Theorem sge0hsphoire 40803

Description: If the generalized sum of dimensional volumes of n-dimensional half-open intervals is finite, then the sum stays finite if every half-open interval is intersected with a half-space. (Contributed by Glauco Siliprandi, 21-Nov-2020.)

Hypotheses

Ref	Expression
sge0hsphoire.l	|- L = ( x e. Fin |-> ( a e. ( RR ^m x ) , b e. ( RR ^m x ) |-> if ( x = (/) , 0 , prod_ k e. x ( vol ` ( ( a ` k ) [,) ( b ` k ) ) ) ) ) )
sge0hsphoire.f	|- ( ph -> Y e. Fin )
sge0hsphoire.z	|- ( ph -> Z e. ( W \ Y ) )
sge0hsphoire.w	|- W = ( Y u. { Z } )
sge0hsphoire.c	|- ( ph -> C : NN --> ( RR ^m W ) )
sge0hsphoire.d	|- ( ph -> D : NN --> ( RR ^m W ) )
sge0hsphoire.r	|- ( ph -> (Σ^ ` ( j e. NN |-> ( ( C ` j ) ( L ` W ) ( D ` j ) ) ) ) e. RR )
sge0hsphoire.h	|- H = ( x e. RR |-> ( c e. ( RR ^m W ) |-> ( j e. W |-> if ( j e. Y , ( c ` j ) , if ( ( c ` j ) <_ x , ( c ` j ) , x ) ) ) ) )
sge0hsphoire.s	|- ( ph -> S e. RR )

Assertion

Ref	Expression
sge0hsphoire	|- ( ph -> (Σ^ ` ( j e. NN |-> ( ( C ` j ) ( L ` W ) ( ( H ` S ) ` ( D ` j ) ) ) ) ) e. RR )

Distinct variable groups:   x, k   C, a, b, k   D, a, b, k   D, c, k   H, a, b, k   S, a, b, k, x   S, c, x   W, a, b, j, k, x   W, c, j   Y, c, j, x   Z, c, k, x   ph, a, b, j, k, x   ph, c

Allowed substitution hints:   C( x, j, c)   D( x, j)   S( j)   H( x, j, c)   L( x, j, k, a, b, c)   Y( k, a, b)   Z( j, a, b)

