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Theorem quartlem1 22395
Description: Lemma for quart 22399. (Contributed by Mario Carneiro, 6-May-2015.)
Hypotheses
Ref Expression
quartlem1.p  |-  ( ph  ->  P  e.  CC )
quartlem1.q  |-  ( ph  ->  Q  e.  CC )
quartlem1.r  |-  ( ph  ->  R  e.  CC )
quartlem1.u  |-  ( ph  ->  U  =  ( ( P ^ 2 )  +  (; 1 2  x.  R
) ) )
quartlem1.v  |-  ( ph  ->  V  =  ( (
-u ( 2  x.  ( P ^ 3 ) )  -  (; 2 7  x.  ( Q ^
2 ) ) )  +  (; 7 2  x.  ( P  x.  R )
) ) )
Assertion
Ref Expression
quartlem1  |-  ( ph  ->  ( U  =  ( ( ( 2  x.  P ) ^ 2 )  -  ( 3  x.  ( ( P ^ 2 )  -  ( 4  x.  R
) ) ) )  /\  V  =  ( ( ( 2  x.  ( ( 2  x.  P ) ^ 3 ) )  -  (
9  x.  ( ( 2  x.  P )  x.  ( ( P ^ 2 )  -  ( 4  x.  R
) ) ) ) )  +  (; 2 7  x.  -u ( Q ^ 2 ) ) ) ) )

Proof of Theorem quartlem1
StepHypRef Expression
1 2cn 10507 . . . . . . . . . 10  |-  2  e.  CC
2 quartlem1.p . . . . . . . . . 10  |-  ( ph  ->  P  e.  CC )
3 sqmul 12050 . . . . . . . . . 10  |-  ( ( 2  e.  CC  /\  P  e.  CC )  ->  ( ( 2  x.  P ) ^ 2 )  =  ( ( 2 ^ 2 )  x.  ( P ^
2 ) ) )
41, 2, 3sylancr 663 . . . . . . . . 9  |-  ( ph  ->  ( ( 2  x.  P ) ^ 2 )  =  ( ( 2 ^ 2 )  x.  ( P ^
2 ) ) )
5 sq2 12083 . . . . . . . . . 10  |-  ( 2 ^ 2 )  =  4
65oveq1i 6213 . . . . . . . . 9  |-  ( ( 2 ^ 2 )  x.  ( P ^
2 ) )  =  ( 4  x.  ( P ^ 2 ) )
74, 6syl6eq 2511 . . . . . . . 8  |-  ( ph  ->  ( ( 2  x.  P ) ^ 2 )  =  ( 4  x.  ( P ^
2 ) ) )
87oveq1d 6218 . . . . . . 7  |-  ( ph  ->  ( ( ( 2  x.  P ) ^
2 )  -  (
3  x.  ( P ^ 2 ) ) )  =  ( ( 4  x.  ( P ^ 2 ) )  -  ( 3  x.  ( P ^ 2 ) ) ) )
9 4cn 10514 . . . . . . . . 9  |-  4  e.  CC
109a1i 11 . . . . . . . 8  |-  ( ph  ->  4  e.  CC )
11 3cn 10511 . . . . . . . . 9  |-  3  e.  CC
1211a1i 11 . . . . . . . 8  |-  ( ph  ->  3  e.  CC )
132sqcld 12127 . . . . . . . 8  |-  ( ph  ->  ( P ^ 2 )  e.  CC )
1410, 12, 13subdird 9916 . . . . . . 7  |-  ( ph  ->  ( ( 4  -  3 )  x.  ( P ^ 2 ) )  =  ( ( 4  x.  ( P ^
2 ) )  -  ( 3  x.  ( P ^ 2 ) ) ) )
158, 14eqtr4d 2498 . . . . . 6  |-  ( ph  ->  ( ( ( 2  x.  P ) ^
2 )  -  (
3  x.  ( P ^ 2 ) ) )  =  ( ( 4  -  3 )  x.  ( P ^
2 ) ) )
16 ax-1cn 9455 . . . . . . . . . 10  |-  1  e.  CC
