Theorem tuslemOLD 23419

Description: Obsolete proof of tuslem 23418 as of 28-Oct-2024. Lemma for tusbas 23420, tusunif 23421, and tustopn 23423. (Contributed by Thierry Arnoux, 5-Dec-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
tuslem.k	⊢ 𝐾 = (toUnifSp‘𝑈)

Assertion

Ref	Expression
tuslemOLD	⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 = (Base‘𝐾) ∧ 𝑈 = (UnifSet‘𝐾) ∧ (unifTop‘𝑈) = (TopOpen‘𝐾)))

Proof of Theorem tuslemOLD

Step	Hyp	Ref	Expression
1		baseid 16915	. . . 4 ⊢ Base = Slot (Base‘ndx)
2		1re 10975	. . . . . 6 ⊢ 1 ∈ ℝ
3		1lt9 12179	. . . . . 6 ⊢ 1 < 9
4	2, 3	ltneii 11088	. . . . 5 ⊢ 1 ≠ 9
5		basendx 16921	. . . . . 6 ⊢ (Base‘ndx) = 1
6		tsetndx 17062	. . . . . 6 ⊢ (TopSet‘ndx) = 9
7	5, 6	neeq12i 3010	. . . . 5 ⊢ ((Base‘ndx) ≠ (TopSet‘ndx) ↔ 1 ≠ 9)
8	4, 7	mpbir 230	. . . 4 ⊢ (Base‘ndx) ≠ (TopSet‘ndx)
9	1, 8	setsnid 16910	. . 3 ⊢ (Base‘{⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩}) = (Base‘({⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩))
10		ustbas2 23377	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = dom ∪ 𝑈)
11		uniexg 7593	. . . . 5 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ∪ 𝑈 ∈ V)
12		dmexg 7750	. . . . 5 ⊢ (∪ 𝑈 ∈ V → dom ∪ 𝑈 ∈ V)
13		eqid 2738	. . . . . 6 ⊢ {⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} = {⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩}
14		df-unif 16985	. . . . . 6 ⊢ UnifSet = Slot ;13
15		1nn 11984	. . . . . . 7 ⊢ 1 ∈ ℕ
16		3nn0 12251	. . . . . . 7 ⊢ 3 ∈ ℕ0
17		1nn0 12249	. . . . . . 7 ⊢ 1 ∈ ℕ0
18		1lt10 12576	. . . . . . 7 ⊢ 1 < ;10
19	15, 16, 17, 18	declti 12475	. . . . . 6 ⊢ 1 < ;13
20		3nn 12052	. . . . . . 7 ⊢ 3 ∈ ℕ
21	17, 20	decnncl 12457	. . . . . 6 ⊢ ;13 ∈ ℕ
22	13, 14, 19, 21	2strbas 16935	. . . . 5 ⊢ (dom ∪ 𝑈 ∈ V → dom ∪ 𝑈 = (Base‘{⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩}))
23	11, 12, 22	3syl 18	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → dom ∪ 𝑈 = (Base‘{⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩}))
24	10, 23	eqtrd 2778	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = (Base‘{⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩}))
25		tuslem.k	. . . . 5 ⊢ 𝐾 = (toUnifSp‘𝑈)
26		tusval 23417	. . . . 5 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (toUnifSp‘𝑈) = ({⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩))
27	25, 26	eqtrid 2790	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝐾 = ({⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩))
28	27	fveq2d 6778	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (Base‘𝐾) = (Base‘({⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩)))
29	9, 24, 28	3eqtr4a 2804	. 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = (Base‘𝐾))
30		unifid 17106	. . . 4 ⊢ UnifSet = Slot (UnifSet‘ndx)
31		9re 12072	. . . . . 6 ⊢ 9 ∈ ℝ
32		9nn0 12257	. . . . . . 7 ⊢ 9 ∈ ℕ0
33		9lt10 12568	. . . . . . 7 ⊢ 9 < ;10
34	15, 16, 32, 33	declti 12475	. . . . . 6 ⊢ 9 < ;13
35	31, 34	gtneii 11087	. . . . 5 ⊢ ;13 ≠ 9
36		unifndx 17105	. . . . . 6 ⊢ (UnifSet‘ndx) = ;13
37	36, 6	neeq12i 3010	. . . . 5 ⊢ ((UnifSet‘ndx) ≠ (TopSet‘ndx) ↔ ;13 ≠ 9)
38	35, 37	mpbir 230	. . . 4 ⊢ (UnifSet‘ndx) ≠ (TopSet‘ndx)
39	30, 38	setsnid 16910	. . 3 ⊢ (UnifSet‘{⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩}) = (UnifSet‘({⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩))
