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Theorem pstmxmet 33857
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 2734 . . . . 5 (𝑥 ∈ (𝑋 / ), 𝑦 ∈ (𝑋 / ) ↦ {𝑧 ∣ ∃𝑎𝑥𝑏𝑦 𝑧 = (𝑎𝐷𝑏)}) = (𝑥 ∈ (𝑋 / ), 𝑦 ∈ (𝑋 / ) ↦ {𝑧 ∣ ∃𝑎𝑥𝑏𝑦 𝑧 = (𝑎𝐷𝑏)})
2 vex 3481 . . . . . . 7 𝑥 ∈ V
3 vex 3481 . . . . . . 7 𝑦 ∈ V
42, 3ab2rexex 8002 . . . . . 6 {𝑧 ∣ ∃𝑎𝑥𝑏𝑦 𝑧 = (𝑎𝐷𝑏)} ∈ V
54uniex 7759 . . . . 5 {𝑧 ∣ ∃𝑎𝑥𝑏𝑦 𝑧 = (𝑎𝐷𝑏)} ∈ V
61, 5fnmpoi 8093 . . . 4 (𝑥 ∈ (𝑋 / ), 𝑦 ∈ (𝑋 / ) ↦ {𝑧 ∣ ∃𝑎𝑥𝑏𝑦 𝑧 = (𝑎𝐷𝑏)}) Fn ((𝑋 / ) × (𝑋 / ))
7 pstmval.1 . . . . . 6 = (~Met𝐷)
87pstmval 33855 . . . . 5 (𝐷 ∈ (PsMet‘𝑋) → (pstoMet‘𝐷) = (𝑥 ∈ (𝑋 / ), 𝑦 ∈ (𝑋 / ) ↦ {𝑧 ∣ ∃𝑎𝑥𝑏𝑦 𝑧 = (𝑎𝐷𝑏)}))
98fneq1d 6661 . . . 4 (𝐷 ∈ (PsMet‘𝑋) → ((pstoMet‘𝐷) Fn ((𝑋 / ) × (𝑋 / )) ↔ (𝑥 ∈ (𝑋 / ), 𝑦 ∈ (𝑋 / ) ↦ {𝑧 ∣ ∃𝑎𝑥𝑏𝑦 𝑧 = (𝑎𝐷𝑏)}) Fn ((𝑋 / ) × (𝑋 / ))))
106, 9mpbiri 258 . . 3 (𝐷 ∈ (PsMet‘𝑋) → (pstoMet‘𝐷) Fn ((𝑋 / ) × (𝑋 / )))
11 simpllr 776 . . . . . . . . 9 ((((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) → 𝑥 = [𝑎] )
12 simpr 484 . . . . . . . . 9 ((((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) → 𝑦 = [𝑏] )
1311, 12oveq12d 7448 . . . . . . . 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 785 . . . . . . . . 9 ((((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) → 𝐷 ∈ (PsMet‘𝑋))
15 simp-4r 784 . . . . . . . . 9 ((((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) → 𝑎𝑋)
16 simplr 769 . . . . . . . . 9 ((((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) → 𝑏𝑋)
177pstmfval 33856 . . . . . . . . 9 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋𝑏𝑋) → ([𝑎] (pstoMet‘𝐷)[𝑏] ) = (𝑎𝐷𝑏))
1814, 15, 16, 17syl3anc 1370 . . . . . . . 8 ((((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) → ([𝑎] (pstoMet‘𝐷)[𝑏] ) = (𝑎𝐷𝑏))
1913, 18eqtrd 2774 . . . . . . 7 ((((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) → (𝑥(pstoMet‘𝐷)𝑦) = (𝑎𝐷𝑏))
20 psmetf 24331 . . . . . . . . 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, 16fovcdmd 7604 . . . . . . 7 ((((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) → (𝑎𝐷𝑏) ∈ ℝ*)
2319, 22eqeltrd 2838 . . . . . 6 ((((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) → (𝑥(pstoMet‘𝐷)𝑦) ∈ ℝ*)
24 elqsi 8808 . . . . . . . 8 (𝑦 ∈ (𝑋 / ) → ∃𝑏𝑋 𝑦 = [𝑏] )
2524ad2antll 729 . . . . . . 7 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) → ∃𝑏𝑋 𝑦 = [𝑏] )
2625ad2antrr 726 . . . . . 6 ((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) → ∃𝑏𝑋 𝑦 = [𝑏] )
2723, 26r19.29a 3159 . . . . 5 ((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) → (𝑥(pstoMet‘𝐷)𝑦) ∈ ℝ*)
28 elqsi 8808 . . . . . 6 (𝑥 ∈ (𝑋 / ) → ∃𝑎𝑋 𝑥 = [𝑎] )
2928ad2antrl 728 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) → ∃𝑎𝑋 𝑥 = [𝑎] )
3027, 29r19.29a 3159 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) → (𝑥(pstoMet‘𝐷)𝑦) ∈ ℝ*)
3130ralrimivva 3199 . . 3 (𝐷 ∈ (PsMet‘𝑋) → ∀𝑥 ∈ (𝑋 / )∀𝑦 ∈ (𝑋 / )(𝑥(pstoMet‘𝐷)𝑦) ∈ ℝ*)
