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Theorem quartlem1 20697
Description: Lemma for quart 20701. (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 10070 . . . . . . . . . 10  |-  2  e.  CC
2 quartlem1.p . . . . . . . . . 10  |-  ( ph  ->  P  e.  CC )
3 sqmul 11445 . . . . . . . . . 10  |-  ( ( 2  e.  CC  /\  P  e.  CC )  ->  ( ( 2  x.  P ) ^ 2 )  =  ( ( 2 ^ 2 )  x.  ( P ^
2 ) ) )
41, 2, 3sylancr 645 . . . . . . . . 9  |-  ( ph  ->  ( ( 2  x.  P ) ^ 2 )  =  ( ( 2 ^ 2 )  x.  ( P ^
2 ) ) )
5 sq2 11477 . . . . . . . . . 10  |-  ( 2 ^ 2 )  =  4
65oveq1i 6091 . . . . . . . . 9  |-  ( ( 2 ^ 2 )  x.  ( P ^
2 ) )  =  ( 4  x.  ( P ^ 2 ) )
74, 6syl6eq 2484 . . . . . . . 8  |-  ( ph  ->  ( ( 2  x.  P ) ^ 2 )  =  ( 4  x.  ( P ^
2 ) ) )
87oveq1d 6096 . . . . . . 7  |-  ( ph  ->  ( ( ( 2  x.  P ) ^
2 )  -  (
3  x.  ( P ^ 2 ) ) )  =  ( ( 4  x.  ( P ^ 2 ) )  -  ( 3  x.  ( P ^ 2 ) ) ) )
9 4cn 10074 . . . . . . . . 9  |-  4  e.  CC
109a1i 11 . . . . . . . 8  |-  ( ph  ->  4  e.  CC )
11 3cn 10072 . . . . . . . . 9  |-  3  e.  CC
1211a1i 11 . . . . . . . 8  |-  ( ph  ->  3  e.  CC )
132sqcld 11521 . . . . . . . 8  |-  ( ph  ->  ( P ^ 2 )  e.  CC )
1410, 12, 13subdird 9490 . . . . . . 7  |-  ( ph  ->  ( ( 4  -  3 )  x.  ( P ^ 2 ) )  =  ( ( 4  x.  ( P ^
2 ) )  -  ( 3  x.  ( P ^ 2 ) ) ) )
158, 14eqtr4d 2471 . . . . . 6  |-  ( ph  ->  ( ( ( 2  x.  P ) ^
2 )  -  (
3  x.  ( P ^ 2 ) ) )  =  ( ( 4  -  3 )  x.  ( P ^
2 ) ) )
16 ax-1cn 9048 . . . . . . . . . 10  |-  1  e.  CC
17 3p1e4 10104 . . . . . . . . . 10  |-  ( 3  +  1 )  =  4
189, 11, 16, 17subaddrii 9389 . . . . . . . . 9  |-  ( 4  -  3 )  =  1
1918oveq1i 6091 . . . . . . . 8  |-  ( ( 4  -  3 )  x.  ( P ^
2 ) )  =  ( 1  x.  ( P ^ 2 ) )
20 mulid2 9089 . . . . . . . 8  |-  ( ( P ^ 2 )  e.  CC  ->  (
1  x.  ( P ^ 2 ) )  =  ( P ^
2 ) )
2119, 20syl5eq 2480 . . . . . . 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 2469 . . . . 5  |-  ( ph  ->  ( P ^ 2 )  =  ( ( ( 2  x.  P
) ^ 2 )  -  ( 3  x.  ( P ^ 2 ) ) ) )
2423oveq1d 6096 . . . 4  |-  ( ph  ->  ( ( P ^
2 )  +  (; 1
2  x.  R ) )  =  ( ( ( ( 2  x.  P ) ^ 2 )  -  ( 3  x.  ( P ^
2 ) ) )  +  (; 1 2  x.  R
) ) )
25 mulcl 9074 . . . . . . 7  |-  ( ( 2  e.  CC  /\  P  e.  CC )  ->  ( 2  x.  P
)  e.  CC )
261, 2, 25sylancr 645 . . . . . 6  |-  ( ph  ->  ( 2  x.  P
)  e.  CC )
2726sqcld 11521 . . . . 5  |-  ( ph  ->  ( ( 2  x.  P ) ^ 2 )  e.  CC )
28 mulcl 9074 . . . . . 6  |-  ( ( 3  e.  CC  /\  ( P ^ 2 )  e.  CC )  -> 
( 3  x.  ( P ^ 2 ) )  e.  CC )
2911, 13, 28sylancr 645 . . . . 5  |-  ( ph  ->  ( 3  x.  ( P ^ 2 ) )  e.  CC )
30 1nn0 10237 . . . . . . . 8  |-  1  e.  NN0
