MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  metustto Structured version   Visualization version   GIF version

Theorem metustto 22405
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 805 . . . . 5 (((𝑎 ∈ ℝ+𝑏 ∈ ℝ+) ∧ (𝐴 = (𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (𝐷 “ (0[,)𝑏)))) → 𝑎 ∈ ℝ+)
21rpred 11910 . . . 4 (((𝑎 ∈ ℝ+𝑏 ∈ ℝ+) ∧ (𝐴 = (𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (𝐷 “ (0[,)𝑏)))) → 𝑎 ∈ ℝ)
3 simplr 807 . . . . 5 (((𝑎 ∈ ℝ+𝑏 ∈ ℝ+) ∧ (𝐴 = (𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (𝐷 “ (0[,)𝑏)))) → 𝑏 ∈ ℝ+)
43rpred 11910 . . . 4 (((𝑎 ∈ ℝ+𝑏 ∈ ℝ+) ∧ (𝐴 = (𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (𝐷 “ (0[,)𝑏)))) → 𝑏 ∈ ℝ)
5 simpllr 815 . . . . . . . 8 ((((𝑎 ∈ ℝ+𝑏 ∈ ℝ+) ∧ (𝐴 = (𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (𝐷 “ (0[,)𝑏)))) ∧ 𝑎𝑏) → 𝑏 ∈ ℝ+)
65rpred 11910 . . . . . . 7 ((((𝑎 ∈ ℝ+𝑏 ∈ ℝ+) ∧ (𝐴 = (𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (𝐷 “ (0[,)𝑏)))) ∧ 𝑎𝑏) → 𝑏 ∈ ℝ)
7 0xr 10124 . . . . . . . . . 10 0 ∈ ℝ*
87a1i 11 . . . . . . . . 9 ((𝑏 ∈ ℝ ∧ 𝑎𝑏) → 0 ∈ ℝ*)
9 simpl 472 . . . . . . . . . 10 ((𝑏 ∈ ℝ ∧ 𝑎𝑏) → 𝑏 ∈ ℝ)
109rexrd 10127 . . . . . . . . 9 ((𝑏 ∈ ℝ ∧ 𝑎𝑏) → 𝑏 ∈ ℝ*)
11 0le0 11148 . . . . . . . . . 10 0 ≤ 0
1211a1i 11 . . . . . . . . 9 ((𝑏 ∈ ℝ ∧ 𝑎𝑏) → 0 ≤ 0)
13 simpr 476 . . . . . . . . 9 ((𝑏 ∈ ℝ ∧ 𝑎𝑏) → 𝑎𝑏)
14 icossico 12281 . . . . . . . . 9 (((0 ∈ ℝ*𝑏 ∈ ℝ*) ∧ (0 ≤ 0 ∧ 𝑎𝑏)) → (0[,)𝑎) ⊆ (0[,)𝑏))
158, 10, 12, 13, 14syl22anc 1367 . . . . . . . 8 ((𝑏 ∈ ℝ ∧ 𝑎𝑏) → (0[,)𝑎) ⊆ (0[,)𝑏))
16 imass2 5536 . . . . . . . 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 702 . . . . . 6 ((((𝑎 ∈ ℝ+𝑏 ∈ ℝ+) ∧ (𝐴 = (𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (𝐷 “ (0[,)𝑏)))) ∧ 𝑎𝑏) → (𝐷 “ (0[,)𝑎)) ⊆ (𝐷 “ (0[,)𝑏)))
19 simplrl 817 . . . . . 6 ((((𝑎 ∈ ℝ+𝑏 ∈ ℝ+) ∧ (𝐴 = (𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (𝐷 “ (0[,)𝑏)))) ∧ 𝑎𝑏) → 𝐴 = (𝐷 “ (0[,)𝑎)))
20 simplrr 818 . . . . . 6 ((((𝑎 ∈ ℝ+𝑏 ∈ ℝ+) ∧ (𝐴 = (𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (𝐷 “ (0[,)𝑏)))) ∧ 𝑎𝑏) → 𝐵 = (𝐷 “ (0[,)𝑏)))
2118, 19, 203sstr4d 3681 . . . . 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 813 . . . . . . . 8 ((((𝑎 ∈ ℝ+𝑏 ∈ ℝ+) ∧ (𝐴 = (𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (𝐷 “ (0[,)𝑏)))) ∧ 𝑏𝑎) → 𝑎 ∈ ℝ+)
2423rpred 11910 . . . . . . 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 472 . . . . . . . . . 10 ((𝑎 ∈ ℝ ∧ 𝑏𝑎) → 𝑎 ∈ ℝ)
2726rexrd 10127 . . . . . . . . 9 ((𝑎 ∈ ℝ ∧ 𝑏𝑎) → 𝑎 ∈ ℝ*)
2811a1i 11 . . . . . . . . 9 ((𝑎 ∈ ℝ ∧ 𝑏𝑎) → 0 ≤ 0)
29 simpr 476 . . . . . . . . 9 ((𝑎 ∈ ℝ ∧ 𝑏𝑎) → 𝑏𝑎)
30 icossico 12281 . . . . . . . . 9 (((0 ∈ ℝ*𝑎 ∈ ℝ*) ∧ (0 ≤ 0 ∧ 𝑏𝑎)) → (0[,)𝑏) ⊆ (0[,)𝑎))
3125, 27, 28, 29, 30syl22anc 1367 . . . . . . . 8 ((𝑎 ∈ ℝ ∧ 𝑏𝑎) → (0[,)𝑏) ⊆ (0[,)𝑎))
