Theorem pstmxmet 34268

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 2763	. . . . 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 3459	. . . . . . 7 ⊢ 𝑥 ∈ V
3		vex 3459	. . . . . . 7 ⊢ 𝑦 ∈ V
4	2, 3	ab2rexex 7977	. . . . . 6 ⊢ {𝑧 ∣ ∃𝑎 ∈ 𝑥 ∃𝑏 ∈ 𝑦 𝑧 = (𝑎𝐷𝑏)} ∈ V
5	4	uniex 7741	. . . . 5 ⊢ ∪ {𝑧 ∣ ∃𝑎 ∈ 𝑥 ∃𝑏 ∈ 𝑦 𝑧 = (𝑎𝐷𝑏)} ∈ V
6	1, 5	fnmpoi 8068	. . . 4 ⊢ (𝑥 ∈ (𝑋 / ∼ ), 𝑦 ∈ (𝑋 / ∼ ) ↦ ∪ {𝑧 ∣ ∃𝑎 ∈ 𝑥 ∃𝑏 ∈ 𝑦 𝑧 = (𝑎𝐷𝑏)}) Fn ((𝑋 / ∼ ) × (𝑋 / ∼ ))
7		pstmval.1	. . . . . 6 ⊢ ∼ = (~Met‘𝐷)
8	7	pstmval 34266	. . . . 5 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (pstoMet‘𝐷) = (𝑥 ∈ (𝑋 / ∼ ), 𝑦 ∈ (𝑋 / ∼ ) ↦ ∪ {𝑧 ∣ ∃𝑎 ∈ 𝑥 ∃𝑏 ∈ 𝑦 𝑧 = (𝑎𝐷𝑏)}))
9	8	fneq1d 6630	. . . 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 261	. . 3 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (pstoMet‘𝐷) Fn ((𝑋 / ∼ ) × (𝑋 / ∼ )))
11		simpllr 787	. . . . . . . . 9 ⊢ ((((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ∼ ) ∧ 𝑦 ∈ (𝑋 / ∼ ))) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) → 𝑥 = [𝑎] ∼ )
12		simpr 489	. . . . . . . . 9 ⊢ ((((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ∼ ) ∧ 𝑦 ∈ (𝑋 / ∼ ))) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) → 𝑦 = [𝑏] ∼ )
13	11, 12	oveq12d 7430	. . . . . . . 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 796	. . . . . . . . 9 ⊢ ((((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ∼ ) ∧ 𝑦 ∈ (𝑋 / ∼ ))) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) → 𝐷 ∈ (PsMet‘𝑋))
15		simp-4r 795	. . . . . . . . 9 ⊢ ((((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ∼ ) ∧ 𝑦 ∈ (𝑋 / ∼ ))) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) → 𝑎 ∈ 𝑋)
16		simplr 780	. . . . . . . . 9 ⊢ ((((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ∼ ) ∧ 𝑦 ∈ (𝑋 / ∼ ))) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) → 𝑏 ∈ 𝑋)
17	7	pstmfval 34267	. . . . . . . . 9 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) → ([𝑎] ∼ (pstoMet‘𝐷)[𝑏] ∼ ) = (𝑎𝐷𝑏))
18	14, 15, 16, 17	syl3anc 1398	. . . . . . . 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 2798	. . . . . . 7 ⊢ ((((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ∼ ) ∧ 𝑦 ∈ (𝑋 / ∼ ))) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) → (𝑥(pstoMet‘𝐷)𝑦) = (𝑎𝐷𝑏))
20		psmetf 24444	. . . . . . . . 9 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
21	14, 20	syl 18	. . . . . . . 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 7584	. . . . . . 7 ⊢ ((((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ∼ ) ∧ 𝑦 ∈ (𝑋 / ∼ ))) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) → (𝑎𝐷𝑏) ∈ ℝ*)
23	19, 22	eqeltrd 2863	. . . . . 6 ⊢ ((((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ∼ ) ∧ 𝑦 ∈ (𝑋 / ∼ ))) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) → (𝑥(pstoMet‘𝐷)𝑦) ∈ ℝ*)
24		elqsi 8764	. . . . . . . 8 ⊢ (𝑦 ∈ (𝑋 / ∼ ) → ∃𝑏 ∈ 𝑋 𝑦 = [𝑏] ∼ )
25	24	ad2antll 741	. . . . . . 7 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ∼ ) ∧ 𝑦 ∈ (𝑋 / ∼ ))) → ∃𝑏 ∈ 𝑋 𝑦 = [𝑏] ∼ )
