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Theorem psmetres2 24324
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 24316 . . . 4 (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
21adantr 480 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
3 simpr 484 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) → 𝑅𝑋)
4 xpss12 5700 . . . 4 ((𝑅𝑋𝑅𝑋) → (𝑅 × 𝑅) ⊆ (𝑋 × 𝑋))
53, 3, 4syl2anc 584 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) → (𝑅 × 𝑅) ⊆ (𝑋 × 𝑋))
62, 5fssresd 6775 . 2 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) → (𝐷 ↾ (𝑅 × 𝑅)):(𝑅 × 𝑅)⟶ℝ*)
7 simpr 484 . . . . . 6 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) → 𝑎𝑅)
87, 7ovresd 7600 . . . . 5 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) → (𝑎(𝐷 ↾ (𝑅 × 𝑅))𝑎) = (𝑎𝐷𝑎))
9 simpll 767 . . . . . 6 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) → 𝐷 ∈ (PsMet‘𝑋))
103sselda 3983 . . . . . 6 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) → 𝑎𝑋)
11 psmet0 24318 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) → (𝑎𝐷𝑎) = 0)
129, 10, 11syl2anc 584 . . . . 5 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) → (𝑎𝐷𝑎) = 0)
138, 12eqtrd 2777 . . . 4 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) → (𝑎(𝐷 ↾ (𝑅 × 𝑅))𝑎) = 0)
149ad2antrr 726 . . . . . . . 8 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) ∧ 𝑏𝑅) ∧ 𝑐𝑅) → 𝐷 ∈ (PsMet‘𝑋))
153ad2antrr 726 . . . . . . . . 9 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) ∧ 𝑏𝑅) → 𝑅𝑋)
1615sselda 3983 . . . . . . . 8 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) ∧ 𝑏𝑅) ∧ 𝑐𝑅) → 𝑐𝑋)
1710ad2antrr 726 . . . . . . . 8 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) ∧ 𝑏𝑅) ∧ 𝑐𝑅) → 𝑎𝑋)
183adantr 480 . . . . . . . . . 10 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) → 𝑅𝑋)
1918sselda 3983 . . . . . . . . 9 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) ∧ 𝑏𝑅) → 𝑏𝑋)
2019adantr 480 . . . . . . . 8 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) ∧ 𝑏𝑅) ∧ 𝑐𝑅) → 𝑏𝑋)
21 psmettri2 24319 . . . . . . . 8 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑐𝑋𝑎𝑋𝑏𝑋)) → (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)))
2214, 16, 17, 20, 21syl13anc 1374 . . . . . . 7 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) ∧ 𝑏𝑅) ∧ 𝑐𝑅) → (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)))
237adantr 480 . . . . . . . . 9 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) ∧ 𝑏𝑅) → 𝑎𝑅)
24 simpr 484 . . . . . . . . 9 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) ∧ 𝑏𝑅) → 𝑏𝑅)
2523, 24ovresd 7600 . . . . . . . 8 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) ∧ 𝑏𝑅) → (𝑎(𝐷 ↾ (𝑅 × 𝑅))𝑏) = (𝑎𝐷𝑏))
2625adantr 480 . . . . . . 7 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) ∧ 𝑏𝑅) ∧ 𝑐𝑅) → (𝑎(𝐷 ↾ (𝑅 × 𝑅))𝑏) = (𝑎𝐷𝑏))
27 simpr 484 . . . . . . . . 9 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) ∧ 𝑏𝑅) ∧ 𝑐𝑅) → 𝑐𝑅)
287ad2antrr 726 . . . . . . . . 9 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) ∧ 𝑏𝑅) ∧ 𝑐𝑅) → 𝑎𝑅)
2927, 28ovresd 7600 . . . . . . . 8 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) ∧ 𝑏𝑅) ∧ 𝑐𝑅) → (𝑐(𝐷 ↾ (𝑅 × 𝑅))𝑎) = (𝑐𝐷𝑎))
3024adantr 480 . . . . . . . . 9 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) ∧ 𝑏𝑅) ∧ 𝑐𝑅) → 𝑏𝑅)
3127, 30ovresd 7600 . . . . . . . 8 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) ∧ 𝑏𝑅) ∧ 𝑐𝑅) → (𝑐(𝐷 ↾ (𝑅 × 𝑅))𝑏) = (𝑐𝐷𝑏))
3229, 31oveq12d 7449 . . . . . . 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 5175 . . . . . 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 3146 . . . . 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 3146 . . . 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 511 . . 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 3146 . 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 6944 . . . . 5 (𝐷 ∈ (PsMet‘𝑋) → 𝑋 ∈ V)
3938adantr 480 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) → 𝑋 ∈ V)
4039, 3ssexd 5324 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) → 𝑅 ∈ V)
41 ispsmet 24314 . . 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 713 1 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) → (𝐷 ↾ (𝑅 × 𝑅)) ∈ (PsMet‘𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wral 3061  Vcvv 3480  wss 3951   class class class wbr 5143   × cxp 5683  cres 5687  wf 6557  cfv 6561  (class class class)co 7431  0cc0 11155  *cxr 11294  cle 11296   +𝑒 cxad 13152  PsMetcpsmet 21348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8868  df-xr 11299  df-psmet 21356
This theorem is referenced by:  restmetu  24583
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