40	13, 14, 19, 21	2strop 16936	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 = (UnifSet‘{⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩}))
41	27	fveq2d 6778	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (UnifSet‘𝐾) = (UnifSet‘({⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩)))
42	39, 40, 41	3eqtr4a 2804	. 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 = (UnifSet‘𝐾))
43		prex 5355	. . . . 5 ⊢ {⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} ∈ V
44		fvex 6787	. . . . 5 ⊢ (unifTop‘𝑈) ∈ V
45		tsetid 17063	. . . . . 6 ⊢ TopSet = Slot (TopSet‘ndx)
46	45	setsid 16909	. . . . 5 ⊢ (({⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} ∈ V ∧ (unifTop‘𝑈) ∈ V) → (unifTop‘𝑈) = (TopSet‘({⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩)))
47	43, 44, 46	mp2an 689	. . . 4 ⊢ (unifTop‘𝑈) = (TopSet‘({⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩))
48	27	fveq2d 6778	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (TopSet‘𝐾) = (TopSet‘({⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩)))
49	47, 48	eqtr4id 2797	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) = (TopSet‘𝐾))
50		utopbas 23387	. . . . . 6 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = ∪ (unifTop‘𝑈))
51	49	unieqd 4853	. . . . . 6 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ∪ (unifTop‘𝑈) = ∪ (TopSet‘𝐾))
52	50, 29, 51	3eqtr3rd 2787	. . . . 5 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ∪ (TopSet‘𝐾) = (Base‘𝐾))
53	52	oveq2d 7291	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ((TopSet‘𝐾) ↾t ∪ (TopSet‘𝐾)) = ((TopSet‘𝐾) ↾t (Base‘𝐾)))
54		fvex 6787	. . . . 5 ⊢ (TopSet‘𝐾) ∈ V
55		eqid 2738	. . . . . 6 ⊢ ∪ (TopSet‘𝐾) = ∪ (TopSet‘𝐾)
56	55	restid 17144	. . . . 5 ⊢ ((TopSet‘𝐾) ∈ V → ((TopSet‘𝐾) ↾t ∪ (TopSet‘𝐾)) = (TopSet‘𝐾))
57	54, 56	ax-mp 5	. . . 4 ⊢ ((TopSet‘𝐾) ↾t ∪ (TopSet‘𝐾)) = (TopSet‘𝐾)
58		eqid 2738	. . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾)
59		eqid 2738	. . . . 5 ⊢ (TopSet‘𝐾) = (TopSet‘𝐾)
60	58, 59	topnval 17145	. . . 4 ⊢ ((TopSet‘𝐾) ↾t (Base‘𝐾)) = (TopOpen‘𝐾)
61	53, 57, 60	3eqtr3g 2801	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (TopSet‘𝐾) = (TopOpen‘𝐾))
62	49, 61	eqtrd 2778	. 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) = (TopOpen‘𝐾))
63	29, 42, 62	3jca 1127	1 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 = (Base‘𝐾) ∧ 𝑈 = (UnifSet‘𝐾) ∧ (unifTop‘𝑈) = (TopOpen‘𝐾)))

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  Vcvv 3432  {cpr 4563  ⟨cop 4567  ∪ cuni 4839  dom cdm 5589  ‘cfv 6433  (class class class)co 7275  1c1 10872  3c3 12029  9c9 12035  ;cdc 12437   sSet csts 16864  ndxcnx 16894  Basecbs 16912  TopSetcts 16968  UnifSetcunif 16972   ↾_t crest 17131  TopOpenctopn 17132  UnifOncust 23351  unifTopcutop 23382  toUnifSpctus 23407

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-2 12036  df-3 12037  df-4 12038  df-5 12039  df-6 12040  df-7 12041  df-8 12042  df-9 12043  df-n0 12234  df-z 12320  df-dec 12438  df-uz 12583  df-fz 13240  df-struct 16848  df-sets 16865  df-slot 16883  df-ndx 16895  df-base 16913  df-tset 16981  df-unif 16985  df-rest 17133  df-topn 17134  df-ust 23352  df-utop 23383  df-tus 23410

This theorem is referenced by: (None)