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Theorem pstmxmet 29746
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 2621 . . . . 5 (𝑥 ∈ (𝑋 / ), 𝑦 ∈ (𝑋 / ) ↦ {𝑧 ∣ ∃𝑎𝑥𝑏𝑦 𝑧 = (𝑎𝐷𝑏)}) = (𝑥 ∈ (𝑋 / ), 𝑦 ∈ (𝑋 / ) ↦ {𝑧 ∣ ∃𝑎𝑥𝑏𝑦 𝑧 = (𝑎𝐷𝑏)})
2 vex 3192 . . . . . . 7 𝑥 ∈ V
3 vex 3192 . . . . . . 7 𝑦 ∈ V
42, 3ab2rexex 7111 . . . . . 6 {𝑧 ∣ ∃𝑎𝑥𝑏𝑦 𝑧 = (𝑎𝐷𝑏)} ∈ V
54uniex 6913 . . . . 5 {𝑧 ∣ ∃𝑎𝑥𝑏𝑦 𝑧 = (𝑎𝐷𝑏)} ∈ V
61, 5fnmpt2i 7191 . . . 4 (𝑥 ∈ (𝑋 / ), 𝑦 ∈ (𝑋 / ) ↦ {𝑧 ∣ ∃𝑎𝑥𝑏𝑦 𝑧 = (𝑎𝐷𝑏)}) Fn ((𝑋 / ) × (𝑋 / ))
7 pstmval.1 . . . . . 6 = (~Met𝐷)
87pstmval 29744 . . . . 5 (𝐷 ∈ (PsMet‘𝑋) → (pstoMet‘𝐷) = (𝑥 ∈ (𝑋 / ), 𝑦 ∈ (𝑋 / ) ↦ {𝑧 ∣ ∃𝑎𝑥𝑏𝑦 𝑧 = (𝑎𝐷𝑏)}))
98fneq1d 5944 . . . 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 798 . . . . . . . . 9 ((((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) → 𝑥 = [𝑎] )
12 simpr 477 . . . . . . . . 9 ((((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) → 𝑦 = [𝑏] )
1311, 12oveq12d 6628 . . . . . . . 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 806 . . . . . . . . 9 ((((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) → 𝑎𝑋)
16 simplr 791 . . . . . . . . 9 ((((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) → 𝑏𝑋)
177pstmfval 29745 . . . . . . . . 9 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋𝑏𝑋) → ([𝑎] (pstoMet‘𝐷)[𝑏] ) = (𝑎𝐷𝑏))
1814, 15, 16, 17syl3anc 1323 . . . . . . . 8 ((((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) → ([𝑎] (pstoMet‘𝐷)[𝑏] ) = (𝑎𝐷𝑏))
1913, 18eqtrd 2655 . . . . . . 7 ((((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) → (𝑥(pstoMet‘𝐷)𝑦) = (𝑎𝐷𝑏))
20 psmetf 22034 . . . . . . . . 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 6766 . . . . . . 7 ((((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) → (𝑎𝐷𝑏) ∈ ℝ*)
2319, 22eqeltrd 2698 . . . . . 6 ((((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) → (𝑥(pstoMet‘𝐷)𝑦) ∈ ℝ*)
24 elqsi 7752 . . . . . . . 8 (𝑦 ∈ (𝑋 / ) → ∃𝑏𝑋 𝑦 = [𝑏] )
2524ad2antll 764 . . . . . . 7 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) → ∃𝑏𝑋 𝑦 = [𝑏] )
2625ad2antrr 761 . . . . . 6 ((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) → ∃𝑏𝑋 𝑦 = [𝑏] )
2723, 26r19.29a 3072 . . . . 5 ((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) → (𝑥(pstoMet‘𝐷)𝑦) ∈ ℝ*)
28 elqsi 7752 . . . . . 6 (𝑥 ∈ (𝑋 / ) → ∃𝑎𝑋 𝑥 = [𝑎] )
2928ad2antrl 763 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) → ∃𝑎𝑋 𝑥 = [𝑎] )
3027, 29r19.29a 3072 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) → (𝑥(pstoMet‘𝐷)𝑦) ∈ ℝ*)
3130ralrimivva 2966 . . 3 (𝐷 ∈ (PsMet‘𝑋) → ∀𝑥 ∈ (𝑋 / )∀𝑦 ∈ (𝑋 / )(𝑥(pstoMet‘𝐷)𝑦) ∈ ℝ*)
32 ffnov 6724 . . 3 ((pstoMet‘𝐷):((𝑋 / ) × (𝑋 / ))⟶ℝ* ↔ ((pstoMet‘𝐷) Fn ((𝑋 / ) × (𝑋 / )) ∧ ∀𝑥 ∈ (𝑋 / )∀𝑦 ∈ (𝑋 / )(𝑥(pstoMet‘𝐷)𝑦) ∈ ℝ*))
