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Theorem psmetres2 13500
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 13492 . . . 4 (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
21adantr 276 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
3 simpr 110 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) → 𝑅𝑋)
4 xpss12 4730 . . . 4 ((𝑅𝑋𝑅𝑋) → (𝑅 × 𝑅) ⊆ (𝑋 × 𝑋))
53, 3, 4syl2anc 411 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) → (𝑅 × 𝑅) ⊆ (𝑋 × 𝑋))
62, 5fssresd 5388 . 2 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) → (𝐷 ↾ (𝑅 × 𝑅)):(𝑅 × 𝑅)⟶ℝ*)
7 simpr 110 . . . . . 6 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) → 𝑎𝑅)
87, 7ovresd 6009 . . . . 5 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) → (𝑎(𝐷 ↾ (𝑅 × 𝑅))𝑎) = (𝑎𝐷𝑎))
9 simpll 527 . . . . . 6 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) → 𝐷 ∈ (PsMet‘𝑋))
103sselda 3155 . . . . . 6 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) → 𝑎𝑋)
11 psmet0 13494 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) → (𝑎𝐷𝑎) = 0)
129, 10, 11syl2anc 411 . . . . 5 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) → (𝑎𝐷𝑎) = 0)
138, 12eqtrd 2210 . . . 4 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) → (𝑎(𝐷 ↾ (𝑅 × 𝑅))𝑎) = 0)
149ad2antrr 488 . . . . . . . 8 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) ∧ 𝑏𝑅) ∧ 𝑐𝑅) → 𝐷 ∈ (PsMet‘𝑋))
153ad2antrr 488 . . . . . . . . 9 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) ∧ 𝑏𝑅) → 𝑅𝑋)
1615sselda 3155 . . . . . . . 8 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) ∧ 𝑏𝑅) ∧ 𝑐𝑅) → 𝑐𝑋)
1710ad2antrr 488 . . . . . . . 8 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) ∧ 𝑏𝑅) ∧ 𝑐𝑅) → 𝑎𝑋)
183adantr 276 . . . . . . . . . 10 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) → 𝑅𝑋)
1918sselda 3155 . . . . . . . . 9 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) ∧ 𝑏𝑅) → 𝑏𝑋)
2019adantr 276 . . . . . . . 8 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) ∧ 𝑏𝑅) ∧ 𝑐𝑅) → 𝑏𝑋)
21 psmettri2 13495 . . . . . . . 8 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑐𝑋𝑎𝑋𝑏𝑋)) → (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)))
2214, 16, 17, 20, 21syl13anc 1240 . . . . . . 7 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) ∧ 𝑏𝑅) ∧ 𝑐𝑅) → (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)))
237adantr 276 . . . . . . . . 9 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) ∧ 𝑏𝑅) → 𝑎𝑅)
24 simpr 110 . . . . . . . . 9 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) ∧ 𝑏𝑅) → 𝑏𝑅)
2523, 24ovresd 6009 . . . . . . . 8 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) ∧ 𝑏𝑅) → (𝑎(𝐷 ↾ (𝑅 × 𝑅))𝑏) = (𝑎𝐷𝑏))
2625adantr 276 . . . . . . 7 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) ∧ 𝑏𝑅) ∧ 𝑐𝑅) → (𝑎(𝐷 ↾ (𝑅 × 𝑅))𝑏) = (𝑎𝐷𝑏))
27 simpr 110 . . . . . . . . 9 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) ∧ 𝑏𝑅) ∧ 𝑐𝑅) → 𝑐𝑅)
287ad2antrr 488 . . . . . . . . 9 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) ∧ 𝑏𝑅) ∧ 𝑐𝑅) → 𝑎𝑅)
2927, 28ovresd 6009 . . . . . . . 8 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) ∧ 𝑏𝑅) ∧ 𝑐𝑅) → (𝑐(𝐷 ↾ (𝑅 × 𝑅))𝑎) = (𝑐𝐷𝑎))
3024adantr 276 . . . . . . . . 9 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) ∧ 𝑏𝑅) ∧ 𝑐𝑅) → 𝑏𝑅)
3127, 30ovresd 6009 . . . . . . . 8 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) ∧ 𝑏𝑅) ∧ 𝑐𝑅) → (𝑐(𝐷 ↾ (𝑅 × 𝑅))𝑏) = (𝑐𝐷𝑏))
3229, 31oveq12d 5887 . . . . . . 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 4032 . . . . . 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 2550 . . . . 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 2550 . . . 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 306 . . 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 2550 . 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 13154 . . . . . 6 PsMet = (𝑎 ∈ V ↦ {𝑏 ∈ (ℝ*𝑚 (𝑎 × 𝑎)) ∣ ∀𝑐𝑎 ((𝑐𝑏𝑐) = 0 ∧ ∀𝑑𝑎𝑒𝑎 (𝑐𝑏𝑑) ≤ ((𝑒𝑏𝑐) +𝑒 (𝑒𝑏𝑑)))})
3938mptrcl 5594 . . . . 5 (𝐷 ∈ (PsMet‘𝑋) → 𝑋 ∈ V)
4039adantr 276 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) → 𝑋 ∈ V)
4140, 3ssexd 4140 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) → 𝑅 ∈ V)
42 ispsmet 13490 . . 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 944 1 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) → (𝐷 ↾ (𝑅 × 𝑅)) ∈ (PsMet‘𝑅))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1353  wcel 2148  wral 2455  {crab 2459  Vcvv 2737  wss 3129   class class class wbr 4000   × cxp 4621  cres 4625  wf 5208  cfv 5212  (class class class)co 5869  𝑚 cmap 6642  0cc0 7802  *cxr 7981  cle 7983   +𝑒 cxad 9757  PsMetcpsmet 13146
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-cnex 7893  ax-resscn 7894
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-fv 5220  df-ov 5872  df-oprab 5873  df-mpo 5874  df-map 6644  df-pnf 7984  df-mnf 7985  df-xr 7986  df-psmet 13154
This theorem is referenced by: (None)
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