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Theorem pstmxmet 30241
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 2752 . . . . 5 (𝑥 ∈ (𝑋 / ), 𝑦 ∈ (𝑋 / ) ↦ {𝑧 ∣ ∃𝑎𝑥𝑏𝑦 𝑧 = (𝑎𝐷𝑏)}) = (𝑥 ∈ (𝑋 / ), 𝑦 ∈ (𝑋 / ) ↦ {𝑧 ∣ ∃𝑎𝑥𝑏𝑦 𝑧 = (𝑎𝐷𝑏)})
2 vex 3335 . . . . . . 7 𝑥 ∈ V
3 vex 3335 . . . . . . 7 𝑦 ∈ V
42, 3ab2rexex 7316 . . . . . 6 {𝑧 ∣ ∃𝑎𝑥𝑏𝑦 𝑧 = (𝑎𝐷𝑏)} ∈ V
54uniex 7110 . . . . 5 {𝑧 ∣ ∃𝑎𝑥𝑏𝑦 𝑧 = (𝑎𝐷𝑏)} ∈ V
61, 5fnmpt2i 7399 . . . 4 (𝑥 ∈ (𝑋 / ), 𝑦 ∈ (𝑋 / ) ↦ {𝑧 ∣ ∃𝑎𝑥𝑏𝑦 𝑧 = (𝑎𝐷𝑏)}) Fn ((𝑋 / ) × (𝑋 / ))
7 pstmval.1 . . . . . 6 = (~Met𝐷)
87pstmval 30239 . . . . 5 (𝐷 ∈ (PsMet‘𝑋) → (pstoMet‘𝐷) = (𝑥 ∈ (𝑋 / ), 𝑦 ∈ (𝑋 / ) ↦ {𝑧 ∣ ∃𝑎𝑥𝑏𝑦 𝑧 = (𝑎𝐷𝑏)}))
98fneq1d 6134 . . . 4 (𝐷 ∈ (PsMet‘𝑋) → ((pstoMet‘𝐷) Fn ((𝑋 / ) × (𝑋 / )) ↔ (𝑥 ∈ (𝑋 / ), 𝑦 ∈ (𝑋 / ) ↦ {𝑧 ∣ ∃𝑎𝑥𝑏𝑦 𝑧 = (𝑎𝐷𝑏)}) Fn ((𝑋 / ) × (𝑋 / ))))
106, 9mpbiri 248 . . 3 (𝐷 ∈ (PsMet‘𝑋) → (pstoMet‘𝐷) Fn ((𝑋 / ) × (𝑋 / )))
11 simpllr 817 . . . . . . . . 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 6823 . . . . . . . 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 829 . . . . . . . . 9 ((((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) → 𝐷 ∈ (PsMet‘𝑋))
15 simp-4r 827 . . . . . . . . 9 ((((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) → 𝑎𝑋)
16 simplr 809 . . . . . . . . 9 ((((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) → 𝑏𝑋)
177pstmfval 30240 . . . . . . . . 9 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋𝑏𝑋) → ([𝑎] (pstoMet‘𝐷)[𝑏] ) = (𝑎𝐷𝑏))
1814, 15, 16, 17syl3anc 1473 . . . . . . . 8 ((((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) → ([𝑎] (pstoMet‘𝐷)[𝑏] ) = (𝑎𝐷𝑏))
1913, 18eqtrd 2786 . . . . . . 7 ((((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) → (𝑥(pstoMet‘𝐷)𝑦) = (𝑎𝐷𝑏))
20 psmetf 22304 . . . . . . . . 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 6963 . . . . . . 7 ((((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) → (𝑎𝐷𝑏) ∈ ℝ*)
2319, 22eqeltrd 2831 . . . . . 6 ((((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) → (𝑥(pstoMet‘𝐷)𝑦) ∈ ℝ*)
24 elqsi 7959 . . . . . . . 8 (𝑦 ∈ (𝑋 / ) → ∃𝑏𝑋 𝑦 = [𝑏] )
2524ad2antll 767 . . . . . . 7 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) → ∃𝑏𝑋 𝑦 = [𝑏] )
2625ad2antrr 764 . . . . . 6 ((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) → ∃𝑏𝑋 𝑦 = [𝑏] )
2723, 26r19.29a 3208 . . . . 5 ((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) → (𝑥(pstoMet‘𝐷)𝑦) ∈ ℝ*)
28 elqsi 7959 . . . . . 6 (𝑥 ∈ (𝑋 / ) → ∃𝑎𝑋 𝑥 = [𝑎] )
2928ad2antrl 766 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) → ∃𝑎𝑋 𝑥 = [𝑎] )
3027, 29r19.29a 3208 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) → (𝑥(pstoMet‘𝐷)𝑦) ∈ ℝ*)
3130ralrimivva 3101 . . 3 (𝐷 ∈ (PsMet‘𝑋) → ∀𝑥 ∈ (𝑋 / )∀𝑦 ∈ (𝑋 / )(𝑥(pstoMet‘𝐷)𝑦) ∈ ℝ*)
32 ffnov 6921 . . 3 ((pstoMet‘𝐷):((𝑋 / ) × (𝑋 / ))⟶ℝ* ↔ ((pstoMet‘𝐷) Fn ((𝑋 / ) × (𝑋 / )) ∧ ∀𝑥 ∈ (𝑋 / )∀𝑦 ∈ (𝑋 / )(𝑥(pstoMet‘𝐷)𝑦) ∈ ℝ*))
3310, 31, 32sylanbrc 701 . 2 (𝐷 ∈ (PsMet‘𝑋) → (pstoMet‘𝐷):((𝑋 / ) × (𝑋 / ))⟶ℝ*)
