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Theorem quartlem1 20153
Description: Lemma for quart 20157. (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 9816 . . . . . . . . . 10  |-  2  e.  CC
2 quartlem1.p . . . . . . . . . 10  |-  ( ph  ->  P  e.  CC )
3 sqmul 11167 . . . . . . . . . 10  |-  ( ( 2  e.  CC  /\  P  e.  CC )  ->  ( ( 2  x.  P ) ^ 2 )  =  ( ( 2 ^ 2 )  x.  ( P ^
2 ) ) )
41, 2, 3sylancr 644 . . . . . . . . 9  |-  ( ph  ->  ( ( 2  x.  P ) ^ 2 )  =  ( ( 2 ^ 2 )  x.  ( P ^
2 ) ) )
5 sq2 11199 . . . . . . . . . 10  |-  ( 2 ^ 2 )  =  4
65oveq1i 5868 . . . . . . . . 9  |-  ( ( 2 ^ 2 )  x.  ( P ^
2 ) )  =  ( 4  x.  ( P ^ 2 ) )
74, 6syl6eq 2331 . . . . . . . 8  |-  ( ph  ->  ( ( 2  x.  P ) ^ 2 )  =  ( 4  x.  ( P ^
2 ) ) )
87oveq1d 5873 . . . . . . 7  |-  ( ph  ->  ( ( ( 2  x.  P ) ^
2 )  -  (
3  x.  ( P ^ 2 ) ) )  =  ( ( 4  x.  ( P ^ 2 ) )  -  ( 3  x.  ( P ^ 2 ) ) ) )
9 4cn 9820 . . . . . . . . 9  |-  4  e.  CC
109a1i 10 . . . . . . . 8  |-  ( ph  ->  4  e.  CC )
11 3cn 9818 . . . . . . . . 9  |-  3  e.  CC
1211a1i 10 . . . . . . . 8  |-  ( ph  ->  3  e.  CC )
132sqcld 11243 . . . . . . . 8  |-  ( ph  ->  ( P ^ 2 )  e.  CC )
1410, 12, 13subdird 9236 . . . . . . 7  |-  ( ph  ->  ( ( 4  -  3 )  x.  ( P ^ 2 ) )  =  ( ( 4  x.  ( P ^
2 ) )  -  ( 3  x.  ( P ^ 2 ) ) ) )
158, 14eqtr4d 2318 . . . . . 6  |-  ( ph  ->  ( ( ( 2  x.  P ) ^
2 )  -  (
3  x.  ( P ^ 2 ) ) )  =  ( ( 4  -  3 )  x.  ( P ^
2 ) ) )
16 ax-1cn 8795 . . . . . . . . . 10  |-  1  e.  CC
17 3p1e4 9848 . . . . . . . . . 10  |-  ( 3  +  1 )  =  4
189, 11, 16, 17subaddrii 9135 . . . . . . . . 9  |-  ( 4  -  3 )  =  1
1918oveq1i 5868 . . . . . . . 8  |-  ( ( 4  -  3 )  x.  ( P ^
2 ) )  =  ( 1  x.  ( P ^ 2 ) )
20 mulid2 8836 . . . . . . . 8  |-  ( ( P ^ 2 )  e.  CC  ->  (
1  x.  ( P ^ 2 ) )  =  ( P ^
2 ) )
2119, 20syl5eq 2327 . . . . . . 7  |-  ( ( P ^ 2 )  e.  CC  ->  (
( 4  -  3 )  x.  ( P ^ 2 ) )  =  ( P ^
2 ) )
2213, 21syl 15 . . . . . 6  |-  ( ph  ->  ( ( 4  -  3 )  x.  ( P ^ 2 ) )  =  ( P ^
2 ) )
2315, 22eqtr2d 2316 . . . . 5  |-  ( ph  ->  ( P ^ 2 )  =  ( ( ( 2  x.  P
) ^ 2 )  -  ( 3  x.  ( P ^ 2 ) ) ) )
2423oveq1d 5873 . . . 4  |-  ( ph  ->  ( ( P ^
2 )  +  (; 1
2  x.  R ) )  =  ( ( ( ( 2  x.  P ) ^ 2 )  -  ( 3  x.  ( P ^
2 ) ) )  +  (; 1 2  x.  R
) ) )
