MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  psmetres2 Structured version   Visualization version   GIF version

Theorem psmetres2 22896
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 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 psmetf 22888 . . . 4 (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
21adantr 483 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
3 simpr 487 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) → 𝑅𝑋)
4 xpss12 5542 . . . 4 ((𝑅𝑋𝑅𝑋) → (𝑅 × 𝑅) ⊆ (𝑋 × 𝑋))
53, 3, 4syl2anc 586 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) → (𝑅 × 𝑅) ⊆ (𝑋 × 𝑋))
62, 5fssresd 6517 . 2 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) → (𝐷 ↾ (𝑅 × 𝑅)):(𝑅 × 𝑅)⟶ℝ*)
7 simpr 487 . . . . . 6 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) → 𝑎𝑅)
87, 7ovresd 7289 . . . . 5 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) → (𝑎(𝐷 ↾ (𝑅 × 𝑅))𝑎) = (𝑎𝐷𝑎))
9 simpll 765 . . . . . 6 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) → 𝐷 ∈ (PsMet‘𝑋))
103sselda 3942 . . . . . 6 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) → 𝑎𝑋)
11 psmet0 22890 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) → (𝑎𝐷𝑎) = 0)
129, 10, 11syl2anc 586 . . . . 5 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) → (𝑎𝐷𝑎) = 0)
138, 12eqtrd 2855 . . . 4 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) → (𝑎(𝐷 ↾ (𝑅 × 𝑅))𝑎) = 0)
149ad2antrr 724 . . . . . . . 8 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) ∧ 𝑏𝑅) ∧ 𝑐𝑅) → 𝐷 ∈ (PsMet‘𝑋))
153ad2antrr 724 . . . . . . . . 9 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) ∧ 𝑏𝑅) → 𝑅𝑋)
1615sselda 3942 . . . . . . . 8 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) ∧ 𝑏𝑅) ∧ 𝑐𝑅) → 𝑐𝑋)
1710ad2antrr 724 . . . . . . . 8 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) ∧ 𝑏𝑅) ∧ 𝑐𝑅) → 𝑎𝑋)
183adantr 483 . . . . . . . . . 10 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) → 𝑅𝑋)
1918sselda 3942 . . . . . . . . 9 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) ∧ 𝑏𝑅) → 𝑏𝑋)
2019adantr 483 . . . . . . . 8 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) ∧ 𝑏𝑅) ∧ 𝑐𝑅) → 𝑏𝑋)
21 psmettri2 22891 . . . . . . . 8 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑐𝑋𝑎𝑋𝑏𝑋)) → (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)))
2214, 16, 17, 20, 21syl13anc 1368 . . . . . . 7 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) ∧ 𝑏𝑅) ∧ 𝑐𝑅) → (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)))
237adantr 483 . . . . . . . . 9 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) ∧ 𝑏𝑅) → 𝑎𝑅)
24 simpr 487 . . . . . . . . 9 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) ∧ 𝑏𝑅) → 𝑏𝑅)
2523, 24ovresd 7289 . . . . . . . 8 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) ∧ 𝑏𝑅) → (𝑎(𝐷 ↾ (𝑅 × 𝑅))𝑏) = (𝑎𝐷𝑏))
2625adantr 483 . . . . . . 7 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) ∧ 𝑏𝑅) ∧ 𝑐𝑅) → (𝑎(𝐷 ↾ (𝑅 × 𝑅))𝑏) = (𝑎𝐷𝑏))
27 simpr 487 . . . . . . . . 9 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) ∧ 𝑏𝑅) ∧ 𝑐𝑅) → 𝑐𝑅)
287ad2antrr 724 . . . . . . . . 9 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) ∧ 𝑏𝑅) ∧ 𝑐𝑅) → 𝑎𝑅)
2927, 28ovresd 7289 . . . . . . . 8 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) ∧ 𝑏𝑅) ∧ 𝑐𝑅) → (𝑐(𝐷 ↾ (𝑅 × 𝑅))𝑎) = (𝑐𝐷𝑎))
3024adantr 483 . . . . . . . . 9 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) ∧ 𝑏𝑅) ∧ 𝑐𝑅) → 𝑏𝑅)
3127, 30ovresd 7289 . . . . . . . 8 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) ∧ 𝑏𝑅) ∧ 𝑐𝑅) → (𝑐(𝐷 ↾ (𝑅 × 𝑅))𝑏) = (𝑐𝐷𝑏))
3229, 31oveq12d 7147 . . . . . . 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 5070 . . . . . 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 3169 . . . . 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 3169 . . . 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 514 . . 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 3169 . 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 elfvex 6675 . . . . 5 (𝐷 ∈ (PsMet‘𝑋) → 𝑋 ∈ V)
3938adantr 483 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) → 𝑋 ∈ V)
4039, 3ssexd 5200 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) → 𝑅 ∈ V)
41 ispsmet 22886 . . 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))))))
4240, 41syl 17 . 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))))))
436, 37, 42mpbir2and 711 1 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) → (𝐷 ↾ (𝑅 × 𝑅)) ∈ (PsMet‘𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398   = wceq 1537  wcel 2114  wral 3125  Vcvv 3470  wss 3909   class class class wbr 5038   × cxp 5525  cres 5529  wf 6323  cfv 6327  (class class class)co 7129  0cc0 10511  *cxr 10648  cle 10650   +𝑒 cxad 12480  PsMetcpsmet 20501
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5175  ax-nul 5182  ax-pow 5238  ax-pr 5302  ax-un 7435  ax-cnex 10567  ax-resscn 10568
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3472  df-sbc 3749  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4811  df-br 5039  df-opab 5101  df-mpt 5119  df-id 5432  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-iota 6286  df-fun 6329  df-fn 6330  df-f 6331  df-fv 6335  df-ov 7132  df-oprab 7133  df-mpo 7134  df-map 8382  df-xr 10653  df-psmet 20509
This theorem is referenced by:  restmetu  23152
  Copyright terms: Public domain W3C validator