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Theorem pstmxmet 31145
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 2820 . . . . 5 (𝑥 ∈ (𝑋 / ), 𝑦 ∈ (𝑋 / ) ↦ {𝑧 ∣ ∃𝑎𝑥𝑏𝑦 𝑧 = (𝑎𝐷𝑏)}) = (𝑥 ∈ (𝑋 / ), 𝑦 ∈ (𝑋 / ) ↦ {𝑧 ∣ ∃𝑎𝑥𝑏𝑦 𝑧 = (𝑎𝐷𝑏)})
2 vex 3476 . . . . . . 7 𝑥 ∈ V
3 vex 3476 . . . . . . 7 𝑦 ∈ V
42, 3ab2rexex 7656 . . . . . 6 {𝑧 ∣ ∃𝑎𝑥𝑏𝑦 𝑧 = (𝑎𝐷𝑏)} ∈ V
54uniex 7443 . . . . 5 {𝑧 ∣ ∃𝑎𝑥𝑏𝑦 𝑧 = (𝑎𝐷𝑏)} ∈ V
61, 5fnmpoi 7744 . . . 4 (𝑥 ∈ (𝑋 / ), 𝑦 ∈ (𝑋 / ) ↦ {𝑧 ∣ ∃𝑎𝑥𝑏𝑦 𝑧 = (𝑎𝐷𝑏)}) Fn ((𝑋 / ) × (𝑋 / ))
7 pstmval.1 . . . . . 6 = (~Met𝐷)
87pstmval 31143 . . . . 5 (𝐷 ∈ (PsMet‘𝑋) → (pstoMet‘𝐷) = (𝑥 ∈ (𝑋 / ), 𝑦 ∈ (𝑋 / ) ↦ {𝑧 ∣ ∃𝑎𝑥𝑏𝑦 𝑧 = (𝑎𝐷𝑏)}))
98fneq1d 6420 . . . 4 (𝐷 ∈ (PsMet‘𝑋) → ((pstoMet‘𝐷) Fn ((𝑋 / ) × (𝑋 / )) ↔ (𝑥 ∈ (𝑋 / ), 𝑦 ∈ (𝑋 / ) ↦ {𝑧 ∣ ∃𝑎𝑥𝑏𝑦 𝑧 = (𝑎𝐷𝑏)}) Fn ((𝑋 / ) × (𝑋 / ))))
106, 9mpbiri 260 . . 3 (𝐷 ∈ (PsMet‘𝑋) → (pstoMet‘𝐷) Fn ((𝑋 / ) × (𝑋 / )))
11 simpllr 774 . . . . . . . . 9 ((((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) → 𝑥 = [𝑎] )
12 simpr 487 . . . . . . . . 9 ((((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) → 𝑦 = [𝑏] )
1311, 12oveq12d 7149 . . . . . . . 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 783 . . . . . . . . 9 ((((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) → 𝐷 ∈ (PsMet‘𝑋))
15 simp-4r 782 . . . . . . . . 9 ((((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) → 𝑎𝑋)
16 simplr 767 . . . . . . . . 9 ((((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) → 𝑏𝑋)
177pstmfval 31144 . . . . . . . . 9 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋𝑏𝑋) → ([𝑎] (pstoMet‘𝐷)[𝑏] ) = (𝑎𝐷𝑏))
1814, 15, 16, 17syl3anc 1367 . . . . . . . 8 ((((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) → ([𝑎] (pstoMet‘𝐷)[𝑏] ) = (𝑎𝐷𝑏))
1913, 18eqtrd 2855 . . . . . . 7 ((((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) → (𝑥(pstoMet‘𝐷)𝑦) = (𝑎𝐷𝑏))
20 psmetf 22889 . . . . . . . . 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 7296 . . . . . . 7 ((((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) → (𝑎𝐷𝑏) ∈ ℝ*)
2319, 22eqeltrd 2911 . . . . . 6 ((((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) → (𝑥(pstoMet‘𝐷)𝑦) ∈ ℝ*)
24 elqsi 8326 . . . . . . . 8 (𝑦 ∈ (𝑋 / ) → ∃𝑏𝑋 𝑦 = [𝑏] )
2524ad2antll 727 . . . . . . 7 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) → ∃𝑏𝑋 𝑦 = [𝑏] )
2625ad2antrr 724 . . . . . 6 ((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) → ∃𝑏𝑋 𝑦 = [𝑏] )
2723, 26r19.29a 3276 . . . . 5 ((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) → (𝑥(pstoMet‘𝐷)𝑦) ∈ ℝ*)
