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Theorem metustto 23909
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 765 . . . . 5 (((𝑎 ∈ ℝ+𝑏 ∈ ℝ+) ∧ (𝐴 = (𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (𝐷 “ (0[,)𝑏)))) → 𝑎 ∈ ℝ+)
21rpred 12957 . . . 4 (((𝑎 ∈ ℝ+𝑏 ∈ ℝ+) ∧ (𝐴 = (𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (𝐷 “ (0[,)𝑏)))) → 𝑎 ∈ ℝ)
3 simplr 767 . . . . 5 (((𝑎 ∈ ℝ+𝑏 ∈ ℝ+) ∧ (𝐴 = (𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (𝐷 “ (0[,)𝑏)))) → 𝑏 ∈ ℝ+)
43rpred 12957 . . . 4 (((𝑎 ∈ ℝ+𝑏 ∈ ℝ+) ∧ (𝐴 = (𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (𝐷 “ (0[,)𝑏)))) → 𝑏 ∈ ℝ)
5 simpllr 774 . . . . . . . 8 ((((𝑎 ∈ ℝ+𝑏 ∈ ℝ+) ∧ (𝐴 = (𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (𝐷 “ (0[,)𝑏)))) ∧ 𝑎𝑏) → 𝑏 ∈ ℝ+)
65rpred 12957 . . . . . . 7 ((((𝑎 ∈ ℝ+𝑏 ∈ ℝ+) ∧ (𝐴 = (𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (𝐷 “ (0[,)𝑏)))) ∧ 𝑎𝑏) → 𝑏 ∈ ℝ)
7 0xr 11202 . . . . . . . . . 10 0 ∈ ℝ*
87a1i 11 . . . . . . . . 9 ((𝑏 ∈ ℝ ∧ 𝑎𝑏) → 0 ∈ ℝ*)
9 simpl 483 . . . . . . . . . 10 ((𝑏 ∈ ℝ ∧ 𝑎𝑏) → 𝑏 ∈ ℝ)
109rexrd 11205 . . . . . . . . 9 ((𝑏 ∈ ℝ ∧ 𝑎𝑏) → 𝑏 ∈ ℝ*)
11 0le0 12254 . . . . . . . . . 10 0 ≤ 0
1211a1i 11 . . . . . . . . 9 ((𝑏 ∈ ℝ ∧ 𝑎𝑏) → 0 ≤ 0)
13 simpr 485 . . . . . . . . 9 ((𝑏 ∈ ℝ ∧ 𝑎𝑏) → 𝑎𝑏)
14 icossico 13334 . . . . . . . . 9 (((0 ∈ ℝ*𝑏 ∈ ℝ*) ∧ (0 ≤ 0 ∧ 𝑎𝑏)) → (0[,)𝑎) ⊆ (0[,)𝑏))
158, 10, 12, 13, 14syl22anc 837 . . . . . . . 8 ((𝑏 ∈ ℝ ∧ 𝑎𝑏) → (0[,)𝑎) ⊆ (0[,)𝑏))
16 imass2 6054 . . . . . . . 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 588 . . . . . 6 ((((𝑎 ∈ ℝ+𝑏 ∈ ℝ+) ∧ (𝐴 = (𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (𝐷 “ (0[,)𝑏)))) ∧ 𝑎𝑏) → (𝐷 “ (0[,)𝑎)) ⊆ (𝐷 “ (0[,)𝑏)))
19 simplrl 775 . . . . . 6 ((((𝑎 ∈ ℝ+𝑏 ∈ ℝ+) ∧ (𝐴 = (𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (𝐷 “ (0[,)𝑏)))) ∧ 𝑎𝑏) → 𝐴 = (𝐷 “ (0[,)𝑎)))
20 simplrr 776 . . . . . 6 ((((𝑎 ∈ ℝ+𝑏 ∈ ℝ+) ∧ (𝐴 = (𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (𝐷 “ (0[,)𝑏)))) ∧ 𝑎𝑏) → 𝐵 = (𝐷 “ (0[,)𝑏)))
2118, 19, 203sstr4d 3991 . . . . 5 ((((𝑎 ∈ ℝ+𝑏 ∈ ℝ+) ∧ (𝐴 = (𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (𝐷 “ (0[,)𝑏)))) ∧ 𝑎𝑏) → 𝐴𝐵)
2221orcd 871 . . . 4 ((((𝑎 ∈ ℝ+𝑏 ∈ ℝ+) ∧ (𝐴 = (𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (𝐷 “ (0[,)𝑏)))) ∧ 𝑎𝑏) → (𝐴𝐵𝐵𝐴))
23 simplll 773 . . . . . . . 8 ((((𝑎 ∈ ℝ+𝑏 ∈ ℝ+) ∧ (𝐴 = (𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (𝐷 “ (0[,)𝑏)))) ∧ 𝑏𝑎) → 𝑎 ∈ ℝ+)
2423rpred 12957 . . . . . . 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 483 . . . . . . . . . 10 ((𝑎 ∈ ℝ ∧ 𝑏𝑎) → 𝑎 ∈ ℝ)
2726rexrd 11205 . . . . . . . . 9 ((𝑎 ∈ ℝ ∧ 𝑏𝑎) → 𝑎 ∈ ℝ*)
2811a1i 11 . . . . . . . . 9 ((𝑎 ∈ ℝ ∧ 𝑏𝑎) → 0 ≤ 0)
29 simpr 485 . . . . . . . . 9 ((𝑎 ∈ ℝ ∧ 𝑏𝑎) → 𝑏𝑎)
30 icossico 13334 . . . . . . . . 9 (((0 ∈ ℝ*𝑎 ∈ ℝ*) ∧ (0 ≤ 0 ∧ 𝑏𝑎)) → (0[,)𝑏) ⊆ (0[,)𝑎))
3125, 27, 28, 29, 30syl22anc 837 . . . . . . . 8 ((𝑎 ∈ ℝ ∧ 𝑏𝑎) → (0[,)𝑏) ⊆ (0[,)𝑎))
32 imass2 6054 . . . . . . . 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 588 . . . . . 6 ((((𝑎 ∈ ℝ+𝑏 ∈ ℝ+) ∧ (𝐴 = (𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (𝐷 “ (0[,)𝑏)))) ∧ 𝑏𝑎) → (𝐷 “ (0[,)𝑏)) ⊆ (𝐷 “ (0[,)𝑎)))
