Theorem pstmxmet 33843

Description: The metric induced by a pseudometric is a full-fledged metric on the equivalence classes of the metric identification. (Contributed by Thierry Arnoux, 11-Feb-2018.)

Hypothesis

Ref	Expression
pstmval.1	⊢ ∼ = (~Met‘𝐷)

Assertion

Ref	Expression
pstmxmet	⊢ (𝐷 ∈ (PsMet‘𝑋) → (pstoMet‘𝐷) ∈ (∞Met‘(𝑋 / ∼ )))

Proof of Theorem pstmxmet

Dummy variables 𝑎 𝑏 𝑐 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2740	. . . . 5 ⊢ (𝑥 ∈ (𝑋 / ∼ ), 𝑦 ∈ (𝑋 / ∼ ) ↦ ∪ {𝑧 ∣ ∃𝑎 ∈ 𝑥 ∃𝑏 ∈ 𝑦 𝑧 = (𝑎𝐷𝑏)}) = (𝑥 ∈ (𝑋 / ∼ ), 𝑦 ∈ (𝑋 / ∼ ) ↦ ∪ {𝑧 ∣ ∃𝑎 ∈ 𝑥 ∃𝑏 ∈ 𝑦 𝑧 = (𝑎𝐷𝑏)})
2		vex 3492	. . . . . . 7 ⊢ 𝑥 ∈ V
3		vex 3492	. . . . . . 7 ⊢ 𝑦 ∈ V
4	2, 3	ab2rexex 8020	. . . . . 6 ⊢ {𝑧 ∣ ∃𝑎 ∈ 𝑥 ∃𝑏 ∈ 𝑦 𝑧 = (𝑎𝐷𝑏)} ∈ V
5	4	uniex 7776	. . . . 5 ⊢ ∪ {𝑧 ∣ ∃𝑎 ∈ 𝑥 ∃𝑏 ∈ 𝑦 𝑧 = (𝑎𝐷𝑏)} ∈ V
6	1, 5	fnmpoi 8111	. . . 4 ⊢ (𝑥 ∈ (𝑋 / ∼ ), 𝑦 ∈ (𝑋 / ∼ ) ↦ ∪ {𝑧 ∣ ∃𝑎 ∈ 𝑥 ∃𝑏 ∈ 𝑦 𝑧 = (𝑎𝐷𝑏)}) Fn ((𝑋 / ∼ ) × (𝑋 / ∼ ))
7		pstmval.1	. . . . . 6 ⊢ ∼ = (~Met‘𝐷)
8	7	pstmval 33841	. . . . 5 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (pstoMet‘𝐷) = (𝑥 ∈ (𝑋 / ∼ ), 𝑦 ∈ (𝑋 / ∼ ) ↦ ∪ {𝑧 ∣ ∃𝑎 ∈ 𝑥 ∃𝑏 ∈ 𝑦 𝑧 = (𝑎𝐷𝑏)}))
9	8	fneq1d 6672	. . . 4 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ((pstoMet‘𝐷) Fn ((𝑋 / ∼ ) × (𝑋 / ∼ )) ↔ (𝑥 ∈ (𝑋 / ∼ ), 𝑦 ∈ (𝑋 / ∼ ) ↦ ∪ {𝑧 ∣ ∃𝑎 ∈ 𝑥 ∃𝑏 ∈ 𝑦 𝑧 = (𝑎𝐷𝑏)}) Fn ((𝑋 / ∼ ) × (𝑋 / ∼ ))))
10	6, 9	mpbiri 258	. . 3 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (pstoMet‘𝐷) Fn ((𝑋 / ∼ ) × (𝑋 / ∼ )))
11		simpllr 775	. . . . . . . . 9 ⊢ ((((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ∼ ) ∧ 𝑦 ∈ (𝑋 / ∼ ))) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) → 𝑥 = [𝑎] ∼ )
12		simpr 484	. . . . . . . . 9 ⊢ ((((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ∼ ) ∧ 𝑦 ∈ (𝑋 / ∼ ))) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) → 𝑦 = [𝑏] ∼ )
13	11, 12	oveq12d 7466	. . . . . . . 8 ⊢ ((((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ∼ ) ∧ 𝑦 ∈ (𝑋 / ∼ ))) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) → (𝑥(pstoMet‘𝐷)𝑦) = ([𝑎] ∼ (pstoMet‘𝐷)[𝑏] ∼ ))
14		simp-5l 784	. . . . . . . . 9 ⊢ ((((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ∼ ) ∧ 𝑦 ∈ (𝑋 / ∼ ))) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) → 𝐷 ∈ (PsMet‘𝑋))
15		simp-4r 783	. . . . . . . . 9 ⊢ ((((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ∼ ) ∧ 𝑦 ∈ (𝑋 / ∼ ))) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) → 𝑎 ∈ 𝑋)
