Theorem nvs 8290

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

Step	Hyp	Ref	Expression
1		opreq2 3969	. . . . . . 7 |- (x = B -> (ySx) = (ySB))
2	1	fveq2d 3728	. . . . . 6 |- (x = B -> (N` (ySx)) = (N` (ySB)))
3		fveq2 3724	. . . . . . 7 |- (x = B -> (N` x) = (N` B))
4	3	opreq2d 3976	. . . . . 6 |- (x = B -> ((abs` y) x. (N` x)) = ((abs` y) x. (N` B)))
5	2, 4	eqeq12d 1489	. . . . 5 |- (x = B -> ((N` (ySx)) = ((abs` y) x. (N` x)) <-> (N` (ySB)) = ((abs` y) x. (N` B))))
6		opreq1 3968	. . . . . . 7 |- (y = A -> (ySB) = (ASB))
7	6	fveq2d 3728	. . . . . 6 |- (y = A -> (N` (ySB)) = (N` (ASB)))
8		fveq2 3724	. . . . . . 7 |- (y = A -> (abs` y) = (abs` A))
9	8	opreq1d 3975	. . . . . 6 |- (y = A -> ((abs` y) x. (N` B)) = ((abs` A) x. (N` B)))
10	7, 9	eqeq12d 1489	. . . . 5 |- (y = A -> ((N` (ySB)) = ((abs` y) x. (N` B)) <-> (N` (ASB)) = ((abs` A) x. (N` B))))
11	5, 10	rcla42v 1880	. . . 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)
17	12, 13, 14, 15, 16	nvi 8233	. . . . 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 790	. . . . 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 789	. . . . . 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)))
20	19	r19.20si 1706	. . . . 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)))
21	17, 18, 20	3syl 20	. . . 4 |- (U e. NrmCVec -> A.x e. X A.y e. CC (N` (ySx)) = ((abs` y) x. (N` x)))
22	11, 21	syl5 21	. . 3 |- ((B e. X /\ A e. CC) -> (U e. NrmCVec -> (N` (ASB)) = ((abs` A) x. (N` B))))
23	22	3impia 830	. 2 |- ((B e. X /\ A e. CC /\ U e. NrmCVec) -> (N` (ASB)) = ((abs` A) x. (N` B)))
24	23	3com13 838	1 |- ((U e. NrmCVec /\ A e. CC /\ B e. X) -> (N` (ASB)) = ((abs` A) x. (N` B)))

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 775   = wceq 956   e. wcel 958  A.wral 1645  <.cop 2411   class class class wbr 2619  -->wf 3178  ` cfv 3182  (class class class)co 3963  CCcc 5232  RRcr 5233  0cc0 5234   + caddc 5237   x. cmul 5239   <_ cle 5295  abscabs 6750  CVeccvc 8164  NrmCVeccnv 8203  +vcpv 8204  Basecba 8205  .scns 8206  0vcn0v 8207  normcnm 8209

This theorem is referenced by:  nvsge0 8291  nvm1 8292  nvpi 8294  nvmtri 8299  sm1cnilem 8347  ipid 8363

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-rab 1652  df-v 1812  df-sbc 1942  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-fo 3196  df-fv 3198  df-opr 3965  df-oprab 3966  df-1st 4079  df-2nd 4080  df-grp 8037  df-gid 8038  df-nv 8211  df-va 8214  df-ba 8215  df-sm 8216  df-0v 8217  df-nm 8219

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