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Theorem pstmxmet 30485
 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 = (~Met𝐷)
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.
StepHypRef Expression
1 eqid 2825 . . . . 5 (𝑥 ∈ (𝑋 / ), 𝑦 ∈ (𝑋 / ) ↦ {𝑧 ∣ ∃𝑎𝑥𝑏𝑦 𝑧 = (𝑎𝐷𝑏)}) = (𝑥 ∈ (𝑋 / ), 𝑦 ∈ (𝑋 / ) ↦ {𝑧 ∣ ∃𝑎𝑥𝑏𝑦 𝑧 = (𝑎𝐷𝑏)})
2 vex 3417 . . . . . . 7 𝑥 ∈ V
3 vex 3417 . . . . . . 7 𝑦 ∈ V
42, 3ab2rexex 7419 . . . . . 6 {𝑧 ∣ ∃𝑎𝑥𝑏𝑦 𝑧 = (𝑎𝐷𝑏)} ∈ V
54uniex 7213 . . . . 5 {𝑧 ∣ ∃𝑎𝑥𝑏𝑦 𝑧 = (𝑎𝐷𝑏)} ∈ V
61, 5fnmpt2i 7502 . . . 4 (𝑥 ∈ (𝑋 / ), 𝑦 ∈ (𝑋 / ) ↦ {𝑧 ∣ ∃𝑎𝑥𝑏𝑦 𝑧 = (𝑎𝐷𝑏)}) Fn ((𝑋 / ) × (𝑋 / ))
7 pstmval.1 . . . . . 6 = (~Met𝐷)
87pstmval 30483 . . . . 5 (𝐷 ∈ (PsMet‘𝑋) → (pstoMet‘𝐷) = (𝑥 ∈ (𝑋 / ), 𝑦 ∈ (𝑋 / ) ↦ {𝑧 ∣ ∃𝑎𝑥𝑏𝑦 𝑧 = (𝑎𝐷𝑏)}))
98fneq1d 6214 . . . 4 (𝐷 ∈ (PsMet‘𝑋) → ((pstoMet‘𝐷) Fn ((𝑋 / ) × (𝑋 / )) ↔ (𝑥 ∈ (𝑋 / ), 𝑦 ∈ (𝑋 / ) ↦ {𝑧 ∣ ∃𝑎𝑥𝑏𝑦 𝑧 = (𝑎𝐷𝑏)}) Fn ((𝑋 / ) × (𝑋 / ))))
106, 9mpbiri 250 . . 3 (𝐷 ∈ (PsMet‘𝑋) → (pstoMet‘𝐷) Fn ((𝑋 / ) × (𝑋 / )))
11 simpllr 795 . . . . . . . . 9 ((((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) → 𝑥 = [𝑎] )
12 simpr 479 . . . . . . . . 9 ((((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) → 𝑦 = [𝑏] )
1311, 12oveq12d 6923 . . . . . . . 8 ((((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) → (𝑥(pstoMet‘𝐷)𝑦) = ([𝑎] (pstoMet‘𝐷)[𝑏] ))
14 simp-5l 807 . . . . . . . . 9 ((((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) → 𝐷 ∈ (PsMet‘𝑋))
15 simp-4r 805 . . . . . . . . 9 ((((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) → 𝑎𝑋)
16 simplr 787 . . . . . . . . 9 ((((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) → 𝑏𝑋)
177pstmfval 30484 . . . . . . . . 9 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋𝑏𝑋) → ([𝑎] (pstoMet‘𝐷)[𝑏] ) = (𝑎𝐷𝑏))
1814, 15, 16, 17syl3anc 1496 . . . . . . . 8 ((((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) → ([𝑎] (pstoMet‘𝐷)[𝑏] ) = (𝑎𝐷𝑏))
1913, 18eqtrd 2861 . . . . . . 7 ((((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) → (𝑥(pstoMet‘𝐷)𝑦) = (𝑎𝐷𝑏))
20 psmetf 22481 . . . . . . . . 9 (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
2114, 20syl 17 . . . . . . . 8 ((((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
2221, 15, 16fovrnd 7066 . . . . . . 7 ((((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) → (𝑎𝐷𝑏) ∈ ℝ*)
2319, 22eqeltrd 2906 . . . . . 6 ((((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) → (𝑥(pstoMet‘𝐷)𝑦) ∈ ℝ*)
24 elqsi 8065 . . . . . . . 8 (𝑦 ∈ (𝑋 / ) → ∃𝑏𝑋 𝑦 = [𝑏] )
2524ad2antll 722 . . . . . . 7 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) → ∃𝑏𝑋 𝑦 = [𝑏] )
2625ad2antrr 719 . . . . . 6 ((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) → ∃𝑏𝑋 𝑦 = [𝑏] )