16		simplr 768	. . . . . . . . 9 ⊢ ((((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ∼ ) ∧ 𝑦 ∈ (𝑋 / ∼ ))) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) → 𝑏 ∈ 𝑋)
17	7	pstmfval 33842	. . . . . . . . 9 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) → ([𝑎] ∼ (pstoMet‘𝐷)[𝑏] ∼ ) = (𝑎𝐷𝑏))
18	14, 15, 16, 17	syl3anc 1371	. . . . . . . 8 ⊢ ((((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ∼ ) ∧ 𝑦 ∈ (𝑋 / ∼ ))) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) → ([𝑎] ∼ (pstoMet‘𝐷)[𝑏] ∼ ) = (𝑎𝐷𝑏))
19	13, 18	eqtrd 2780	. . . . . . 7 ⊢ ((((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ∼ ) ∧ 𝑦 ∈ (𝑋 / ∼ ))) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) → (𝑥(pstoMet‘𝐷)𝑦) = (𝑎𝐷𝑏))
20		psmetf 24337	. . . . . . . . 9 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
21	14, 20	syl 17	. . . . . . . 8 ⊢ ((((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ∼ ) ∧ 𝑦 ∈ (𝑋 / ∼ ))) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
22	21, 15, 16	fovcdmd 7622	. . . . . . 7 ⊢ ((((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ∼ ) ∧ 𝑦 ∈ (𝑋 / ∼ ))) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) → (𝑎𝐷𝑏) ∈ ℝ*)
23	19, 22	eqeltrd 2844	. . . . . 6 ⊢ ((((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ∼ ) ∧ 𝑦 ∈ (𝑋 / ∼ ))) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) → (𝑥(pstoMet‘𝐷)𝑦) ∈ ℝ*)
24		elqsi 8828	. . . . . . . 8 ⊢ (𝑦 ∈ (𝑋 / ∼ ) → ∃𝑏 ∈ 𝑋 𝑦 = [𝑏] ∼ )
25	24	ad2antll 728	. . . . . . 7 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ∼ ) ∧ 𝑦 ∈ (𝑋 / ∼ ))) → ∃𝑏 ∈ 𝑋 𝑦 = [𝑏] ∼ )
26	25	ad2antrr 725	. . . . . 6 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ∼ ) ∧ 𝑦 ∈ (𝑋 / ∼ ))) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) → ∃𝑏 ∈ 𝑋 𝑦 = [𝑏] ∼ )
27	23, 26	r19.29a 3168	. . . . 5 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ∼ ) ∧ 𝑦 ∈ (𝑋 / ∼ ))) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) → (𝑥(pstoMet‘𝐷)𝑦) ∈ ℝ*)
28		elqsi 8828	. . . . . 6 ⊢ (𝑥 ∈ (𝑋 / ∼ ) → ∃𝑎 ∈ 𝑋 𝑥 = [𝑎] ∼ )
29	28	ad2antrl 727	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ∼ ) ∧ 𝑦 ∈ (𝑋 / ∼ ))) → ∃𝑎 ∈ 𝑋 𝑥 = [𝑎] ∼ )
30	27, 29	r19.29a 3168	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ∼ ) ∧ 𝑦 ∈ (𝑋 / ∼ ))) → (𝑥(pstoMet‘𝐷)𝑦) ∈ ℝ*)
31	30	ralrimivva 3208	. . 3 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ∀𝑥 ∈ (𝑋 / ∼ )∀𝑦 ∈ (𝑋 / ∼ )(𝑥(pstoMet‘𝐷)𝑦) ∈ ℝ*)
