Theorem iunhoiioo 40890

Description: A n-dimensional open interval expressed as the indexed union of half-open intervals. (Contributed by Glauco Siliprandi, 8-Apr-2021.)

Hypotheses

Ref	Expression
iunhoiioo.k	|- F/ k ph
iunhoiioo.x	|- ( ph -> X e. Fin )
iunhoiioo.a	|- ( ( ph /\ k e. X ) -> A e. RR )
iunhoiioo.b	|- ( ( ph /\ k e. X ) -> B e. RR* )

Assertion

Ref	Expression
iunhoiioo	|- ( ph -> U_ n e. NN X_ k e. X ( ( A + ( 1 / n ) ) [,) B ) = X_ k e. X ( A (,) B ) )

Distinct variable groups:   A, n   B, n   k, X, n   ph, n

Allowed substitution hints:   ph( k)   A( k)   B( k)

Proof of Theorem iunhoiioo

Dummy variable f is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nnn0 39595	. . . . . 6 |- NN =/= (/)
2		iunconst 4529	. . . . . 6 |- ( NN =/= (/) -> U_ n e. NN { (/) } = { (/) } )
3	1, 2	ax-mp 5	. . . . 5 |- U_ n e. NN { (/) } = { (/) }
4	3	a1i 11	. . . 4 |- ( X = (/) -> U_ n e. NN { (/) } = { (/) } )
5		ixpeq1 7919	. . . . . . 7 |- ( X = (/) -> X_ k e. X ( ( A + ( 1 / n ) ) [,) B ) = X_ k e. (/) ( ( A + ( 1 / n ) ) [,) B ) )
6		ixp0x 7936	. . . . . . . 8 |- X_ k e. (/) ( ( A + ( 1 / n ) ) [,) B ) = { (/) }
7	6	a1i 11	. . . . . . 7 |- ( X = (/) -> X_ k e. (/) ( ( A + ( 1 / n ) ) [,) B ) = { (/) } )
8	5, 7	eqtrd 2656	. . . . . 6 |- ( X = (/) -> X_ k e. X ( ( A + ( 1 / n ) ) [,) B ) = { (/) } )
9	8	adantr 481	. . . . 5 |- ( ( X = (/) /\ n e. NN ) -> X_ k e. X ( ( A + ( 1 / n ) ) [,) B ) = { (/) } )
10	9	iuneq2dv 4542	. . . 4 |- ( X = (/) -> U_ n e. NN X_ k e. X ( ( A + ( 1 / n ) ) [,) B ) = U_ n e. NN { (/) } )
11		ixpeq1 7919	. . . . 5 |- ( X = (/) -> X_ k e. X ( A (,) B ) = X_ k e. (/) ( A (,) B ) )
12		ixp0x 7936	. . . . . 6 |- X_ k e. (/) ( A (,) B ) = { (/) }
13	12	a1i 11	. . . . 5 |- ( X = (/) -> X_ k e. (/) ( A (,) B ) = { (/) } )
14	11, 13	eqtrd 2656	. . . 4 |- ( X = (/) -> X_ k e. X ( A (,) B ) = { (/) } )
15	4, 10, 14	3eqtr4d 2666	. . 3 |- ( X = (/) -> U_ n e. NN X_ k e. X ( ( A + ( 1 / n ) ) [,) B ) = X_ k e. X ( A (,) B ) )
16	15	adantl 482	. 2 |- ( ( ph /\ X = (/) ) -> U_ n e. NN X_ k e. X ( ( A + ( 1 / n ) ) [,) B ) = X_ k e. X ( A (,) B ) )
17		iunhoiioo.k	. . . . . . . 8 |- F/ k ph
18		nfv 1843	. . . . . . . 8 |- F/ k n e. NN
19	17, 18	nfan 1828	. . . . . . 7 |- F/ k ( ph /\ n e. NN )
20		iunhoiioo.a	. . . . . . . . . 10 |- ( ( ph /\ k e. X ) -> A e. RR )
21	20	rexrd 10089	. . . . . . . . 9 |- ( ( ph /\ k e. X ) -> A e. RR* )