2723, 26r19.29a 3288 . . . . 5 ((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) → (𝑥(pstoMet‘𝐷)𝑦) ∈ ℝ*)
28 elqsi 8065 . . . . . 6 (𝑥 ∈ (𝑋 / ) → ∃𝑎𝑋 𝑥 = [𝑎] )
2928ad2antrl 721 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) → ∃𝑎𝑋 𝑥 = [𝑎] )
3027, 29r19.29a 3288 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) → (𝑥(pstoMet‘𝐷)𝑦) ∈ ℝ*)
3130ralrimivva 3180 . . 3 (𝐷 ∈ (PsMet‘𝑋) → ∀𝑥 ∈ (𝑋 / )∀𝑦 ∈ (𝑋 / )(𝑥(pstoMet‘𝐷)𝑦) ∈ ℝ*)
32 ffnov 7024 . . 3 ((pstoMet‘𝐷):((𝑋 / ) × (𝑋 / ))⟶ℝ* ↔ ((pstoMet‘𝐷) Fn ((𝑋 / ) × (𝑋 / )) ∧ ∀𝑥 ∈ (𝑋 / )∀𝑦 ∈ (𝑋 / )(𝑥(pstoMet‘𝐷)𝑦) ∈ ℝ*))
3310, 31, 32sylanbrc 580 . 2 (𝐷 ∈ (PsMet‘𝑋) → (pstoMet‘𝐷):((𝑋 / ) × (𝑋 / ))⟶ℝ*)
34173expa 1153 . . . . . . . . . . . 12 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑏𝑋) → ([𝑎] (pstoMet‘𝐷)[𝑏] ) = (𝑎𝐷𝑏))
3534eqeq1d 2827 . . . . . . . . . . 11 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑏𝑋) → (([𝑎] (pstoMet‘𝐷)[𝑏] ) = 0 ↔ (𝑎𝐷𝑏) = 0))
367breqi 4879 . . . . . . . . . . . 12 (𝑎 𝑏𝑎(~Met𝐷)𝑏)
37 metidv 30480 . . . . . . . . . . . . 13 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑎𝑋𝑏𝑋)) → (𝑎(~Met𝐷)𝑏 ↔ (𝑎𝐷𝑏) = 0))
3837anassrs 461 . . . . . . . . . . . 12 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑏𝑋) → (𝑎(~Met𝐷)𝑏 ↔ (𝑎𝐷𝑏) = 0))
3936, 38syl5bb 275 . . . . . . . . . . 11 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑏𝑋) → (𝑎 𝑏 ↔ (𝑎𝐷𝑏) = 0))
40 metider 30482 . . . . . . . . . . . . . 14 (𝐷 ∈ (PsMet‘𝑋) → (~Met𝐷) Er 𝑋)
4140ad2antrr 719 . . . . . . . . . . . . 13 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑏𝑋) → (~Met𝐷) Er 𝑋)
42 ereq1 8016 . . . . . . . . . . . . . 14 ( = (~Met𝐷) → ( Er 𝑋 ↔ (~Met𝐷) Er 𝑋))
437, 42ax-mp 5 . . . . . . . . . . . . 13 ( Er 𝑋 ↔ (~Met𝐷) Er 𝑋)
4441, 43sylibr 226 . . . . . . . . . . . 12 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑏𝑋) → Er 𝑋)
45 simplr 787 . . . . . . . . . . . 12 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑏𝑋) → 𝑎𝑋)
4644, 45erth 8056 . . . . . . . . . . 11 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑏𝑋) → (𝑎 𝑏 ↔ [𝑎] = [𝑏] ))
4735, 39, 463bitr2d 299 . . . . . . . . . 10 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑏𝑋) → (([𝑎] (pstoMet‘𝐷)[𝑏] ) = 0 ↔ [𝑎] = [𝑏] ))
4847adantllr 712 . . . . . . . . 9 ((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) ∧ 𝑎𝑋) ∧ 𝑏𝑋) → (([𝑎] (pstoMet‘𝐷)[𝑏] ) = 0 ↔ [𝑎] = [𝑏] ))
4948adantlr 708 . . . . . . . 8 (((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) → (([𝑎] (pstoMet‘𝐷)[𝑏] ) = 0 ↔ [𝑎] = [𝑏] ))
5049adantr 474 . . . . . . 7 ((((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) → (([𝑎] (pstoMet‘𝐷)[𝑏] ) = 0 ↔ [𝑎] = [𝑏] ))
5113eqeq1d 2827 . . . . . . 7 ((((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) → ((𝑥(pstoMet‘𝐷)𝑦) = 0 ↔ ([𝑎] (pstoMet‘𝐷)[𝑏] ) = 0))
