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Theorem metustto 22551
 Description: Any two elements of the filter base generated by the metric 𝐷 can be compared, like for RR+ (i.e. it's totally ordered). (Contributed by Thierry Arnoux, 22-Nov-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
Hypothesis
Ref Expression
metust.1 𝐹 = ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))
Assertion
Ref Expression
metustto ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹𝐵𝐹) → (𝐴𝐵𝐵𝐴))
Distinct variable groups:   𝐵,𝑎   𝐷,𝑎   𝑋,𝑎   𝐴,𝑎   𝐹,𝑎

Proof of Theorem metustto
Dummy variable 𝑏 is distinct from all other variables.
StepHypRef Expression
1 simpll 807 . . . . 5 (((𝑎 ∈ ℝ+𝑏 ∈ ℝ+) ∧ (𝐴 = (𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (𝐷 “ (0[,)𝑏)))) → 𝑎 ∈ ℝ+)
21rpred 12057 . . . 4 (((𝑎 ∈ ℝ+𝑏 ∈ ℝ+) ∧ (𝐴 = (𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (𝐷 “ (0[,)𝑏)))) → 𝑎 ∈ ℝ)
3 simplr 809 . . . . 5 (((𝑎 ∈ ℝ+𝑏 ∈ ℝ+) ∧ (𝐴 = (𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (𝐷 “ (0[,)𝑏)))) → 𝑏 ∈ ℝ+)
43rpred 12057 . . . 4 (((𝑎 ∈ ℝ+𝑏 ∈ ℝ+) ∧ (𝐴 = (𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (𝐷 “ (0[,)𝑏)))) → 𝑏 ∈ ℝ)
5 simpllr 817 . . . . . . . 8 ((((𝑎 ∈ ℝ+𝑏 ∈ ℝ+) ∧ (𝐴 = (𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (𝐷 “ (0[,)𝑏)))) ∧ 𝑎𝑏) → 𝑏 ∈ ℝ+)
65rpred 12057 . . . . . . 7 ((((𝑎 ∈ ℝ+𝑏 ∈ ℝ+) ∧ (𝐴 = (𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (𝐷 “ (0[,)𝑏)))) ∧ 𝑎𝑏) → 𝑏 ∈ ℝ)
7 0xr 10270 . . . . . . . . . 10 0 ∈ ℝ*
87a1i 11 . . . . . . . . 9 ((𝑏 ∈ ℝ ∧ 𝑎𝑏) → 0 ∈ ℝ*)
9 simpl 474 . . . . . . . . . 10 ((𝑏 ∈ ℝ ∧ 𝑎𝑏) → 𝑏 ∈ ℝ)
109rexrd 10273 . . . . . . . . 9 ((𝑏 ∈ ℝ ∧ 𝑎𝑏) → 𝑏 ∈ ℝ*)
11 0le0 11294 . . . . . . . . . 10 0 ≤ 0
1211a1i 11 . . . . . . . . 9 ((𝑏 ∈ ℝ ∧ 𝑎𝑏) → 0 ≤ 0)
13 simpr 479 . . . . . . . . 9 ((𝑏 ∈ ℝ ∧ 𝑎𝑏) → 𝑎𝑏)
14 icossico 12428 . . . . . . . . 9 (((0 ∈ ℝ*𝑏 ∈ ℝ*) ∧ (0 ≤ 0 ∧ 𝑎𝑏)) → (0[,)𝑎) ⊆ (0[,)𝑏))
158, 10, 12, 13, 14syl22anc 1474 . . . . . . . 8 ((𝑏 ∈ ℝ ∧ 𝑎𝑏) → (0[,)𝑎) ⊆ (0[,)𝑏))
16 imass2 5651 . . . . . . . 8 ((0[,)𝑎) ⊆ (0[,)𝑏) → (𝐷 “ (0[,)𝑎)) ⊆ (𝐷 “ (0[,)𝑏)))
1715, 16syl 17 . . . . . . 7 ((𝑏 ∈ ℝ ∧ 𝑎𝑏) → (𝐷 “ (0[,)𝑎)) ⊆ (𝐷 “ (0[,)𝑏)))
186, 17sylancom 704 . . . . . 6 ((((𝑎 ∈ ℝ+𝑏 ∈ ℝ+) ∧ (𝐴 = (𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (𝐷 “ (0[,)𝑏)))) ∧ 𝑎𝑏) → (𝐷 “ (0[,)𝑎)) ⊆ (𝐷 “ (0[,)𝑏)))
19 simplrl 819 . . . . . 6 ((((𝑎 ∈ ℝ+𝑏 ∈ ℝ+) ∧ (𝐴 = (𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (𝐷 “ (0[,)𝑏)))) ∧ 𝑎𝑏) → 𝐴 = (𝐷 “ (0[,)𝑎)))