22	21	adantlr 751	. . . . . . . 8 |- ( ( ( ph /\ n e. NN ) /\ k e. X ) -> A e. RR* )
23		iunhoiioo.b	. . . . . . . . 9 |- ( ( ph /\ k e. X ) -> B e. RR* )
24	23	adantlr 751	. . . . . . . 8 |- ( ( ( ph /\ n e. NN ) /\ k e. X ) -> B e. RR* )
25	20	adantlr 751	. . . . . . . . 9 |- ( ( ( ph /\ n e. NN ) /\ k e. X ) -> A e. RR )
26		nnrp 11842	. . . . . . . . . . 11 |- ( n e. NN -> n e. RR+ )
27	26	ad2antlr 763	. . . . . . . . . 10 |- ( ( ( ph /\ n e. NN ) /\ k e. X ) -> n e. RR+ )
28	27	rpreccld 11882	. . . . . . . . 9 |- ( ( ( ph /\ n e. NN ) /\ k e. X ) -> ( 1 / n ) e. RR+ )
29	25, 28	ltaddrpd 11905	. . . . . . . 8 |- ( ( ( ph /\ n e. NN ) /\ k e. X ) -> A < ( A + ( 1 / n ) ) )
30		xrleid 11983	. . . . . . . . . 10 |- ( B e. RR* -> B <_ B )
31	23, 30	syl 17	. . . . . . . . 9 |- ( ( ph /\ k e. X ) -> B <_ B )
32	31	adantlr 751	. . . . . . . 8 |- ( ( ( ph /\ n e. NN ) /\ k e. X ) -> B <_ B )
33		icossioo 12264	. . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( A < ( A + ( 1 / n ) ) /\ B <_ B ) ) -> ( ( A + ( 1 / n ) ) [,) B ) C_ ( A (,) B ) )
34	22, 24, 29, 32, 33	syl22anc 1327	. . . . . . 7 |- ( ( ( ph /\ n e. NN ) /\ k e. X ) -> ( ( A + ( 1 / n ) ) [,) B ) C_ ( A (,) B ) )
35	19, 34	ixpssixp 39269	. . . . . 6 |- ( ( ph /\ n e. NN ) -> X_ k e. X ( ( A + ( 1 / n ) ) [,) B ) C_ X_ k e. X ( A (,) B ) )
36	35	ralrimiva 2966	. . . . 5 |- ( ph -> A. n e. NN X_ k e. X ( ( A + ( 1 / n ) ) [,) B ) C_ X_ k e. X ( A (,) B ) )
37		iunss 4561	. . . . 5 |- ( U_ n e. NN X_ k e. X ( ( A + ( 1 / n ) ) [,) B ) C_ X_ k e. X ( A (,) B ) <-> A. n e. NN X_ k e. X ( ( A + ( 1 / n ) ) [,) B ) C_ X_ k e. X ( A (,) B ) )
38	36, 37	sylibr 224	. . . 4 |- ( ph -> U_ n e. NN X_ k e. X ( ( A + ( 1 / n ) ) [,) B ) C_ X_ k e. X ( A (,) B ) )
39	38	adantr 481	. . 3 |- ( ( ph /\ -. X = (/) ) -> U_ n e. NN X_ k e. X ( ( A + ( 1 / n ) ) [,) B ) C_ X_ k e. X ( A (,) B ) )
40		nfv 1843	. . . . . . . 8 |- F/ k -. X = (/)
41	17, 40	nfan 1828	. . . . . . 7 |- F/ k ( ph /\ -. X = (/) )
42		nfcv 2764	. . . . . . . 8 |- F/_ k f
43		nfixp1 7928	. . . . . . . 8 |- F/_ k X_ k e. X ( A (,) B )
44	42, 43	nfel 2777	. . . . . . 7 |- F/ k f e. X_ k e. X ( A (,) B )
45	41, 44	nfan 1828	. . . . . 6 |- F/ k ( ( ph /\ -. X = (/) ) /\ f e. X_ k e. X ( A (,) B ) )
46		iunhoiioo.x	. . . . . . 7 |- ( ph -> X e. Fin )
47	46	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ -. X = (/) ) /\ f e. X_ k e. X ( A (,) B ) ) -> X e. Fin )