5211, 12eqeq12d 2840 . . . . . . 7 ((((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) → (𝑥 = 𝑦 ↔ [𝑎] = [𝑏] ))
5350, 51, 523bitr4d 303 . . . . . 6 ((((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) → ((𝑥(pstoMet‘𝐷)𝑦) = 0 ↔ 𝑥 = 𝑦))
5453, 26r19.29a 3288 . . . . 5 ((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) → ((𝑥(pstoMet‘𝐷)𝑦) = 0 ↔ 𝑥 = 𝑦))
5554, 29r19.29a 3288 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) → ((𝑥(pstoMet‘𝐷)𝑦) = 0 ↔ 𝑥 = 𝑦))
56 simp-6l 811 . . . . . . . . . . . . . 14 (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) ∧ 𝑐𝑋) ∧ 𝑧 = [𝑐] ) → 𝐷 ∈ (PsMet‘𝑋))
57 simplr 787 . . . . . . . . . . . . . 14 (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) ∧ 𝑐𝑋) ∧ 𝑧 = [𝑐] ) → 𝑐𝑋)
58 simp-6r 813 . . . . . . . . . . . . . 14 (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) ∧ 𝑐𝑋) ∧ 𝑧 = [𝑐] ) → 𝑎𝑋)
59 simp-4r 805 . . . . . . . . . . . . . 14 (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) ∧ 𝑐𝑋) ∧ 𝑧 = [𝑐] ) → 𝑏𝑋)
60 psmettri2 22484 . . . . . . . . . . . . . 14 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑐𝑋𝑎𝑋𝑏𝑋)) → (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)))
6156, 57, 58, 59, 60syl13anc 1497 . . . . . . . . . . . . 13 (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) ∧ 𝑐𝑋) ∧ 𝑧 = [𝑐] ) → (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)))
62 simp-5r 809 . . . . . . . . . . . . . . 15 (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) ∧ 𝑐𝑋) ∧ 𝑧 = [𝑐] ) → 𝑥 = [𝑎] )
63 simpllr 795 . . . . . . . . . . . . . . 15 (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) ∧ 𝑐𝑋) ∧ 𝑧 = [𝑐] ) → 𝑦 = [𝑏] )
6462, 63oveq12d 6923 . . . . . . . . . . . . . 14 (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) ∧ 𝑐𝑋) ∧ 𝑧 = [𝑐] ) → (𝑥(pstoMet‘𝐷)𝑦) = ([𝑎] (pstoMet‘𝐷)[𝑏] ))
6556, 58, 59, 17syl3anc 1496 . . . . . . . . . . . . . 14 (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) ∧ 𝑐𝑋) ∧ 𝑧 = [𝑐] ) → ([𝑎] (pstoMet‘𝐷)[𝑏] ) = (𝑎𝐷𝑏))
6664, 65eqtrd 2861 . . . . . . . . . . . . 13 (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) ∧ 𝑐𝑋) ∧ 𝑧 = [𝑐] ) → (𝑥(pstoMet‘𝐷)𝑦) = (𝑎𝐷𝑏))
67 simpr 479 . . . . . . . . . . . . . . . 16 (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) ∧ 𝑐𝑋) ∧ 𝑧 = [𝑐] ) → 𝑧 = [𝑐] )
6867, 62oveq12d 6923 . . . . . . . . . . . . . . 15 (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) ∧ 𝑐𝑋) ∧ 𝑧 = [𝑐] ) → (𝑧(pstoMet‘𝐷)𝑥) = ([𝑐] (pstoMet‘𝐷)[𝑎] ))
697pstmfval 30484 . . . . . . . . . . . . . . . 16 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑐𝑋𝑎𝑋) → ([𝑐] (pstoMet‘𝐷)[𝑎] ) = (𝑐𝐷𝑎))
7056, 57, 58, 69syl3anc 1496 . . . . . . . . . . . . . . 15 (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) ∧ 𝑐𝑋) ∧ 𝑧 = [𝑐] ) → ([𝑐] (pstoMet‘𝐷)[𝑎] ) = (𝑐𝐷𝑎))
