Theorem vonhoire 40886

Description: The Lebesgue measure of a n-dimensional half-open interval is a real number. (Contributed by Glauco Siliprandi, 8-Apr-2021.)

Hypotheses

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

Assertion

Ref	Expression
vonhoire	|- ( ph -> ( (voln ` X ) ` X_ k e. X ( A [,) B ) ) e. RR )

Distinct variable group:   k, X

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

Proof of Theorem vonhoire

Dummy variable j is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . . . 6 |- ( X = (/) -> (voln ` X ) = (voln ` (/) ) )
2	1	fveq1d 6193	. . . . 5 |- ( X = (/) -> ( (voln ` X ) ` X_ k e. X ( A [,) B ) ) = ( (voln ` (/) ) ` X_ k e. X ( A [,) B ) ) )
3	2	adantl 482	. . . 4 |- ( ( ph /\ X = (/) ) -> ( (voln ` X ) ` X_ k e. X ( A [,) B ) ) = ( (voln ` (/) ) ` X_ k e. X ( A [,) B ) ) )
4		ixpeq1 7919	. . . . . . 7 |- ( X = (/) -> X_ k e. X ( A [,) B ) = X_ k e. (/) ( A [,) B ) )
5	4	adantl 482	. . . . . 6 |- ( ( ph /\ X = (/) ) -> X_ k e. X ( A [,) B ) = X_ k e. (/) ( A [,) B ) )
6		vonhoire.n	. . . . . . . 8 |- F/ k ph
7		0fin 8188	. . . . . . . . 9 |- (/) e. Fin
8	7	a1i 11	. . . . . . . 8 |- ( ph -> (/) e. Fin )
9		eqid 2622	. . . . . . . 8 |- dom (voln ` (/) ) = dom (voln ` (/) )
10		noel 3919	. . . . . . . . . 10 |- -. k e. (/)
11	10	pm2.21i 116	. . . . . . . . 9 |- ( k e. (/) -> A e. RR )
12	11	adantl 482	. . . . . . . 8 |- ( ( ph /\ k e. (/) ) -> A e. RR )
13	10	pm2.21i 116	. . . . . . . . 9 |- ( k e. (/) -> B e. RR )
14	13	adantl 482	. . . . . . . 8 |- ( ( ph /\ k e. (/) ) -> B e. RR )
15	6, 8, 9, 12, 14	hoimbl2 40879	. . . . . . 7 |- ( ph -> X_ k e. (/) ( A [,) B ) e. dom (voln ` (/) ) )
16	15	adantr 481	. . . . . 6 |- ( ( ph /\ X = (/) ) -> X_ k e. (/) ( A [,) B ) e. dom (voln ` (/) ) )
17	5, 16	eqeltrd 2701	. . . . 5 |- ( ( ph /\ X = (/) ) -> X_ k e. X ( A [,) B ) e. dom (voln ` (/) ) )
18	17	von0val 40885	. . . 4 |- ( ( ph /\ X = (/) ) -> ( (voln ` (/) ) ` X_ k e. X ( A [,) B ) ) = 0 )
19	3, 18	eqtrd 2656	. . 3 |- ( ( ph /\ X = (/) ) -> ( (voln ` X ) ` X_ k e. X ( A [,) B ) ) = 0 )
20		0red 10041	. . 3 |- ( ( ph /\ X = (/) ) -> 0 e. RR )
21	19, 20	eqeltrd 2701	. 2 |- ( ( ph /\ X = (/) ) -> ( (voln ` X ) ` X_ k e. X ( A [,) B ) ) e. RR )
22		neqne 2802	. . . 4 |- ( -. X = (/) -> X =/= (/) )
23	22	adantl 482	. . 3 |- ( ( ph /\ -. X = (/) ) -> X =/= (/) )
24		simpr 477	. . . . . . . . . . 11 |- ( ( ph /\ j e. X ) -> j e. X )
25		nfv 1843	. . . . . . . . . . . . . 14 |- F/ k j e. X
26	6, 25	nfan 1828	. . . . . . . . . . . . 13 |- F/ k ( ph /\ j e. X )
27		nfcv 2764	. . . . . . . . . . . . . . 15 |- F/_ k j
