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Theorem nvtri 29911
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 𝑋 = (BaseSetβ€˜π‘ˆ)
nvtri.2 𝐺 = ( +𝑣 β€˜π‘ˆ)
nvtri.6 𝑁 = (normCVβ€˜π‘ˆ)
Assertion
Ref Expression
nvtri ((π‘ˆ ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (π‘β€˜(𝐴𝐺𝐡)) ≀ ((π‘β€˜π΄) + (π‘β€˜π΅)))

Proof of Theorem nvtri
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nvtri.1 . . . . . . 7 𝑋 = (BaseSetβ€˜π‘ˆ)
2 nvtri.2 . . . . . . 7 𝐺 = ( +𝑣 β€˜π‘ˆ)
3 eqid 2733 . . . . . . . . 9 ( ·𝑠OLD β€˜π‘ˆ) = ( ·𝑠OLD β€˜π‘ˆ)
43smfval 29846 . . . . . . . 8 ( ·𝑠OLD β€˜π‘ˆ) = (2nd β€˜(1st β€˜π‘ˆ))
54eqcomi 2742 . . . . . . 7 (2nd β€˜(1st β€˜π‘ˆ)) = ( ·𝑠OLD β€˜π‘ˆ)
6 eqid 2733 . . . . . . 7 (0vecβ€˜π‘ˆ) = (0vecβ€˜π‘ˆ)
7 nvtri.6 . . . . . . 7 𝑁 = (normCVβ€˜π‘ˆ)
81, 2, 5, 6, 7nvi 29855 . . . . . 6 (π‘ˆ ∈ NrmCVec β†’ (⟨𝐺, (2nd β€˜(1st β€˜π‘ˆ))⟩ ∈ CVecOLD ∧ 𝑁:π‘‹βŸΆβ„ ∧ βˆ€π‘₯ ∈ 𝑋 (((π‘β€˜π‘₯) = 0 β†’ π‘₯ = (0vecβ€˜π‘ˆ)) ∧ βˆ€π‘¦ ∈ β„‚ (π‘β€˜(𝑦(2nd β€˜(1st β€˜π‘ˆ))π‘₯)) = ((absβ€˜π‘¦) Β· (π‘β€˜π‘₯)) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯𝐺𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦)))))
98simp3d 1145 . . . . 5 (π‘ˆ ∈ NrmCVec β†’ βˆ€π‘₯ ∈ 𝑋 (((π‘β€˜π‘₯) = 0 β†’ π‘₯ = (0vecβ€˜π‘ˆ)) ∧ βˆ€π‘¦ ∈ β„‚ (π‘β€˜(𝑦(2nd β€˜(1st β€˜π‘ˆ))π‘₯)) = ((absβ€˜π‘¦) Β· (π‘β€˜π‘₯)) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯𝐺𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦))))
10 simp3 1139 . . . . . 6 ((((π‘β€˜π‘₯) = 0 β†’ π‘₯ = (0vecβ€˜π‘ˆ)) ∧ βˆ€π‘¦ ∈ β„‚ (π‘β€˜(𝑦(2nd β€˜(1st β€˜π‘ˆ))π‘₯)) = ((absβ€˜π‘¦) Β· (π‘β€˜π‘₯)) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯𝐺𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦))) β†’ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯𝐺𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦)))
1110ralimi 3084 . . . . 5 (βˆ€π‘₯ ∈ 𝑋 (((π‘β€˜π‘₯) = 0 β†’ π‘₯ = (0vecβ€˜π‘ˆ)) ∧ βˆ€π‘¦ ∈ β„‚ (π‘β€˜(𝑦(2nd β€˜(1st β€˜π‘ˆ))π‘₯)) = ((absβ€˜π‘¦) Β· (π‘β€˜π‘₯)) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯𝐺𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦))) β†’ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯𝐺𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦)))
