Theorem iinhoiicc 40888

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

Hypotheses

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

Assertion

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

Dummy variables f m are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 6658	. . . . . . . . . . . 12 |- ( n = m -> ( 1 / n ) = ( 1 / m ) )
2	1	oveq2d 6666	. . . . . . . . . . 11 |- ( n = m -> ( B + ( 1 / n ) ) = ( B + ( 1 / m ) ) )
3	2	oveq2d 6666	. . . . . . . . . 10 |- ( n = m -> ( A [,) ( B + ( 1 / n ) ) ) = ( A [,) ( B + ( 1 / m ) ) ) )
4	3	ixpeq2dv 7924	. . . . . . . . 9 |- ( n = m -> X_ k e. X ( A [,) ( B + ( 1 / n ) ) ) = X_ k e. X ( A [,) ( B + ( 1 / m ) ) ) )
5	4	cbviinv 4560	. . . . . . . 8 |- |^|_ n e. NN X_ k e. X ( A [,) ( B + ( 1 / n ) ) ) = |^|_ m e. NN X_ k e. X ( A [,) ( B + ( 1 / m ) ) )
6	5	eleq2i 2693	. . . . . . 7 |- ( f e. |^|_ n e. NN X_ k e. X ( A [,) ( B + ( 1 / n ) ) ) <-> f e. |^|_ m e. NN X_ k e. X ( A [,) ( B + ( 1 / m ) ) ) )
7	6	biimpi 206	. . . . . 6 |- ( f e. |^|_ n e. NN X_ k e. X ( A [,) ( B + ( 1 / n ) ) ) -> f e. |^|_ m e. NN X_ k e. X ( A [,) ( B + ( 1 / m ) ) ) )
8	7	adantl 482	. . . . 5 |- ( ( ph /\ f e. |^|_ n e. NN X_ k e. X ( A [,) ( B + ( 1 / n ) ) ) ) -> f e. |^|_ m e. NN X_ k e. X ( A [,) ( B + ( 1 / m ) ) ) )
9		iunhoiicc.k	. . . . . . 7 |- F/ k ph
10		nfcv 2764	. . . . . . . 8 |- F/_ k f
11		nfcv 2764	. . . . . . . . 9 |- F/_ k NN
12		nfixp1 7928	. . . . . . . . 9 |- F/_ k X_ k e. X ( A [,) ( B + ( 1 / m ) ) )
13	11, 12	nfiin 4549	. . . . . . . 8 |- F/_ k |^|_ m e. NN X_ k e. X ( A [,) ( B + ( 1 / m ) ) )
14	10, 13	nfel 2777	. . . . . . 7 |- F/ k f e. |^|_ m e. NN X_ k e. X ( A [,) ( B + ( 1 / m ) ) )
15	9, 14	nfan 1828	. . . . . 6 |- F/ k ( ph /\ f e. |^|_ m e. NN X_ k e. X ( A [,) ( B + ( 1 / m ) ) ) )
16		iunhoiicc.a	. . . . . . 7 |- ( ( ph /\ k e. X ) -> A e. RR )
17	16	adantlr 751	. . . . . 6 |- ( ( ( ph /\ f e. |^|_ m e. NN X_ k e. X ( A [,) ( B + ( 1 / m ) ) ) ) /\ k e. X ) -> A e. RR )
18		iunhoiicc.b	. . . . . . 7 |- ( ( ph /\ k e. X ) -> B e. RR )
19	18	adantlr 751	. . . . . 6 |- ( ( ( ph /\ f e. |^|_ m e. NN X_ k e. X ( A [,) ( B + ( 1 / m ) ) ) ) /\ k e. X ) -> B e. RR )
20	6	biimpri 218	. . . . . . 7 |- ( f e. |^|_ m e. NN X_ k e. X ( A [,) ( B + ( 1 / m ) ) ) -> f e. |^|_ n e. NN X_ k e. X ( A [,) ( B + ( 1 / n ) ) ) )
21	20	adantl 482	. . . . . 6 |- ( ( ph /\ f e. |^|_ m e. NN X_ k e. X ( A [,) ( B + ( 1 / m ) ) ) ) -> f e. |^|_ n e. NN X_ k e. X ( A [,) ( B + ( 1 / n ) ) ) )
22	15, 17, 19, 21	iinhoiicclem 40887	. . . . 5 |- ( ( ph /\ f e. |^|_ m e. NN X_ k e. X ( A [,) ( B + ( 1 / m ) ) ) ) -> f e. X_ k e. X ( A [,] B ) )
23	8, 22	syldan 487	. . . 4 |- ( ( ph /\ f e. |^|_ n e. NN X_ k e. X ( A [,) ( B + ( 1 / n ) ) ) ) -> f e. X_ k e. X ( A [,] B ) )
24	23	ralrimiva 2966	. . 3 |- ( ph -> A. f e. |^|_ n e. NN X_ k e. X ( A [,) ( B + ( 1 / n ) ) ) f e. X_ k e. X ( A [,] B ) )
25		dfss3 3592	. . 3 |- ( |^|_ n e. NN X_ k e. X ( A [,) ( B + ( 1 / n ) ) ) C_ X_ k e. X ( A [,] B ) <-> A. f e. |^|_ n e. NN X_ k e. X ( A [,) ( B + ( 1 / n ) ) ) f e. X_ k e. X ( A [,] B ) )
