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Theorem nvtri 22151
 Description: Triangle inequality for the norm of a normed complex vector space. (Contributed by NM, 11-Nov-2006.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
nvtri.1
nvtri.2
nvtri.6 CV
Assertion
Ref Expression
nvtri

Proof of Theorem nvtri
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nvtri.1 . . . . . . 7
2 nvtri.2 . . . . . . 7
3 eqid 2435 . . . . . . . . 9
43smfval 22076 . . . . . . . 8
54eqcomi 2439 . . . . . . 7
6 eqid 2435 . . . . . . 7
7 nvtri.6 . . . . . . 7 CV
81, 2, 5, 6, 7nvi 22085 . . . . . 6
98simp3d 971 . . . . 5
10 simp3 959 . . . . . 6
1110ralimi 2773 . . . . 5
129, 11syl 16 . . . 4
13 oveq1 6080 . . . . . . 7
1413fveq2d 5724 . . . . . 6
15 fveq2 5720 . . . . . . 7
1615oveq1d 6088 . . . . . 6
1714, 16breq12d 4217 . . . . 5
18 oveq2 6081 . . . . . . 7
1918fveq2d 5724 . . . . . 6
20 fveq2 5720 . . . . . . 7
2120oveq2d 6089 . . . . . 6
2219, 21breq12d 4217 . . . . 5
2317, 22rspc2v 3050 . . . 4
2412, 23syl5 30 . . 3
25243impia 1150 . 2
26253comr 1161 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   w3a 936   wceq 1652   wcel 1725  wral 2697  cop 3809   class class class wbr 4204  wf 5442  cfv 5446  (class class class)co 6073  c1st 6339  c2nd 6340  cc 8980  cr 8981  cc0 8982   caddc 8985   cmul 8987   cle 9113  cabs 12031  cvc 22016  cnv 22055  cpv 22056  cba 22057  cns 22058  cn0v 22059  CVcnmcv 22061 This theorem is referenced by:  nvmtri  22152  nvmtri2  22153  nvabs  22154  nvge0  22155  imsmetlem  22174  vacn  22182 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-1st 6341  df-2nd 6342  df-vc 22017  df-nv 22063  df-va 22066  df-ba 22067  df-sm 22068  df-0v 22069  df-nmcv 22071
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