17 3p1e4 10562 . . . . . . . . . 10  |-  ( 3  +  1 )  =  4
189, 11, 16, 17subaddrii 9812 . . . . . . . . 9  |-  ( 4  -  3 )  =  1
1918oveq1i 6213 . . . . . . . 8  |-  ( ( 4  -  3 )  x.  ( P ^
2 ) )  =  ( 1  x.  ( P ^ 2 ) )
20 mulid2 9499 . . . . . . . 8  |-  ( ( P ^ 2 )  e.  CC  ->  (
1  x.  ( P ^ 2 ) )  =  ( P ^
2 ) )
2119, 20syl5eq 2507 . . . . . . 7  |-  ( ( P ^ 2 )  e.  CC  ->  (
( 4  -  3 )  x.  ( P ^ 2 ) )  =  ( P ^
2 ) )
2213, 21syl 16 . . . . . 6  |-  ( ph  ->  ( ( 4  -  3 )  x.  ( P ^ 2 ) )  =  ( P ^
2 ) )
2315, 22eqtr2d 2496 . . . . 5  |-  ( ph  ->  ( P ^ 2 )  =  ( ( ( 2  x.  P
) ^ 2 )  -  ( 3  x.  ( P ^ 2 ) ) ) )
2423oveq1d 6218 . . . 4  |-  ( ph  ->  ( ( P ^
2 )  +  (; 1
2  x.  R ) )  =  ( ( ( ( 2  x.  P ) ^ 2 )  -  ( 3  x.  ( P ^
2 ) ) )  +  (; 1 2  x.  R
) ) )
25 mulcl 9481 . . . . . . 7  |-  ( ( 2  e.  CC  /\  P  e.  CC )  ->  ( 2  x.  P
)  e.  CC )
261, 2, 25sylancr 663 . . . . . 6  |-  ( ph  ->  ( 2  x.  P
)  e.  CC )
2726sqcld 12127 . . . . 5  |-  ( ph  ->  ( ( 2  x.  P ) ^ 2 )  e.  CC )
28 mulcl 9481 . . . . . 6  |-  ( ( 3  e.  CC  /\  ( P ^ 2 )  e.  CC )  -> 
( 3  x.  ( P ^ 2 ) )  e.  CC )
2911, 13, 28sylancr 663 . . . . 5  |-  ( ph  ->  ( 3  x.  ( P ^ 2 ) )  e.  CC )
30 1nn0 10710 . . . . . . . 8  |-  1  e.  NN0
31 2nn 10594 . . . . . . . 8  |-  2  e.  NN
3230, 31decnncl 10883 . . . . . . 7  |- ; 1 2  e.  NN
3332nncni 10447 . . . . . 6  |- ; 1 2  e.  CC
34 quartlem1.r . . . . . 6  |-  ( ph  ->  R  e.  CC )
35 mulcl 9481 . . . . . 6  |-  ( (; 1
2  e.  CC  /\  R  e.  CC )  ->  (; 1 2  x.  R
)  e.  CC )
3633, 34, 35sylancr 663 . . . . 5  |-  ( ph  ->  (; 1 2  x.  R
)  e.  CC )
3727, 29, 36subsubd 9862 . . . 4  |-  ( ph  ->  ( ( ( 2  x.  P ) ^
2 )  -  (
( 3  x.  ( P ^ 2 ) )  -  (; 1 2  x.  R
) ) )  =  ( ( ( ( 2  x.  P ) ^ 2 )  -  ( 3  x.  ( P ^ 2 ) ) )  +  (; 1 2  x.  R
) ) )
3824, 37eqtr4d 2498 . . 3  |-  ( ph  ->  ( ( P ^
2 )  +  (; 1
2  x.  R ) )  =  ( ( ( 2  x.  P
) ^ 2 )  -  ( ( 3  x.  ( P ^
2 ) )  -  (; 1 2  x.  R ) ) ) )
39 quartlem1.u . . 3  |-  ( ph  ->  U  =  ( ( P ^ 2 )  +  (; 1 2  x.  R
) ) )
40 mulcl 9481 . . . . . . 7  |-  ( ( 4  e.  CC  /\  R  e.  CC )  ->  ( 4  x.  R
)  e.  CC )
419, 34, 40sylancr 663 . . . . . 6  |-  ( ph  ->  ( 4  x.  R
