Theorem metustto 24852

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 779	. . . . 5 ⊢ (((𝑎 ∈ ℝ+ ∧ 𝑏 ∈ ℝ+) ∧ (𝐴 = (◡𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (◡𝐷 “ (0[,)𝑏)))) → 𝑎 ∈ ℝ+)
2	1	rpred 13145	. . . 4 ⊢ (((𝑎 ∈ ℝ+ ∧ 𝑏 ∈ ℝ+) ∧ (𝐴 = (◡𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (◡𝐷 “ (0[,)𝑏)))) → 𝑎 ∈ ℝ)
3		simplr 781	. . . . 5 ⊢ (((𝑎 ∈ ℝ+ ∧ 𝑏 ∈ ℝ+) ∧ (𝐴 = (◡𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (◡𝐷 “ (0[,)𝑏)))) → 𝑏 ∈ ℝ+)
4	3	rpred 13145	. . . 4 ⊢ (((𝑎 ∈ ℝ+ ∧ 𝑏 ∈ ℝ+) ∧ (𝐴 = (◡𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (◡𝐷 “ (0[,)𝑏)))) → 𝑏 ∈ ℝ)
5		simpllr 788	. . . . . . . 8 ⊢ ((((𝑎 ∈ ℝ+ ∧ 𝑏 ∈ ℝ+) ∧ (𝐴 = (◡𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (◡𝐷 “ (0[,)𝑏)))) ∧ 𝑎 ≤ 𝑏) → 𝑏 ∈ ℝ+)
6	5	rpred 13145	. . . . . . 7 ⊢ ((((𝑎 ∈ ℝ+ ∧ 𝑏 ∈ ℝ+) ∧ (𝐴 = (◡𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (◡𝐷 “ (0[,)𝑏)))) ∧ 𝑎 ≤ 𝑏) → 𝑏 ∈ ℝ)
7		0xr 11337	. . . . . . . . . 10 ⊢ 0 ∈ ℝ*
8	7	a1i 11	. . . . . . . . 9 ⊢ ((𝑏 ∈ ℝ ∧ 𝑎 ≤ 𝑏) → 0 ∈ ℝ*)
9		simpl 488	. . . . . . . . . 10 ⊢ ((𝑏 ∈ ℝ ∧ 𝑎 ≤ 𝑏) → 𝑏 ∈ ℝ)
10	9	rexrd 11340	. . . . . . . . 9 ⊢ ((𝑏 ∈ ℝ ∧ 𝑎 ≤ 𝑏) → 𝑏 ∈ ℝ*)
11		0le0 12425	. . . . . . . . . 10 ⊢ 0 ≤ 0
12	11	a1i 11	. . . . . . . . 9 ⊢ ((𝑏 ∈ ℝ ∧ 𝑎 ≤ 𝑏) → 0 ≤ 0)
13		simpr 490	. . . . . . . . 9 ⊢ ((𝑏 ∈ ℝ ∧ 𝑎 ≤ 𝑏) → 𝑎 ≤ 𝑏)
14		icossico 13528	. . . . . . . . 9 ⊢ (((0 ∈ ℝ* ∧ 𝑏 ∈ ℝ*) ∧ (0 ≤ 0 ∧ 𝑎 ≤ 𝑏)) → (0[,)𝑎) ⊆ (0[,)𝑏))
15	8, 10, 12, 13, 14	syl22anc 852	. . . . . . . 8 ⊢ ((𝑏 ∈ ℝ ∧ 𝑎 ≤ 𝑏) → (0[,)𝑎) ⊆ (0[,)𝑏))
16		imass2 6096	. . . . . . . 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 600	. . . . . 6 ⊢ ((((𝑎 ∈ ℝ+ ∧ 𝑏 ∈ ℝ+) ∧ (𝐴 = (◡𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (◡𝐷 “ (0[,)𝑏)))) ∧ 𝑎 ≤ 𝑏) → (◡𝐷 “ (0[,)𝑎)) ⊆ (◡𝐷 “ (0[,)𝑏)))
19		simplrl 789	. . . . . 6 ⊢ ((((𝑎 ∈ ℝ+ ∧ 𝑏 ∈ ℝ+) ∧ (𝐴 = (◡𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (◡𝐷 “ (0[,)𝑏)))) ∧ 𝑎 ≤ 𝑏) → 𝐴 = (◡𝐷 “ (0[,)𝑎)))
20		simplrr 790	. . . . . 6 ⊢ ((((𝑎 ∈ ℝ+ ∧ 𝑏 ∈ ℝ+) ∧ (𝐴 = (◡𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (◡𝐷 “ (0[,)𝑏)))) ∧ 𝑎 ≤ 𝑏) → 𝐵 = (◡𝐷 “ (0[,)𝑏)))
21	18, 19, 20	3sstr4d 3986	. . . . 5 ⊢ ((((𝑎 ∈ ℝ+ ∧ 𝑏 ∈ ℝ+) ∧ (𝐴 = (◡𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (◡𝐷 “ (0[,)𝑏)))) ∧ 𝑎 ≤ 𝑏) → 𝐴 ⊆ 𝐵)
22	21	orcd 887	. . . 4 ⊢ ((((𝑎 ∈ ℝ+ ∧ 𝑏 ∈ ℝ+) ∧ (𝐴 = (◡𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (◡𝐷 “ (0[,)𝑏)))) ∧ 𝑎 ≤ 𝑏) → (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴))
23		simplll 787	. . . . . . . 8 ⊢ ((((𝑎 ∈ ℝ+ ∧ 𝑏 ∈ ℝ+) ∧ (𝐴 = (◡𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (◡𝐷 “ (0[,)𝑏)))) ∧ 𝑏 ≤ 𝑎) → 𝑎 ∈ ℝ+)
24	23	rpred 13145	. . . . . . 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 488	. . . . . . . . . 10 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ≤ 𝑎) → 𝑎 ∈ ℝ)
27	26	rexrd 11340	. . . . . . . . 9 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ≤ 𝑎) → 𝑎 ∈ ℝ*)
28	11	a1i 11	. . . . . . . . 9 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ≤ 𝑎) → 0 ≤ 0)
29		simpr 490	. . . . . . . . 9 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ≤ 𝑎) → 𝑏 ≤ 𝑎)
30		icossico 13528	. . . . . . . . 9 ⊢ (((0 ∈ ℝ* ∧ 𝑎 ∈ ℝ*) ∧ (0 ≤ 0 ∧ 𝑏 ≤ 𝑎)) → (0[,)𝑏) ⊆ (0[,)𝑎))