20 simplrr 820 . . . . . 6 ((((𝑎 ∈ ℝ+𝑏 ∈ ℝ+) ∧ (𝐴 = (𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (𝐷 “ (0[,)𝑏)))) ∧ 𝑎𝑏) → 𝐵 = (𝐷 “ (0[,)𝑏)))
2118, 19, 203sstr4d 3781 . . . . 5 ((((𝑎 ∈ ℝ+𝑏 ∈ ℝ+) ∧ (𝐴 = (𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (𝐷 “ (0[,)𝑏)))) ∧ 𝑎𝑏) → 𝐴𝐵)
2221orcd 406 . . . 4 ((((𝑎 ∈ ℝ+𝑏 ∈ ℝ+) ∧ (𝐴 = (𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (𝐷 “ (0[,)𝑏)))) ∧ 𝑎𝑏) → (𝐴𝐵𝐵𝐴))
23 simplll 815 . . . . . . . 8 ((((𝑎 ∈ ℝ+𝑏 ∈ ℝ+) ∧ (𝐴 = (𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (𝐷 “ (0[,)𝑏)))) ∧ 𝑏𝑎) → 𝑎 ∈ ℝ+)
2423rpred 12057 . . . . . . 7 ((((𝑎 ∈ ℝ+𝑏 ∈ ℝ+) ∧ (𝐴 = (𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (𝐷 “ (0[,)𝑏)))) ∧ 𝑏𝑎) → 𝑎 ∈ ℝ)
257a1i 11 . . . . . . . . 9 ((𝑎 ∈ ℝ ∧ 𝑏𝑎) → 0 ∈ ℝ*)
26 simpl 474 . . . . . . . . . 10 ((𝑎 ∈ ℝ ∧ 𝑏𝑎) → 𝑎 ∈ ℝ)
2726rexrd 10273 . . . . . . . . 9 ((𝑎 ∈ ℝ ∧ 𝑏𝑎) → 𝑎 ∈ ℝ*)
2811a1i 11 . . . . . . . . 9 ((𝑎 ∈ ℝ ∧ 𝑏𝑎) → 0 ≤ 0)
29 simpr 479 . . . . . . . . 9 ((𝑎 ∈ ℝ ∧ 𝑏𝑎) → 𝑏𝑎)
30 icossico 12428 . . . . . . . . 9 (((0 ∈ ℝ*𝑎 ∈ ℝ*) ∧ (0 ≤ 0 ∧ 𝑏𝑎)) → (0[,)𝑏) ⊆ (0[,)𝑎))
3125, 27, 28, 29, 30syl22anc 1474 . . . . . . . 8 ((𝑎 ∈ ℝ ∧ 𝑏𝑎) → (0[,)𝑏) ⊆ (0[,)𝑎))
32 imass2 5651 . . . . . . . 8 ((0[,)𝑏) ⊆ (0[,)𝑎) → (𝐷 “ (0[,)𝑏)) ⊆ (𝐷 “ (0[,)𝑎)))
3331, 32syl 17 . . . . . . 7 ((𝑎 ∈ ℝ ∧ 𝑏𝑎) → (𝐷 “ (0[,)𝑏)) ⊆ (𝐷 “ (0[,)𝑎)))
3424, 33sylancom 704 . . . . . 6 ((((𝑎 ∈ ℝ+𝑏 ∈ ℝ+) ∧ (𝐴 = (𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (𝐷 “ (0[,)𝑏)))) ∧ 𝑏𝑎) → (𝐷 “ (0[,)𝑏)) ⊆ (𝐷 “ (0[,)𝑎)))
35 simplrr 820 . . . . . 6 ((((𝑎 ∈ ℝ+𝑏 ∈ ℝ+) ∧ (𝐴 = (𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (𝐷 “ (0[,)𝑏)))) ∧ 𝑏𝑎) → 𝐵 = (𝐷 “ (0[,)𝑏)))
36 simplrl 819 . . . . . 6 ((((𝑎 ∈ ℝ+𝑏 ∈ ℝ+) ∧ (𝐴 = (𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (𝐷 “ (0[,)𝑏)))) ∧ 𝑏𝑎) → 𝐴 = (𝐷 “ (0[,)𝑎)))
3734, 35, 363sstr4d 3781 . . . . 5 ((((𝑎 ∈ ℝ+𝑏 ∈ ℝ+) ∧ (𝐴 = (𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (𝐷 “ (0[,)𝑏)))) ∧ 𝑏𝑎) → 𝐵𝐴)
3837olcd 407 . . . 4 ((((𝑎 ∈ ℝ+𝑏 ∈ ℝ+) ∧ (𝐴 = (𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (𝐷 “ (0[,)𝑏)))) ∧ 𝑏𝑎) → (𝐴𝐵𝐵𝐴))
392, 4, 22, 38lecasei 10327 . . 3 (((𝑎 ∈ ℝ+𝑏 ∈ ℝ+) ∧ (𝐴 = (𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (𝐷 “ (0[,)𝑏)))) → (𝐴𝐵𝐵𝐴))
4039adantlll 756 . 2 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹𝐵𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝑏 ∈ ℝ+) ∧ (𝐴 = (𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (𝐷 “ (0[,)𝑏)))) → (𝐴𝐵𝐵𝐴))
41 metust.1 . . . . . 6 𝐹 = ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))
