Theorem pstmval 29266

Description: Value of the metric induced by a pseudometric 𝐷. (Contributed by Thierry Arnoux, 7-Feb-2018.)

Hypothesis

Ref	Expression
pstmval.1	⊢ ∼ = (~Met‘𝐷)

Assertion

Ref	Expression
pstmval	⊢ (𝐷 ∈ (PsMet‘𝑋) → (pstoMet‘𝐷) = (𝑎 ∈ (𝑋 / ∼ ), 𝑏 ∈ (𝑋 / ∼ ) ↦ ∪ {𝑧 ∣ ∃𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝑧 = (𝑥𝐷𝑦)}))

Distinct variable groups:   𝑎,𝑏,𝑥,𝑦,𝑧,𝐷   𝑋,𝑎,𝑏,𝑥,𝑦,𝑧   ∼ ,𝑎,𝑏,𝑥,𝑦,𝑧

Proof of Theorem pstmval

Dummy variables 𝑐 𝑑 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-pstm 29260	. . 3 ⊢ pstoMet = (𝑑 ∈ ∪ ran PsMet ↦ (𝑎 ∈ (dom dom 𝑑 / (~Met‘𝑑)), 𝑏 ∈ (dom dom 𝑑 / (~Met‘𝑑)) ↦ ∪ {𝑧 ∣ ∃𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝑧 = (𝑥𝑑𝑦)}))
2	1	a1i 11	. 2 ⊢ (𝐷 ∈ (PsMet‘𝑋) → pstoMet = (𝑑 ∈ ∪ ran PsMet ↦ (𝑎 ∈ (dom dom 𝑑 / (~Met‘𝑑)), 𝑏 ∈ (dom dom 𝑑 / (~Met‘𝑑)) ↦ ∪ {𝑧 ∣ ∃𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝑧 = (𝑥𝑑𝑦)})))
3		psmetdmdm 21920	. . . . . . . 8 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝑋 = dom dom 𝐷)
4	3	adantr 480	. . . . . . 7 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → 𝑋 = dom dom 𝐷)
5		dmeq 5246	. . . . . . . . 9 ⊢ (𝑑 = 𝐷 → dom 𝑑 = dom 𝐷)
6	5	dmeqd 5248	. . . . . . . 8 ⊢ (𝑑 = 𝐷 → dom dom 𝑑 = dom dom 𝐷)
7	6	adantl 481	. . . . . . 7 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → dom dom 𝑑 = dom dom 𝐷)
8	4, 7	eqtr4d 2647	. . . . . 6 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → 𝑋 = dom dom 𝑑)
9		qseq1 7683	. . . . . 6 ⊢ (𝑋 = dom dom 𝑑 → (𝑋 / ∼ ) = (dom dom 𝑑 / ∼ ))
10	8, 9	syl 17	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → (𝑋 / ∼ ) = (dom dom 𝑑 / ∼ ))
11		fveq2 6103	. . . . . . . 8 ⊢ (𝑑 = 𝐷 → (~Met‘𝑑) = (~Met‘𝐷))
12		pstmval.1	. . . . . . . 8 ⊢ ∼ = (~Met‘𝐷)
13	11, 12	syl6reqr 2663	. . . . . . 7 ⊢ (𝑑 = 𝐷 → ∼ = (~Met‘𝑑))
14		qseq2 7684	. . . . . . 7 ⊢ ( ∼ = (~Met‘𝑑) → (dom dom 𝑑 / ∼ ) = (dom dom 𝑑 / (~Met‘𝑑)))
15	13, 14	syl 17	. . . . . 6 ⊢ (𝑑 = 𝐷 → (dom dom 𝑑 / ∼ ) = (dom dom 𝑑 / (~Met‘𝑑)))
16	15	adantl 481	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → (dom dom 𝑑 / ∼ ) = (dom dom 𝑑 / (~Met‘𝑑)))
17	10, 16	eqtr2d 2645	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → (dom dom 𝑑 / (~Met‘𝑑)) = (𝑋 / ∼ ))
