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Theorem nmparlem 18669
Description: Lemma for nmpar 18670. (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 18642 . . . 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 18645 . . . 4  |-  ( ph  ->  ( ( A  .-  B )  .,  ( A  .-  B ) )  =  ( ( ( A  .,  A )  +  ( B  .,  B ) )  -  ( ( A  .,  B )  +  ( B  .,  A ) ) ) )
107, 9oveq12d 5876 . . 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 18625 . . . . . . 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 18577 . . . . . 6  |-  ( W  e. CMod  ->  K  C_  CC )
1612, 15syl 15 . . . . 5  |-  ( ph  ->  K  C_  CC )
17 cphphl 18607 . . . . . . . 8  |-  ( W  e.  CPreHil  ->  W  e.  PreHil )
184, 17syl 15 . . . . . . 7  |-  ( ph  ->  W  e.  PreHil )
1913, 1, 2, 14ipcl 16537 . . . . . . 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 16537 . . . . . . 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 18581 . . . . . 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 3181 . . . 4  |-  ( ph  ->  ( ( A  .,  A )  +  ( B  .,  B ) )  e.  CC )
2613, 1, 2, 14ipcl 16537 . . . . . . 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 16537 . . . . . . 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 18581 . . . . . 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 3181 . . . 4  |-  ( ph  ->  ( ( A  .,  B )  +  ( B  .,  A ) )  e.  CC )
3325, 32, 25ppncand 9197 . . 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 2315 . 2  |-  ( ph  ->  ( ( ( A 
.+  B )  .,  ( A  .+  B ) )  +  ( ( A  .-  B ) 
.,  ( A  .-  B ) ) )  =  ( ( ( A  .,  A )  +  ( B  .,  B ) )  +  ( ( A  .,  A )  +  ( B  .,  B ) ) ) )
35 cphlmod 18610 . . . . . 6  |-  ( W  e.  CPreHil  ->  W  e.  LMod )
364, 35syl 15 . . . . 5  |-  ( ph  ->  W  e.  LMod )
372, 3lmodvacl 15641 . . . . 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 18630 . . . 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 15670 . . . . 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 18630 . . . 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 5876 . 2  |-  ( ph  ->  ( ( ( N `
 ( A  .+  B ) ) ^
2 )  +  ( ( N `  ( A  .-  B ) ) ^ 2 ) )  =  ( ( ( A  .+  B ) 
.,  ( A  .+  B ) )  +  ( ( A  .-  B )  .,  ( A  .-  B ) ) ) )
472, 1, 39nmsq 18630 . . . . . 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 18630 . . . . . 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 5876 . . . 4  |-  ( ph  ->  ( ( ( N `
 A ) ^
2 )  +  ( ( N `  B
) ^ 2 ) )  =  ( ( A  .,  A )  +  ( B  .,  B ) ) )
5251oveq2d 5874 . . 3  |-  ( ph  ->  ( 2  x.  (
( ( N `  A ) ^ 2 )  +  ( ( N `  B ) ^ 2 ) ) )  =  ( 2  x.  ( ( A 
.,  A )  +  ( B  .,  B
) ) ) )
53252timesd 9954 . . 3  |-  ( ph  ->  ( 2  x.  (
( A  .,  A
)  +  ( B 
.,  B ) ) )  =  ( ( ( A  .,  A
)  +  ( B 
.,  B ) )  +  ( ( A 
.,  A )  +  ( B  .,  B
) ) ) )
5452, 53eqtrd 2315 . 2  |-  ( ph  ->  ( 2  x.  (
( ( N `  A ) ^ 2 )  +  ( ( N `  B ) ^ 2 ) ) )  =  ( ( ( A  .,  A
)  +  ( B 
.,  B ) )  +  ( ( A 
.,  A )  +  ( B  .,  B
) ) ) )
5534, 46, 543eqtr4d 2325 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 1623    e. wcel 1684    C_ wss 3152   ` cfv 5255  (class class class)co 5858   CCcc 8735    + caddc 8740    x. cmul 8742    - cmin 9037   2c2 9795   ^cexp 11104   Basecbs 13148   +g cplusg 13208  Scalarcsca 13211   .icip 13213   -gcsg 14365   LModclmod 15627   PreHilcphl 16528   normcnm 18099  CModcclm 18560   CPreHilccph 18602
This theorem is referenced by:  nmpar  18670
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-rep 4131  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  ax-addf 8816  ax-mulf 8817
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-int 3863  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-1st 6122  df-2nd 6123  df-tpos 6234  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  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-fz 10783  df-seq 11047  df-exp 11105  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-0g 13404  df-mnd 14367  df-mhm 14415  df-grp 14489  df-minusg 14490  df-sbg 14491  df-subg 14618  df-ghm 14681  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-cring 15341  df-ur 15342  df-oppr 15405  df-dvdsr 15423  df-unit 15424  df-rnghom 15496  df-drng 15514  df-subrg 15543  df-staf 15610  df-srng 15611  df-lmod 15629  df-lmhm 15779  df-lvec 15856  df-sra 15925  df-rgmod 15926  df-cnfld 16378  df-phl 16530  df-nlm 18109  df-clm 18561  df-cph 18604
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