Proof of Theorem sge0hsphoire

Dummy variable h is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nnex 11026	. . . 4 |- NN e. _V
2	1	a1i 11	. . 3 |- ( ph -> NN e. _V )
3		sge0hsphoire.l	. . . . . 6 |- L = ( x e. Fin |-> ( a e. ( RR ^m x ) , b e. ( RR ^m x ) |-> if ( x = (/) , 0 , prod_ k e. x ( vol ` ( ( a ` k ) [,) ( b ` k ) ) ) ) ) )
4		sge0hsphoire.w	. . . . . . . 8 |- W = ( Y u. { Z } )
5		sge0hsphoire.f	. . . . . . . . 9 |- ( ph -> Y e. Fin )
6		snfi 8038	. . . . . . . . . 10 |- { Z } e. Fin
7	6	a1i 11	. . . . . . . . 9 |- ( ph -> { Z } e. Fin )
8		unfi 8227	. . . . . . . . 9 |- ( ( Y e. Fin /\ { Z } e. Fin ) -> ( Y u. { Z } ) e. Fin )
9	5, 7, 8	syl2anc 693	. . . . . . . 8 |- ( ph -> ( Y u. { Z } ) e. Fin )
10	4, 9	syl5eqel 2705	. . . . . . 7 |- ( ph -> W e. Fin )
11	10	adantr 481	. . . . . 6 |- ( ( ph /\ j e. NN ) -> W e. Fin )
12		sge0hsphoire.c	. . . . . . . 8 |- ( ph -> C : NN --> ( RR ^m W ) )
13	12	ffvelrnda 6359	. . . . . . 7 |- ( ( ph /\ j e. NN ) -> ( C ` j ) e. ( RR ^m W ) )
14		elmapi 7879	. . . . . . 7 |- ( ( C ` j ) e. ( RR ^m W ) -> ( C ` j ) : W --> RR )
15	13, 14	syl 17	. . . . . 6 |- ( ( ph /\ j e. NN ) -> ( C ` j ) : W --> RR )
16		sge0hsphoire.h	. . . . . . . 8 |- H = ( x e. RR |-> ( c e. ( RR ^m W ) |-> ( j e. W |-> if ( j e. Y , ( c ` j ) , if ( ( c ` j ) <_ x , ( c ` j ) , x ) ) ) ) )
17		eleq1 2689	. . . . . . . . . . . 12 |- ( j = h -> ( j e. Y <-> h e. Y ) )
18		fveq2 6191	. . . . . . . . . . . 12 |- ( j = h -> ( c ` j ) = ( c ` h ) )
19	18	breq1d 4663	. . . . . . . . . . . . 13 |- ( j = h -> ( ( c ` j ) <_ x <-> ( c ` h ) <_ x ) )
20	19, 18	ifbieq1d 4109	. . . . . . . . . . . 12 |- ( j = h -> if ( ( c ` j ) <_ x , ( c ` j ) , x ) = if ( ( c ` h ) <_ x , ( c ` h ) , x ) )
21	17, 18, 20	ifbieq12d 4113	. . . . . . . . . . 11 |- ( j = h -> if ( j e. Y , ( c ` j ) , if ( ( c ` j ) <_ x , ( c ` j ) , x ) ) = if ( h e. Y , ( c ` h ) , if ( ( c ` h ) <_ x , ( c ` h ) , x ) ) )
22	21	cbvmptv 4750	. . . . . . . . . 10 |- ( j e. W |-> if ( j e. Y , ( c ` j ) , if ( ( c ` j ) <_ x , ( c ` j ) , x ) ) ) = ( h e. W |-> if ( h e. Y , ( c ` h ) , if ( ( c ` h ) <_ x , ( c ` h ) , x ) ) )
23	22	mpteq2i 4741	. . . . . . . . 9 |- ( c e. ( RR ^m W ) |-> ( j e. W |-> if ( j e. Y , ( c ` j ) , if ( ( c ` j ) <_ x , ( c ` j ) , x ) ) ) ) = ( c e. ( RR ^m W ) |-> ( h e. W |-> if ( h e. Y , ( c ` h ) , if ( ( c ` h ) <_ x , ( c ` h ) , x ) ) ) )
24	23	mpteq2i 4741	. . . . . . . 8 |- ( x e. RR |-> ( c e. ( RR ^m W ) |-> ( j e. W |-> if ( j e. Y , ( c ` j ) , if ( ( c ` j ) <_ x , ( c ` j ) , x ) ) ) ) ) = ( x e. RR |-> ( c e. ( RR ^m W ) |-> ( h e. W |-> if ( h e. Y , ( c ` h ) , if ( ( c ` h ) <_ x , ( c ` h ) , x ) ) ) ) )
25	16, 24	eqtri 2644	. . . . . . 7 |- H = ( x e. RR |-> ( c e. ( RR ^m W ) |-> ( h e. W |-> if ( h e. Y , ( c ` h ) , if ( ( c ` h ) <_ x , ( c ` h ) , x ) ) ) ) )
26		sge0hsphoire.s	. . . . . . . 8 |- ( ph -> S e. RR )
27	26	adantr 481	. . . . . . 7 |- ( ( ph /\ j e. NN ) -> S e. RR )
28		sge0hsphoire.d	. . . . . . . . 9 |- ( ph -> D : NN --> ( RR ^m W ) )
29	28	ffvelrnda 6359	. . . . . . . 8 |- ( ( ph /\ j e. NN ) -> ( D ` j ) e. ( RR ^m W ) )
30		elmapi 7879	. . . . . . . 8 |- ( ( D ` j ) e. ( RR ^m W ) -> ( D ` j ) : W --> RR )
31	29, 30	syl 17	. . . . . . 7 |- ( ( ph /\ j e. NN ) -> ( D ` j ) : W --> RR )
32	25, 27, 11, 31	hsphoif 40790	. . . . . 6 |- ( ( ph /\ j e. NN ) -> ( ( H ` S ) ` ( D ` j ) ) : W --> RR )
33	3, 11, 15, 32	hoidmvcl 40796	. . . . 5 |- ( ( ph /\ j e. NN ) -> ( ( C ` j ) ( L ` W ) ( ( H ` S ) ` ( D ` j ) ) ) e. ( 0 [,) +oo ) )
34		eqid 2622	. . . . 5 |- ( j e. NN |-> ( ( C ` j ) ( L ` W ) ( ( H ` S ) ` ( D ` j ) ) ) ) = ( j e. NN |-> ( ( C ` j ) ( L ` W ) ( ( H ` S ) ` ( D ` j ) ) ) )
