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Theorem pstmxmet 32866

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 3479	. . . . . . 7 ⊢ 𝑥 ∈ V
3		vex 3479	. . . . . . 7 ⊢ 𝑦 ∈ V
4	2, 3	ab2rexex 7963	. . . . . 6 ⊢ {𝑧 ∣ ∃𝑎 ∈ 𝑥 ∃𝑏 ∈ 𝑦 𝑧 = (𝑎𝐷𝑏)} ∈ V
5	4	uniex 7728	. . . . 5 ⊢ ∪ {𝑧 ∣ ∃𝑎 ∈ 𝑥 ∃𝑏 ∈ 𝑦 𝑧 = (𝑎𝐷𝑏)} ∈ V
6	1, 5	fnmpoi 8053	. . . 4 ⊢ (𝑥 ∈ (𝑋 / ∼ ), 𝑦 ∈ (𝑋 / ∼ ) ↦ ∪ {𝑧 ∣ ∃𝑎 ∈ 𝑥 ∃𝑏 ∈ 𝑦 𝑧 = (𝑎𝐷𝑏)}) Fn ((𝑋 / ∼ ) × (𝑋 / ∼ ))
7		pstmval.1	. . . . . 6 ⊢ ∼ = (~_Met‘𝐷)
8	7	pstmval 32864	. . . . 5 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (pstoMet‘𝐷) = (𝑥 ∈ (𝑋 / ∼ ), 𝑦 ∈ (𝑋 / ∼ ) ↦ ∪ {𝑧 ∣ ∃𝑎 ∈ 𝑥 ∃𝑏 ∈ 𝑦 𝑧 = (𝑎𝐷𝑏)}))
9	8	fneq1d 6640	. . . 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 486	. . . . . . . . 9 ⊢ ((((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ∼ ) ∧ 𝑦 ∈ (𝑋 / ∼ ))) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) → 𝑦 = [𝑏] ∼ )
13	11, 12	oveq12d 7424	. . . . . . . 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 32865	. . . . . . . . 9 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) → ([𝑎] ∼ (pstoMet‘𝐷)[𝑏] ∼ ) = (𝑎𝐷𝑏))
18	14, 15, 16, 17	syl3anc 1372	. . . . . . . 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 2773	. . . . . . 7 ⊢ ((((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ∼ ) ∧ 𝑦 ∈ (𝑋 / ∼ ))) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) → (𝑥(pstoMet‘𝐷)𝑦) = (𝑎𝐷𝑏))
20		psmetf 23804	. . . . . . . . 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 7576	. . . . . . 7 ⊢ ((((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ∼ ) ∧ 𝑦 ∈ (𝑋 / ∼ ))) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) → (𝑎𝐷𝑏) ∈ ℝ^*)
23	19, 22	eqeltrd 2834	. . . . . 6 ⊢ ((((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ∼ ) ∧ 𝑦 ∈ (𝑋 / ∼ ))) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) → (𝑥(pstoMet‘𝐷)𝑦) ∈ ℝ^*)
24		elqsi 8761	. . . . . . . 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 3163	. . . . 5 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ∼ ) ∧ 𝑦 ∈ (𝑋 / ∼ ))) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) → (𝑥(pstoMet‘𝐷)𝑦) ∈ ℝ^*)
28		elqsi 8761	. . . . . 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 3163	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ∼ ) ∧ 𝑦 ∈ (𝑋 / ∼ ))) → (𝑥(pstoMet‘𝐷)𝑦) ∈ ℝ^*)
31	30	ralrimivva 3201	. . 3 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ∀𝑥 ∈ (𝑋 / ∼ )∀𝑦 ∈ (𝑋 / ∼ )(𝑥(pstoMet‘𝐷)𝑦) ∈ ℝ^*)
32		ffnov 7532	. . 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 584	. 2 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (pstoMet‘𝐷):((𝑋 / ∼ ) × (𝑋 / ∼ ))⟶ℝ^*)
34	17	3expa 1119	. . . . . . . . . . . 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 5154	. . . . . . . . . . . 12 ⊢ (𝑎 ∼ 𝑏 ↔ 𝑎(~_Met‘𝐷)𝑏)
37		metidv 32861	. . . . . . . . . . . . 13 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (𝑎(~_Met‘𝐷)𝑏 ↔ (𝑎𝐷𝑏) = 0))
38	37	anassrs 469	. . . . . . . . . . . 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 32863	. . . . . . . . . . . . . 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 8707	. . . . . . . . . . . . . 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 233	. . . . . . . . . . . 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 8749	. . . . . . . . . . 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 482	. . . . . . 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 3163	. . . . 5 ⊢ ((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ∼ ) ∧ 𝑦 ∈ (𝑋 / ∼ ))) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) → ((𝑥(pstoMet‘𝐷)𝑦) = 0 ↔ 𝑥 = 𝑦))
