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Theorem nmparlem 20881
Description: Lemma for nmpar 20882. (Contributed by Mario Carneiro, 7-Oct-2015.)
Hypotheses
Ref Expression
nmpar.v  |-  V  =  ( Base `  W
)
nmpar.p  |-  .+  =  ( +g  `  W )
nmpar.m  |-  .-  =  ( -g `  W )
nmpar.n  |-  N  =  ( norm `  W
)
nmpar.h  |-  .,  =  ( .i `  W )
nmpar.f  |-  F  =  (Scalar `  W )
nmpar.k  |-  K  =  ( Base `  F
)
nmpar.1  |-  ( ph  ->  W  e.  CPreHil )
nmpar.2  |-  ( ph  ->  A  e.  V )
nmpar.3  |-  ( ph  ->  B  e.  V )
Assertion
Ref Expression
nmparlem  |-  ( ph  ->  ( ( ( N `
 ( A  .+  B ) ) ^
2 )  +  ( ( N `  ( A  .-  B ) ) ^ 2 ) )  =  ( 2  x.  ( ( ( N `
 A ) ^
2 )  +  ( ( N `  B
) ^ 2 ) ) ) )

Proof of Theorem nmparlem
StepHypRef Expression
1 nmpar.h . . . . 5  |-  .,  =  ( .i `  W )
2 nmpar.v . . . . 5  |-  V  =  ( Base `  W
)
3 nmpar.p . . . . 5  |-  .+  =  ( +g  `  W )
4 nmpar.1 . . . . 5  |-  ( ph  ->  W  e.  CPreHil )
5 nmpar.2 . . . . 5  |-  ( ph  ->  A  e.  V )
6 nmpar.3 . . . . 5  |-  ( ph  ->  B  e.  V )
71, 2, 3, 4, 5, 6, 5, 6cph2di 20852 . . . 4  |-  ( ph  ->  ( ( A  .+  B )  .,  ( A  .+  B ) )  =  ( ( ( A  .,  A )  +  ( B  .,  B ) )  +  ( ( A  .,  B )  +  ( B  .,  A ) ) ) )
8 nmpar.m . . . . 5  |-  .-  =  ( -g `  W )
91, 2, 8, 4, 5, 6, 5, 6cph2subdi 20855 . . . 4  |-  ( ph  ->  ( ( A  .-  B )  .,  ( A  .-  B ) )  =  ( ( ( A  .,  A )  +  ( B  .,  B ) )  -  ( ( A  .,  B )  +  ( B  .,  A ) ) ) )
107, 9oveq12d 6213 . . 3  |-  ( ph  ->  ( ( ( A 
.+  B )  .,  ( A  .+  B ) )  +  ( ( A  .-  B ) 
.,  ( A  .-  B ) ) )  =  ( ( ( ( A  .,  A
)  +  ( B 
.,  B ) )  +  ( ( A 
.,  B )  +  ( B  .,  A
) ) )  +  ( ( ( A 
.,  A )  +  ( B  .,  B
) )  -  (
( A  .,  B
)  +  ( B 
.,  A ) ) ) ) )
11 cphclm 20835 . . . . . . 7  |-  ( W  e.  CPreHil  ->  W  e. CMod )
124, 11syl 16 . . . . . 6  |-  ( ph  ->  W  e. CMod )
13 nmpar.f . . . . . . 7  |-  F  =  (Scalar `  W )
14 nmpar.k . . . . . . 7  |-  K  =  ( Base `  F
)
1513, 14clmsscn 20778 . . . . . 6  |-  ( W  e. CMod  ->  K  C_  CC )
1612, 15syl 16 . . . . 5  |-  ( ph  ->  K  C_  CC )
17 cphphl 20817 . . . . . . . 8  |-  ( W  e.  CPreHil  ->  W  e.  PreHil )
184, 17syl 16 . . . . . . 7  |-  ( ph  ->  W  e.  PreHil )
1913, 1, 2, 14ipcl 18182 . . . . . . 7  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  A  e.  V )  ->  ( A  .,  A )  e.  K )
2018, 5, 5, 19syl3anc 1219 . . . . . 6  |-  ( ph  ->  ( A  .,  A
)  e.  K )
2113, 1, 2, 14ipcl 18182 . . . . . . 7  |-  ( ( W  e.  PreHil  /\  B  e.  V  /\  B  e.  V )  ->  ( B  .,  B )  e.  K )
2218, 6, 6, 21syl3anc 1219 . . . . . 6  |-  ( ph  ->  ( B  .,  B
)  e.  K )
2313, 14clmacl 20782 . . . . . 6  |-  ( ( W  e. CMod  /\  ( A  .,  A )  e.  K  /\  ( B 
.,  B )  e.  K )  ->  (
( A  .,  A
)  +  ( B 
.,  B ) )  e.  K )
2412, 20, 22, 23syl3anc 1219 . . . . 5  |-  ( ph  ->  ( ( A  .,  A )  +  ( B  .,  B ) )  e.  K )
2516, 24sseldd 3460 . . . 4  |-  ( ph  ->  ( ( A  .,  A )  +  ( B  .,  B ) )  e.  CC )
2613, 1, 2, 14ipcl 18182 . . . . . . 7  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  B  e.  V )  ->  ( A  .,  B )  e.  K )