31 2nn 10133 . . . . . . . 8  |-  2  e.  NN
3230, 31decnncl 10395 . . . . . . 7  |- ; 1 2  e.  NN
3332nncni 10010 . . . . . 6  |- ; 1 2  e.  CC
34 quartlem1.r . . . . . 6  |-  ( ph  ->  R  e.  CC )
35 mulcl 9074 . . . . . 6  |-  ( (; 1
2  e.  CC  /\  R  e.  CC )  ->  (; 1 2  x.  R
)  e.  CC )
3633, 34, 35sylancr 645 . . . . 5  |-  ( ph  ->  (; 1 2  x.  R
)  e.  CC )
3727, 29, 36subsubd 9439 . . . 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 2471 . . 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 9074 . . . . . . 7  |-  ( ( 4  e.  CC  /\  R  e.  CC )  ->  ( 4  x.  R
)  e.  CC )
419, 34, 40sylancr 645 . . . . . 6  |-  ( ph  ->  ( 4  x.  R
)  e.  CC )
4212, 13, 41subdid 9489 . . . . 5  |-  ( ph  ->  ( 3  x.  (
( P ^ 2 )  -  ( 4  x.  R ) ) )  =  ( ( 3  x.  ( P ^ 2 ) )  -  ( 3  x.  ( 4  x.  R
) ) ) )
43 4t3e12 10454 . . . . . . . . 9  |-  ( 4  x.  3 )  = ; 1
2
449, 11, 43mulcomli 9097 . . . . . . . 8  |-  ( 3  x.  4 )  = ; 1
2
4544oveq1i 6091 . . . . . . 7  |-  ( ( 3  x.  4 )  x.  R )  =  (; 1 2  x.  R
)
4612, 10, 34mulassd 9111 . . . . . . 7  |-  ( ph  ->  ( ( 3  x.  4 )  x.  R
)  =  ( 3  x.  ( 4  x.  R ) ) )
4745, 46syl5eqr 2482 . . . . . 6  |-  ( ph  ->  (; 1 2  x.  R
)  =  ( 3  x.  ( 4  x.  R ) ) )
4847oveq2d 6097 . . . . 5  |-  ( ph  ->  ( ( 3  x.  ( P ^ 2 ) )  -  (; 1 2  x.  R ) )  =  ( ( 3  x.  ( P ^
2 ) )  -  ( 3  x.  (
4  x.  R ) ) ) )
4942, 48eqtr4d 2471 . . . 4  |-  ( ph  ->  ( 3  x.  (
( P ^ 2 )  -  ( 4  x.  R ) ) )  =  ( ( 3  x.  ( P ^ 2 ) )  -  (; 1 2  x.  R
) ) )
5049oveq2d 6097 . . 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 2478 . 2  |-  ( ph  ->  U  =  ( ( ( 2  x.  P
) ^ 2 )  -  ( 3  x.  ( ( P ^
2 )  -  (
4  x.  R ) ) ) ) )
521a1i 11 . . . . . . . . . 10  |-  ( ph  ->  2  e.  CC )
53 3nn0 10239 . . . . . . . . . . 11  |-  3  e.  NN0
5453a1i 11 . . . . . . . . . 10  |-  ( ph  ->  3  e.  NN0 )
5552, 2, 54mulexpd 11538 . . . . . . . . 9  |-  ( ph  ->  ( ( 2  x.  P ) ^ 3 )  =  ( ( 2 ^ 3 )  x.  ( P ^
3 ) ) )
56 cu2 11479 . . . . . . . . . 10  |-  ( 2 ^ 3 )  =  8
5756oveq1i 6091 . . . . . . . . 9  |-  ( ( 2 ^ 3 )  x.  ( P ^
3 ) )  =  ( 8  x.  ( P ^ 3 ) )
5855, 57syl6eq 2484 . . . . . . . 8  |-  ( ph  ->  ( ( 2  x.  P ) ^ 3 )  =  ( 8  x.  ( P ^
3 ) ) )
5958oveq2d 6097 . . . . . . 7  |-  ( ph  ->  ( 2  x.  (
( 2  x.  P
) ^ 3 ) )  =  ( 2  x.  ( 8  x.  ( P ^ 3 ) ) ) )
60 8nn 10139 . . . . . . . . . 10  |-  8  e.  NN
6160nncni 10010 . . . . . . . . 9  |-  8  e.  CC
6261a1i 11 . . . . . . . 8  |-  ( ph  ->  8  e.  CC )
63 expcl 11399 . . . . . . . . 9  |-  ( ( P  e.  CC  /\  3  e.  NN0 )  -> 
( P ^ 3 )  e.  CC )
