Theorem metustto 24691

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) ∧ 𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐹) → (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴))

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.

Step	Hyp	Ref	Expression
1		simpll 778	. . . . 5 ⊢ (((𝑎 ∈ ℝ+ ∧ 𝑏 ∈ ℝ+) ∧ (𝐴 = (◡𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (◡𝐷 “ (0[,)𝑏)))) → 𝑎 ∈ ℝ+)
2	1	rpred 13061	. . . 4 ⊢ (((𝑎 ∈ ℝ+ ∧ 𝑏 ∈ ℝ+) ∧ (𝐴 = (◡𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (◡𝐷 “ (0[,)𝑏)))) → 𝑎 ∈ ℝ)
3		simplr 780	. . . . 5 ⊢ (((𝑎 ∈ ℝ+ ∧ 𝑏 ∈ ℝ+) ∧ (𝐴 = (◡𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (◡𝐷 “ (0[,)𝑏)))) → 𝑏 ∈ ℝ+)
4	3	rpred 13061	. . . 4 ⊢ (((𝑎 ∈ ℝ+ ∧ 𝑏 ∈ ℝ+) ∧ (𝐴 = (◡𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (◡𝐷 “ (0[,)𝑏)))) → 𝑏 ∈ ℝ)
5		simpllr 787	. . . . . . . 8 ⊢ ((((𝑎 ∈ ℝ+ ∧ 𝑏 ∈ ℝ+) ∧ (𝐴 = (◡𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (◡𝐷 “ (0[,)𝑏)))) ∧ 𝑎 ≤ 𝑏) → 𝑏 ∈ ℝ+)
6	5	rpred 13061	. . . . . . 7 ⊢ ((((𝑎 ∈ ℝ+ ∧ 𝑏 ∈ ℝ+) ∧ (𝐴 = (◡𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (◡𝐷 “ (0[,)𝑏)))) ∧ 𝑎 ≤ 𝑏) → 𝑏 ∈ ℝ)
7		0xr 11257	. . . . . . . . . 10 ⊢ 0 ∈ ℝ*
8	7	a1i 11	. . . . . . . . 9 ⊢ ((𝑏 ∈ ℝ ∧ 𝑎 ≤ 𝑏) → 0 ∈ ℝ*)
9		simpl 487	. . . . . . . . . 10 ⊢ ((𝑏 ∈ ℝ ∧ 𝑎 ≤ 𝑏) → 𝑏 ∈ ℝ)
10	9	rexrd 11260	. . . . . . . . 9 ⊢ ((𝑏 ∈ ℝ ∧ 𝑎 ≤ 𝑏) → 𝑏 ∈ ℝ*)
11		0le0 12343	. . . . . . . . . 10 ⊢ 0 ≤ 0
12	11	a1i 11	. . . . . . . . 9 ⊢ ((𝑏 ∈ ℝ ∧ 𝑎 ≤ 𝑏) → 0 ≤ 0)
13		simpr 489	. . . . . . . . 9 ⊢ ((𝑏 ∈ ℝ ∧ 𝑎 ≤ 𝑏) → 𝑎 ≤ 𝑏)
14		icossico 13444	. . . . . . . . 9 ⊢ (((0 ∈ ℝ* ∧ 𝑏 ∈ ℝ*) ∧ (0 ≤ 0 ∧ 𝑎 ≤ 𝑏)) → (0[,)𝑎) ⊆ (0[,)𝑏))
15	8, 10, 12, 13, 14	syl22anc 851	. . . . . . . 8 ⊢ ((𝑏 ∈ ℝ ∧ 𝑎 ≤ 𝑏) → (0[,)𝑎) ⊆ (0[,)𝑏))
16		imass2 6106	. . . . . . . 8 ⊢ ((0[,)𝑎) ⊆ (0[,)𝑏) → (◡𝐷 “ (0[,)𝑎)) ⊆ (◡𝐷 “ (0[,)𝑏)))
17	15, 16	syl 18	. . . . . . 7 ⊢ ((𝑏 ∈ ℝ ∧ 𝑎 ≤ 𝑏) → (◡𝐷 “ (0[,)𝑎)) ⊆ (◡𝐷 “ (0[,)𝑏)))
18	6, 17	sylancom 599	. . . . . 6 ⊢ ((((𝑎 ∈ ℝ+ ∧ 𝑏 ∈ ℝ+) ∧ (𝐴 = (◡𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (◡𝐷 “ (0[,)𝑏)))) ∧ 𝑎 ≤ 𝑏) → (◡𝐷 “ (0[,)𝑎)) ⊆ (◡𝐷 “ (0[,)𝑏)))
19		simplrl 788	. . . . . 6 ⊢ ((((𝑎 ∈ ℝ+ ∧ 𝑏 ∈ ℝ+) ∧ (𝐴 = (◡𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (◡𝐷 “ (0[,)𝑏)))) ∧ 𝑎 ≤ 𝑏) → 𝐴 = (◡𝐷 “ (0[,)𝑎)))
20		simplrr 789	. . . . . 6 ⊢ ((((𝑎 ∈ ℝ+ ∧ 𝑏 ∈ ℝ+) ∧ (𝐴 = (◡𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (◡𝐷 “ (0[,)𝑏)))) ∧ 𝑎 ≤ 𝑏) → 𝐵 = (◡𝐷 “ (0[,)𝑏)))
21	18, 19, 20	3sstr4d 3993	. . . . 5 ⊢ ((((𝑎 ∈ ℝ+ ∧ 𝑏 ∈ ℝ+) ∧ (𝐴 = (◡𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (◡𝐷 “ (0[,)𝑏)))) ∧ 𝑎 ≤ 𝑏) → 𝐴 ⊆ 𝐵)