25 mulcl 8821 . . . . . . 7  |-  ( ( 2  e.  CC  /\  P  e.  CC )  ->  ( 2  x.  P
)  e.  CC )
261, 2, 25sylancr 644 . . . . . 6  |-  ( ph  ->  ( 2  x.  P
)  e.  CC )
2726sqcld 11243 . . . . 5  |-  ( ph  ->  ( ( 2  x.  P ) ^ 2 )  e.  CC )
28 mulcl 8821 . . . . . 6  |-  ( ( 3  e.  CC  /\  ( P ^ 2 )  e.  CC )  -> 
( 3  x.  ( P ^ 2 ) )  e.  CC )
2911, 13, 28sylancr 644 . . . . 5  |-  ( ph  ->  ( 3  x.  ( P ^ 2 ) )  e.  CC )
30 1nn0 9981 . . . . . . . 8  |-  1  e.  NN0
31 2nn 9877 . . . . . . . 8  |-  2  e.  NN
3230, 31decnncl 10137 . . . . . . 7  |- ; 1 2  e.  NN
3332nncni 9756 . . . . . 6  |- ; 1 2  e.  CC
34 quartlem1.r . . . . . 6  |-  ( ph  ->  R  e.  CC )
35 mulcl 8821 . . . . . 6  |-  ( (; 1
2  e.  CC  /\  R  e.  CC )  ->  (; 1 2  x.  R
)  e.  CC )
3633, 34, 35sylancr 644 . . . . 5  |-  ( ph  ->  (; 1 2  x.  R
)  e.  CC )
3727, 29, 36subsubd 9185 . . . 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 2318 . . 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 8821 . . . . . . 7  |-  ( ( 4  e.  CC  /\  R  e.  CC )  ->  ( 4  x.  R
)  e.  CC )
419, 34, 40sylancr 644 . . . . . 6  |-  ( ph  ->  ( 4  x.  R
)  e.  CC )
4212, 13, 41subdid 9235 . . . . 5  |-  ( ph  ->  ( 3  x.  (
( P ^ 2 )  -  ( 4  x.  R ) ) )  =  ( ( 3  x.  ( P ^ 2 ) )  -  ( 3  x.  ( 4  x.  R
) ) ) )
43 4t3e12 10196 . . . . . . . . 9  |-  ( 4  x.  3 )  = ; 1
2
449, 11, 43mulcomli 8844 . . . . . . . 8  |-  ( 3  x.  4 )  = ; 1
2
4544oveq1i 5868 . . . . . . 7  |-  ( ( 3  x.  4 )  x.  R )  =  (; 1 2  x.  R
)
4612, 10, 34mulassd 8858 . . . . . . 7  |-  ( ph  ->  ( ( 3  x.  4 )  x.  R
)  =  ( 3  x.  ( 4  x.  R ) ) )
4745, 46syl5eqr 2329 . . . . . 6  |-  ( ph  ->  (; 1 2  x.  R
)  =  ( 3  x.  ( 4  x.  R ) ) )
4847oveq2d 5874 . . . . 5  |-  ( ph  ->  ( ( 3  x.  ( P ^ 2 ) )  -  (; 1 2  x.  R ) )  =  ( ( 3  x.  ( P ^
2 ) )  -  ( 3  x.  (
4  x.  R ) ) ) )
4942, 48eqtr4d 2318 . . . 4  |-  ( ph  ->  ( 3  x.  (
( P ^ 2 )  -  ( 4  x.  R ) ) )  =  ( ( 3  x.  ( P ^ 2 ) )  -  (; 1 2  x.  R
) ) )
5049oveq2d 5874 . . 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 2325 . 2  |-  ( ph  ->  U  =  ( ( ( 2  x.  P
) ^ 2 )  -  ( 3  x.  ( ( P ^
2 )  -  (
4  x.  R ) ) ) ) )
521a1i 10 . . . . . . . . . 10  |-  ( ph  ->  2  e.  CC )
53 3nn0 9983 . . . . . . . . . . 11  |-  3  e.  NN0
5453a1i 10 . . . . . . . . . 10  |-  ( ph  ->  3  e.  NN0 )
