Theorem psmettri2 24319

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 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elfvex 6944	. . . . . . . 8 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝑋 ∈ V)
2		ispsmet 24314	. . . . . . . 8 ⊢ (𝑋 ∈ V → (𝐷 ∈ (PsMet‘𝑋) ↔ (𝐷:(𝑋 × 𝑋)⟶ℝ* ∧ ∀𝑎 ∈ 𝑋 ((𝑎𝐷𝑎) = 0 ∧ ∀𝑏 ∈ 𝑋 ∀𝑐 ∈ 𝑋 (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏))))))
3	1, 2	syl 17	. . . . . . 7 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (𝐷 ∈ (PsMet‘𝑋) ↔ (𝐷:(𝑋 × 𝑋)⟶ℝ* ∧ ∀𝑎 ∈ 𝑋 ((𝑎𝐷𝑎) = 0 ∧ ∀𝑏 ∈ 𝑋 ∀𝑐 ∈ 𝑋 (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏))))))
4	3	ibi 267	. . . . . 6 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (𝐷:(𝑋 × 𝑋)⟶ℝ* ∧ ∀𝑎 ∈ 𝑋 ((𝑎𝐷𝑎) = 0 ∧ ∀𝑏 ∈ 𝑋 ∀𝑐 ∈ 𝑋 (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)))))
5	4	simprd 495	. . . . 5 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ∀𝑎 ∈ 𝑋 ((𝑎𝐷𝑎) = 0 ∧ ∀𝑏 ∈ 𝑋 ∀𝑐 ∈ 𝑋 (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏))))
6	5	r19.21bi 3251	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) → ((𝑎𝐷𝑎) = 0 ∧ ∀𝑏 ∈ 𝑋 ∀𝑐 ∈ 𝑋 (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏))))
7	6	simprd 495	. . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) → ∀𝑏 ∈ 𝑋 ∀𝑐 ∈ 𝑋 (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)))
8	7	ralrimiva 3146	. 2 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ∀𝑐 ∈ 𝑋 (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)))
9		oveq1 7438	. . . . 5 ⊢ (𝑎 = 𝐴 → (𝑎𝐷𝑏) = (𝐴𝐷𝑏))
10		oveq2 7439	. . . . . 6 ⊢ (𝑎 = 𝐴 → (𝑐𝐷𝑎) = (𝑐𝐷𝐴))
11	10	oveq1d 7446	. . . . 5 ⊢ (𝑎 = 𝐴 → ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)) = ((𝑐𝐷𝐴) +𝑒 (𝑐𝐷𝑏)))
12	9, 11	breq12d 5156	. . . 4 ⊢ (𝑎 = 𝐴 → ((𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)) ↔ (𝐴𝐷𝑏) ≤ ((𝑐𝐷𝐴) +𝑒 (𝑐𝐷𝑏))))
13		oveq2 7439	. . . . 5 ⊢ (𝑏 = 𝐵 → (𝐴𝐷𝑏) = (𝐴𝐷𝐵))
14		oveq2 7439	. . . . . 6 ⊢ (𝑏 = 𝐵 → (𝑐𝐷𝑏) = (𝑐𝐷𝐵))
15	14	oveq2d 7447	. . . . 5 ⊢ (𝑏 = 𝐵 → ((𝑐𝐷𝐴) +𝑒 (𝑐𝐷𝑏)) = ((𝑐𝐷𝐴) +𝑒 (𝑐𝐷𝐵)))
16	13, 15	breq12d 5156	. . . 4 ⊢ (𝑏 = 𝐵 → ((𝐴𝐷𝑏) ≤ ((𝑐𝐷𝐴) +𝑒 (𝑐𝐷𝑏)) ↔ (𝐴𝐷𝐵) ≤ ((𝑐𝐷𝐴) +𝑒 (𝑐𝐷𝐵))))
17		oveq1 7438	. . . . . 6 ⊢ (𝑐 = 𝐶 → (𝑐𝐷𝐴) = (𝐶𝐷𝐴))
18		oveq1 7438	. . . . . 6 ⊢ (𝑐 = 𝐶 → (𝑐𝐷𝐵) = (𝐶𝐷𝐵))
19	17, 18	oveq12d 7449	. . . . 5 ⊢ (𝑐 = 𝐶 → ((𝑐𝐷𝐴) +𝑒 (𝑐𝐷𝐵)) = ((𝐶𝐷𝐴) +𝑒 (𝐶𝐷𝐵)))
20	19	breq2d 5155	. . . 4 ⊢ (𝑐 = 𝐶 → ((𝐴𝐷𝐵) ≤ ((𝑐𝐷𝐴) +𝑒 (𝑐𝐷𝐵)) ↔ (𝐴𝐷𝐵) ≤ ((𝐶𝐷𝐴) +𝑒 (𝐶𝐷𝐵))))
21	12, 16, 20	rspc3v 3638	. . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ∀𝑐 ∈ 𝑋 (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)) → (𝐴𝐷𝐵) ≤ ((𝐶𝐷𝐴) +𝑒 (𝐶𝐷𝐵))))
22	21	3comr 1126	. 2 ⊢ ((𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ∀𝑐 ∈ 𝑋 (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)) → (𝐴𝐷𝐵) ≤ ((𝐶𝐷𝐴) +𝑒 (𝐶𝐷𝐵))))
23	8, 22	mpan9 506	1 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐷𝐵) ≤ ((𝐶𝐷𝐴) +𝑒 (𝐶𝐷𝐵)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108  ∀wral 3061  Vcvv 3480   class class class wbr 5143   × cxp 5683  ⟶wf 6557  ‘cfv 6561  (class class class)co 7431  0cc0 11155  ℝ^*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-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:  psmetsym  24320  psmettri  24321  psmetge0  24322  psmetres2  24324  xblss2ps  24411  metideq  33892  metider  33893  pstmxmet  33896