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Theorem quartlem2 20690
Description: Closure lemmas for quart 20693. (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 20686 . . . . . 6  |-  ( ph  ->  ( P  e.  CC  /\  Q  e.  CC  /\  R  e.  CC )
)
109simp1d 969 . . . . 5  |-  ( ph  ->  P  e.  CC )
1110sqcld 11513 . . . 4  |-  ( ph  ->  ( P ^ 2 )  e.  CC )
12 1nn0 10229 . . . . . . 7  |-  1  e.  NN0
13 2nn 10125 . . . . . . 7  |-  2  e.  NN
1412, 13decnncl 10387 . . . . . 6  |- ; 1 2  e.  NN
1514nncni 10002 . . . . 5  |- ; 1 2  e.  CC
169simp3d 971 . . . . 5  |-  ( ph  ->  R  e.  CC )
17 mulcl 9066 . . . . 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 9099 . . 3  |-  ( ph  ->  ( ( P ^
2 )  +  (; 1
2  x.  R ) )  e.  CC )
201, 19eqeltrd 2509 . 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 10062 . . . . . . 7  |-  2  e.  CC
23 3nn0 10231 . . . . . . . 8  |-  3  e.  NN0
24 expcl 11391 . . . . . . . 8  |-  ( ( P  e.  CC  /\  3  e.  NN0 )  -> 
( P ^ 3 )  e.  CC )
2510, 23, 24sylancl 644 . . . . . . 7  |-  ( ph  ->  ( P ^ 3 )  e.  CC )
26 mulcl 9066 . . . . . . 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 9390 . . . . 5  |-  ( ph  -> 
-u ( 2  x.  ( P ^ 3 ) )  e.  CC )
29 2nn0 10230 . . . . . . . 8  |-  2  e.  NN0
30 7nn 10130 . . . . . . . 8  |-  7  e.  NN
3129, 30decnncl 10387 . . . . . . 7  |- ; 2 7  e.  NN
3231nncni 10002 . . . . . 6  |- ; 2 7  e.  CC
339simp2d 970 . . . . . . 7  |-  ( ph  ->  Q  e.  CC )
3433sqcld 11513 . . . . . 6  |-  ( ph  ->  ( Q ^ 2 )  e.  CC )
35 mulcl 9066 . . . . . 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 9403 . . . 4  |-  ( ph  ->  ( -u ( 2  x.  ( P ^
3 ) )  -  (; 2 7  x.  ( Q ^ 2 ) ) )  e.  CC )
38 7nn0 10235 . . . . . . 7  |-  7  e.  NN0
3938, 13decnncl 10387 . . . . . 6  |- ; 7 2  e.  NN
4039nncni 10002 . . . . 5  |- ; 7 2  e.  CC
4110, 16mulcld 9100 . . . . 5  |-  ( ph  ->  ( P  x.  R
)  e.  CC )
42 mulcl 9066 . . . . 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 9099 . . 3  |-  ( ph  ->  ( ( -u (
2  x.  ( P ^ 3 ) )  -  (; 2 7  x.  ( Q ^ 2 ) ) )  +  (; 7 2  x.  ( P  x.  R )
) )  e.  CC )
4521, 44eqeltrd 2509 . 2  |-  ( ph  ->  V  e.  CC )
46 quart.w . . 3  |-  ( ph  ->  W  =  ( sqr `  ( ( V ^
2 )  -  (
4  x.  ( U ^ 3 ) ) ) ) )
4745sqcld 11513 . . . . 5  |-  ( ph  ->  ( V ^ 2 )  e.  CC )
48 4cn 10066 . . . . . 6  |-  4  e.  CC
49 expcl 11391 . . . . . . 7  |-  ( ( U  e.  CC  /\  3  e.  NN0 )  -> 
( U ^ 3 )  e.  CC )
5020, 23, 49sylancl 644 . . . . . 6  |-  ( ph  ->  ( U ^ 3 )  e.  CC )
51 mulcl 9066 . . . . . 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 9403 . . . 4  |-  ( ph  ->  ( ( V ^
2 )  -  (
4  x.  ( U ^ 3 ) ) )  e.  CC )
5453sqrcld 12231 . . 3  |-  ( ph  ->  ( sqr `  (
( V ^ 2 )  -  ( 4  x.  ( U ^
3 ) ) ) )  e.  CC )
5546, 54eqeltrd 2509 . 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 1652    e. wcel 1725   ` cfv 5446  (class class class)co 6073   CCcc 8980   1c1 8983    + caddc 8985    x. cmul 8987    - cmin 9283   -ucneg 9284    / cdiv 9669   2c2 10041   3c3 10042   4c4 10043   5c5 10044   6c6 10045   7c7 10046   8c8 10047   NN0cn0 10213  ;cdc 10374   ^cexp 11374   sqrcsqr 12030
This theorem is referenced by:  quartlem3  20691  quart  20693
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 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-sup 7438  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-7 10055  df-8 10056  df-9 10057  df-10 10058  df-n0 10214  df-z 10275  df-dec 10375  df-uz 10481  df-rp 10605  df-seq 11316  df-exp 11375  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033
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