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Theorem quartlem1 23769
Description: Lemma for quart 23773. (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 10680 . . . . . . . . . 10  |-  2  e.  CC
2 quartlem1.p . . . . . . . . . 10  |-  ( ph  ->  P  e.  CC )
3 sqmul 12337 . . . . . . . . . 10  |-  ( ( 2  e.  CC  /\  P  e.  CC )  ->  ( ( 2  x.  P ) ^ 2 )  =  ( ( 2 ^ 2 )  x.  ( P ^
2 ) ) )
41, 2, 3sylancr 667 . . . . . . . . 9  |-  ( ph  ->  ( ( 2  x.  P ) ^ 2 )  =  ( ( 2 ^ 2 )  x.  ( P ^
2 ) ) )
5 sq2 12370 . . . . . . . . . 10  |-  ( 2 ^ 2 )  =  4
65oveq1i 6311 . . . . . . . . 9  |-  ( ( 2 ^ 2 )  x.  ( P ^
2 ) )  =  ( 4  x.  ( P ^ 2 ) )
74, 6syl6eq 2479 . . . . . . . 8  |-  ( ph  ->  ( ( 2  x.  P ) ^ 2 )  =  ( 4  x.  ( P ^
2 ) ) )
87oveq1d 6316 . . . . . . 7  |-  ( ph  ->  ( ( ( 2  x.  P ) ^
2 )  -  (
3  x.  ( P ^ 2 ) ) )  =  ( ( 4  x.  ( P ^ 2 ) )  -  ( 3  x.  ( P ^ 2 ) ) ) )
9 4cn 10687 . . . . . . . . 9  |-  4  e.  CC
109a1i 11 . . . . . . . 8  |-  ( ph  ->  4  e.  CC )
11 3cn 10684 . . . . . . . . 9  |-  3  e.  CC
1211a1i 11 . . . . . . . 8  |-  ( ph  ->  3  e.  CC )
132sqcld 12413 . . . . . . . 8  |-  ( ph  ->  ( P ^ 2 )  e.  CC )
1410, 12, 13subdird 10075 . . . . . . 7  |-  ( ph  ->  ( ( 4  -  3 )  x.  ( P ^ 2 ) )  =  ( ( 4  x.  ( P ^
2 ) )  -  ( 3  x.  ( P ^ 2 ) ) ) )
158, 14eqtr4d 2466 . . . . . 6  |-  ( ph  ->  ( ( ( 2  x.  P ) ^
2 )  -  (
3  x.  ( P ^ 2 ) ) )  =  ( ( 4  -  3 )  x.  ( P ^
2 ) ) )
16 ax-1cn 9597 . . . . . . . . . 10  |-  1  e.  CC
17 3p1e4 10735 . . . . . . . . . 10  |-  ( 3  +  1 )  =  4
189, 11, 16, 17subaddrii 9964 . . . . . . . . 9  |-  ( 4  -  3 )  =  1
1918oveq1i 6311 . . . . . . . 8  |-  ( ( 4  -  3 )  x.  ( P ^
2 ) )  =  ( 1  x.  ( P ^ 2 ) )
20 mulid2 9641 . . . . . . . 8  |-  ( ( P ^ 2 )  e.  CC  ->  (
1  x.  ( P ^ 2 ) )  =  ( P ^
2 ) )
2119, 20syl5eq 2475 . . . . . . 7  |-  ( ( P ^ 2 )  e.  CC  ->  (
( 4  -  3 )  x.  ( P ^ 2 ) )  =  ( P ^
2 ) )
2213, 21syl 17 . . . . . 6  |-  ( ph  ->  ( ( 4  -  3 )  x.  ( P ^ 2 ) )  =  ( P ^
2 ) )
2315, 22eqtr2d 2464 . . . . 5  |-  ( ph  ->  ( P ^ 2 )  =  ( ( ( 2  x.  P
) ^ 2 )  -  ( 3  x.  ( P ^ 2 ) ) ) )
2423oveq1d 6316 . . . 4  |-  ( ph  ->  ( ( P ^
2 )  +  (; 1
2  x.  R ) )  =  ( ( ( ( 2  x.  P ) ^ 2 )  -  ( 3  x.  ( P ^
2 ) ) )  +  (; 1 2  x.  R
) ) )
25 mulcl 9623 . . . . . . 7  |-  ( ( 2  e.  CC  /\  P  e.  CC )  ->  ( 2  x.  P
)  e.  CC )
261, 2, 25sylancr 667 . . . . . 6  |-  ( ph  ->  ( 2  x.  P
)  e.  CC )
2726sqcld 12413 . . . . 5  |-  ( ph  ->  ( ( 2  x.  P ) ^ 2 )  e.  CC )
28 mulcl 9623 . . . . . 6  |-  ( ( 3  e.  CC  /\  ( P ^ 2 )  e.  CC )  -> 
( 3  x.  ( P ^ 2 ) )  e.  CC )
2911, 13, 28sylancr 667 . . . . 5  |-  ( ph  ->  ( 3  x.  ( P ^ 2 ) )  e.  CC )