28	27	nfcsb1 3548	. . . . . . . . . . . . . 14 |- F/_ k [_ j / k ]_ A
29	28	nfel1 2779	. . . . . . . . . . . . 13 |- F/ k [_ j / k ]_ A e. RR
30	26, 29	nfim 1825	. . . . . . . . . . . 12 |- F/ k ( ( ph /\ j e. X ) -> [_ j / k ]_ A e. RR )
31		eleq1 2689	. . . . . . . . . . . . . 14 |- ( k = j -> ( k e. X <-> j e. X ) )
32	31	anbi2d 740	. . . . . . . . . . . . 13 |- ( k = j -> ( ( ph /\ k e. X ) <-> ( ph /\ j e. X ) ) )
33		csbeq1a 3542	. . . . . . . . . . . . . 14 |- ( k = j -> A = [_ j / k ]_ A )
34	33	eleq1d 2686	. . . . . . . . . . . . 13 |- ( k = j -> ( A e. RR <-> [_ j / k ]_ A e. RR ) )
35	32, 34	imbi12d 334	. . . . . . . . . . . 12 |- ( k = j -> ( ( ( ph /\ k e. X ) -> A e. RR ) <-> ( ( ph /\ j e. X ) -> [_ j / k ]_ A e. RR ) ) )
36		vonhoire.a	. . . . . . . . . . . 12 |- ( ( ph /\ k e. X ) -> A e. RR )
37	30, 35, 36	chvar 2262	. . . . . . . . . . 11 |- ( ( ph /\ j e. X ) -> [_ j / k ]_ A e. RR )
38		eqid 2622	. . . . . . . . . . . 12 |- ( k e. X |-> A ) = ( k e. X |-> A )
39	27, 28, 33, 38	fvmptf 6301	. . . . . . . . . . 11 |- ( ( j e. X /\ [_ j / k ]_ A e. RR ) -> ( ( k e. X |-> A ) ` j ) = [_ j / k ]_ A )
40	24, 37, 39	syl2anc 693	. . . . . . . . . 10 |- ( ( ph /\ j e. X ) -> ( ( k e. X |-> A ) ` j ) = [_ j / k ]_ A )
41	27	nfcsb1 3548	. . . . . . . . . . . . . 14 |- F/_ k [_ j / k ]_ B
42		nfcv 2764	. . . . . . . . . . . . . 14 |- F/_ k RR
43	41, 42	nfel 2777	. . . . . . . . . . . . 13 |- F/ k [_ j / k ]_ B e. RR
44	26, 43	nfim 1825	. . . . . . . . . . . 12 |- F/ k ( ( ph /\ j e. X ) -> [_ j / k ]_ B e. RR )
45		csbeq1a 3542	. . . . . . . . . . . . . 14 |- ( k = j -> B = [_ j / k ]_ B )
46	45	eleq1d 2686	. . . . . . . . . . . . 13 |- ( k = j -> ( B e. RR <-> [_ j / k ]_ B e. RR ) )
47	32, 46	imbi12d 334	. . . . . . . . . . . 12 |- ( k = j -> ( ( ( ph /\ k e. X ) -> B e. RR ) <-> ( ( ph /\ j e. X ) -> [_ j / k ]_ B e. RR ) ) )
48		vonhoire.b	. . . . . . . . . . . 12 |- ( ( ph /\ k e. X ) -> B e. RR )
49	44, 47, 48	chvar 2262	. . . . . . . . . . 11 |- ( ( ph /\ j e. X ) -> [_ j / k ]_ B e. RR )
50		eqid 2622	. . . . . . . . . . . 12 |- ( k e. X |-> B ) = ( k e. X |-> B )
51	27, 41, 45, 50	fvmptf 6301	. . . . . . . . . . 11 |- ( ( j e. X /\ [_ j / k ]_ B e. RR ) -> ( ( k e. X |-> B ) ` j ) = [_ j / k ]_ B )
52	24, 49, 51	syl2anc 693	. . . . . . . . . 10 |- ( ( ph /\ j e. X ) -> ( ( k e. X |-> B ) ` j ) = [_ j / k ]_ B )
53	40, 52	oveq12d 6668	. . . . . . . . 9 |- ( ( ph /\ j e. X ) -> ( ( ( k e. X |-> A ) ` j ) [,) ( ( k e. X |-> B ) ` j ) ) = ( [_ j / k ]_ A [,) [_ j / k ]_ B ) )
