Theorem tuslem 22071

Description: Lemma for tusbas 22072, tusunif 22073, and tustopn 22075. (Contributed by Thierry Arnoux, 5-Dec-2017.)

Hypothesis

Ref	Expression
tuslem.k	|- K = (toUnifSp ` U )

Assertion

Ref	Expression
tuslem	|- ( U e. (UnifOn ` X ) -> ( X = ( Base ` K ) /\ U = ( UnifSet ` K ) /\ (unifTop ` U ) = ( TopOpen ` K ) ) )

Proof of Theorem tuslem

Step	Hyp	Ref	Expression
1		baseid 15919	. . . 4 |- Base = Slot ( Base ` ndx )
2		1re 10039	. . . . . 6 |- 1 e. RR
3		1lt9 11229	. . . . . 6 |- 1 < 9
4	2, 3	ltneii 10150	. . . . 5 |- 1 =/= 9
5		basendx 15923	. . . . . 6 |- ( Base ` ndx ) = 1
6		tsetndx 16040	. . . . . 6 |- (TopSet ` ndx ) = 9
7	5, 6	neeq12i 2860	. . . . 5 |- ( ( Base ` ndx ) =/= (TopSet ` ndx ) <-> 1 =/= 9 )
8	4, 7	mpbir 221	. . . 4 |- ( Base ` ndx ) =/= (TopSet ` ndx )
9	1, 8	setsnid 15915	. . 3 |- ( Base ` { <. ( Base ` ndx ) , dom U. U >. , <. ( UnifSet ` ndx ) , U >. } ) = ( Base ` ( { <. ( Base ` ndx ) , dom U. U >. , <. ( UnifSet ` ndx ) , U >. } sSet <. (TopSet ` ndx ) , (unifTop ` U ) >. ) )
10		ustbas2 22029	. . . 4 |- ( U e. (UnifOn ` X ) -> X = dom U. U )
11		uniexg 6955	. . . . 5 |- ( U e. (UnifOn ` X ) -> U. U e. _V )
12		dmexg 7097	. . . . 5 |- ( U. U e. _V -> dom U. U e. _V )
13		eqid 2622	. . . . . 6 |- { <. ( Base ` ndx ) , dom U. U >. , <. ( UnifSet ` ndx ) , U >. } = { <. ( Base ` ndx ) , dom U. U >. , <. ( UnifSet ` ndx ) , U >. }
14		df-unif 15965	. . . . . 6 |- UnifSet = Slot ; 1 3
15		1nn 11031	. . . . . . 7 |- 1 e. NN
16		3nn0 11310	. . . . . . 7 |- 3 e. NN0
17		1nn0 11308	. . . . . . 7 |- 1 e. NN0
18		1lt10 11681	. . . . . . 7 |- 1 < ; 1 0
19	15, 16, 17, 18	declti 11546	. . . . . 6 |- 1 < ; 1 3
20		3nn 11186	. . . . . . 7 |- 3 e. NN
21	17, 20	decnncl 11518	. . . . . 6 |- ; 1 3 e. NN
22	13, 14, 19, 21	2strbas 15984	. . . . 5 |- ( dom U. U e. _V -> dom U. U = ( Base ` { <. ( Base ` ndx ) , dom U. U >. , <. ( UnifSet ` ndx ) , U >. } ) )
23	11, 12, 22	3syl 18	. . . 4 |- ( U e. (UnifOn ` X ) -> dom U. U = ( Base ` { <. ( Base ` ndx ) , dom U. U >. , <. ( UnifSet ` ndx ) , U >. } ) )
24	10, 23	eqtrd 2656	. . 3 |- ( U e. (UnifOn ` X ) -> X = ( Base ` { <. ( Base ` ndx ) , dom U. U >. , <. ( UnifSet ` ndx ) , U >. } ) )
25		tuslem.k	. . . . 5 |- K = (toUnifSp ` U )