3310, 31, 32sylanbrc 697 . 2 (𝐷 ∈ (PsMet‘𝑋) → (pstoMet‘𝐷):((𝑋 / ) × (𝑋 / ))⟶ℝ*)
34173expa 1262 . . . . . . . . . . . 12 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑏𝑋) → ([𝑎] (pstoMet‘𝐷)[𝑏] ) = (𝑎𝐷𝑏))
3534eqeq1d 2623 . . . . . . . . . . 11 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑏𝑋) → (([𝑎] (pstoMet‘𝐷)[𝑏] ) = 0 ↔ (𝑎𝐷𝑏) = 0))
367breqi 4624 . . . . . . . . . . . 12 (𝑎 𝑏𝑎(~Met𝐷)𝑏)
37 metidv 29741 . . . . . . . . . . . . 13 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑎𝑋𝑏𝑋)) → (𝑎(~Met𝐷)𝑏 ↔ (𝑎𝐷𝑏) = 0))
3837anassrs 679 . . . . . . . . . . . 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 29743 . . . . . . . . . . . . . 14 (𝐷 ∈ (PsMet‘𝑋) → (~Met𝐷) Er 𝑋)
4140ad2antrr 761 . . . . . . . . . . . . 13 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑏𝑋) → (~Met𝐷) Er 𝑋)
42 ereq1 7701 . . . . . . . . . . . . . 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 791 . . . . . . . . . . . 12 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑏𝑋) → 𝑎𝑋)
4644, 45erth 7743 . . . . . . . . . . 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 754 . . . . . . . . 9 ((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) ∧ 𝑎𝑋) ∧ 𝑏𝑋) → (([𝑎] (pstoMet‘𝐷)[𝑏] ) = 0 ↔ [𝑎] = [𝑏] ))
4948adantlr 750 . . . . . . . 8 (((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) → (([𝑎] (pstoMet‘𝐷)[𝑏] ) = 0 ↔ [𝑎] = [𝑏] ))
5049adantr 481 . . . . . . 7 ((((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) → (([𝑎] (pstoMet‘𝐷)[𝑏] ) = 0 ↔ [𝑎] = [𝑏] ))
5113eqeq1d 2623 . . . . . . 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 2636 . . . . . . 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 3072 . . . . 5 ((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) → ((𝑥(pstoMet‘𝐷)𝑦) = 0 ↔ 𝑥 = 𝑦))
5554, 29r19.29a 3072 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) → ((𝑥(pstoMet‘𝐷)𝑦) = 0 ↔ 𝑥 = 𝑦))
56 simp-6l 809 . . . . . . . . . . . . . 14 (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) ∧ 𝑐𝑋) ∧ 𝑧 = [𝑐] ) → 𝐷 ∈ (PsMet‘𝑋))
57 simplr 791 . . . . . . . . . . . . . 14 (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) ∧ 𝑐𝑋) ∧ 𝑧 = [𝑐] ) → 𝑐𝑋)
58 simp-6r 810 . . . . . . . . . . . . . 14 (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) ∧ 𝑐𝑋) ∧ 𝑧 = [𝑐] ) → 𝑎𝑋)
59 simp-4r 806 . . . . . . . . . . . . . 14 (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) ∧ 𝑐𝑋) ∧ 𝑧 = [𝑐] ) → 𝑏𝑋)
60 psmettri2 22037 . . . . . . . . . . . . . 14 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑐𝑋𝑎𝑋𝑏𝑋)) → (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)))
6156, 57, 58, 59, 60syl13anc 1325 . . . . . . . . . . . . 13 (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) ∧ 𝑐𝑋) ∧ 𝑧 = [𝑐] ) → (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)))