2718, 5, 6, 26syl3anc 1219 . . . . . 6  |-  ( ph  ->  ( A  .,  B
)  e.  K )
2813, 1, 2, 14ipcl 18182 . . . . . . 7  |-  ( ( W  e.  PreHil  /\  B  e.  V  /\  A  e.  V )  ->  ( B  .,  A )  e.  K )
2918, 6, 5, 28syl3anc 1219 . . . . . 6  |-  ( ph  ->  ( B  .,  A
)  e.  K )
3013, 14clmacl 20782 . . . . . 6  |-  ( ( W  e. CMod  /\  ( A  .,  B )  e.  K  /\  ( B 
.,  A )  e.  K )  ->  (
( A  .,  B
)  +  ( B 
.,  A ) )  e.  K )
3112, 27, 29, 30syl3anc 1219 . . . . 5  |-  ( ph  ->  ( ( A  .,  B )  +  ( B  .,  A ) )  e.  K )
3216, 31sseldd 3460 . . . 4  |-  ( ph  ->  ( ( A  .,  B )  +  ( B  .,  A ) )  e.  CC )
3325, 32, 25ppncand 9865 . . 3  |-  ( ph  ->  ( ( ( ( A  .,  A )  +  ( B  .,  B ) )  +  ( ( A  .,  B )  +  ( B  .,  A ) ) )  +  ( ( ( A  .,  A )  +  ( B  .,  B ) )  -  ( ( A  .,  B )  +  ( B  .,  A ) ) ) )  =  ( ( ( A  .,  A
)  +  ( B 
.,  B ) )  +  ( ( A 
.,  A )  +  ( B  .,  B
) ) ) )
3410, 33eqtrd 2493 . 2  |-  ( ph  ->  ( ( ( A 
.+  B )  .,  ( A  .+  B ) )  +  ( ( A  .-  B ) 
.,  ( A  .-  B ) ) )  =  ( ( ( A  .,  A )  +  ( B  .,  B ) )  +  ( ( A  .,  A )  +  ( B  .,  B ) ) ) )
35 cphlmod 20820 . . . . . 6  |-  ( W  e.  CPreHil  ->  W  e.  LMod )
364, 35syl 16 . . . . 5  |-  ( ph  ->  W  e.  LMod )
372, 3lmodvacl 17080 . . . . 5  |-  ( ( W  e.  LMod  /\  A  e.  V  /\  B  e.  V )  ->  ( A  .+  B )  e.  V )
3836, 5, 6, 37syl3anc 1219 . . . 4  |-  ( ph  ->  ( A  .+  B
)  e.  V )
39 nmpar.n . . . . 5  |-  N  =  ( norm `  W
)
402, 1, 39nmsq 20840 . . . 4  |-  ( ( W  e.  CPreHil  /\  ( A  .+  B )  e.  V )  ->  (
( N `  ( A  .+  B ) ) ^ 2 )  =  ( ( A  .+  B )  .,  ( A  .+  B ) ) )
414, 38, 40syl2anc 661 . . 3  |-  ( ph  ->  ( ( N `  ( A  .+  B ) ) ^ 2 )  =  ( ( A 
.+  B )  .,  ( A  .+  B ) ) )
422, 8lmodvsubcl 17108 . . . . 5  |-  ( ( W  e.  LMod  /\  A  e.  V  /\  B  e.  V )  ->  ( A  .-  B )  e.  V )
4336, 5, 6, 42syl3anc 1219 . . . 4  |-  ( ph  ->  ( A  .-  B
)  e.  V )
442, 1, 39nmsq 20840 . . . 4  |-  ( ( W  e.  CPreHil  /\  ( A  .-  B )  e.  V )  ->  (
( N `  ( A  .-  B ) ) ^ 2 )  =  ( ( A  .-  B )  .,  ( A  .-  B ) ) )
454, 43, 44syl2anc 661 . . 3  |-  ( ph  ->  ( ( N `  ( A  .-  B ) ) ^ 2 )  =  ( ( A 
.-  B )  .,  ( A  .-  B ) ) )
4641, 45oveq12d 6213 . 2  |-  ( ph  ->  ( ( ( N `
 ( A  .+  B ) ) ^
2 )  +  ( ( N `  ( A  .-  B ) ) ^ 2 ) )  =  ( ( ( A  .+  B ) 
.,  ( A  .+  B ) )  +  ( ( A  .-  B )  .,  ( A  .-  B ) ) ) )
472, 1, 39nmsq 20840 . . . . . 6  |-  ( ( W  e.  CPreHil  /\  A  e.  V )  ->  (
( N `  A
) ^ 2 )  =  ( A  .,  A ) )
484, 5, 47syl2anc 661 . . . . 5  |-  ( ph  ->  ( ( N `  A ) ^ 2 )  =  ( A 
.,  A ) )
492, 1, 39nmsq 20840 . . . . . 6  |-  ( ( W  e.  CPreHil  /\  B  e.  V )  ->  (
( N `  B
) ^ 2 )  =  ( B  .,  B ) )
504, 6, 49syl2anc 661 . . . . 5  |-  ( ph  ->  ( ( N `  B ) ^ 2 )  =  ( B 
.,  B ) )
5148, 50oveq12d 6213 . . . 4  |-  ( ph  ->  ( ( ( N `
 A ) ^
2 )  +  ( ( N `  B
) ^ 2 ) )  =  ( ( A  .,  A )  +  ( B  .,  B ) ) )
5251oveq2d 6211 . . 3  |-  ( ph  ->  ( 2  x.  (