26		tusval 22070	. . . . 5 |- ( U e. (UnifOn ` X ) -> (toUnifSp ` U ) = ( { <. ( Base ` ndx ) , dom U. U >. , <. ( UnifSet ` ndx ) , U >. } sSet <. (TopSet ` ndx ) , (unifTop ` U ) >. ) )
27	25, 26	syl5eq 2668	. . . 4 |- ( U e. (UnifOn ` X ) -> K = ( { <. ( Base ` ndx ) , dom U. U >. , <. ( UnifSet ` ndx ) , U >. } sSet <. (TopSet ` ndx ) , (unifTop ` U ) >. ) )
28	27	fveq2d 6195	. . 3 |- ( U e. (UnifOn ` X ) -> ( Base ` K ) = ( Base ` ( { <. ( Base ` ndx ) , dom U. U >. , <. ( UnifSet ` ndx ) , U >. } sSet <. (TopSet ` ndx ) , (unifTop ` U ) >. ) ) )
29	9, 24, 28	3eqtr4a 2682	. 2 |- ( U e. (UnifOn ` X ) -> X = ( Base ` K ) )
30		unifid 16065	. . . 4 |- UnifSet = Slot ( UnifSet ` ndx )
31		9re 11107	. . . . . 6 |- 9 e. RR
32		9nn0 11316	. . . . . . 7 |- 9 e. NN0
33		9lt10 11673	. . . . . . 7 |- 9 < ; 1 0
34	15, 16, 32, 33	declti 11546	. . . . . 6 |- 9 < ; 1 3
35	31, 34	gtneii 10149	. . . . 5 |- ; 1 3 =/= 9
36		unifndx 16064	. . . . . 6 |- ( UnifSet ` ndx ) = ; 1 3
37	36, 6	neeq12i 2860	. . . . 5 |- ( ( UnifSet ` ndx ) =/= (TopSet ` ndx ) <-> ; 1 3 =/= 9 )
38	35, 37	mpbir 221	. . . 4 |- ( UnifSet ` ndx ) =/= (TopSet ` ndx )
39	30, 38	setsnid 15915	. . 3 |- ( UnifSet ` { <. ( Base ` ndx ) , dom U. U >. , <. ( UnifSet ` ndx ) , U >. } ) = ( UnifSet ` ( { <. ( Base ` ndx ) , dom U. U >. , <. ( UnifSet ` ndx ) , U >. } sSet <. (TopSet ` ndx ) , (unifTop ` U ) >. ) )
40	13, 14, 19, 21	2strop 15985	. . 3 |- ( U e. (UnifOn ` X ) -> U = ( UnifSet ` { <. ( Base ` ndx ) , dom U. U >. , <. ( UnifSet ` ndx ) , U >. } ) )
41	27	fveq2d 6195	. . 3 |- ( U e. (UnifOn ` X ) -> ( UnifSet ` K ) = ( UnifSet ` ( { <. ( Base ` ndx ) , dom U. U >. , <. ( UnifSet ` ndx ) , U >. } sSet <. (TopSet ` ndx ) , (unifTop ` U ) >. ) ) )
42	39, 40, 41	3eqtr4a 2682	. 2 |- ( U e. (UnifOn ` X ) -> U = ( UnifSet ` K ) )
43	27	fveq2d 6195	. . . 4 |- ( U e. (UnifOn ` X ) -> (TopSet ` K ) = (TopSet ` ( { <. ( Base ` ndx ) , dom U. U >. , <. ( UnifSet ` ndx ) , U >. } sSet <. (TopSet ` ndx ) , (unifTop ` U ) >. ) ) )
44		prex 4909	. . . . 5 |- { <. ( Base ` ndx ) , dom U. U >. , <. ( UnifSet ` ndx ) , U >. } e. _V
45		fvex 6201	. . . . 5 |- (unifTop ` U ) e. _V
46		tsetid 16041	. . . . . 6 |- TopSet = Slot (TopSet ` ndx )
47	46	setsid 15914	. . . . 5 |- ( ( { <. ( Base ` ndx ) , dom U. U >. , <. ( UnifSet ` ndx ) , U >. } e. _V /\ (unifTop ` U ) e. _V ) -> (unifTop ` U ) = (TopSet ` ( { <. ( Base ` ndx ) , dom U. U >. , <. ( UnifSet ` ndx ) , U >. } sSet <. (TopSet ` ndx ) , (unifTop ` U ) >. ) ) )