)  e.  CC )
4212, 13, 41subdid 9915 . . . . 5  |-  ( ph  ->  ( 3  x.  (
( P ^ 2 )  -  ( 4  x.  R ) ) )  =  ( ( 3  x.  ( P ^ 2 ) )  -  ( 3  x.  ( 4  x.  R
) ) ) )
43 4t3e12 10942 . . . . . . . . 9  |-  ( 4  x.  3 )  = ; 1
2
449, 11, 43mulcomli 9508 . . . . . . . 8  |-  ( 3  x.  4 )  = ; 1
2
4544oveq1i 6213 . . . . . . 7  |-  ( ( 3  x.  4 )  x.  R )  =  (; 1 2  x.  R
)
4612, 10, 34mulassd 9524 . . . . . . 7  |-  ( ph  ->  ( ( 3  x.  4 )  x.  R
)  =  ( 3  x.  ( 4  x.  R ) ) )
4745, 46syl5eqr 2509 . . . . . 6  |-  ( ph  ->  (; 1 2  x.  R
)  =  ( 3  x.  ( 4  x.  R ) ) )
4847oveq2d 6219 . . . . 5  |-  ( ph  ->  ( ( 3  x.  ( P ^ 2 ) )  -  (; 1 2  x.  R ) )  =  ( ( 3  x.  ( P ^
2 ) )  -  ( 3  x.  (
4  x.  R ) ) ) )
4942, 48eqtr4d 2498 . . . 4  |-  ( ph  ->  ( 3  x.  (
( P ^ 2 )  -  ( 4  x.  R ) ) )  =  ( ( 3  x.  ( P ^ 2 ) )  -  (; 1 2  x.  R
) ) )
5049oveq2d 6219 . . 3  |-  ( ph  ->  ( ( ( 2  x.  P ) ^
2 )  -  (
3  x.  ( ( P ^ 2 )  -  ( 4  x.  R ) ) ) )  =  ( ( ( 2  x.  P
) ^ 2 )  -  ( ( 3  x.  ( P ^
2 ) )  -  (; 1 2  x.  R ) ) ) )
5138, 39, 503eqtr4d 2505 . 2  |-  ( ph  ->  U  =  ( ( ( 2  x.  P
) ^ 2 )  -  ( 3  x.  ( ( P ^
2 )  -  (
4  x.  R ) ) ) ) )
521a1i 11 . . . . . . . . . 10  |-  ( ph  ->  2  e.  CC )
53 3nn0 10712 . . . . . . . . . . 11  |-  3  e.  NN0
5453a1i 11 . . . . . . . . . 10  |-  ( ph  ->  3  e.  NN0 )
5552, 2, 54mulexpd 12144 . . . . . . . . 9  |-  ( ph  ->  ( ( 2  x.  P ) ^ 3 )  =  ( ( 2 ^ 3 )  x.  ( P ^
3 ) ) )
56 cu2 12085 . . . . . . . . . 10  |-  ( 2 ^ 3 )  =  8
5756oveq1i 6213 . . . . . . . . 9  |-  ( ( 2 ^ 3 )  x.  ( P ^
3 ) )  =  ( 8  x.  ( P ^ 3 ) )
5855, 57syl6eq 2511 . . . . . . . 8  |-  ( ph  ->  ( ( 2  x.  P ) ^ 3 )  =  ( 8  x.  ( P ^
3 ) ) )
5958oveq2d 6219 . . . . . . 7  |-  ( ph  ->  ( 2  x.  (
( 2  x.  P
) ^ 3 ) )  =  ( 2  x.  ( 8  x.  ( P ^ 3 ) ) ) )
60 8cn 10522 . . . . . . . . 9  |-  8  e.  CC
6160a1i 11 . . . . . . . 8  |-  ( ph  ->  8  e.  CC )
62 expcl 12004 . . . . . . . . 9  |-  ( ( P  e.  CC  /\  3  e.  NN0 )  -> 
( P ^ 3 )  e.  CC )
632, 53, 62sylancl 662 . . . . . . . 8  |-  ( ph  ->  ( P ^ 3 )  e.  CC )
6452, 61, 63mul12d 9693 . . . . . . 7  |-  ( ph  ->  ( 2  x.  (
