Theorem pstmxmet 33982

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 2733	. . . . 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 3441	. . . . . . 7 ⊢ 𝑥 ∈ V
3		vex 3441	. . . . . . 7 ⊢ 𝑦 ∈ V
4	2, 3	ab2rexex 7920	. . . . . 6 ⊢ {𝑧 ∣ ∃𝑎 ∈ 𝑥 ∃𝑏 ∈ 𝑦 𝑧 = (𝑎𝐷𝑏)} ∈ V
5	4	uniex 7683	. . . . 5 ⊢ ∪ {𝑧 ∣ ∃𝑎 ∈ 𝑥 ∃𝑏 ∈ 𝑦 𝑧 = (𝑎𝐷𝑏)} ∈ V
6	1, 5	fnmpoi 8011	. . . 4 ⊢ (𝑥 ∈ (𝑋 / ∼ ), 𝑦 ∈ (𝑋 / ∼ ) ↦ ∪ {𝑧 ∣ ∃𝑎 ∈ 𝑥 ∃𝑏 ∈ 𝑦 𝑧 = (𝑎𝐷𝑏)}) Fn ((𝑋 / ∼ ) × (𝑋 / ∼ ))
7		pstmval.1	. . . . . 6 ⊢ ∼ = (~Met‘𝐷)
8	7	pstmval 33980	. . . . 5 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (pstoMet‘𝐷) = (𝑥 ∈ (𝑋 / ∼ ), 𝑦 ∈ (𝑋 / ∼ ) ↦ ∪ {𝑧 ∣ ∃𝑎 ∈ 𝑥 ∃𝑏 ∈ 𝑦 𝑧 = (𝑎𝐷𝑏)}))
9	8	fneq1d 6582	. . . 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 7373	. . . . . . . 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 33981	. . . . . . . . 9 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) → ([𝑎] ∼ (pstoMet‘𝐷)[𝑏] ∼ ) = (𝑎𝐷𝑏))
18	14, 15, 16, 17	syl3anc 1373	. . . . . . . 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 2768	. . . . . . 7 ⊢ ((((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ∼ ) ∧ 𝑦 ∈ (𝑋 / ∼ ))) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) → (𝑥(pstoMet‘𝐷)𝑦) = (𝑎𝐷𝑏))
20		psmetf 24241	. . . . . . . . 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 7527	. . . . . . 7 ⊢ ((((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ∼ ) ∧ 𝑦 ∈ (𝑋 / ∼ ))) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) → (𝑎𝐷𝑏) ∈ ℝ*)
23	19, 22	eqeltrd 2833	. . . . . 6 ⊢ ((((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ∼ ) ∧ 𝑦 ∈ (𝑋 / ∼ ))) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) → (𝑥(pstoMet‘𝐷)𝑦) ∈ ℝ*)
24		elqsi 8699	. . . . . . . 8 ⊢ (𝑦 ∈ (𝑋 / ∼ ) → ∃𝑏 ∈ 𝑋 𝑦 = [𝑏] ∼ )
25	24	ad2antll 729	. . . . . . 7 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ∼ ) ∧ 𝑦 ∈ (𝑋 / ∼ ))) → ∃𝑏 ∈ 𝑋 𝑦 = [𝑏] ∼ )
26	25	ad2antrr 726	. . . . . 6 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ∼ ) ∧ 𝑦 ∈ (𝑋 / ∼ ))) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) → ∃𝑏 ∈ 𝑋 𝑦 = [𝑏] ∼ )
27	23, 26	r19.29a 3141	. . . . 5 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ∼ ) ∧ 𝑦 ∈ (𝑋 / ∼ ))) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) → (𝑥(pstoMet‘𝐷)𝑦) ∈ ℝ*)
28		elqsi 8699	. . . . . 6 ⊢ (𝑥 ∈ (𝑋 / ∼ ) → ∃𝑎 ∈ 𝑋 𝑥 = [𝑎] ∼ )
29	28	ad2antrl 728	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ∼ ) ∧ 𝑦 ∈ (𝑋 / ∼ ))) → ∃𝑎 ∈ 𝑋 𝑥 = [𝑎] ∼ )
30	27, 29	r19.29a 3141	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ∼ ) ∧ 𝑦 ∈ (𝑋 / ∼ ))) → (𝑥(pstoMet‘𝐷)𝑦) ∈ ℝ*)
31	30	ralrimivva 3176	. . 3 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ∀𝑥 ∈ (𝑋 / ∼ )∀𝑦 ∈ (𝑋 / ∼ )(𝑥(pstoMet‘𝐷)𝑦) ∈ ℝ*)
32		ffnov 7481	. . 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 583	. 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 2735	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) ∧ 𝑏 ∈ 𝑋) → (([𝑎] ∼ (pstoMet‘𝐷)[𝑏] ∼ ) = 0 ↔ (𝑎𝐷𝑏) = 0))
36	7	breqi 5101	. . . . . . . . . . . 12 ⊢ (𝑎 ∼ 𝑏 ↔ 𝑎(~Met‘𝐷)𝑏)
37		metidv 33977	. . . . . . . . . . . . 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 33979	. . . . . . . . . . . . . 14 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (~Met‘𝐷) Er 𝑋)
41	40	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) ∧ 𝑏 ∈ 𝑋) → (~Met‘𝐷) Er 𝑋)
42		ereq1 8638	. . . . . . . . . . . . . 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 8685	. . . . . . . . . . 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 719	. . . . . . . . 9 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ∼ ) ∧ 𝑦 ∈ (𝑋 / ∼ ))) ∧ 𝑎 ∈ 𝑋) ∧ 𝑏 ∈ 𝑋) → (([𝑎] ∼ (pstoMet‘𝐷)[𝑏] ∼ ) = 0 ↔ [𝑎] ∼ = [𝑏] ∼ ))
49	48	adantlr 715	. . . . . . . 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 2735	. . . . . . 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 2749	. . . . . . 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 3141	. . . . 5 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ∼ ) ∧ 𝑦 ∈ (𝑋 / ∼ ))) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) → ((𝑥(pstoMet‘𝐷)𝑦) = 0 ↔ 𝑥 = 𝑦))
55	54, 29	r19.29a 3141	. . . 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 24244	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑐 ∈ 𝑋 ∧ 𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)))
61	56, 57, 58, 59, 60	syl13anc 1374	. . . . . . . . . . . . 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 7373	. . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . 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 2768	. . . . . . . . . . . . 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 7373	. . . . . . . . . . . . . . 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 33981	. . . . . . . . . . . . . . . 16 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑐 ∈ 𝑋 ∧ 𝑎 ∈ 𝑋) → ([𝑐] ∼ (pstoMet‘𝐷)[𝑎] ∼ ) = (𝑐𝐷𝑎))
70	56, 57, 58, 69	syl3anc 1373	. . . . . . . . . . . . . . 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 2768	. . . . . . . . . . . . . 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 7373	. . . . . . . . . . . . . . 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 33981	. . . . . . . . . . . . . . . 16 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑐 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) → ([𝑐] ∼ (pstoMet‘𝐷)[𝑏] ∼ ) = (𝑐𝐷𝑏))
74	56, 57, 59, 73	syl3anc 1373	. . . . . . . . . . . . . . 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 2768	. . . . . . . . . . . . . 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 7373	. . . . . . . . . . . . 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 5127	. . . . . . . . . . . 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 8699	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (𝑋 / ∼ ) → ∃𝑐 ∈ 𝑋 𝑧 = [𝑐] ∼ )
80	79	ad5antlr 735	. . . . . . . . . . 11 ⊢ ((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑧 ∈ (𝑋 / ∼ )) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) → ∃𝑐 ∈ 𝑋 𝑧 = [𝑐] ∼ )
81	78, 80	r19.29a 3141	. . . . . . . . . 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 733	. . . . . . . . 9 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦 ∈ (𝑋 / ∼ )) ∧ 𝑧 ∈ (𝑋 / ∼ )) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) → ∃𝑏 ∈ 𝑋 𝑦 = [𝑏] ∼ )
84	82, 83	r19.29a 3141	. . . . . . . 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 755	. . . . . . 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 731	. . . . . . 7 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 ∈ (𝑋 / ∼ )) ∧ 𝑦 ∈ (𝑋 / ∼ )) ∧ 𝑧 ∈ (𝑋 / ∼ )) → ∃𝑎 ∈ 𝑋 𝑥 = [𝑎] ∼ )
87	85, 86	r19.29a 3141	. . . . . 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 3125	. . . . 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 3176	. 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 6866	. . 3 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝑋 ∈ V)
93		qsexg 8705	. . 3 ⊢ (𝑋 ∈ V → (𝑋 / ∼ ) ∈ V)
94		isxmet 24259	. . 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 713	1 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (pstoMet‘𝐷) ∈ (∞Met‘(𝑋 / ∼ )))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  {cab 2711  ∀wral 3048  ∃wrex 3057  Vcvv 3437  ∪ cuni 4860   class class class wbr 5095   × cxp 5619   Fn wfn 6484  ⟶wf 6485  ‘cfv 6489  (class class class)co 7355   ∈ cmpo 7357   Er wer 8628  [cec 8629   / cqs 8630  0cc0 11017  ℝ^*cxr 11156   ≤ cle 11158   +_𝑒 cxad 13015  PsMetcpsmet 21284  ∞Metcxmet 21285  ~_Metcmetid 33971  pstoMetcpstm 33972

