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Theorem axsegconlem8 23175
Description: Lemma for axsegcon 23178. Show that a particular mapping generates a point. (Contributed by Scott Fenton, 18-Sep-2013.)
Hypotheses
Ref Expression
axsegconlem2.1  |-  S  = 
sum_ p  e.  (
1 ... N ) ( ( ( A `  p )  -  ( B `  p )
) ^ 2 )
axsegconlem7.2  |-  T  = 
sum_ p  e.  (
1 ... N ) ( ( ( C `  p )  -  ( D `  p )
) ^ 2 )
axsegconlem8.3  |-  F  =  ( k  e.  ( 1 ... N ) 
|->  ( ( ( ( ( sqr `  S
)  +  ( sqr `  T ) )  x.  ( B `  k
) )  -  (
( sqr `  T
)  x.  ( A `
 k ) ) )  /  ( sqr `  S ) ) )
Assertion
Ref Expression
axsegconlem8  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  ->  F  e.  ( EE `  N
) )
Distinct variable groups:    A, p    B, p    C, p    D, p    N, p    A, k    B, k    C, k    D, k   
k, N    S, k    T, k
Allowed substitution hints:    S( p)    T( p)    F( k, p)

Proof of Theorem axsegconlem8
StepHypRef Expression
1 axsegconlem8.3 . 2  |-  F  =  ( k  e.  ( 1 ... N ) 
|->  ( ( ( ( ( sqr `  S
)  +  ( sqr `  T ) )  x.  ( B `  k
) )  -  (
( sqr `  T
)  x.  ( A `
 k ) ) )  /  ( sqr `  S ) ) )
2 axsegconlem2.1 . . . . . . . . . . 11  |-  S  = 
sum_ p  e.  (
1 ... N ) ( ( ( A `  p )  -  ( B `  p )
) ^ 2 )
32axsegconlem4 23171 . . . . . . . . . 10  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  -> 
( sqr `  S
)  e.  RR )
433adant3 1008 . . . . . . . . 9  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N )  /\  A  =/=  B )  ->  ( sqr `  S )  e.  RR )
54ad2antrr 725 . . . . . . . 8  |-  ( ( ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  /\  k  e.  ( 1 ... N
) )  ->  ( sqr `  S )  e.  RR )
6 axsegconlem7.2 . . . . . . . . . 10  |-  T  = 
sum_ p  e.  (
1 ... N ) ( ( ( C `  p )  -  ( D `  p )
) ^ 2 )
76axsegconlem4 23171 . . . . . . . . 9  |-  ( ( C  e.  ( EE
`  N )  /\  D  e.  ( EE `  N ) )  -> 
( sqr `  T
)  e.  RR )
87ad2antlr 726 . . . . . . . 8  |-  ( ( ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  /\  k  e.  ( 1 ... N
) )  ->  ( sqr `  T )  e.  RR )
95, 8readdcld 9418 . . . . . . 7  |-  ( ( ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  /\  k  e.  ( 1 ... N
) )  ->  (
( sqr `  S
)  +  ( sqr `  T ) )  e.  RR )
10 simpl2 992 . . . . . . . 8  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  ->  B  e.  ( EE `  N
) )
11 fveere 23152 . . . . . . . 8  |-  ( ( B  e.  ( EE
`  N )  /\  k  e.  ( 1 ... N ) )  ->  ( B `  k )  e.  RR )
1210, 11sylan 471 . . . . . . 7  |-  ( ( ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  /\  k  e.  ( 1 ... N
) )  ->  ( B `  k )  e.  RR )
139, 12remulcld 9419 . . . . . 6  |-  ( ( ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  /\  k  e.  ( 1 ... N
) )  ->  (
( ( sqr `  S
)  +  ( sqr `  T ) )  x.  ( B `  k
) )  e.  RR )
14 simpl1 991 . . . . . . . 8  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  ->  A  e.  ( EE `  N
) )
15 fveere 23152 . . . . . . . 8  |-  ( ( A  e.  ( EE
`  N )  /\  k  e.  ( 1 ... N ) )  ->  ( A `  k )  e.  RR )
1614, 15sylan 471 . . . . . . 7  |-  ( ( ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  /\  k  e.  ( 1 ... N
) )  ->  ( A `  k )  e.  RR )
178, 16remulcld 9419 . . . . . 6  |-  ( ( ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  /\  k  e.  ( 1 ... N
) )  ->  (
( sqr `  T
)  x.  ( A `
 k ) )  e.  RR )
1813, 17resubcld 9781 . . . . 5  |-  ( ( ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  /\  k  e.  ( 1 ... N
) )  ->  (
( ( ( sqr `  S )  +  ( sqr `  T ) )  x.  ( B `
 k ) )  -  ( ( sqr `  T )  x.  ( A `  k )
) )  e.  RR )
192axsegconlem6 23173 . . . . . . 7  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N )  /\  A  =/=  B )  ->  0  <  ( sqr `  S