22	21	orcd 886	. . . 4 ⊢ ((((𝑎 ∈ ℝ+ ∧ 𝑏 ∈ ℝ+) ∧ (𝐴 = (◡𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (◡𝐷 “ (0[,)𝑏)))) ∧ 𝑎 ≤ 𝑏) → (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴))
23		simplll 786	. . . . . . . 8 ⊢ ((((𝑎 ∈ ℝ+ ∧ 𝑏 ∈ ℝ+) ∧ (𝐴 = (◡𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (◡𝐷 “ (0[,)𝑏)))) ∧ 𝑏 ≤ 𝑎) → 𝑎 ∈ ℝ+)
24	23	rpred 13061	. . . . . . 7 ⊢ ((((𝑎 ∈ ℝ+ ∧ 𝑏 ∈ ℝ+) ∧ (𝐴 = (◡𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (◡𝐷 “ (0[,)𝑏)))) ∧ 𝑏 ≤ 𝑎) → 𝑎 ∈ ℝ)
25	7	a1i 11	. . . . . . . . 9 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ≤ 𝑎) → 0 ∈ ℝ*)
26		simpl 487	. . . . . . . . . 10 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ≤ 𝑎) → 𝑎 ∈ ℝ)
27	26	rexrd 11260	. . . . . . . . 9 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ≤ 𝑎) → 𝑎 ∈ ℝ*)
28	11	a1i 11	. . . . . . . . 9 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ≤ 𝑎) → 0 ≤ 0)
29		simpr 489	. . . . . . . . 9 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ≤ 𝑎) → 𝑏 ≤ 𝑎)
30		icossico 13444	. . . . . . . . 9 ⊢ (((0 ∈ ℝ* ∧ 𝑎 ∈ ℝ*) ∧ (0 ≤ 0 ∧ 𝑏 ≤ 𝑎)) → (0[,)𝑏) ⊆ (0[,)𝑎))
31	25, 27, 28, 29, 30	syl22anc 851	. . . . . . . 8 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ≤ 𝑎) → (0[,)𝑏) ⊆ (0[,)𝑎))
32		imass2 6106	. . . . . . . 8 ⊢ ((0[,)𝑏) ⊆ (0[,)𝑎) → (◡𝐷 “ (0[,)𝑏)) ⊆ (◡𝐷 “ (0[,)𝑎)))
33	31, 32	syl 18	. . . . . . 7 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ≤ 𝑎) → (◡𝐷 “ (0[,)𝑏)) ⊆ (◡𝐷 “ (0[,)𝑎)))
34	24, 33	sylancom 599	. . . . . 6 ⊢ ((((𝑎 ∈ ℝ+ ∧ 𝑏 ∈ ℝ+) ∧ (𝐴 = (◡𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (◡𝐷 “ (0[,)𝑏)))) ∧ 𝑏 ≤ 𝑎) → (◡𝐷 “ (0[,)𝑏)) ⊆ (◡𝐷 “ (0[,)𝑎)))
35		simplrr 789	. . . . . 6 ⊢ ((((𝑎 ∈ ℝ+ ∧ 𝑏 ∈ ℝ+) ∧ (𝐴 = (◡𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (◡𝐷 “ (0[,)𝑏)))) ∧ 𝑏 ≤ 𝑎) → 𝐵 = (◡𝐷 “ (0[,)𝑏)))
36		simplrl 788	. . . . . 6 ⊢ ((((𝑎 ∈ ℝ+ ∧ 𝑏 ∈ ℝ+) ∧ (𝐴 = (◡𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (◡𝐷 “ (0[,)𝑏)))) ∧ 𝑏 ≤ 𝑎) → 𝐴 = (◡𝐷 “ (0[,)𝑎)))
37	34, 35, 36	3sstr4d 3993	. . . . 5 ⊢ ((((𝑎 ∈ ℝ+ ∧ 𝑏 ∈ ℝ+) ∧ (𝐴 = (◡𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (◡𝐷 “ (0[,)𝑏)))) ∧ 𝑏 ≤ 𝑎) → 𝐵 ⊆ 𝐴)
38	37	olcd 887	. . . 4 ⊢ ((((𝑎 ∈ ℝ+ ∧ 𝑏 ∈ ℝ+) ∧ (𝐴 = (◡𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (◡𝐷 “ (0[,)𝑏)))) ∧ 𝑏 ≤ 𝑎) → (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴))
39	2, 4, 22, 38	lecasei 11317	. . 3 ⊢ (((𝑎 ∈ ℝ+ ∧ 𝑏 ∈ ℝ+) ∧ (𝐴 = (◡𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (◡𝐷 “ (0[,)𝑏)))) → (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴))
40	39	adantlll 730	. 2 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝑏 ∈ ℝ+) ∧ (𝐴 = (◡𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (◡𝐷 “ (0[,)𝑏)))) → (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴))