34173expa 1111 . . . . . . . . . . . 12 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑏𝑋) → ([𝑎] (pstoMet‘𝐷)[𝑏] ) = (𝑎𝐷𝑏))
3534eqeq1d 2754 . . . . . . . . . . 11 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑏𝑋) → (([𝑎] (pstoMet‘𝐷)[𝑏] ) = 0 ↔ (𝑎𝐷𝑏) = 0))
367breqi 4802 . . . . . . . . . . . 12 (𝑎 𝑏𝑎(~Met𝐷)𝑏)
37 metidv 30236 . . . . . . . . . . . . 13 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑎𝑋𝑏𝑋)) → (𝑎(~Met𝐷)𝑏 ↔ (𝑎𝐷𝑏) = 0))
3837anassrs 683 . . . . . . . . . . . 12 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑏𝑋) → (𝑎(~Met𝐷)𝑏 ↔ (𝑎𝐷𝑏) = 0))
3936, 38syl5bb 272 . . . . . . . . . . 11 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑏𝑋) → (𝑎 𝑏 ↔ (𝑎𝐷𝑏) = 0))
40 metider 30238 . . . . . . . . . . . . . 14 (𝐷 ∈ (PsMet‘𝑋) → (~Met𝐷) Er 𝑋)
4140ad2antrr 764 . . . . . . . . . . . . 13 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑏𝑋) → (~Met𝐷) Er 𝑋)
42 ereq1 7910 . . . . . . . . . . . . . 14 ( = (~Met𝐷) → ( Er 𝑋 ↔ (~Met𝐷) Er 𝑋))
437, 42ax-mp 5 . . . . . . . . . . . . 13 ( Er 𝑋 ↔ (~Met𝐷) Er 𝑋)
4441, 43sylibr 224 . . . . . . . . . . . 12 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑏𝑋) → Er 𝑋)
45 simplr 809 . . . . . . . . . . . 12 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑏𝑋) → 𝑎𝑋)
4644, 45erth 7950 . . . . . . . . . . 11 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑏𝑋) → (𝑎 𝑏 ↔ [𝑎] = [𝑏] ))
4735, 39, 463bitr2d 296 . . . . . . . . . 10 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑏𝑋) → (([𝑎] (pstoMet‘𝐷)[𝑏] ) = 0 ↔ [𝑎] = [𝑏] ))
4847adantllr 757 . . . . . . . . 9 ((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) ∧ 𝑎𝑋) ∧ 𝑏𝑋) → (([𝑎] (pstoMet‘𝐷)[𝑏] ) = 0 ↔ [𝑎] = [𝑏] ))
4948adantlr 753 . . . . . . . 8 (((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) → (([𝑎] (pstoMet‘𝐷)[𝑏] ) = 0 ↔ [𝑎] = [𝑏] ))
5049adantr 472 . . . . . . 7 ((((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) → (([𝑎] (pstoMet‘𝐷)[𝑏] ) = 0 ↔ [𝑎] = [𝑏] ))
5113eqeq1d 2754 . . . . . . 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 2767 . . . . . . 7 ((((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) → (𝑥 = 𝑦 ↔ [𝑎] = [𝑏] ))
5350, 51, 523bitr4d 300 . . . . . 6 ((((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) → ((𝑥(pstoMet‘𝐷)𝑦) = 0 ↔ 𝑥 = 𝑦))
5453, 26r19.29a 3208 . . . . 5 ((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) → ((𝑥(pstoMet‘𝐷)𝑦) = 0 ↔ 𝑥 = 𝑦))
5554, 29r19.29a 3208 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) → ((𝑥(pstoMet‘𝐷)𝑦) = 0 ↔ 𝑥 = 𝑦))
56 simp-6l 833 . . . . . . . . . . . . . 14 (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) ∧ 𝑐𝑋) ∧ 𝑧 = [𝑐] ) → 𝐷 ∈ (PsMet‘𝑋))
57 simplr 809 . . . . . . . . . . . . . 14 (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) ∧ 𝑐𝑋) ∧ 𝑧 = [𝑐] ) → 𝑐𝑋)
58 simp-6r 835 . . . . . . . . . . . . . 14 (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) ∧ 𝑐𝑋) ∧ 𝑧 = [𝑐] ) → 𝑎𝑋)