54	53	ixpeq2dva 7923	. . . . . . . 8 |- ( ph -> X_ j e. X ( ( ( k e. X |-> A ) ` j ) [,) ( ( k e. X |-> B ) ` j ) ) = X_ j e. X ( [_ j / k ]_ A [,) [_ j / k ]_ B ) )
55		nfcv 2764	. . . . . . . . . . 11 |- F/_ j ( A [,) B )
56		nfcv 2764	. . . . . . . . . . . 12 |- F/_ k [,)
57	28, 56, 41	nfov 6676	. . . . . . . . . . 11 |- F/_ k ( [_ j / k ]_ A [,) [_ j / k ]_ B )
58	33, 45	oveq12d 6668	. . . . . . . . . . 11 |- ( k = j -> ( A [,) B ) = ( [_ j / k ]_ A [,) [_ j / k ]_ B ) )
59	55, 57, 58	cbvixp 7925	. . . . . . . . . 10 |- X_ k e. X ( A [,) B ) = X_ j e. X ( [_ j / k ]_ A [,) [_ j / k ]_ B )
60	59	eqcomi 2631	. . . . . . . . 9 |- X_ j e. X ( [_ j / k ]_ A [,) [_ j / k ]_ B ) = X_ k e. X ( A [,) B )
61	60	a1i 11	. . . . . . . 8 |- ( ph -> X_ j e. X ( [_ j / k ]_ A [,) [_ j / k ]_ B ) = X_ k e. X ( A [,) B ) )
62	54, 61	eqtr2d 2657	. . . . . . 7 |- ( ph -> X_ k e. X ( A [,) B ) = X_ j e. X ( ( ( k e. X |-> A ) ` j ) [,) ( ( k e. X |-> B ) ` j ) ) )
63	62	fveq2d 6195	. . . . . 6 |- ( ph -> ( (voln ` X ) ` X_ k e. X ( A [,) B ) ) = ( (voln ` X ) ` X_ j e. X ( ( ( k e. X |-> A ) ` j ) [,) ( ( k e. X |-> B ) ` j ) ) ) )
64	63	adantr 481	. . . . 5 |- ( ( ph /\ X =/= (/) ) -> ( (voln ` X ) ` X_ k e. X ( A [,) B ) ) = ( (voln ` X ) ` X_ j e. X ( ( ( k e. X |-> A ) ` j ) [,) ( ( k e. X |-> B ) ` j ) ) ) )
65		vonhoire.x	. . . . . . 7 |- ( ph -> X e. Fin )
66	65	adantr 481	. . . . . 6 |- ( ( ph /\ X =/= (/) ) -> X e. Fin )
67		simpr 477	. . . . . 6 |- ( ( ph /\ X =/= (/) ) -> X =/= (/) )
68	6, 36, 38	fmptdf 6387	. . . . . . 7 |- ( ph -> ( k e. X |-> A ) : X --> RR )
69	68	adantr 481	. . . . . 6 |- ( ( ph /\ X =/= (/) ) -> ( k e. X |-> A ) : X --> RR )
70	6, 48, 50	fmptdf 6387	. . . . . . 7 |- ( ph -> ( k e. X |-> B ) : X --> RR )
71	70	adantr 481	. . . . . 6 |- ( ( ph /\ X =/= (/) ) -> ( k e. X |-> B ) : X --> RR )
72		eqid 2622	. . . . . 6 |- X_ j e. X ( ( ( k e. X |-> A ) ` j ) [,) ( ( k e. X |-> B ) ` j ) ) = X_ j e. X ( ( ( k e. X |-> A ) ` j ) [,) ( ( k e. X |-> B ) ` j ) )
73	66, 67, 69, 71, 72	vonn0hoi 40884	. . . . 5 |- ( ( ph /\ X =/= (/) ) -> ( (voln ` X ) ` X_ j e. X ( ( ( k e. X |-> A ) ` j ) [,) ( ( k e. X |-> B ) ` j ) ) ) = prod_ j e. X ( vol ` ( ( ( k e. X |-> A ) ` j ) [,) ( ( k e. X |-> B ) ` j ) ) ) )
74	64, 73	eqtrd 2656	. . . 4 |- ( ( ph /\ X =/= (/) ) -> ( (voln ` X ) ` X_ k e. X ( A [,) B ) ) = prod_ j e. X ( vol ` ( ( ( k e. X |-> A ) ` j ) [,) ( ( k e. X |-> B ) ` j ) ) ) )
75	40, 37	eqeltrd 2701	. . . . . . 7 |- ( ( ph /\ j e. X ) -> ( ( k e. X |-> A ) ` j ) e. RR )
76	52, 49	eqeltrd 2701	. . . . . . 7 |- ( ( ph /\ j e. X ) -> ( ( k e. X |-> B ) ` j ) e. RR )
