Step |
Hyp |
Ref |
Expression |
1 |
|
pythagtriplem13.1 |
|
2 |
1
|
oveq1i |
|
3 |
|
nncn |
|
4 |
|
nncn |
|
5 |
|
addcl |
|
6 |
3 4 5
|
syl2anr |
|
7 |
6
|
sqrtcld |
|
8 |
|
subcl |
|
9 |
3 4 8
|
syl2anr |
|
10 |
9
|
sqrtcld |
|
11 |
7 10
|
subcld |
|
12 |
11
|
3adant1 |
|
13 |
12
|
3ad2ant1 |
|
14 |
|
2cn |
|
15 |
|
2ne0 |
|
16 |
|
sqdiv |
|
17 |
14 15 16
|
mp3an23 |
|
18 |
13 17
|
syl |
|
19 |
14
|
sqvali |
|
20 |
19
|
oveq2i |
|
21 |
13
|
sqcld |
|
22 |
|
2cnne0 |
|
23 |
|
divdiv1 |
|
24 |
22 22 23
|
mp3an23 |
|
25 |
21 24
|
syl |
|
26 |
|
simp12 |
|
27 |
|
simp13 |
|
28 |
26 27 7
|
syl2anc |
|
29 |
26 27 10
|
syl2anc |
|
30 |
|
binom2sub |
|
31 |
28 29 30
|
syl2anc |
|
32 |
|
nnre |
|
33 |
|
nnre |
|
34 |
|
readdcl |
|
35 |
32 33 34
|
syl2anr |
|
36 |
35
|
3adant1 |
|
37 |
36
|
3ad2ant1 |
|
38 |
37
|
recnd |
|
39 |
|
resubcl |
|
40 |
32 33 39
|
syl2anr |
|
41 |
40
|
3adant1 |
|
42 |
41
|
3ad2ant1 |
|
43 |
42
|
recnd |
|
44 |
7
|
3adant1 |
|
45 |
10
|
3adant1 |
|
46 |
44 45
|
mulcld |
|
47 |
|
mulcl |
|
48 |
14 46 47
|
sylancr |
|
49 |
48
|
3ad2ant1 |
|
50 |
38 43 49
|
addsubd |
|
51 |
27
|
nncnd |
|
52 |
|
simp11 |
|
53 |
52
|
nncnd |
|
54 |
|
subdi |
|
55 |
14 51 53 54
|
mp3an2i |
|
56 |
|
ppncan |
|
57 |
56
|
3anidm13 |
|
58 |
|
2times |
|
59 |
58
|
adantr |
|
60 |
57 59
|
eqtr4d |
|
61 |
3 4 60
|
syl2anr |
|
62 |
61
|
3adant1 |
|
63 |
62
|
3ad2ant1 |
|
64 |
26
|
nncnd |
|
65 |
|
subsq |
|
66 |
51 64 65
|
syl2anc |
|
67 |
|
oveq1 |
|
68 |
67
|
3ad2ant2 |
|
69 |
|
nncn |
|
70 |
69
|
sqcld |
|
71 |
70
|
3ad2ant1 |
|
72 |
4
|
sqcld |
|
73 |
72
|
3ad2ant2 |
|
74 |
71 73
|
pncand |
|
75 |
74
|
3ad2ant1 |
|
76 |
68 75
|
eqtr3d |
|
77 |
66 76
|
eqtr3d |
|
78 |
77
|
fveq2d |
|
79 |
32
|
adantl |
|
80 |
33
|
adantr |
|
81 |
|
nngt0 |
|
82 |
81
|
adantl |
|
83 |
|
nngt0 |
|
84 |
83
|
adantr |
|
85 |
79 80 82 84
|
addgt0d |
|
86 |
|
0re |
|
87 |
|
ltle |
|
88 |
86 87
|
mpan |
|
89 |
35 85 88
|
sylc |
|
90 |
89
|
3adant1 |
|
91 |
90
|
3ad2ant1 |
|
92 |
|
pythagtriplem10 |
|
93 |
92
|
3adant3 |
|
94 |
|
ltle |
|
95 |
86 94
|
mpan |
|
96 |
42 93 95
|
sylc |
|
97 |
37 91 42 96
|
sqrtmuld |
|
98 |
78 97
|
eqtr3d |
|
99 |
|
nnre |
|
100 |
99
|
3ad2ant1 |
|
101 |
100
|
3ad2ant1 |
|
102 |
|
nnnn0 |
|
103 |
102
|
nn0ge0d |
|
104 |
103
|
3ad2ant1 |
|
105 |
104
|
3ad2ant1 |
|
106 |
101 105
|
sqrtsqd |
|
107 |
98 106
|
eqtr3d |
|
108 |
107
|
oveq2d |
|
109 |
63 108
|
oveq12d |
|
110 |
55 109
|
eqtr4d |
|
111 |
|
resqrtth |
|
112 |
37 91 111
|
syl2anc |
|
113 |
112
|
oveq1d |
|
114 |
|
resqrtth |
|
115 |
42 96 114
|
syl2anc |
|
116 |
113 115
|
oveq12d |
|
117 |
50 110 116
|
3eqtr4rd |
|
118 |
31 117
|
eqtrd |
|
119 |
118
|
oveq1d |
|
120 |
|
subcl |
|
121 |
3 69 120
|
syl2anr |
|
122 |
121
|
3adant2 |
|
123 |
122
|
3ad2ant1 |
|
124 |
|
divcan3 |
|
125 |
14 15 124
|
mp3an23 |
|
126 |
123 125
|
syl |
|
127 |
119 126
|
eqtrd |
|
128 |
127
|
oveq1d |
|
129 |
25 128
|
eqtr3d |
|
130 |
20 129
|
syl5eq |
|
131 |
18 130
|
eqtrd |
|
132 |
2 131
|
syl5eq |
|