8  x.  ( P ^ 3 ) ) )  =  ( 8  x.  ( 2  x.  ( P ^ 3 ) ) ) )
6559, 64eqtrd 2495 . . . . . 6  |-  ( ph  ->  ( 2  x.  (
( 2  x.  P
) ^ 3 ) )  =  ( 8  x.  ( 2  x.  ( P ^ 3 ) ) ) )
66 9cn 10524 . . . . . . . . 9  |-  9  e.  CC
6766a1i 11 . . . . . . . 8  |-  ( ph  ->  9  e.  CC )
68 mulcl 9481 . . . . . . . . 9  |-  ( ( 2  e.  CC  /\  ( P ^ 3 )  e.  CC )  -> 
( 2  x.  ( P ^ 3 ) )  e.  CC )
691, 63, 68sylancr 663 . . . . . . . 8  |-  ( ph  ->  ( 2  x.  ( P ^ 3 ) )  e.  CC )
702, 34mulcld 9521 . . . . . . . . 9  |-  ( ph  ->  ( P  x.  R
)  e.  CC )
71 mulcl 9481 . . . . . . . . 9  |-  ( ( 8  e.  CC  /\  ( P  x.  R
)  e.  CC )  ->  ( 8  x.  ( P  x.  R
) )  e.  CC )
7260, 70, 71sylancr 663 . . . . . . . 8  |-  ( ph  ->  ( 8  x.  ( P  x.  R )
)  e.  CC )
7367, 69, 72subdid 9915 . . . . . . 7  |-  ( ph  ->  ( 9  x.  (
( 2  x.  ( P ^ 3 ) )  -  ( 8  x.  ( P  x.  R
) ) ) )  =  ( ( 9  x.  ( 2  x.  ( P ^ 3 ) ) )  -  ( 9  x.  (
8  x.  ( P  x.  R ) ) ) ) )
7426, 13, 41subdid 9915 . . . . . . . . 9  |-  ( ph  ->  ( ( 2  x.  P )  x.  (
( P ^ 2 )  -  ( 4  x.  R ) ) )  =  ( ( ( 2  x.  P
)  x.  ( P ^ 2 ) )  -  ( ( 2  x.  P )  x.  ( 4  x.  R
) ) ) )
7552, 2, 13mulassd 9524 . . . . . . . . . . 11  |-  ( ph  ->  ( ( 2  x.  P )  x.  ( P ^ 2 ) )  =  ( 2  x.  ( P  x.  ( P ^ 2 ) ) ) )
762, 13mulcomd 9522 . . . . . . . . . . . . 13  |-  ( ph  ->  ( P  x.  ( P ^ 2 ) )  =  ( ( P ^ 2 )  x.  P ) )
77 df-3 10496 . . . . . . . . . . . . . . 15  |-  3  =  ( 2  +  1 )
7877oveq2i 6214 . . . . . . . . . . . . . 14  |-  ( P ^ 3 )  =  ( P ^ (
2  +  1 ) )
79 2nn0 10711 . . . . . . . . . . . . . . 15  |-  2  e.  NN0
80 expp1 11993 . . . . . . . . . . . . . . 15  |-  ( ( P  e.  CC  /\  2  e.  NN0 )  -> 
( P ^ (
2  +  1 ) )  =  ( ( P ^ 2 )  x.  P ) )
812, 79, 80sylancl 662 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( P ^ (
2  +  1 ) )  =  ( ( P ^ 2 )  x.  P ) )
8278, 81syl5eq 2507 . . . . . . . . . . . . 13  |-  ( ph  ->  ( P ^ 3 )  =  ( ( P ^ 2 )  x.  P ) )
8376, 82eqtr4d 2498 . . . . . . . . . . . 12  |-  ( ph  ->  ( P  x.  ( P ^ 2 ) )  =  ( P ^
3 ) )
8483oveq2d 6219 . . . . . . . . . . 11  |-  ( ph  ->  ( 2  x.  ( P  x.  ( P ^ 2 ) ) )  =  ( 2  x.  ( P ^
3 ) ) )
