Theorem difelcarsg 30372

Description: The Caratheodory measurable sets are closed under complement. (Contributed by Thierry Arnoux, 17-May-2020.)

Hypotheses

Ref	Expression
carsgval.1	|- ( ph -> O e. V )
carsgval.2	|- ( ph -> M : ~P O --> ( 0 [,] +oo ) )
difelcarsg.1	|- ( ph -> A e. (toCaraSiga ` M ) )

Assertion

Ref	Expression
difelcarsg	|- ( ph -> ( O \ A ) e. (toCaraSiga ` M ) )

Proof of Theorem difelcarsg

Dummy variable e is distinct from all other variables.

Step	Hyp	Ref	Expression
1		difssd 3738	. . 3 |- ( ph -> ( O \ A ) C_ O )
2		indif2 3870	. . . . . . . 8 |- ( e i^i ( O \ A ) ) = ( ( e i^i O ) \ A )
3		elpwi 4168	. . . . . . . . . . 11 |- ( e e. ~P O -> e C_ O )
4	3	adantl 482	. . . . . . . . . 10 |- ( ( ph /\ e e. ~P O ) -> e C_ O )
5		df-ss 3588	. . . . . . . . . 10 |- ( e C_ O <-> ( e i^i O ) = e )
6	4, 5	sylib 208	. . . . . . . . 9 |- ( ( ph /\ e e. ~P O ) -> ( e i^i O ) = e )
7	6	difeq1d 3727	. . . . . . . 8 |- ( ( ph /\ e e. ~P O ) -> ( ( e i^i O ) \ A ) = ( e \ A ) )
8	2, 7	syl5eq 2668	. . . . . . 7 |- ( ( ph /\ e e. ~P O ) -> ( e i^i ( O \ A ) ) = ( e \ A ) )
9	8	fveq2d 6195	. . . . . 6 |- ( ( ph /\ e e. ~P O ) -> ( M ` ( e i^i ( O \ A ) ) ) = ( M ` ( e \ A ) ) )
10		difdif2 3884	. . . . . . . 8 |- ( e \ ( O \ A ) ) = ( ( e \ O ) u. ( e i^i A ) )
11		ssdif0 3942	. . . . . . . . . . 11 |- ( e C_ O <-> ( e \ O ) = (/) )
12	4, 11	sylib 208	. . . . . . . . . 10 |- ( ( ph /\ e e. ~P O ) -> ( e \ O ) = (/) )
13	12	uneq1d 3766	. . . . . . . . 9 |- ( ( ph /\ e e. ~P O ) -> ( ( e \ O ) u. ( e i^i A ) ) = ( (/) u. ( e i^i A ) ) )
14		uncom 3757	. . . . . . . . . 10 |- ( ( e i^i A ) u. (/) ) = ( (/) u. ( e i^i A ) )
15		un0 3967	. . . . . . . . . 10 |- ( ( e i^i A ) u. (/) ) = ( e i^i A )
16	14, 15	eqtr3i 2646	. . . . . . . . 9 |- ( (/) u. ( e i^i A ) ) = ( e i^i A )
17	13, 16	syl6eq 2672	. . . . . . . 8 |- ( ( ph /\ e e. ~P O ) -> ( ( e \ O ) u. ( e i^i A ) ) = ( e i^i A ) )
18	10, 17	syl5eq 2668	. . . . . . 7 |- ( ( ph /\ e e. ~P O ) -> ( e \ ( O \ A ) ) = ( e i^i A ) )
19	18	fveq2d 6195	. . . . . 6 |- ( ( ph /\ e e. ~P O ) -> ( M ` ( e \ ( O \ A ) ) ) = ( M ` ( e i^i A ) ) )
20	9, 19	oveq12d 6668	. . . . 5 |- ( ( ph /\ e e. ~P O ) -> ( ( M ` ( e i^i ( O \ A ) ) ) +e ( M ` ( e \ ( O \ A ) ) ) ) = ( ( M ` ( e \ A ) ) +e ( M ` ( e i^i A ) ) ) )
21		iccssxr 12256	. . . . . . 7 |- ( 0 [,] +oo ) C_ RR*
22		carsgval.2	. . . . . . . . 9 |- ( ph -> M : ~P O --> ( 0 [,] +oo ) )
23	22	adantr 481	. . . . . . . 8 |- ( ( ph /\ e e. ~P O ) -> M : ~P O --> ( 0 [,] +oo ) )
24		simpr 477	. . . . . . . . 9 |- ( ( ph /\ e e. ~P O ) -> e e. ~P O )
25	24	elpwdifcl 29358	. . . . . . . 8 |- ( ( ph /\ e e. ~P O ) -> ( e \ A ) e. ~P O )
26	23, 25	ffvelrnd 6360	. . . . . . 7 |- ( ( ph /\ e e. ~P O ) -> ( M ` ( e \ A ) ) e. ( 0 [,] +oo ) )
