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Theorem psmettri2 24170
Description: Triangle inequality for the distance function of a pseudometric. (Contributed by Thierry Arnoux, 11-Feb-2018.)
Assertion
Ref Expression
psmettri2 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐢 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ (𝐴𝐷𝐡) ≀ ((𝐢𝐷𝐴) +𝑒 (𝐢𝐷𝐡)))

Proof of Theorem psmettri2
Dummy variables π‘Ž 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elfvex 6923 . . . . . . . 8 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ 𝑋 ∈ V)
2 ispsmet 24165 . . . . . . . 8 (𝑋 ∈ V β†’ (𝐷 ∈ (PsMetβ€˜π‘‹) ↔ (𝐷:(𝑋 Γ— 𝑋)βŸΆβ„* ∧ βˆ€π‘Ž ∈ 𝑋 ((π‘Žπ·π‘Ž) = 0 ∧ βˆ€π‘ ∈ 𝑋 βˆ€π‘ ∈ 𝑋 (π‘Žπ·π‘) ≀ ((π‘π·π‘Ž) +𝑒 (𝑐𝐷𝑏))))))
31, 2syl 17 . . . . . . 7 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ (𝐷 ∈ (PsMetβ€˜π‘‹) ↔ (𝐷:(𝑋 Γ— 𝑋)βŸΆβ„* ∧ βˆ€π‘Ž ∈ 𝑋 ((π‘Žπ·π‘Ž) = 0 ∧ βˆ€π‘ ∈ 𝑋 βˆ€π‘ ∈ 𝑋 (π‘Žπ·π‘) ≀ ((π‘π·π‘Ž) +𝑒 (𝑐𝐷𝑏))))))
43ibi 267 . . . . . 6 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ (𝐷:(𝑋 Γ— 𝑋)βŸΆβ„* ∧ βˆ€π‘Ž ∈ 𝑋 ((π‘Žπ·π‘Ž) = 0 ∧ βˆ€π‘ ∈ 𝑋 βˆ€π‘ ∈ 𝑋 (π‘Žπ·π‘) ≀ ((π‘π·π‘Ž) +𝑒 (𝑐𝐷𝑏)))))
54simprd 495 . . . . 5 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ βˆ€π‘Ž ∈ 𝑋 ((π‘Žπ·π‘Ž) = 0 ∧ βˆ€π‘ ∈ 𝑋 βˆ€π‘ ∈ 𝑋 (π‘Žπ·π‘) ≀ ((π‘π·π‘Ž) +𝑒 (𝑐𝐷𝑏))))
65r19.21bi 3242 . . . 4 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ π‘Ž ∈ 𝑋) β†’ ((π‘Žπ·π‘Ž) = 0 ∧ βˆ€π‘ ∈ 𝑋 βˆ€π‘ ∈ 𝑋 (π‘Žπ·π‘) ≀ ((π‘π·π‘Ž) +𝑒 (𝑐𝐷𝑏))))
76simprd 495 . . 3 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ π‘Ž ∈ 𝑋) β†’ βˆ€π‘ ∈ 𝑋 βˆ€π‘ ∈ 𝑋 (π‘Žπ·π‘) ≀ ((π‘π·π‘Ž) +𝑒 (𝑐𝐷𝑏)))
87ralrimiva 3140 . 2 (𝐷 ∈ (PsMetβ€˜π‘‹) β†’ βˆ€π‘Ž ∈ 𝑋 βˆ€π‘ ∈ 𝑋 βˆ€π‘ ∈ 𝑋 (π‘Žπ·π‘) ≀ ((π‘π·π‘Ž) +𝑒 (𝑐𝐷𝑏)))
9 oveq1 7412 . . . . 5 (π‘Ž = 𝐴 β†’ (π‘Žπ·π‘) = (𝐴𝐷𝑏))
10 oveq2 7413 . . . . . 6 (π‘Ž = 𝐴 β†’ (π‘π·π‘Ž) = (𝑐𝐷𝐴))