28 elqsi 8326 . . . . . 6 (𝑥 ∈ (𝑋 / ) → ∃𝑎𝑋 𝑥 = [𝑎] )
2928ad2antrl 726 . . . . 5 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) → ∃𝑎𝑋 𝑥 = [𝑎] )
3027, 29r19.29a 3276 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) → (𝑥(pstoMet‘𝐷)𝑦) ∈ ℝ*)
3130ralrimivva 3178 . . 3 (𝐷 ∈ (PsMet‘𝑋) → ∀𝑥 ∈ (𝑋 / )∀𝑦 ∈ (𝑋 / )(𝑥(pstoMet‘𝐷)𝑦) ∈ ℝ*)
32 ffnov 7253 . . 3 ((pstoMet‘𝐷):((𝑋 / ) × (𝑋 / ))⟶ℝ* ↔ ((pstoMet‘𝐷) Fn ((𝑋 / ) × (𝑋 / )) ∧ ∀𝑥 ∈ (𝑋 / )∀𝑦 ∈ (𝑋 / )(𝑥(pstoMet‘𝐷)𝑦) ∈ ℝ*))
3310, 31, 32sylanbrc 585 . 2 (𝐷 ∈ (PsMet‘𝑋) → (pstoMet‘𝐷):((𝑋 / ) × (𝑋 / ))⟶ℝ*)
34173expa 1114 . . . . . . . . . . . 12 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑏𝑋) → ([𝑎] (pstoMet‘𝐷)[𝑏] ) = (𝑎𝐷𝑏))
3534eqeq1d 2822 . . . . . . . . . . 11 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑏𝑋) → (([𝑎] (pstoMet‘𝐷)[𝑏] ) = 0 ↔ (𝑎𝐷𝑏) = 0))
367breqi 5046 . . . . . . . . . . . 12 (𝑎 𝑏𝑎(~Met𝐷)𝑏)
37 metidv 31140 . . . . . . . . . . . . 13 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑎𝑋𝑏𝑋)) → (𝑎(~Met𝐷)𝑏 ↔ (𝑎𝐷𝑏) = 0))
3837anassrs 470 . . . . . . . . . . . 12 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑏𝑋) → (𝑎(~Met𝐷)𝑏 ↔ (𝑎𝐷𝑏) = 0))
3936, 38syl5bb 285 . . . . . . . . . . 11 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑏𝑋) → (𝑎 𝑏 ↔ (𝑎𝐷𝑏) = 0))
40 metider 31142 . . . . . . . . . . . . . 14 (𝐷 ∈ (PsMet‘𝑋) → (~Met𝐷) Er 𝑋)
4140ad2antrr 724 . . . . . . . . . . . . 13 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑏𝑋) → (~Met𝐷) Er 𝑋)
42 ereq1 8272 . . . . . . . . . . . . . 14 ( = (~Met𝐷) → ( Er 𝑋 ↔ (~Met𝐷) Er 𝑋))
437, 42ax-mp 5 . . . . . . . . . . . . 13 ( Er 𝑋 ↔ (~Met𝐷) Er 𝑋)
4441, 43sylibr 236 . . . . . . . . . . . 12 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑏𝑋) → Er 𝑋)
45 simplr 767 . . . . . . . . . . . 12 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑏𝑋) → 𝑎𝑋)
4644, 45erth 8314 . . . . . . . . . . 11 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑏𝑋) → (𝑎 𝑏 ↔ [𝑎] = [𝑏] ))
4735, 39, 463bitr2d 309 . . . . . . . . . 10 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑏𝑋) → (([𝑎] (pstoMet‘𝐷)[𝑏] ) = 0 ↔ [𝑎] = [𝑏] ))
4847adantllr 717 . . . . . . . . 9 ((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) ∧ 𝑎𝑋) ∧ 𝑏𝑋) → (([𝑎] (pstoMet‘𝐷)[𝑏] ) = 0 ↔ [𝑎] = [𝑏] ))
4948adantlr 713 . . . . . . . 8 (((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) → (([𝑎] (pstoMet‘𝐷)[𝑏] ) = 0 ↔ [𝑎] = [𝑏] ))
