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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  cr 10119  0cc0 10120  *cxr 10257  cle 10259  +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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