( ( N `  A ) ^ 2 )  +  ( ( N `  B ) ^ 2 ) ) )  =  ( 2  x.  ( ( A 
.,  A )  +  ( B  .,  B
) ) ) )
53252timesd 10673 . . 3  |-  ( ph  ->  ( 2  x.  (
( A  .,  A
)  +  ( B 
.,  B ) ) )  =  ( ( ( A  .,  A
)  +  ( B 
.,  B ) )  +  ( ( A 
.,  A )  +  ( B  .,  B
) ) ) )
5452, 53eqtrd 2493 . 2  |-  ( ph  ->  ( 2  x.  (
( ( N `  A ) ^ 2 )  +  ( ( N `  B ) ^ 2 ) ) )  =  ( ( ( A  .,  A
)  +  ( B 
.,  B ) )  +  ( ( A 
.,  A )  +  ( B  .,  B
) ) ) )
5534, 46, 543eqtr4d 2503 1  |-  ( ph  ->  ( ( ( N `
 ( A  .+  B ) ) ^
2 )  +  ( ( N `  ( A  .-  B ) ) ^ 2 ) )  =  ( 2  x.  ( ( ( N `
 A ) ^
2 )  +  ( ( N `  B
) ^ 2 ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758    C_ wss 3431   ` cfv 5521  (class class class)co 6195   CCcc 9386    + caddc 9391    x. cmul 9393    - cmin 9701   2c2 10477   ^cexp 11977   Basecbs 14287   +g cplusg 14352  Scalarcsca 14355   .icip 14357   -gcsg 15527   LModclmod 17066   PreHilcphl 18173   normcnm 20296  CModcclm 20761   CPreHilccph 20812
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-cnex 9444  ax-resscn 9445  ax-1cn 9446  ax-icn 9447  ax-addcl 9448  ax-addrcl 9449  ax-mulcl 9450  ax-mulrcl 9451  ax-mulcom 9452  ax-addass 9453  ax-mulass 9454  ax-distr 9455  ax-i2m1 9456  ax-1ne0 9457  ax-1rid 9458  ax-rnegex 9459  ax-rrecex 9460  ax-cnre 9461  ax-pre-lttri 9462  ax-pre-lttrn 9463  ax-pre-ltadd 9464  ax-pre-mulgt0 9465  ax-pre-sup 9466  ax-addf 9467  ax-mulf 9468
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-reu 2803  df-rmo 2804  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-int 4232  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-om 6582  df-1st 6682  df-2nd 6683  df-tpos 6850  df-recs 6937  df-rdg 6971  df-1o 7025  df-oadd 7029  df-er 7206  df-map 7321  df-en 7416  df-dom 7417  df-sdom 7418  df-fin 7419  df-sup 7797  df-pnf 9526  df-mnf 9527  df-xr 9528  df-ltxr 9529  df-le 9530  df-sub 9703  df-neg 9704  df-div 10100  df-nn 10429  df-2 10486  df-3 10487  df-4 10488  df-5 10489  df-6 10490  df-7 10491  df-8 10492  df-9 10493  df-10 10494  df-n0 10686  df-z 10753  df-dec 10862  df-uz 10968  df-rp 11098  df-fz 11550  df-seq 11919  df-exp 11978  df-cj 12701  df-re 12702  df-im 12703  df-sqr 12837  df-abs 12838  df-struct 14289  df-ndx 14290  df-slot 14291  df-base 14292  df-sets 14293  df-ress 14294  df-plusg 14365  df-mulr 14366  df-starv 14367  df-sca 14368  df-vsca 14369  df-ip 14370  df-tset 14371  df-ple 14372  df-ds 14374  df-unif 14375  df-0g 14494  df-mnd 15529  df-mhm 15578  df-grp 15659  df-minusg 15660  df-sbg 15661  df-subg 15792  df-ghm 15859  df-cmn 16395  df-abl 16396  df-mgp 16709  df-ur 16721  df-rng 16765  df-cring 16766  df-oppr 16833  df-dvdsr 16851  df-unit 16852  df-rnghom 16924  df-drng 16952  df-subrg 16981  df-staf 17048  df-srng 17049  df-lmod 17068  df-lmhm 17221  df-lvec 17302  df-sra 17371  df-rgmod 17372  df-cnfld 17939  df-phl 18175  df-nlm 20306  df-clm 20762  df-cph 20814
This theorem is referenced by:  nmpar  20882
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