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Theorem nvtri 8298
Description: Triangle inequality for the norm of a normed complex vector space.
Hypotheses
Ref Expression
nvtri.1 |- X = (Base` U)
nvtri.2 |- G = (+v` U)
nvtri.6 |- N = (norm` U)
Assertion
Ref Expression
nvtri |- ((U e. NrmCVec /\ A e. X /\ B e. X) -> (N` (AGB)) <_ ((N` A) + (N` B)))
Proof of Theorem nvtri
StepHypRef Expression
1 opreq1 3968 . . . . . . 7 |- (x = A -> (xGy) = (AGy))
21fveq2d 3728 . . . . . 6 |- (x = A -> (N` (xGy)) = (N` (AGy)))
3 fveq2 3724 . . . . . . 7 |- (x = A -> (N` x) = (N` A))
43opreq1d 3975 . . . . . 6 |- (x = A -> ((N` x) + (N` y)) = ((N` A) + (N` y)))
52, 4breq12d 2631 . . . . 5 |- (x = A -> ((N` (xGy)) <_ ((N` x) + (N` y)) <-> (N` (AGy)) <_ ((N` A) + (N` y))))
6 opreq2 3969 . . . . . . 7 |- (y = B -> (AGy) = (AGB))
76fveq2d 3728 . . . . . 6 |- (y = B -> (N` (AGy)) = (N` (AGB)))
8 fveq2 3724 . . . . . . 7 |- (y = B -> (N` y) = (N` B))
98opreq2d 3976 . . . . . 6 |- (y = B -> ((N` A) + (N` y)) = ((N` A) + (N` B)))
107, 9breq12d 2631 . . . . 5 |- (y = B -> ((N` (AGy)) <_ ((N` A) + (N` y)) <-> (N` (AGB)) <_ ((N` A) + (N` B))))
115, 10rcla42v 1880 . . . 4 |- ((A e. X /\ B e. X) -> (A.x e. X A.y e. X (N` (xGy)) <_ ((N` x) + (N` y)) -> (N` (AGB)) <_ ((N` A) + (N` B))))
12 nvtri.1 . . . . . . 7 |- X = (Base` U)
13 nvtri.2 . . . . . . 7 |- G = (+v` U)
14 eqid 1475 . . . . . . . . 9 |- (.s` U) = (.s` U)
1514smfval 8224 . . . . . . . 8 |- (.s` U) = (2nd` (1st` U))
1615eqcomi 1479 . . . . . . 7 |- (2nd` (1st` U)) = (.s` U)
17 eqid 1475 . . . . . . . . 9 |- (0v` U) = (0v` U)
1813, 170vfval 8225 . . . . . . . 8 |- (0v` U) = (Id` G)
1918eqcomi 1479 . . . . . . 7 |- (Id` G) = (0v` U)
20 nvtri.6 . . . . . . 7 |- N = (norm` U)
2112, 13, 16, 19, 20nvi 8233 . . . . . 6 |- (U e. NrmCVec -> (<.G, (2nd` (1st` U))>. e. CVec /\ N:X-->RR /\ A.x e. X (((N` x) = 0 -> x = (Id` G)) /\ A.y e. CC (N` (y(2nd` (1st` U))x)) = ((abs` y) x. (N` x)) /\ A.y e. X (N` (xGy)) <_ ((N` x) + (N` y)))))
22213simp3d 796 . . . . 5 |- (U e. NrmCVec -> A.x e. X (((N` x) = 0 -> x = (Id` G)) /\ A.y e. CC (N` (y(2nd` (1st` U))x)) = ((abs` y) x. (N` x)) /\ A.y e. X (N` (xGy)) <_ ((N` x) + (N` y))))
23 3simp3 790 . . . . . 6 |- ((((N` x) = 0 -> x = (Id` G)) /\ A.y e. CC (N` (y(2nd` (1st` U))x)) = ((abs` y) x. (N` x)) /\ A.y e. X (N` (xGy)) <_ ((N` x) + (N` y))) -> A.y e. X (N` (xGy)) <_ ((N` x) + (N` y)))
2423r19.20si 1706 . . . . 5 |- (A.x e. X (((N` x) = 0 -> x = (Id` G)) /\ A.y e. CC (N` (y(2nd` (1st` U))x)) = ((abs` y) x. (N` x)) /\ A.y e. X (N` (xGy)) <_ ((N` x) + (N` y))) -> A.x e. X A.y e. X (N` (xGy)) <_ ((N` x) + (N` y)))
2522, 24syl 10 . . . 4 |- (U e. NrmCVec -> A.x e. X A.y e. X (N` (xGy)) <_ ((N` x) + (N` y)))
2611, 25syl5 21 . . 3 |- ((A e. X /\ B e. X) -> (U e. NrmCVec -> (N` (AGB)) <_ ((N` A) + (N` B))))
27263impia 830 . 2 |- ((A e. X /\ B e. X /\ U e. NrmCVec) -> (N` (AGB)) <_ ((N` A) + (N` B)))
28273comr 841 1 |- ((U e. NrmCVec /\ A e. X /\ B e. X) -> (N` (AGB)) <_ ((N` A) + (N` B)))
Colors of variables: wff set class
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  1stc1st 4077  2ndc2nd 4078  CCcc 5232  RRcr 5233  0cc0 5234   + caddc 5237   x. cmul 5239   <_ cle 5295  abscabs 6750  Idcgi 8034  CVeccvc 8164  NrmCVeccnv 8203  +vcpv 8204  Basecba 8205  .scns 8206  0vcn0v 8207  normcnm 8209
This theorem is referenced by:  nvmtri 8299  nvmtri2 8300  nvabs 8301  nvge0 8302  imsmetlem 8323
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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