Theorem psmettri2 24204

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 6899	. . . . . . . 8 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝑋 ∈ V)
2		ispsmet 24199	. . . . . . . 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 3230	. . . 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 3126	. 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 5123	. . . 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 5123	. . . 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 5122	. . . 4 ⊢ (𝑐 = 𝐶 → ((𝐴𝐷𝐵) ≤ ((𝑐𝐷𝐴) +𝑒 (𝑐𝐷𝐵)) ↔ (𝐴𝐷𝐵) ≤ ((𝐶𝐷𝐴) +𝑒 (𝐶𝐷𝐵))))
21	12, 16, 20	rspc3v 3607	. . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ∀𝑐 ∈ 𝑋 (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)) → (𝐴𝐷𝐵) ≤ ((𝐶𝐷𝐴) +𝑒 (𝐶𝐷𝐵))))
22	21	3comr 1125	. 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 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3045  Vcvv 3450   class class class wbr 5110   × cxp 5639  ⟶wf 6510  ‘cfv 6514  (class class class)co 7390  0cc0 11075  ℝ^*cxr 11214   ≤ cle 11216   +_𝑒 cxad 13077  PsMetcpsmet 21255

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-resscn 11132

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-map 8804  df-xr 11219  df-psmet 21263

This theorem is referenced by:  psmetsym  24205  psmettri  24206  psmetge0  24207  psmetres2  24209  xblss2ps  24296  metideq  33890  metider  33891  pstmxmet  33894