) )
2019gt0ne0d 9909 . . . . . 6  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N )  /\  A  =/=  B )  ->  ( sqr `  S )  =/=  0 )
2120ad2antrr 725 . . . . 5  |-  ( ( ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  /\  k  e.  ( 1 ... N
) )  ->  ( sqr `  S )  =/=  0 )
2218, 5, 21redivcld 10164 . . . 4  |-  ( ( ( ( A  e.  ( EE `  N
)  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  /\  k  e.  ( 1 ... N
) )  ->  (
( ( ( ( sqr `  S )  +  ( sqr `  T
) )  x.  ( B `  k )
)  -  ( ( sqr `  T )  x.  ( A `  k ) ) )  /  ( sqr `  S
) )  e.  RR )
2322ralrimiva 2804 . . 3  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  ->  A. k  e.  ( 1 ... N
) ( ( ( ( ( sqr `  S
)  +  ( sqr `  T ) )  x.  ( B `  k
) )  -  (
( sqr `  T
)  x.  ( A `
 k ) ) )  /  ( sqr `  S ) )  e.  RR )
24 eleenn 23147 . . . . 5  |-  ( D  e.  ( EE `  N )  ->  N  e.  NN )
2524ad2antll 728 . . . 4  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  ->  N  e.  NN )
26 mptelee 23146 . . . 4  |-  ( N  e.  NN  ->  (
( k  e.  ( 1 ... N ) 
|->  ( ( ( ( ( sqr `  S
)  +  ( sqr `  T ) )  x.  ( B `  k
) )  -  (
( sqr `  T
)  x.  ( A `
 k ) ) )  /  ( sqr `  S ) ) )  e.  ( EE `  N )  <->  A. k  e.  ( 1 ... N
) ( ( ( ( ( sqr `  S
)  +  ( sqr `  T ) )  x.  ( B `  k
) )  -  (
( sqr `  T
)  x.  ( A `
 k ) ) )  /  ( sqr `  S ) )  e.  RR ) )
2725, 26syl 16 . . 3  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  ->  (
( k  e.  ( 1 ... N ) 
|->  ( ( ( ( ( sqr `  S
)  +  ( sqr `  T ) )  x.  ( B `  k
) )  -  (
( sqr `  T
)  x.  ( A `
 k ) ) )  /  ( sqr `  S ) ) )  e.  ( EE `  N )  <->  A. k  e.  ( 1 ... N
) ( ( ( ( ( sqr `  S
)  +  ( sqr `  T ) )  x.  ( B `  k
) )  -  (
( sqr `  T
)  x.  ( A `
 k ) ) )  /  ( sqr `  S ) )  e.  RR ) )
2823, 27mpbird 232 . 2  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  ->  (
k  e.  ( 1 ... N )  |->  ( ( ( ( ( sqr `  S )  +  ( sqr `  T
) )  x.  ( B `  k )
)  -  ( ( sqr `  T )  x.  ( A `  k ) ) )  /  ( sqr `  S
) ) )  e.  ( EE `  N
) )
291, 28syl5eqel 2527 1  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N )  /\  A  =/=  B
)  /\  ( C  e.  ( EE `  N
)  /\  D  e.  ( EE `  N ) ) )  ->  F  e.  ( EE `  N
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2611   A.wral 2720    e. cmpt 4355   ` cfv 5423  (class class class)co 6096   RRcr 9286   0cc0 9287   1c1 9288    + caddc 9290    x. cmul 9292    - cmin 9600    / cdiv 9998   NNcn 10327   2c2 10376   ...cfz 11442   ^cexp 11870   sqrcsqr 12727   sum_csu 13168   EEcee 23139
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-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-inf2 7852  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-pre-sup 9365
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  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 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-se 4685  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-isom 5432  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-oadd 6929  df-er 7106  df-map 7221  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-sup 7696  df-oi 7729  df-card 8114  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-nn 10328  df-2 10385  df-3 10386  df-n0 10585  df-z 10652  df-uz 10867  df-rp 10997  df-ico 11311  df-fz 11443  df-fzo 11554  df-seq 11812  df-exp 11871  df-hash 12109  df-cj 12593  df-re 12594  df-im 12595  df-sqr 12729  df-abs 12730  df-clim 12971  df-sum 13169  df-ee 23142
This theorem is referenced by:  axsegconlem10  23177  axsegcon  23178
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