32 ffnov 7558 . . 3 ((pstoMet‘𝐷):((𝑋 / ) × (𝑋 / ))⟶ℝ* ↔ ((pstoMet‘𝐷) Fn ((𝑋 / ) × (𝑋 / )) ∧ ∀𝑥 ∈ (𝑋 / )∀𝑦 ∈ (𝑋 / )(𝑥(pstoMet‘𝐷)𝑦) ∈ ℝ*))
3310, 31, 32sylanbrc 583 . 2 (𝐷 ∈ (PsMet‘𝑋) → (pstoMet‘𝐷):((𝑋 / ) × (𝑋 / ))⟶ℝ*)
34173expa 1117 . . . . . . . . . . . 12 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑏𝑋) → ([𝑎] (pstoMet‘𝐷)[𝑏] ) = (𝑎𝐷𝑏))
3534eqeq1d 2736 . . . . . . . . . . 11 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑏𝑋) → (([𝑎] (pstoMet‘𝐷)[𝑏] ) = 0 ↔ (𝑎𝐷𝑏) = 0))
367breqi 5153 . . . . . . . . . . . 12 (𝑎 𝑏𝑎(~Met𝐷)𝑏)
37 metidv 33852 . . . . . . . . . . . . 13 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑎𝑋𝑏𝑋)) → (𝑎(~Met𝐷)𝑏 ↔ (𝑎𝐷𝑏) = 0))
3837anassrs 467 . . . . . . . . . . . 12 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑏𝑋) → (𝑎(~Met𝐷)𝑏 ↔ (𝑎𝐷𝑏) = 0))
3936, 38bitrid 283 . . . . . . . . . . 11 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑏𝑋) → (𝑎 𝑏 ↔ (𝑎𝐷𝑏) = 0))
40 metider 33854 . . . . . . . . . . . . . 14 (𝐷 ∈ (PsMet‘𝑋) → (~Met𝐷) Er 𝑋)
4140ad2antrr 726 . . . . . . . . . . . . 13 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑏𝑋) → (~Met𝐷) Er 𝑋)
42 ereq1 8750 . . . . . . . . . . . . . 14 ( = (~Met𝐷) → ( Er 𝑋 ↔ (~Met𝐷) Er 𝑋))
437, 42ax-mp 5 . . . . . . . . . . . . 13 ( Er 𝑋 ↔ (~Met𝐷) Er 𝑋)
4441, 43sylibr 234 . . . . . . . . . . . 12 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑏𝑋) → Er 𝑋)
45 simplr 769 . . . . . . . . . . . 12 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑏𝑋) → 𝑎𝑋)
4644, 45erth 8794 . . . . . . . . . . 11 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑏𝑋) → (𝑎 𝑏 ↔ [𝑎] = [𝑏] ))
4735, 39, 463bitr2d 307 . . . . . . . . . 10 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑏𝑋) → (([𝑎] (pstoMet‘𝐷)[𝑏] ) = 0 ↔ [𝑎] = [𝑏] ))
4847adantllr 719 . . . . . . . . 9 ((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) ∧ 𝑎𝑋) ∧ 𝑏𝑋) → (([𝑎] (pstoMet‘𝐷)[𝑏] ) = 0 ↔ [𝑎] = [𝑏] ))
4948adantlr 715 . . . . . . . 8 (((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) → (([𝑎] (pstoMet‘𝐷)[𝑏] ) = 0 ↔ [𝑎] = [𝑏] ))
5049adantr 480 . . . . . . 7 ((((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) → (([𝑎] (pstoMet‘𝐷)[𝑏] ) = 0 ↔ [𝑎] = [𝑏] ))
5113eqeq1d 2736 . . . . . . 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 2750 . . . . . . 7 ((((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) → (𝑥 = 𝑦 ↔ [𝑎] = [𝑏] ))
5350, 51, 523bitr4d 311 . . . . . 6 ((((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) → ((𝑥(pstoMet‘𝐷)𝑦) = 0 ↔ 𝑥 = 𝑦))
5453, 26r19.29a 3159 . . . . 5 ((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) → ((𝑥(pstoMet‘𝐷)𝑦) = 0 ↔ 𝑥 = 𝑦))
5554, 29r19.29a 3159 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) → ((𝑥(pstoMet‘𝐷)𝑦) = 0 ↔ 𝑥 = 𝑦))
56 simp-6l 787 . . . . . . . . . . . . . 14 (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) ∧ 𝑐𝑋) ∧ 𝑧 = [𝑐] ) → 𝐷 ∈ (PsMet‘𝑋))
