Theorem metustto 24497

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 766	. . . . 5 ⊢ (((𝑎 ∈ ℝ+ ∧ 𝑏 ∈ ℝ+) ∧ (𝐴 = (◡𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (◡𝐷 “ (0[,)𝑏)))) → 𝑎 ∈ ℝ+)
2	1	rpred 12949	. . . 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+)
4	3	rpred 12949	. . . 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)))) ∧ 𝑎 ≤ 𝑏) → 𝑏 ∈ ℝ+)
6	5	rpred 12949	. . . . . . 7 ⊢ ((((𝑎 ∈ ℝ+ ∧ 𝑏 ∈ ℝ+) ∧ (𝐴 = (◡𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (◡𝐷 “ (0[,)𝑏)))) ∧ 𝑎 ≤ 𝑏) → 𝑏 ∈ ℝ)
7		0xr 11179	. . . . . . . . . 10 ⊢ 0 ∈ ℝ*
8	7	a1i 11	. . . . . . . . 9 ⊢ ((𝑏 ∈ ℝ ∧ 𝑎 ≤ 𝑏) → 0 ∈ ℝ*)
9		simpl 482	. . . . . . . . . 10 ⊢ ((𝑏 ∈ ℝ ∧ 𝑎 ≤ 𝑏) → 𝑏 ∈ ℝ)
10	9	rexrd 11182	. . . . . . . . 9 ⊢ ((𝑏 ∈ ℝ ∧ 𝑎 ≤ 𝑏) → 𝑏 ∈ ℝ*)
11		0le0 12246	. . . . . . . . . 10 ⊢ 0 ≤ 0
12	11	a1i 11	. . . . . . . . 9 ⊢ ((𝑏 ∈ ℝ ∧ 𝑎 ≤ 𝑏) → 0 ≤ 0)
13		simpr 484	. . . . . . . . 9 ⊢ ((𝑏 ∈ ℝ ∧ 𝑎 ≤ 𝑏) → 𝑎 ≤ 𝑏)
14		icossico 13332	. . . . . . . . 9 ⊢ (((0 ∈ ℝ* ∧ 𝑏 ∈ ℝ*) ∧ (0 ≤ 0 ∧ 𝑎 ≤ 𝑏)) → (0[,)𝑎) ⊆ (0[,)𝑏))
15	8, 10, 12, 13, 14	syl22anc 838	. . . . . . . 8 ⊢ ((𝑏 ∈ ℝ ∧ 𝑎 ≤ 𝑏) → (0[,)𝑎) ⊆ (0[,)𝑏))
16		imass2 6061	. . . . . . . 8 ⊢ ((0[,)𝑎) ⊆ (0[,)𝑏) → (◡𝐷 “ (0[,)𝑎)) ⊆ (◡𝐷 “ (0[,)𝑏)))
17	15, 16	syl 17	. . . . . . 7 ⊢ ((𝑏 ∈ ℝ ∧ 𝑎 ≤ 𝑏) → (◡𝐷 “ (0[,)𝑎)) ⊆ (◡𝐷 “ (0[,)𝑏)))
18	6, 17	sylancom 588	. . . . . 6 ⊢ ((((𝑎 ∈ ℝ+ ∧ 𝑏 ∈ ℝ+) ∧ (𝐴 = (◡𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (◡𝐷 “ (0[,)𝑏)))) ∧ 𝑎 ≤ 𝑏) → (◡𝐷 “ (0[,)𝑎)) ⊆ (◡𝐷 “ (0[,)𝑏)))
19		simplrl 776	. . . . . 6 ⊢ ((((𝑎 ∈ ℝ+ ∧ 𝑏 ∈ ℝ+) ∧ (𝐴 = (◡𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (◡𝐷 “ (0[,)𝑏)))) ∧ 𝑎 ≤ 𝑏) → 𝐴 = (◡𝐷 “ (0[,)𝑎)))
20		simplrr 777	. . . . . 6 ⊢ ((((𝑎 ∈ ℝ+ ∧ 𝑏 ∈ ℝ+) ∧ (𝐴 = (◡𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (◡𝐷 “ (0[,)𝑏)))) ∧ 𝑎 ≤ 𝑏) → 𝐵 = (◡𝐷 “ (0[,)𝑏)))
21	18, 19, 20	3sstr4d 3989	. . . . 5 ⊢ ((((𝑎 ∈ ℝ+ ∧ 𝑏 ∈ ℝ+) ∧ (𝐴 = (◡𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (◡𝐷 “ (0[,)𝑏)))) ∧ 𝑎 ≤ 𝑏) → 𝐴 ⊆ 𝐵)
22	21	orcd 873	. . . 4 ⊢ ((((𝑎 ∈ ℝ+ ∧ 𝑏 ∈ ℝ+) ∧ (𝐴 = (◡𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (◡𝐷 “ (0[,)𝑏)))) ∧ 𝑎 ≤ 𝑏) → (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴))
23		simplll 774	. . . . . . . 8 ⊢ ((((𝑎 ∈ ℝ+ ∧ 𝑏 ∈ ℝ+) ∧ (𝐴 = (◡𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (◡𝐷 “ (0[,)𝑏)))) ∧ 𝑏 ≤ 𝑎) → 𝑎 ∈ ℝ+)
24	23	rpred 12949	. . . . . . 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 482	. . . . . . . . . 10 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ≤ 𝑎) → 𝑎 ∈ ℝ)
27	26	rexrd 11182	. . . . . . . . 9 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ≤ 𝑎) → 𝑎 ∈ ℝ*)
28	11	a1i 11	. . . . . . . . 9 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ≤ 𝑎) → 0 ≤ 0)
29		simpr 484	. . . . . . . . 9 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ≤ 𝑎) → 𝑏 ≤ 𝑎)
30		icossico 13332	. . . . . . . . 9 ⊢ (((0 ∈ ℝ* ∧ 𝑎 ∈ ℝ*) ∧ (0 ≤ 0 ∧ 𝑏 ≤ 𝑎)) → (0[,)𝑏) ⊆ (0[,)𝑎))
31	25, 27, 28, 29, 30	syl22anc 838	. . . . . . . 8 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ≤ 𝑎) → (0[,)𝑏) ⊆ (0[,)𝑎))
32		imass2 6061	. . . . . . . 8 ⊢ ((0[,)𝑏) ⊆ (0[,)𝑎) → (◡𝐷 “ (0[,)𝑏)) ⊆ (◡𝐷 “ (0[,)𝑎)))