32		ffnov 7576	. . 3 ⊢ ((pstoMet‘𝐷):((𝑋 / ∼ ) × (𝑋 / ∼ ))⟶ℝ* ↔ ((pstoMet‘𝐷) Fn ((𝑋 / ∼ ) × (𝑋 / ∼ )) ∧ ∀𝑥 ∈ (𝑋 / ∼ )∀𝑦 ∈ (𝑋 / ∼ )(𝑥(pstoMet‘𝐷)𝑦) ∈ ℝ*))
33	10, 31, 32	sylanbrc 582	. 2 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (pstoMet‘𝐷):((𝑋 / ∼ ) × (𝑋 / ∼ ))⟶ℝ*)
34	17	3expa 1118	. . . . . . . . . . . 12 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) ∧ 𝑏 ∈ 𝑋) → ([𝑎] ∼ (pstoMet‘𝐷)[𝑏] ∼ ) = (𝑎𝐷𝑏))
35	34	eqeq1d 2742	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) ∧ 𝑏 ∈ 𝑋) → (([𝑎] ∼ (pstoMet‘𝐷)[𝑏] ∼ ) = 0 ↔ (𝑎𝐷𝑏) = 0))
36	7	breqi 5172	. . . . . . . . . . . 12 ⊢ (𝑎 ∼ 𝑏 ↔ 𝑎(~Met‘𝐷)𝑏)
37		metidv 33838	. . . . . . . . . . . . 13 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (𝑎(~Met‘𝐷)𝑏 ↔ (𝑎𝐷𝑏) = 0))
38	37	anassrs 467	. . . . . . . . . . . 12 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) ∧ 𝑏 ∈ 𝑋) → (𝑎(~Met‘𝐷)𝑏 ↔ (𝑎𝐷𝑏) = 0))
39	36, 38	bitrid 283	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) ∧ 𝑏 ∈ 𝑋) → (𝑎 ∼ 𝑏 ↔ (𝑎𝐷𝑏) = 0))
40		metider 33840	. . . . . . . . . . . . . 14 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (~Met‘𝐷) Er 𝑋)
41	40	ad2antrr 725	. . . . . . . . . . . . 13 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) ∧ 𝑏 ∈ 𝑋) → (~Met‘𝐷) Er 𝑋)
42		ereq1 8770	. . . . . . . . . . . . . 14 ⊢ ( ∼ = (~Met‘𝐷) → ( ∼ Er 𝑋 ↔ (~Met‘𝐷) Er 𝑋))
43	7, 42	ax-mp 5	. . . . . . . . . . . . 13 ⊢ ( ∼ Er 𝑋 ↔ (~Met‘𝐷) Er 𝑋)
44	41, 43	sylibr 234	. . . . . . . . . . . 12 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) ∧ 𝑏 ∈ 𝑋) → ∼ Er 𝑋)
45		simplr 768	. . . . . . . . . . . 12 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) ∧ 𝑏 ∈ 𝑋) → 𝑎 ∈ 𝑋)
46	44, 45	erth 8814	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) ∧ 𝑏 ∈ 𝑋) → (𝑎 ∼ 𝑏 ↔ [𝑎] ∼ = [𝑏] ∼ ))
47	35, 39, 46	3bitr2d 307	. . . . . . . . . 10 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) ∧ 𝑏 ∈ 𝑋) → (([𝑎] ∼ (pstoMet‘𝐷)[𝑏] ∼ ) = 0 ↔ [𝑎] ∼ = [𝑏] ∼ ))
48	47	adantllr 718	. . . . . . . . 9 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ∼ ) ∧ 𝑦 ∈ (𝑋 / ∼ ))) ∧ 𝑎 ∈ 𝑋) ∧ 𝑏 ∈ 𝑋) → (([𝑎] ∼ (pstoMet‘𝐷)[𝑏] ∼ ) = 0 ↔ [𝑎] ∼ = [𝑏] ∼ ))
49	48	adantlr 714	. . . . . . . 8 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ∼ ) ∧ 𝑦 ∈ (𝑋 / ∼ ))) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) → (([𝑎] ∼ (pstoMet‘𝐷)[𝑏] ∼ ) = 0 ↔ [𝑎] ∼ = [𝑏] ∼ ))