57 simplr 769 . . . . . . . . . . . . . 14 (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) ∧ 𝑐𝑋) ∧ 𝑧 = [𝑐] ) → 𝑐𝑋)
58 simp-6r 788 . . . . . . . . . . . . . 14 (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) ∧ 𝑐𝑋) ∧ 𝑧 = [𝑐] ) → 𝑎𝑋)
59 simp-4r 784 . . . . . . . . . . . . . 14 (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) ∧ 𝑐𝑋) ∧ 𝑧 = [𝑐] ) → 𝑏𝑋)
60 psmettri2 24334 . . . . . . . . . . . . . 14 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑐𝑋𝑎𝑋𝑏𝑋)) → (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)))
6156, 57, 58, 59, 60syl13anc 1371 . . . . . . . . . . . . 13 (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) ∧ 𝑐𝑋) ∧ 𝑧 = [𝑐] ) → (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)))
62 simp-5r 786 . . . . . . . . . . . . . . 15 (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) ∧ 𝑐𝑋) ∧ 𝑧 = [𝑐] ) → 𝑥 = [𝑎] )
63 simpllr 776 . . . . . . . . . . . . . . 15 (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) ∧ 𝑐𝑋) ∧ 𝑧 = [𝑐] ) → 𝑦 = [𝑏] )
6462, 63oveq12d 7448 . . . . . . . . . . . . . 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 1370 . . . . . . . . . . . . . 14 (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) ∧ 𝑐𝑋) ∧ 𝑧 = [𝑐] ) → ([𝑎] (pstoMet‘𝐷)[𝑏] ) = (𝑎𝐷𝑏))
6664, 65eqtrd 2774 . . . . . . . . . . . . 13 (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) ∧ 𝑐𝑋) ∧ 𝑧 = [𝑐] ) → (𝑥(pstoMet‘𝐷)𝑦) = (𝑎𝐷𝑏))
67 simpr 484 . . . . . . . . . . . . . . . 16 (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) ∧ 𝑐𝑋) ∧ 𝑧 = [𝑐] ) → 𝑧 = [𝑐] )
6867, 62oveq12d 7448 . . . . . . . . . . . . . . 15 (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) ∧ 𝑐𝑋) ∧ 𝑧 = [𝑐] ) → (𝑧(pstoMet‘𝐷)𝑥) = ([𝑐] (pstoMet‘𝐷)[𝑎] ))
697pstmfval 33856 . . . . . . . . . . . . . . . 16 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑐𝑋𝑎𝑋) → ([𝑐] (pstoMet‘𝐷)[𝑎] ) = (𝑐𝐷𝑎))
7056, 57, 58, 69syl3anc 1370 . . . . . . . . . . . . . . 15 (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) ∧ 𝑐𝑋) ∧ 𝑧 = [𝑐] ) → ([𝑐] (pstoMet‘𝐷)[𝑎] ) = (𝑐𝐷𝑎))
7168, 70eqtrd 2774 . . . . . . . . . . . . . 14 (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) ∧ 𝑐𝑋) ∧ 𝑧 = [𝑐] ) → (𝑧(pstoMet‘𝐷)𝑥) = (𝑐𝐷𝑎))
7267, 63oveq12d 7448 . . . . . . . . . . . . . . 15 (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) ∧ 𝑐𝑋) ∧ 𝑧 = [𝑐] ) → (𝑧(pstoMet‘𝐷)𝑦) = ([𝑐] (pstoMet‘𝐷)[𝑏] ))
737pstmfval 33856 . . . . . . . . . . . . . . . 16 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑐𝑋𝑏𝑋) → ([𝑐] (pstoMet‘𝐷)[𝑏] ) = (𝑐𝐷𝑏))
7456, 57, 59, 73syl3anc 1370 . . . . . . . . . . . . . . 15 (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) ∧ 𝑐𝑋) ∧ 𝑧 = [𝑐] ) → ([𝑐] (pstoMet‘𝐷)[𝑏] ) = (𝑐𝐷𝑏))
7572, 74eqtrd 2774 . . . . . . . . . . . . . 14 (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) ∧ 𝑐𝑋) ∧ 𝑧 = [𝑐] ) → (𝑧(pstoMet‘𝐷)𝑦) = (𝑐𝐷𝑏))
