Theorem caratheodory 40742

Description: Caratheodory's construction of a measure given an outer measure. Proof of Theorem 113C of [Fremlin1] p. 19. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypotheses

Ref	Expression
caratheodory.o	|- ( ph -> O e. OutMeas )
caratheodory.s	|- S = (CaraGen ` O )

Assertion

Ref	Expression
caratheodory	|- ( ph -> ( O |` S ) e. Meas )

Proof of Theorem caratheodory

Dummy variables a e j n m are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		caratheodory.o	. . 3 |- ( ph -> O e. OutMeas )
2		caratheodory.s	. . 3 |- S = (CaraGen ` O )
3	1, 2	caragensal 40739	. 2 |- ( ph -> S e. SAlg )
4		eqid 2622	. . . 4 |- U. dom O = U. dom O
5	1, 4	omef 40710	. . 3 |- ( ph -> O : ~P U. dom O --> ( 0 [,] +oo ) )
6		caragenval 40707	. . . . . . 7 |- ( O e. OutMeas -> (CaraGen ` O ) = { e e. ~P U. dom O | A. a e. ~P U. dom O ( ( O ` ( a i^i e ) ) +e ( O ` ( a \ e ) ) ) = ( O ` a ) } )
7	1, 6	syl 17	. . . . . 6 |- ( ph -> (CaraGen ` O ) = { e e. ~P U. dom O | A. a e. ~P U. dom O ( ( O ` ( a i^i e ) ) +e ( O ` ( a \ e ) ) ) = ( O ` a ) } )
8	7	eqcomd 2628	. . . . 5 |- ( ph -> { e e. ~P U. dom O | A. a e. ~P U. dom O ( ( O ` ( a i^i e ) ) +e ( O ` ( a \ e ) ) ) = ( O ` a ) } = (CaraGen ` O ) )
9	2	eqcomi 2631	. . . . . 6 |- (CaraGen ` O ) = S
10	9	a1i 11	. . . . 5 |- ( ph -> (CaraGen ` O ) = S )
11	8, 10	eqtr2d 2657	. . . 4 |- ( ph -> S = { e e. ~P U. dom O | A. a e. ~P U. dom O ( ( O ` ( a i^i e ) ) +e ( O ` ( a \ e ) ) ) = ( O ` a ) } )
12		ssrab2 3687	. . . 4 |- { e e. ~P U. dom O | A. a e. ~P U. dom O ( ( O ` ( a i^i e ) ) +e ( O ` ( a \ e ) ) ) = ( O ` a ) } C_ ~P U. dom O
13	11, 12	syl6eqss 3655	. . 3 |- ( ph -> S C_ ~P U. dom O )
14	5, 13	fssresd 6071	. 2 |- ( ph -> ( O |` S ) : S --> ( 0 [,] +oo ) )
15	1, 2	caragen0 40720	. . . 4 |- ( ph -> (/) e. S )
16		fvres 6207	. . . 4 |- ( (/) e. S -> ( ( O |` S ) ` (/) ) = ( O ` (/) ) )
17	15, 16	syl 17	. . 3 |- ( ph -> ( ( O |` S ) ` (/) ) = ( O ` (/) ) )
18	1	ome0 40711	. . 3 |- ( ph -> ( O ` (/) ) = 0 )
19	17, 18	eqtrd 2656	. 2 |- ( ph -> ( ( O |` S ) ` (/) ) = 0 )
20		simp1 1061	. . . 4 |- ( ( ph /\ e : NN --> S /\ Disj n e. NN ( e ` n ) ) -> ph )
21		simp2 1062	. . . 4 |- ( ( ph /\ e : NN --> S /\ Disj n e. NN ( e ` n ) ) -> e : NN --> S )
22		fveq2 6191	. . . . . . 7 |- ( n = m -> ( e ` n ) = ( e ` m ) )
23	22	cbvdisjv 4631	. . . . . 6 |- (Disj n e. NN ( e ` n ) <-> Disj m e. NN ( e ` m ) )
24	23	biimpi 206	. . . . 5 |- (Disj n e. NN ( e ` n ) -> Disj m e. NN ( e ` m ) )
25	24	3ad2ant3 1084	. . . 4 |- ( ( ph /\ e : NN --> S /\ Disj n e. NN ( e ` n ) ) -> Disj m e. NN ( e ` m ) )
26	1	3ad2ant1 1082	. . . . 5 |- ( ( ph /\ e : NN --> S /\ Disj m e. NN ( e ` m ) ) -> O e. OutMeas )
27		simp2 1062	. . . . 5 |- ( ( ph /\ e : NN --> S /\ Disj m e. NN ( e ` m ) ) -> e : NN --> S )
28	23	biimpri 218	. . . . . 6 |- (Disj m e. NN ( e ` m ) -> Disj n e. NN ( e ` n ) )
29	28	3ad2ant3 1084	. . . . 5 |- ( ( ph /\ e : NN --> S /\ Disj m e. NN ( e ` m ) ) -> Disj n e. NN ( e ` n ) )
30		fveq2 6191	. . . . . . 7 |- ( m = n -> ( e ` m ) = ( e ` n ) )
31	30	cbviunv 4559	. . . . . 6 |- U_ m e. ( 1 ... j ) ( e ` m ) = U_ n e. ( 1 ... j ) ( e ` n )
32	31	mpteq2i 4741	. . . . 5 |- ( j e. NN |-> U_ m e. ( 1 ... j ) ( e ` m ) ) = ( j e. NN |-> U_ n e. ( 1 ... j ) ( e ` n ) )