59 simp-4r 827 . . . . . . . . . . . . . 14 (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) ∧ 𝑐𝑋) ∧ 𝑧 = [𝑐] ) → 𝑏𝑋)
60 psmettri2 22307 . . . . . . . . . . . . . 14 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑐𝑋𝑎𝑋𝑏𝑋)) → (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)))
6156, 57, 58, 59, 60syl13anc 1475 . . . . . . . . . . . . 13 (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) ∧ 𝑐𝑋) ∧ 𝑧 = [𝑐] ) → (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)))
62 simp-5r 831 . . . . . . . . . . . . . . 15 (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) ∧ 𝑐𝑋) ∧ 𝑧 = [𝑐] ) → 𝑥 = [𝑎] )
63 simpllr 817 . . . . . . . . . . . . . . 15 (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) ∧ 𝑐𝑋) ∧ 𝑧 = [𝑐] ) → 𝑦 = [𝑏] )
6462, 63oveq12d 6823 . . . . . . . . . . . . . 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 1473 . . . . . . . . . . . . . 14 (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) ∧ 𝑐𝑋) ∧ 𝑧 = [𝑐] ) → ([𝑎] (pstoMet‘𝐷)[𝑏] ) = (𝑎𝐷𝑏))
6664, 65eqtrd 2786 . . . . . . . . . . . . 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 6823 . . . . . . . . . . . . . . 15 (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) ∧ 𝑐𝑋) ∧ 𝑧 = [𝑐] ) → (𝑧(pstoMet‘𝐷)𝑥) = ([𝑐] (pstoMet‘𝐷)[𝑎] ))
697pstmfval 30240 . . . . . . . . . . . . . . . 16 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑐𝑋𝑎𝑋) → ([𝑐] (pstoMet‘𝐷)[𝑎] ) = (𝑐𝐷𝑎))
7056, 57, 58, 69syl3anc 1473 . . . . . . . . . . . . . . 15 (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) ∧ 𝑐𝑋) ∧ 𝑧 = [𝑐] ) → ([𝑐] (pstoMet‘𝐷)[𝑎] ) = (𝑐𝐷𝑎))
7168, 70eqtrd 2786 . . . . . . . . . . . . . 14 (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) ∧ 𝑐𝑋) ∧ 𝑧 = [𝑐] ) → (𝑧(pstoMet‘𝐷)𝑥) = (𝑐𝐷𝑎))
7267, 63oveq12d 6823 . . . . . . . . . . . . . . 15 (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) ∧ 𝑐𝑋) ∧ 𝑧 = [𝑐] ) → (𝑧(pstoMet‘𝐷)𝑦) = ([𝑐] (pstoMet‘𝐷)[𝑏] ))
737pstmfval 30240 . . . . . . . . . . . . . . . 16 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑐𝑋𝑏𝑋) → ([𝑐] (pstoMet‘𝐷)[𝑏] ) = (𝑐𝐷𝑏))
7456, 57, 59, 73syl3anc 1473 . . . . . . . . . . . . . . 15 (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) ∧ 𝑐𝑋) ∧ 𝑧 = [𝑐] ) → ([𝑐] (pstoMet‘𝐷)[𝑏] ) = (𝑐𝐷𝑏))
7572, 74eqtrd 2786 . . . . . . . . . . . . . 14 (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) ∧ 𝑐𝑋) ∧ 𝑧 = [𝑐] ) → (𝑧(pstoMet‘𝐷)𝑦) = (𝑐𝐷𝑏))
7671, 75oveq12d 6823 . . . . . . . . . . . . 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 4828 . . . . . . . . . . . 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 806 . . . . . . . . . . 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 7959 . . . . . . . . . . . 12 (𝑧 ∈ (𝑋 / ) → ∃𝑐𝑋 𝑧 = [𝑐] )
8079ad5antlr 777 . . . . . . . . . . 11 ((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑧 ∈ (𝑋 / )) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) → ∃𝑐𝑋 𝑧 = [𝑐] )