55	54, 29	r19.29a 3163	. . . 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 23807	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑐 ∈ 𝑋 ∧ 𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +_𝑒 (𝑐𝐷𝑏)))
61	56, 57, 58, 59, 60	syl13anc 1373	. . . . . . . . . . . . 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 7424	. . . . . . . . . . . . . 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 1372	. . . . . . . . . . . . . 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 2773	. . . . . . . . . . . . 13 ⊢ (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) ∧ 𝑐 ∈ 𝑋) ∧ 𝑧 = [𝑐] ∼ ) → (𝑥(pstoMet‘𝐷)𝑦) = (𝑎𝐷𝑏))
67		simpr 486	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ 𝑋) ∧ 𝑥 = [𝑎] ∼ ) ∧ 𝑏 ∈ 𝑋) ∧ 𝑦 = [𝑏] ∼ ) ∧ 𝑐 ∈ 𝑋) ∧ 𝑧 = [𝑐] ∼ ) → 𝑧 = [𝑐] ∼ )
68	67, 62	oveq12d 7424	. . . . . . . . . . . . . . 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 32865	. . . . . . . . . . . . . . . 16 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑐 ∈ 𝑋 ∧ 𝑎 ∈ 𝑋) → ([𝑐] ∼ (pstoMet‘𝐷)[𝑎] ∼ ) = (𝑐𝐷𝑎))
70	56, 57, 58, 69	syl3anc 1372	. . . . . . . . . . . . . . 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 2773	. . . . . . . . . . . . . 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 7424	. . . . . . . . . . . . . . 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 32865	. . . . . . . . . . . . . . . 16 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑐 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) → ([𝑐] ∼ (pstoMet‘𝐷)[𝑏] ∼ ) = (𝑐𝐷𝑏))
74	56, 57, 59, 73	syl3anc 1372	. . . . . . . . . . . . . . 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 2773	. . . . . . . . . . . . . 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 7424	. . . . . . . . . . . . 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 5180	. . . . . . . . . . . 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 8761	. . . . . . . . . . . 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 3163	. . . . . . . . . 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 3163	. . . . . . . 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 3163	. . . . . 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 3147	. . . . 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 468	. . . 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 513	. . 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 3201	. 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 6927	. . 3 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝑋 ∈ V)
93		qsexg 8766	. . 3 ⊢ (𝑋 ∈ V → (𝑋 / ∼ ) ∈ V)
94		isxmet 23822	. . 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 / ∼ )))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 {cab 2710 ∀wral 3062 ∃wrex 3071 Vcvv 3475 ∪ cuni 4908 class class class wbr 5148 × cxp 5674 Fn wfn 6536 ⟶wf 6537 ‘cfv 6541 (class class class)co 7406 ∈ cmpo 7408 Er wer 8697 [cec 8698 / cqs 8699 0cc0 11107 ℝ^*cxr 11244 ≤ cle 11246 +_𝑒 cxad 13087 PsMetcpsmet 20921 ∞Metcxmet 20922 ~_Metcmetid 32855 pstoMetcpstm 32856

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-po 5588 df-so 5589 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-1st 7972 df-2nd 7973 df-er 8700 df-ec 8702 df-qs 8706 df-map 8819 df-en 8937 df-dom 8938 df-sdom 8939 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-div 11869 df-2 12272 df-rp 12972 df-xneg 13089 df-xadd 13090 df-xmul 13091 df-psmet 20929 df-xmet 20930 df-metid 32857 df-pstm 32858

This theorem is referenced by: (None)

W3C validator