18		mpt2eq12 6613	. . . 4 ⊢ (((dom dom 𝑑 / (~Met‘𝑑)) = (𝑋 / ∼ ) ∧ (dom dom 𝑑 / (~Met‘𝑑)) = (𝑋 / ∼ )) → (𝑎 ∈ (dom dom 𝑑 / (~Met‘𝑑)), 𝑏 ∈ (dom dom 𝑑 / (~Met‘𝑑)) ↦ ∪ {𝑧 ∣ ∃𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝑧 = (𝑥𝑑𝑦)}) = (𝑎 ∈ (𝑋 / ∼ ), 𝑏 ∈ (𝑋 / ∼ ) ↦ ∪ {𝑧 ∣ ∃𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝑧 = (𝑥𝑑𝑦)}))
19	17, 17, 18	syl2anc 691	. . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → (𝑎 ∈ (dom dom 𝑑 / (~Met‘𝑑)), 𝑏 ∈ (dom dom 𝑑 / (~Met‘𝑑)) ↦ ∪ {𝑧 ∣ ∃𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝑧 = (𝑥𝑑𝑦)}) = (𝑎 ∈ (𝑋 / ∼ ), 𝑏 ∈ (𝑋 / ∼ ) ↦ ∪ {𝑧 ∣ ∃𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝑧 = (𝑥𝑑𝑦)}))
20		simp1r 1079	. . . . . . . . 9 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) ∧ 𝑎 ∈ (𝑋 / ∼ ) ∧ 𝑏 ∈ (𝑋 / ∼ )) → 𝑑 = 𝐷)
21	20	oveqd 6566	. . . . . . . 8 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) ∧ 𝑎 ∈ (𝑋 / ∼ ) ∧ 𝑏 ∈ (𝑋 / ∼ )) → (𝑥𝑑𝑦) = (𝑥𝐷𝑦))
22	21	eqeq2d 2620	. . . . . . 7 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) ∧ 𝑎 ∈ (𝑋 / ∼ ) ∧ 𝑏 ∈ (𝑋 / ∼ )) → (𝑧 = (𝑥𝑑𝑦) ↔ 𝑧 = (𝑥𝐷𝑦)))
23	22	2rexbidv 3039	. . . . . 6 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) ∧ 𝑎 ∈ (𝑋 / ∼ ) ∧ 𝑏 ∈ (𝑋 / ∼ )) → (∃𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝑧 = (𝑥𝑑𝑦) ↔ ∃𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝑧 = (𝑥𝐷𝑦)))
24	23	abbidv 2728	. . . . 5 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) ∧ 𝑎 ∈ (𝑋 / ∼ ) ∧ 𝑏 ∈ (𝑋 / ∼ )) → {𝑧 ∣ ∃𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝑧 = (𝑥𝑑𝑦)} = {𝑧 ∣ ∃𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝑧 = (𝑥𝐷𝑦)})
25	24	unieqd 4382	. . . 4 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) ∧ 𝑎 ∈ (𝑋 / ∼ ) ∧ 𝑏 ∈ (𝑋 / ∼ )) → ∪ {𝑧 ∣ ∃𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝑧 = (𝑥𝑑𝑦)} = ∪ {𝑧 ∣ ∃𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝑧 = (𝑥𝐷𝑦)})
26	25	mpt2eq3dva 6617	. . 3 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → (𝑎 ∈ (𝑋 / ∼ ), 𝑏 ∈ (𝑋 / ∼ ) ↦ ∪ {𝑧 ∣ ∃𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝑧 = (𝑥𝑑𝑦)}) = (𝑎 ∈ (𝑋 / ∼ ), 𝑏 ∈ (𝑋 / ∼ ) ↦ ∪ {𝑧 ∣ ∃𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝑧 = (𝑥𝐷𝑦)}))
27	19, 26	eqtrd 2644	. 2 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → (𝑎 ∈ (dom dom 𝑑 / (~Met‘𝑑)), 𝑏 ∈ (dom dom 𝑑 / (~Met‘𝑑)) ↦ ∪ {𝑧 ∣ ∃𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝑧 = (𝑥𝑑𝑦)}) = (𝑎 ∈ (𝑋 / ∼ ), 𝑏 ∈ (𝑋 / ∼ ) ↦ ∪ {𝑧 ∣ ∃𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝑧 = (𝑥𝐷𝑦)}))
