Theorem psmetres2 24633

Description: Restriction of a pseudometric. (Contributed by Thierry Arnoux, 11-Feb-2018.)

Assertion

Ref	Expression
psmetres2	⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) → (𝐷 ↾ (𝑅 × 𝑅)) ∈ (PsMet‘𝑅))

Proof of Theorem psmetres2

Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		psmetf 24625	. . . 4 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
2	1	adantr 486	. . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
3		simpr 490	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) → 𝑅 ⊆ 𝑋)
4		xpss12 5665	. . . 4 ⊢ ((𝑅 ⊆ 𝑋 ∧ 𝑅 ⊆ 𝑋) → (𝑅 × 𝑅) ⊆ (𝑋 × 𝑋))
5	3, 3, 4	syl2anc 596	. . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) → (𝑅 × 𝑅) ⊆ (𝑋 × 𝑋))
6	2, 5	fssresd 6749	. 2 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) → (𝐷 ↾ (𝑅 × 𝑅)):(𝑅 × 𝑅)⟶ℝ*)
7		simpr 490	. . . . . 6 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) → 𝑎 ∈ 𝑅)
8	7, 7	ovresd 7587	. . . . 5 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) → (𝑎(𝐷 ↾ (𝑅 × 𝑅))𝑎) = (𝑎𝐷𝑎))
9		simpll 779	. . . . . 6 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) → 𝐷 ∈ (PsMet‘𝑋))
10	3	sselda 3908	. . . . . 6 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) → 𝑎 ∈ 𝑋)
11		psmet0 24627	. . . . . 6 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) → (𝑎𝐷𝑎) = 0)
12	9, 10, 11	syl2anc 596	. . . . 5 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) → (𝑎𝐷𝑎) = 0)
13	8, 12	eqtrd 2795	. . . 4 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) → (𝑎(𝐷 ↾ (𝑅 × 𝑅))𝑎) = 0)
14	9	ad2antrr 739	. . . . . . . 8 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) ∧ 𝑏 ∈ 𝑅) ∧ 𝑐 ∈ 𝑅) → 𝐷 ∈ (PsMet‘𝑋))
15	3	ad2antrr 739	. . . . . . . . 9 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) ∧ 𝑏 ∈ 𝑅) → 𝑅 ⊆ 𝑋)
16	15	sselda 3908	. . . . . . . 8 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) ∧ 𝑏 ∈ 𝑅) ∧ 𝑐 ∈ 𝑅) → 𝑐 ∈ 𝑋)
17	10	ad2antrr 739	. . . . . . . 8 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) ∧ 𝑏 ∈ 𝑅) ∧ 𝑐 ∈ 𝑅) → 𝑎 ∈ 𝑋)
18	3	adantr 486	. . . . . . . . . 10 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) → 𝑅 ⊆ 𝑋)
19	18	sselda 3908	. . . . . . . . 9 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) ∧ 𝑏 ∈ 𝑅) → 𝑏 ∈ 𝑋)
20	19	adantr 486	. . . . . . . 8 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) ∧ 𝑏 ∈ 𝑅) ∧ 𝑐 ∈ 𝑅) → 𝑏 ∈ 𝑋)
21		psmettri2 24628	. . . . . . . 8 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑐 ∈ 𝑋 ∧ 𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +e (𝑐𝐷𝑏)))
22	14, 16, 17, 20, 21	syl13anc 1399	. . . . . . 7 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) ∧ 𝑏 ∈ 𝑅) ∧ 𝑐 ∈ 𝑅) → (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +e (𝑐𝐷𝑏)))
23	7	adantr 486	. . . . . . . . 9 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) ∧ 𝑏 ∈ 𝑅) → 𝑎 ∈ 𝑅)
24		simpr 490	. . . . . . . . 9 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) ∧ 𝑏 ∈ 𝑅) → 𝑏 ∈ 𝑅)
25	23, 24	ovresd 7587	. . . . . . . 8 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) ∧ 𝑏 ∈ 𝑅) → (𝑎(𝐷 ↾ (𝑅 × 𝑅))𝑏) = (𝑎𝐷𝑏))
