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Theorem nmparlem 18685
Description: Lemma for nmpar 18686. (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 18658 . . . 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 18661 . . . 4  |-  ( ph  ->  ( ( A  .-  B )  .,  ( A  .-  B ) )  =  ( ( ( A  .,  A )  +  ( B  .,  B ) )  -  ( ( A  .,  B )  +  ( B  .,  A ) ) ) )
107, 9oveq12d 5892 . . 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 18641 . . . . . . 7  |-  ( W  e.  CPreHil  ->  W  e. CMod )
124, 11syl 15 . . . . . 6  |-  ( ph  ->  W  e. CMod )
13 nmpar.f . . . . . . 7  |-  F  =  (Scalar `  W )
14 nmpar.k . . . . . . 7  |-  K  =  ( Base `  F
)
1513, 14clmsscn 18593 . . . . . 6  |-  ( W  e. CMod  ->  K  C_  CC )
1612, 15syl 15 . . . . 5  |-  ( ph  ->  K  C_  CC )
17 cphphl 18623 . . . . . . . 8  |-  ( W  e.  CPreHil  ->  W  e.  PreHil )
184, 17syl 15 . . . . . . 7  |-  ( ph  ->  W  e.  PreHil )
1913, 1, 2, 14ipcl 16553 . . . . . . 7  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  A  e.  V )  ->  ( A  .,  A )  e.  K )
2018, 5, 5, 19syl3anc 1182 . . . . . 6  |-  ( ph  ->  ( A  .,  A
)  e.  K )
2113, 1, 2, 14ipcl 16553 . . . . . . 7  |-  ( ( W  e.  PreHil  /\  B  e.  V  /\  B  e.  V )  ->  ( B  .,  B )  e.  K )
2218, 6, 6, 21syl3anc 1182 . . . . . 6  |-  ( ph  ->  ( B  .,  B
)  e.  K )
2313, 14clmacl 18597 . . . . . 6  |-  ( ( W  e. CMod  /\  ( A  .,  A )  e.  K  /\  ( B 
.,  B )  e.  K )  ->  (
( A  .,  A
)  +  ( B 
.,  B ) )  e.  K )
2412, 20, 22, 23syl3anc 1182 . . . . 5  |-  ( ph  ->  ( ( A  .,  A )  +  ( B  .,  B ) )  e.  K )
2516, 24sseldd 3194 . . . 4  |-  ( ph  ->  ( ( A  .,  A )  +  ( B  .,  B ) )  e.  CC )
2613, 1, 2, 14ipcl 16553 . . . . . . 7  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  B  e.  V )  ->  ( A  .,  B )  e.  K )
2718, 5, 6, 26syl3anc 1182 . . . . . 6  |-  ( ph  ->  ( A  .,  B
)  e.  K )
2813, 1, 2, 14ipcl 16553 . . . . . . 7  |-  ( ( W  e.  PreHil  /\  B  e.  V  /\  A  e.  V )  ->  ( B  .,  A )  e.  K )
2918, 6, 5, 28syl3anc 1182 . . . . . 6  |-  ( ph  ->  ( B  .,  A
)  e.  K )
3013, 14clmacl 18597 . . . . . 6  |-  ( ( W  e. CMod  /\  ( A  .,  B )  e.  K  /\  ( B 
.,  A )  e.  K )  ->  (
( A  .,  B
)  +  ( B 
.,  A ) )  e.  K )
3112, 27, 29, 30syl3anc 1182 . . . . 5  |-  ( ph  ->  ( ( A  .,  B )  +  ( B  .,  A ) )  e.  K )
3216, 31sseldd 3194 . . . 4  |-  ( ph  ->  ( ( A  .,  B )  +  ( B  .,  A ) )  e.  CC )
3325, 32, 25ppncand 9213 . . 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 2328 . 2  |-  ( ph  ->  ( ( ( A 
.+  B )  .,  ( A  .+  B ) )  +  ( ( A  .-  B ) 
.,  ( A  .-  B ) ) )  =  ( ( ( A  .,  A )  +  ( B  .,  B ) )  +  ( ( A  .,  A )  +  ( B  .,  B ) ) ) )
35 cphlmod 18626 . . . . . 6  |-  ( W  e.  CPreHil  ->  W  e.  LMod )
364, 35syl 15 . . . . 5  |-  ( ph  ->  W  e.  LMod )
372, 3lmodvacl 15657 . . . . 5  |-  ( ( W  e.  LMod  /\  A  e.  V  /\  B  e.  V )  ->  ( A  .+  B )  e.  V )
3836, 5, 6, 37syl3anc 1182 . . . 4  |-  ( ph  ->  ( A  .+  B
)  e.  V )
39 nmpar.n . . . . 5  |-  N  =  ( norm `  W
)
402, 1, 39nmsq 18646 . . . 4  |-  ( ( W  e.  CPreHil  /\  ( A  .+  B )  e.  V )  ->  (
( N `  ( A  .+  B ) ) ^ 2 )  =  ( ( A  .+  B )  .,  ( A  .+  B ) ) )
