Metamath Proof Explorer

Theorem flt4lem5e

Description: Satisfy the hypotheses of flt4lem4 . (Contributed by SN, 23-Aug-2024)

Ref Expression
Hypotheses flt4lem5a.m 
|- M = ( ( ( sqrt ` ( C + ( B ^ 2 ) ) ) + ( sqrt ` ( C - ( B ^ 2 ) ) ) ) / 2 )
flt4lem5a.n 
|- N = ( ( ( sqrt ` ( C + ( B ^ 2 ) ) ) - ( sqrt ` ( C - ( B ^ 2 ) ) ) ) / 2 )
flt4lem5a.r 
|- R = ( ( ( sqrt ` ( M + N ) ) + ( sqrt ` ( M - N ) ) ) / 2 )
flt4lem5a.s 
|- S = ( ( ( sqrt ` ( M + N ) ) - ( sqrt ` ( M - N ) ) ) / 2 )
flt4lem5a.a 
|- ( ph -> A e. NN )
flt4lem5a.b 
|- ( ph -> B e. NN )
flt4lem5a.c 
|- ( ph -> C e. NN )
flt4lem5a.1 
|- ( ph -> -. 2 || A )
flt4lem5a.2 
|- ( ph -> ( A gcd B ) = 1 )
flt4lem5a.3 
|- ( ph -> ( ( A ^ 4 ) + ( B ^ 4 ) ) = ( C ^ 2 ) )
Assertion flt4lem5e 
|- ( ph -> ( ( ( R gcd S ) = 1 /\ ( R gcd M ) = 1 /\ ( S gcd M ) = 1 ) /\ ( R e. NN /\ S e. NN /\ M e. NN ) /\ ( ( M x. ( R x. S ) ) = ( ( B / 2 ) ^ 2 ) /\ ( B / 2 ) e. NN ) ) )

Proof

Step Hyp Ref Expression
1 flt4lem5a.m 
 |-  M = ( ( ( sqrt ` ( C + ( B ^ 2 ) ) ) + ( sqrt ` ( C - ( B ^ 2 ) ) ) ) / 2 )
2 flt4lem5a.n 
 |-  N = ( ( ( sqrt ` ( C + ( B ^ 2 ) ) ) - ( sqrt ` ( C - ( B ^ 2 ) ) ) ) / 2 )
3 flt4lem5a.r 
 |-  R = ( ( ( sqrt ` ( M + N ) ) + ( sqrt ` ( M - N ) ) ) / 2 )
4 flt4lem5a.s 
 |-  S = ( ( ( sqrt ` ( M + N ) ) - ( sqrt ` ( M - N ) ) ) / 2 )
5 flt4lem5a.a 
 |-  ( ph -> A e. NN )
6 flt4lem5a.b 
 |-  ( ph -> B e. NN )
7 flt4lem5a.c 
 |-  ( ph -> C e. NN )
8 flt4lem5a.1 
 |-  ( ph -> -. 2 || A )
9 flt4lem5a.2 
 |-  ( ph -> ( A gcd B ) = 1 )
10 flt4lem5a.3 
 |-  ( ph -> ( ( A ^ 4 ) + ( B ^ 4 ) ) = ( C ^ 2 ) )
11 5 nnsqcld 
 |-  ( ph -> ( A ^ 2 ) e. NN )
12 6 nnsqcld 
 |-  ( ph -> ( B ^ 2 ) e. NN )
13 2prm 
 |-  2 e. Prime
14 5 nnzd 
 |-  ( ph -> A e. ZZ )
15 prmdvdssq 
 |-  ( ( 2 e. Prime /\ A e. ZZ ) -> ( 2 || A <-> 2 || ( A ^ 2 ) ) )
16 13 14 15 sylancr 
 |-  ( ph -> ( 2 || A <-> 2 || ( A ^ 2 ) ) )
17 8 16 mtbid 
 |-  ( ph -> -. 2 || ( A ^ 2 ) )
18 2nn 
 |-  2 e. NN
19 18 a1i 
 |-  ( ph -> 2 e. NN )
20 rppwr 