8575, 84eqtrd 2495 . . . . . . . . . 10  |-  ( ph  ->  ( ( 2  x.  P )  x.  ( P ^ 2 ) )  =  ( 2  x.  ( P ^ 3 ) ) )
8652, 2, 10, 34mul4d 9696 . . . . . . . . . . 11  |-  ( ph  ->  ( ( 2  x.  P )  x.  (
4  x.  R ) )  =  ( ( 2  x.  4 )  x.  ( P  x.  R ) ) )
87 4t2e8 10590 . . . . . . . . . . . . 13  |-  ( 4  x.  2 )  =  8
889, 1, 87mulcomli 9508 . . . . . . . . . . . 12  |-  ( 2  x.  4 )  =  8
8988oveq1i 6213 . . . . . . . . . . 11  |-  ( ( 2  x.  4 )  x.  ( P  x.  R ) )  =  ( 8  x.  ( P  x.  R )
)
9086, 89syl6eq 2511 . . . . . . . . . 10  |-  ( ph  ->  ( ( 2  x.  P )  x.  (
4  x.  R ) )  =  ( 8  x.  ( P  x.  R ) ) )
9185, 90oveq12d 6221 . . . . . . . . 9  |-  ( ph  ->  ( ( ( 2  x.  P )  x.  ( P ^ 2 ) )  -  (
( 2  x.  P
)  x.  ( 4  x.  R ) ) )  =  ( ( 2  x.  ( P ^ 3 ) )  -  ( 8  x.  ( P  x.  R
) ) ) )
9274, 91eqtrd 2495 . . . . . . . 8  |-  ( ph  ->  ( ( 2  x.  P )  x.  (
( P ^ 2 )  -  ( 4  x.  R ) ) )  =  ( ( 2  x.  ( P ^ 3 ) )  -  ( 8  x.  ( P  x.  R
) ) ) )
9392oveq2d 6219 . . . . . . 7  |-  ( ph  ->  ( 9  x.  (
( 2  x.  P
)  x.  ( ( P ^ 2 )  -  ( 4  x.  R ) ) ) )  =  ( 9  x.  ( ( 2  x.  ( P ^
3 ) )  -  ( 8  x.  ( P  x.  R )
) ) ) )
94 9t8e72 10971 . . . . . . . . . 10  |-  ( 9  x.  8 )  = ; 7
2
9594oveq1i 6213 . . . . . . . . 9  |-  ( ( 9  x.  8 )  x.  ( P  x.  R ) )  =  (; 7 2  x.  ( P  x.  R )
)
9667, 61, 70mulassd 9524 . . . . . . . . 9  |-  ( ph  ->  ( ( 9  x.  8 )  x.  ( P  x.  R )
)  =  ( 9  x.  ( 8  x.  ( P  x.  R
) ) ) )
9795, 96syl5eqr 2509 . . . . . . . 8  |-  ( ph  ->  (; 7 2  x.  ( P  x.  R )
)  =  ( 9  x.  ( 8  x.  ( P  x.  R
) ) ) )
9897oveq2d 6219 . . . . . . 7  |-  ( ph  ->  ( ( 9  x.  ( 2  x.  ( P ^ 3 ) ) )  -  (; 7 2  x.  ( P  x.  R )
) )  =  ( ( 9  x.  (
2  x.  ( P ^ 3 ) ) )  -  ( 9  x.  ( 8  x.  ( P  x.  R
) ) ) ) )
9973, 93, 983eqtr4d 2505 . . . . . 6  |-  ( ph  ->  ( 9  x.  (
( 2  x.  P
)  x.  ( ( P ^ 2 )  -  ( 4  x.  R ) ) ) )  =  ( ( 9  x.  ( 2  x.  ( P ^
3 ) ) )  -  (; 7 2  x.  ( P  x.  R )
) ) )
10065, 99oveq12d 6221 . . . . 5  |-  ( ph  ->  ( ( 2  x.  ( ( 2  x.  P ) ^ 3 ) )  -  (
9  x.  ( ( 2  x.  P )  x.  ( ( P ^ 2 )  -  ( 4  x.  R
) ) ) ) )  =  ( ( 8  x.  ( 2  x.  ( P ^
3 ) ) )  -  ( ( 9  x.  ( 2  x.  ( P ^ 3 ) ) )  -  (; 7 2  x.  ( P  x.  R ) ) ) ) )