32 imass2 5536 . . . . . . . 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 702 . . . . . 6 ((((𝑎 ∈ ℝ+𝑏 ∈ ℝ+) ∧ (𝐴 = (𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (𝐷 “ (0[,)𝑏)))) ∧ 𝑏𝑎) → (𝐷 “ (0[,)𝑏)) ⊆ (𝐷 “ (0[,)𝑎)))
35 simplrr 818 . . . . . 6 ((((𝑎 ∈ ℝ+𝑏 ∈ ℝ+) ∧ (𝐴 = (𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (𝐷 “ (0[,)𝑏)))) ∧ 𝑏𝑎) → 𝐵 = (𝐷 “ (0[,)𝑏)))
36 simplrl 817 . . . . . 6 ((((𝑎 ∈ ℝ+𝑏 ∈ ℝ+) ∧ (𝐴 = (𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (𝐷 “ (0[,)𝑏)))) ∧ 𝑏𝑎) → 𝐴 = (𝐷 “ (0[,)𝑎)))
3734, 35, 363sstr4d 3681 . . . . 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 10181 . . 3 (((𝑎 ∈ ℝ+𝑏 ∈ ℝ+) ∧ (𝐴 = (𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (𝐷 “ (0[,)𝑏)))) → (𝐴𝐵𝐵𝐴))
4039adantlll 754 . 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 22402 . . . . 5 (𝐷 ∈ (PsMet‘𝑋) → (𝐴𝐹 ↔ ∃𝑎 ∈ ℝ+ 𝐴 = (𝐷 “ (0[,)𝑎))))
4342biimpa 500 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) → ∃𝑎 ∈ ℝ+ 𝐴 = (𝐷 “ (0[,)𝑎)))
44433adant3 1101 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹𝐵𝐹) → ∃𝑎 ∈ ℝ+ 𝐴 = (𝐷 “ (0[,)𝑎)))
45 oveq2 6698 . . . . . . . . . 10 (𝑎 = 𝑏 → (0[,)𝑎) = (0[,)𝑏))
4645imaeq2d 5501 . . . . . . . . 9 (𝑎 = 𝑏 → (𝐷 “ (0[,)𝑎)) = (𝐷 “ (0[,)𝑏)))
4746cbvmptv 4783 . . . . . . . 8 (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎))) = (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏)))
4847rneqi 5384 . . . . . . 7 ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎))) = ran (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏)))
4941, 48eqtri 2673 . . . . . 6 𝐹 = ran (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏)))
5049metustel 22402 . . . . 5 (𝐷 ∈ (PsMet‘𝑋) → (𝐵𝐹 ↔ ∃𝑏 ∈ ℝ+ 𝐵 = (𝐷 “ (0[,)𝑏))))
5150biimpa 500 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐵𝐹) → ∃𝑏 ∈ ℝ+ 𝐵 = (𝐷 “ (0[,)𝑏)))
52513adant2 1100 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹𝐵𝐹) → ∃𝑏 ∈ ℝ+ 𝐵 = (𝐷 “ (0[,)𝑏)))
53 reeanv 3136 . . 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 699 . 2 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹𝐵𝐹) → ∃𝑎 ∈ ℝ+𝑏 ∈ ℝ+ (𝐴 = (𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (𝐷 “ (0[,)𝑏))))
5540, 54r19.29vva 3110 1 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹𝐵𝐹) → (𝐴𝐵𝐵𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 382  wa 383  w3a 1054   = wceq 1523  wcel 2030  wrex 2942  wss 3607   class class class wbr 4685  cmpt 4762  ccnv 5142  ran crn 5144  cima 5146  cfv 5926  (class class class)co 6690  cr 9973  0cc0 9974  *cxr 10111  cle 10113  +crp 11870  [,)cico 12215  PsMetcpsmet 19778
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-i2m1 10042  ax-1ne0 10043  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-po 5064  df-so 5065  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-rp 11871  df-ico 12219
This theorem is referenced by:  metustfbas  22409
  Copyright terms: Public domain W3C validator