 |-  ( ( A e. NN /\ B e. NN /\ 2 e. NN ) -> ( ( A gcd B ) = 1 -> ( ( A ^ 2 ) gcd ( B ^ 2 ) ) = 1 ) )
21 5 6 19 20 syl3anc 
 |-  ( ph -> ( ( A gcd B ) = 1 -> ( ( A ^ 2 ) gcd ( B ^ 2 ) ) = 1 ) )
22 9 21 mpd 
 |-  ( ph -> ( ( A ^ 2 ) gcd ( B ^ 2 ) ) = 1 )
23 5 nncnd 
 |-  ( ph -> A e. CC )
24 23 exp4sqsq 
 |-  ( ph -> ( A ^ 4 ) = ( ( A ^ 2 ) ^ 2 ) )
25 6 nncnd 
 |-  ( ph -> B e. CC )
26 25 exp4sqsq 
 |-  ( ph -> ( B ^ 4 ) = ( ( B ^ 2 ) ^ 2 ) )
27 24 26 oveq12d 
 |-  ( ph -> ( ( A ^ 4 ) + ( B ^ 4 ) ) = ( ( ( A ^ 2 ) ^ 2 ) + ( ( B ^ 2 ) ^ 2 ) ) )
28 27 10 eqtr3d 
 |-  ( ph -> ( ( ( A ^ 2 ) ^ 2 ) + ( ( B ^ 2 ) ^ 2 ) ) = ( C ^ 2 ) )
29 11 12 7 17 22 28 flt4lem1 
 |-  ( ph -> ( ( ( A ^ 2 ) e. NN /\ ( B ^ 2 ) e. NN /\ C e. NN ) /\ ( ( ( A ^ 2 ) ^ 2 ) + ( ( B ^ 2 ) ^ 2 ) ) = ( C ^ 2 ) /\ ( ( ( A ^ 2 ) gcd ( B ^ 2 ) ) = 1 /\ -. 2 || ( A ^ 2 ) ) ) )
30 2 pythagtriplem13 
 |-  ( ( ( ( A ^ 2 ) e. NN /\ ( B ^ 2 ) e. NN /\ C e. NN ) /\ ( ( ( A ^ 2 ) ^ 2 ) + ( ( B ^ 2 ) ^ 2 ) ) = ( C ^ 2 ) /\ ( ( ( A ^ 2 ) gcd ( B ^ 2 ) ) = 1 /\ -. 2 || ( A ^ 2 ) ) ) -> N e. NN )
31 29 30 syl 
 |-  ( ph -> N e. NN )
32 1 pythagtriplem11 
 |-  ( ( ( ( A ^ 2 ) e. NN /\ ( B ^ 2 ) e. NN /\ C e. NN ) /\ ( ( ( A ^ 2 ) ^ 2 ) + ( ( B ^ 2 ) ^ 2 ) ) = ( C ^ 2 ) /\ ( ( ( A ^ 2 ) gcd ( B ^ 2 ) ) = 1 /\ -. 2 || ( A ^ 2 ) ) ) -> M e. NN )
33 29 32 syl 
 |-  ( ph -> M e. NN )
34 31 nnzd 
 |-  ( ph -> N e. ZZ )
35 14 34 gcdcomd 
 |-  ( ph -> ( A gcd N ) = ( N gcd A ) )
36 33 nnzd 
 |-  ( ph -> M e. ZZ )
37 34 36 gcdcomd 
 |-  ( ph -> ( N gcd M ) = ( M gcd N ) )
38 1 2 flt4lem5 
 |-  ( ( ( ( A ^ 2 ) e. NN /\ ( B ^ 2 ) e. NN /\ C e. NN ) /\ ( ( ( A ^ 2 ) ^ 2 ) + ( ( B ^ 2 ) ^ 2 ) ) = ( C ^ 2 ) /\ ( ( ( A ^ 2 ) gcd ( B ^ 2 ) ) = 1 /\ -. 2 || ( A ^ 2 ) ) ) -> ( M gcd N ) = 1 )
39 29 38 syl 
 |-  ( ph -> ( M gcd N ) = 1 )
40 37 39 eqtrd 
 |-  ( ph -> ( N gcd M ) = 1 )
41 31 nnsqcld 
 |-  ( ph -> ( N ^ 2 ) e. NN )
42 41 nncnd 
 |-  ( ph -> ( N ^ 2 ) e. CC )
43 11 nncnd 
 |-  ( ph -> ( A ^ 2 ) e. CC )
44 42 43 addcomd 