5552, 2, 54mulexpd 11260 . . . . . . . . 9  |-  ( ph  ->  ( ( 2  x.  P ) ^ 3 )  =  ( ( 2 ^ 3 )  x.  ( P ^
3 ) ) )
56 cu2 11201 . . . . . . . . . 10  |-  ( 2 ^ 3 )  =  8
5756oveq1i 5868 . . . . . . . . 9  |-  ( ( 2 ^ 3 )  x.  ( P ^
3 ) )  =  ( 8  x.  ( P ^ 3 ) )
5855, 57syl6eq 2331 . . . . . . . 8  |-  ( ph  ->  ( ( 2  x.  P ) ^ 3 )  =  ( 8  x.  ( P ^
3 ) ) )
5958oveq2d 5874 . . . . . . 7  |-  ( ph  ->  ( 2  x.  (
( 2  x.  P
) ^ 3 ) )  =  ( 2  x.  ( 8  x.  ( P ^ 3 ) ) ) )
60 8nn 9883 . . . . . . . . . 10  |-  8  e.  NN
6160nncni 9756 . . . . . . . . 9  |-  8  e.  CC
6261a1i 10 . . . . . . . 8  |-  ( ph  ->  8  e.  CC )
63 expcl 11121 . . . . . . . . 9  |-  ( ( P  e.  CC  /\  3  e.  NN0 )  -> 
( P ^ 3 )  e.  CC )
642, 53, 63sylancl 643 . . . . . . . 8  |-  ( ph  ->  ( P ^ 3 )  e.  CC )
6552, 62, 64mul12d 9021 . . . . . . 7  |-  ( ph  ->  ( 2  x.  (
8  x.  ( P ^ 3 ) ) )  =  ( 8  x.  ( 2  x.  ( P ^ 3 ) ) ) )
6659, 65eqtrd 2315 . . . . . 6  |-  ( ph  ->  ( 2  x.  (
( 2  x.  P
) ^ 3 ) )  =  ( 8  x.  ( 2  x.  ( P ^ 3 ) ) ) )
67 9nn 9884 . . . . . . . . . 10  |-  9  e.  NN
6867nncni 9756 . . . . . . . . 9  |-  9  e.  CC
6968a1i 10 . . . . . . . 8  |-  ( ph  ->  9  e.  CC )
70 mulcl 8821 . . . . . . . . 9  |-  ( ( 2  e.  CC  /\  ( P ^ 3 )  e.  CC )  -> 
( 2  x.  ( P ^ 3 ) )  e.  CC )
711, 64, 70sylancr 644 . . . . . . . 8  |-  ( ph  ->  ( 2  x.  ( P ^ 3 ) )  e.  CC )
722, 34mulcld 8855 . . . . . . . . 9  |-  ( ph  ->  ( P  x.  R
)  e.  CC )
73 mulcl 8821 . . . . . . . . 9  |-  ( ( 8  e.  CC  /\  ( P  x.  R
)  e.  CC )  ->  ( 8  x.  ( P  x.  R
) )  e.  CC )
7461, 72, 73sylancr 644 . . . . . . . 8  |-  ( ph  ->  ( 8  x.  ( P  x.  R )
)  e.  CC )
7569, 71, 74subdid 9235 . . . . . . 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 ) ) ) ) )
7626, 13, 41subdid 9235 . . . . . . . . 9  |-  ( ph  ->  ( ( 2  x.  P )  x.  (
( P ^ 2 )  -  ( 4  x.  R ) ) )  =  ( ( ( 2  x.  P
)  x.  ( P ^ 2 ) )  -  ( ( 2  x.  P )  x.  ( 4  x.  R
) ) ) )
7752, 2, 13mulassd 8858 . . . . . . . . . . 11  |-  ( ph  ->  ( ( 2  x.  P )  x.  ( P ^ 2 ) )  =  ( 2  x.  ( P  x.  ( P ^ 2 ) ) ) )
782, 13mulcomd 8856 . . . . . . . . . . . . 13  |-  ( ph  ->  ( P  x.  ( P ^ 2 ) )  =  ( ( P ^ 2 )  x.  P ) )
79 df-3 9805 . . . . . . . . . . . . . . 15  |-  3  =  ( 2  +  1 )
8079oveq2i 5869 . . . . . . . . . . . . . 14  |-  ( P ^ 3 )  =  ( P ^ (