4241metustel 22548 . . . . 5 (𝐷 ∈ (PsMet‘𝑋) → (𝐴𝐹 ↔ ∃𝑎 ∈ ℝ+ 𝐴 = (𝐷 “ (0[,)𝑎))))
4342biimpa 502 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) → ∃𝑎 ∈ ℝ+ 𝐴 = (𝐷 “ (0[,)𝑎)))
44433adant3 1126 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹𝐵𝐹) → ∃𝑎 ∈ ℝ+ 𝐴 = (𝐷 “ (0[,)𝑎)))
45 oveq2 6813 . . . . . . . . . 10 (𝑎 = 𝑏 → (0[,)𝑎) = (0[,)𝑏))
4645imaeq2d 5616 . . . . . . . . 9 (𝑎 = 𝑏 → (𝐷 “ (0[,)𝑎)) = (𝐷 “ (0[,)𝑏)))
4746cbvmptv 4894 . . . . . . . 8 (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎))) = (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏)))
4847rneqi 5499 . . . . . . 7 ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎))) = ran (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏)))
4941, 48eqtri 2774 . . . . . 6 𝐹 = ran (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏)))
5049metustel 22548 . . . . 5 (𝐷 ∈ (PsMet‘𝑋) → (𝐵𝐹 ↔ ∃𝑏 ∈ ℝ+ 𝐵 = (𝐷 “ (0[,)𝑏))))
5150biimpa 502 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐵𝐹) → ∃𝑏 ∈ ℝ+ 𝐵 = (𝐷 “ (0[,)𝑏)))
52513adant2 1125 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹𝐵𝐹) → ∃𝑏 ∈ ℝ+ 𝐵 = (𝐷 “ (0[,)𝑏)))
53 reeanv 3237 . . 3 (∃𝑎 ∈ ℝ+𝑏 ∈ ℝ+ (𝐴 = (𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (𝐷 “ (0[,)𝑏))) ↔ (∃𝑎 ∈ ℝ+ 𝐴 = (𝐷 “ (0[,)𝑎)) ∧ ∃𝑏 ∈ ℝ+ 𝐵 = (𝐷 “ (0[,)𝑏))))
5444, 52, 53sylanbrc 701 . 2 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹𝐵𝐹) → ∃𝑎 ∈ ℝ+𝑏 ∈ ℝ+ (𝐴 = (𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (𝐷 “ (0[,)𝑏))))
5540, 54r19.29vva 3211 1 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹𝐵𝐹) → (𝐴𝐵𝐵𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∨ wo 382   ∧ wa 383   ∧ w3a 1072   = wceq 1624   ∈ wcel 2131  ∃wrex 3043   ⊆ wss 3707   class class class wbr 4796   ↦ cmpt 4873  ◡ccnv 5257  ran crn 5259   “ cima 5261  ‘cfv 6041  (class class class)co 6805  ℝcr 10119  0cc0 10120  ℝ*cxr 10257   ≤ cle 10259  ℝ+crp 12017  [,)cico 12362  PsMetcpsmet 19924 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-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-i2m1 10188  ax-1ne0 10189  ax-rnegex 10191  ax-rrecex 10192  ax-cnre 10193  ax-pre-lttri 10194  ax-pre-lttrn 10195 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-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-ov 6808  df-oprab 6809  df-mpt2 6810  df-1st 7325  df-2nd 7326  df-er 7903  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-rp 12018  df-ico 12366 This theorem is referenced by:  metustfbas  22555
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