Theorem pstmval 29938

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 29932	. . 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 22110	. . . . . . . 8 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝑋 = dom dom 𝐷)
4	3	adantr 481	. . . . . . 7 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → 𝑋 = dom dom 𝐷)
5		dmeq 5324	. . . . . . . . 9 ⊢ (𝑑 = 𝐷 → dom 𝑑 = dom 𝐷)
6	5	dmeqd 5326	. . . . . . . 8 ⊢ (𝑑 = 𝐷 → dom dom 𝑑 = dom dom 𝐷)
7	6	adantl 482	. . . . . . 7 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → dom dom 𝑑 = dom dom 𝐷)
8	4, 7	eqtr4d 2659	. . . . . 6 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → 𝑋 = dom dom 𝑑)
9		qseq1 7796	. . . . . 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 6191	. . . . . . . 8 ⊢ (𝑑 = 𝐷 → (~Met‘𝑑) = (~Met‘𝐷))
12		pstmval.1	. . . . . . . 8 ⊢ ∼ = (~Met‘𝐷)
13	11, 12	syl6reqr 2675	. . . . . . 7 ⊢ (𝑑 = 𝐷 → ∼ = (~Met‘𝑑))
14		qseq2 7797	. . . . . . 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 482	. . . . 5 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → (dom dom 𝑑 / ∼ ) = (dom dom 𝑑 / (~Met‘𝑑)))
17	10, 16	eqtr2d 2657	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) → (dom dom 𝑑 / (~Met‘𝑑)) = (𝑋 / ∼ ))
18		mpt2eq12 6715	. . . 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 693	. . 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 1086	. . . . . . . . 9 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) ∧ 𝑎 ∈ (𝑋 / ∼ ) ∧ 𝑏 ∈ (𝑋 / ∼ )) → 𝑑 = 𝐷)
21	20	oveqd 6667	. . . . . . . 8 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) ∧ 𝑎 ∈ (𝑋 / ∼ ) ∧ 𝑏 ∈ (𝑋 / ∼ )) → (𝑥𝑑𝑦) = (𝑥𝐷𝑦))
22	21	eqeq2d 2632	. . . . . . 7 ⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑑 = 𝐷) ∧ 𝑎 ∈ (𝑋 / ∼ ) ∧ 𝑏 ∈ (𝑋 / ∼ )) → (𝑧 = (𝑥𝑑𝑦) ↔ 𝑧 = (𝑥𝐷𝑦)))
23	22	2rexbidv 3057	. . . . . 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 2741	. . . . 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 4446	. . . 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 6719	. . 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 2656	. 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 6220	. . . 4 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝑋 ∈ dom PsMet)
29		fveq2 6191	. . . . . 6 ⊢ (𝑥 = 𝑋 → (PsMet‘𝑥) = (PsMet‘𝑋))
30	29	eleq2d 2687	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝐷 ∈ (PsMet‘𝑥) ↔ 𝐷 ∈ (PsMet‘𝑋)))
31	30	rspcev 3309	. . . 4 ⊢ ((𝑋 ∈ dom PsMet ∧ 𝐷 ∈ (PsMet‘𝑋)) → ∃𝑥 ∈ dom PsMet𝐷 ∈ (PsMet‘𝑥))
32	28, 31	mpancom 703	. . 3 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ∃𝑥 ∈ dom PsMet𝐷 ∈ (PsMet‘𝑥))
33		df-psmet 19738	. . . . 5 ⊢ PsMet = (𝑥 ∈ V ↦ {𝑑 ∈ (ℝ* ↑𝑚 (𝑥 × 𝑥)) ∣ ∀𝑎 ∈ 𝑥 ((𝑎𝑑𝑎) = 0 ∧ ∀𝑏 ∈ 𝑥 ∀𝑐 ∈ 𝑥 (𝑎𝑑𝑏) ≤ ((𝑐𝑑𝑎) +𝑒 (𝑐𝑑𝑏)))})
34	33	funmpt2 5927	. . . 4 ⊢ Fun PsMet
35		elunirn 6509	. . . 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 224	. 2 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝐷 ∈ ∪ ran PsMet)
38		elfvex 6221	. . . 4 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝑋 ∈ V)
39		qsexg 7805	. . . 4 ⊢ (𝑋 ∈ V → (𝑋 / ∼ ) ∈ V)
40	38, 39	syl 17	. . 3 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (𝑋 / ∼ ) ∈ V)
41		mpt2exga 7246	. . 3 ⊢ (((𝑋 / ∼ ) ∈ V ∧ (𝑋 / ∼ ) ∈ V) → (𝑎 ∈ (𝑋 / ∼ ), 𝑏 ∈ (𝑋 / ∼ ) ↦ ∪ {𝑧 ∣ ∃𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝑧 = (𝑥𝐷𝑦)}) ∈ V)
42	40, 40, 41	syl2anc 693	. 2 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (𝑎 ∈ (𝑋 / ∼ ), 𝑏 ∈ (𝑋 / ∼ ) ↦ ∪ {𝑧 ∣ ∃𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝑧 = (𝑥𝐷𝑦)}) ∈ V)
43	2, 27, 37, 42	fvmptd 6288	1 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (pstoMet‘𝐷) = (𝑎 ∈ (𝑋 / ∼ ), 𝑏 ∈ (𝑋 / ∼ ) ↦ ∪ {𝑧 ∣ ∃𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑏 𝑧 = (𝑥𝐷𝑦)}))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  {cab 2608  ∀wral 2912  ∃wrex 2913  {crab 2916  Vcvv 3200  ∪ cuni 4436   class class class wbr 4653   ↦ cmpt 4729   × cxp 5112  dom cdm 5114  ran crn 5115  Fun wfun 5882  ‘cfv 5888  (class class class)co 6650   ↦ cmpt2 6652   / cqs 7741   ↑_𝑚 cmap 7857  0cc0 9936  ℝ^*cxr 10073   ≤ cle 10075   +_𝑒 cxad 11944  PsMetcpsmet 19730  ~_Metcmetid 29929  pstoMetcpstm 29930

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-ec 7744  df-qs 7748  df-map 7859  df-xr 10078  df-psmet 19738  df-pstm 29932

This theorem is referenced by:  pstmfval  29939  pstmxmet  29940