33	31, 32	syl 17	. . . . . . 7 ⊢ ((𝑎 ∈ ℝ ∧ 𝑏 ≤ 𝑎) → (◡𝐷 “ (0[,)𝑏)) ⊆ (◡𝐷 “ (0[,)𝑎)))
34	24, 33	sylancom 588	. . . . . 6 ⊢ ((((𝑎 ∈ ℝ+ ∧ 𝑏 ∈ ℝ+) ∧ (𝐴 = (◡𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (◡𝐷 “ (0[,)𝑏)))) ∧ 𝑏 ≤ 𝑎) → (◡𝐷 “ (0[,)𝑏)) ⊆ (◡𝐷 “ (0[,)𝑎)))
35		simplrr 777	. . . . . 6 ⊢ ((((𝑎 ∈ ℝ+ ∧ 𝑏 ∈ ℝ+) ∧ (𝐴 = (◡𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (◡𝐷 “ (0[,)𝑏)))) ∧ 𝑏 ≤ 𝑎) → 𝐵 = (◡𝐷 “ (0[,)𝑏)))
36		simplrl 776	. . . . . 6 ⊢ ((((𝑎 ∈ ℝ+ ∧ 𝑏 ∈ ℝ+) ∧ (𝐴 = (◡𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (◡𝐷 “ (0[,)𝑏)))) ∧ 𝑏 ≤ 𝑎) → 𝐴 = (◡𝐷 “ (0[,)𝑎)))
37	34, 35, 36	3sstr4d 3989	. . . . 5 ⊢ ((((𝑎 ∈ ℝ+ ∧ 𝑏 ∈ ℝ+) ∧ (𝐴 = (◡𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (◡𝐷 “ (0[,)𝑏)))) ∧ 𝑏 ≤ 𝑎) → 𝐵 ⊆ 𝐴)
38	37	olcd 874	. . . 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 11239	. . 3 ⊢ (((𝑎 ∈ ℝ+ ∧ 𝑏 ∈ ℝ+) ∧ (𝐴 = (◡𝐷 “ (0[,)𝑎)) ∧ 𝐵 = (◡𝐷 “ (0[,)𝑏)))) → (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴))
40	39	adantlll 718	. 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 24494	. . . . 5 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (𝐴 ∈ 𝐹 ↔ ∃𝑎 ∈ ℝ+ 𝐴 = (◡𝐷 “ (0[,)𝑎))))
43	42	biimpa 476	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹) → ∃𝑎 ∈ ℝ+ 𝐴 = (◡𝐷 “ (0[,)𝑎)))
44	43	3adant3 1132	. . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐹) → ∃𝑎 ∈ ℝ+ 𝐴 = (◡𝐷 “ (0[,)𝑎)))
45		oveq2 7366	. . . . . . . . . 10 ⊢ (𝑎 = 𝑏 → (0[,)𝑎) = (0[,)𝑏))
46	45	imaeq2d 6019	. . . . . . . . 9 ⊢ (𝑎 = 𝑏 → (◡𝐷 “ (0[,)𝑎)) = (◡𝐷 “ (0[,)𝑏)))
47	46	cbvmptv 5202	. . . . . . . 8 ⊢ (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) = (𝑏 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑏)))
48	47	rneqi 5886	. . . . . . 7 ⊢ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) = ran (𝑏 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑏)))
49	41, 48	eqtri 2759	. . . . . 6 ⊢ 𝐹 = ran (𝑏 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑏)))
50	49	metustel 24494	. . . . 5 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (𝐵 ∈ 𝐹 ↔ ∃𝑏 ∈ ℝ+ 𝐵 = (◡𝐷 “ (0[,)𝑏))))
51	50	biimpa 476	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐵 ∈ 𝐹) → ∃𝑏 ∈ ℝ+ 𝐵 = (◡𝐷 “ (0[,)𝑏)))
52	51	3adant2 1131	. . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐹) → ∃𝑏 ∈ ℝ+ 𝐵 = (◡𝐷 “ (0[,)𝑏)))
53		reeanv 3208	. . 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 583	. 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 3196	1 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴 ∈ 𝐹 ∧ 𝐵 ∈ 𝐹) → (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  ∃wrex 3060   ⊆ wss 3901   class class class wbr 5098   ↦ cmpt 5179  ^◡ccnv 5623  ran crn 5625   “ cima 5627  ‘cfv 6492  (class class class)co 7358  ℝcr 11025  0cc0 11026  ℝ^*cxr 11165   ≤ cle 11167  ℝ⁺crp 12905  [,)cico 13263  PsMetcpsmet 21293

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-addrcl 11087  ax-rnegex 11097  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-po 5532  df-so 5533  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7933  df-2nd 7934  df-er 8635  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-rp 12906  df-ico 13267

This theorem is referenced by:  metustfbas  24501