7168, 70eqtrd 2861 . . . . . . . . . . . . . 14 (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) ∧ 𝑐𝑋) ∧ 𝑧 = [𝑐] ) → (𝑧(pstoMet‘𝐷)𝑥) = (𝑐𝐷𝑎))
7267, 63oveq12d 6923 . . . . . . . . . . . . . . 15 (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) ∧ 𝑐𝑋) ∧ 𝑧 = [𝑐] ) → (𝑧(pstoMet‘𝐷)𝑦) = ([𝑐] (pstoMet‘𝐷)[𝑏] ))
737pstmfval 30484 . . . . . . . . . . . . . . . 16 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑐𝑋𝑏𝑋) → ([𝑐] (pstoMet‘𝐷)[𝑏] ) = (𝑐𝐷𝑏))
7456, 57, 59, 73syl3anc 1496 . . . . . . . . . . . . . . 15 (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) ∧ 𝑐𝑋) ∧ 𝑧 = [𝑐] ) → ([𝑐] (pstoMet‘𝐷)[𝑏] ) = (𝑐𝐷𝑏))
7572, 74eqtrd 2861 . . . . . . . . . . . . . 14 (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) ∧ 𝑐𝑋) ∧ 𝑧 = [𝑐] ) → (𝑧(pstoMet‘𝐷)𝑦) = (𝑐𝐷𝑏))
7671, 75oveq12d 6923 . . . . . . . . . . . . 13 (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) ∧ 𝑐𝑋) ∧ 𝑧 = [𝑐] ) → ((𝑧(pstoMet‘𝐷)𝑥) +𝑒 (𝑧(pstoMet‘𝐷)𝑦)) = ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)))
7761, 66, 763brtr4d 4905 . . . . . . . . . . . 12 (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) ∧ 𝑐𝑋) ∧ 𝑧 = [𝑐] ) → (𝑥(pstoMet‘𝐷)𝑦) ≤ ((𝑧(pstoMet‘𝐷)𝑥) +𝑒 (𝑧(pstoMet‘𝐷)𝑦)))
7877adantl6r 782 . . . . . . . . . . 11 ((((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑧 ∈ (𝑋 / )) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) ∧ 𝑐𝑋) ∧ 𝑧 = [𝑐] ) → (𝑥(pstoMet‘𝐷)𝑦) ≤ ((𝑧(pstoMet‘𝐷)𝑥) +𝑒 (𝑧(pstoMet‘𝐷)𝑦)))
79 elqsi 8065 . . . . . . . . . . . 12 (𝑧 ∈ (𝑋 / ) → ∃𝑐𝑋 𝑧 = [𝑐] )
8079ad5antlr 732 . . . . . . . . . . 11 ((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑧 ∈ (𝑋 / )) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) → ∃𝑐𝑋 𝑧 = [𝑐] )
8178, 80r19.29a 3288 . . . . . . . . . 10 ((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑧 ∈ (𝑋 / )) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) → (𝑥(pstoMet‘𝐷)𝑦) ≤ ((𝑧(pstoMet‘𝐷)𝑥) +𝑒 (𝑧(pstoMet‘𝐷)𝑦)))
8281adantl5r 781 . . . . . . . . 9 (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦 ∈ (𝑋 / )) ∧ 𝑧 ∈ (𝑋 / )) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) → (𝑥(pstoMet‘𝐷)𝑦) ≤ ((𝑧(pstoMet‘𝐷)𝑥) +𝑒 (𝑧(pstoMet‘𝐷)𝑦)))
8324ad4antlr 728 . . . . . . . . 9 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦 ∈ (𝑋 / )) ∧ 𝑧 ∈ (𝑋 / )) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) → ∃𝑏𝑋 𝑦 = [𝑏] )
8482, 83r19.29a 3288 . . . . . . . 8 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦 ∈ (𝑋 / )) ∧ 𝑧 ∈ (𝑋 / )) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) → (𝑥(pstoMet‘𝐷)𝑦) ≤ ((𝑧(pstoMet‘𝐷)𝑥) +𝑒 (𝑧(pstoMet‘𝐷)𝑦)))
8584adantl4r 767 . . . . . . 7 ((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 ∈ (𝑋 / )) ∧ 𝑦 ∈ (𝑋 / )) ∧ 𝑧 ∈ (𝑋 / )) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) → (𝑥(pstoMet‘𝐷)𝑦) ≤ ((𝑧(pstoMet‘𝐷)𝑥) +𝑒 (𝑧(pstoMet‘𝐷)𝑦)))
