Theorem psmeasure 40688

Description: Point supported measure, Remark 112B (d) of [Fremlin1] p. 15. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypotheses

Ref	Expression
psmeasure.x	|- ( ph -> X e. V )
psmeasure.h	|- ( ph -> H : X --> ( 0 [,] +oo ) )
psmeasure.m	|- M = ( x e. ~P X |-> (Σ^ ` ( H |` x ) ) )

Assertion

Ref	Expression
psmeasure	|- ( ph -> M e. Meas )

Distinct variable groups:   x, H   x, X   ph, x

Allowed substitution hints:   M( x)   V( x)

Proof of Theorem psmeasure

Dummy variables z y w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 477	. . . . . . 7 |- ( ( ph /\ x e. ~P X ) -> x e. ~P X )
2		psmeasure.h	. . . . . . . . 9 |- ( ph -> H : X --> ( 0 [,] +oo ) )
3	2	adantr 481	. . . . . . . 8 |- ( ( ph /\ x e. ~P X ) -> H : X --> ( 0 [,] +oo ) )
4	1	elpwid 4170	. . . . . . . 8 |- ( ( ph /\ x e. ~P X ) -> x C_ X )
5		fssres 6070	. . . . . . . 8 |- ( ( H : X --> ( 0 [,] +oo ) /\ x C_ X ) -> ( H |` x ) : x --> ( 0 [,] +oo ) )
6	3, 4, 5	syl2anc 693	. . . . . . 7 |- ( ( ph /\ x e. ~P X ) -> ( H |` x ) : x --> ( 0 [,] +oo ) )
7	1, 6	sge0cl 40598	. . . . . 6 |- ( ( ph /\ x e. ~P X ) -> (Σ^ ` ( H |` x ) ) e. ( 0 [,] +oo ) )
8		psmeasure.m	. . . . . 6 |- M = ( x e. ~P X |-> (Σ^ ` ( H |` x ) ) )
9	7, 8	fmptd 6385	. . . . 5 |- ( ph -> M : ~P X --> ( 0 [,] +oo ) )
10	8, 7	dmmptd 6024	. . . . . 6 |- ( ph -> dom M = ~P X )
11	10	feq2d 6031	. . . . 5 |- ( ph -> ( M : dom M --> ( 0 [,] +oo ) <-> M : ~P X --> ( 0 [,] +oo ) ) )
12	9, 11	mpbird 247	. . . 4 |- ( ph -> M : dom M --> ( 0 [,] +oo ) )
13		psmeasure.x	. . . . . 6 |- ( ph -> X e. V )
14		pwsal 40535	. . . . . 6 |- ( X e. V -> ~P X e. SAlg )
15	13, 14	syl 17	. . . . 5 |- ( ph -> ~P X e. SAlg )
16	10, 15	eqeltrd 2701	. . . 4 |- ( ph -> dom M e. SAlg )
17	12, 16	jca 554	. . 3 |- ( ph -> ( M : dom M --> ( 0 [,] +oo ) /\ dom M e. SAlg ) )
18	8	a1i 11	. . . . 5 |- ( ph -> M = ( x e. ~P X |-> (Σ^ ` ( H |` x ) ) ) )
19		reseq2 5391	. . . . . . 7 |- ( x = (/) -> ( H |` x ) = ( H |` (/) ) )
20	19	fveq2d 6195	. . . . . 6 |- ( x = (/) -> (Σ^ ` ( H |` x ) ) = (Σ^ ` ( H |` (/) ) ) )
21	20	adantl 482	. . . . 5 |- ( ( ph /\ x = (/) ) -> (Σ^ ` ( H |` x ) ) = (Σ^ ` ( H |` (/) ) ) )
22		0elpw 4834	. . . . . 6 |- (/) e. ~P X
23	22	a1i 11	. . . . 5 |- ( ph -> (/) e. ~P X )
24		fvexd 6203	. . . . 5 |- ( ph -> (Σ^ ` ( H |` (/) ) ) e. _V )
25	18, 21, 23, 24	fvmptd 6288	. . . 4 |- ( ph -> ( M ` (/) ) = (Σ^ ` ( H |` (/) ) ) )
26		res0 5400	. . . . . . 7 |- ( H |` (/) ) = (/)
27	26	fveq2i 6194	. . . . . 6 |- (Σ^ ` ( H |` (/) ) ) = (Σ^ ` (/) )
28		sge00 40593	. . . . . 6 |- (Σ^ ` (/) ) = 0
29	27, 28	eqtri 2644	. . . . 5 |- (Σ^ ` ( H |` (/) ) ) = 0
30	29	a1i 11	. . . 4 |- ( ph -> (Σ^ ` ( H |` (/) ) ) = 0 )
31	25, 30	eqtrd 2656	. . 3 |- ( ph -> ( M ` (/) ) = 0 )
32		simpl 473	. . . . 5 |- ( ( ph /\ y e. ~P dom M ) -> ph )