35 simplrr 776 . . . . . 6 ((((𝑎 ∈ ℝ+𝑏 ∈ ℝ+) ∧ (𝐴 = (𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (𝐷 “ (0[,)𝑏)))) ∧ 𝑏𝑎) → 𝐵 = (𝐷 “ (0[,)𝑏)))
36 simplrl 775 . . . . . 6 ((((𝑎 ∈ ℝ+𝑏 ∈ ℝ+) ∧ (𝐴 = (𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (𝐷 “ (0[,)𝑏)))) ∧ 𝑏𝑎) → 𝐴 = (𝐷 “ (0[,)𝑎)))
3734, 35, 363sstr4d 3991 . . . . 5 ((((𝑎 ∈ ℝ+𝑏 ∈ ℝ+) ∧ (𝐴 = (𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (𝐷 “ (0[,)𝑏)))) ∧ 𝑏𝑎) → 𝐵𝐴)
3837olcd 872 . . . 4 ((((𝑎 ∈ ℝ+𝑏 ∈ ℝ+) ∧ (𝐴 = (𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (𝐷 “ (0[,)𝑏)))) ∧ 𝑏𝑎) → (𝐴𝐵𝐵𝐴))
392, 4, 22, 38lecasei 11261 . . 3 (((𝑎 ∈ ℝ+𝑏 ∈ ℝ+) ∧ (𝐴 = (𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (𝐷 “ (0[,)𝑏)))) → (𝐴𝐵𝐵𝐴))
4039adantlll 716 . 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 23906 . . . . 5 (𝐷 ∈ (PsMet‘𝑋) → (𝐴𝐹 ↔ ∃𝑎 ∈ ℝ+ 𝐴 = (𝐷 “ (0[,)𝑎))))
4342biimpa 477 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) → ∃𝑎 ∈ ℝ+ 𝐴 = (𝐷 “ (0[,)𝑎)))
44433adant3 1132 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹𝐵𝐹) → ∃𝑎 ∈ ℝ+ 𝐴 = (𝐷 “ (0[,)𝑎)))
45 oveq2 7365 . . . . . . . . . 10 (𝑎 = 𝑏 → (0[,)𝑎) = (0[,)𝑏))
4645imaeq2d 6013 . . . . . . . . 9 (𝑎 = 𝑏 → (𝐷 “ (0[,)𝑎)) = (𝐷 “ (0[,)𝑏)))
4746cbvmptv 5218 . . . . . . . 8 (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎))) = (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏)))
4847rneqi 5892 . . . . . . 7 ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎))) = ran (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏)))
4941, 48eqtri 2764 . . . . . 6 𝐹 = ran (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏)))
5049metustel 23906 . . . . 5 (𝐷 ∈ (PsMet‘𝑋) → (𝐵𝐹 ↔ ∃𝑏 ∈ ℝ+ 𝐵 = (𝐷 “ (0[,)𝑏))))
5150biimpa 477 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐵𝐹) → ∃𝑏 ∈ ℝ+ 𝐵 = (𝐷 “ (0[,)𝑏)))
52513adant2 1131 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹𝐵𝐹) → ∃𝑏 ∈ ℝ+ 𝐵 = (𝐷 “ (0[,)𝑏)))
53 reeanv 3217 . . 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 583 . 2 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹𝐵𝐹) → ∃𝑎 ∈ ℝ+𝑏 ∈ ℝ+ (𝐴 = (𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (𝐷 “ (0[,)𝑏))))
5540, 54r19.29vva 3207 1 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹𝐵𝐹) → (𝐴𝐵𝐵𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wo 845  w3a 1087   = wceq 1541  wcel 2106  wrex 3073  wss 3910   class class class wbr 5105  cmpt 5188  ccnv 5632  ran crn 5634  cima 5636  cfv 6496  (class class class)co 7357  cr 11050  0cc0 11051  *cxr 11188  cle 11190  +crp 12915  [,)cico 13266  PsMetcpsmet 20780
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-addrcl 11112  ax-rnegex 11122  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-po 5545  df-so 5546  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7921  df-2nd 7922  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-rp 12916  df-ico 13270
This theorem is referenced by:  metustfbas  23913
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