642, 53, 63sylancl 644 . . . . . . . 8  |-  ( ph  ->  ( P ^ 3 )  e.  CC )
6552, 62, 64mul12d 9275 . . . . . . 7  |-  ( ph  ->  ( 2  x.  (
8  x.  ( P ^ 3 ) ) )  =  ( 8  x.  ( 2  x.  ( P ^ 3 ) ) ) )
6659, 65eqtrd 2468 . . . . . 6  |-  ( ph  ->  ( 2  x.  (
( 2  x.  P
) ^ 3 ) )  =  ( 8  x.  ( 2  x.  ( P ^ 3 ) ) ) )
67 9nn 10140 . . . . . . . . . 10  |-  9  e.  NN
6867nncni 10010 . . . . . . . . 9  |-  9  e.  CC
6968a1i 11 . . . . . . . 8  |-  ( ph  ->  9  e.  CC )
70 mulcl 9074 . . . . . . . . 9  |-  ( ( 2  e.  CC  /\  ( P ^ 3 )  e.  CC )  -> 
( 2  x.  ( P ^ 3 ) )  e.  CC )
711, 64, 70sylancr 645 . . . . . . . 8  |-  ( ph  ->  ( 2  x.  ( P ^ 3 ) )  e.  CC )
722, 34mulcld 9108 . . . . . . . . 9  |-  ( ph  ->  ( P  x.  R
)  e.  CC )
73 mulcl 9074 . . . . . . . . 9  |-  ( ( 8  e.  CC  /\  ( P  x.  R
)  e.  CC )  ->  ( 8  x.  ( P  x.  R
) )  e.  CC )
7461, 72, 73sylancr 645 . . . . . . . 8  |-  ( ph  ->  ( 8  x.  ( P  x.  R )
)  e.  CC )
7569, 71, 74subdid 9489 . . . . . . 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 9489 . . . . . . . . 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 9111 . . . . . . . . . . 11  |-  ( ph  ->  ( ( 2  x.  P )  x.  ( P ^ 2 ) )  =  ( 2  x.  ( P  x.  ( P ^ 2 ) ) ) )
782, 13mulcomd 9109 . . . . . . . . . . . . 13  |-  ( ph  ->  ( P  x.  ( P ^ 2 ) )  =  ( ( P ^ 2 )  x.  P ) )
79 df-3 10059 . . . . . . . . . . . . . . 15  |-  3  =  ( 2  +  1 )
8079oveq2i 6092 . . . . . . . . . . . . . 14  |-  ( P ^ 3 )  =  ( P ^ (
2  +  1 ) )
81 2nn0 10238 . . . . . . . . . . . . . . 15  |-  2  e.  NN0
82 expp1 11388 . . . . . . . . . . . . . . 15  |-  ( ( P  e.  CC  /\  2  e.  NN0 )  -> 
( P ^ (
2  +  1 ) )  =  ( ( P ^ 2 )  x.  P ) )
832, 81, 82sylancl 644 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( P ^ (
2  +  1 ) )  =  ( ( P ^ 2 )  x.  P ) )
8480, 83syl5eq 2480 . . . . . . . . . . . . 13  |-  ( ph  ->  ( P ^ 3 )  =  ( ( P ^ 2 )  x.  P ) )
8578, 84eqtr4d 2471 . . . . . . . . . . . 12  |-  ( ph  ->  ( P  x.  ( P ^ 2 ) )  =  ( P ^
3 ) )
8685oveq2d 6097 . . . . . . . . . . 11  |-  ( ph  ->  ( 2  x.  ( P  x.  ( P ^ 2 ) ) )  =  ( 2  x.  ( P ^
3 ) ) )
8777, 86eqtrd 2468 . . . . . . . . . 10  |-  ( ph  ->  ( ( 2  x.  P )  x.  ( P ^ 2 ) )  =  ( 2  x.  ( P ^ 3 ) ) )
8852, 2, 10, 34mul4d 9278 . . . . . . . . . . 11  |-  ( ph  ->  ( ( 2  x.  P )  x.  (
4  x.  R ) )  =  ( ( 2  x.  4 )  x.  ( P  x.  R ) ) )
89 4t2e8 10130 . . . . . . . . . . . . 13  |-  ( 4  x.  2 )  =  8
909, 1, 89mulcomli 9097 . . . . . . . . . . . 12  |-  ( 2  x.  4 )  =  8
9190oveq1i 6091 . . . . . . . . . . 11  |-  ( ( 2  x.  4 )  x.  ( P  x.  R ) )  =  ( 8  x.  ( P  x.  R )
)