 |-  ( ph -> ( ( N ^ 2 ) + ( A ^ 2 ) ) = ( ( A ^ 2 ) + ( N ^ 2 ) ) )
45 1 2 3 4 5 6 7 8 9 10 flt4lem5a 
 |-  ( ph -> ( ( A ^ 2 ) + ( N ^ 2 ) ) = ( M ^ 2 ) )
46 44 45 eqtrd 
 |-  ( ph -> ( ( N ^ 2 ) + ( A ^ 2 ) ) = ( M ^ 2 ) )
47 31 5 33 40 46 fltabcoprm 
 |-  ( ph -> ( N gcd A ) = 1 )
48 35 47 eqtrd 
 |-  ( ph -> ( A gcd N ) = 1 )
49 5 31 33 8 48 45 flt4lem1 
 |-  ( ph -> ( ( A e. NN /\ N e. NN /\ M e. NN ) /\ ( ( A ^ 2 ) + ( N ^ 2 ) ) = ( M ^ 2 ) /\ ( ( A gcd N ) = 1 /\ -. 2 || A ) ) )
50 3 4 flt4lem5 
 |-  ( ( ( A e. NN /\ N e. NN /\ M e. NN ) /\ ( ( A ^ 2 ) + ( N ^ 2 ) ) = ( M ^ 2 ) /\ ( ( A gcd N ) = 1 /\ -. 2 || A ) ) -> ( R gcd S ) = 1 )
51 49 50 syl 
 |-  ( ph -> ( R gcd S ) = 1 )
52 3 pythagtriplem11 
 |-  ( ( ( A e. NN /\ N e. NN /\ M e. NN ) /\ ( ( A ^ 2 ) + ( N ^ 2 ) ) = ( M ^ 2 ) /\ ( ( A gcd N ) = 1 /\ -. 2 || A ) ) -> R e. NN )
53 49 52 syl 
 |-  ( ph -> R e. NN )
54 4 pythagtriplem13 
 |-  ( ( ( A e. NN /\ N e. NN /\ M e. NN ) /\ ( ( A ^ 2 ) + ( N ^ 2 ) ) = ( M ^ 2 ) /\ ( ( A gcd N ) = 1 /\ -. 2 || A ) ) -> S e. NN )
55 49 54 syl 
 |-  ( ph -> S e. NN )
56 1 2 3 4 5 6 7 8 9 10 flt4lem5d 
 |-  ( ph -> M = ( ( R ^ 2 ) + ( S ^ 2 ) ) )
57 33 53 55 56 51 flt4lem5elem 
 |-  ( ph -> ( ( R gcd M ) = 1 /\ ( S gcd M ) = 1 ) )
58 3anass 
 |-  ( ( ( R gcd S ) = 1 /\ ( R gcd M ) = 1 /\ ( S gcd M ) = 1 ) <-> ( ( R gcd S ) = 1 /\ ( ( R gcd M ) = 1 /\ ( S gcd M ) = 1 ) ) )
59 51 57 58 sylanbrc 
 |-  ( ph -> ( ( R gcd S ) = 1 /\ ( R gcd M ) = 1 /\ ( S gcd M ) = 1 ) )
60 53 55 33 3jca 
 |-  ( ph -> ( R e. NN /\ S e. NN /\ M e. NN ) )
61 2cnd 
 |-  ( ph -> 2 e. CC )
62 61 sqcld 
 |-  ( ph -> ( 2 ^ 2 ) e. CC )
63 53 55 nnmulcld 
 |-  ( ph -> ( R x. S ) e. NN )
64 33 63 nnmulcld 
 |-  ( ph -> ( M x. ( R x. S ) ) e. NN )
65 64 nncnd 
 |-  ( ph -> ( M x. ( R x. S ) ) e. CC )
66 2ne0 
 |-  2 =/= 0
67 66 a1i 
 |-  ( ph -> 2 =/= 0 )
68 2z 
 |-  2 e. ZZ
69 68 a1i 
 |-  ( ph -> 2 e. ZZ )
70 61 67 69 expne0d 
 |-  ( ph -> ( 2 ^ 2 ) =/= 0 )
71 2cn 
 |-  2 e. CC
72 71 sqvali 
