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Theorem quartlem2 20170
Description: Closure lemmas for quart 20173. (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 20166 . . . . . 6  |-  ( ph  ->  ( P  e.  CC  /\  Q  e.  CC  /\  R  e.  CC )
)
109simp1d 967 . . . . 5  |-  ( ph  ->  P  e.  CC )
1110sqcld 11259 . . . 4  |-  ( ph  ->  ( P ^ 2 )  e.  CC )
12 1nn0 9997 . . . . . . 7  |-  1  e.  NN0
13 2nn 9893 . . . . . . 7  |-  2  e.  NN
1412, 13decnncl 10153 . . . . . 6  |- ; 1 2  e.  NN
1514nncni 9772 . . . . 5  |- ; 1 2  e.  CC
169simp3d 969 . . . . 5  |-  ( ph  ->  R  e.  CC )
17 mulcl 8837 . . . . 5  |-  ( (; 1
2  e.  CC  /\  R  e.  CC )  ->  (; 1 2  x.  R
)  e.  CC )
1815, 16, 17sylancr 644 . . . 4  |-  ( ph  ->  (; 1 2  x.  R
)  e.  CC )
1911, 18addcld 8870 . . 3  |-  ( ph  ->  ( ( P ^
2 )  +  (; 1
2  x.  R ) )  e.  CC )
201, 19eqeltrd 2370 . 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 9832 . . . . . . 7  |-  2  e.  CC
23 3nn0 9999 . . . . . . . 8  |-  3  e.  NN0
24 expcl 11137 . . . . . . . 8  |-  ( ( P  e.  CC  /\  3  e.  NN0 )  -> 
( P ^ 3 )  e.  CC )
2510, 23, 24sylancl 643 . . . . . . 7  |-  ( ph  ->  ( P ^ 3 )  e.  CC )
26 mulcl 8837 . . . . . . 7  |-  ( ( 2  e.  CC  /\  ( P ^ 3 )  e.  CC )  -> 
( 2  x.  ( P ^ 3 ) )  e.  CC )
2722, 25, 26sylancr 644 . . . . . 6  |-  ( ph  ->  ( 2  x.  ( P ^ 3 ) )  e.  CC )
2827negcld 9160 . . . . 5  |-  ( ph  -> 
-u ( 2  x.  ( P ^ 3 ) )  e.  CC )
29 2nn0 9998 . . . . . . . 8  |-  2  e.  NN0
30 7nn 9898 . . . . . . . 8  |-  7  e.  NN
3129, 30decnncl 10153 . . . . . . 7  |- ; 2 7  e.  NN
3231nncni 9772 . . . . . 6  |- ; 2 7  e.  CC
339simp2d 968 . . . . . . 7  |-  ( ph  ->  Q  e.  CC )
3433sqcld 11259 . . . . . 6  |-  ( ph  ->  ( Q ^ 2 )  e.  CC )
35 mulcl 8837 . . . . . 6  |-  ( (; 2
7  e.  CC  /\  ( Q ^ 2 )  e.  CC )  -> 
(; 2 7  x.  ( Q ^ 2 ) )  e.  CC )
3632, 34, 35sylancr 644 . . . . 5  |-  ( ph  ->  (; 2 7  x.  ( Q ^ 2 ) )  e.  CC )
3728, 36subcld 9173 . . . 4  |-  ( ph  ->  ( -u ( 2  x.  ( P ^
3 ) )  -  (; 2 7  x.  ( Q ^ 2 ) ) )  e.  CC )
38 7nn0 10003 . . . . . . 7  |-  7  e.  NN0
3938, 13decnncl 10153 . . . . . 6  |- ; 7 2  e.  NN
4039nncni 9772 . . . . 5  |- ; 7 2  e.  CC
4110, 16mulcld 8871 . . . . 5  |-  ( ph  ->  ( P  x.  R
)  e.  CC )
42 mulcl 8837 . . . . 5  |-  ( (; 7
2  e.  CC  /\  ( P  x.  R
)  e.  CC )  ->  (; 7 2  x.  ( P  x.  R )
)  e.  CC )
4340, 41, 42sylancr 644 . . . 4  |-  ( ph  ->  (; 7 2  x.  ( P  x.  R )
)  e.  CC )
4437, 43addcld 8870 . . 3  |-  ( ph  ->  ( ( -u (
2  x.  ( P ^ 3 ) )  -  (; 2 7  x.  ( Q ^ 2 ) ) )  +  (; 7 2  x.  ( P  x.  R )
) )  e.  CC )
4521, 44eqeltrd 2370 . 2  |-  ( ph  ->  V  e.  CC )
46 quart.w . . 3  |-  ( ph  ->  W  =  ( sqr `  ( ( V ^
2 )  -  (
4  x.  ( U ^ 3 ) ) ) ) )
4745sqcld 11259 . . . . 5  |-  ( ph  ->  ( V ^ 2 )  e.  CC )
48 4cn 9836 . . . . . 6  |-  4  e.  CC
49 expcl 11137 . . . . . . 7  |-  ( ( U  e.  CC  /\  3  e.  NN0 )  -> 
( U ^ 3 )  e.  CC )
5020, 23, 49sylancl 643 . . . . . 6  |-  ( ph  ->  ( U ^ 3 )  e.  CC )
51 mulcl 8837 . . . . . 6  |-  ( ( 4  e.  CC  /\  ( U ^ 3 )  e.  CC )  -> 
( 4  x.  ( U ^ 3 ) )  e.  CC )
5248, 50, 51sylancr 644 . . . . 5  |-  ( ph  ->  ( 4  x.  ( U ^ 3 ) )  e.  CC )
5347, 52subcld 9173 . . . 4  |-  ( ph  ->  ( ( V ^
2 )  -  (
4  x.  ( U ^ 3 ) ) )  e.  CC )
5453sqrcld 11935 . . 3  |-  ( ph  ->  ( sqr `  (
( V ^ 2 )  -  ( 4  x.  ( U ^
3 ) ) ) )  e.  CC )
5546, 54eqeltrd 2370 . 2  |-  ( ph  ->  W  e.  CC )
5620, 45, 553jca 1132 1  |-  ( ph  ->  ( U  e.  CC  /\  V  e.  CC  /\  W  e.  CC )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1632    e. wcel 1696   ` cfv 5271  (class class class)co 5874   CCcc 8751   1c1 8754    + caddc 8756    x. cmul 8758    - cmin 9053   -ucneg 9054    / cdiv 9439   2c2 9811   3c3 9812   4c4 9813   5c5 9814   6c6 9815   7c7 9816   8c8 9817   NN0cn0 9981  ;cdc 10140   ^cexp 11120   sqrcsqr 11734
This theorem is referenced by:  quartlem3  20171  quart  20173
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-rp 10371  df-seq 11063  df-exp 11121  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737
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