Theorem psmettri2 15058

Description: Triangle inequality for the distance function of a pseudometric. (Contributed by Thierry Arnoux, 11-Feb-2018.)

Assertion

Ref	Expression
psmettri2	⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐷𝐵) ≤ ((𝐶𝐷𝐴) +𝑒 (𝐶𝐷𝐵)))

Proof of Theorem psmettri2

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

Step	Hyp	Ref	Expression
1		df-psmet 14563	. . . . . . . . 9 ⊢ PsMet = (𝑑 ∈ V ↦ {𝑒 ∈ (ℝ* ↑𝑚 (𝑑 × 𝑑)) ∣ ∀𝑎 ∈ 𝑑 ((𝑎𝑒𝑎) = 0 ∧ ∀𝑏 ∈ 𝑑 ∀𝑐 ∈ 𝑑 (𝑎𝑒𝑏) ≤ ((𝑐𝑒𝑎) +𝑒 (𝑐𝑒𝑏)))})
2	1	mptrcl 5729	. . . . . . . 8 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝑋 ∈ V)
3		ispsmet 15053	. . . . . . . 8 ⊢ (𝑋 ∈ V → (𝐷 ∈ (PsMet‘𝑋) ↔ (𝐷:(𝑋 × 𝑋)⟶ℝ* ∧ ∀𝑎 ∈ 𝑋 ((𝑎𝐷𝑎) = 0 ∧ ∀𝑏 ∈ 𝑋 ∀𝑐 ∈ 𝑋 (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏))))))
4	2, 3	syl 14	. . . . . . 7 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (𝐷 ∈ (PsMet‘𝑋) ↔ (𝐷:(𝑋 × 𝑋)⟶ℝ* ∧ ∀𝑎 ∈ 𝑋 ((𝑎𝐷𝑎) = 0 ∧ ∀𝑏 ∈ 𝑋 ∀𝑐 ∈ 𝑋 (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏))))))
5	4	ibi 176	. . . . . 6 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (𝐷:(𝑋 × 𝑋)⟶ℝ* ∧ ∀𝑎 ∈ 𝑋 ((𝑎𝐷𝑎) = 0 ∧ ∀𝑏 ∈ 𝑋 ∀𝑐 ∈ 𝑋 (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)))))
6	5	simprd 114	. . . . 5 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ∀𝑎 ∈ 𝑋 ((𝑎𝐷𝑎) = 0 ∧ ∀𝑏 ∈ 𝑋 ∀𝑐 ∈ 𝑋 (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏))))
7	6	r19.21bi 2620	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) → ((𝑎𝐷𝑎) = 0 ∧ ∀𝑏 ∈ 𝑋 ∀𝑐 ∈ 𝑋 (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏))))
8	7	simprd 114	. . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) → ∀𝑏 ∈ 𝑋 ∀𝑐 ∈ 𝑋 (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)))
9	8	ralrimiva 2605	. 2 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ∀𝑐 ∈ 𝑋 (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)))
10		oveq1 6025	. . . . 5 ⊢ (𝑎 = 𝐴 → (𝑎𝐷𝑏) = (𝐴𝐷𝑏))
11		oveq2 6026	. . . . . 6 ⊢ (𝑎 = 𝐴 → (𝑐𝐷𝑎) = (𝑐𝐷𝐴))
12	11	oveq1d 6033	. . . . 5 ⊢ (𝑎 = 𝐴 → ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)) = ((𝑐𝐷𝐴) +𝑒 (𝑐𝐷𝑏)))
13	10, 12	breq12d 4101	. . . 4 ⊢ (𝑎 = 𝐴 → ((𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)) ↔ (𝐴𝐷𝑏) ≤ ((𝑐𝐷𝐴) +𝑒 (𝑐𝐷𝑏))))
14		oveq2 6026	. . . . 5 ⊢ (𝑏 = 𝐵 → (𝐴𝐷𝑏) = (𝐴𝐷𝐵))
15		oveq2 6026	. . . . . 6 ⊢ (𝑏 = 𝐵 → (𝑐𝐷𝑏) = (𝑐𝐷𝐵))
16	15	oveq2d 6034	. . . . 5 ⊢ (𝑏 = 𝐵 → ((𝑐𝐷𝐴) +𝑒 (𝑐𝐷𝑏)) = ((𝑐𝐷𝐴) +𝑒 (𝑐𝐷𝐵)))
17	14, 16	breq12d 4101	. . . 4 ⊢ (𝑏 = 𝐵 → ((𝐴𝐷𝑏) ≤ ((𝑐𝐷𝐴) +𝑒 (𝑐𝐷𝑏)) ↔ (𝐴𝐷𝐵) ≤ ((𝑐𝐷𝐴) +𝑒 (𝑐𝐷𝐵))))
18		oveq1 6025	. . . . . 6 ⊢ (𝑐 = 𝐶 → (𝑐𝐷𝐴) = (𝐶𝐷𝐴))
19		oveq1 6025	. . . . . 6 ⊢ (𝑐 = 𝐶 → (𝑐𝐷𝐵) = (𝐶𝐷𝐵))
20	18, 19	oveq12d 6036	. . . . 5 ⊢ (𝑐 = 𝐶 → ((𝑐𝐷𝐴) +𝑒 (𝑐𝐷𝐵)) = ((𝐶𝐷𝐴) +𝑒 (𝐶𝐷𝐵)))
21	20	breq2d 4100	. . . 4 ⊢ (𝑐 = 𝐶 → ((𝐴𝐷𝐵) ≤ ((𝑐𝐷𝐴) +𝑒 (𝑐𝐷𝐵)) ↔ (𝐴𝐷𝐵) ≤ ((𝐶𝐷𝐴) +𝑒 (𝐶𝐷𝐵))))
22	13, 17, 21	rspc3v 2926	. . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ∀𝑐 ∈ 𝑋 (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)) → (𝐴𝐷𝐵) ≤ ((𝐶𝐷𝐴) +𝑒 (𝐶𝐷𝐵))))
23	22	3comr 1237	. 2 ⊢ ((𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ∀𝑐 ∈ 𝑋 (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)) → (𝐴𝐷𝐵) ≤ ((𝐶𝐷𝐴) +𝑒 (𝐶𝐷𝐵))))
24	9, 23	mpan9 281	1 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐷𝐵) ≤ ((𝐶𝐷𝐴) +𝑒 (𝐶𝐷𝐵)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1004   = wceq 1397   ∈ wcel 2202  ∀wral 2510  {crab 2514  Vcvv 2802   class class class wbr 4088   × cxp 4723  ⟶wf 5322  ‘cfv 5326  (class class class)co 6018   ↑_𝑚 cmap 6817  0cc0 8032  ℝ^*cxr 8213   ≤ cle 8215   +_𝑒 cxad 10005  PsMetcpsmet 14555

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8123  ax-resscn 8124

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fv 5334  df-ov 6021  df-oprab 6022  df-mpo 6023  df-map 6819  df-pnf 8216  df-mnf 8217  df-xr 8218  df-psmet 14563

This theorem is referenced by:  psmetsym  15059  psmettri  15060  psmetge0  15061  psmetres2  15063  xblss2ps  15134