Theorem ipval3 27564

Description: Expansion of the inner product value ipval 27558. (Contributed by NM, 17-Nov-2007.) (New usage is discouraged.)

Hypotheses

Ref	Expression
dipfval.1	|- X = ( BaseSet ` U )
dipfval.2	|- G = ( +v ` U )
dipfval.4	|- S = ( .sOLD ` U )
dipfval.6	|- N = ( normCV ` U )
dipfval.7	|- P = ( .iOLD ` U )
ipval3.3	|- M = ( -v ` U )

Assertion

Ref	Expression
ipval3	|- ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( A P B ) = ( ( ( ( ( N ` ( A G B ) ) ^ 2 ) - ( ( N ` ( A M B ) ) ^ 2 ) ) + ( _i x. ( ( ( N ` ( A G ( _i S B ) ) ) ^ 2 ) - ( ( N ` ( A M ( _i S B ) ) ) ^ 2 ) ) ) ) / 4 ) )

Proof of Theorem ipval3

Step	Hyp	Ref	Expression
1		dipfval.1	. . 3 |- X = ( BaseSet ` U )
2		dipfval.2	. . 3 |- G = ( +v ` U )
3		dipfval.4	. . 3 |- S = ( .sOLD ` U )
4		dipfval.6	. . 3 |- N = ( normCV ` U )
5		dipfval.7	. . 3 |- P = ( .iOLD ` U )
6	1, 2, 3, 4, 5	ipval2 27562	. 2 |- ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( A P B ) = ( ( ( ( ( N ` ( A G B ) ) ^ 2 ) - ( ( N ` ( A G ( -u 1 S B ) ) ) ^ 2 ) ) + ( _i x. ( ( ( N ` ( A G ( _i S B ) ) ) ^ 2 ) - ( ( N ` ( A G ( -u _i S B ) ) ) ^ 2 ) ) ) ) / 4 ) )
7		ipval3.3	. . . . . . . 8 |- M = ( -v ` U )
8	1, 2, 3, 7	nvmval 27497	. . . . . . 7 |- ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( A M B ) = ( A G ( -u 1 S B ) ) )
9	8	fveq2d 6195	. . . . . 6 |- ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( N ` ( A M B ) ) = ( N ` ( A G ( -u 1 S B ) ) ) )
10	9	oveq1d 6665	. . . . 5 |- ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( ( N ` ( A M B ) ) ^ 2 ) = ( ( N ` ( A G ( -u 1 S B ) ) ) ^ 2 ) )
11	10	oveq2d 6666	. . . 4 |- ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( ( ( N ` ( A G B ) ) ^ 2 ) - ( ( N ` ( A M B ) ) ^ 2 ) ) = ( ( ( N ` ( A G B ) ) ^ 2 ) - ( ( N ` ( A G ( -u 1 S B ) ) ) ^ 2 ) ) )
12		ax-icn 9995	. . . . . . . . . . . 12 |- _i e. CC
13	1, 3	nvscl 27481	. . . . . . . . . . . 12 |- ( ( U e. NrmCVec /\ _i e. CC /\ B e. X ) -> ( _i S B ) e. X )
14	12, 13	mp3an2 1412	. . . . . . . . . . 11 |- ( ( U e. NrmCVec /\ B e. X ) -> ( _i S B ) e. X )
15	14	3adant2 1080	. . . . . . . . . 10 |- ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( _i S B ) e. X )
16	1, 2, 3, 7	nvmval 27497	. . . . . . . . . 10 |- ( ( U e. NrmCVec /\ A e. X /\ ( _i S B ) e. X ) -> ( A M ( _i S B ) ) = ( A G ( -u 1 S ( _i S B ) ) ) )
17	15, 16	syld3an3 1371	. . . . . . . . 9 |- ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( A M ( _i S B ) ) = ( A G ( -u 1 S ( _i S B ) ) ) )
18	12	mulm1i 10475	. . . . . . . . . . . . 13 |- ( -u 1 x. _i ) = -u _i
19	18	oveq1i 6660	. . . . . . . . . . . 12 |- ( ( -u 1 x. _i ) S B ) = ( -u _i S B )
20		neg1cn 11124	. . . . . . . . . . . . . 14 |- -u 1 e. CC
21	1, 3	nvsass 27483	. . . . . . . . . . . . . 14 |- ( ( U e. NrmCVec /\ ( -u 1 e. CC /\ _i e. CC /\ B e. X ) ) -> ( ( -u 1 x. _i ) S B ) = ( -u 1 S ( _i S B ) ) )
22	20, 21	mp3anr1 1421	. . . . . . . . . . . . 13 |- ( ( U e. NrmCVec /\ ( _i e. CC /\ B e. X ) ) -> ( ( -u 1 x. _i ) S B ) = ( -u 1 S ( _i S B ) ) )
23	12, 22	mpanr1 719	. . . . . . . . . . . 12 |- ( ( U e. NrmCVec /\ B e. X ) -> ( ( -u 1 x. _i ) S B ) = ( -u 1 S ( _i S B ) ) )
