
|
81 |
4
|
releqi |
|
82 |
80 81
|
sylibr |
|
83 |
|
1stdm |
|
84 |
82 83
|
sylan |
|
85 |
14
|
eqcomd |
|
86 |
85
|
adantr |
|
87 |
84 86
|
eleqtrrd |
|
88 |
87
|
ad4ant13 |
|
89 |
|
oveq2 |
|
90 |
89
|
eqeq2d |
|
91 |
90
|
adantl |
|
92 |
|
simpr |
|
93 |
88 91 92
|
rspcedvd |
|
94 |
93
|
rexlimdva2 |
|
95 |
94
|
orim1d |
|
96 |
95
|
imp |
|
97 |
64 74 96
|
rspcedvd |
|
98 |
97
|
rexlimdva2 |
|
99 |
55
|
adantr |
|
100 |
|
satfrel |
|
101 |
99 100
|
syl |
|
102 |
42
|
releqi |
|
103 |
101 102
|
sylibr |
|
104 |
|
1stdm |
|
105 |
103 104
|
sylan |
|
106 |
42
|
dmeqi |
|
107 |
99 56
|
syl |
|
108 |
106 107
|
syl5eq |
|
109 |
108
|
eqcomd |
|
110 |
109
|
adantr |
|
111 |
105 110
|
eleqtrrd |
|
112 |
111
|
adantr |
|
113 |
66
|
rexbidv |
|
114 |
113
|
adantl |
|
115 |
|
simpr |
|
116 |
|
fveqeq2 |
|
117 |
116
|
adantl |
|
118 |
|
eqidd |
|
119 |
115 117 118
|
rspcedvd |
|
120 |
44
|
adantr |
|
121 |
|
funeldmdif |
|
122 |
120 121
|
syl |
|
123 |
119 122
|
mpbird |
|
124 |
123
|
ex |
|
125 |
50
|
eleq2i |
|
126 |
125
|
a1i |
|
127 |
12
|
eqcomd |
|
128 |
127 58
|
difeq12d |
|
129 |
128
|
eleq2d |
|
130 |
129
|
adantr |
|
131 |
124 126 130
|
3imtr4d |
|
132 |
131
|
adantr |
|
133 |
132
|
imp |
|
134 |
133
|
adantr |
|
135 |
90
|
adantl |
|
136 |
|
simpr |
|
137 |
134 135 136
|
rspcedvd |
|
138 |
137
|
r19.29an |
|
139 |
112 114 138
|
rspcedvd |
|
140 |
139
|
rexlimdva2 |
|
141 |
98 140
|
orim12d |
|
142 |
10
|
adantr |
|
143 |
11
|
eqcomd |
|
144 |
142 143
|
syl |
|
145 |
107
|
eqcomd |
|
146 |
144 145
|
difeq12d |
|
147 |
146
|
eleq2d |
|
148 |
|
eqid |
|
149 |
148
|
satfsschain |
|
150 |
28 29 149
|
mpisyl |
|
151 |
|
releldmdifi |
|
152 |
80 150 151
|
syl2anc |
|
153 |
147 152
|
sylbid |
|
154 |
50
|
eqcomi |
|
155 |
154
|
rexeqi |
|
156 |
|
r19.41v |
|
157 |
|
oveq1 |
|
158 |
157
|
eqeq2d |
|
159 |
158
|
rexbidv |
|
160 |
|
eqidd |
|
161 |
|
id |
|
162 |
160 161
|
goaleq12d |
|
163 |
162
|
eqeq2d |
|
164 |
163
|
rexbidv |
|
165 |
159 164
|
orbi12d |
|
166 |
165
|
adantl |
|
167 |
142 11
|
syl |
|
168 |
167
|
eqcomd |
|
169 |
168
|
eleq2d |
|
170 |
|
releldm2 |
|
171 |
80 170
|
syl |
|
172 |
169 171
|
bitrd |
|
173 |
|
r19.41v |
|
174 |
1
|
eqcomi |
|
175 |
174
|
fveq1i |
|
176 |
175
|
rexeqi |
|
177 |
89
|
eqcoms |
|
178 |
177
|
eqeq2d |
|
179 |
178
|
biimpa |
|
180 |
179
|
a1i |
|
181 |
180
|
reximdv |
|
182 |
176 181
|
syl5bi |
|
183 |
173 182
|
syl5bir |
|
184 |
183
|
expd |
|
185 |
172 184
|
sylbid |
|
186 |
185
|
rexlimdv |
|
187 |
186
|
ad2antrr |
|
188 |
187
|
orim1d |
|
189 |
166 188
|
sylbird |
|
190 |
189
|
expimpd |
|
191 |
190
|
reximdva |
|
192 |
156 191
|
syl5bir |
|
193 |
192
|
expd |
|
194 |
155 193
|
syl5bi |
|
195 |
153 194
|
syld |
|
196 |
195
|
rexlimdv |
|
197 |
145
|
eleq2d |
|
198 |
55 100
|
syl |
|
199 |
198
|
adantr |
|
200 |
|
releldm2 |
|
201 |
199 200
|
syl |
|
202 |
197 201
|
bitrd |
|
203 |
|
r19.41v |
|
204 |
42
|
eqcomi |
|
205 |
204
|
rexeqi |
|
206 |
158
|
rexbidv |
|
207 |
206
|
adantl |
|
208 |
146
|
eleq2d |
|
209 |
|
releldmdifi |
|
210 |
80 150 209
|
syl2anc |
|
211 |
208 210
|
sylbid |
|
212 |
154
|
rexeqi |
|
213 |
178
|
biimpcd |
|
214 |
213
|
adantl |
|
215 |
214
|
reximdv |
|
216 |
215
|
ex |
|
217 |
216
|
com23 |
|
218 |
212 217
|
syl5bi |
|
219 |
211 218
|
syld |
|
220 |
219
|
rexlimdv |
|
221 |
220
|
adantr |
|
222 |
207 221
|
sylbird |
|
223 |
222
|
expimpd |
|
224 |
223
|
reximdv |
|
225 |
205 224
|
syl5bi |
|
226 |
203 225
|
syl5bir |
|
227 |
226
|
expd |
|
228 |
202 227
|
sylbid |
|
229 |
228
|
rexlimdv |
|
230 |
196 229
|
orim12d |
|
231 |
141 230
|
impbid |
|
232 |
231
|
abbidv |
|
233 |
33 232
|
eqtrd |
|
234 |
14 233
|
ineq12d |
|
235 |
|
fmlasucdisj |
|
236 |
235
|
ad2antrr |
|
237 |
234 236
|
eqtrd |
|