MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  quartlem2 Unicode version

Theorem quartlem2 20154
Description: Closure lemmas for quart 20157. (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 20150 . . . . . 6  |-  ( ph  ->  ( P  e.  CC  /\  Q  e.  CC  /\  R  e.  CC )
)
109simp1d 967 . . . . 5  |-  ( ph  ->  P  e.  CC )
1110sqcld 11243 . . . 4  |-  ( ph  ->  ( P ^ 2 )  e.  CC )
12 1nn0 9981 . . . . . . 7  |-  1  e.  NN0
13 2nn 9877 . . . . . . 7  |-  2  e.  NN
1412, 13decnncl 10137 . . . . . 6  |- ; 1 2  e.  NN
1514nncni 9756 . . . . 5  |- ; 1 2  e.  CC
169simp3d 969 . . . . 5  |-  ( ph  ->  R  e.  CC )
17 mulcl 8821 . . . . 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 8854 . . 3  |-  ( ph  ->  ( ( P ^
2 )  +  (; 1
2  x.  R ) )  e.  CC )
201, 19eqeltrd 2357 . 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 9816 . . . . . . 7  |-  2  e.  CC
23 3nn0 9983 . . . . . . . 8  |-  3  e.  NN0
24 expcl 11121 . . . . . . . 8  |-  ( ( P  e.  CC  /\  3  e.  NN0 )  -> 
( P ^ 3 )  e.  CC )
2510, 23, 24sylancl 643 . . . . . . 7  |-  ( ph  ->  ( P ^ 3 )  e.  CC )
26 mulcl 8821 . . . . . . 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 9144 . . . . 5  |-  ( ph  -> 
-u ( 2  x.  ( P ^ 3 ) )  e.  CC )
29 2nn0 9982 . . . . . . . 8  |-  2  e.  NN0
30 7nn 9882 . . . . . . . 8  |-  7  e.  NN
3129, 30decnncl 10137 . . . . . . 7  |- ; 2 7  e.  NN
3231nncni 9756 . . . . . 6  |- ; 2 7  e.  CC
339simp2d 968 . . . . . . 7  |-  ( ph  ->  Q  e.  CC )
3433sqcld 11243 . . . . . 6  |-  ( ph  ->  ( Q ^ 2 )  e.  CC )
35 mulcl 8821 . . . . . 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 9157 . . . 4  |-  ( ph  ->  ( -u ( 2  x.  ( P ^
3 ) )  -  (; 2 7  x.  ( Q ^ 2 ) ) )  e.  CC )
38 7nn0 9987 . . . . . . 7  |-  7  e.  NN0
3938, 13decnncl 10137 . . . . . 6  |- ; 7 2  e.  NN
4039nncni 9756 . . . . 5  |- ; 7 2  e.  CC
4110, 16mulcld 8855 . . . . 5  |-  ( ph  ->  ( P  x.  R
)  e.  CC )
42 mulcl 8821 . . . . 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 8854 . . 3  |-  ( ph  ->  ( ( -u (
2  x.  ( P ^ 3 ) )  -  (; 2 7  x.  ( Q ^ 2 ) ) )  +  (; 7 2  x.  ( P  x.  R )
) )  e.  CC )
4521, 44eqeltrd 2357 . 2  |-  ( ph  ->  V  e.  CC )
46 quart.w . . 3  |-  ( ph  ->  W  =  ( sqr `  ( ( V ^
2 )  -  (
4  x.  ( U ^ 3 ) ) ) ) )
4745sqcld 11243 . . . . 5  |-  ( ph  ->  ( V ^ 2 )  e.  CC )
48 4cn 9820 . . . . . 6  |-  4  e.  CC
49 expcl 11121 . . . . . . 7  |-  ( ( U  e.  CC  /\  3  e.  NN0 )  -> 
( U ^ 3 )  e.  CC )
5020, 23, 49sylancl 643 . . . . . 6  |-  ( ph  ->  ( U ^ 3 )  e.  CC )
51 mulcl 8821 . . . . . 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 9157 . . . 4  |-  ( ph  ->  ( ( V ^
2 )  -  (
4  x.  ( U ^ 3 ) ) )  e.  CC )
5453sqrcld 11919 . . 3  |-  ( ph  ->  ( sqr `  (
( V ^ 2 )  -  ( 4  x.  ( U ^
3 ) ) ) )  e.  CC )
5546, 54eqeltrd 2357 . 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 1623    e. wcel 1684   ` cfv 5255  (class class class)co 5858   CCcc 8735   1c1 8738    + caddc 8740    x. cmul 8742    - cmin 9037   -ucneg 9038    / cdiv 9423   2c2 9795   3c3 9796   4c4 9797   5c5 9798   6c6 9799   7c7 9800   8c8 9801   NN0cn0 9965  ;cdc 10124   ^cexp 11104   sqrcsqr 11718
This theorem is referenced by:  quartlem3  20155  quart  20157
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-rp 10355  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721
  Copyright terms: Public domain W3C validator