8628ad3antlr 724 . . . . . . 7 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 ∈ (𝑋 / )) ∧ 𝑦 ∈ (𝑋 / )) ∧ 𝑧 ∈ (𝑋 / )) → ∃𝑎𝑋 𝑥 = [𝑎] )
8785, 86r19.29a 3288 . . . . . 6 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 ∈ (𝑋 / )) ∧ 𝑦 ∈ (𝑋 / )) ∧ 𝑧 ∈ (𝑋 / )) → (𝑥(pstoMet‘𝐷)𝑦) ≤ ((𝑧(pstoMet‘𝐷)𝑥) +𝑒 (𝑧(pstoMet‘𝐷)𝑦)))
8887ralrimiva 3175 . . . . 5 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 ∈ (𝑋 / )) ∧ 𝑦 ∈ (𝑋 / )) → ∀𝑧 ∈ (𝑋 / )(𝑥(pstoMet‘𝐷)𝑦) ≤ ((𝑧(pstoMet‘𝐷)𝑥) +𝑒 (𝑧(pstoMet‘𝐷)𝑦)))
8988anasss 460 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) → ∀𝑧 ∈ (𝑋 / )(𝑥(pstoMet‘𝐷)𝑦) ≤ ((𝑧(pstoMet‘𝐷)𝑥) +𝑒 (𝑧(pstoMet‘𝐷)𝑦)))
9055, 89jca 509 . . 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))))
9190ralrimivva 3180 . 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 6467 . . 3 (𝐷 ∈ (PsMet‘𝑋) → 𝑋 ∈ V)
93 qsexg 8070 . . 3 (𝑋 ∈ V → (𝑋 / ) ∈ V)
94 isxmet 22499 . . 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))))))
9592, 93, 943syl 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))))))
9633, 91, 95mpbir2and 706 1 (𝐷 ∈ (PsMet‘𝑋) → (pstoMet‘𝐷) ∈ (∞Met‘(𝑋 / )))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   = wceq 1658   ∈ wcel 2166  {cab 2811  ∀wral 3117  ∃wrex 3118  Vcvv 3414  ∪ cuni 4658   class class class wbr 4873   × cxp 5340   Fn wfn 6118  ⟶wf 6119  ‘cfv 6123  (class class class)co 6905   ↦ cmpt2 6907   Er wer 8006  [cec 8007   / cqs 8008  0cc0 10252  ℝ*cxr 10390   ≤ cle 10392   +𝑒 cxad 12230  PsMetcpsmet 20090  ∞Metcxmet 20091  ~Metcmetid 30474  pstoMetcpstm 30475 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209  ax-cnex 10308  ax-resscn 10309  ax-1cn 10310  ax-icn 10311  ax-addcl 10312  ax-addrcl 10313  ax-mulcl 10314  ax-mulrcl 10315  ax-mulcom 10316  ax-addass 10317  ax-mulass 10318  ax-distr 10319  ax-i2m1 10320  ax-1ne0 10321  ax-1rid 10322  ax-rnegex 10323  ax-rrecex 10324  ax-cnre 10325  ax-pre-lttri 10326  ax-pre-lttrn 10327  ax-pre-ltadd 10328  ax-pre-mulgt0 10329 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-po 5263  df-so 5264  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-riota 6866  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-1st 7428  df-2nd 7429  df-er 8009  df-ec 8011  df-qs 8015  df-map 8124  df-en 8223  df-dom 8224  df-sdom 8225  df-pnf 10393  df-mnf 10394  df-xr 10395  df-ltxr 10396  df-le 10397  df-sub 10587  df-neg 10588  df-div 11010  df-2 11414  df-rp 12113  df-xneg 12232  df-xadd 12233  df-xmul 12234  df-psmet 20098  df-xmet 20099  df-metid 30476  df-pstm 30477 This theorem is referenced by: (None)
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