41		metust.1	. . . . . 6 ⊢ 𝐹 = ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎)))
42	41	metustel 24688	. . . . 5 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (𝐴 ∈ 𝐹 ↔ ∃𝑎 ∈ ℝ+ 𝐴 = (◡𝐷 “ (0[,)𝑎))))
43	42	biimpa 481	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) → ∃𝑎 ∈ ℝ+ 𝐴 = (◡𝐷 “ (0[,)𝑎)))
44	43	3adant3 1150	. . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐹) → ∃𝑎 ∈ ℝ+ 𝐴 = (◡𝐷 “ (0[,)𝑎)))
45		oveq2 7420	. . . . . . . . . 10 ⊢ (𝑎 = 𝑏 → (0[,)𝑎) = (0[,)𝑏))
46	45	imaeq2d 6064	. . . . . . . . 9 ⊢ (𝑎 = 𝑏 → (◡𝐷 “ (0[,)𝑎)) = (◡𝐷 “ (0[,)𝑏)))
47	46	cbvmptv 5216	. . . . . . . 8 ⊢ (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) = (𝑏 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑏)))
48	47	rneqi 5929	. . . . . . 7 ⊢ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) = ran (𝑏 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑏)))
49	41, 48	eqtri 2786	. . . . . 6 ⊢ 𝐹 = ran (𝑏 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑏)))
50	49	metustel 24688	. . . . 5 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (𝐵 ∈ 𝐹 ↔ ∃𝑏 ∈ ℝ+ 𝐵 = (◡𝐷 “ (0[,)𝑏))))
51	50	biimpa 481	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐵 ∈ 𝐹) → ∃𝑏 ∈ ℝ+ 𝐵 = (◡𝐷 “ (0[,)𝑏)))
52	51	3adant2 1149	. . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐹) → ∃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))))
54	44, 52, 53	sylanbrc 594	. 2 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐹) → ∃𝑎 ∈ ℝ+ ∃𝑏 ∈ ℝ+ (𝐴 = (◡𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (◡𝐷 “ (0[,)𝑏))))
55	40, 54	r19.29vva 3225	1 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐹) → (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴))

Syntax hints:   → wi 4   ∧ wa 400   ∨ wo 860   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  ∃wrex 3089   ⊆ wss 3906   class class class wbr 5110   ↦ cmpt 5193  ^◡ccnv 5662  ran crn 5664   “ cima 5666  ‘cfv 6538  (class class class)co 7412  ℝcr 11100  0cc0 11101  ℝ^*cxr 11243   ≤ cle 11245  ℝ⁺crp 13017  [,)cico 13375  PsMetcpsmet 21487

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734  ax-cnex 11157  ax-resscn 11158  ax-1cn 11159  ax-addrcl 11162  ax-rnegex 11172  ax-cnre 11174  ax-pre-lttri 11175  ax-pre-lttrn 11176

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-po 5571  df-so 5572  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7987  df-2nd 7988  df-er 8695  df-en 8945  df-dom 8946  df-sdom 8947  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-rp 13018  df-ico 13379

This theorem is referenced by:  metustfbas  24695