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Theorem psmet0 23369

Description: The distance function of a pseudometric space is zero if its arguments are equal. (Contributed by Thierry Arnoux, 7-Feb-2018.)

**Assertion**
Ref	Expression
psmet0	⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋) → (𝐴𝐷𝐴) = 0)

Proof of Theorem psmet0 Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elfvex 6789	. . . . . . . 8 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝑋 ∈ V)
2		ispsmet 23365	. . . . . . . 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 266	. . . . . 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 3132	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) → ((𝑎𝐷𝑎) = 0 ∧ ∀𝑏 ∈ 𝑋 ∀𝑐 ∈ 𝑋 (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +_𝑒 (𝑐𝐷𝑏))))
7	6	simpld 494	. . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) → (𝑎𝐷𝑎) = 0)
8	7	ralrimiva 3107	. 2 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ∀𝑎 ∈ 𝑋 (𝑎𝐷𝑎) = 0)
9		id 22	. . . . 5 ⊢ (𝑎 = 𝐴 → 𝑎 = 𝐴)
10	9, 9	oveq12d 7273	. . . 4 ⊢ (𝑎 = 𝐴 → (𝑎𝐷𝑎) = (𝐴𝐷𝐴))
11	10	eqeq1d 2740	. . 3 ⊢ (𝑎 = 𝐴 → ((𝑎𝐷𝑎) = 0 ↔ (𝐴𝐷𝐴) = 0))
12	11	rspcv 3547	. 2 ⊢ (𝐴 ∈ 𝑋 → (∀𝑎 ∈ 𝑋 (𝑎𝐷𝑎) = 0 → (𝐴𝐷𝐴) = 0))
13	8, 12	mpan9 506	1 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝑋) → (𝐴𝐷𝐴) = 0)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1539 ∈ wcel 2108 ∀wral 3063 Vcvv 3422 class class class wbr 5070 × cxp 5578 ⟶wf 6414 ‘cfv 6418 (class class class)co 7255 0cc0 10802 ℝ^*cxr 10939 ≤ cle 10941 +_𝑒 cxad 12775 PsMetcpsmet 20494

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-ral 3068 df-rex 3069 df-rab 3072 df-v 3424 df-sbc 3712 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4837 df-br 5071 df-opab 5133 df-mpt 5154 df-id 5480 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-fv 6426 df-ov 7258 df-oprab 7259 df-mpo 7260 df-map 8575 df-xr 10944 df-psmet 20502

This theorem is referenced by: psmetsym 23371 psmetge0 23373 psmetres2 23375 distspace 23377 xblcntrps 23471 ssblps 23483 metustid 23616 metider 31746 pstmfval 31748

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