5049adantr 483 . . . . . . 7 ((((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) → (([𝑎] (pstoMet‘𝐷)[𝑏] ) = 0 ↔ [𝑎] = [𝑏] ))
5113eqeq1d 2822 . . . . . . 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 2836 . . . . . . 7 ((((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) → (𝑥 = 𝑦 ↔ [𝑎] = [𝑏] ))
5350, 51, 523bitr4d 313 . . . . . 6 ((((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) → ((𝑥(pstoMet‘𝐷)𝑦) = 0 ↔ 𝑥 = 𝑦))
5453, 26r19.29a 3276 . . . . 5 ((((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) → ((𝑥(pstoMet‘𝐷)𝑦) = 0 ↔ 𝑥 = 𝑦))
5554, 29r19.29a 3276 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑥 ∈ (𝑋 / ) ∧ 𝑦 ∈ (𝑋 / ))) → ((𝑥(pstoMet‘𝐷)𝑦) = 0 ↔ 𝑥 = 𝑦))
56 simp-6l 785 . . . . . . . . . . . . . 14 (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) ∧ 𝑐𝑋) ∧ 𝑧 = [𝑐] ) → 𝐷 ∈ (PsMet‘𝑋))
57 simplr 767 . . . . . . . . . . . . . 14 (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) ∧ 𝑐𝑋) ∧ 𝑧 = [𝑐] ) → 𝑐𝑋)
58 simp-6r 786 . . . . . . . . . . . . . 14 (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) ∧ 𝑐𝑋) ∧ 𝑧 = [𝑐] ) → 𝑎𝑋)
59 simp-4r 782 . . . . . . . . . . . . . 14 (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) ∧ 𝑐𝑋) ∧ 𝑧 = [𝑐] ) → 𝑏𝑋)
60 psmettri2 22892 . . . . . . . . . . . . . 14 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑐𝑋𝑎𝑋𝑏𝑋)) → (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)))
6156, 57, 58, 59, 60syl13anc 1368 . . . . . . . . . . . . 13 (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) ∧ 𝑐𝑋) ∧ 𝑧 = [𝑐] ) → (𝑎𝐷𝑏) ≤ ((𝑐𝐷𝑎) +𝑒 (𝑐𝐷𝑏)))
62 simp-5r 784 . . . . . . . . . . . . . . 15 (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) ∧ 𝑐𝑋) ∧ 𝑧 = [𝑐] ) → 𝑥 = [𝑎] )
63 simpllr 774 . . . . . . . . . . . . . . 15 (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) ∧ 𝑐𝑋) ∧ 𝑧 = [𝑐] ) → 𝑦 = [𝑏] )
6462, 63oveq12d 7149 . . . . . . . . . . . . . 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 1367 . . . . . . . . . . . . . 14 (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) ∧ 𝑐𝑋) ∧ 𝑧 = [𝑐] ) → ([𝑎] (pstoMet‘𝐷)[𝑏] ) = (𝑎𝐷𝑏))
6664, 65eqtrd 2855 . . . . . . . . . . . . 13 (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) ∧ 𝑐𝑋) ∧ 𝑧 = [𝑐] ) → (𝑥(pstoMet‘𝐷)𝑦) = (𝑎𝐷𝑏))
67 simpr 487 . . . . . . . . . . . . . . . 16 (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) ∧ 𝑐𝑋) ∧ 𝑧 = [𝑐] ) → 𝑧 = [𝑐] )
6867, 62oveq12d 7149 . . . . . . . . . . . . . . 15 (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) ∧ 𝑐𝑋) ∧ 𝑧 = [𝑐] ) → (𝑧(pstoMet‘𝐷)𝑥) = ([𝑐] (pstoMet‘𝐷)[𝑎] ))