101 mulcl 9481 . . . . . . 7  |-  ( ( 8  e.  CC  /\  ( 2  x.  ( P ^ 3 ) )  e.  CC )  -> 
( 8  x.  (
2  x.  ( P ^ 3 ) ) )  e.  CC )
10260, 69, 101sylancr 663 . . . . . 6  |-  ( ph  ->  ( 8  x.  (
2  x.  ( P ^ 3 ) ) )  e.  CC )
103 mulcl 9481 . . . . . . 7  |-  ( ( 9  e.  CC  /\  ( 2  x.  ( P ^ 3 ) )  e.  CC )  -> 
( 9  x.  (
2  x.  ( P ^ 3 ) ) )  e.  CC )
10466, 69, 103sylancr 663 . . . . . 6  |-  ( ph  ->  ( 9  x.  (
2  x.  ( P ^ 3 ) ) )  e.  CC )
105 7nn0 10716 . . . . . . . . 9  |-  7  e.  NN0
106105, 31decnncl 10883 . . . . . . . 8  |- ; 7 2  e.  NN
107106nncni 10447 . . . . . . 7  |- ; 7 2  e.  CC
108 mulcl 9481 . . . . . . 7  |-  ( (; 7
2  e.  CC  /\  ( P  x.  R
)  e.  CC )  ->  (; 7 2  x.  ( P  x.  R )
)  e.  CC )
109107, 70, 108sylancr 663 . . . . . 6  |-  ( ph  ->  (; 7 2  x.  ( P  x.  R )
)  e.  CC )
110102, 104, 109subsubd 9862 . . . . 5  |-  ( ph  ->  ( ( 8  x.  ( 2  x.  ( P ^ 3 ) ) )  -  ( ( 9  x.  ( 2  x.  ( P ^
3 ) ) )  -  (; 7 2  x.  ( P  x.  R )
) ) )  =  ( ( ( 8  x.  ( 2  x.  ( P ^ 3 ) ) )  -  ( 9  x.  (
2  x.  ( P ^ 3 ) ) ) )  +  (; 7
2  x.  ( P  x.  R ) ) ) )
111104, 102negsubdi2d 9850 . . . . . . 7  |-  ( ph  -> 
-u ( ( 9  x.  ( 2  x.  ( P ^ 3 ) ) )  -  ( 8  x.  (
2  x.  ( P ^ 3 ) ) ) )  =  ( ( 8  x.  (
2  x.  ( P ^ 3 ) ) )  -  ( 9  x.  ( 2  x.  ( P ^ 3 ) ) ) ) )
11267, 61, 69subdird 9916 . . . . . . . . 9  |-  ( ph  ->  ( ( 9  -  8 )  x.  (
2  x.  ( P ^ 3 ) ) )  =  ( ( 9  x.  ( 2  x.  ( P ^
3 ) ) )  -  ( 8  x.  ( 2  x.  ( P ^ 3 ) ) ) ) )
113 8p1e9 10567 . . . . . . . . . . . 12  |-  ( 8  +  1 )  =  9
11466, 60, 16, 113subaddrii 9812 . . . . . . . . . . 11  |-  ( 9  -  8 )  =  1
115114oveq1i 6213 . . . . . . . . . 10  |-  ( ( 9  -  8 )  x.  ( 2  x.  ( P ^ 3 ) ) )  =  ( 1  x.  (
2  x.  ( P ^ 3 ) ) )
11669mulid2d 9519 . . . . . . . . . 10  |-  ( ph  ->  ( 1  x.  (
2  x.  ( P ^ 3 ) ) )  =  ( 2  x.  ( P ^
3 ) ) )
117115, 116syl5eq 2507 . . . . . . . . 9  |-  ( ph  ->  ( ( 9  -  8 )  x.  (
2  x.  ( P ^ 3 ) ) )  =  ( 2  x.  ( P ^
3 ) ) )
118112, 117eqtr3d 2497 . . . . . . . 8  |-  ( ph  ->  ( ( 9  x.  ( 2  x.  ( P ^ 3 ) ) )  -  ( 8  x.  ( 2  x.  ( P ^ 3 ) ) ) )  =  ( 2  x.  ( P ^ 3 ) ) )