30 1nn0 10885 . . . . . . . 8  |-  1  e.  NN0
31 2nn 10767 . . . . . . . 8  |-  2  e.  NN
3230, 31decnncl 11064 . . . . . . 7  |- ; 1 2  e.  NN
3332nncni 10619 . . . . . 6  |- ; 1 2  e.  CC
34 quartlem1.r . . . . . 6  |-  ( ph  ->  R  e.  CC )
35 mulcl 9623 . . . . . 6  |-  ( (; 1
2  e.  CC  /\  R  e.  CC )  ->  (; 1 2  x.  R
)  e.  CC )
3633, 34, 35sylancr 667 . . . . 5  |-  ( ph  ->  (; 1 2  x.  R
)  e.  CC )
3727, 29, 36subsubd 10014 . . . 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 2466 . . 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 9623 . . . . . . 7  |-  ( ( 4  e.  CC  /\  R  e.  CC )  ->  ( 4  x.  R
)  e.  CC )
419, 34, 40sylancr 667 . . . . . 6  |-  ( ph  ->  ( 4  x.  R
)  e.  CC )
4212, 13, 41subdid 10074 . . . . 5  |-  ( ph  ->  ( 3  x.  (
( P ^ 2 )  -  ( 4  x.  R ) ) )  =  ( ( 3  x.  ( P ^ 2 ) )  -  ( 3  x.  ( 4  x.  R
) ) ) )
43 4t3e12 11123 . . . . . . . . 9  |-  ( 4  x.  3 )  = ; 1
2
449, 11, 43mulcomli 9650 . . . . . . . 8  |-  ( 3  x.  4 )  = ; 1
2
4544oveq1i 6311 . . . . . . 7  |-  ( ( 3  x.  4 )  x.  R )  =  (; 1 2  x.  R
)
4612, 10, 34mulassd 9666 . . . . . . 7  |-  ( ph  ->  ( ( 3  x.  4 )  x.  R
)  =  ( 3  x.  ( 4  x.  R ) ) )
4745, 46syl5eqr 2477 . . . . . 6  |-  ( ph  ->  (; 1 2  x.  R
)  =  ( 3  x.  ( 4  x.  R ) ) )
4847oveq2d 6317 . . . . 5  |-  ( ph  ->  ( ( 3  x.  ( P ^ 2 ) )  -  (; 1 2  x.  R ) )  =  ( ( 3  x.  ( P ^
2 ) )  -  ( 3  x.  (
4  x.  R ) ) ) )
4942, 48eqtr4d 2466 . . . 4  |-  ( ph  ->  ( 3  x.  (
( P ^ 2 )  -  ( 4  x.  R ) ) )  =  ( ( 3  x.  ( P ^ 2 ) )  -  (; 1 2  x.  R
) ) )
5049oveq2d 6317 . . 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 2473 . 2  |-  ( ph  ->  U  =  ( ( ( 2  x.  P
) ^ 2 )  -  ( 3  x.  ( ( P ^
2 )  -  (
4  x.  R ) ) ) ) )
521a1i 11 . . . . . . . . . 10  |-  ( ph  ->  2  e.  CC )
53 3nn0 10887 . . . . . . . . . . 11  |-  3  e.  NN0
5453a1i 11 . . . . . . . . . 10  |-  ( ph  ->  3  e.  NN0 )
5552, 2, 54mulexpd 12430 . . . . . . . . 9  |-  ( ph  ->  ( ( 2  x.  P ) ^ 3 )  =  ( ( 2 ^ 3 )  x.  ( P ^
3 ) ) )
56 cu2 12372 . . . . . . . . . 10  |-  ( 2 ^ 3 )  =  8
5756oveq1i 6311 . . . . . . . . 9  |-  ( ( 2 ^ 3 )  x.  ( P ^
3 ) )  =  ( 8  x.  ( P ^ 3 ) )
5855, 57syl6eq 2479 . . . . . . . 8  |-  ( ph  ->  ( ( 2  x.  P ) ^ 3 )  =  ( 8  x.  ( P ^
3 ) ) )
5958oveq2d 6317 . . . . . . 7  |-  ( ph  ->  ( 2  x.  (
( 2  x.  P
) ^ 3 ) )  =  ( 2  x.  ( 8  x.  ( P ^ 3 ) ) ) )
60 8cn 10695 . . . . . . . . 9  |-  8  e.  CC
6160a1i 11 . . . . . . . 8  |-  ( ph  ->  8  e.  CC )
62 expcl 12289 . . . . . . . . 9  |-  ( ( P  e.  CC  /\  3  e.  NN0 )  -> 
( P ^ 3 )  e.  CC )
632, 53, 62sylancl 666 . . . . . . . 8  |-  ( ph  ->  ( P ^ 3 )  e.  CC )
6452, 61, 63mul12d 9842 . . . . . . 7  |-  ( ph  ->  ( 2  x.  (
