Theorem psmettri2 24356

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 6896	. . . . . . . 8 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝑋 ∈ V)
2		ispsmet 24351	. . . . . . . 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 269	. . . . . 6 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (𝐷:(𝑋 × 𝑋)⟶ℝ* ∧ ∀𝑎 ∈ 𝑋 ((𝑎𝐷𝑎) = 0 ∧ ∀𝑏 ∈ 𝑋 ∀𝑐 ∈ 𝑋 (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)))))
5	4	simprd 499	. . . . 5 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ∀𝑎 ∈ 𝑋 ((𝑎𝐷𝑎) = 0 ∧ ∀𝑏 ∈ 𝑋 ∀𝑐 ∈ 𝑋 (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏))))
6	5	r19.21bi 3253	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) → ((𝑎𝐷𝑎) = 0 ∧ ∀𝑏 ∈ 𝑋 ∀𝑐 ∈ 𝑋 (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏))))
7	6	simprd 499	. . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) → ∀𝑏 ∈ 𝑋 ∀𝑐 ∈ 𝑋 (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)))
8	7	ralrimiva 3153	. 2 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ∀𝑐 ∈ 𝑋 (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)))
9		oveq1 7397	. . . . 5 ⊢ (𝑎 = 𝐴 → (𝑎𝐷𝑏) = (𝐴𝐷𝑏))
10		oveq2 7398	. . . . . 6 ⊢ (𝑎 = 𝐴 → (𝑐𝐷𝑎) = (𝑐𝐷𝐴))
11	10	oveq1d 7405	. . . . 5 ⊢ (𝑎 = 𝐴 → ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)) = ((𝑐𝐷𝐴) +𝑒 (𝑐𝐷𝑏)))
12	9, 11	breq12d 5110	. . . 4 ⊢ (𝑎 = 𝐴 → ((𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)) ↔ (𝐴𝐷𝑏) ≤ ((𝑐𝐷𝐴) +𝑒 (𝑐𝐷𝑏))))
13		oveq2 7398	. . . . 5 ⊢ (𝑏 = 𝐵 → (𝐴𝐷𝑏) = (𝐴𝐷𝐵))
14		oveq2 7398	. . . . . 6 ⊢ (𝑏 = 𝐵 → (𝑐𝐷𝑏) = (𝑐𝐷𝐵))
15	14	oveq2d 7406	. . . . 5 ⊢ (𝑏 = 𝐵 → ((𝑐𝐷𝐴) +𝑒 (𝑐𝐷𝑏)) = ((𝑐𝐷𝐴) +𝑒 (𝑐𝐷𝐵)))
16	13, 15	breq12d 5110	. . . 4 ⊢ (𝑏 = 𝐵 → ((𝐴𝐷𝑏) ≤ ((𝑐𝐷𝐴) +𝑒 (𝑐𝐷𝑏)) ↔ (𝐴𝐷𝐵) ≤ ((𝑐𝐷𝐴) +𝑒 (𝑐𝐷𝐵))))
17		oveq1 7397	. . . . . 6 ⊢ (𝑐 = 𝐶 → (𝑐𝐷𝐴) = (𝐶𝐷𝐴))
18		oveq1 7397	. . . . . 6 ⊢ (𝑐 = 𝐶 → (𝑐𝐷𝐵) = (𝐶𝐷𝐵))
19	17, 18	oveq12d 7408	. . . . 5 ⊢ (𝑐 = 𝐶 → ((𝑐𝐷𝐴) +𝑒 (𝑐𝐷𝐵)) = ((𝐶𝐷𝐴) +𝑒 (𝐶𝐷𝐵)))
20	19	breq2d 5109	. . . 4 ⊢ (𝑐 = 𝐶 → ((𝐴𝐷𝐵) ≤ ((𝑐𝐷𝐴) +𝑒 (𝑐𝐷𝐵)) ↔ (𝐴𝐷𝐵) ≤ ((𝐶𝐷𝐴) +𝑒 (𝐶𝐷𝐵))))
21	12, 16, 20	rspc3v 3596	. . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ∀𝑐 ∈ 𝑋 (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)) → (𝐴𝐷𝐵) ≤ ((𝐶𝐷𝐴) +𝑒 (𝐶𝐷𝐵))))
22	21	3comr 1137	. 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 514	1 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐷𝐵) ≤ ((𝐶𝐷𝐴) +𝑒 (𝐶𝐷𝐵)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141  ∀wral 3075  Vcvv 3453   class class class wbr 5097   × cxp 5641  ⟶wf 6511  ‘cfv 6515  (class class class)co 7390  0cc0 11066  ℝ^*cxr 11208   ≤ cle 11210   +_𝑒 cxad 13105  PsMetcpsmet 21395

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7712  ax-cnex 11122  ax-resscn 11123

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-sbc 3743  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-iota 6471  df-fun 6517  df-fn 6518  df-f 6519  df-fv 6523  df-ov 7393  df-oprab 7394  df-mpo 7395  df-map 8803  df-xr 11213  df-psmet 21403

This theorem is referenced by:  psmetsym  24357  psmettri  24358  psmetge0  24359  psmetres2  24361  xblss2ps  24448  metideq  34150  metider  34151  pstmxmet  34154