1110oveq1d 7420 . . . . 5 (π‘Ž = 𝐴 β†’ ((π‘π·π‘Ž) +𝑒 (𝑐𝐷𝑏)) = ((𝑐𝐷𝐴) +𝑒 (𝑐𝐷𝑏)))
129, 11breq12d 5154 . . . 4 (π‘Ž = 𝐴 β†’ ((π‘Žπ·π‘) ≀ ((π‘π·π‘Ž) +𝑒 (𝑐𝐷𝑏)) ↔ (𝐴𝐷𝑏) ≀ ((𝑐𝐷𝐴) +𝑒 (𝑐𝐷𝑏))))
13 oveq2 7413 . . . . 5 (𝑏 = 𝐡 β†’ (𝐴𝐷𝑏) = (𝐴𝐷𝐡))
14 oveq2 7413 . . . . . 6 (𝑏 = 𝐡 β†’ (𝑐𝐷𝑏) = (𝑐𝐷𝐡))
1514oveq2d 7421 . . . . 5 (𝑏 = 𝐡 β†’ ((𝑐𝐷𝐴) +𝑒 (𝑐𝐷𝑏)) = ((𝑐𝐷𝐴) +𝑒 (𝑐𝐷𝐡)))
1613, 15breq12d 5154 . . . 4 (𝑏 = 𝐡 β†’ ((𝐴𝐷𝑏) ≀ ((𝑐𝐷𝐴) +𝑒 (𝑐𝐷𝑏)) ↔ (𝐴𝐷𝐡) ≀ ((𝑐𝐷𝐴) +𝑒 (𝑐𝐷𝐡))))
17 oveq1 7412 . . . . . 6 (𝑐 = 𝐢 β†’ (𝑐𝐷𝐴) = (𝐢𝐷𝐴))
18 oveq1 7412 . . . . . 6 (𝑐 = 𝐢 β†’ (𝑐𝐷𝐡) = (𝐢𝐷𝐡))
1917, 18oveq12d 7423 . . . . 5 (𝑐 = 𝐢 β†’ ((𝑐𝐷𝐴) +𝑒 (𝑐𝐷𝐡)) = ((𝐢𝐷𝐴) +𝑒 (𝐢𝐷𝐡)))
2019breq2d 5153 . . . 4 (𝑐 = 𝐢 β†’ ((𝐴𝐷𝐡) ≀ ((𝑐𝐷𝐴) +𝑒 (𝑐𝐷𝐡)) ↔ (𝐴𝐷𝐡) ≀ ((𝐢𝐷𝐴) +𝑒 (𝐢𝐷𝐡))))
2112, 16, 20rspc3v 3622 . . 3 ((𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋) β†’ (βˆ€π‘Ž ∈ 𝑋 βˆ€π‘ ∈ 𝑋 βˆ€π‘ ∈ 𝑋 (π‘Žπ·π‘) ≀ ((π‘π·π‘Ž) +𝑒 (𝑐𝐷𝑏)) β†’ (𝐴𝐷𝐡) ≀ ((𝐢𝐷𝐴) +𝑒 (𝐢𝐷𝐡))))
22213comr 1122 . 2 ((𝐢 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (βˆ€π‘Ž ∈ 𝑋 βˆ€π‘ ∈ 𝑋 βˆ€π‘ ∈ 𝑋 (π‘Žπ·π‘) ≀ ((π‘π·π‘Ž) +𝑒 (𝑐𝐷𝑏)) β†’ (𝐴𝐷𝐡) ≀ ((𝐢𝐷𝐴) +𝑒 (𝐢𝐷𝐡))))
238, 22mpan9 506 1 ((𝐷 ∈ (PsMetβ€˜π‘‹) ∧ (𝐢 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ (𝐴𝐷𝐡) ≀ ((𝐢𝐷𝐴) +𝑒 (𝐢𝐷𝐡)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆ€wral 3055  Vcvv 3468   class class class wbr 5141   Γ— cxp 5667  βŸΆwf 6533  β€˜cfv 6537  (class class class)co 7405  0cc0 11112  β„*cxr 11251   ≀ cle 11253   +𝑒 cxad 13096  PsMetcpsmet 21224
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-cnex 11168  ax-resscn 11169
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-map 8824  df-xr 11256  df-psmet 21232
This theorem is referenced by:  psmetsym  24171  psmettri  24172  psmetge0  24173  psmetres2  24175  xblss2ps  24262  metideq  33403  metider  33404  pstmxmet  33407
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