9288, 91syl6eq 2484 . . . . . . . . . 10  |-  ( ph  ->  ( ( 2  x.  P )  x.  (
4  x.  R ) )  =  ( 8  x.  ( P  x.  R ) ) )
9387, 92oveq12d 6099 . . . . . . . . 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 2468 . . . . . . . 8  |-  ( ph  ->  ( ( 2  x.  P )  x.  (
( P ^ 2 )  -  ( 4  x.  R ) ) )  =  ( ( 2  x.  ( P ^ 3 ) )  -  ( 8  x.  ( P  x.  R
) ) ) )
9594oveq2d 6097 . . . . . . 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 10483 . . . . . . . . . 10  |-  ( 9  x.  8 )  = ; 7
2
9796oveq1i 6091 . . . . . . . . 9  |-  ( ( 9  x.  8 )  x.  ( P  x.  R ) )  =  (; 7 2  x.  ( P  x.  R )
)
9869, 62, 72mulassd 9111 . . . . . . . . 9  |-  ( ph  ->  ( ( 9  x.  8 )  x.  ( P  x.  R )
)  =  ( 9  x.  ( 8  x.  ( P  x.  R
) ) ) )
9997, 98syl5eqr 2482 . . . . . . . 8  |-  ( ph  ->  (; 7 2  x.  ( P  x.  R )
)  =  ( 9  x.  ( 8  x.  ( P  x.  R
) ) ) )
10099oveq2d 6097 . . . . . . 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 2478 . . . . . 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 6099 . . . . 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 9074 . . . . . . 7  |-  ( ( 8  e.  CC  /\  ( 2  x.  ( P ^ 3 ) )  e.  CC )  -> 
( 8  x.  (
2  x.  ( P ^ 3 ) ) )  e.  CC )
10461, 71, 103sylancr 645 . . . . . 6  |-  ( ph  ->  ( 8  x.  (
2  x.  ( P ^ 3 ) ) )  e.  CC )
105 mulcl 9074 . . . . . . 7  |-  ( ( 9  e.  CC  /\  ( 2  x.  ( P ^ 3 ) )  e.  CC )  -> 
( 9  x.  (
2  x.  ( P ^ 3 ) ) )  e.  CC )
10668, 71, 105sylancr 645 . . . . . 6  |-  ( ph  ->  ( 9  x.  (
2  x.  ( P ^ 3 ) ) )  e.  CC )
107 7nn0 10243 . . . . . . . . 9  |-  7  e.  NN0
108107, 31decnncl 10395 . . . . . . . 8  |- ; 7 2  e.  NN
109108nncni 10010 . . . . . . 7  |- ; 7 2  e.  CC
110 mulcl 9074 . . . . . . 7  |-  ( (; 7
2  e.  CC  /\  ( P  x.  R
)  e.  CC )  ->  (; 7 2  x.  ( P  x.  R )
)  e.  CC )
111109, 72, 110sylancr 645 . . . . . 6  |-  ( ph  ->  (; 7 2  x.  ( P  x.  R )
)  e.  CC )
112104, 106, 111subsubd 9439 . . . . 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 9427 . . . . . . 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 9490 . . . . . . . . 9  |-  ( ph  ->  ( ( 9  -  8 )  x.  (
2  x.  ( P ^ 3 ) ) )  =  ( ( 9  x.  ( 2  x.  ( P ^
3 ) ) )  -  ( 8  x.  ( 2  x.  ( P ^ 3 ) ) ) ) )
115 8p1e9 10109 . . . . . . . . . . . 12  |-  ( 8  +  1 )  =  9
11668, 61, 16, 115subaddrii 9389 . . . . . . . . . . 11  |-  ( 9  -  8 )  =  1
117116oveq1i 6091 . . . . . . . . . 10  |-  ( ( 9  -  8 )  x.  ( 2  x.  ( P ^ 3 ) ) )  =  ( 1  x.  (
2  x.  ( P ^ 3 ) ) )
11871mulid2d 9106 . . . . . . . . . 10  |-  ( ph  ->  ( 1  x.  (
2  x.  ( P ^ 3 ) ) )  =  ( 2  x.  ( P ^
3 ) ) )
119117, 118syl5eq 2480 . . . . . . . . 9  |-  ( ph  ->  ( ( 9  -  8 )  x.  (