8178, 80r19.29a 3208 . . . . . . . . . 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 805 . . . . . . . . 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 773 . . . . . . . . 9 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦 ∈ (𝑋 / )) ∧ 𝑧 ∈ (𝑋 / )) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) → ∃𝑏𝑋 𝑦 = [𝑏] )
8482, 83r19.29a 3208 . . . . . . . 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 804 . . . . . . 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 769 . . . . . . 7 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 ∈ (𝑋 / )) ∧ 𝑦 ∈ (𝑋 / )) ∧ 𝑧 ∈ (𝑋 / )) → ∃𝑎𝑋 𝑥 = [𝑎] )
8785, 86r19.29a 3208 . . . . . 6 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 ∈ (𝑋 / )) ∧ 𝑦 ∈ (𝑋 / )) ∧ 𝑧 ∈ (𝑋 / )) → (𝑥(pstoMet‘𝐷)𝑦) ≤ ((𝑧(pstoMet‘𝐷)𝑥) +𝑒 (𝑧(pstoMet‘𝐷)𝑦)))
8887ralrimiva 3096 . . . . 5 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 ∈ (𝑋 / )) ∧ 𝑦 ∈ (𝑋 / )) → ∀𝑧 ∈ (𝑋 / )(𝑥(pstoMet‘𝐷)𝑦) ≤ ((𝑧(pstoMet‘𝐷)𝑥) +𝑒 (𝑧(pstoMet‘𝐷)𝑦)))
8988anasss 682 . . . 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 555 . . 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 3101 . 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 6374 . . 3 (𝐷 ∈ (PsMet‘𝑋) → 𝑋 ∈ V)
93 qsexg 7964 . . 3 (𝑋 ∈ V → (𝑋 / ) ∈ V)
94 isxmet 22322 . . 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 995 1 (𝐷 ∈ (PsMet‘𝑋) → (pstoMet‘𝐷) ∈ (∞Met‘(𝑋 / )))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1624  wcel 2131  {cab 2738  wral 3042  wrex 3043  Vcvv 3332   cuni 4580   class class class wbr 4796   × cxp 5256   Fn wfn 6036  wf 6037  cfv 6041  (class class class)co 6805  cmpt2 6807   Er wer 7900  [cec 7901   / cqs 7902  0cc0 10120  *cxr 10257  cle 10259   +𝑒 cxad 12129  PsMetcpsmet 19924  ∞Metcxmt 19925  ~Metcmetid 30230  pstoMetcpstm 30231
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-rep 4915  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106  ax-cnex 10176  ax-resscn 10177  ax-1cn 10178  ax-icn 10179  ax-addcl 10180  ax-addrcl 10181  ax-mulcl 10182  ax-mulrcl 10183  ax-mulcom 10184  ax-addass 10185  ax-mulass 10186  ax-distr 10187  ax-i2m1 10188  ax-1ne0 10189  ax-1rid 10190  ax-rnegex 10191  ax-rrecex 10192  ax-cnre 10193  ax-pre-lttri 10194  ax-pre-lttrn 10195  ax-pre-ltadd 10196  ax-pre-mulgt0 10197
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-nel 3028  df-ral 3047  df-rex 3048  df-reu 3049  df-rmo 3050  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-po 5179  df-so 5180  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-riota 6766  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-1st 7325  df-2nd 7326  df-er 7903  df-ec 7905  df-qs 7909  df-map 8017  df-en 8114  df-dom 8115  df-sdom 8116  df-pnf 10260  df-mnf 10261  df-xr 10262  df-ltxr 10263  df-le 10264  df-sub 10452  df-neg 10453  df-div 10869  df-2 11263  df-rp 12018  df-xneg 12131  df-xadd 12132  df-xmul 12133  df-psmet 19932  df-xmet 19933  df-metid 30232  df-pstm 30233
This theorem is referenced by: (None)
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