2  +  1 ) )
81 2nn0 9982 . . . . . . . . . . . . . . 15  |-  2  e.  NN0
82 expp1 11110 . . . . . . . . . . . . . . 15  |-  ( ( P  e.  CC  /\  2  e.  NN0 )  -> 
( P ^ (
2  +  1 ) )  =  ( ( P ^ 2 )  x.  P ) )
832, 81, 82sylancl 643 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( P ^ (
2  +  1 ) )  =  ( ( P ^ 2 )  x.  P ) )
8480, 83syl5eq 2327 . . . . . . . . . . . . 13  |-  ( ph  ->  ( P ^ 3 )  =  ( ( P ^ 2 )  x.  P ) )
8578, 84eqtr4d 2318 . . . . . . . . . . . 12  |-  ( ph  ->  ( P  x.  ( P ^ 2 ) )  =  ( P ^
3 ) )
8685oveq2d 5874 . . . . . . . . . . 11  |-  ( ph  ->  ( 2  x.  ( P  x.  ( P ^ 2 ) ) )  =  ( 2  x.  ( P ^
3 ) ) )
8777, 86eqtrd 2315 . . . . . . . . . 10  |-  ( ph  ->  ( ( 2  x.  P )  x.  ( P ^ 2 ) )  =  ( 2  x.  ( P ^ 3 ) ) )
8852, 2, 10, 34mul4d 9024 . . . . . . . . . . 11  |-  ( ph  ->  ( ( 2  x.  P )  x.  (
4  x.  R ) )  =  ( ( 2  x.  4 )  x.  ( P  x.  R ) ) )
89 4t2e8 9874 . . . . . . . . . . . . 13  |-  ( 4  x.  2 )  =  8
909, 1, 89mulcomli 8844 . . . . . . . . . . . 12  |-  ( 2  x.  4 )  =  8
9190oveq1i 5868 . . . . . . . . . . 11  |-  ( ( 2  x.  4 )  x.  ( P  x.  R ) )  =  ( 8  x.  ( P  x.  R )
)
9288, 91syl6eq 2331 . . . . . . . . . 10  |-  ( ph  ->  ( ( 2  x.  P )  x.  (
4  x.  R ) )  =  ( 8  x.  ( P  x.  R ) ) )
9387, 92oveq12d 5876 . . . . . . . . 9  |-  ( ph  ->  ( ( ( 2  x.  P )  x.  ( P ^ 2 ) )  -  (
( 2  x.  P
)  x.  ( 4  x.  R ) ) )  =  ( ( 2  x.  ( P ^ 3 ) )  -  ( 8  x.  ( P  x.  R
) ) ) )
9476, 93eqtrd 2315 . . . . . . . 8  |-  ( ph  ->  ( ( 2  x.  P )  x.  (
( P ^ 2 )  -  ( 4  x.  R ) ) )  =  ( ( 2  x.  ( P ^ 3 ) )  -  ( 8  x.  ( P  x.  R
) ) ) )
9594oveq2d 5874 . . . . . . 7  |-  ( ph  ->  ( 9  x.  (
( 2  x.  P
)  x.  ( ( P ^ 2 )  -  ( 4  x.  R ) ) ) )  =  ( 9  x.  ( ( 2  x.  ( P ^
3 ) )  -  ( 8  x.  ( P  x.  R )
) ) ) )
96 9t8e72 10225 . . . . . . . . . 10  |-  ( 9  x.  8 )  = ; 7
2
9796oveq1i 5868 . . . . . . . . 9  |-  ( ( 9  x.  8 )  x.  ( P  x.  R ) )  =  (; 7 2  x.  ( P  x.  R )
)
9869, 62, 72mulassd 8858 . . . . . . . . 9  |-  ( ph  ->  ( ( 9  x.  8 )  x.  ( P  x.  R )
)  =  ( 9  x.  ( 8  x.  ( P  x.  R
) ) ) )
9997, 98syl5eqr 2329 . . . . . . . 8  |-  ( ph  ->  (; 7 2  x.  ( P  x.  R )
)  =  ( 9  x.  ( 8  x.  ( P  x.  R
) ) ) )
10099oveq2d 5874 . . . . . . 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
) ) ) ) )