129, 11syl 17 . . . 4 (π‘ˆ ∈ NrmCVec β†’ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯𝐺𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦)))
13 fvoveq1 7429 . . . . . 6 (π‘₯ = 𝐴 β†’ (π‘β€˜(π‘₯𝐺𝑦)) = (π‘β€˜(𝐴𝐺𝑦)))
14 fveq2 6889 . . . . . . 7 (π‘₯ = 𝐴 β†’ (π‘β€˜π‘₯) = (π‘β€˜π΄))
1514oveq1d 7421 . . . . . 6 (π‘₯ = 𝐴 β†’ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦)) = ((π‘β€˜π΄) + (π‘β€˜π‘¦)))
1613, 15breq12d 5161 . . . . 5 (π‘₯ = 𝐴 β†’ ((π‘β€˜(π‘₯𝐺𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦)) ↔ (π‘β€˜(𝐴𝐺𝑦)) ≀ ((π‘β€˜π΄) + (π‘β€˜π‘¦))))
17 oveq2 7414 . . . . . . 7 (𝑦 = 𝐡 β†’ (𝐴𝐺𝑦) = (𝐴𝐺𝐡))
1817fveq2d 6893 . . . . . 6 (𝑦 = 𝐡 β†’ (π‘β€˜(𝐴𝐺𝑦)) = (π‘β€˜(𝐴𝐺𝐡)))
19 fveq2 6889 . . . . . . 7 (𝑦 = 𝐡 β†’ (π‘β€˜π‘¦) = (π‘β€˜π΅))
2019oveq2d 7422 . . . . . 6 (𝑦 = 𝐡 β†’ ((π‘β€˜π΄) + (π‘β€˜π‘¦)) = ((π‘β€˜π΄) + (π‘β€˜π΅)))
2118, 20breq12d 5161 . . . . 5 (𝑦 = 𝐡 β†’ ((π‘β€˜(𝐴𝐺𝑦)) ≀ ((π‘β€˜π΄) + (π‘β€˜π‘¦)) ↔ (π‘β€˜(𝐴𝐺𝐡)) ≀ ((π‘β€˜π΄) + (π‘β€˜π΅))))
2216, 21rspc2v 3622 . . . 4 ((𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯𝐺𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦)) β†’ (π‘β€˜(𝐴𝐺𝐡)) ≀ ((π‘β€˜π΄) + (π‘β€˜π΅))))
2312, 22syl5 34 . . 3 ((𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (π‘ˆ ∈ NrmCVec β†’ (π‘β€˜(𝐴𝐺𝐡)) ≀ ((π‘β€˜π΄) + (π‘β€˜π΅))))
24233impia 1118 . 2 ((𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ π‘ˆ ∈ NrmCVec) β†’ (π‘β€˜(𝐴𝐺𝐡)) ≀ ((π‘β€˜π΄) + (π‘β€˜π΅)))
25243comr 1126 1 ((π‘ˆ ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (π‘β€˜(𝐴𝐺𝐡)) ≀ ((π‘β€˜π΄) + (π‘β€˜π΅)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062  βŸ¨cop 4634   class class class wbr 5148  βŸΆwf 6537  β€˜cfv 6541  (class class class)co 7406  1st c1st 7970  2nd c2nd 7971  β„‚cc 11105  β„cr 11106  0cc0 11107   + caddc 11110   Β· cmul 11112   ≀ cle 11246  abscabs 15178  CVecOLDcvc 29799  NrmCVeccnv 29825   +𝑣 cpv 29826  BaseSetcba 29827   ·𝑠OLD cns 29828  0veccn0v 29829  normCVcnmcv 29831
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-ov 7409  df-oprab 7410  df-1st 7972  df-2nd 7973  df-vc 29800  df-nv 29833  df-va 29836  df-ba 29837  df-sm 29838  df-0v 29839  df-nmcv 29841
This theorem is referenced by:  nvmtri  29912  nvabs  29913  nvge0  29914  imsmetlem  29931  vacn  29935
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