Theorem tuslemOLD 23502

Description: Obsolete proof of tuslem 23501 as of 28-Oct-2024. Lemma for tusbas 23503, tusunif 23504, and tustopn 23506. (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 16992	. . . 4 ⊢ Base = Slot (Base‘ndx)
2		1re 11055	. . . . . 6 ⊢ 1 ∈ ℝ
3		1lt9 12259	. . . . . 6 ⊢ 1 < 9
4	2, 3	ltneii 11168	. . . . 5 ⊢ 1 ≠ 9
5		basendx 16998	. . . . . 6 ⊢ (Base‘ndx) = 1
6		tsetndx 17139	. . . . . 6 ⊢ (TopSet‘ndx) = 9
7	5, 6	neeq12i 3008	. . . . 5 ⊢ ((Base‘ndx) ≠ (TopSet‘ndx) ↔ 1 ≠ 9)
8	4, 7	mpbir 230	. . . 4 ⊢ (Base‘ndx) ≠ (TopSet‘ndx)
9	1, 8	setsnid 16987	. . 3 ⊢ (Base‘{⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩}) = (Base‘({⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩))
10		ustbas2 23460	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = dom ∪ 𝑈)
11		uniexg 7635	. . . . 5 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ∪ 𝑈 ∈ V)
12		dmexg 7797	. . . . 5 ⊢ (∪ 𝑈 ∈ V → dom ∪ 𝑈 ∈ V)
13		eqid 2737	. . . . . 6 ⊢ {⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} = {⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩}
14		df-unif 17062	. . . . . 6 ⊢ UnifSet = Slot ;13
15		1nn 12064	. . . . . . 7 ⊢ 1 ∈ ℕ
16		3nn0 12331	. . . . . . 7 ⊢ 3 ∈ ℕ0
17		1nn0 12329	. . . . . . 7 ⊢ 1 ∈ ℕ0
18		1lt10 12656	. . . . . . 7 ⊢ 1 < ;10
19	15, 16, 17, 18	declti 12555	. . . . . 6 ⊢ 1 < ;13
20		3nn 12132	. . . . . . 7 ⊢ 3 ∈ ℕ
21	17, 20	decnncl 12537	. . . . . 6 ⊢ ;13 ∈ ℕ
22	13, 14, 19, 21	2strbas 17012	. . . . 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 2777	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = (Base‘{⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩}))
25		tuslem.k	. . . . 5 ⊢ 𝐾 = (toUnifSp‘𝑈)
26		tusval 23500	. . . . 5 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (toUnifSp‘𝑈) = ({⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩))
27	25, 26	eqtrid 2789	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝐾 = ({⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩))
28	27	fveq2d 6816	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (Base‘𝐾) = (Base‘({⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩)))
29	9, 24, 28	3eqtr4a 2803	. 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = (Base‘𝐾))
30		unifid 17183	. . . 4 ⊢ UnifSet = Slot (UnifSet‘ndx)
31		9re 12152	. . . . . 6 ⊢ 9 ∈ ℝ
32		9nn0 12337	. . . . . . 7 ⊢ 9 ∈ ℕ0
33		9lt10 12648	. . . . . . 7 ⊢ 9 < ;10
34	15, 16, 32, 33	declti 12555	. . . . . 6 ⊢ 9 < ;13
35	31, 34	gtneii 11167	. . . . 5 ⊢ ;13 ≠ 9
36		unifndx 17182	. . . . . 6 ⊢ (UnifSet‘ndx) = ;13
37	36, 6	neeq12i 3008	. . . . 5 ⊢ ((UnifSet‘ndx) ≠ (TopSet‘ndx) ↔ ;13 ≠ 9)
38	35, 37	mpbir 230	. . . 4 ⊢ (UnifSet‘ndx) ≠ (TopSet‘ndx)
39	30, 38	setsnid 16987	. . 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 17013	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 = (UnifSet‘{⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩}))
41	27	fveq2d 6816	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (UnifSet‘𝐾) = (UnifSet‘({⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩)))
42	39, 40, 41	3eqtr4a 2803	. 2 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑈 = (UnifSet‘𝐾))
43		prex 5370	. . . . 5 ⊢ {⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} ∈ V
44		fvex 6825	. . . . 5 ⊢ (unifTop‘𝑈) ∈ V
45		tsetid 17140	. . . . . 6 ⊢ TopSet = Slot (TopSet‘ndx)
46	45	setsid 16986	. . . . 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 6816	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (TopSet‘𝐾) = (TopSet‘({⟨(Base‘ndx), dom ∪ 𝑈⟩, ⟨(UnifSet‘ndx), 𝑈⟩} sSet ⟨(TopSet‘ndx), (unifTop‘𝑈)⟩)))
49	47, 48	eqtr4id 2796	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) = (TopSet‘𝐾))
50		utopbas 23470	. . . . . 6 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = ∪ (unifTop‘𝑈))
51	49	unieqd 4864	. . . . . 6 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ∪ (unifTop‘𝑈) = ∪ (TopSet‘𝐾))
52	50, 29, 51	3eqtr3rd 2786	. . . . 5 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ∪ (TopSet‘𝐾) = (Base‘𝐾))
53	52	oveq2d 7333	. . . 4 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → ((TopSet‘𝐾) ↾t ∪ (TopSet‘𝐾)) = ((TopSet‘𝐾) ↾t (Base‘𝐾)))
54		fvex 6825	. . . . 5 ⊢ (TopSet‘𝐾) ∈ V
55		eqid 2737	. . . . . 6 ⊢ ∪ (TopSet‘𝐾) = ∪ (TopSet‘𝐾)
56	55	restid 17221	. . . . 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 2737	. . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾)
59		eqid 2737	. . . . 5 ⊢ (TopSet‘𝐾) = (TopSet‘𝐾)
60	58, 59	topnval 17222	. . . 4 ⊢ ((TopSet‘𝐾) ↾t (Base‘𝐾)) = (TopOpen‘𝐾)
61	53, 57, 60	3eqtr3g 2800	. . 3 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (TopSet‘𝐾) = (TopOpen‘𝐾))
62	49, 61	eqtrd 2777	. 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 1540   ∈ wcel 2105   ≠ wne 2941  Vcvv 3441  {cpr 4573  ⟨cop 4577  ∪ cuni 4850  dom cdm 5608  ‘cfv 6466  (class class class)co 7317  1c1 10952  3c3 12109  9c9 12115  ;cdc 12517   sSet csts 16941  ndxcnx 16971  Basecbs 16989  TopSetcts 17045  UnifSetcunif 17049   ↾_t crest 17208  TopOpenctopn 17209  UnifOncust 23434  unifTopcutop 23465  toUnifSpctus 23490