119118negeqd 9719 . . . . . . 7  |-  ( ph  -> 
-u ( ( 9  x.  ( 2  x.  ( P ^ 3 ) ) )  -  ( 8  x.  (
2  x.  ( P ^ 3 ) ) ) )  =  -u ( 2  x.  ( P ^ 3 ) ) )
120111, 119eqtr3d 2497 . . . . . 6  |-  ( ph  ->  ( ( 8  x.  ( 2  x.  ( P ^ 3 ) ) )  -  ( 9  x.  ( 2  x.  ( P ^ 3 ) ) ) )  =  -u ( 2  x.  ( P ^ 3 ) ) )
121120oveq1d 6218 . . . . 5  |-  ( ph  ->  ( ( ( 8  x.  ( 2  x.  ( P ^ 3 ) ) )  -  ( 9  x.  (
2  x.  ( P ^ 3 ) ) ) )  +  (; 7
2  x.  ( P  x.  R ) ) )  =  ( -u ( 2  x.  ( P ^ 3 ) )  +  (; 7 2  x.  ( P  x.  R )
) ) )
122100, 110, 1213eqtrd 2499 . . . 4  |-  ( ph  ->  ( ( 2  x.  ( ( 2  x.  P ) ^ 3 ) )  -  (
9  x.  ( ( 2  x.  P )  x.  ( ( P ^ 2 )  -  ( 4  x.  R
) ) ) ) )  =  ( -u ( 2  x.  ( P ^ 3 ) )  +  (; 7 2  x.  ( P  x.  R )
) ) )
123 7nn 10599 . . . . . . 7  |-  7  e.  NN
12479, 123decnncl 10883 . . . . . 6  |- ; 2 7  e.  NN
125124nncni 10447 . . . . 5  |- ; 2 7  e.  CC
126 quartlem1.q . . . . . 6  |-  ( ph  ->  Q  e.  CC )
127126sqcld 12127 . . . . 5  |-  ( ph  ->  ( Q ^ 2 )  e.  CC )
128 mulneg2 9897 . . . . 5  |-  ( (; 2
7  e.  CC  /\  ( Q ^ 2 )  e.  CC )  -> 
(; 2 7  x.  -u ( Q ^ 2 ) )  =  -u (; 2 7  x.  ( Q ^ 2 ) ) )
129125, 127, 128sylancr 663 . . . 4  |-  ( ph  ->  (; 2 7  x.  -u ( Q ^ 2 ) )  =  -u (; 2 7  x.  ( Q ^ 2 ) ) )
130122, 129oveq12d 6221 . . 3  |-  ( ph  ->  ( ( ( 2  x.  ( ( 2  x.  P ) ^
3 ) )  -  ( 9  x.  (
( 2  x.  P
)  x.  ( ( P ^ 2 )  -  ( 4  x.  R ) ) ) ) )  +  (; 2
7  x.  -u ( Q ^ 2 ) ) )  =  ( (
-u ( 2  x.  ( P ^ 3 ) )  +  (; 7
2  x.  ( P  x.  R ) ) )  +  -u (; 2 7  x.  ( Q ^
2 ) ) ) )
13169negcld 9821 . . . . 5  |-  ( ph  -> 
-u ( 2  x.  ( P ^ 3 ) )  e.  CC )
132 mulcl 9481 . . . . . 6  |-  ( (; 2
7  e.  CC  /\  ( Q ^ 2 )  e.  CC )  -> 
(; 2 7  x.  ( Q ^ 2 ) )  e.  CC )
133125, 127, 132sylancr 663 . . . . 5  |-  ( ph  ->  (; 2 7  x.  ( Q ^ 2 ) )  e.  CC )
134131, 109, 133addsubd 9855 . . . 4  |-  ( ph  ->  ( ( -u (
2  x.  ( P ^ 3 ) )  +  (; 7 2  x.  ( P  x.  R )
) )  -  (; 2 7  x.  ( Q ^
2 ) ) )  =  ( ( -u ( 2  x.  ( P ^ 3 ) )  -  (; 2 7  x.  ( Q ^ 2 ) ) )  +  (; 7 2  x.  ( P  x.  R )
) ) )
135131, 109addcld 9520 . . . . 5  |-  ( ph  ->  ( -u ( 2  x.  ( P ^
3 ) )  +  (; 7 2  x.  ( P  x.  R )
) )  e.  CC )