50	49	adantr 480	. . . . . . 7 ⊢ ((((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ∼ ) ∧ 𝑦 ∈ (𝑋 / ∼ ))) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) → (([𝑎] ∼ (pstoMet‘𝐷)[𝑏] ∼ ) = 0 ↔ [𝑎] ∼ = [𝑏] ∼ ))
51	13	eqeq1d 2742	. . . . . . 7 ⊢ ((((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ∼ ) ∧ 𝑦 ∈ (𝑋 / ∼ ))) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) → ((𝑥(pstoMet‘𝐷)𝑦) = 0 ↔ ([𝑎] ∼ (pstoMet‘𝐷)[𝑏] ∼ ) = 0))
52	11, 12	eqeq12d 2756	. . . . . . 7 ⊢ ((((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ∼ ) ∧ 𝑦 ∈ (𝑋 / ∼ ))) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) → (𝑥 = 𝑦 ↔ [𝑎] ∼ = [𝑏] ∼ ))
53	50, 51, 52	3bitr4d 311	. . . . . 6 ⊢ ((((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ∼ ) ∧ 𝑦 ∈ (𝑋 / ∼ ))) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) → ((𝑥(pstoMet‘𝐷)𝑦) = 0 ↔ 𝑥 = 𝑦))
54	53, 26	r19.29a 3168	. . . . 5 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ∼ ) ∧ 𝑦 ∈ (𝑋 / ∼ ))) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) → ((𝑥(pstoMet‘𝐷)𝑦) = 0 ↔ 𝑥 = 𝑦))
55	54, 29	r19.29a 3168	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ∼ ) ∧ 𝑦 ∈ (𝑋 / ∼ ))) → ((𝑥(pstoMet‘𝐷)𝑦) = 0 ↔ 𝑥 = 𝑦))
56		simp-6l 786	. . . . . . . . . . . . . 14 ⊢ (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) ∧ 𝑐 ∈ 𝑋) ∧ 𝑧 = [𝑐] ∼ ) → 𝐷 ∈ (PsMet‘𝑋))
57		simplr 768	. . . . . . . . . . . . . 14 ⊢ (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) ∧ 𝑐 ∈ 𝑋) ∧ 𝑧 = [𝑐] ∼ ) → 𝑐 ∈ 𝑋)
58		simp-6r 787	. . . . . . . . . . . . . 14 ⊢ (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) ∧ 𝑐 ∈ 𝑋) ∧ 𝑧 = [𝑐] ∼ ) → 𝑎 ∈ 𝑋)
59		simp-4r 783	. . . . . . . . . . . . . 14 ⊢ (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) ∧ 𝑐 ∈ 𝑋) ∧ 𝑧 = [𝑐] ∼ ) → 𝑏 ∈ 𝑋)
60		psmettri2 24340	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑐 ∈ 𝑋 ∧ 𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)))
61	56, 57, 58, 59, 60	syl13anc 1372	. . . . . . . . . . . . 13 ⊢ (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) ∧ 𝑐 ∈ 𝑋) ∧ 𝑧 = [𝑐] ∼ ) → (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)))
62		simp-5r 785	. . . . . . . . . . . . . . 15 ⊢ (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) ∧ 𝑐 ∈ 𝑋) ∧ 𝑧 = [𝑐] ∼ ) → 𝑥 = [𝑎] ∼ )
63		simpllr 775	. . . . . . . . . . . . . . 15 ⊢ (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) ∧ 𝑐 ∈ 𝑋) ∧ 𝑧 = [𝑐] ∼ ) → 𝑦 = [𝑏] ∼ )