10175, 95, 1003eqtr4d 2325 . . . . . 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 )
) ) )
10266, 101oveq12d 5876 . . . . 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 ) ) ) ) )
103 mulcl 8821 . . . . . . 7  |-  ( ( 8  e.  CC  /\  ( 2  x.  ( P ^ 3 ) )  e.  CC )  -> 
( 8  x.  (
2  x.  ( P ^ 3 ) ) )  e.  CC )
10461, 71, 103sylancr 644 . . . . . 6  |-  ( ph  ->  ( 8  x.  (
2  x.  ( P ^ 3 ) ) )  e.  CC )
105 mulcl 8821 . . . . . . 7  |-  ( ( 9  e.  CC  /\  ( 2  x.  ( P ^ 3 ) )  e.  CC )  -> 
( 9  x.  (
2  x.  ( P ^ 3 ) ) )  e.  CC )
10668, 71, 105sylancr 644 . . . . . 6  |-  ( ph  ->  ( 9  x.  (
2  x.  ( P ^ 3 ) ) )  e.  CC )
107 7nn0 9987 . . . . . . . . 9  |-  7  e.  NN0
108107, 31decnncl 10137 . . . . . . . 8  |- ; 7 2  e.  NN
109108nncni 9756 . . . . . . 7  |- ; 7 2  e.  CC
110 mulcl 8821 . . . . . . 7  |-  ( (; 7
2  e.  CC  /\  ( P  x.  R
)  e.  CC )  ->  (; 7 2  x.  ( P  x.  R )
)  e.  CC )
111109, 72, 110sylancr 644 . . . . . 6  |-  ( ph  ->  (; 7 2  x.  ( P  x.  R )
)  e.  CC )
112104, 106, 111subsubd 9185 . . . . 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 ) ) ) )
113106, 104negsubdi2d 9173 . . . . . . 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 ) ) ) ) )
11469, 62, 71subdird 9236 . . . . . . . . 9  |-  ( ph  ->  ( ( 9  -  8 )  x.  (
2  x.  ( P ^ 3 ) ) )  =  ( ( 9  x.  ( 2  x.  ( P ^
3 ) ) )  -  ( 8  x.  ( 2  x.  ( P ^ 3 ) ) ) ) )
115 8p1e9 9853 . . . . . . . . . . . 12  |-  ( 8  +  1 )  =  9
11668, 61, 16, 115subaddrii 9135 . . . . . . . . . . 11  |-  ( 9  -  8 )  =  1
117116oveq1i 5868 . . . . . . . . . 10  |-  ( ( 9  -  8 )  x.  ( 2  x.  ( P ^ 3 ) ) )  =  ( 1  x.  (
2  x.  ( P ^ 3 ) ) )
11871mulid2d 8853 . . . . . . . . . 10  |-  ( ph  ->  ( 1  x.  (
2  x.  ( P ^ 3 ) ) )  =  ( 2  x.  ( P ^
3 ) ) )
119117, 118syl5eq 2327 . . . . . . . . 9  |-  ( ph  ->  ( ( 9  -  8 )  x.  (
2  x.  ( P ^ 3 ) ) )  =  ( 2  x.  ( P ^
3 ) ) )
120114, 119eqtr3d 2317 . . . . . . . 8  |-  ( ph  ->  ( ( 9  x.  ( 2  x.  ( P ^ 3 ) ) )  -  ( 8  x.  ( 2  x.  ( P ^ 3 ) ) ) )  =  ( 2  x.  ( P ^ 3 ) ) )
121120negeqd 9046 . . . . . . 7  |-  ( ph  -> 
-u ( ( 9  x.  ( 2  x.  ( P ^ 3 ) ) )  -  ( 8  x.  (
2  x.  ( P ^ 3 ) ) ) )  =  -u ( 2  x.  ( P ^ 3 ) ) )
122113, 121eqtr3d 2317 . . . . . 6  |-  ( ph  ->  ( ( 8  x.  ( 2  x.  ( P ^ 3 ) ) )  -  ( 9  x.  ( 2  x.  ( P ^ 3 ) ) ) )  =  -u ( 2  x.  ( P ^ 3 ) ) )