8  x.  ( P ^ 3 ) ) )  =  ( 8  x.  ( 2  x.  ( P ^ 3 ) ) ) )
6559, 64eqtrd 2463 . . . . . 6  |-  ( ph  ->  ( 2  x.  (
( 2  x.  P
) ^ 3 ) )  =  ( 8  x.  ( 2  x.  ( P ^ 3 ) ) ) )
66 9cn 10697 . . . . . . . . 9  |-  9  e.  CC
6766a1i 11 . . . . . . . 8  |-  ( ph  ->  9  e.  CC )
68 mulcl 9623 . . . . . . . . 9  |-  ( ( 2  e.  CC  /\  ( P ^ 3 )  e.  CC )  -> 
( 2  x.  ( P ^ 3 ) )  e.  CC )
691, 63, 68sylancr 667 . . . . . . . 8  |-  ( ph  ->  ( 2  x.  ( P ^ 3 ) )  e.  CC )
702, 34mulcld 9663 . . . . . . . . 9  |-  ( ph  ->  ( P  x.  R
)  e.  CC )
71 mulcl 9623 . . . . . . . . 9  |-  ( ( 8  e.  CC  /\  ( P  x.  R
)  e.  CC )  ->  ( 8  x.  ( P  x.  R
) )  e.  CC )
7260, 70, 71sylancr 667 . . . . . . . 8  |-  ( ph  ->  ( 8  x.  ( P  x.  R )
)  e.  CC )
7367, 69, 72subdid 10074 . . . . . . 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 10074 . . . . . . . . 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 9666 . . . . . . . . . . 11  |-  ( ph  ->  ( ( 2  x.  P )  x.  ( P ^ 2 ) )  =  ( 2  x.  ( P  x.  ( P ^ 2 ) ) ) )
762, 13mulcomd 9664 . . . . . . . . . . . . 13  |-  ( ph  ->  ( P  x.  ( P ^ 2 ) )  =  ( ( P ^ 2 )  x.  P ) )
77 df-3 10669 . . . . . . . . . . . . . . 15  |-  3  =  ( 2  +  1 )
7877oveq2i 6312 . . . . . . . . . . . . . 14  |-  ( P ^ 3 )  =  ( P ^ (
2  +  1 ) )
79 2nn0 10886 . . . . . . . . . . . . . . 15  |-  2  e.  NN0
80 expp1 12278 . . . . . . . . . . . . . . 15  |-  ( ( P  e.  CC  /\  2  e.  NN0 )  -> 
( P ^ (
2  +  1 ) )  =  ( ( P ^ 2 )  x.  P ) )
812, 79, 80sylancl 666 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( P ^ (
2  +  1 ) )  =  ( ( P ^ 2 )  x.  P ) )
8278, 81syl5eq 2475 . . . . . . . . . . . . 13  |-  ( ph  ->  ( P ^ 3 )  =  ( ( P ^ 2 )  x.  P ) )
8376, 82eqtr4d 2466 . . . . . . . . . . . 12  |-  ( ph  ->  ( P  x.  ( P ^ 2 ) )  =  ( P ^
3 ) )
8483oveq2d 6317 . . . . . . . . . . 11  |-  ( ph  ->  ( 2  x.  ( P  x.  ( P ^ 2 ) ) )  =  ( 2  x.  ( P ^
3 ) ) )
8575, 84eqtrd 2463 . . . . . . . . . 10  |-  ( ph  ->  ( ( 2  x.  P )  x.  ( P ^ 2 ) )  =  ( 2  x.  ( P ^ 3 ) ) )
8652, 2, 10, 34mul4d 9845 . . . . . . . . . . 11  |-  ( ph  ->  ( ( 2  x.  P )  x.  (
4  x.  R ) )  =  ( ( 2  x.  4 )  x.  ( P  x.  R ) ) )
87 4t2e8 10763 . . . . . . . . . . . . 13  |-  ( 4  x.  2 )  =  8
889, 1, 87mulcomli 9650 . . . . . . . . . . . 12  |-  ( 2  x.  4 )  =  8
8988oveq1i 6311 . . . . . . . . . . 11  |-  ( ( 2  x.  4 )  x.  ( P  x.  R ) )  =  ( 8  x.  ( P  x.  R )
)
9086, 89syl6eq 2479 . . . . . . . . . 10  |-  ( ph  ->  ( ( 2  x.  P )  x.  (
4  x.  R ) )  =  ( 8  x.  ( P  x.  R ) ) )
9185, 90oveq12d 6319 . . . . . . . . 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 2463 . . . . . . . 8  |-  ( ph  ->  ( ( 2  x.  P )  x.  (