136135, 133negsubd 9840 . . . 4  |-  ( ph  ->  ( ( -u (
2  x.  ( P ^ 3 ) )  +  (; 7 2  x.  ( P  x.  R )
) )  +  -u (; 2 7  x.  ( Q ^ 2 ) ) )  =  ( (
-u ( 2  x.  ( P ^ 3 ) )  +  (; 7
2  x.  ( P  x.  R ) ) )  -  (; 2 7  x.  ( Q ^ 2 ) ) ) )
137 quartlem1.v . . . 4  |-  ( ph  ->  V  =  ( (
-u ( 2  x.  ( P ^ 3 ) )  -  (; 2 7  x.  ( Q ^
2 ) ) )  +  (; 7 2  x.  ( P  x.  R )
) ) )
138134, 136, 1373eqtr4d 2505 . . 3  |-  ( ph  ->  ( ( -u (
2  x.  ( P ^ 3 ) )  +  (; 7 2  x.  ( P  x.  R )
) )  +  -u (; 2 7  x.  ( Q ^ 2 ) ) )  =  V )
139130, 138eqtr2d 2496 . 2  |-  ( ph  ->  V  =  ( ( ( 2  x.  (
( 2  x.  P
) ^ 3 ) )  -  ( 9  x.  ( ( 2  x.  P )  x.  ( ( P ^
2 )  -  (
4  x.  R ) ) ) ) )  +  (; 2 7  x.  -u ( Q ^ 2 ) ) ) )
14051, 139jca 532 1  |-  ( ph  ->  ( U  =  ( ( ( 2  x.  P ) ^ 2 )  -  ( 3  x.  ( ( P ^ 2 )  -  ( 4  x.  R
) ) ) )  /\  V  =  ( ( ( 2  x.  ( ( 2  x.  P ) ^ 3 ) )  -  (
9  x.  ( ( 2  x.  P )  x.  ( ( P ^ 2 )  -  ( 4  x.  R
) ) ) ) )  +  (; 2 7  x.  -u ( Q ^ 2 ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758  (class class class)co 6203   CCcc 9395   1c1 9398    + caddc 9400    x. cmul 9402    - cmin 9710   -ucneg 9711   2c2 10486   3c3 10487   4c4 10488   7c7 10491   8c8 10492   9c9 10493   NN0cn0 10694  ;cdc 10870   ^cexp 11986
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9453  ax-resscn 9454  ax-1cn 9455  ax-icn 9456  ax-addcl 9457  ax-addrcl 9458  ax-mulcl 9459  ax-mulrcl 9460  ax-mulcom 9461  ax-addass 9462  ax-mulass 9463  ax-distr 9464  ax-i2m1 9465  ax-1ne0 9466  ax-1rid 9467  ax-rnegex 9468  ax-rrecex 9469  ax-cnre 9470  ax-pre-lttri 9471  ax-pre-lttrn 9472  ax-pre-ltadd 9473  ax-pre-mulgt0 9474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-2nd 6691  df-recs 6945  df-rdg 6979  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-pnf 9535  df-mnf 9536  df-xr 9537  df-ltxr 9538  df-le 9539  df-sub 9712  df-neg 9713  df-nn 10438  df-2 10495  df-3 10496  df-4 10497  df-5 10498  df-6 10499  df-7 10500  df-8 10501  df-9 10502  df-10 10503  df-n0 10695  df-z 10762  df-dec 10871  df-uz 10977  df-seq 11928  df-exp 11987
This theorem is referenced by:  quart  22399
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