64	62, 63	oveq12d 7466	. . . . . . . . . . . . . 14 ⊢ (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) ∧ 𝑐 ∈ 𝑋) ∧ 𝑧 = [𝑐] ∼ ) → (𝑥(pstoMet‘𝐷)𝑦) = ([𝑎] ∼ (pstoMet‘𝐷)[𝑏] ∼ ))
65	56, 58, 59, 17	syl3anc 1371	. . . . . . . . . . . . . 14 ⊢ (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) ∧ 𝑐 ∈ 𝑋) ∧ 𝑧 = [𝑐] ∼ ) → ([𝑎] ∼ (pstoMet‘𝐷)[𝑏] ∼ ) = (𝑎𝐷𝑏))
66	64, 65	eqtrd 2780	. . . . . . . . . . . . 13 ⊢ (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) ∧ 𝑐 ∈ 𝑋) ∧ 𝑧 = [𝑐] ∼ ) → (𝑥(pstoMet‘𝐷)𝑦) = (𝑎𝐷𝑏))
67		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) ∧ 𝑐 ∈ 𝑋) ∧ 𝑧 = [𝑐] ∼ ) → 𝑧 = [𝑐] ∼ )
68	67, 62	oveq12d 7466	. . . . . . . . . . . . . . 15 ⊢ (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) ∧ 𝑐 ∈ 𝑋) ∧ 𝑧 = [𝑐] ∼ ) → (𝑧(pstoMet‘𝐷)𝑥) = ([𝑐] ∼ (pstoMet‘𝐷)[𝑎] ∼ ))
69	7	pstmfval 33842	. . . . . . . . . . . . . . . 16 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑐 ∈ 𝑋 ∧ 𝑎 ∈ 𝑋) → ([𝑐] ∼ (pstoMet‘𝐷)[𝑎] ∼ ) = (𝑐𝐷𝑎))
70	56, 57, 58, 69	syl3anc 1371	. . . . . . . . . . . . . . 15 ⊢ (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) ∧ 𝑐 ∈ 𝑋) ∧ 𝑧 = [𝑐] ∼ ) → ([𝑐] ∼ (pstoMet‘𝐷)[𝑎] ∼ ) = (𝑐𝐷𝑎))
71	68, 70	eqtrd 2780	. . . . . . . . . . . . . 14 ⊢ (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) ∧ 𝑐 ∈ 𝑋) ∧ 𝑧 = [𝑐] ∼ ) → (𝑧(pstoMet‘𝐷)𝑥) = (𝑐𝐷𝑎))
72	67, 63	oveq12d 7466	. . . . . . . . . . . . . . 15 ⊢ (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) ∧ 𝑐 ∈ 𝑋) ∧ 𝑧 = [𝑐] ∼ ) → (𝑧(pstoMet‘𝐷)𝑦) = ([𝑐] ∼ (pstoMet‘𝐷)[𝑏] ∼ ))
73	7	pstmfval 33842	. . . . . . . . . . . . . . . 16 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑐 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) → ([𝑐] ∼ (pstoMet‘𝐷)[𝑏] ∼ ) = (𝑐𝐷𝑏))
74	56, 57, 59, 73	syl3anc 1371	. . . . . . . . . . . . . . 15 ⊢ (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) ∧ 𝑐 ∈ 𝑋) ∧ 𝑧 = [𝑐] ∼ ) → ([𝑐] ∼ (pstoMet‘𝐷)[𝑏] ∼ ) = (𝑐𝐷𝑏))
75	72, 74	eqtrd 2780	. . . . . . . . . . . . . 14 ⊢ (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) ∧ 𝑐 ∈ 𝑋) ∧ 𝑧 = [𝑐] ∼ ) → (𝑧(pstoMet‘𝐷)𝑦) = (𝑐𝐷𝑏))
76	71, 75	oveq12d 7466	. . . . . . . . . . . . 13 ⊢ (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) ∧ 𝑐 ∈ 𝑋) ∧ 𝑧 = [𝑐] ∼ ) → ((𝑧(pstoMet‘𝐷)𝑥) +𝑒 (𝑧(pstoMet‘𝐷)𝑦)) = ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)))
