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Theorem psmetres2 12408
Description: Restriction of a pseudometric. (Contributed by Thierry Arnoux, 11-Feb-2018.)
Assertion
Ref Expression
psmetres2 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) → (𝐷 ↾ (𝑅 × 𝑅)) ∈ (PsMet‘𝑅))

Proof of Theorem psmetres2
Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑒 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 psmetf 12400 . . . 4 (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
21adantr 272 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
3 simpr 109 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) → 𝑅𝑋)
4 xpss12 4614 . . . 4 ((𝑅𝑋𝑅𝑋) → (𝑅 × 𝑅) ⊆ (𝑋 × 𝑋))
53, 3, 4syl2anc 406 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) → (𝑅 × 𝑅) ⊆ (𝑋 × 𝑋))
62, 5fssresd 5267 . 2 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) → (𝐷 ↾ (𝑅 × 𝑅)):(𝑅 × 𝑅)⟶ℝ*)
7 simpr 109 . . . . . 6 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) → 𝑎𝑅)
87, 7ovresd 5877 . . . . 5 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) → (𝑎(𝐷 ↾ (𝑅 × 𝑅))𝑎) = (𝑎𝐷𝑎))
9 simpll 501 . . . . . 6 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) → 𝐷 ∈ (PsMet‘𝑋))
103sselda 3065 . . . . . 6 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) → 𝑎𝑋)
11 psmet0 12402 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) → (𝑎𝐷𝑎) = 0)
129, 10, 11syl2anc 406 . . . . 5 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) → (𝑎𝐷𝑎) = 0)
138, 12eqtrd 2148 . . . 4 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) → (𝑎(𝐷 ↾ (𝑅 × 𝑅))𝑎) = 0)
149ad2antrr 477 . . . . . . . 8 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) ∧ 𝑏𝑅) ∧ 𝑐𝑅) → 𝐷 ∈ (PsMet‘𝑋))
153ad2antrr 477 . . . . . . . . 9 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) ∧ 𝑏𝑅) → 𝑅𝑋)
1615sselda 3065 . . . . . . . 8 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) ∧ 𝑏𝑅) ∧ 𝑐𝑅) → 𝑐𝑋)
1710ad2antrr 477 . . . . . . . 8 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) ∧ 𝑏𝑅) ∧ 𝑐𝑅) → 𝑎𝑋)
183adantr 272 . . . . . . . . . 10 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) → 𝑅𝑋)
1918sselda 3065 . . . . . . . . 9 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) ∧ 𝑏𝑅) → 𝑏𝑋)
2019adantr 272 . . . . . . . 8 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) ∧ 𝑏𝑅) ∧ 𝑐𝑅) → 𝑏𝑋)
21 psmettri2 12403 . . . . . . . 8 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑐𝑋𝑎𝑋𝑏𝑋)) → (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)))
2214, 16, 17, 20, 21syl13anc 1201 . . . . . . 7 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) ∧ 𝑏𝑅) ∧ 𝑐𝑅) → (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)))
237adantr 272 . . . . . . . . 9 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) ∧ 𝑏𝑅) → 𝑎𝑅)
24 simpr 109 . . . . . . . . 9 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) ∧ 𝑏𝑅) → 𝑏𝑅)
2523, 24ovresd 5877 . . . . . . . 8 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) ∧ 𝑏𝑅) → (𝑎(𝐷 ↾ (𝑅 × 𝑅))𝑏) = (𝑎𝐷𝑏))
2625adantr 272 . . . . . . 7 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) ∧ 𝑏𝑅) ∧ 𝑐𝑅) → (𝑎(𝐷 ↾ (𝑅 × 𝑅))𝑏) = (𝑎𝐷𝑏))
