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Theorem quartlem2 22253
Description: Closure lemmas for quart 22256. (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 22249 . . . . . 6  |-  ( ph  ->  ( P  e.  CC  /\  Q  e.  CC  /\  R  e.  CC )
)
109simp1d 1000 . . . . 5  |-  ( ph  ->  P  e.  CC )
1110sqcld 12006 . . . 4  |-  ( ph  ->  ( P ^ 2 )  e.  CC )
12 1nn0 10595 . . . . . . 7  |-  1  e.  NN0
13 2nn 10479 . . . . . . 7  |-  2  e.  NN
1412, 13decnncl 10768 . . . . . 6  |- ; 1 2  e.  NN
1514nncni 10332 . . . . 5  |- ; 1 2  e.  CC
169simp3d 1002 . . . . 5  |-  ( ph  ->  R  e.  CC )
17 mulcl 9366 . . . . 5  |-  ( (; 1
2  e.  CC  /\  R  e.  CC )  ->  (; 1 2  x.  R
)  e.  CC )
1815, 16, 17sylancr 663 . . . 4  |-  ( ph  ->  (; 1 2  x.  R
)  e.  CC )
1911, 18addcld 9405 . . 3  |-  ( ph  ->  ( ( P ^
2 )  +  (; 1
2  x.  R ) )  e.  CC )
201, 19eqeltrd 2517 . 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 10392 . . . . . . 7  |-  2  e.  CC
23 3nn0 10597 . . . . . . . 8  |-  3  e.  NN0
24 expcl 11883 . . . . . . . 8  |-  ( ( P  e.  CC  /\  3  e.  NN0 )  -> 
( P ^ 3 )  e.  CC )
2510, 23, 24sylancl 662 . . . . . . 7  |-  ( ph  ->  ( P ^ 3 )  e.  CC )
26 mulcl 9366 . . . . . . 7  |-  ( ( 2  e.  CC  /\  ( P ^ 3 )  e.  CC )  -> 
( 2  x.  ( P ^ 3 ) )  e.  CC )
2722, 25, 26sylancr 663 . . . . . 6  |-  ( ph  ->  ( 2  x.  ( P ^ 3 ) )  e.  CC )
2827negcld 9706 . . . . 5  |-  ( ph  -> 
-u ( 2  x.  ( P ^ 3 ) )  e.  CC )
29 2nn0 10596 . . . . . . . 8  |-  2  e.  NN0
30 7nn 10484 . . . . . . . 8  |-  7  e.  NN
3129, 30decnncl 10768 . . . . . . 7  |- ; 2 7  e.  NN
3231nncni 10332 . . . . . 6  |- ; 2 7  e.  CC
339simp2d 1001 . . . . . . 7  |-  ( ph  ->  Q  e.  CC )
3433sqcld 12006 . . . . . 6  |-  ( ph  ->  ( Q ^ 2 )  e.  CC )
35 mulcl 9366 . . . . . 6  |-  ( (; 2
7  e.  CC  /\  ( Q ^ 2 )  e.  CC )  -> 
(; 2 7  x.  ( Q ^ 2 ) )  e.  CC )
3632, 34, 35sylancr 663 . . . . 5  |-  ( ph  ->  (; 2 7  x.  ( Q ^ 2 ) )  e.  CC )
3728, 36subcld 9719 . . . 4  |-  ( ph  ->  ( -u ( 2  x.  ( P ^
3 ) )  -  (; 2 7  x.  ( Q ^ 2 ) ) )  e.  CC )
38 7nn0 10601 . . . . . . 7  |-  7  e.  NN0
3938, 13decnncl 10768 . . . . . 6  |- ; 7 2  e.  NN
4039nncni 10332 . . . . 5  |- ; 7 2  e.  CC
4110, 16mulcld 9406 . . . . 5  |-  ( ph  ->  ( P  x.  R
)  e.  CC )
42 mulcl 9366 . . . . 5  |-  ( (; 7
2  e.  CC  /\  ( P  x.  R
)  e.  CC )  ->  (; 7 2  x.  ( P  x.  R )
)  e.  CC )
4340, 41, 42sylancr 663 . . . 4  |-  ( ph  ->  (; 7 2  x.  ( P  x.  R )
)  e.  CC )
4437, 43addcld 9405 . . 3  |-  ( ph  ->  ( ( -u (
2  x.  ( P ^ 3 ) )  -  (; 2 7  x.  ( Q ^ 2 ) ) )  +  (; 7 2  x.  ( P  x.  R )
) )  e.  CC )
4521, 44eqeltrd 2517 . 2  |-  ( ph  ->  V  e.  CC )
46 quart.w . . 3  |-  ( ph  ->  W  =  ( sqr `  ( ( V ^
2 )  -  (
4  x.  ( U ^ 3 ) ) ) ) )
4745sqcld 12006 . . . . 5  |-  ( ph  ->  ( V ^ 2 )  e.  CC )
48 4cn 10399 . . . . . 6  |-  4  e.  CC
49 expcl 11883 . . . . . . 7  |-  ( ( U  e.  CC  /\  3  e.  NN0 )  -> 
( U ^ 3 )  e.  CC )
5020, 23, 49sylancl 662 . . . . . 6  |-  ( ph  ->  ( U ^ 3 )  e.  CC )
51 mulcl 9366 . . . . . 6  |-  ( ( 4  e.  CC  /\  ( U ^ 3 )  e.  CC )  -> 
( 4  x.  ( U ^ 3 ) )  e.  CC )
5248, 50, 51sylancr 663 . . . . 5  |-  ( ph  ->  ( 4  x.  ( U ^ 3 ) )  e.  CC )
5347, 52subcld 9719 . . . 4  |-  ( ph  ->  ( ( V ^
2 )  -  (
4  x.  ( U ^ 3 ) ) )  e.  CC )
5453sqrcld 12923 . . 3  |-  ( ph  ->  ( sqr `  (
( V ^ 2 )  -  ( 4  x.  ( U ^
3 ) ) ) )  e.  CC )
5546, 54eqeltrd 2517 . 2  |-  ( ph  ->  W  e.  CC )
5620, 45, 553jca 1168 1  |-  ( ph  ->  ( U  e.  CC  /\  V  e.  CC  /\  W  e.  CC )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 965    = wceq 1369    e. wcel 1756   ` cfv 5418  (class class class)co 6091   CCcc 9280   1c1 9283    + caddc 9285    x. cmul 9287    - cmin 9595   -ucneg 9596    / cdiv 9993   2c2 10371   3c3 10372   4c4 10373   5c5 10374   6c6 10375   7c7 10376   8c8 10377   NN0cn0 10579  ;cdc 10755   ^cexp 11865   sqrcsqr 12722
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-2nd 6578  df-recs 6832  df-rdg 6866  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-sup 7691  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-3 10381  df-4 10382  df-5 10383  df-6 10384  df-7 10385  df-8 10386  df-9 10387  df-10 10388  df-n0 10580  df-z 10647  df-dec 10756  df-uz 10862  df-rp 10992  df-seq 11807  df-exp 11866  df-cj 12588  df-re 12589  df-im 12590  df-sqr 12724  df-abs 12725
This theorem is referenced by:  quartlem3  22254  quart  22256
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