62 simp-5r 808 . . . . . . . . . . . . . . 15 (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) ∧ 𝑐𝑋) ∧ 𝑧 = [𝑐] ) → 𝑥 = [𝑎] )
63 simpllr 798 . . . . . . . . . . . . . . 15 (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) ∧ 𝑐𝑋) ∧ 𝑧 = [𝑐] ) → 𝑦 = [𝑏] )
6462, 63oveq12d 6628 . . . . . . . . . . . . . 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 1323 . . . . . . . . . . . . . 14 (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) ∧ 𝑐𝑋) ∧ 𝑧 = [𝑐] ) → ([𝑎] (pstoMet‘𝐷)[𝑏] ) = (𝑎𝐷𝑏))
6664, 65eqtrd 2655 . . . . . . . . . . . . 13 (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) ∧ 𝑐𝑋) ∧ 𝑧 = [𝑐] ) → (𝑥(pstoMet‘𝐷)𝑦) = (𝑎𝐷𝑏))
67 simpr 477 . . . . . . . . . . . . . . . 16 (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) ∧ 𝑐𝑋) ∧ 𝑧 = [𝑐] ) → 𝑧 = [𝑐] )
6867, 62oveq12d 6628 . . . . . . . . . . . . . . 15 (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) ∧ 𝑐𝑋) ∧ 𝑧 = [𝑐] ) → (𝑧(pstoMet‘𝐷)𝑥) = ([𝑐] (pstoMet‘𝐷)[𝑎] ))
697pstmfval 29745 . . . . . . . . . . . . . . . 16 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑐𝑋𝑎𝑋) → ([𝑐] (pstoMet‘𝐷)[𝑎] ) = (𝑐𝐷𝑎))
7056, 57, 58, 69syl3anc 1323 . . . . . . . . . . . . . . 15 (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) ∧ 𝑐𝑋) ∧ 𝑧 = [𝑐] ) → ([𝑐] (pstoMet‘𝐷)[𝑎] ) = (𝑐𝐷𝑎))
7168, 70eqtrd 2655 . . . . . . . . . . . . . 14 (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) ∧ 𝑐𝑋) ∧ 𝑧 = [𝑐] ) → (𝑧(pstoMet‘𝐷)𝑥) = (𝑐𝐷𝑎))
7267, 63oveq12d 6628 . . . . . . . . . . . . . . 15 (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) ∧ 𝑐𝑋) ∧ 𝑧 = [𝑐] ) → (𝑧(pstoMet‘𝐷)𝑦) = ([𝑐] (pstoMet‘𝐷)[𝑏] ))
737pstmfval 29745 . . . . . . . . . . . . . . . 16 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑐𝑋𝑏𝑋) → ([𝑐] (pstoMet‘𝐷)[𝑏] ) = (𝑐𝐷𝑏))
7456, 57, 59, 73syl3anc 1323 . . . . . . . . . . . . . . 15 (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) ∧ 𝑐𝑋) ∧ 𝑧 = [𝑐] ) → ([𝑐] (pstoMet‘𝐷)[𝑏] ) = (𝑐𝐷𝑏))
7572, 74eqtrd 2655 . . . . . . . . . . . . . 14 (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) ∧ 𝑐𝑋) ∧ 𝑧 = [𝑐] ) → (𝑧(pstoMet‘𝐷)𝑦) = (𝑐𝐷𝑏))
7671, 75oveq12d 6628 . . . . . . . . . . . . 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 4650 . . . . . . . . . . . 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 788 . . . . . . . . . . 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 7752 . . . . . . . . . . . 12 (𝑧 ∈ (𝑋 / ) → ∃𝑐𝑋 𝑧 = [𝑐] )
8079ad5antlr 770 . . . . . . . . . . 11 ((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑧 ∈ (𝑋 / )) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) → ∃𝑐𝑋 𝑧 = [𝑐] )