697pstmfval 31144 . . . . . . . . . . . . . . . 16 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑐𝑋𝑎𝑋) → ([𝑐] (pstoMet‘𝐷)[𝑎] ) = (𝑐𝐷𝑎))
7056, 57, 58, 69syl3anc 1367 . . . . . . . . . . . . . . 15 (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) ∧ 𝑐𝑋) ∧ 𝑧 = [𝑐] ) → ([𝑐] (pstoMet‘𝐷)[𝑎] ) = (𝑐𝐷𝑎))
7168, 70eqtrd 2855 . . . . . . . . . . . . . 14 (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) ∧ 𝑐𝑋) ∧ 𝑧 = [𝑐] ) → (𝑧(pstoMet‘𝐷)𝑥) = (𝑐𝐷𝑎))
7267, 63oveq12d 7149 . . . . . . . . . . . . . . 15 (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) ∧ 𝑐𝑋) ∧ 𝑧 = [𝑐] ) → (𝑧(pstoMet‘𝐷)𝑦) = ([𝑐] (pstoMet‘𝐷)[𝑏] ))
737pstmfval 31144 . . . . . . . . . . . . . . . 16 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑐𝑋𝑏𝑋) → ([𝑐] (pstoMet‘𝐷)[𝑏] ) = (𝑐𝐷𝑏))
7456, 57, 59, 73syl3anc 1367 . . . . . . . . . . . . . . 15 (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) ∧ 𝑐𝑋) ∧ 𝑧 = [𝑐] ) → ([𝑐] (pstoMet‘𝐷)[𝑏] ) = (𝑐𝐷𝑏))
7572, 74eqtrd 2855 . . . . . . . . . . . . . 14 (((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) ∧ 𝑐𝑋) ∧ 𝑧 = [𝑐] ) → (𝑧(pstoMet‘𝐷)𝑦) = (𝑐𝐷𝑏))
7671, 75oveq12d 7149 . . . . . . . . . . . . 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 5072 . . . . . . . . . . . 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 762 . . . . . . . . . . 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 8326 . . . . . . . . . . . 12 (𝑧 ∈ (𝑋 / ) → ∃𝑐𝑋 𝑧 = [𝑐] )
8079ad5antlr 733 . . . . . . . . . . 11 ((((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑧 ∈ (𝑋 / )) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) ∧ 𝑏𝑋) ∧ 𝑦 = [𝑏] ) → ∃𝑐𝑋 𝑧 = [𝑐] )
8178, 80r19.29a 3276 . . . . . . . . . 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 761 . . . . . . . . 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 731 . . . . . . . . 9 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑦 ∈ (𝑋 / )) ∧ 𝑧 ∈ (𝑋 / )) ∧ 𝑎𝑋) ∧ 𝑥 = [𝑎] ) → ∃𝑏𝑋 𝑦 = [𝑏] )
8482, 83r19.29a 3276 . . . . . . . 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 753 . . . . . . 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 729 . . . . . . 7 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 ∈ (𝑋 / )) ∧ 𝑦 ∈ (𝑋 / )) ∧ 𝑧 ∈ (𝑋 / )) → ∃𝑎𝑋 𝑥 = [𝑎] )
8785, 86r19.29a 3276 . . . . . 6 ((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 ∈ (𝑋 / )) ∧ 𝑦 ∈ (𝑋 / )) ∧ 𝑧 ∈ (𝑋 / )) → (𝑥(pstoMet‘𝐷)𝑦) ≤ ((𝑧(pstoMet‘𝐷)𝑥) +𝑒 (𝑧(pstoMet‘𝐷)𝑦)))
8887ralrimiva 3169 . . . . 5 (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑥 ∈ (𝑋 / )) ∧ 𝑦 ∈ (𝑋 / )) → ∀𝑧 ∈ (𝑋 / )(𝑥(pstoMet‘𝐷)𝑦) ≤ ((𝑧(pstoMet‘𝐷)𝑥) +𝑒 (𝑧(pstoMet‘𝐷)𝑦)))