77	61, 66, 76	3brtr4d 5198	. . . . . . . . . . . 12 ⊢ (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) ∧ 𝑐 ∈ 𝑋) ∧ 𝑧 = [𝑐] ∼ ) → (𝑥(pstoMet‘𝐷)𝑦) ≤ ((𝑧(pstoMet‘𝐷)𝑥) +𝑒 (𝑧(pstoMet‘𝐷)𝑦)))
78	77	adantl6r 763	. . . . . . . . . . 11 ⊢ ((((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑧 ∈ (𝑋 / ∼ )) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) ∧ 𝑐 ∈ 𝑋) ∧ 𝑧 = [𝑐] ∼ ) → (𝑥(pstoMet‘𝐷)𝑦) ≤ ((𝑧(pstoMet‘𝐷)𝑥) +𝑒 (𝑧(pstoMet‘𝐷)𝑦)))
79		elqsi 8828	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (𝑋 / ∼ ) → ∃𝑐 ∈ 𝑋 𝑧 = [𝑐] ∼ )
80	79	ad5antlr 734	. . . . . . . . . . 11 ⊢ ((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑧 ∈ (𝑋 / ∼ )) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) → ∃𝑐 ∈ 𝑋 𝑧 = [𝑐] ∼ )
81	78, 80	r19.29a 3168	. . . . . . . . . 10 ⊢ ((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑧 ∈ (𝑋 / ∼ )) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) → (𝑥(pstoMet‘𝐷)𝑦) ≤ ((𝑧(pstoMet‘𝐷)𝑥) +𝑒 (𝑧(pstoMet‘𝐷)𝑦)))
82	81	adantl5r 762	. . . . . . . . 9 ⊢ (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦 ∈ (𝑋 / ∼ )) ∧ 𝑧 ∈ (𝑋 / ∼ )) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) → (𝑥(pstoMet‘𝐷)𝑦) ≤ ((𝑧(pstoMet‘𝐷)𝑥) +𝑒 (𝑧(pstoMet‘𝐷)𝑦)))
83	24	ad4antlr 732	. . . . . . . . 9 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦 ∈ (𝑋 / ∼ )) ∧ 𝑧 ∈ (𝑋 / ∼ )) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) → ∃𝑏 ∈ 𝑋 𝑦 = [𝑏] ∼ )
84	82, 83	r19.29a 3168	. . . . . . . 8 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦 ∈ (𝑋 / ∼ )) ∧ 𝑧 ∈ (𝑋 / ∼ )) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) → (𝑥(pstoMet‘𝐷)𝑦) ≤ ((𝑧(pstoMet‘𝐷)𝑥) +𝑒 (𝑧(pstoMet‘𝐷)𝑦)))
85	84	adantl4r 754	. . . . . . 7 ⊢ ((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 ∈ (𝑋 / ∼ )) ∧ 𝑦 ∈ (𝑋 / ∼ )) ∧ 𝑧 ∈ (𝑋 / ∼ )) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) → (𝑥(pstoMet‘𝐷)𝑦) ≤ ((𝑧(pstoMet‘𝐷)𝑥) +𝑒 (𝑧(pstoMet‘𝐷)𝑦)))
86	28	ad3antlr 730	. . . . . . 7 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 ∈ (𝑋 / ∼ )) ∧ 𝑦 ∈ (𝑋 / ∼ )) ∧ 𝑧 ∈ (𝑋 / ∼ )) → ∃𝑎 ∈ 𝑋 𝑥 = [𝑎] ∼ )
87	85, 86	r19.29a 3168	. . . . . 6 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 ∈ (𝑋 / ∼ )) ∧ 𝑦 ∈ (𝑋 / ∼ )) ∧ 𝑧 ∈ (𝑋 / ∼ )) → (𝑥(pstoMet‘𝐷)𝑦) ≤ ((𝑧(pstoMet‘𝐷)𝑥) +𝑒 (𝑧(pstoMet‘𝐷)𝑦)))
88	87	ralrimiva 3152	. . . . 5 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 ∈ (𝑋 / ∼ )) ∧ 𝑦 ∈ (𝑋 / ∼ )) → ∀𝑧 ∈ (𝑋 / ∼ )(𝑥(pstoMet‘𝐷)𝑦) ≤ ((𝑧(pstoMet‘𝐷)𝑥) +𝑒 (𝑧(pstoMet‘𝐷)𝑦)))
