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Theorem quartlem2 20565
Description: Closure lemmas for quart 20568. (Contributed by Mario Carneiro, 7-May-2015.)
Hypotheses
Ref Expression
quart.a  |-  ( ph  ->  A  e.  CC )
quart.b  |-  ( ph  ->  B  e.  CC )
quart.c  |-  ( ph  ->  C  e.  CC )
quart.d  |-  ( ph  ->  D  e.  CC )
quart.x  |-  ( ph  ->  X  e.  CC )
quart.e  |-  ( ph  ->  E  =  -u ( A  /  4 ) )
quart.p  |-  ( ph  ->  P  =  ( B  -  ( ( 3  /  8 )  x.  ( A ^ 2 ) ) ) )
quart.q  |-  ( ph  ->  Q  =  ( ( C  -  ( ( A  x.  B )  /  2 ) )  +  ( ( A ^ 3 )  / 
8 ) ) )
quart.r  |-  ( ph  ->  R  =  ( ( D  -  ( ( C  x.  A )  /  4 ) )  +  ( ( ( ( A ^ 2 )  x.  B )  / ; 1 6 )  -  ( ( 3  / ;; 2 5 6 )  x.  ( A ^
4 ) ) ) ) )
quart.u  |-  ( ph  ->  U  =  ( ( P ^ 2 )  +  (; 1 2  x.  R
) ) )
quart.v  |-  ( ph  ->  V  =  ( (
-u ( 2  x.  ( P ^ 3 ) )  -  (; 2 7  x.  ( Q ^
2 ) ) )  +  (; 7 2  x.  ( P  x.  R )
) ) )
quart.w  |-  ( ph  ->  W  =  ( sqr `  ( ( V ^
2 )  -  (
4  x.  ( U ^ 3 ) ) ) ) )
Assertion
Ref Expression
quartlem2  |-  ( ph  ->  ( U  e.  CC  /\  V  e.  CC  /\  W  e.  CC )
)

Proof of Theorem quartlem2
StepHypRef Expression
1 quart.u . . 3  |-  ( ph  ->  U  =  ( ( P ^ 2 )  +  (; 1 2  x.  R
) ) )
2 quart.a . . . . . . 7  |-  ( ph  ->  A  e.  CC )
3 quart.b . . . . . . 7  |-  ( ph  ->  B  e.  CC )
4 quart.c . . . . . . 7  |-  ( ph  ->  C  e.  CC )
5 quart.d . . . . . . 7  |-  ( ph  ->  D  e.  CC )
6 quart.p . . . . . . 7  |-  ( ph  ->  P  =  ( B  -  ( ( 3  /  8 )  x.  ( A ^ 2 ) ) ) )
7 quart.q . . . . . . 7  |-  ( ph  ->  Q  =  ( ( C  -  ( ( A  x.  B )  /  2 ) )  +  ( ( A ^ 3 )  / 
8 ) ) )
8 quart.r . . . . . . 7  |-  ( ph  ->  R  =  ( ( D  -  ( ( C  x.  A )  /  4 ) )  +  ( ( ( ( A ^ 2 )  x.  B )  / ; 1 6 )  -  ( ( 3  / ;; 2 5 6 )  x.  ( A ^
4 ) ) ) ) )
92, 3, 4, 5, 6, 7, 8quart1cl 20561 . . . . . 6  |-  ( ph  ->  ( P  e.  CC  /\  Q  e.  CC  /\  R  e.  CC )
)
109simp1d 969 . . . . 5  |-  ( ph  ->  P  e.  CC )
1110sqcld 11448 . . . 4  |-  ( ph  ->  ( P ^ 2 )  e.  CC )
12 1nn0 10169 . . . . . . 7  |-  1  e.  NN0
13 2nn 10065 . . . . . . 7  |-  2  e.  NN
1412, 13decnncl 10327 . . . . . 6  |- ; 1 2  e.  NN
1514nncni 9942 . . . . 5  |- ; 1 2  e.  CC
169simp3d 971 . . . . 5  |-  ( ph  ->  R  e.  CC )
17 mulcl 9007 . . . . 5  |-  ( (; 1
2  e.  CC  /\  R  e.  CC )  ->  (; 1 2  x.  R
)  e.  CC )
1815, 16, 17sylancr 645 . . . 4  |-  ( ph  ->  (; 1 2  x.  R
)  e.  CC )
1911, 18addcld 9040 . . 3  |-  ( ph  ->  ( ( P ^
2 )  +  (; 1
2  x.  R ) )  e.  CC )
201, 19eqeltrd 2461 . 2  |-  ( ph  ->  U  e.  CC )
21 quart.v . . 3  |-  ( ph  ->  V  =  ( (
-u ( 2  x.  ( P ^ 3 ) )  -  (; 2 7  x.  ( Q ^
2 ) ) )  +  (; 7 2  x.  ( P  x.  R )
) ) )
22 2cn 10002 . . . . . . 7  |-  2  e.  CC
23 3nn0 10171 . . . . . . . 8  |-  3  e.  NN0
24 expcl 11326 . . . . . . . 8  |-  ( ( P  e.  CC  /\  3  e.  NN0 )  -> 
( P ^ 3 )  e.  CC )
2510, 23, 24sylancl 644 . . . . . . 7  |-  ( ph  ->  ( P ^ 3 )  e.  CC )
26 mulcl 9007 . . . . . . 7  |-  ( ( 2  e.  CC  /\  ( P ^ 3 )  e.  CC )  -> 
( 2  x.  ( P ^ 3 ) )  e.  CC )
2722, 25, 26sylancr 645 . . . . . 6  |-  ( ph  ->  ( 2  x.  ( P ^ 3 ) )  e.  CC )