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2708  ax-rep 5224  ax-sep 5238  ax-nul 5245  ax-pow 5303  ax-pr 5367  ax-un 7630  ax-cnex 11007  ax-resscn 11008  ax-1cn 11009  ax-icn 11010  ax-addcl 11011  ax-addrcl 11012  ax-mulcl 11013  ax-mulrcl 11014  ax-mulcom 11015  ax-addass 11016  ax-mulass 11017  ax-distr 11018  ax-i2m1 11019  ax-1ne0 11020  ax-1rid 11021  ax-rnegex 11022  ax-rrecex 11023  ax-cnre 11024  ax-pre-lttri 11025  ax-pre-lttrn 11026  ax-pre-ltadd 11027  ax-pre-mulgt0 11028

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3351  df-rab 3405  df-v 3443  df-sbc 3727  df-csb 3843  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3916  df-nul 4268  df-if 4472  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4851  df-iun 4939  df-br 5088  df-opab 5150  df-mpt 5171  df-tr 5205  df-id 5507  df-eprel 5513  df-po 5521  df-so 5522  df-fr 5563  df-we 5565  df-xp 5614  df-rel 5615  df-cnv 5616  df-co 5617  df-dm 5618  df-rn 5619  df-res 5620  df-ima 5621  df-pred 6225  df-ord 6292  df-on 6293  df-lim 6294  df-suc 6295  df-iota 6418  df-fun 6468  df-fn 6469  df-f 6470  df-f1 6471  df-fo 6472  df-f1o 6473  df-fv 6474  df-riota 7274  df-ov 7320  df-oprab 7321  df-mpo 7322  df-om 7760  df-1st 7878  df-2nd 7879  df-frecs 8146  df-wrecs 8177  df-recs 8251  df-rdg 8290  df-1o 8346  df-er 8548  df-en 8784  df-dom 8785  df-sdom 8786  df-fin 8787  df-pnf 11091  df-mnf 11092  df-xr 11093  df-ltxr 11094  df-le 11095  df-sub 11287  df-neg 11288  df-nn 12054  df-2 12116  df-3 12117  df-4 12118  df-5 12119  df-6 12120  df-7 12121  df-8 12122  df-9 12123  df-n0 12314  df-z 12400  df-dec 12518  df-uz 12663  df-fz 13320  df-struct 16925  df-sets 16942  df-slot 16960  df-ndx 16972  df-base 16990  df-tset 17058  df-unif 17062  df-rest 17210  df-topn 17211  df-ust 23435  df-utop 23466  df-tus 23493

This theorem is referenced by: (None)