48	44, 45, 47	mp2an 708	. . . 4 |- (unifTop ` U ) = (TopSet ` ( { <. ( Base ` ndx ) , dom U. U >. , <. ( UnifSet ` ndx ) , U >. } sSet <. (TopSet ` ndx ) , (unifTop ` U ) >. ) )
49	43, 48	syl6reqr 2675	. . 3 |- ( U e. (UnifOn ` X ) -> (unifTop ` U ) = (TopSet ` K ) )
50		utopbas 22039	. . . . . 6 |- ( U e. (UnifOn ` X ) -> X = U. (unifTop ` U ) )
51	49	unieqd 4446	. . . . . 6 |- ( U e. (UnifOn ` X ) -> U. (unifTop ` U ) = U. (TopSet ` K ) )
52	50, 29, 51	3eqtr3rd 2665	. . . . 5 |- ( U e. (UnifOn ` X ) -> U. (TopSet ` K ) = ( Base ` K ) )
53	52	oveq2d 6666	. . . 4 |- ( U e. (UnifOn ` X ) -> ( (TopSet ` K ) ↾t U. (TopSet ` K ) ) = ( (TopSet ` K ) ↾t ( Base ` K ) ) )
54		fvex 6201	. . . . 5 |- (TopSet ` K ) e. _V
55		eqid 2622	. . . . . 6 |- U. (TopSet ` K ) = U. (TopSet ` K )
56	55	restid 16094	. . . . 5 |- ( (TopSet ` K ) e. _V -> ( (TopSet ` K ) ↾t U. (TopSet ` K ) ) = (TopSet ` K ) )
57	54, 56	ax-mp 5	. . . 4 |- ( (TopSet ` K ) ↾t U. (TopSet ` K ) ) = (TopSet ` K )
58		eqid 2622	. . . . 5 |- ( Base ` K ) = ( Base ` K )
59		eqid 2622	. . . . 5 |- (TopSet ` K ) = (TopSet ` K )
60	58, 59	topnval 16095	. . . 4 |- ( (TopSet ` K ) ↾t ( Base ` K ) ) = ( TopOpen ` K )
61	53, 57, 60	3eqtr3g 2679	. . 3 |- ( U e. (UnifOn ` X ) -> (TopSet ` K ) = ( TopOpen ` K ) )
62	49, 61	eqtrd 2656	. 2 |- ( U e. (UnifOn ` X ) -> (unifTop ` U ) = ( TopOpen ` K ) )
63	29, 42, 62	3jca 1242	1 |- ( U e. (UnifOn ` X ) -> ( X = ( Base ` K ) /\ U = ( UnifSet ` K ) /\ (unifTop ` U ) = ( TopOpen ` K ) ) )

Syntax hints:   -> wi 4   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   {cpr 4179   <.cop 4183   U.cuni 4436   dom cdm 5114   ` cfv 5888  (class class class)co 6650   1c1 9937   3c3 11071   9c9 11077  ;cdc 11493   ndxcnx 15854   sSet csts 15855   Basecbs 15857  TopSetcts 15947   UnifSetcunif 15951   ↾_t crest 16081   TopOpenctopn 16082  UnifOncust 22003  unifTopcutop 22034  toUnifSpctus 22059

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  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-nel 2898  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-tset 15960  df-unif 15965  df-rest 16083  df-topn 16084  df-ust 22004  df-utop 22035  df-tus 22062

This theorem is referenced by:  tusbas  22072  tusunif  22073  tustopn  22075  tususp  22076