27	21, 26	sseldi 3601	. . . . . 6 |- ( ( ph /\ e e. ~P O ) -> ( M ` ( e \ A ) ) e. RR* )
28	24	elpwincl1 29357	. . . . . . . 8 |- ( ( ph /\ e e. ~P O ) -> ( e i^i A ) e. ~P O )
29	23, 28	ffvelrnd 6360	. . . . . . 7 |- ( ( ph /\ e e. ~P O ) -> ( M ` ( e i^i A ) ) e. ( 0 [,] +oo ) )
30	21, 29	sseldi 3601	. . . . . 6 |- ( ( ph /\ e e. ~P O ) -> ( M ` ( e i^i A ) ) e. RR* )
31		xaddcom 12071	. . . . . 6 |- ( ( ( M ` ( e \ A ) ) e. RR* /\ ( M ` ( e i^i A ) ) e. RR* ) -> ( ( M ` ( e \ A ) ) +e ( M ` ( e i^i A ) ) ) = ( ( M ` ( e i^i A ) ) +e ( M ` ( e \ A ) ) ) )
32	27, 30, 31	syl2anc 693	. . . . 5 |- ( ( ph /\ e e. ~P O ) -> ( ( M ` ( e \ A ) ) +e ( M ` ( e i^i A ) ) ) = ( ( M ` ( e i^i A ) ) +e ( M ` ( e \ A ) ) ) )
33		difelcarsg.1	. . . . . . . 8 |- ( ph -> A e. (toCaraSiga ` M ) )
34		carsgval.1	. . . . . . . . 9 |- ( ph -> O e. V )
35	34, 22	elcarsg 30367	. . . . . . . 8 |- ( ph -> ( A e. (toCaraSiga ` M ) <-> ( A C_ O /\ A. e e. ~P O ( ( M ` ( e i^i A ) ) +e ( M ` ( e \ A ) ) ) = ( M ` e ) ) ) )
36	33, 35	mpbid 222	. . . . . . 7 |- ( ph -> ( A C_ O /\ A. e e. ~P O ( ( M ` ( e i^i A ) ) +e ( M ` ( e \ A ) ) ) = ( M ` e ) ) )
37	36	simprd 479	. . . . . 6 |- ( ph -> A. e e. ~P O ( ( M ` ( e i^i A ) ) +e ( M ` ( e \ A ) ) ) = ( M ` e ) )
38	37	r19.21bi 2932	. . . . 5 |- ( ( ph /\ e e. ~P O ) -> ( ( M ` ( e i^i A ) ) +e ( M ` ( e \ A ) ) ) = ( M ` e ) )
39	20, 32, 38	3eqtrd 2660	. . . 4 |- ( ( ph /\ e e. ~P O ) -> ( ( M ` ( e i^i ( O \ A ) ) ) +e ( M ` ( e \ ( O \ A ) ) ) ) = ( M ` e ) )
40	39	ralrimiva 2966	. . 3 |- ( ph -> A. e e. ~P O ( ( M ` ( e i^i ( O \ A ) ) ) +e ( M ` ( e \ ( O \ A ) ) ) ) = ( M ` e ) )
41	1, 40	jca 554	. 2 |- ( ph -> ( ( O \ A ) C_ O /\ A. e e. ~P O ( ( M ` ( e i^i ( O \ A ) ) ) +e ( M ` ( e \ ( O \ A ) ) ) ) = ( M ` e ) ) )
42	34, 22	elcarsg 30367	. 2 |- ( ph -> ( ( O \ A ) e. (toCaraSiga ` M ) <-> ( ( O \ A ) C_ O /\ A. e e. ~P O ( ( M ` ( e i^i ( O \ A ) ) ) +e ( M ` ( e \ ( O \ A ) ) ) ) = ( M ` e ) ) ) )
43	41, 42	mpbird 247	1 |- ( ph -> ( O \ A ) e. (toCaraSiga ` M ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   \ cdif 3571   u. cun 3572   i^i cin 3573   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   -->wf 5884   ` cfv 5888  (class class class)co 6650   0cc0 9936   +oocpnf 10071   RR*cxr 10073   +ecxad 11944   [,]cicc 12178  toCaraSigaccarsg 30363

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-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

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-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  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-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-xadd 11947  df-icc 12182  df-carsg 30364

This theorem is referenced by:  unelcarsg  30374  difelcarsg2  30375  fiunelcarsg  30378  carsgsiga  30384