24	19, 23	syl5reqr 2671	. . . . . . . . . . 11 |- ( ( U e. NrmCVec /\ B e. X ) -> ( -u 1 S ( _i S B ) ) = ( -u _i S B ) )
25	24	3adant2 1080	. . . . . . . . . 10 |- ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( -u 1 S ( _i S B ) ) = ( -u _i S B ) )
26	25	oveq2d 6666	. . . . . . . . 9 |- ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( A G ( -u 1 S ( _i S B ) ) ) = ( A G ( -u _i S B ) ) )
27	17, 26	eqtrd 2656	. . . . . . . 8 |- ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( A M ( _i S B ) ) = ( A G ( -u _i S B ) ) )
28	27	fveq2d 6195	. . . . . . 7 |- ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( N ` ( A M ( _i S B ) ) ) = ( N ` ( A G ( -u _i S B ) ) ) )
29	28	oveq1d 6665	. . . . . 6 |- ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( ( N ` ( A M ( _i S B ) ) ) ^ 2 ) = ( ( N ` ( A G ( -u _i S B ) ) ) ^ 2 ) )
30	29	oveq2d 6666	. . . . 5 |- ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( ( ( N ` ( A G ( _i S B ) ) ) ^ 2 ) - ( ( N ` ( A M ( _i S B ) ) ) ^ 2 ) ) = ( ( ( N ` ( A G ( _i S B ) ) ) ^ 2 ) - ( ( N ` ( A G ( -u _i S B ) ) ) ^ 2 ) ) )
31	30	oveq2d 6666	. . . 4 |- ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( _i x. ( ( ( N ` ( A G ( _i S B ) ) ) ^ 2 ) - ( ( N ` ( A M ( _i S B ) ) ) ^ 2 ) ) ) = ( _i x. ( ( ( N ` ( A G ( _i S B ) ) ) ^ 2 ) - ( ( N ` ( A G ( -u _i S B ) ) ) ^ 2 ) ) ) )
32	11, 31	oveq12d 6668	. . 3 |- ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( ( ( ( N ` ( A G B ) ) ^ 2 ) - ( ( N ` ( A M B ) ) ^ 2 ) ) + ( _i x. ( ( ( N ` ( A G ( _i S B ) ) ) ^ 2 ) - ( ( N ` ( A M ( _i S B ) ) ) ^ 2 ) ) ) ) = ( ( ( ( N ` ( A G B ) ) ^ 2 ) - ( ( N ` ( A G ( -u 1 S B ) ) ) ^ 2 ) ) + ( _i x. ( ( ( N ` ( A G ( _i S B ) ) ) ^ 2 ) - ( ( N ` ( A G ( -u _i S B ) ) ) ^ 2 ) ) ) ) )
33	32	oveq1d 6665	. 2 |- ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( ( ( ( ( N ` ( A G B ) ) ^ 2 ) - ( ( N ` ( A M B ) ) ^ 2 ) ) + ( _i x. ( ( ( N ` ( A G ( _i S B ) ) ) ^ 2 ) - ( ( N ` ( A M ( _i S B ) ) ) ^ 2 ) ) ) ) / 4 ) = ( ( ( ( ( N ` ( A G B ) ) ^ 2 ) - ( ( N ` ( A G ( -u 1 S B ) ) ) ^ 2 ) ) + ( _i x. ( ( ( N ` ( A G ( _i S B ) ) ) ^ 2 ) - ( ( N ` ( A G ( -u _i S B ) ) ) ^ 2 ) ) ) ) / 4 ) )
34	6, 33	eqtr4d 2659	1 |- ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( A P B ) = ( ( ( ( ( N ` ( A G B ) ) ^ 2 ) - ( ( N ` ( A M B ) ) ^ 2 ) ) + ( _i x. ( ( ( N ` ( A G ( _i S B ) ) ) ^ 2 ) - ( ( N ` ( A M ( _i S B ) ) ) ^ 2 ) ) ) ) / 4 ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   ` cfv 5888  (class class class)co 6650   CCcc 9934   1c1 9937   _ici 9938   + caddc 9939   x. cmul 9941   - cmin 10266   -ucneg 10267   / cdiv 10684   2c2 11070   4c4 11072   ^cexp 12860   NrmCVeccnv 27439   +vcpv 27440   BaseSetcba 27441   .sOLDcns 27442   -vcnsb 27444   norm_CVcnmcv 27445   .iOLDcdip 27555

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417  df-grpo 27347  df-gid 27348  df-ginv 27349  df-gdiv 27350  df-ablo 27399  df-vc 27414  df-nv 27447  df-va 27450  df-ba 27451  df-sm 27452  df-0v 27453  df-vs 27454  df-nmcv 27455  df-dip 27556

This theorem is referenced by:  hhip  28034