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Theorem psmetres2 22029
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 22021 . . . 4 (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
21adantr 481 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
3 simpr 477 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) → 𝑅𝑋)
4 xpss12 5186 . . . 4 ((𝑅𝑋𝑅𝑋) → (𝑅 × 𝑅) ⊆ (𝑋 × 𝑋))
53, 3, 4syl2anc 692 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) → (𝑅 × 𝑅) ⊆ (𝑋 × 𝑋))
62, 5fssresd 6028 . 2 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) → (𝐷 ↾ (𝑅 × 𝑅)):(𝑅 × 𝑅)⟶ℝ*)
7 simpr 477 . . . . . 6 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) → 𝑎𝑅)
87, 7ovresd 6754 . . . . 5 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) → (𝑎(𝐷 ↾ (𝑅 × 𝑅))𝑎) = (𝑎𝐷𝑎))
9 simpll 789 . . . . . 6 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) → 𝐷 ∈ (PsMet‘𝑋))
103sselda 3583 . . . . . 6 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) → 𝑎𝑋)
11 psmet0 22023 . . . . . 6 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) → (𝑎𝐷𝑎) = 0)
129, 10, 11syl2anc 692 . . . . 5 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) → (𝑎𝐷𝑎) = 0)
138, 12eqtrd 2655 . . . 4 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) → (𝑎(𝐷 ↾ (𝑅 × 𝑅))𝑎) = 0)
149ad2antrr 761 . . . . . . . 8 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) ∧ 𝑏𝑅) ∧ 𝑐𝑅) → 𝐷 ∈ (PsMet‘𝑋))
153ad2antrr 761 . . . . . . . . 9 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) ∧ 𝑏𝑅) → 𝑅𝑋)
1615sselda 3583 . . . . . . . 8 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) ∧ 𝑏𝑅) ∧ 𝑐𝑅) → 𝑐𝑋)
1710ad2antrr 761 . . . . . . . 8 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) ∧ 𝑏𝑅) ∧ 𝑐𝑅) → 𝑎𝑋)
183adantr 481 . . . . . . . . . 10 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) → 𝑅𝑋)
1918sselda 3583 . . . . . . . . 9 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) ∧ 𝑏𝑅) → 𝑏𝑋)
2019adantr 481 . . . . . . . 8 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) ∧ 𝑏𝑅) ∧ 𝑐𝑅) → 𝑏𝑋)
21 psmettri2 22024 . . . . . . . 8 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑐𝑋𝑎𝑋𝑏𝑋)) → (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)))
2214, 16, 17, 20, 21syl13anc 1325 . . . . . . 7 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) ∧ 𝑏𝑅) ∧ 𝑐𝑅) → (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)))
237adantr 481 . . . . . . . . 9 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) ∧ 𝑏𝑅) → 𝑎𝑅)
24 simpr 477 . . . . . . . . 9 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) ∧ 𝑏𝑅) → 𝑏𝑅)
2523, 24ovresd 6754 . . . . . . . 8 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) ∧ 𝑏𝑅) → (𝑎(𝐷 ↾ (𝑅 × 𝑅))𝑏) = (𝑎𝐷𝑏))
2625adantr 481 . . . . . . 7 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) ∧ 𝑏𝑅) ∧ 𝑐𝑅) → (𝑎(𝐷 ↾ (𝑅 × 𝑅))𝑏) = (𝑎𝐷𝑏))
27 simpr 477 . . . . . . . . 9 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) ∧ 𝑏𝑅) ∧ 𝑐𝑅) → 𝑐𝑅)
287ad2antrr 761 . . . . . . . . 9 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) ∧ 𝑏𝑅) ∧ 𝑐𝑅) → 𝑎𝑅)
2927, 28ovresd 6754 . . . . . . . 8 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) ∧ 𝑏𝑅) ∧ 𝑐𝑅) → (𝑐(𝐷 ↾ (𝑅 × 𝑅))𝑎) = (𝑐𝐷𝑎))
3024adantr 481 . . . . . . . . 9 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) ∧ 𝑏𝑅) ∧ 𝑐𝑅) → 𝑏𝑅)
3127, 30ovresd 6754 . . . . . . . 8 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) ∧ 𝑎𝑅) ∧ 𝑏𝑅) ∧ 𝑐𝑅) → (𝑐(𝐷 ↾ (𝑅 × 𝑅))𝑏) = (𝑐𝐷𝑏))
3229, 31oveq12d 6622 . . . . . . 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 4645 . . . . . 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 2960 . . . . 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 2960 . . . 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 554 . . 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 2960 . 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 6178 . . . . 5 (𝐷 ∈ (PsMet‘𝑋) → 𝑋 ∈ V)
3938adantr 481 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) → 𝑋 ∈ V)
4039, 3ssexd 4765 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) → 𝑅 ∈ V)
41 ispsmet 22019 . . 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 956 1 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅𝑋) → (𝐷 ↾ (𝑅 × 𝑅)) ∈ (PsMet‘𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2907  Vcvv 3186  wss 3555   class class class wbr 4613   × cxp 5072  cres 5076  wf 5843  cfv 5847  (class class class)co 6604  0cc0 9880  *cxr 10017  cle 10019   +𝑒 cxad 11888  PsMetcpsmet 19649
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-map 7804  df-xr 10022  df-psmet 19657
This theorem is referenced by:  restmetu  22285
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