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677  ax-cnex 11073  ax-resscn 11074  ax-1cn 11075  ax-icn 11076  ax-addcl 11077  ax-addrcl 11078  ax-mulcl 11079  ax-mulrcl 11080  ax-mulcom 11081  ax-addass 11082  ax-mulass 11083  ax-distr 11084  ax-i2m1 11085  ax-1ne0 11086  ax-1rid 11087  ax-rnegex 11088  ax-rrecex 11089  ax-cnre 11090  ax-pre-lttri 11091  ax-pre-lttrn 11092  ax-pre-ltadd 11093  ax-pre-mulgt0 11094

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-nel 3034  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-po 5529  df-so 5530  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-riota 7312  df-ov 7358  df-oprab 7359  df-mpo 7360  df-1st 7930  df-2nd 7931  df-er 8631  df-ec 8633  df-qs 8637  df-map 8761  df-en 8880  df-dom 8881  df-sdom 8882  df-pnf 11159  df-mnf 11160  df-xr 11161  df-ltxr 11162  df-le 11163  df-sub 11357  df-neg 11358  df-div 11786  df-2 12199  df-rp 12897  df-xneg 13017  df-xadd 13018  df-xmul 13019  df-psmet 21292  df-xmet 21293  df-metid 33973  df-pstm 33974

This theorem is referenced by: (None)