31	25, 27, 28, 29, 30	syl22anc 852	. . . . . . . 8 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ≤ 𝑎) → (0[,)𝑏) ⊆ (0[,)𝑎))
32		imass2 6096	. . . . . . . 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 600	. . . . . 6 ⊢ ((((𝑎 ∈ ℝ+ ∧ 𝑏 ∈ ℝ+) ∧ (𝐴 = (◡𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (◡𝐷 “ (0[,)𝑏)))) ∧ 𝑏 ≤ 𝑎) → (◡𝐷 “ (0[,)𝑏)) ⊆ (◡𝐷 “ (0[,)𝑎)))
35		simplrr 790	. . . . . 6 ⊢ ((((𝑎 ∈ ℝ+ ∧ 𝑏 ∈ ℝ+) ∧ (𝐴 = (◡𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (◡𝐷 “ (0[,)𝑏)))) ∧ 𝑏 ≤ 𝑎) → 𝐵 = (◡𝐷 “ (0[,)𝑏)))
36		simplrl 789	. . . . . 6 ⊢ ((((𝑎 ∈ ℝ+ ∧ 𝑏 ∈ ℝ+) ∧ (𝐴 = (◡𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (◡𝐷 “ (0[,)𝑏)))) ∧ 𝑏 ≤ 𝑎) → 𝐴 = (◡𝐷 “ (0[,)𝑎)))
37	34, 35, 36	3sstr4d 3986	. . . . 5 ⊢ ((((𝑎 ∈ ℝ+ ∧ 𝑏 ∈ ℝ+) ∧ (𝐴 = (◡𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (◡𝐷 “ (0[,)𝑏)))) ∧ 𝑏 ≤ 𝑎) → 𝐵 ⊆ 𝐴)
38	37	olcd 888	. . . 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 11397	. . 3 ⊢ (((𝑎 ∈ ℝ+ ∧ 𝑏 ∈ ℝ+) ∧ (𝐴 = (◡𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (◡𝐷 “ (0[,)𝑏)))) → (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴))
40	39	adantlll 731	. 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 24849	. . . . 5 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (𝐴 ∈ 𝐹 ↔ ∃𝑎 ∈ ℝ+ 𝐴 = (◡𝐷 “ (0[,)𝑎))))
43	42	biimpa 482	. . . 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 6054	. . . . . . . . 9 ⊢ (𝑎 = 𝑏 → (◡𝐷 “ (0[,)𝑎)) = (◡𝐷 “ (0[,)𝑏)))
47	46	cbvmptv 5209	. . . . . . . 8 ⊢ (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) = (𝑏 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑏)))
48	47	rneqi 5919	. . . . . . 7 ⊢ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) = ran (𝑏 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑏)))
49	41, 48	eqtri 2784	. . . . . 6 ⊢ 𝐹 = ran (𝑏 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑏)))
50	49	metustel 24849	. . . . 5 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (𝐵 ∈ 𝐹 ↔ ∃𝑏 ∈ ℝ+ 𝐵 = (◡𝐷 “ (0[,)𝑏))))
51	50	biimpa 482	. . . 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 3235	. . 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 595	. 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 3223	1 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐹) → (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∃wrex 3087   ⊆ wss 3899   class class class wbr 5103   ↦ cmpt 5186  ^◡ccnv 5650  ran crn 5652   “ cima 5654  ‘cfv 6531  (class class class)co 7412  ℝcr 11180  0cc0 11181  ℝ^*cxr 11323   ≤ cle 11325  ℝ⁺crp 13101  [,)cico 13459  PsMetcpsmet 21642

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740  ax-cnex 11237  ax-resscn 11238  ax-1cn 11239  ax-addrcl 11242  ax-rnegex 11252  ax-cnre 11254  ax-pre-lttri 11255  ax-pre-lttrn 11256

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-po 5559  df-so 5560  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7990  df-2nd 7991  df-er 8701  df-en 8958  df-dom 8959  df-sdom 8960  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-rp 13102  df-ico 13463

This theorem is used by:  metustfbas  24856