89	88	anasss 466	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ∼ ) ∧ 𝑦 ∈ (𝑋 / ∼ ))) → ∀𝑧 ∈ (𝑋 / ∼ )(𝑥(pstoMet‘𝐷)𝑦) ≤ ((𝑧(pstoMet‘𝐷)𝑥) +𝑒 (𝑧(pstoMet‘𝐷)𝑦)))
90	55, 89	jca 511	. . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ∼ ) ∧ 𝑦 ∈ (𝑋 / ∼ ))) → (((𝑥(pstoMet‘𝐷)𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ (𝑋 / ∼ )(𝑥(pstoMet‘𝐷)𝑦) ≤ ((𝑧(pstoMet‘𝐷)𝑥) +𝑒 (𝑧(pstoMet‘𝐷)𝑦))))
91	90	ralrimivva 3208	. 2 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ∀𝑥 ∈ (𝑋 / ∼ )∀𝑦 ∈ (𝑋 / ∼ )(((𝑥(pstoMet‘𝐷)𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ (𝑋 / ∼ )(𝑥(pstoMet‘𝐷)𝑦) ≤ ((𝑧(pstoMet‘𝐷)𝑥) +𝑒 (𝑧(pstoMet‘𝐷)𝑦))))
92		elfvex 6958	. . 3 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝑋 ∈ V)
93		qsexg 8833	. . 3 ⊢ (𝑋 ∈ V → (𝑋 / ∼ ) ∈ V)
94		isxmet 24355	. . 3 ⊢ ((𝑋 / ∼ ) ∈ V → ((pstoMet‘𝐷) ∈ (∞Met‘(𝑋 / ∼ )) ↔ ((pstoMet‘𝐷):((𝑋 / ∼ ) × (𝑋 / ∼ ))⟶ℝ* ∧ ∀𝑥 ∈ (𝑋 / ∼ )∀𝑦 ∈ (𝑋 / ∼ )(((𝑥(pstoMet‘𝐷)𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ (𝑋 / ∼ )(𝑥(pstoMet‘𝐷)𝑦) ≤ ((𝑧(pstoMet‘𝐷)𝑥) +𝑒 (𝑧(pstoMet‘𝐷)𝑦))))))
95	92, 93, 94	3syl 18	. 2 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ((pstoMet‘𝐷) ∈ (∞Met‘(𝑋 / ∼ )) ↔ ((pstoMet‘𝐷):((𝑋 / ∼ ) × (𝑋 / ∼ ))⟶ℝ* ∧ ∀𝑥 ∈ (𝑋 / ∼ )∀𝑦 ∈ (𝑋 / ∼ )(((𝑥(pstoMet‘𝐷)𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ (𝑋 / ∼ )(𝑥(pstoMet‘𝐷)𝑦) ≤ ((𝑧(pstoMet‘𝐷)𝑥) +𝑒 (𝑧(pstoMet‘𝐷)𝑦))))))
96	33, 91, 95	mpbir2and 712	1 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (pstoMet‘𝐷) ∈ (∞Met‘(𝑋 / ∼ )))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  {cab 2717  ∀wral 3067  ∃wrex 3076  Vcvv 3488  ∪ cuni 4931   class class class wbr 5166   × cxp 5698   Fn wfn 6568  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448   ∈ cmpo 7450   Er wer 8760  [cec 8761   / cqs 8762  0cc0 11184  ℝ^*cxr 11323   ≤ cle 11325   +_𝑒 cxad 13173  PsMetcpsmet 21371  ∞Metcxmet 21372  ~_Metcmetid 33832  pstoMetcpstm 33833

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-po 5607  df-so 5608  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-1st 8030  df-2nd 8031  df-er 8763  df-ec 8765  df-qs 8769  df-map 8886  df-en 9004  df-dom 9005  df-sdom 9006  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-div 11948  df-2 12356  df-rp 13058  df-xneg 13175  df-xadd 13176  df-xmul 13177  df-psmet 21379  df-xmet 21380  df-metid 33834  df-pstm 33835

This theorem is referenced by: (None)