 |-  ( 2 ^ 2 ) = ( 2 x. 2 )
73 72 oveq1i 
 |-  ( ( 2 ^ 2 ) x. ( M x. ( R x. S ) ) ) = ( ( 2 x. 2 ) x. ( M x. ( R x. S ) ) )
74 33 nncnd 
 |-  ( ph -> M e. CC )
75 63 nncnd 
 |-  ( ph -> ( R x. S ) e. CC )
76 61 61 74 75 mul4d 
 |-  ( ph -> ( ( 2 x. 2 ) x. ( M x. ( R x. S ) ) ) = ( ( 2 x. M ) x. ( 2 x. ( R x. S ) ) ) )
77 1 2 3 4 5 6 7 8 9 10 flt4lem5c 
 |-  ( ph -> N = ( 2 x. ( R x. S ) ) )
78 77 oveq2d 
 |-  ( ph -> ( ( 2 x. M ) x. N ) = ( ( 2 x. M ) x. ( 2 x. ( R x. S ) ) ) )
79 31 nncnd 
 |-  ( ph -> N e. CC )
80 61 74 79 mulassd 
 |-  ( ph -> ( ( 2 x. M ) x. N ) = ( 2 x. ( M x. N ) ) )
81 78 80 eqtr3d 
 |-  ( ph -> ( ( 2 x. M ) x. ( 2 x. ( R x. S ) ) ) = ( 2 x. ( M x. N ) ) )
82 1 2 3 4 5 6 7 8 9 10 flt4lem5b 
 |-  ( ph -> ( 2 x. ( M x. N ) ) = ( B ^ 2 ) )
83 76 81 82 3eqtrd 
 |-  ( ph -> ( ( 2 x. 2 ) x. ( M x. ( R x. S ) ) ) = ( B ^ 2 ) )
84 73 83 eqtrid 
 |-  ( ph -> ( ( 2 ^ 2 ) x. ( M x. ( R x. S ) ) ) = ( B ^ 2 ) )
85 62 65 70 84 mvllmuld 
 |-  ( ph -> ( M x. ( R x. S ) ) = ( ( B ^ 2 ) / ( 2 ^ 2 ) ) )
86 25 61 67 sqdivd 
 |-  ( ph -> ( ( B / 2 ) ^ 2 ) = ( ( B ^ 2 ) / ( 2 ^ 2 ) ) )
87 85 86 eqtr4d 
 |-  ( ph -> ( M x. ( R x. S ) ) = ( ( B / 2 ) ^ 2 ) )
88 64 nnzd 
 |-  ( ph -> ( M x. ( R x. S ) ) e. ZZ )
89 87 88 eqeltrrd 
 |-  ( ph -> ( ( B / 2 ) ^ 2 ) e. ZZ )
90 6 nnzd 
 |-  ( ph -> B e. ZZ )
91 znq 
 |-  ( ( B e. ZZ /\ 2 e. NN ) -> ( B / 2 ) e. QQ )
92 90 18 91 sylancl 
 |-  ( ph -> ( B / 2 ) e. QQ )
93 6 nngt0d 
 |-  ( ph -> 0 < B )
94 6 nnred 
 |-  ( ph -> B e. RR )
95 halfpos2 
 |-  ( B e. RR -> ( 0 < B <-> 0 < ( B / 2 ) ) )
96 94 95 syl 
 |-  ( ph -> ( 0 < B <-> 0 < ( B / 2 ) ) )
97 93 96 mpbid 
 |-  ( ph -> 0 < ( B / 2 ) )
98 89 92 97 posqsqznn 
 |-  ( ph -> ( B / 2 ) e. NN )
99 87 98 jca 
 |-  ( ph -> ( ( M x. ( R x. S ) ) = ( ( B / 2 ) ^ 2 ) /\ ( B / 2 ) e. NN ) )
100 59 60 99 3jca 
 |-  ( ph -> ( ( ( R gcd S ) = 1 /\ ( R gcd M ) = 1 /\ ( S gcd M ) = 1 ) /\ ( R e. NN /\ S e. NN /\ M e. NN ) /\ ( ( M x. ( R x. S ) ) = ( ( B / 2 ) ^ 2 ) /\ ( B / 2 ) e. NN ) ) )