26	25	ad2antrr 738	. . . . . 6 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ∼ ) ∧ 𝑦 ∈ (𝑋 / ∼ ))) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) → ∃𝑏 ∈ 𝑋 𝑦 = [𝑏] ∼ )
27	23, 26	r19.29a 3173	. . . . 5 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ∼ ) ∧ 𝑦 ∈ (𝑋 / ∼ ))) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) → (𝑥(pstoMet‘𝐷)𝑦) ∈ ℝ*)
28		elqsi 8764	. . . . . 6 ⊢ (𝑥 ∈ (𝑋 / ∼ ) → ∃𝑎 ∈ 𝑋 𝑥 = [𝑎] ∼ )
29	28	ad2antrl 740	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ∼ ) ∧ 𝑦 ∈ (𝑋 / ∼ ))) → ∃𝑎 ∈ 𝑋 𝑥 = [𝑎] ∼ )
30	27, 29	r19.29a 3173	. . . 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 7538	. . 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 594	. 2 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (pstoMet‘𝐷):((𝑋 / ∼ ) × (𝑋 / ∼ ))⟶ℝ*)
34	17	3expa 1136	. . . . . . . . . . . 12 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) ∧ 𝑏 ∈ 𝑋) → ([𝑎] ∼ (pstoMet‘𝐷)[𝑏] ∼ ) = (𝑎𝐷𝑏))
35	34	eqeq1d 2765	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) ∧ 𝑏 ∈ 𝑋) → (([𝑎] ∼ (pstoMet‘𝐷)[𝑏] ∼ ) = 0 ↔ (𝑎𝐷𝑏) = 0))
36	7	breqi 5116	. . . . . . . . . . . 12 ⊢ (𝑎 ∼ 𝑏 ↔ 𝑎(~Met‘𝐷)𝑏)
37		metidv 34263	. . . . . . . . . . . . 13 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (𝑎(~Met‘𝐷)𝑏 ↔ (𝑎𝐷𝑏) = 0))
38	37	anassrs 472	. . . . . . . . . . . 12 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) ∧ 𝑏 ∈ 𝑋) → (𝑎(~Met‘𝐷)𝑏 ↔ (𝑎𝐷𝑏) = 0))
39	36, 38	bitrid 286	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) ∧ 𝑏 ∈ 𝑋) → (𝑎 ∼ 𝑏 ↔ (𝑎𝐷𝑏) = 0))
40		metider 34265	. . . . . . . . . . . . . 14 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (~Met‘𝐷) Er 𝑋)
41	40	ad2antrr 738	. . . . . . . . . . . . 13 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) ∧ 𝑏 ∈ 𝑋) → (~Met‘𝐷) Er 𝑋)
42		ereq1 8703	. . . . . . . . . . . . . 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 237	. . . . . . . . . . . 12 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) ∧ 𝑏 ∈ 𝑋) → ∼ Er 𝑋)
45		simplr 780	. . . . . . . . . . . 12 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) ∧ 𝑏 ∈ 𝑋) → 𝑎 ∈ 𝑋)
46	44, 45	erth 8750	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) ∧ 𝑏 ∈ 𝑋) → (𝑎 ∼ 𝑏 ↔ [𝑎] ∼ = [𝑏] ∼ ))
47	35, 39, 46	3bitr2d 310	. . . . . . . . . 10 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) ∧ 𝑏 ∈ 𝑋) → (([𝑎] ∼ (pstoMet‘𝐷)[𝑏] ∼ ) = 0 ↔ [𝑎] ∼ = [𝑏] ∼ ))
48	47	adantllr 731	. . . . . . . . 9 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ∼ ) ∧ 𝑦 ∈ (𝑋 / ∼ ))) ∧ 𝑎 ∈ 𝑋) ∧ 𝑏 ∈ 𝑋) → (([𝑎] ∼ (pstoMet‘𝐷)[𝑏] ∼ ) = 0 ↔ [𝑎] ∼ = [𝑏] ∼ ))
49	48	adantlr 727	. . . . . . . 8 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ∼ ) ∧ 𝑦 ∈ (𝑋 / ∼ ))) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) → (([𝑎] ∼ (pstoMet‘𝐷)[𝑏] ∼ ) = 0 ↔ [𝑎] ∼ = [𝑏] ∼ ))
50	49	adantr 485	. . . . . . 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 2765	. . . . . . 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 2779	. . . . . . 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 314	. . . . . 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 3173	. . . . 5 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ∼ ) ∧ 𝑦 ∈ (𝑋 / ∼ ))) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) → ((𝑥(pstoMet‘𝐷)𝑦) = 0 ↔ 𝑥 = 𝑦))