27 simpr 109 . . . . . . . . 9 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) ∧ 𝑏𝑅) ∧ 𝑐𝑅) → 𝑐𝑅)
287ad2antrr 477 . . . . . . . . 9 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) ∧ 𝑏𝑅) ∧ 𝑐𝑅) → 𝑎𝑅)
2927, 28ovresd 5877 . . . . . . . 8 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) ∧ 𝑏𝑅) ∧ 𝑐𝑅) → (𝑐(𝐷 ↾ (𝑅 × 𝑅))𝑎) = (𝑐𝐷𝑎))
3024adantr 272 . . . . . . . . 9 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) ∧ 𝑏𝑅) ∧ 𝑐𝑅) → 𝑏𝑅)
3127, 30ovresd 5877 . . . . . . . 8 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) ∧ 𝑏𝑅) ∧ 𝑐𝑅) → (𝑐(𝐷 ↾ (𝑅 × 𝑅))𝑏) = (𝑐𝐷𝑏))
3229, 31oveq12d 5758 . . . . . . 7 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) ∧ 𝑏𝑅) ∧ 𝑐𝑅) → ((𝑐(𝐷 ↾ (𝑅 × 𝑅))𝑎) +𝑒 (𝑐(𝐷 ↾ (𝑅 × 𝑅))𝑏)) = ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)))
3322, 26, 323brtr4d 3928 . . . . . 6 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) ∧ 𝑏𝑅) ∧ 𝑐𝑅) → (𝑎(𝐷 ↾ (𝑅 × 𝑅))𝑏) ≤ ((𝑐(𝐷 ↾ (𝑅 × 𝑅))𝑎) +𝑒 (𝑐(𝐷 ↾ (𝑅 × 𝑅))𝑏)))
3433ralrimiva 2480 . . . . 5 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) ∧ 𝑏𝑅) → ∀𝑐𝑅 (𝑎(𝐷 ↾ (𝑅 × 𝑅))𝑏) ≤ ((𝑐(𝐷 ↾ (𝑅 × 𝑅))𝑎) +𝑒 (𝑐(𝐷 ↾ (𝑅 × 𝑅))𝑏)))
3534ralrimiva 2480 . . . 4 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) → ∀𝑏𝑅𝑐𝑅 (𝑎(𝐷 ↾ (𝑅 × 𝑅))𝑏) ≤ ((𝑐(𝐷 ↾ (𝑅 × 𝑅))𝑎) +𝑒 (𝑐(𝐷 ↾ (𝑅 × 𝑅))𝑏)))
3613, 35jca 302 . . 3 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) → ((𝑎(𝐷 ↾ (𝑅 × 𝑅))𝑎) = 0 ∧ ∀𝑏𝑅𝑐𝑅 (𝑎(𝐷 ↾ (𝑅 × 𝑅))𝑏) ≤ ((𝑐(𝐷 ↾ (𝑅 × 𝑅))𝑎) +𝑒 (𝑐(𝐷 ↾ (𝑅 × 𝑅))𝑏))))
3736ralrimiva 2480 . 2 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) → ∀𝑎𝑅 ((𝑎(𝐷 ↾ (𝑅 × 𝑅))𝑎) = 0 ∧ ∀𝑏𝑅𝑐𝑅 (𝑎(𝐷 ↾ (𝑅 × 𝑅))𝑏) ≤ ((𝑐(𝐷 ↾ (𝑅 × 𝑅))𝑎) +𝑒 (𝑐(𝐷 ↾ (𝑅 × 𝑅))𝑏))))
38 df-psmet 12062 . . . . . 6 PsMet = (𝑎 ∈ V ↦ {𝑏 ∈ (ℝ*𝑚 (𝑎 × 𝑎)) ∣ ∀𝑐𝑎 ((𝑐𝑏𝑐) = 0 ∧ ∀𝑑𝑎𝑒𝑎 (𝑐𝑏𝑑) ≤ ((𝑒𝑏𝑐) +𝑒 (𝑒𝑏𝑑)))})
3938mptrcl 5469 . . . . 5 (𝐷 ∈ (PsMet‘𝑋) → 𝑋 ∈ V)
4039adantr 272 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) → 𝑋 ∈ V)
4140, 3ssexd 4036 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) → 𝑅 ∈ V)
42 ispsmet 12398 . . 3 (𝑅 ∈ V → ((𝐷 ↾ (𝑅 × 𝑅)) ∈ (PsMet‘𝑅) ↔ ((𝐷 ↾ (𝑅 × 𝑅)):(𝑅 × 𝑅)⟶ℝ* ∧ ∀𝑎𝑅 ((𝑎(𝐷 ↾ (𝑅 × 𝑅))𝑎) = 0 ∧ ∀𝑏𝑅𝑐𝑅 (𝑎(𝐷 ↾ (𝑅 × 𝑅))𝑏) ≤ ((𝑐(𝐷 ↾ (𝑅 × 𝑅))𝑎) +𝑒 (𝑐(𝐷 ↾ (𝑅 × 𝑅))𝑏))))))
4341, 42syl 14 . 2 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) → ((𝐷 ↾ (𝑅 × 𝑅)) ∈ (PsMet‘𝑅) ↔ ((𝐷 ↾ (𝑅 × 𝑅)):(𝑅 × 𝑅)⟶ℝ* ∧ ∀𝑎𝑅 ((𝑎(𝐷 ↾ (𝑅 × 𝑅))𝑎) = 0 ∧ ∀𝑏𝑅𝑐𝑅 (𝑎(𝐷 ↾ (𝑅 × 𝑅))𝑏) ≤ ((𝑐(𝐷 ↾ (𝑅 × 𝑅))𝑎) +𝑒 (𝑐(𝐷 ↾ (𝑅 × 𝑅))𝑏))))))
446, 37, 43mpbir2and 911 1 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) → (𝐷 ↾ (𝑅 × 𝑅)) ∈ (PsMet‘𝑅))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1314  wcel 1463  wral 2391  {crab 2395  Vcvv 2658  wss 3039   class class class wbr 3897   × cxp 4505  cres 4509  wf 5087  cfv 5091  (class class class)co 5740  𝑚 cmap 6508  0cc0 7584  *cxr 7763  cle 7765   +𝑒 cxad 9508  PsMetcpsmet 12054
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-cnex 7675  ax-resscn 7676
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-mpt 3959  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-fv 5099  df-ov 5743  df-oprab 5744  df-mpo 5745  df-map 6510  df-pnf 7766  df-mnf 7767  df-xr 7768  df-psmet 12062
This theorem is referenced by: (None)
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