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Theorem nvs 8275
Description: Proportionality property of the norm of a scalar product in a normed complex vector space.
Hypotheses
Ref Expression
nvs.1 |- X = (Base` U)
nvs.4 |- S = (.s` U)
nvs.6 |- N = (norm` U)
Assertion
Ref Expression
nvs |- ((U e. NrmCVec /\ A e. CC /\ B e. X) -> (N` (ASB)) = ((abs` A) x. (N` B)))
Proof of Theorem nvs
StepHypRef Expression
1 opreq2 3966 . . . . . . 7 |- (x = B -> (ySx) = (ySB))
21fveq2d 3725 . . . . . 6 |- (x = B -> (N` (ySx)) = (N` (ySB)))
3 fveq2 3721 . . . . . . 7 |- (x = B -> (N` x) = (N` B))
43opreq2d 3973 . . . . . 6 |- (x = B -> ((abs` y) x. (N` x)) = ((abs` y) x. (N` B)))
52, 4eqeq12d 1488 . . . . 5 |- (x = B -> ((N` (ySx)) = ((abs` y) x. (N` x)) <-> (N` (ySB)) = ((abs` y) x. (N` B))))
6 opreq1 3965 . . . . . . 7 |- (y = A -> (ySB) = (ASB))
76fveq2d 3725 . . . . . 6 |- (y = A -> (N` (ySB)) = (N` (ASB)))
8 fveq2 3721 . . . . . . 7 |- (y = A -> (abs` y) = (abs` A))
98opreq1d 3972 . . . . . 6 |- (y = A -> ((abs` y) x. (N` B)) = ((abs` A) x. (N` B)))
107, 9eqeq12d 1488 . . . . 5 |- (y = A -> ((N` (ySB)) = ((abs` y) x. (N` B)) <-> (N` (ASB)) = ((abs` A) x. (N` B))))
115, 10rcla42v 1878 . . . 4 |- ((B e. X /\ A e. CC) -> (A.x e. X A.y e. CC (N` (ySx)) = ((abs` y) x. (N` x)) -> (N` (ASB)) = ((abs` A) x. (N` B))))
12 nvs.1 . . . . . 6 |- X = (Base` U)
13 eqid 1475 . . . . . 6 |- (+v` U) = (+v` U)
14 nvs.4 . . . . . 6 |- S = (.s` U)
15 eqid 1475 . . . . . 6 |- (0v` U) = (0v` U)
16 nvs.6 . . . . . 6 |- N = (norm` U)
1712, 13, 14, 15, 16nvi 8218 . . . . 5 |- (U e. NrmCVec -> (<.(+v` U), S>. e. CVec /\ N:X-->RR /\ A.x e. X (((N` x) = 0 -> x = (0v` U)) /\ A.y e. CC (N` (ySx)) = ((abs` y) x. (N` x)) /\ A.y e. X (N` (x(+v` U)y)) <_ ((N` x) + (N` y)))))
18 3simp3 789 . . . . 5 |- ((<.(+v` U), S>. e. CVec /\ N:X-->RR /\ A.x e. X (((N` x) = 0 -> x = (0v` U)) /\ A.y e. CC (N` (ySx)) = ((abs` y) x. (N` x)) /\ A.y e. X (N` (x(+v` U)y)) <_ ((N` x) + (N` y)))) -> A.x e. X (((N` x) = 0 -> x = (0v` U)) /\ A.y e. CC (N` (ySx)) = ((abs` y) x. (N` x)) /\ A.y e. X (N` (x(+v` U)y)) <_ ((N` x) + (N` y))))
19 3simp2 788 . . . . . 6 |- ((((N` x) = 0 -> x = (0v` U)) /\ A.y e. CC (N` (ySx)) = ((abs` y) x. (N` x)) /\ A.y e. X (N` (x(+v` U)y)) <_ ((N` x) + (N` y))) -> A.y e. CC (N` (ySx)) = ((abs` y) x. (N` x)))
2019r19.20si 1705 . . . . 5 |- (A.x e. X (((N` x) = 0 -> x = (0v` U)) /\ A.y e. CC (N` (ySx)) = ((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` (ySx)) = ((abs` y) x. (N` x)))
2117, 18, 203syl 20 . . . 4 |- (U e. NrmCVec -> A.x e. X A.y e. CC (N` (ySx)) = ((abs` y) x. (N` x)))
2211, 21syl5 21 . . 3 |- ((B e. X /\ A e. CC) -> (U e. NrmCVec -> (N` (ASB)) = ((abs` A) x. (N` B))))
23223impia 829 . 2 |- ((B e. X /\ A e. CC /\ U e. NrmCVec) -> (N` (ASB)) = ((abs` A) x. (N` B)))
24233com13 837 1 |- ((U e. NrmCVec /\ A e. CC /\ B e. X) -> (N` (ASB)) = ((abs` A) x. (N` B)))
Colors of variables: wff set class
Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 774   = wceq 955   e. wcel 957  A.wral 1644  <.cop 2409   class class class wbr 2616  -->wf 3175  ` cfv 3179  (class class class)co 3960  CCcc 5219  RRcr 5220  0cc0 5221   + caddc 5224   x. cmul 5226   <_ cle 5282  abscabs 6702  CVeccvc 8149  NrmCVeccnv 8188  +vcpv 8189  Basecba 8190  .scns 8191  0vcn0v 8192  normcnm 8194
This theorem is referenced by:  nvsge0 8276  nvm1 8277  nvpi 8279  nvmtri 8284  sm1cnilem 8333  ipid 8349
This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2700  ax-nul 2707  ax-pow 2739  ax-pr 2776  ax-un 2863
This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 980  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1586  df-ral 1648  df-rex 1649  df-rab 1651  df-v 1810  df-sbc 1940  df-dif 2047  df-un 2048  df-in 2049  df-ss 2051  df-nul 2279  df-pw 2400  df-sn 2410  df-pr 2411  df-op 2414  df-uni 2501  df-br 2617  df-opab 2664  df-id 2832  df-xp 3181  df-rel 3182  df-cnv 3183  df-co 3184  df-dm 3185  df-rn 3186  df-res 3187  df-ima 3188  df-fun 3189  df-fn 3190  df-f 3191  df-fo 3193  df-fv 3195  df-opr 3962  df-oprab 3963  df-1st 4076  df-2nd 4077  df-grp 8020  df-gid 8021  df-nv 8196  df-va 8199  df-ba 8200  df-sm 8201  df-0v 8202  df-nm 8204
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