55	54, 29	r19.29a 3173	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ∼ ) ∧ 𝑦 ∈ (𝑋 / ∼ ))) → ((𝑥(pstoMet‘𝐷)𝑦) = 0 ↔ 𝑥 = 𝑦))
56		simp-6l 798	. . . . . . . . . . . . . 14 ⊢ (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) ∧ 𝑐 ∈ 𝑋) ∧ 𝑧 = [𝑐] ∼ ) → 𝐷 ∈ (PsMet‘𝑋))
57		simplr 780	. . . . . . . . . . . . . 14 ⊢ (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) ∧ 𝑐 ∈ 𝑋) ∧ 𝑧 = [𝑐] ∼ ) → 𝑐 ∈ 𝑋)
58		simp-6r 799	. . . . . . . . . . . . . 14 ⊢ (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) ∧ 𝑐 ∈ 𝑋) ∧ 𝑧 = [𝑐] ∼ ) → 𝑎 ∈ 𝑋)
59		simp-4r 795	. . . . . . . . . . . . . 14 ⊢ (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) ∧ 𝑐 ∈ 𝑋) ∧ 𝑧 = [𝑐] ∼ ) → 𝑏 ∈ 𝑋)
60		psmettri2 24447	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑐 ∈ 𝑋 ∧ 𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)))
61	56, 57, 58, 59, 60	syl13anc 1399	. . . . . . . . . . . . 13 ⊢ (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) ∧ 𝑐 ∈ 𝑋) ∧ 𝑧 = [𝑐] ∼ ) → (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)))
62		simp-5r 797	. . . . . . . . . . . . . . 15 ⊢ (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) ∧ 𝑐 ∈ 𝑋) ∧ 𝑧 = [𝑐] ∼ ) → 𝑥 = [𝑎] ∼ )
63		simpllr 787	. . . . . . . . . . . . . . 15 ⊢ (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) ∧ 𝑐 ∈ 𝑋) ∧ 𝑧 = [𝑐] ∼ ) → 𝑦 = [𝑏] ∼ )
64	62, 63	oveq12d 7430	. . . . . . . . . . . . . 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 1398	. . . . . . . . . . . . . 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 2798	. . . . . . . . . . . . 13 ⊢ (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) ∧ 𝑐 ∈ 𝑋) ∧ 𝑧 = [𝑐] ∼ ) → (𝑥(pstoMet‘𝐷)𝑦) = (𝑎𝐷𝑏))
67		simpr 489	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) ∧ 𝑐 ∈ 𝑋) ∧ 𝑧 = [𝑐] ∼ ) → 𝑧 = [𝑐] ∼ )
68	67, 62	oveq12d 7430	. . . . . . . . . . . . . . 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 34267	. . . . . . . . . . . . . . . 16 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑐 ∈ 𝑋 ∧ 𝑎 ∈ 𝑋) → ([𝑐] ∼ (pstoMet‘𝐷)[𝑎] ∼ ) = (𝑐𝐷𝑎))
70	56, 57, 58, 69	syl3anc 1398	. . . . . . . . . . . . . . 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 2798	. . . . . . . . . . . . . 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 7430	. . . . . . . . . . . . . . 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 34267	. . . . . . . . . . . . . . . 16 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑐 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) → ([𝑐] ∼ (pstoMet‘𝐷)[𝑏] ∼ ) = (𝑐𝐷𝑏))
74	56, 57, 59, 73	syl3anc 1398	. . . . . . . . . . . . . . 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 2798	. . . . . . . . . . . . . 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 7430	. . . . . . . . . . . . 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 5144	. . . . . . . . . . . 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 775	. . . . . . . . . . 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 8764	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (𝑋 / ∼ ) → ∃𝑐 ∈ 𝑋 𝑧 = [𝑐] ∼ )