26	24, 25	sylibr 224	. 2 |- ( ph -> |^|_ n e. NN X_ k e. X ( A [,) ( B + ( 1 / n ) ) ) C_ X_ k e. X ( A [,] B ) )
27		nfv 1843	. . . . . 6 |- F/ k n e. NN
28	9, 27	nfan 1828	. . . . 5 |- F/ k ( ph /\ n e. NN )
29	16	rexrd 10089	. . . . . . 7 |- ( ( ph /\ k e. X ) -> A e. RR* )
30	29	adantlr 751	. . . . . 6 |- ( ( ( ph /\ n e. NN ) /\ k e. X ) -> A e. RR* )
31	18	adantlr 751	. . . . . . . 8 |- ( ( ( ph /\ n e. NN ) /\ k e. X ) -> B e. RR )
32		nnrp 11842	. . . . . . . . . . 11 |- ( n e. NN -> n e. RR+ )
33	32	ad2antlr 763	. . . . . . . . . 10 |- ( ( ( ph /\ n e. NN ) /\ k e. X ) -> n e. RR+ )
34	33	rpreccld 11882	. . . . . . . . 9 |- ( ( ( ph /\ n e. NN ) /\ k e. X ) -> ( 1 / n ) e. RR+ )
35	34	rpred 11872	. . . . . . . 8 |- ( ( ( ph /\ n e. NN ) /\ k e. X ) -> ( 1 / n ) e. RR )
36	31, 35	readdcld 10069	. . . . . . 7 |- ( ( ( ph /\ n e. NN ) /\ k e. X ) -> ( B + ( 1 / n ) ) e. RR )
37	36	rexrd 10089	. . . . . 6 |- ( ( ( ph /\ n e. NN ) /\ k e. X ) -> ( B + ( 1 / n ) ) e. RR* )
38	16	adantlr 751	. . . . . . 7 |- ( ( ( ph /\ n e. NN ) /\ k e. X ) -> A e. RR )
39	38	leidd 10594	. . . . . 6 |- ( ( ( ph /\ n e. NN ) /\ k e. X ) -> A <_ A )
40	31, 34	ltaddrpd 11905	. . . . . 6 |- ( ( ( ph /\ n e. NN ) /\ k e. X ) -> B < ( B + ( 1 / n ) ) )
41		iccssico 12245	. . . . . 6 |- ( ( ( A e. RR* /\ ( B + ( 1 / n ) ) e. RR* ) /\ ( A <_ A /\ B < ( B + ( 1 / n ) ) ) ) -> ( A [,] B ) C_ ( A [,) ( B + ( 1 / n ) ) ) )
42	30, 37, 39, 40, 41	syl22anc 1327	. . . . 5 |- ( ( ( ph /\ n e. NN ) /\ k e. X ) -> ( A [,] B ) C_ ( A [,) ( B + ( 1 / n ) ) ) )
43	28, 42	ixpssixp 39269	. . . 4 |- ( ( ph /\ n e. NN ) -> X_ k e. X ( A [,] B ) C_ X_ k e. X ( A [,) ( B + ( 1 / n ) ) ) )
44	43	ralrimiva 2966	. . 3 |- ( ph -> A. n e. NN X_ k e. X ( A [,] B ) C_ X_ k e. X ( A [,) ( B + ( 1 / n ) ) ) )
45		ssiin 4570	. . 3 |- ( X_ k e. X ( A [,] B ) C_ |^|_ n e. NN X_ k e. X ( A [,) ( B + ( 1 / n ) ) ) <-> A. n e. NN X_ k e. X ( A [,] B ) C_ X_ k e. X ( A [,) ( B + ( 1 / n ) ) ) )
46	44, 45	sylibr 224	. 2 |- ( ph -> X_ k e. X ( A [,] B ) C_ |^|_ n e. NN X_ k e. X ( A [,) ( B + ( 1 / n ) ) ) )
47	26, 46	eqssd 3620	1 |- ( ph -> |^|_ n e. NN X_ k e. X ( A [,) ( B + ( 1 / n ) ) ) = X_ k e. X ( A [,] B ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   F/wnf 1708   e. wcel 1990   A.wral 2912   C_ wss 3574   |^|_ciin 4521   class class class wbr 4653  (class class class)co 6650   X_cixp 7908   RRcr 9935   1c1 9937   + caddc 9939   RR*cxr 10073   < clt 10074   <_ cle 10075   / cdiv 10684   NNcn 11020   RR+crp 11832   [,)cico 12177   [,]cicc 12178

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-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-iun 4522  df-iin 4523  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-er 7742  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  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-q 11789  df-rp 11833  df-ico 12181  df-icc 12182  df-fl 12593

This theorem is referenced by:  vonicclem2  40898