7671, 75oveq12d 7448 . . . . . . . . . . . . 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 5179 . . . . . . . . . . . 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 764 . . . . . . . . . . 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 8808 . . . . . . . . . . . 12 (𝑧 ∈ (𝑋 / ) → ∃𝑐𝑋 𝑧 = [𝑐] )
8079ad5antlr 735 . . . . . . . . . . 11 ((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑧 ∈ (𝑋 / )) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) → ∃𝑐𝑋 𝑧 = [𝑐] )
8178, 80r19.29a 3159 . . . . . . . . . 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 763 . . . . . . . . 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 733 . . . . . . . . 9 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦 ∈ (𝑋 / )) ∧ 𝑧 ∈ (𝑋 / )) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) → ∃𝑏𝑋 𝑦 = [𝑏] )
8482, 83r19.29a 3159 . . . . . . . 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 755 . . . . . . 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 731 . . . . . . 7 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 ∈ (𝑋 / )) ∧ 𝑦 ∈ (𝑋 / )) ∧ 𝑧 ∈ (𝑋 / )) → ∃𝑎𝑋 𝑥 = [𝑎] )
8785, 86r19.29a 3159 . . . . . 6 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 ∈ (𝑋 / )) ∧ 𝑦 ∈ (𝑋 / )) ∧ 𝑧 ∈ (𝑋 / )) → (𝑥(pstoMet‘𝐷)𝑦) ≤ ((𝑧(pstoMet‘𝐷)𝑥) +𝑒 (𝑧(pstoMet‘𝐷)𝑦)))
8887ralrimiva 3143 . . . . 5 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 ∈ (𝑋 / )) ∧ 𝑦 ∈ (𝑋 / )) → ∀𝑧 ∈ (𝑋 / )(𝑥(pstoMet‘𝐷)𝑦) ≤ ((𝑧(pstoMet‘𝐷)𝑥) +𝑒 (𝑧(pstoMet‘𝐷)𝑦)))
8988anasss 466 . . . 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 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))))
9190ralrimivva 3199 . 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 6944 . . 3 (𝐷 ∈ (PsMet‘𝑋) → 𝑋 ∈ V)
93 qsexg 8813 . . 3 (𝑋 ∈ V → (𝑋 / ) ∈ V)
94 isxmet 24349 . . 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 713 1 (𝐷 ∈ (PsMet‘𝑋) → (pstoMet‘𝐷) ∈ (∞Met‘(𝑋 / )))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1536  wcel 2105  {cab 2711  wral 3058  wrex 3067  Vcvv 3477   cuni 4911   class class class wbr 5147   × cxp 5686   Fn wfn 6557  wf 6558  cfv 6562  (class class class)co 7430  cmpo 7432   Er wer 8740  [cec 8741   / cqs 8742  0cc0 11152  *cxr 11291  cle 11293   +𝑒 cxad 13149  PsMetcpsmet 21365  ∞Metcxmet 21366  ~Metcmetid 33846  pstoMetcpstm 33847
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-po 5596  df-so 5597  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-1st 8012  df-2nd 8013  df-er 8743  df-ec 8745  df-qs 8749  df-map 8866  df-en 8984  df-dom 8985  df-sdom 8986  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-div 11918  df-2 12326  df-rp 13032  df-xneg 13151  df-xadd 13152  df-xmul 13153  df-psmet 21373  df-xmet 21374  df-metid 33848  df-pstm 33849
This theorem is referenced by: (None)
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