2827negcld 9330 . . . . 5  |-  ( ph  -> 
-u ( 2  x.  ( P ^ 3 ) )  e.  CC )
29 2nn0 10170 . . . . . . . 8  |-  2  e.  NN0
30 7nn 10070 . . . . . . . 8  |-  7  e.  NN
3129, 30decnncl 10327 . . . . . . 7  |- ; 2 7  e.  NN
3231nncni 9942 . . . . . 6  |- ; 2 7  e.  CC
339simp2d 970 . . . . . . 7  |-  ( ph  ->  Q  e.  CC )
3433sqcld 11448 . . . . . 6  |-  ( ph  ->  ( Q ^ 2 )  e.  CC )
35 mulcl 9007 . . . . . 6  |-  ( (; 2
7  e.  CC  /\  ( Q ^ 2 )  e.  CC )  -> 
(; 2 7  x.  ( Q ^ 2 ) )  e.  CC )
3632, 34, 35sylancr 645 . . . . 5  |-  ( ph  ->  (; 2 7  x.  ( Q ^ 2 ) )  e.  CC )
3728, 36subcld 9343 . . . 4  |-  ( ph  ->  ( -u ( 2  x.  ( P ^
3 ) )  -  (; 2 7  x.  ( Q ^ 2 ) ) )  e.  CC )
38 7nn0 10175 . . . . . . 7  |-  7  e.  NN0
3938, 13decnncl 10327 . . . . . 6  |- ; 7 2  e.  NN
4039nncni 9942 . . . . 5  |- ; 7 2  e.  CC
4110, 16mulcld 9041 . . . . 5  |-  ( ph  ->  ( P  x.  R
)  e.  CC )
42 mulcl 9007 . . . . 5  |-  ( (; 7
2  e.  CC  /\  ( P  x.  R
)  e.  CC )  ->  (; 7 2  x.  ( P  x.  R )
)  e.  CC )
4340, 41, 42sylancr 645 . . . 4  |-  ( ph  ->  (; 7 2  x.  ( P  x.  R )
)  e.  CC )
4437, 43addcld 9040 . . 3  |-  ( ph  ->  ( ( -u (
2  x.  ( P ^ 3 ) )  -  (; 2 7  x.  ( Q ^ 2 ) ) )  +  (; 7 2  x.  ( P  x.  R )
) )  e.  CC )
4521, 44eqeltrd 2461 . 2  |-  ( ph  ->  V  e.  CC )
46 quart.w . . 3  |-  ( ph  ->  W  =  ( sqr `  ( ( V ^
2 )  -  (
4  x.  ( U ^ 3 ) ) ) ) )
4745sqcld 11448 . . . . 5  |-  ( ph  ->  ( V ^ 2 )  e.  CC )
48 4cn 10006 . . . . . 6  |-  4  e.  CC
49 expcl 11326 . . . . . . 7  |-  ( ( U  e.  CC  /\  3  e.  NN0 )  -> 
( U ^ 3 )  e.  CC )
5020, 23, 49sylancl 644 . . . . . 6  |-  ( ph  ->  ( U ^ 3 )  e.  CC )
51 mulcl 9007 . . . . . 6  |-  ( ( 4  e.  CC  /\  ( U ^ 3 )  e.  CC )  -> 
( 4  x.  ( U ^ 3 ) )  e.  CC )
5248, 50, 51sylancr 645 . . . . 5  |-  ( ph  ->  ( 4  x.  ( U ^ 3 ) )  e.  CC )
5347, 52subcld 9343 . . . 4  |-  ( ph  ->  ( ( V ^
2 )  -  (
4  x.  ( U ^ 3 ) ) )  e.  CC )
5453sqrcld 12166 . . 3  |-  ( ph  ->  ( sqr `  (
( V ^ 2 )  -  ( 4  x.  ( U ^
3 ) ) ) )  e.  CC )
5546, 54eqeltrd 2461 . 2  |-  ( ph  ->  W  e.  CC )
5620, 45, 553jca 1134 1  |-  ( ph  ->  ( U  e.  CC  /\  V  e.  CC  /\  W  e.  CC )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 936    = wceq 1649    e. wcel 1717   ` cfv 5394  (class class class)co 6020   CCcc 8921   1c1 8924    + caddc 8926    x. cmul 8928    - cmin 9223   -ucneg 9224    / cdiv 9609   2c2 9981   3c3 9982   4c4 9983   5c5 9984   6c6 9985   7c7 9986   8c8 9987   NN0cn0 10153  ;cdc 10314   ^cexp 11309   sqrcsqr 11965
This theorem is referenced by:  quartlem3  20566  quart  20568
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000  ax-pre-sup 9001
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-sup 7381  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-div 9610  df-nn 9933  df-2 9990  df-3 9991  df-4 9992  df-5 9993  df-6 9994  df-7 9995  df-8 9996  df-9 9997  df-10 9998  df-n0 10154  df-z 10215  df-dec 10315  df-uz 10421  df-rp 10545  df-seq 11251  df-exp 11310  df-cj 11831  df-re 11832  df-im 11833  df-sqr 11967  df-abs 11968
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