123122oveq1d 5873 . . . . 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 )
) ) )
124102, 112, 1233eqtrd 2319 . . . 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 )
) ) )
125 7nn 9882 . . . . . . 7  |-  7  e.  NN
12681, 125decnncl 10137 . . . . . 6  |- ; 2 7  e.  NN
127126nncni 9756 . . . . 5  |- ; 2 7  e.  CC
128 quartlem1.q . . . . . 6  |-  ( ph  ->  Q  e.  CC )
129128sqcld 11243 . . . . 5  |-  ( ph  ->  ( Q ^ 2 )  e.  CC )
130 mulneg2 9217 . . . . 5  |-  ( (; 2
7  e.  CC  /\  ( Q ^ 2 )  e.  CC )  -> 
(; 2 7  x.  -u ( Q ^ 2 ) )  =  -u (; 2 7  x.  ( Q ^ 2 ) ) )
131127, 129, 130sylancr 644 . . . 4  |-  ( ph  ->  (; 2 7  x.  -u ( Q ^ 2 ) )  =  -u (; 2 7  x.  ( Q ^ 2 ) ) )
132124, 131oveq12d 5876 . . 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 ) ) ) )
13371negcld 9144 . . . . 5  |-  ( ph  -> 
-u ( 2  x.  ( P ^ 3 ) )  e.  CC )
134 mulcl 8821 . . . . . 6  |-  ( (; 2
7  e.  CC  /\  ( Q ^ 2 )  e.  CC )  -> 
(; 2 7  x.  ( Q ^ 2 ) )  e.  CC )
135127, 129, 134sylancr 644 . . . . 5  |-  ( ph  ->  (; 2 7  x.  ( Q ^ 2 ) )  e.  CC )
136133, 111, 135addsubd 9178 . . . 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 )
) ) )
137133, 111addcld 8854 . . . . 5  |-  ( ph  ->  ( -u ( 2  x.  ( P ^
3 ) )  +  (; 7 2  x.  ( P  x.  R )
) )  e.  CC )
138137, 135negsubd 9163 . . . 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 ) ) ) )
139 quartlem1.v . . . 4  |-  ( ph  ->  V  =  ( (
-u ( 2  x.  ( P ^ 3 ) )  -  (; 2 7  x.  ( Q ^
2 ) ) )  +  (; 7 2  x.  ( P  x.  R )
) ) )
140136, 138, 1393eqtr4d 2325 . . 3  |-  ( ph  ->  ( ( -u (
2  x.  ( P ^ 3 ) )  +  (; 7 2  x.  ( P  x.  R )
) )  +  -u (; 2 7  x.  ( Q ^ 2 ) ) )  =  V )
141132, 140eqtr2d 2316 . 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 ) ) ) )
14251, 141jca 518 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 set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684  (class class class)co 5858   CCcc 8735   1c1 8738    + caddc 8740    x. cmul 8742    - cmin 9037   -ucneg 9038   2c2 9795   3c3 9796   4c4 9797   7c7 9800   8c8 9801   9c9 9802   NN0cn0 9965  ;cdc 10124   ^cexp 11104
This theorem is referenced by:  quart  20157
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-seq 11047  df-exp 11105
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