33		simpr 477	. . . . . . 7 |- ( ( ph /\ y e. ~P dom M ) -> y e. ~P dom M )
34	10	pweqd 4163	. . . . . . . 8 |- ( ph -> ~P dom M = ~P ~P X )
35	34	adantr 481	. . . . . . 7 |- ( ( ph /\ y e. ~P dom M ) -> ~P dom M = ~P ~P X )
36	33, 35	eleqtrd 2703	. . . . . 6 |- ( ( ph /\ y e. ~P dom M ) -> y e. ~P ~P X )
37		elpwi 4168	. . . . . 6 |- ( y e. ~P ~P X -> y C_ ~P X )
38	36, 37	syl 17	. . . . 5 |- ( ( ph /\ y e. ~P dom M ) -> y C_ ~P X )
39	13	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ y C_ ~P X ) /\ Disj w e. y w ) -> X e. V )
40	2	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ y C_ ~P X ) /\ Disj w e. y w ) -> H : X --> ( 0 [,] +oo ) )
41	9	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ y C_ ~P X ) /\ Disj w e. y w ) -> M : ~P X --> ( 0 [,] +oo ) )
42		simplr 792	. . . . . . . 8 |- ( ( ( ph /\ y C_ ~P X ) /\ Disj w e. y w ) -> y C_ ~P X )
43		id 22	. . . . . . . . . . 11 |- ( w = z -> w = z )
44	43	cbvdisjv 4631	. . . . . . . . . 10 |- (Disj w e. y w <-> Disj z e. y z )
45	44	biimpi 206	. . . . . . . . 9 |- (Disj w e. y w -> Disj z e. y z )
46	45	adantl 482	. . . . . . . 8 |- ( ( ( ph /\ y C_ ~P X ) /\ Disj w e. y w ) -> Disj z e. y z )
47	39, 40, 8, 41, 42, 46	psmeasurelem 40687	. . . . . . 7 |- ( ( ( ph /\ y C_ ~P X ) /\ Disj w e. y w ) -> ( M ` U. y ) = (Σ^ ` ( M |` y ) ) )
48	47	adantrl 752	. . . . . 6 |- ( ( ( ph /\ y C_ ~P X ) /\ ( y ~<_ om /\ Disj w e. y w ) ) -> ( M ` U. y ) = (Σ^ ` ( M |` y ) ) )
49	48	ex 450	. . . . 5 |- ( ( ph /\ y C_ ~P X ) -> ( ( y ~<_ om /\ Disj w e. y w ) -> ( M ` U. y ) = (Σ^ ` ( M |` y ) ) ) )
50	32, 38, 49	syl2anc 693	. . . 4 |- ( ( ph /\ y e. ~P dom M ) -> ( ( y ~<_ om /\ Disj w e. y w ) -> ( M ` U. y ) = (Σ^ ` ( M |` y ) ) ) )
51	50	ralrimiva 2966	. . 3 |- ( ph -> A. y e. ~P dom M ( ( y ~<_ om /\ Disj w e. y w ) -> ( M ` U. y ) = (Σ^ ` ( M |` y ) ) ) )
52	17, 31, 51	jca31 557	. 2 |- ( ph -> ( ( ( M : dom M --> ( 0 [,] +oo ) /\ dom M e. SAlg ) /\ ( M ` (/) ) = 0 ) /\ A. y e. ~P dom M ( ( y ~<_ om /\ Disj w e. y w ) -> ( M ` U. y ) = (Σ^ ` ( M |` y ) ) ) ) )
53		ismea 40668	. 2 |- ( M e. Meas <-> ( ( ( M : dom M --> ( 0 [,] +oo ) /\ dom M e. SAlg ) /\ ( M ` (/) ) = 0 ) /\ A. y e. ~P dom M ( ( y ~<_ om /\ Disj w e. y w ) -> ( M ` U. y ) = (Σ^ ` ( M |` y ) ) ) ) )
54	52, 53	sylibr 224	1 |- ( ph -> M e. Meas )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   U.cuni 4436  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   ~<_ cdom 7953   0cc0 9936   +oocpnf 10071   [,]cicc 12178  SAlgcsalg 40528  Σ^{^}csumge0 40579  Meascmea 40666

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-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  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-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

This theorem is referenced by: (None)