33	26, 4, 2, 27, 29, 32	caratheodorylem2 40741	. . . 4 |- ( ( ph /\ e : NN --> S /\ Disj m e. NN ( e ` m ) ) -> ( O ` U_ n e. NN ( e ` n ) ) = (Σ^ ` ( n e. NN |-> ( O ` ( e ` n ) ) ) ) )
34	20, 21, 25, 33	syl3anc 1326	. . 3 |- ( ( ph /\ e : NN --> S /\ Disj n e. NN ( e ` n ) ) -> ( O ` U_ n e. NN ( e ` n ) ) = (Σ^ ` ( n e. NN |-> ( O ` ( e ` n ) ) ) ) )
35	3	adantr 481	. . . . . 6 |- ( ( ph /\ e : NN --> S ) -> S e. SAlg )
36		nnenom 12779	. . . . . . . 8 |- NN ~~ om
37		endom 7982	. . . . . . . 8 |- ( NN ~~ om -> NN ~<_ om )
38	36, 37	ax-mp 5	. . . . . . 7 |- NN ~<_ om
39	38	a1i 11	. . . . . 6 |- ( ( ph /\ e : NN --> S ) -> NN ~<_ om )
40		ffvelrn 6357	. . . . . . 7 |- ( ( e : NN --> S /\ n e. NN ) -> ( e ` n ) e. S )
41	40	adantll 750	. . . . . 6 |- ( ( ( ph /\ e : NN --> S ) /\ n e. NN ) -> ( e ` n ) e. S )
42	35, 39, 41	saliuncl 40542	. . . . 5 |- ( ( ph /\ e : NN --> S ) -> U_ n e. NN ( e ` n ) e. S )
43		fvres 6207	. . . . 5 |- ( U_ n e. NN ( e ` n ) e. S -> ( ( O |` S ) ` U_ n e. NN ( e ` n ) ) = ( O ` U_ n e. NN ( e ` n ) ) )
44	42, 43	syl 17	. . . 4 |- ( ( ph /\ e : NN --> S ) -> ( ( O |` S ) ` U_ n e. NN ( e ` n ) ) = ( O ` U_ n e. NN ( e ` n ) ) )
45	44	3adant3 1081	. . 3 |- ( ( ph /\ e : NN --> S /\ Disj n e. NN ( e ` n ) ) -> ( ( O |` S ) ` U_ n e. NN ( e ` n ) ) = ( O ` U_ n e. NN ( e ` n ) ) )
46		fvres 6207	. . . . . . 7 |- ( ( e ` n ) e. S -> ( ( O |` S ) ` ( e ` n ) ) = ( O ` ( e ` n ) ) )
47	41, 46	syl 17	. . . . . 6 |- ( ( ( ph /\ e : NN --> S ) /\ n e. NN ) -> ( ( O |` S ) ` ( e ` n ) ) = ( O ` ( e ` n ) ) )
48	47	mpteq2dva 4744	. . . . 5 |- ( ( ph /\ e : NN --> S ) -> ( n e. NN |-> ( ( O |` S ) ` ( e ` n ) ) ) = ( n e. NN |-> ( O ` ( e ` n ) ) ) )
49	48	fveq2d 6195	. . . 4 |- ( ( ph /\ e : NN --> S ) -> (Σ^ ` ( n e. NN |-> ( ( O |` S ) ` ( e ` n ) ) ) ) = (Σ^ ` ( n e. NN |-> ( O ` ( e ` n ) ) ) ) )
50	49	3adant3 1081	. . 3 |- ( ( ph /\ e : NN --> S /\ Disj n e. NN ( e ` n ) ) -> (Σ^ ` ( n e. NN |-> ( ( O |` S ) ` ( e ` n ) ) ) ) = (Σ^ ` ( n e. NN |-> ( O ` ( e ` n ) ) ) ) )
51	34, 45, 50	3eqtr4d 2666	. 2 |- ( ( ph /\ e : NN --> S /\ Disj n e. NN ( e ` n ) ) -> ( ( O |` S ) ` U_ n e. NN ( e ` n ) ) = (Σ^ ` ( n e. NN |-> ( ( O |` S ) ` ( e ` n ) ) ) ) )
52	3, 14, 19, 51	ismeannd 40684	1 |- ( ph -> ( O |` S ) e. Meas )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   \ cdif 3571   i^i cin 3573   (/)c0 3915   ~Pcpw 4158   U.cuni 4436   U_ciun 4520  Disj wdisj 4620   class class class wbr 4653   |-> cmpt 4729   dom cdm 5114   |` cres 5116   -->wf 5884   ` cfv 5888  (class class class)co 6650   omcom 7065   ~~ cen 7952   ~<_ cdom 7953   0cc0 9936   1c1 9937   +oocpnf 10071   NNcn 11020   +ecxad 11944   [,]cicc 12178   ...cfz 12326  SAlgcsalg 40528  Σ^{^}csumge0 40579  Meascmea 40666  OutMeascome 40703  CaraGenccaragen 40705

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-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

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-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-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-omul 7565  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-acn 8768  df-ac 8939  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-xadd 11947  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  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-sum 14417  df-salg 40529  df-sumge0 40580  df-mea 40667  df-ome 40704  df-caragen 40706

This theorem is referenced by:  vonmea  40788