8988anasss 469 . . . 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 514 . . 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 3178 . 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 6677 . . 3 (𝐷 ∈ (PsMet‘𝑋) → 𝑋 ∈ V)
93 qsexg 8331 . . 3 (𝑋 ∈ V → (𝑋 / ) ∈ V)
94 isxmet 22907 . . 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 711 1 (𝐷 ∈ (PsMet‘𝑋) → (pstoMet‘𝐷) ∈ (∞Met‘(𝑋 / )))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398   = wceq 1537  wcel 2114  {cab 2798  wral 3125  wrex 3126  Vcvv 3473   cuni 4812   class class class wbr 5040   × cxp 5527   Fn wfn 6324  wf 6325  cfv 6329  (class class class)co 7131  cmpo 7133   Er wer 8262  [cec 8263   / cqs 8264  0cc0 10513  *cxr 10650  cle 10652   +𝑒 cxad 12482  PsMetcpsmet 20502  ∞Metcxmet 20503  ~Metcmetid 31134  pstoMetcpstm 31135
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-rep 5164  ax-sep 5177  ax-nul 5184  ax-pow 5240  ax-pr 5304  ax-un 7437  ax-cnex 10569  ax-resscn 10570  ax-1cn 10571  ax-icn 10572  ax-addcl 10573  ax-addrcl 10574  ax-mulcl 10575  ax-mulrcl 10576  ax-mulcom 10577  ax-addass 10578  ax-mulass 10579  ax-distr 10580  ax-i2m1 10581  ax-1ne0 10582  ax-1rid 10583  ax-rnegex 10584  ax-rrecex 10585  ax-cnre 10586  ax-pre-lttri 10587  ax-pre-lttrn 10588  ax-pre-ltadd 10589  ax-pre-mulgt0 10590
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-nel 3111  df-ral 3130  df-rex 3131  df-reu 3132  df-rmo 3133  df-rab 3134  df-v 3475  df-sbc 3752  df-csb 3860  df-dif 3915  df-un 3917  df-in 3919  df-ss 3928  df-nul 4268  df-if 4442  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4813  df-iun 4895  df-br 5041  df-opab 5103  df-mpt 5121  df-id 5434  df-po 5448  df-so 5449  df-xp 5535  df-rel 5536  df-cnv 5537  df-co 5538  df-dm 5539  df-rn 5540  df-res 5541  df-ima 5542  df-iota 6288  df-fun 6331  df-fn 6332  df-f 6333  df-f1 6334  df-fo 6335  df-f1o 6336  df-fv 6337  df-riota 7089  df-ov 7134  df-oprab 7135  df-mpo 7136  df-1st 7665  df-2nd 7666  df-er 8265  df-ec 8267  df-qs 8271  df-map 8384  df-en 8486  df-dom 8487  df-sdom 8488  df-pnf 10653  df-mnf 10654  df-xr 10655  df-ltxr 10656  df-le 10657  df-sub 10848  df-neg 10849  df-div 11274  df-2 11677  df-rp 12367  df-xneg 12484  df-xadd 12485  df-xmul 12486  df-psmet 20510  df-xmet 20511  df-metid 31136  df-pstm 31137
This theorem is referenced by: (None)
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