( P ^ 2 )  -  ( 4  x.  R ) ) )  =  ( ( 2  x.  ( P ^ 3 ) )  -  ( 8  x.  ( P  x.  R
) ) ) )
9392oveq2d 6317 . . . . . . 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 11152 . . . . . . . . . 10  |-  ( 9  x.  8 )  = ; 7
2
9594oveq1i 6311 . . . . . . . . 9  |-  ( ( 9  x.  8 )  x.  ( P  x.  R ) )  =  (; 7 2  x.  ( P  x.  R )
)
9667, 61, 70mulassd 9666 . . . . . . . . 9  |-  ( ph  ->  ( ( 9  x.  8 )  x.  ( P  x.  R )
)  =  ( 9  x.  ( 8  x.  ( P  x.  R
) ) ) )
9795, 96syl5eqr 2477 . . . . . . . 8  |-  ( ph  ->  (; 7 2  x.  ( P  x.  R )
)  =  ( 9  x.  ( 8  x.  ( P  x.  R
) ) ) )
9897oveq2d 6317 . . . . . . 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 2473 . . . . . 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 6319 . . . . 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 9623 . . . . . . 7  |-  ( ( 8  e.  CC  /\  ( 2  x.  ( P ^ 3 ) )  e.  CC )  -> 
( 8  x.  (
2  x.  ( P ^ 3 ) ) )  e.  CC )
10260, 69, 101sylancr 667 . . . . . 6  |-  ( ph  ->  ( 8  x.  (
2  x.  ( P ^ 3 ) ) )  e.  CC )
103 mulcl 9623 . . . . . . 7  |-  ( ( 9  e.  CC  /\  ( 2  x.  ( P ^ 3 ) )  e.  CC )  -> 
( 9  x.  (
2  x.  ( P ^ 3 ) ) )  e.  CC )
10466, 69, 103sylancr 667 . . . . . 6  |-  ( ph  ->  ( 9  x.  (
2  x.  ( P ^ 3 ) ) )  e.  CC )
105 7nn0 10891 . . . . . . . . 9  |-  7  e.  NN0
106105, 31decnncl 11064 . . . . . . . 8  |- ; 7 2  e.  NN
107106nncni 10619 . . . . . . 7  |- ; 7 2  e.  CC
108 mulcl 9623 . . . . . . 7  |-  ( (; 7
2  e.  CC  /\  ( P  x.  R
)  e.  CC )  ->  (; 7 2  x.  ( P  x.  R )
)  e.  CC )
109107, 70, 108sylancr 667 . . . . . 6  |-  ( ph  ->  (; 7 2  x.  ( P  x.  R )
)  e.  CC )
110102, 104, 109subsubd 10014 . . . . 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 10002 . . . . . . 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 10075 . . . . . . . . 9  |-  ( ph  ->  ( ( 9  -  8 )  x.  (
2  x.  ( P ^ 3 ) ) )  =  ( ( 9  x.  ( 2  x.  ( P ^
3 ) ) )  -  ( 8  x.  ( 2  x.  ( P ^ 3 ) ) ) ) )
113 8p1e9 10740 . . . . . . . . . . . 12  |-  ( 8  +  1 )  =  9
11466, 60, 16, 113subaddrii 9964 . . . . . . . . . . 11  |-  ( 9  -  8 )  =  1
115114oveq1i 6311 . . . . . . . . . 10  |-  ( ( 9  -  8 )  x.  ( 2  x.  ( P ^ 3 ) ) )  =  ( 1  x.  (
2  x.  ( P ^ 3 ) ) )
11669mulid2d 9661 . . . . . . . . . 10  |-  ( ph  ->  ( 1  x.  (
2  x.  ( P ^ 3 ) ) )  =  ( 2  x.  ( P ^
3 ) ) )
117115, 116syl5eq 2475 . . . . . . . . 9  |-  ( ph  ->  ( ( 9  -  8 )  x.  (
2  x.  ( P ^ 3 ) ) )  =  ( 2  x.  ( P ^
3 ) ) )
118112, 117eqtr3d 2465 . . . . . . . 8  |-  ( ph  ->  ( ( 9  x.  ( 2  x.  ( P ^ 3 ) ) )  -  ( 8  x.  ( 2  x.  ( P ^ 3 ) ) ) )  =  ( 2  x.  ( P ^ 3 ) ) )
119118negeqd 9869 . . . . . . 7  |-  ( ph  -> 
-u ( ( 9  x.  ( 2  x.  ( P ^ 3 ) ) )  -  ( 8  x.  (
2  x.  ( P ^ 3 ) ) ) )  =  -u ( 2  x.  ( P ^ 3 ) ) )