28		elfvdm 6130	. . . 4 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝑋 ∈ dom PsMet)
29		fveq2 6103	. . . . . 6 ⊢ (𝑥 = 𝑋 → (PsMet‘𝑥) = (PsMet‘𝑋))
30	29	eleq2d 2673	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝐷 ∈ (PsMet‘𝑥) ↔ 𝐷 ∈ (PsMet‘𝑋)))
31	30	rspcev 3282	. . . 4 ⊢ ((𝑋 ∈ dom PsMet ∧ 𝐷 ∈ (PsMet‘𝑋)) → ∃𝑥 ∈ dom PsMet𝐷 ∈ (PsMet‘𝑥))
32	28, 31	mpancom 700	. . 3 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ∃𝑥 ∈ dom PsMet𝐷 ∈ (PsMet‘𝑥))
33		df-psmet 19559	. . . . 5 ⊢ PsMet = (𝑥 ∈ V ↦ {𝑑 ∈ (ℝ* ↑𝑚 (𝑥 × 𝑥)) ∣ ∀𝑎 ∈ 𝑥 ((𝑎𝑑𝑎) = 0 ∧ ∀𝑏 ∈ 𝑥 ∀𝑐 ∈ 𝑥 (𝑎𝑑𝑏) ≤ ((𝑐𝑑𝑎) +𝑒 (𝑐𝑑𝑏)))})
34	33	funmpt2 5841	. . . 4 ⊢ Fun PsMet
35		elunirn 6413	. . . 4 ⊢ (Fun PsMet → (𝐷 ∈ ∪ ran PsMet ↔ ∃𝑥 ∈ dom PsMet𝐷 ∈ (PsMet‘𝑥)))
36	34, 35	ax-mp 5	. . 3 ⊢ (𝐷 ∈ ∪ ran PsMet ↔ ∃𝑥 ∈ dom PsMet𝐷 ∈ (PsMet‘𝑥))
37	32, 36	sylibr 223	. 2 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝐷 ∈ ∪ ran PsMet)
38		elfvex 6131	. . . 4 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝑋 ∈ V)
39		qsexg 7692	. . . 4 ⊢ (𝑋 ∈ V → (𝑋 / ∼ ) ∈ V)
40	38, 39	syl 17	. . 3 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (𝑋 / ∼ ) ∈ V)
41		mpt2exga 7135	. . 3 ⊢ (((𝑋 / ∼ ) ∈ V ∧ (𝑋 / ∼ ) ∈ V) → (𝑎 ∈ (𝑋 / ∼ ), 𝑏 ∈ (𝑋 / ∼ ) ↦ ∪ {𝑧 ∣ ∃𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝑧 = (𝑥𝐷𝑦)}) ∈ V)
42	40, 40, 41	syl2anc 691	. 2 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (𝑎 ∈ (𝑋 / ∼ ), 𝑏 ∈ (𝑋 / ∼ ) ↦ ∪ {𝑧 ∣ ∃𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝑧 = (𝑥𝐷𝑦)}) ∈ V)
43	2, 27, 37, 42	fvmptd 6197	1 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (pstoMet‘𝐷) = (𝑎 ∈ (𝑋 / ∼ ), 𝑏 ∈ (𝑋 / ∼ ) ↦ ∪ {𝑧 ∣ ∃𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝑧 = (𝑥𝐷𝑦)}))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  {cab 2596  ∀wral 2896  ∃wrex 2897  {crab 2900  Vcvv 3173  ∪ cuni 4372   class class class wbr 4583   ↦ cmpt 4643   × cxp 5036  dom cdm 5038  ran crn 5039  Fun wfun 5798  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551   / cqs 7628   ↑_𝑚 cmap 7744  0cc0 9815  ℝ^*cxr 9952   ≤ cle 9954   +_𝑒 cxad 11820  PsMetcpsmet 19551  ~_Metcmetid 29257  pstoMetcpstm 29258

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-ec 7631  df-qs 7635  df-map 7746  df-xr 9957  df-psmet 19559  df-pstm 29260

This theorem is referenced by:  pstmfval  29267  pstmxmet  29268