35	33, 34	fmptd 6385	. . . 4 |- ( ph -> ( j e. NN |-> ( ( C ` j ) ( L ` W ) ( ( H ` S ) ` ( D ` j ) ) ) ) : NN --> ( 0 [,) +oo ) )
36		icossicc 12260	. . . . 5 |- ( 0 [,) +oo ) C_ ( 0 [,] +oo )
37	36	a1i 11	. . . 4 |- ( ph -> ( 0 [,) +oo ) C_ ( 0 [,] +oo ) )
38	35, 37	fssd 6057	. . 3 |- ( ph -> ( j e. NN |-> ( ( C ` j ) ( L ` W ) ( ( H ` S ) ` ( D ` j ) ) ) ) : NN --> ( 0 [,] +oo ) )
39	2, 38	sge0cl 40598	. 2 |- ( ph -> (Σ^ ` ( j e. NN |-> ( ( C ` j ) ( L ` W ) ( ( H ` S ) ` ( D ` j ) ) ) ) ) e. ( 0 [,] +oo ) )
40	2, 38	sge0xrcl 40602	. . 3 |- ( ph -> (Σ^ ` ( j e. NN |-> ( ( C ` j ) ( L ` W ) ( ( H ` S ) ` ( D ` j ) ) ) ) ) e. RR* )
41		pnfxr 10092	. . . 4 |- +oo e. RR*
42	41	a1i 11	. . 3 |- ( ph -> +oo e. RR* )
43		sge0hsphoire.r	. . . . 5 |- ( ph -> (Σ^ ` ( j e. NN |-> ( ( C ` j ) ( L ` W ) ( D ` j ) ) ) ) e. RR )
44	43	rexrd 10089	. . . 4 |- ( ph -> (Σ^ ` ( j e. NN |-> ( ( C ` j ) ( L ` W ) ( D ` j ) ) ) ) e. RR* )
45		nfv 1843	. . . . 5 |- F/ j ph
46	36, 33	sseldi 3601	. . . . 5 |- ( ( ph /\ j e. NN ) -> ( ( C ` j ) ( L ` W ) ( ( H ` S ) ` ( D ` j ) ) ) e. ( 0 [,] +oo ) )
47	3, 11, 15, 31	hoidmvcl 40796	. . . . . 6 |- ( ( ph /\ j e. NN ) -> ( ( C ` j ) ( L ` W ) ( D ` j ) ) e. ( 0 [,) +oo ) )
48	36, 47	sseldi 3601	. . . . 5 |- ( ( ph /\ j e. NN ) -> ( ( C ` j ) ( L ` W ) ( D ` j ) ) e. ( 0 [,] +oo ) )
49		sge0hsphoire.z	. . . . . . 7 |- ( ph -> Z e. ( W \ Y ) )
50	49	adantr 481	. . . . . 6 |- ( ( ph /\ j e. NN ) -> Z e. ( W \ Y ) )
51	3, 11, 50, 4, 27, 25, 15, 31	hsphoidmvle 40800	. . . . 5 |- ( ( ph /\ j e. NN ) -> ( ( C ` j ) ( L ` W ) ( ( H ` S ) ` ( D ` j ) ) ) <_ ( ( C ` j ) ( L ` W ) ( D ` j ) ) )
52	45, 2, 46, 48, 51	sge0lempt 40627	. . . 4 |- ( ph -> (Σ^ ` ( j e. NN |-> ( ( C ` j ) ( L ` W ) ( ( H ` S ) ` ( D ` j ) ) ) ) ) <_ (Σ^ ` ( j e. NN |-> ( ( C ` j ) ( L ` W ) ( D ` j ) ) ) ) )
53	43	ltpnfd 11955	. . . 4 |- ( ph -> (Σ^ ` ( j e. NN |-> ( ( C ` j ) ( L ` W ) ( D ` j ) ) ) ) < +oo )
54	40, 44, 42, 52, 53	xrlelttrd 11991	. . 3 |- ( ph -> (Σ^ ` ( j e. NN |-> ( ( C ` j ) ( L ` W ) ( ( H ` S ) ` ( D ` j ) ) ) ) ) < +oo )
55	40, 42, 54	xrltned 39573	. 2 |- ( ph -> (Σ^ ` ( j e. NN |-> ( ( C ` j ) ( L ` W ) ( ( H ` S ) ` ( D ` j ) ) ) ) ) =/= +oo )
56		ge0xrre 39758	. 2 |- ( ( (Σ^ ` ( j e. NN |-> ( ( C ` j ) ( L ` W ) ( ( H ` S ) ` ( D ` j ) ) ) ) ) e. ( 0 [,] +oo ) /\ (Σ^ ` ( j e. NN |-> ( ( C ` j ) ( L ` W ) ( ( H ` S ) ` ( D ` j ) ) ) ) ) =/= +oo ) -> (Σ^ ` ( j e. NN |-> ( ( C ` j ) ( L ` W ) ( ( H ` S ) ` ( D ` j ) ) ) ) ) e. RR )
57	39, 55, 56	syl2anc 693	1 |- ( ph -> (Σ^ ` ( j e. NN |-> ( ( C ` j ) ( L ` W ) ( ( H ` S ) ` ( D ` j ) ) ) ) ) e. RR )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   \ cdif 3571   u. cun 3572   C_ wss 3574   (/)c0 3915   ifcif 4086   {csn 4177   class class class wbr 4653   |-> cmpt 4729   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   ^m cmap 7857   Fincfn 7955   RRcr 9935   0cc0 9936   +oocpnf 10071   RR*cxr 10073   <_ cle 10075   NNcn 11020   [,)cico 12177   [,]cicc 12178   prod_cprod 14635   volcvol 23232  Σ^{^}csumge0 40579

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-inf2 8538  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-fal 1489  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-se 5074  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-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  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-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-rlim 14220  df-sum 14417  df-prod 14636  df-rest 16083  df-topgen 16104  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-top 20699  df-topon 20716  df-bases 20750  df-cmp 21190  df-ovol 23233  df-vol 23234  df-sumge0 40580

This theorem is referenced by:  hoidmvlelem1  40809  hoidmvlelem2  40810