80	79	ad5antlr 747	. . . . . . . . . . 11 ⊢ ((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑧 ∈ (𝑋 / ∼ )) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) → ∃𝑐 ∈ 𝑋 𝑧 = [𝑐] ∼ )
81	78, 80	r19.29a 3173	. . . . . . . . . 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 774	. . . . . . . . 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 745	. . . . . . . . 9 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦 ∈ (𝑋 / ∼ )) ∧ 𝑧 ∈ (𝑋 / ∼ )) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) → ∃𝑏 ∈ 𝑋 𝑦 = [𝑏] ∼ )
84	82, 83	r19.29a 3173	. . . . . . . 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 767	. . . . . . 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 743	. . . . . . 7 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 ∈ (𝑋 / ∼ )) ∧ 𝑦 ∈ (𝑋 / ∼ )) ∧ 𝑧 ∈ (𝑋 / ∼ )) → ∃𝑎 ∈ 𝑋 𝑥 = [𝑎] ∼ )
87	85, 86	r19.29a 3173	. . . . . 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 3157	. . . . 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 471	. . . 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 520	. . 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 6918	. . 3 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝑋 ∈ V)
93		qsexg 8770	. . 3 ⊢ (𝑋 ∈ V → (𝑋 / ∼ ) ∈ V)
94		isxmet 24462	. . 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 19	. 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 725	1 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (pstoMet‘𝐷) ∈ (∞Met‘(𝑋 / ∼ )))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143  {cab 2741  ∀wral 3079  ∃wrex 3089  Vcvv 3455  ∪ cuni 4873   class class class wbr 5110   × cxp 5661   Fn wfn 6533  ⟶wf 6534  ‘cfv 6538  (class class class)co 7412   ∈ cmpo 7414   Er wer 8692  [cec 8693   / cqs 8694  0cc0 11101  ℝ^*cxr 11243   ≤ cle 11245   +_𝑒 cxad 13136  PsMetcpsmet 21487  ∞Metcxmet 21488  ~_Metcmetid 34257  pstoMetcpstm 34258

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734  ax-cnex 11157  ax-resscn 11158  ax-1cn 11159  ax-icn 11160  ax-addcl 11161  ax-addrcl 11162  ax-mulcl 11163  ax-mulrcl 11164  ax-mulcom 11165  ax-addass 11166  ax-mulass 11167  ax-distr 11168  ax-i2m1 11169  ax-1ne0 11170  ax-1rid 11171  ax-rnegex 11172  ax-rrecex 11173  ax-cnre 11174  ax-pre-lttri 11175  ax-pre-lttrn 11176  ax-pre-ltadd 11177  ax-pre-mulgt0 11178

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7864  df-1st 7987  df-2nd 7988  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-rdg 8398  df-er 8695  df-ec 8697  df-qs 8701  df-map 8827  df-en 8945  df-dom 8946  df-sdom 8947  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11444  df-neg 11445  df-div 11873  df-nn 12235  df-2 12304  df-rp 13018  df-xneg 13138  df-xadd 13139  df-xmul 13140  df-psmet 21495  df-xmet 21496  df-metid 34259  df-pstm 34260

This theorem is referenced by: (None)