77		volicore 40795	. . . . . . 7 |- ( ( ( ( k e. X |-> A ) ` j ) e. RR /\ ( ( k e. X |-> B ) ` j ) e. RR ) -> ( vol ` ( ( ( k e. X |-> A ) ` j ) [,) ( ( k e. X |-> B ) ` j ) ) ) e. RR )
78	75, 76, 77	syl2anc 693	. . . . . 6 |- ( ( ph /\ j e. X ) -> ( vol ` ( ( ( k e. X |-> A ) ` j ) [,) ( ( k e. X |-> B ) ` j ) ) ) e. RR )
79	65, 78	fprodrecl 14683	. . . . 5 |- ( ph -> prod_ j e. X ( vol ` ( ( ( k e. X |-> A ) ` j ) [,) ( ( k e. X |-> B ) ` j ) ) ) e. RR )
80	79	adantr 481	. . . 4 |- ( ( ph /\ X =/= (/) ) -> prod_ j e. X ( vol ` ( ( ( k e. X |-> A ) ` j ) [,) ( ( k e. X |-> B ) ` j ) ) ) e. RR )
81	74, 80	eqeltrd 2701	. . 3 |- ( ( ph /\ X =/= (/) ) -> ( (voln ` X ) ` X_ k e. X ( A [,) B ) ) e. RR )
82	23, 81	syldan 487	. 2 |- ( ( ph /\ -. X = (/) ) -> ( (voln ` X ) ` X_ k e. X ( A [,) B ) ) e. RR )
83	21, 82	pm2.61dan 832	1 |- ( ph -> ( (voln ` X ) ` X_ k e. X ( A [,) B ) ) e. RR )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   F/wnf 1708   e. wcel 1990   =/= wne 2794   [_csb 3533   (/)c0 3915   |-> cmpt 4729   dom cdm 5114   -->wf 5884   ` cfv 5888  (class class class)co 6650   X_cixp 7908   Fincfn 7955   RRcr 9935   0cc0 9936   [,)cico 12177   prod_cprod 14635   volcvol 23232  volncvoln 40752

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-cc 9257  ax-ac2 9285  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  ax-addf 10015  ax-mulf 10016

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-iin 4523  df-disj 4621  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-tpos 7352  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-omul 7565  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  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-acn 8768  df-ac 8939  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-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-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-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-rest 16083  df-0g 16102  df-topgen 16104  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-subg 17591  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-cring 18550  df-oppr 18623  df-dvdsr 18641  df-unit 18642  df-invr 18672  df-dvr 18683  df-drng 18749  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-cnfld 19747  df-top 20699  df-topon 20716  df-bases 20750  df-cmp 21190  df-ovol 23233  df-vol 23234  df-salg 40529  df-sumge0 40580  df-mea 40667  df-ome 40704  df-caragen 40706  df-ovoln 40751  df-voln 40753

This theorem is referenced by:  vonioolem2  40895  vonicclem2  40898