414, 38, 40syl2anc 642 . . 3  |-  ( ph  ->  ( ( N `  ( A  .+  B ) ) ^ 2 )  =  ( ( A 
.+  B )  .,  ( A  .+  B ) ) )
422, 8lmodvsubcl 15686 . . . . 5  |-  ( ( W  e.  LMod  /\  A  e.  V  /\  B  e.  V )  ->  ( A  .-  B )  e.  V )
4336, 5, 6, 42syl3anc 1182 . . . 4  |-  ( ph  ->  ( A  .-  B
)  e.  V )
442, 1, 39nmsq 18646 . . . 4  |-  ( ( W  e.  CPreHil  /\  ( A  .-  B )  e.  V )  ->  (
( N `  ( A  .-  B ) ) ^ 2 )  =  ( ( A  .-  B )  .,  ( A  .-  B ) ) )
454, 43, 44syl2anc 642 . . 3  |-  ( ph  ->  ( ( N `  ( A  .-  B ) ) ^ 2 )  =  ( ( A 
.-  B )  .,  ( A  .-  B ) ) )
4641, 45oveq12d 5892 . 2  |-  ( ph  ->  ( ( ( N `
 ( A  .+  B ) ) ^
2 )  +  ( ( N `  ( A  .-  B ) ) ^ 2 ) )  =  ( ( ( A  .+  B ) 
.,  ( A  .+  B ) )  +  ( ( A  .-  B )  .,  ( A  .-  B ) ) ) )
472, 1, 39nmsq 18646 . . . . . 6  |-  ( ( W  e.  CPreHil  /\  A  e.  V )  ->  (
( N `  A
) ^ 2 )  =  ( A  .,  A ) )
484, 5, 47syl2anc 642 . . . . 5  |-  ( ph  ->  ( ( N `  A ) ^ 2 )  =  ( A 
.,  A ) )
492, 1, 39nmsq 18646 . . . . . 6  |-  ( ( W  e.  CPreHil  /\  B  e.  V )  ->  (
( N `  B
) ^ 2 )  =  ( B  .,  B ) )
504, 6, 49syl2anc 642 . . . . 5  |-  ( ph  ->  ( ( N `  B ) ^ 2 )  =  ( B 
.,  B ) )
5148, 50oveq12d 5892 . . . 4  |-  ( ph  ->  ( ( ( N `
 A ) ^
2 )  +  ( ( N `  B
) ^ 2 ) )  =  ( ( A  .,  A )  +  ( B  .,  B ) ) )
5251oveq2d 5890 . . 3  |-  ( ph  ->  ( 2  x.  (
( ( N `  A ) ^ 2 )  +  ( ( N `  B ) ^ 2 ) ) )  =  ( 2  x.  ( ( A 
.,  A )  +  ( B  .,  B
) ) ) )
53252timesd 9970 . . 3  |-  ( ph  ->  ( 2  x.  (
( A  .,  A
)  +  ( B 
.,  B ) ) )  =  ( ( ( A  .,  A
)  +  ( B 
.,  B ) )  +  ( ( A 
.,  A )  +  ( B  .,  B
) ) ) )
5452, 53eqtrd 2328 . 2  |-  ( ph  ->  ( 2  x.  (
( ( N `  A ) ^ 2 )  +  ( ( N `  B ) ^ 2 ) ) )  =  ( ( ( A  .,  A
)  +  ( B 
.,  B ) )  +  ( ( A 
.,  A )  +  ( B  .,  B
) ) ) )
5534, 46, 543eqtr4d 2338 1  |-  ( ph  ->  ( ( ( N `
 ( A  .+  B ) ) ^
2 )  +  ( ( N `  ( A  .-  B ) ) ^ 2 ) )  =  ( 2  x.  ( ( ( N `
 A ) ^
2 )  +  ( ( N `  B
) ^ 2 ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696    C_ wss 3165   ` cfv 5271  (class class class)co 5874   CCcc 8751    + caddc 8756    x. cmul 8758    - cmin 9053   2c2 9811   ^cexp 11120   Basecbs 13164   +g cplusg 13224  Scalarcsca 13227   .icip 13229   -gcsg 14381   LModclmod 15643   PreHilcphl 16544   normcnm 18115  CModcclm 18576   CPreHilccph 18618
This theorem is referenced by:  nmpar  18686
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832  ax-mulf 8833
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-tpos 6250  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-rp 10371  df-fz 10799  df-seq 11063  df-exp 11121  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-0g 13420  df-mnd 14383  df-mhm 14431  df-grp 14505  df-minusg 14506  df-sbg 14507  df-subg 14634  df-ghm 14697  df-cmn 15107  df-abl 15108  df-mgp 15342  df-rng 15356  df-cring 15357  df-ur 15358  df-oppr 15421  df-dvdsr 15439  df-unit 15440  df-rnghom 15512  df-drng 15530  df-subrg 15559  df-staf 15626  df-srng 15627  df-lmod 15645  df-lmhm 15795  df-lvec 15872  df-sra 15941  df-rgmod 15942  df-cnfld 16394  df-phl 16546  df-nlm 18125  df-clm 18577  df-cph 18620
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