Theorem tuslemOLD 24127

Description: Obsolete proof of tuslem 24126 as of 28-Oct-2024. Lemma for tusbas 24128, tusunif 24129, and tustopn 24131. (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 17156	. . . 4 ⊢ Base = Slot (Base‘ndx)
2		1re 11218	. . . . . 6 ⊢ 1 ∈ ℝ
3		1lt9 12422	. . . . . 6 ⊢ 1 < 9
4	2, 3	ltneii 11331	. . . . 5 ⊢ 1 ≠ 9
5		basendx 17162	. . . . . 6 ⊢ (Base‘ndx) = 1
6		tsetndx 17306	. . . . . 6 ⊢ (TopSet‘ndx) = 9
7	5, 6	neeq12i 3001	. . . . 5 ⊢ ((Base‘ndx) ≠ (TopSet‘ndx) ↔ 1 ≠ 9)
8	4, 7	mpbir 230	. . . 4 ⊢ (Base‘ndx) ≠ (TopSet‘ndx)
9	1, 8	setsnid 17151	. . 3 ⊢ (Base‘{⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩}) = (Base‘({⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩))
10		ustbas2 24085	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = dom ∪ 𝑈)
11		uniexg 7727	. . . . 5 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ∪ 𝑈 ∈ V)
12		dmexg 7891	. . . . 5 ⊢ (∪ 𝑈 ∈ V → dom ∪ 𝑈 ∈ V)
13		eqid 2726	. . . . . 6 ⊢ {⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} = {⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩}
14		df-unif 17229	. . . . . 6 ⊢ UnifSet = Slot ;13
15		1nn 12227	. . . . . . 7 ⊢ 1 ∈ ℕ
16		3nn0 12494	. . . . . . 7 ⊢ 3 ∈ ℕ0
17		1nn0 12492	. . . . . . 7 ⊢ 1 ∈ ℕ0
18		1lt10 12820	. . . . . . 7 ⊢ 1 < ;10
19	15, 16, 17, 18	declti 12719	. . . . . 6 ⊢ 1 < ;13
20		3nn 12295	. . . . . . 7 ⊢ 3 ∈ ℕ
21	17, 20	decnncl 12701	. . . . . 6 ⊢ ;13 ∈ ℕ
22	13, 14, 19, 21	2strbas 17176	. . . . 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 2766	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = (Base‘{⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩}))
25		tuslem.k	. . . . 5 ⊢ 𝐾 = (toUnifSp‘𝑈)
26		tusval 24125	. . . . 5 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (toUnifSp‘𝑈) = ({⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩))
27	25, 26	eqtrid 2778	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝐾 = ({⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩))
28	27	fveq2d 6889	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (Base‘𝐾) = (Base‘({⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩)))
29	9, 24, 28	3eqtr4a 2792	. 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = (Base‘𝐾))
30		unifid 17350	. . . 4 ⊢ UnifSet = Slot (UnifSet‘ndx)
31		9re 12315	. . . . . 6 ⊢ 9 ∈ ℝ
32		9nn0 12500	. . . . . . 7 ⊢ 9 ∈ ℕ0
33		9lt10 12812	. . . . . . 7 ⊢ 9 < ;10
34	15, 16, 32, 33	declti 12719	. . . . . 6 ⊢ 9 < ;13
35	31, 34	gtneii 11330	. . . . 5 ⊢ ;13 ≠ 9
36		unifndx 17349	. . . . . 6 ⊢ (UnifSet‘ndx) = ;13
37	36, 6	neeq12i 3001	. . . . 5 ⊢ ((UnifSet‘ndx) ≠ (TopSet‘ndx) ↔ ;13 ≠ 9)
38	35, 37	mpbir 230	. . . 4 ⊢ (UnifSet‘ndx) ≠ (TopSet‘ndx)
39	30, 38	setsnid 17151	. . 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 17177	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 = (UnifSet‘{⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩}))
41	27	fveq2d 6889	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (UnifSet‘𝐾) = (UnifSet‘({⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩)))
42	39, 40, 41	3eqtr4a 2792	. 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 = (UnifSet‘𝐾))
43		prex 5425	. . . . 5 ⊢ {⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} ∈ V
44		fvex 6898	. . . . 5 ⊢ (unifTop‘𝑈) ∈ V
45		tsetid 17307	. . . . . 6 ⊢ TopSet = Slot (TopSet‘ndx)
46	45	setsid 17150	. . . . 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 6889	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (TopSet‘𝐾) = (TopSet‘({⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩)))
49	47, 48	eqtr4id 2785	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) = (TopSet‘𝐾))
50		utopbas 24095	. . . . . 6 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = ∪ (unifTop‘𝑈))
51	49	unieqd 4915	. . . . . 6 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ∪ (unifTop‘𝑈) = ∪ (TopSet‘𝐾))
52	50, 29, 51	3eqtr3rd 2775	. . . . 5 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ∪ (TopSet‘𝐾) = (Base‘𝐾))
53	52	oveq2d 7421	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ((TopSet‘𝐾) ↾t ∪ (TopSet‘𝐾)) = ((TopSet‘𝐾) ↾t (Base‘𝐾)))
54		fvex 6898	. . . . 5 ⊢ (TopSet‘𝐾) ∈ V
55		eqid 2726	. . . . . 6 ⊢ ∪ (TopSet‘𝐾) = ∪ (TopSet‘𝐾)
56	55	restid 17388	. . . . 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 2726	. . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾)
59		eqid 2726	. . . . 5 ⊢ (TopSet‘𝐾) = (TopSet‘𝐾)
60	58, 59	topnval 17389	. . . 4 ⊢ ((TopSet‘𝐾) ↾t (Base‘𝐾)) = (TopOpen‘𝐾)
61	53, 57, 60	3eqtr3g 2789	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (TopSet‘𝐾) = (TopOpen‘𝐾))
62	49, 61	eqtrd 2766	. 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) = (TopOpen‘𝐾))
63	29, 42, 62	3jca 1125	1 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 = (Base‘𝐾) ∧ 𝑈 = (UnifSet‘𝐾) ∧ (unifTop‘𝑈) = (TopOpen‘𝐾)))

Syntax hints:   → wi 4   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   ≠ wne 2934  Vcvv 3468  {cpr 4625  ⟨cop 4629  ∪ cuni 4902  dom cdm 5669  ‘cfv 6537  (class class class)co 7405  1c1 11113  3c3 12272  9c9 12278  ;cdc 12681   sSet csts 17105  ndxcnx 17135  Basecbs 17153  TopSetcts 17212  UnifSetcunif 17216   ↾_t crest 17375  TopOpenctopn 17376  UnifOncust 24059  unifTopcutop 24090  toUnifSpctus 24115

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7853  df-1st 7974  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-1o 8467  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-2 12279  df-3 12280  df-4 12281  df-5 12282  df-6 12283  df-7 12284  df-8 12285  df-9 12286  df-n0 12477  df-z 12563  df-dec 12682  df-uz 12827  df-fz 13491  df-struct 17089  df-sets 17106  df-slot 17124  df-ndx 17136  df-base 17154  df-tset 17225  df-unif 17229  df-rest 17377  df-topn 17378  df-ust 24060  df-utop 24091  df-tus 24118

This theorem is referenced by: (None)