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Theorem nvs 27518
Description: Proportionality property of the norm of a scalar product in a normed complex vector space. (Contributed by NM, 11-Nov-2006.) (New usage is discouraged.)
Hypotheses
Ref Expression
nvs.1 |- X = ( BaseSet ` U )
nvs.4 |- S = ( .sOLD ` U )
nvs.6 |- N = ( normCV ` U )
Assertion
Ref Expression
nvs |- ( ( U e. NrmCVec /\ A e. CC /\ B e. X ) -> ( N ` ( A S B ) ) = ( ( abs ` A ) x. ( N ` B ) ) )
Proof of Theorem nvs
Dummy variables y x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nvs.1 . . . . . . 7 |- X = ( BaseSet ` U )
2 eqid 2622 . . . . . . 7 |- ( +v ` U ) = ( +v ` U )
3 nvs.4 . . . . . . 7 |- S = ( .sOLD ` U )
4 eqid 2622 . . . . . . 7 |- ( 0vec ` U ) = ( 0vec ` U )
5 nvs.6 . . . . . . 7 |- N = ( normCV ` U )
61, 2, 3, 4, 5nvi 27469 . . . . . 6 |- ( U e. NrmCVec -> ( <. ( +v ` U ) , S >. e. CVecOLD /\ N : X --> RR /\ A. x e. X ( ( ( N ` x ) = 0 -> x = ( 0vec ` U ) ) /\ A. y e. CC ( N ` ( y S x ) ) = ( ( abs ` y ) x. ( N ` x ) ) /\ A. y e. X ( N ` ( x ( +v ` U ) y ) ) <_ ( ( N ` x ) + ( N ` y ) ) ) ) )
76simp3d 1075 . . . . 5 |- ( U e. NrmCVec -> A. x e. X ( ( ( N ` x ) = 0 -> x = ( 0vec ` U ) ) /\ A. y e. CC ( N ` ( y S x ) ) = ( ( abs ` y ) x. ( N ` x ) ) /\ A. y e. X ( N ` ( x ( +v ` U ) y ) ) <_ ( ( N ` x ) + ( N ` y ) ) ) )
8 simp2 1062 . . . . . 6 |- ( ( ( ( N ` x ) = 0 -> x = ( 0vec ` U ) ) /\ A. y e. CC ( N ` ( y S x ) ) = ( ( abs ` y ) x. ( N ` x ) ) /\ A. y e. X ( N ` ( x ( +v ` U ) y ) ) <_ ( ( N ` x ) + ( N ` y ) ) ) -> A. y e. CC ( N ` ( y S x ) ) = ( ( abs ` y ) x. ( N ` x ) ) )
98ralimi 2952 . . . . 5 |- ( A. x e. X ( ( ( N ` x ) = 0 -> x = ( 0vec ` U ) ) /\ A. y e. CC ( N ` ( y S x ) ) = ( ( abs ` y ) x. ( N ` x ) ) /\ A. y e. X ( N ` ( x ( +v ` U ) y ) ) <_ ( ( N ` x ) + ( N ` y ) ) ) -> A. x e. X A. y e. CC ( N ` ( y S x ) ) = ( ( abs ` y ) x. ( N ` x ) ) )
107, 9syl 17 . . . 4 |- ( U e. NrmCVec -> A. x e. X A. y e. CC ( N ` ( y S x ) ) = ( ( abs ` y ) x. ( N ` x ) ) )
11 oveq2 6658 . . . . . . 7 |- ( x = B -> ( y S x ) = ( y S B ) )
1211fveq2d 6195 . . . . . 6 |- ( x = B -> ( N ` ( y S x ) ) = ( N ` ( y S B ) ) )
13 fveq2 6191 . . . . . . 7 |- ( x = B -> ( N ` x ) = ( N ` B ) )
1413oveq2d 6666 . . . . . 6 |- ( x = B -> ( ( abs ` y ) x. ( N ` x ) ) = ( ( abs ` y ) x. ( N ` B ) ) )
1512, 14eqeq12d 2637 . . . . 5 |- ( x = B -> ( ( N ` ( y S x ) ) = ( ( abs ` y ) x. ( N ` x ) ) <-> ( N ` ( y S B ) ) = ( ( abs ` y ) x. ( N ` B ) ) ) )
16 oveq1 6657 . . . . . . 7 |- ( y = A -> ( y S B ) = ( A S B ) )
1716fveq2d 6195 . . . . . 6 |- ( y = A -> ( N ` ( y S B ) ) = ( N ` ( A S B ) ) )
18 fveq2 6191 . . . . . . 7 |- ( y = A -> ( abs ` y ) = ( abs ` A ) )
1918oveq1d 6665 . . . . . 6 |- ( y = A -> ( ( abs ` y ) x. ( N ` B ) ) = ( ( abs ` A ) x. ( N ` B ) ) )
2017, 19eqeq12d 2637 . . . . 5 |- ( y = A -> ( ( N ` ( y S B ) ) = ( ( abs ` y ) x. ( N ` B ) ) <-> ( N ` ( A S B ) ) = ( ( abs ` A ) x. ( N ` B ) ) ) )
2115, 20rspc2v 3322 . . . 4 |- ( ( B e. X /\ A e. CC ) -> ( A. x e. X A. y e. CC ( N ` ( y S x ) ) = ( ( abs ` y ) x. ( N ` x ) ) -> ( N ` ( A S B ) ) = ( ( abs ` A ) x. ( N ` B ) ) ) )
2210, 21syl5 34 . . 3 |- ( ( B e. X /\ A e. CC ) -> ( U e. NrmCVec -> ( N ` ( A S B ) ) = ( ( abs ` A ) x. ( N ` B ) ) ) )
23223impia 1261 . 2 |- ( ( B e. X /\ A e. CC /\ U e. NrmCVec ) -> ( N ` ( A S B ) ) = ( ( abs ` A ) x. ( N ` B ) ) )
24233com13 1270 1 |- ( ( U e. NrmCVec /\ A e. CC /\ B e. X ) -> ( N ` ( A S B ) ) = ( ( abs ` A ) x. ( N ` B ) ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   <.cop 4183   class class class wbr 4653   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   + caddc 9939   x. cmul 9941   <_ cle 10075   abscabs 13974   CVecOLDcvc 27413   NrmCVeccnv 27439   +vcpv 27440   BaseSetcba 27441   .sOLDcns 27442   0veccn0v 27443   normCVcnmcv 27445
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
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  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-ral 2917  df-rex 2918  df-reu 2919  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-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  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-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-1st 7168  df-2nd 7169  df-vc 27414  df-nv 27447  df-va 27450  df-ba 27451  df-sm 27452  df-0v 27453  df-nmcv 27455
This theorem is referenced by:  nvsge0  27519  nvm1  27520  nvpi  27522  nvmtri  27526  smcnlem  27552  ipidsq  27565  minvecolem2  27731
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