Step |
Hyp |
Ref |
Expression |
1 |
|
xrsxmet.1 |
|
2 |
|
xrex |
|
3 |
2
|
a1i |
|
4 |
|
id |
|
5 |
|
xnegcl |
|
6 |
|
xaddcl |
|
7 |
4 5 6
|
syl2anr |
|
8 |
|
xnegcl |
|
9 |
|
xaddcl |
|
10 |
8 9
|
sylan2 |
|
11 |
7 10
|
ifcld |
|
12 |
11
|
rgen2 |
|
13 |
1
|
xrsds |
|
14 |
13
|
fmpo |
|
15 |
12 14
|
mpbi |
|
16 |
15
|
a1i |
|
17 |
|
breq2 |
|
18 |
|
breq2 |
|
19 |
|
xsubge0 |
|
20 |
19
|
ancoms |
|
21 |
20
|
biimpar |
|
22 |
|
xrletri |
|
23 |
22
|
orcanai |
|
24 |
|
xsubge0 |
|
25 |
24
|
biimpar |
|
26 |
23 25
|
syldan |
|
27 |
17 18 21 26
|
ifbothda |
|
28 |
1
|
xrsdsval |
|
29 |
27 28
|
breqtrrd |
|
30 |
29
|
adantl |
|
31 |
29
|
biantrud |
|
32 |
28 11
|
eqeltrd |
|
33 |
|
0xr |
|
34 |
|
xrletri3 |
|
35 |
32 33 34
|
sylancl |
|
36 |
|
simpr |
|
37 |
|
simplr |
|
38 |
|
0re |
|
39 |
37 38
|
eqeltrdi |
|
40 |
1
|
xrsdsreclb |
|
41 |
40
|
ad4ant124 |
|
42 |
39 41
|
mpbid |
|
43 |
42
|
simpld |
|
44 |
43
|
recnd |
|
45 |
42
|
simprd |
|
46 |
45
|
recnd |
|
47 |
|
rexsub |
|
48 |
42 47
|
syl |
|
49 |
28
|
eqeq1d |
|
50 |
49
|
biimpa |
|
51 |
50
|
adantr |
|
52 |
|
xneg11 |
|
53 |
7 33 52
|
sylancl |
|
54 |
|
simpr |
|
55 |
5
|
adantr |
|
56 |
|
xnegdi |
|
57 |
54 55 56
|
syl2anc |
|
58 |
|
xnegneg |
|
59 |
58
|
adantr |
|
60 |
59
|
oveq2d |
|
61 |
8
|
adantl |
|
62 |
|
simpl |
|
63 |
|
xaddcom |
|
64 |
61 62 63
|
syl2anc |
|
65 |
57 60 64
|
3eqtrd |
|
66 |
|
xneg0 |
|
67 |
66
|
a1i |
|
68 |
65 67
|
eqeq12d |
|
69 |
53 68
|
bitr3d |
|
70 |
69
|
ad2antrr |
|
71 |
|
biidd |
|
72 |
|
eqeq1 |
|
73 |
72
|
bibi1d |
|
74 |
|
eqeq1 |
|
75 |
74
|
bibi1d |
|
76 |
73 75
|
ifboth |
|
77 |
70 71 76
|
syl2anc |
|
78 |
51 77
|
mpbid |
|
79 |
48 78
|
eqtr3d |
|
80 |
44 46 79
|
subeq0d |
|
81 |
36 80
|
pm2.61dane |
|
82 |
81
|
ex |
|
83 |
1
|
xrsdsval |
|
84 |
83
|
anidms |
|
85 |
|
xrleid |
|
86 |
85
|
iftrued |
|
87 |
|
xnegid |
|
88 |
84 86 87
|
3eqtrd |
|
89 |
88
|
adantl |
|
90 |
|
oveq1 |
|
91 |
90
|
eqeq1d |
|
92 |
89 91
|
syl5ibrcom |
|
93 |
82 92
|
impbid |
|
94 |
31 35 93
|
3bitr2d |
|
95 |
94
|
adantl |
|
96 |
|
simplrr |
|
97 |
96
|
leidd |
|
98 |
|
simpr |
|
99 |
98
|
oveq1d |
|
100 |
98
|
oveq1d |
|
101 |
|
simpll1 |
|
102 |
|
oveq12 |