120111, 119eqtr3d 2465 . . . . . 6  |-  ( ph  ->  ( ( 8  x.  ( 2  x.  ( P ^ 3 ) ) )  -  ( 9  x.  ( 2  x.  ( P ^ 3 ) ) ) )  =  -u ( 2  x.  ( P ^ 3 ) ) )
121120oveq1d 6316 . . . . 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 2467 . . . 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 10772 . . . . . . 7  |-  7  e.  NN
12479, 123decnncl 11064 . . . . . 6  |- ; 2 7  e.  NN
125124nncni 10619 . . . . 5  |- ; 2 7  e.  CC
126 quartlem1.q . . . . . 6  |-  ( ph  ->  Q  e.  CC )
127126sqcld 12413 . . . . 5  |-  ( ph  ->  ( Q ^ 2 )  e.  CC )
128 mulneg2 10056 . . . . 5  |-  ( (; 2
7  e.  CC  /\  ( Q ^ 2 )  e.  CC )  -> 
(; 2 7  x.  -u ( Q ^ 2 ) )  =  -u (; 2 7  x.  ( Q ^ 2 ) ) )
129125, 127, 128sylancr 667 . . . 4  |-  ( ph  ->  (; 2 7  x.  -u ( Q ^ 2 ) )  =  -u (; 2 7  x.  ( Q ^ 2 ) ) )
130122, 129oveq12d 6319 . . 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 9973 . . . . 5  |-  ( ph  -> 
-u ( 2  x.  ( P ^ 3 ) )  e.  CC )
132 mulcl 9623 . . . . . 6  |-  ( (; 2
7  e.  CC  /\  ( Q ^ 2 )  e.  CC )  -> 
(; 2 7  x.  ( Q ^ 2 ) )  e.  CC )
133125, 127, 132sylancr 667 . . . . 5  |-  ( ph  ->  (; 2 7  x.  ( Q ^ 2 ) )  e.  CC )
134131, 109, 133addsubd 10007 . . . 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 9662 . . . . 5  |-  ( ph  ->  ( -u ( 2  x.  ( P ^
3 ) )  +  (; 7 2  x.  ( P  x.  R )
) )  e.  CC )
136135, 133negsubd 9992 . . . 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 2473 . . 3  |-  ( ph  ->  ( ( -u (
2  x.  ( P ^ 3 ) )  +  (; 7 2  x.  ( P  x.  R )
) )  +  -u (; 2 7  x.  ( Q ^ 2 ) ) )  =  V )
139130, 138eqtr2d 2464 . 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 534 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 370    = wceq 1437    e. wcel 1868  (class class class)co 6301   CCcc 9537   1c1 9540    + caddc 9542    x. cmul 9544    - cmin 9860   -ucneg 9861   2c2 10659   3c3 10660   4c4 10661   7c7 10664   8c8 10665   9c9 10666   NN0cn0 10869  ;cdc 11051   ^cexp 12271
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656  ax-un 6593  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4760  df-id 4764  df-po 4770  df-so 4771  df-fr 4808  df-we 4810  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-res 4861  df-ima 4862  df-pred 5395  df-ord 5441  df-on 5442  df-lim 5443  df-suc 5444  df-iota 5561  df-fun 5599  df-fn 5600  df-f 5601  df-f1 5602  df-fo 5603  df-f1o 5604  df-fv 5605  df-riota 6263  df-ov 6304  df-oprab 6305  df-mpt2 6306  df-om 6703  df-2nd 6804  df-wrecs 7032  df-recs 7094  df-rdg 7132  df-er 7367  df-en 7574  df-dom 7575  df-sdom 7576  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-seq 12213  df-exp 12272
This theorem is referenced by:  quart  23773
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