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Theorem metustto 23138
 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 766 . . . . 5 (((𝑎 ∈ ℝ+𝑏 ∈ ℝ+) ∧ (𝐴 = (𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (𝐷 “ (0[,)𝑏)))) → 𝑎 ∈ ℝ+)
21rpred 12409 . . . 4 (((𝑎 ∈ ℝ+𝑏 ∈ ℝ+) ∧ (𝐴 = (𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (𝐷 “ (0[,)𝑏)))) → 𝑎 ∈ ℝ)
3 simplr 768 . . . . 5 (((𝑎 ∈ ℝ+𝑏 ∈ ℝ+) ∧ (𝐴 = (𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (𝐷 “ (0[,)𝑏)))) → 𝑏 ∈ ℝ+)
43rpred 12409 . . . 4 (((𝑎 ∈ ℝ+𝑏 ∈ ℝ+) ∧ (𝐴 = (𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (𝐷 “ (0[,)𝑏)))) → 𝑏 ∈ ℝ)
5 simpllr 775 . . . . . . . 8 ((((𝑎 ∈ ℝ+𝑏 ∈ ℝ+) ∧ (𝐴 = (𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (𝐷 “ (0[,)𝑏)))) ∧ 𝑎𝑏) → 𝑏 ∈ ℝ+)
65rpred 12409 . . . . . . 7 ((((𝑎 ∈ ℝ+𝑏 ∈ ℝ+) ∧ (𝐴 = (𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (𝐷 “ (0[,)𝑏)))) ∧ 𝑎𝑏) → 𝑏 ∈ ℝ)
7 0xr 10665 . . . . . . . . . 10 0 ∈ ℝ*
87a1i 11 . . . . . . . . 9 ((𝑏 ∈ ℝ ∧ 𝑎𝑏) → 0 ∈ ℝ*)
9 simpl 486 . . . . . . . . . 10 ((𝑏 ∈ ℝ ∧ 𝑎𝑏) → 𝑏 ∈ ℝ)
109rexrd 10668 . . . . . . . . 9 ((𝑏 ∈ ℝ ∧ 𝑎𝑏) → 𝑏 ∈ ℝ*)
11 0le0 11716 . . . . . . . . . 10 0 ≤ 0
1211a1i 11 . . . . . . . . 9 ((𝑏 ∈ ℝ ∧ 𝑎𝑏) → 0 ≤ 0)
13 simpr 488 . . . . . . . . 9 ((𝑏 ∈ ℝ ∧ 𝑎𝑏) → 𝑎𝑏)
14 icossico 12785 . . . . . . . . 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 5938 . . . . . . . 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 591 . . . . . 6 ((((𝑎 ∈ ℝ+𝑏 ∈ ℝ+) ∧ (𝐴 = (𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (𝐷 “ (0[,)𝑏)))) ∧ 𝑎𝑏) → (𝐷 “ (0[,)𝑎)) ⊆ (𝐷 “ (0[,)𝑏)))
19 simplrl 776 . . . . . 6 ((((𝑎 ∈ ℝ+𝑏 ∈ ℝ+) ∧ (𝐴 = (𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (𝐷 “ (0[,)𝑏)))) ∧ 𝑎𝑏) → 𝐴 = (𝐷 “ (0[,)𝑎)))
20 simplrr 777 . . . . . 6 ((((𝑎 ∈ ℝ+𝑏 ∈ ℝ+) ∧ (𝐴 = (𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (𝐷 “ (0[,)𝑏)))) ∧ 𝑎𝑏) → 𝐵 = (𝐷 “ (0[,)𝑏)))
2118, 19, 203sstr4d 3990 . . . . 5 ((((𝑎 ∈ ℝ+𝑏 ∈ ℝ+) ∧ (𝐴 = (𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (𝐷 “ (0[,)𝑏)))) ∧ 𝑎𝑏) → 𝐴𝐵)
2221orcd 870 . . . 4 ((((𝑎 ∈ ℝ+𝑏 ∈ ℝ+) ∧ (𝐴 = (𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (𝐷 “ (0[,)𝑏)))) ∧ 𝑎𝑏) → (𝐴𝐵𝐵𝐴))
23 simplll 774 . . . . . . . 8 ((((𝑎 ∈ ℝ+𝑏 ∈ ℝ+) ∧ (𝐴 = (𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (𝐷 “ (0[,)𝑏)))) ∧ 𝑏𝑎) → 𝑎 ∈ ℝ+)
2423rpred 12409 . . . . . . 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 486 . . . . . . . . . 10 ((𝑎 ∈ ℝ ∧ 𝑏𝑎) → 𝑎 ∈ ℝ)
2726rexrd 10668 . . . . . . . . 9 ((𝑎 ∈ ℝ ∧ 𝑏𝑎) → 𝑎 ∈ ℝ*)
2811a1i 11 . . . . . . . . 9 ((𝑎 ∈ ℝ ∧ 𝑏𝑎) → 0 ≤ 0)
29 simpr 488 . . . . . . . . 9 ((𝑎 ∈ ℝ ∧ 𝑏𝑎) → 𝑏𝑎)
30 icossico 12785 . . . . . . . . 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 5938 . . . . . . . 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 591 . . . . . 6 ((((𝑎 ∈ ℝ+𝑏 ∈ ℝ+) ∧ (𝐴 = (𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (𝐷 “ (0[,)𝑏)))) ∧ 𝑏𝑎) → (𝐷 “ (0[,)𝑏)) ⊆ (𝐷 “ (0[,)𝑎)))
35 simplrr 777 . . . . . 6 ((((𝑎 ∈ ℝ+𝑏 ∈ ℝ+) ∧ (𝐴 = (𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (𝐷 “ (0[,)𝑏)))) ∧ 𝑏𝑎) → 𝐵 = (𝐷 “ (0[,)𝑏)))