26	25	adantr 486	. . . . . . 7 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) ∧ 𝑏 ∈ 𝑅) ∧ 𝑐 ∈ 𝑅) → (𝑎(𝐷 ↾ (𝑅 × 𝑅))𝑏) = (𝑎𝐷𝑏))
27		simpr 490	. . . . . . . . 9 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) ∧ 𝑏 ∈ 𝑅) ∧ 𝑐 ∈ 𝑅) → 𝑐 ∈ 𝑅)
28	7	ad2antrr 739	. . . . . . . . 9 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) ∧ 𝑏 ∈ 𝑅) ∧ 𝑐 ∈ 𝑅) → 𝑎 ∈ 𝑅)
29	27, 28	ovresd 7587	. . . . . . . 8 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) ∧ 𝑏 ∈ 𝑅) ∧ 𝑐 ∈ 𝑅) → (𝑐(𝐷 ↾ (𝑅 × 𝑅))𝑎) = (𝑐𝐷𝑎))
30	24	adantr 486	. . . . . . . . 9 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) ∧ 𝑏 ∈ 𝑅) ∧ 𝑐 ∈ 𝑅) → 𝑏 ∈ 𝑅)
31	27, 30	ovresd 7587	. . . . . . . 8 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) ∧ 𝑏 ∈ 𝑅) ∧ 𝑐 ∈ 𝑅) → (𝑐(𝐷 ↾ (𝑅 × 𝑅))𝑏) = (𝑐𝐷𝑏))
32	29, 31	oveq12d 7438	. . . . . . 7 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) ∧ 𝑏 ∈ 𝑅) ∧ 𝑐 ∈ 𝑅) → ((𝑐(𝐷 ↾ (𝑅 × 𝑅))𝑎) +e (𝑐(𝐷 ↾ (𝑅 × 𝑅))𝑏)) = ((𝑐𝐷𝑎) +e (𝑐𝐷𝑏)))
33	22, 26, 32	3brtr4d 5136	. . . . . 6 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) ∧ 𝑏 ∈ 𝑅) ∧ 𝑐 ∈ 𝑅) → (𝑎(𝐷 ↾ (𝑅 × 𝑅))𝑏) ≤ ((𝑐(𝐷 ↾ (𝑅 × 𝑅))𝑎) +e (𝑐(𝐷 ↾ (𝑅 × 𝑅))𝑏)))
34	33	ralrimiva 3154	. . . . 5 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) ∧ 𝑏 ∈ 𝑅) → ∀𝑐 ∈ 𝑅 (𝑎(𝐷 ↾ (𝑅 × 𝑅))𝑏) ≤ ((𝑐(𝐷 ↾ (𝑅 × 𝑅))𝑎) +e (𝑐(𝐷 ↾ (𝑅 × 𝑅))𝑏)))
35	34	ralrimiva 3154	. . . 4 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) → ∀𝑏 ∈ 𝑅 ∀𝑐 ∈ 𝑅 (𝑎(𝐷 ↾ (𝑅 × 𝑅))𝑏) ≤ ((𝑐(𝐷 ↾ (𝑅 × 𝑅))𝑎) +e (𝑐(𝐷 ↾ (𝑅 × 𝑅))𝑏)))
36	13, 35	jca 521	. . 3 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) ∧ 𝑎 ∈ 𝑅) → ((𝑎(𝐷 ↾ (𝑅 × 𝑅))𝑎) = 0 ∧ ∀𝑏 ∈ 𝑅 ∀𝑐 ∈ 𝑅 (𝑎(𝐷 ↾ (𝑅 × 𝑅))𝑏) ≤ ((𝑐(𝐷 ↾ (𝑅 × 𝑅))𝑎) +e (𝑐(𝐷 ↾ (𝑅 × 𝑅))𝑏))))
37	36	ralrimiva 3154	. 2 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) → ∀𝑎 ∈ 𝑅 ((𝑎(𝐷 ↾ (𝑅 × 𝑅))𝑎) = 0 ∧ ∀𝑏 ∈ 𝑅 ∀𝑐 ∈ 𝑅 (𝑎(𝐷 ↾ (𝑅 × 𝑅))𝑏) ≤ ((𝑐(𝐷 ↾ (𝑅 × 𝑅))𝑎) +e (𝑐(𝐷 ↾ (𝑅 × 𝑅))𝑏))))
38		elfvex 6920	. . . . 5 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝑋 ∈ V)
39	38	adantr 486	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) → 𝑋 ∈ V)
40	39, 3	ssexd 5285	. . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) → 𝑅 ∈ V)
41		ispsmet 24623	. . 3 ⊢ (𝑅 ∈ V → ((𝐷 ↾ (𝑅 × 𝑅)) ∈ (PsMet‘𝑅) ↔ ((𝐷 ↾ (𝑅 × 𝑅)):(𝑅 × 𝑅)⟶ℝ* ∧ ∀𝑎 ∈ 𝑅 ((𝑎(𝐷 ↾ (𝑅 × 𝑅))𝑎) = 0 ∧ ∀𝑏 ∈ 𝑅 ∀𝑐 ∈ 𝑅 (𝑎(𝐷 ↾ (𝑅 × 𝑅))𝑏) ≤ ((𝑐(𝐷 ↾ (𝑅 × 𝑅))𝑎) +e (𝑐(𝐷 ↾ (𝑅 × 𝑅))𝑏))))))
42	40, 41	syl 18	. 2 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) → ((𝐷 ↾ (𝑅 × 𝑅)) ∈ (PsMet‘𝑅) ↔ ((𝐷 ↾ (𝑅 × 𝑅)):(𝑅 × 𝑅)⟶ℝ* ∧ ∀𝑎 ∈ 𝑅 ((𝑎(𝐷 ↾ (𝑅 × 𝑅))𝑎) = 0 ∧ ∀𝑏 ∈ 𝑅 ∀𝑐 ∈ 𝑅 (𝑎(𝐷 ↾ (𝑅 × 𝑅))𝑏) ≤ ((𝑐(𝐷 ↾ (𝑅 × 𝑅))𝑎) +e (𝑐(𝐷 ↾ (𝑅 × 𝑅))𝑏))))))
43	6, 37, 42	mpbir2and 726	1 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ⊆ 𝑋) → (𝐷 ↾ (𝑅 × 𝑅)) ∈ (PsMet‘𝑅))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∀wral 3076  Vcvv 3450   ⊆ wss 3895   class class class wbr 5102   × cxp 5648   ↾ cres 5652  ⟶wf 6534  ‘cfv 6538  (class class class)co 7420  0cc0 11200  ℝ^*cxr 11342   ≤ cle 11344   +_e cxad 13239  PsMetcpsmet 21662

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-cnex 11256  ax-resscn 11257

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-map 8849  df-xr 11347  df-psmet 21670

This theorem is used by:  restmetu  24889