Theorem dipcl 25826

Description: An inner product is a complex number. (Contributed by NM, 1-Feb-2007.) (Revised by Mario Carneiro, 5-May-2014.) (New usage is discouraged.)

Hypotheses

Ref	Expression
ipcl.1	|- X = ( BaseSet ` U )
ipcl.7	|- P = ( .iOLD ` U )

Assertion

Ref	Expression
dipcl	|- ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( A P B ) e. CC )

Proof of Theorem dipcl

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ipcl.1	. . 3 |- X = ( BaseSet ` U )
2		eqid 2454	. . 3 |- ( +v ` U ) = ( +v ` U )
3		eqid 2454	. . 3 |- ( .sOLD ` U ) = ( .sOLD ` U )
4		eqid 2454	. . 3 |- ( normCV ` U ) = ( normCV ` U )
5		ipcl.7	. . 3 |- P = ( .iOLD ` U )
6	1, 2, 3, 4, 5	ipval 25814	. 2 |- ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( A P B ) = ( sum_ k e. ( 1 ... 4 ) ( ( _i ^ k ) x. ( ( ( normCV ` U ) ` ( A ( +v ` U ) ( ( _i ^ k ) ( .sOLD ` U ) B ) ) ) ^ 2 ) ) / 4 ) )
7		fzfid 12068	. . . 4 |- ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( 1 ... 4 ) e. Fin )
8		ax-icn 9540	. . . . . . 7 |- _i e. CC
9		elfznn 11717	. . . . . . . 8 |- ( k e. ( 1 ... 4 ) -> k e. NN )
10	9	nnnn0d 10848	. . . . . . 7 |- ( k e. ( 1 ... 4 ) -> k e. NN0 )
11		expcl 12169	. . . . . . 7 |- ( ( _i e. CC /\ k e. NN0 ) -> ( _i ^ k ) e. CC )
12	8, 10, 11	sylancr 661	. . . . . 6 |- ( k e. ( 1 ... 4 ) -> ( _i ^ k ) e. CC )
13	12	adantl 464	. . . . 5 |- ( ( ( U e. NrmCVec /\ A e. X /\ B e. X ) /\ k e. ( 1 ... 4 ) ) -> ( _i ^ k ) e. CC )
14	1, 2, 3, 4, 5	ipval2lem4 25817	. . . . . 6 |- ( ( ( U e. NrmCVec /\ A e. X /\ B e. X ) /\ ( _i ^ k ) e. CC ) -> ( ( ( normCV ` U ) ` ( A ( +v ` U ) ( ( _i ^ k ) ( .sOLD ` U ) B ) ) ) ^ 2 ) e. CC )
15	12, 14	sylan2 472	. . . . 5 |- ( ( ( U e. NrmCVec /\ A e. X /\ B e. X ) /\ k e. ( 1 ... 4 ) ) -> ( ( ( normCV ` U ) ` ( A ( +v ` U ) ( ( _i ^ k ) ( .sOLD ` U ) B ) ) ) ^ 2 ) e. CC )
16	13, 15	mulcld 9605	. . . 4 |- ( ( ( U e. NrmCVec /\ A e. X /\ B e. X ) /\ k e. ( 1 ... 4 ) ) -> ( ( _i ^ k ) x. ( ( ( normCV ` U ) ` ( A ( +v ` U ) ( ( _i ^ k ) ( .sOLD ` U ) B ) ) ) ^ 2 ) ) e. CC )
17	7, 16	fsumcl 13640	. . 3 |- ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> sum_ k e. ( 1 ... 4 ) ( ( _i ^ k ) x. ( ( ( normCV ` U ) ` ( A ( +v ` U ) ( ( _i ^ k ) ( .sOLD ` U ) B ) ) ) ^ 2 ) ) e. CC )
18		4cn 10609	. . . 4 |- 4 e. CC
19		4ne0 10628	. . . 4 |- 4 =/= 0
20		divcl 10209	. . . 4 |- ( ( sum_ k e. ( 1 ... 4 ) ( ( _i ^ k ) x. ( ( ( normCV ` U ) ` ( A ( +v ` U ) ( ( _i ^ k ) ( .sOLD ` U ) B ) ) ) ^ 2 ) ) e. CC /\ 4 e. CC /\ 4 =/= 0 ) -> ( sum_ k e. ( 1 ... 4 ) ( ( _i ^ k ) x. ( ( ( normCV ` U ) ` ( A ( +v ` U ) ( ( _i ^ k ) ( .sOLD ` U ) B ) ) ) ^ 2 ) ) / 4 ) e. CC )
21	18, 19, 20	mp3an23 1314	. . 3 |- ( sum_ k e. ( 1 ... 4 ) ( ( _i ^ k ) x. ( ( ( normCV ` U ) ` ( A ( +v ` U ) ( ( _i ^ k ) ( .sOLD ` U ) B ) ) ) ^ 2 ) ) e. CC -> ( sum_ k e. ( 1 ... 4 ) ( ( _i ^ k ) x. ( ( ( normCV ` U ) ` ( A ( +v ` U ) ( ( _i ^ k ) ( .sOLD ` U ) B ) ) ) ^ 2 ) ) / 4 ) e. CC )
22	17, 21	syl 16	. 2 |- ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( sum_ k e. ( 1 ... 4 ) ( ( _i ^ k ) x. ( ( ( normCV ` U ) ` ( A ( +v ` U ) ( ( _i ^ k ) ( .sOLD ` U ) B ) ) ) ^ 2 ) ) / 4 ) e. CC )
23	6, 22	eqeltrd 2542	1 |- ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( A P B ) e. CC )

Syntax hints:   -> wi 4   /\ wa 367   /\ w3a 971   = wceq 1398   e. wcel 1823   =/= wne 2649   ` cfv 5570  (class class class)co 6270   CCcc 9479   0cc0 9481   1c1 9482   _ici 9483   x. cmul 9486   / cdiv 10202   2c2 10581   4c4 10583   NN0cn0 10791   ...cfz 11675   ^cexp 12151   sum_csu 13593   NrmCVeccnv 25678   +vcpv 25679   BaseSetcba 25680   .sOLDcns 25681   norm_CVcnmcv 25684   .iOLDcdip 25811

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-sup 7893  df-oi 7927  df-card 8311  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-n0 10792  df-z 10861  df-uz 11083  df-rp 11222  df-fz 11676  df-fzo 11800  df-seq 12093  df-exp 12152  df-hash 12391  df-cj 13017  df-re 13018  df-im 13019  df-sqrt 13153  df-abs 13154  df-clim 13396  df-sum 13594  df-grpo 25394  df-ablo 25485  df-vc 25640  df-nv 25686  df-va 25689  df-ba 25690  df-sm 25691  df-0v 25692  df-nmcv 25694  df-dip 25812

This theorem is referenced by:  ipf  25827  ipipcj  25829  ip1ilem  25942  ip2i  25944  ipasslem1  25947  ipasslem2  25948  ipasslem4  25950  ipasslem5  25951  ipasslem7  25952  ipasslem8  25953  ipasslem9  25954  ipasslem10  25955  ipasslem11  25956  dipdi  25959  ip2dii  25960  dipassr  25962  dipsubdir  25964  dipsubdi  25965  pythi  25966  siilem1  25967  siilem2  25968  siii  25969  ipblnfi  25972  ip2eqi  25973  htthlem  26035