36 simplrl 776 . . . . . 6 ((((𝑎 ∈ ℝ+𝑏 ∈ ℝ+) ∧ (𝐴 = (𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (𝐷 “ (0[,)𝑏)))) ∧ 𝑏𝑎) → 𝐴 = (𝐷 “ (0[,)𝑎)))
3734, 35, 363sstr4d 3990 . . . . 5 ((((𝑎 ∈ ℝ+𝑏 ∈ ℝ+) ∧ (𝐴 = (𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (𝐷 “ (0[,)𝑏)))) ∧ 𝑏𝑎) → 𝐵𝐴)
3837olcd 871 . . . 4 ((((𝑎 ∈ ℝ+𝑏 ∈ ℝ+) ∧ (𝐴 = (𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (𝐷 “ (0[,)𝑏)))) ∧ 𝑏𝑎) → (𝐴𝐵𝐵𝐴))
392, 4, 22, 38lecasei 10723 . . 3 (((𝑎 ∈ ℝ+𝑏 ∈ ℝ+) ∧ (𝐴 = (𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (𝐷 “ (0[,)𝑏)))) → (𝐴𝐵𝐵𝐴))
4039adantlll 717 . 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 23135 . . . . 5 (𝐷 ∈ (PsMet‘𝑋) → (𝐴𝐹 ↔ ∃𝑎 ∈ ℝ+ 𝐴 = (𝐷 “ (0[,)𝑎))))
4342biimpa 480 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) → ∃𝑎 ∈ ℝ+ 𝐴 = (𝐷 “ (0[,)𝑎)))
44433adant3 1129 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹𝐵𝐹) → ∃𝑎 ∈ ℝ+ 𝐴 = (𝐷 “ (0[,)𝑎)))
45 oveq2 7138 . . . . . . . . . 10 (𝑎 = 𝑏 → (0[,)𝑎) = (0[,)𝑏))
4645imaeq2d 5902 . . . . . . . . 9 (𝑎 = 𝑏 → (𝐷 “ (0[,)𝑎)) = (𝐷 “ (0[,)𝑏)))
4746cbvmptv 5142 . . . . . . . 8 (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎))) = (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏)))
4847rneqi 5780 . . . . . . 7 ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎))) = ran (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏)))
4941, 48eqtri 2844 . . . . . 6 𝐹 = ran (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏)))
5049metustel 23135 . . . . 5 (𝐷 ∈ (PsMet‘𝑋) → (𝐵𝐹 ↔ ∃𝑏 ∈ ℝ+ 𝐵 = (𝐷 “ (0[,)𝑏))))
5150biimpa 480 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐵𝐹) → ∃𝑏 ∈ ℝ+ 𝐵 = (𝐷 “ (0[,)𝑏)))
52513adant2 1128 . . 3 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹𝐵𝐹) → ∃𝑏 ∈ ℝ+ 𝐵 = (𝐷 “ (0[,)𝑏)))
53 reeanv 3352 . . 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 586 . 2 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹𝐵𝐹) → ∃𝑎 ∈ ℝ+𝑏 ∈ ℝ+ (𝐴 = (𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (𝐷 “ (0[,)𝑏))))
5540, 54r19.29vva 3321 1 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹𝐵𝐹) → (𝐴𝐵𝐵𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∨ wo 844   ∧ w3a 1084   = wceq 1538   ∈ wcel 2115  ∃wrex 3127   ⊆ wss 3910   class class class wbr 5039   ↦ cmpt 5119  ◡ccnv 5527  ran crn 5529   “ cima 5531  ‘cfv 6328  (class class class)co 7130  ℝcr 10513  0cc0 10514  ℝ*cxr 10651   ≤ cle 10653  ℝ+crp 12367  [,)cico 12718  PsMetcpsmet 20504 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436  ax-cnex 10570  ax-resscn 10571  ax-1cn 10572  ax-addrcl 10575  ax-rnegex 10585  ax-cnre 10587  ax-pre-lttri 10588  ax-pre-lttrn 10589 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-nel 3112  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-po 5447  df-so 5448  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7133  df-oprab 7134  df-mpo 7135  df-1st 7664  df-2nd 7665  df-er 8264  df-en 8485  df-dom 8486  df-sdom 8487  df-pnf 10654  df-mnf 10655  df-xr 10656  df-ltxr 10657  df-le 10658  df-rp 12368  df-ico 12722 This theorem is referenced by:  metustfbas  23142
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