2  x.  ( P ^ 3 ) ) )  =  ( 2  x.  ( P ^
3 ) ) )
120114, 119eqtr3d 2470 . . . . . . . 8  |-  ( ph  ->  ( ( 9  x.  ( 2  x.  ( P ^ 3 ) ) )  -  ( 8  x.  ( 2  x.  ( P ^ 3 ) ) ) )  =  ( 2  x.  ( P ^ 3 ) ) )
121120negeqd 9300 . . . . . . 7  |-  ( ph  -> 
-u ( ( 9  x.  ( 2  x.  ( P ^ 3 ) ) )  -  ( 8  x.  (
2  x.  ( P ^ 3 ) ) ) )  =  -u ( 2  x.  ( P ^ 3 ) ) )
122113, 121eqtr3d 2470 . . . . . 6  |-  ( ph  ->  ( ( 8  x.  ( 2  x.  ( P ^ 3 ) ) )  -  ( 9  x.  ( 2  x.  ( P ^ 3 ) ) ) )  =  -u ( 2  x.  ( P ^ 3 ) ) )
123122oveq1d 6096 . . . . 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 2472 . . . 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 10138 . . . . . . 7  |-  7  e.  NN
12681, 125decnncl 10395 . . . . . 6  |- ; 2 7  e.  NN
127126nncni 10010 . . . . 5  |- ; 2 7  e.  CC
128 quartlem1.q . . . . . 6  |-  ( ph  ->  Q  e.  CC )
129128sqcld 11521 . . . . 5  |-  ( ph  ->  ( Q ^ 2 )  e.  CC )
130 mulneg2 9471 . . . . 5  |-  ( (; 2
7  e.  CC  /\  ( Q ^ 2 )  e.  CC )  -> 
(; 2 7  x.  -u ( Q ^ 2 ) )  =  -u (; 2 7  x.  ( Q ^ 2 ) ) )
131127, 129, 130sylancr 645 . . . 4  |-  ( ph  ->  (; 2 7  x.  -u ( Q ^ 2 ) )  =  -u (; 2 7  x.  ( Q ^ 2 ) ) )
132124, 131oveq12d 6099 . . 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 9398 . . . . 5  |-  ( ph  -> 
-u ( 2  x.  ( P ^ 3 ) )  e.  CC )
134 mulcl 9074 . . . . . 6  |-  ( (; 2
7  e.  CC  /\  ( Q ^ 2 )  e.  CC )  -> 
(; 2 7  x.  ( Q ^ 2 ) )  e.  CC )
135127, 129, 134sylancr 645 . . . . 5  |-  ( ph  ->  (; 2 7  x.  ( Q ^ 2 ) )  e.  CC )
136133, 111, 135addsubd 9432 . . . 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 9107 . . . . 5  |-  ( ph  ->  ( -u ( 2  x.  ( P ^
3 ) )  +  (; 7 2  x.  ( P  x.  R )
) )  e.  CC )
138137, 135negsubd 9417 . . . 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 2478 . . 3  |-  ( ph  ->  ( ( -u (
2  x.  ( P ^ 3 ) )  +  (; 7 2  x.  ( P  x.  R )
) )  +  -u (; 2 7  x.  ( Q ^ 2 ) ) )  =  V )
141132, 140eqtr2d 2469 . 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 519 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 359    = wceq 1652    e. wcel 1725  (class class class)co 6081   CCcc 8988   1c1 8991    + caddc 8993    x. cmul 8995    - cmin 9291   -ucneg 9292   2c2 10049   3c3 10050   4c4 10051   7c7 10054   8c8 10055   9c9 10056   NN0cn0 10221  ;cdc 10382   ^cexp 11382
This theorem is referenced by:  quart  20701
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-5 10061  df-6 10062  df-7 10063  df-8 10064  df-9 10065  df-10 10066  df-n0 10222  df-z 10283  df-dec 10383  df-uz 10489  df-seq 11324  df-exp 11383
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