48		neqne 2802	. . . . . . 7 |- ( -. X = (/) -> X =/= (/) )
49	48	ad2antlr 763	. . . . . 6 |- ( ( ( ph /\ -. X = (/) ) /\ f e. X_ k e. X ( A (,) B ) ) -> X =/= (/) )
50	20	ad4ant14 1293	. . . . . 6 |- ( ( ( ( ph /\ -. X = (/) ) /\ f e. X_ k e. X ( A (,) B ) ) /\ k e. X ) -> A e. RR )
51	23	ad4ant14 1293	. . . . . 6 |- ( ( ( ( ph /\ -. X = (/) ) /\ f e. X_ k e. X ( A (,) B ) ) /\ k e. X ) -> B e. RR* )
52		simpr 477	. . . . . 6 |- ( ( ( ph /\ -. X = (/) ) /\ f e. X_ k e. X ( A (,) B ) ) -> f e. X_ k e. X ( A (,) B ) )
53		eqid 2622	. . . . . 6 |- inf ( ran ( k e. X |-> ( ( f ` k ) - A ) ) , RR , < ) = inf ( ran ( k e. X |-> ( ( f ` k ) - A ) ) , RR , < )
54	45, 47, 49, 50, 51, 52, 53	iunhoiioolem 40889	. . . . 5 |- ( ( ( ph /\ -. X = (/) ) /\ f e. X_ k e. X ( A (,) B ) ) -> f e. U_ n e. NN X_ k e. X ( ( A + ( 1 / n ) ) [,) B ) )
55	54	ralrimiva 2966	. . . 4 |- ( ( ph /\ -. X = (/) ) -> A. f e. X_ k e. X ( A (,) B ) f e. U_ n e. NN X_ k e. X ( ( A + ( 1 / n ) ) [,) B ) )
56		dfss3 3592	. . . 4 |- ( X_ k e. X ( A (,) B ) C_ U_ n e. NN X_ k e. X ( ( A + ( 1 / n ) ) [,) B ) <-> A. f e. X_ k e. X ( A (,) B ) f e. U_ n e. NN X_ k e. X ( ( A + ( 1 / n ) ) [,) B ) )
57	55, 56	sylibr 224	. . 3 |- ( ( ph /\ -. X = (/) ) -> X_ k e. X ( A (,) B ) C_ U_ n e. NN X_ k e. X ( ( A + ( 1 / n ) ) [,) B ) )
58	39, 57	eqssd 3620	. 2 |- ( ( ph /\ -. X = (/) ) -> U_ n e. NN X_ k e. X ( ( A + ( 1 / n ) ) [,) B ) = X_ k e. X ( A (,) B ) )
59	16, 58	pm2.61dan 832	1 |- ( ph -> U_ n e. NN X_ k e. X ( ( A + ( 1 / n ) ) [,) B ) = X_ k e. X ( A (,) B ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   F/wnf 1708   e. wcel 1990   =/= wne 2794   A.wral 2912   C_ wss 3574   (/)c0 3915   {csn 4177   U_ciun 4520   class class class wbr 4653   |-> cmpt 4729   ran crn 5115   ` cfv 5888  (class class class)co 6650   X_cixp 7908   Fincfn 7955  infcinf 8347   RRcr 9935   1c1 9937   + caddc 9939   RR*cxr 10073   < clt 10074   <_ cle 10075   - cmin 10266   / cdiv 10684   NNcn 11020   RR+crp 11832   (,)cioo 12175   [,)cico 12177

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  ax-pre-sup 10014

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-rmo 2920  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-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-ioo 12179  df-ico 12181  df-fl 12593

This theorem is referenced by:  vonioolem2  40895