8178, 80r19.29a 3072 . . . . . . . . . 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 787 . . . . . . . . 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 768 . . . . . . . . 9 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦 ∈ (𝑋 / )) ∧ 𝑧 ∈ (𝑋 / )) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) → ∃𝑏𝑋 𝑦 = [𝑏] )
8482, 83r19.29a 3072 . . . . . . . 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 786 . . . . . . 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 766 . . . . . . 7 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 ∈ (𝑋 / )) ∧ 𝑦 ∈ (𝑋 / )) ∧ 𝑧 ∈ (𝑋 / )) → ∃𝑎𝑋 𝑥 = [𝑎] )
8785, 86r19.29a 3072 . . . . . 6 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 ∈ (𝑋 / )) ∧ 𝑦 ∈ (𝑋 / )) ∧ 𝑧 ∈ (𝑋 / )) → (𝑥(pstoMet‘𝐷)𝑦) ≤ ((𝑧(pstoMet‘𝐷)𝑥) +𝑒 (𝑧(pstoMet‘𝐷)𝑦)))
8887ralrimiva 2961 . . . . 5 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 ∈ (𝑋 / )) ∧ 𝑦 ∈ (𝑋 / )) → ∀𝑧 ∈ (𝑋 / )(𝑥(pstoMet‘𝐷)𝑦) ≤ ((𝑧(pstoMet‘𝐷)𝑥) +𝑒 (𝑧(pstoMet‘𝐷)𝑦)))
8988anasss 678 . . . 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 554 . . 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 2966 . 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 6183 . . 3 (𝐷 ∈ (PsMet‘𝑋) → 𝑋 ∈ V)
93 qsexg 7757 . . 3 (𝑋 ∈ V → (𝑋 / ) ∈ V)
94 isxmet 22052 . . 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 956 1 (𝐷 ∈ (PsMet‘𝑋) → (pstoMet‘𝐷) ∈ (∞Met‘(𝑋 / )))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  {cab 2607  wral 2907  wrex 2908  Vcvv 3189   cuni 4407   class class class wbr 4618   × cxp 5077   Fn wfn 5847  wf 5848  cfv 5852  (class class class)co 6610  cmpt2 6612   Er wer 7691  [cec 7692   / cqs 7693  0cc0 9888  *cxr 10025  cle 10027   +𝑒 cxad 11896  PsMetcpsmet 19662  ∞Metcxmt 19663  ~Metcmetid 29735  pstoMetcpstm 29736
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9944  ax-resscn 9945  ax-1cn 9946  ax-icn 9947  ax-addcl 9948  ax-addrcl 9949  ax-mulcl 9950  ax-mulrcl 9951  ax-mulcom 9952  ax-addass 9953  ax-mulass 9954  ax-distr 9955  ax-i2m1 9956  ax-1ne0 9957  ax-1rid 9958  ax-rnegex 9959  ax-rrecex 9960  ax-cnre 9961  ax-pre-lttri 9962  ax-pre-lttrn 9963  ax-pre-ltadd 9964  ax-pre-mulgt0 9965
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-po 5000  df-so 5001  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-1st 7120  df-2nd 7121  df-er 7694  df-ec 7696  df-qs 7700  df-map 7811  df-en 7908  df-dom 7909  df-sdom 7910  df-pnf 10028  df-mnf 10029  df-xr 10030  df-ltxr 10031  df-le 10032  df-sub 10220  df-neg 10221  df-div 10637  df-2 11031  df-rp 11785  df-xneg 11898  df-xadd 11899  df-xmul 11900  df-psmet 19670  df-xmet 19671  df-metid 29737  df-pstm 29738
This theorem is referenced by: (None)
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