|
103 |
102
|
anidms |
|
104 |
103
|
eqeq1d |
|
105 |
104 88
|
vtoclga |
|
106 |
101 105
|
syl |
|
107 |
100 106
|
eqtrd |
|
108 |
107
|
oveq1d |
|
109 |
96
|
recnd |
|
110 |
109
|
addid2d |
|
111 |
108 110
|
eqtr2d |
|
112 |
97 99 111
|
3brtr3d |
|
113 |
|
simpr |
|
114 |
113
|
oveq1d |
|
115 |
|
simplrl |
|
116 |
114 115
|
eqeltrrd |
|
117 |
116
|
leidd |
|
118 |
|
simpll1 |
|
119 |
|
simpll2 |
|
120 |
|
oveq2 |
|
121 |
90 120
|
eqtr4d |
|
122 |
121
|
adantl |
|
123 |
|
eqeq2 |
|
124 |
|
eqeq2 |
|
125 |
|
xrleloe |
|
126 |
125
|
adantr |
|
127 |
|
simpr |
|
128 |
127
|
neneqd |
|
129 |
|
biorf |
|
130 |
|
orcom |
|
131 |
129 130
|
bitrdi |
|
132 |
128 131
|
syl |
|
133 |
|
xrltnle |
|
134 |
133
|
adantr |
|
135 |
126 132 134
|
3bitr2d |
|
136 |
135
|
con2bid |
|
137 |
136
|
biimpa |
|
138 |
137
|
iffalsed |
|
139 |
135
|
biimpar |
|
140 |
139
|
iftrued |
|
141 |
123 124 138 140
|
ifbothda |
|
142 |
28
|
adantr |
|
143 |
1
|
xrsdsval |
|
144 |
143
|
ancoms |
|
145 |
144
|
adantr |
|
146 |
141 142 145
|
3eqtr4d |
|
147 |
122 146
|
pm2.61dane |
|
148 |
118 119 147
|
syl2anc |
|
149 |
113
|
oveq1d |
|
150 |
119 88
|
syl |
|
151 |
149 150
|
eqtrd |
|
152 |
114 151
|
oveq12d |
|
153 |
116
|
recnd |
|
154 |
153
|
addid1d |
|
155 |
152 154
|
eqtrd |
|
156 |
117 148 155
|
3brtr4d |
|
157 |
|
simplrl |
|
158 |
|
simpll3 |
|
159 |
|
simpll1 |
|
160 |
|
simprl |
|
161 |
1
|
xrsdsreclb |
|
162 |
158 159 160 161
|
syl3anc |
|
163 |
157 162
|
mpbid |
|
164 |
163
|
simprd |
|
165 |
164
|
recnd |
|
166 |
|
simplrr |
|
167 |
|
simpll2 |
|
168 |
|
simprr |
|
169 |
1
|
xrsdsreclb |
|
170 |
158 167 168 169
|
syl3anc |
|
171 |
166 170
|
mpbid |
|
172 |
171
|
simprd |
|
173 |
172
|
recnd |
|
174 |
163
|
simpld |
|
175 |
174
|
recnd |
|
176 |
165 173 175
|
abs3difd |
|
177 |
1
|
xrsdsreval |
|
178 |
164 172 177
|
syl2anc |
|
179 |
1
|
xrsdsreval |
|
180 |
163 179
|
syl |
|
181 |
175 165
|
abssubd |
|
182 |
180 181
|
eqtrd |
|
183 |
1
|
xrsdsreval |
|
184 |
171 183
|
syl |
|
185 |
182 184
|
oveq12d |
|
186 |
176 178 185
|
3brtr4d |
|
187 |
112 156 186
|
pm2.61da2ne |
|
188 |
187
|
3adant1 |
|
189 |
3 16 30 95 188
|
isxmet2d |
|
190 |
189
|
mptru |
|