Theorem smfrec 40996

Description: The reciprocal of a sigma-measurable functions is sigma-measurable. First part of Proposition 121E (e) of [Fremlin1] p. 38 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypotheses

Ref	Expression
smfrec.x	|- F/ x ph
smfrec.s	|- ( ph -> S e. SAlg )
smfrec.a	|- ( ph -> A e. V )
smfrec.b	|- ( ( ph /\ x e. A ) -> B e. RR )
smfrec.m	|- ( ph -> ( x e. A |-> B ) e. (SMblFn ` S ) )
smfrec.e	|- C = { x e. A | B =/= 0 }

Assertion

Ref	Expression
smfrec	|- ( ph -> ( x e. C |-> ( 1 / B ) ) e. (SMblFn ` S ) )

Distinct variable groups:   x, A   x, C

Allowed substitution hints:   ph( x)   B( x)   S( x)   V( x)

Proof of Theorem smfrec

Dummy variable a is distinct from all other variables.

Step	Hyp	Ref	Expression
1		smfrec.x	. 2 |- F/ x ph
2		nfv 1843	. 2 |- F/ a ph
3		smfrec.s	. 2 |- ( ph -> S e. SAlg )
4		smfrec.e	. . . 4 |- C = { x e. A | B =/= 0 }
5		ssrab2 3687	. . . 4 |- { x e. A | B =/= 0 } C_ A
6	4, 5	eqsstri 3635	. . 3 |- C C_ A
7		eqid 2622	. . . . . 6 |- ( x e. A |-> B ) = ( x e. A |-> B )
8		smfrec.b	. . . . . 6 |- ( ( ph /\ x e. A ) -> B e. RR )
9	1, 7, 8	dmmptdf 39417	. . . . 5 |- ( ph -> dom ( x e. A |-> B ) = A )
10	9	eqcomd 2628	. . . 4 |- ( ph -> A = dom ( x e. A |-> B ) )
11		smfrec.m	. . . . 5 |- ( ph -> ( x e. A |-> B ) e. (SMblFn ` S ) )
12		eqid 2622	. . . . 5 |- dom ( x e. A |-> B ) = dom ( x e. A |-> B )
13	3, 11, 12	smfdmss 40942	. . . 4 |- ( ph -> dom ( x e. A |-> B ) C_ U. S )
14	10, 13	eqsstrd 3639	. . 3 |- ( ph -> A C_ U. S )
15	6, 14	syl5ss 3614	. 2 |- ( ph -> C C_ U. S )
16		1red 10055	. . 3 |- ( ( ph /\ x e. C ) -> 1 e. RR )
17	6	sseli 3599	. . . . 5 |- ( x e. C -> x e. A )
18	17	adantl 482	. . . 4 |- ( ( ph /\ x e. C ) -> x e. A )
19	18, 8	syldan 487	. . 3 |- ( ( ph /\ x e. C ) -> B e. RR )
20	4	eleq2i 2693	. . . . . 6 |- ( x e. C <-> x e. { x e. A | B =/= 0 } )
21	20	biimpi 206	. . . . 5 |- ( x e. C -> x e. { x e. A | B =/= 0 } )
22		rabidim2 39284	. . . . 5 |- ( x e. { x e. A | B =/= 0 } -> B =/= 0 )
23	21, 22	syl 17	. . . 4 |- ( x e. C -> B =/= 0 )
24	23	adantl 482	. . 3 |- ( ( ph /\ x e. C ) -> B =/= 0 )
25	16, 19, 24	redivcld 10853	. 2 |- ( ( ph /\ x e. C ) -> ( 1 / B ) e. RR )
26		nfv 1843	. . . . . . 7 |- F/ x a e. RR
27	1, 26	nfan 1828	. . . . . 6 |- F/ x ( ph /\ a e. RR )
28		nfv 1843	. . . . . 6 |- F/ x 0 < a
29	27, 28	nfan 1828	. . . . 5 |- F/ x ( ( ph /\ a e. RR ) /\ 0 < a )
30	19	ad4ant14 1293	. . . . 5 |- ( ( ( ( ph /\ a e. RR ) /\ 0 < a ) /\ x e. C ) -> B e. RR )
31	23	adantl 482	. . . . 5 |- ( ( ( ( ph /\ a e. RR ) /\ 0 < a ) /\ x e. C ) -> B =/= 0 )
32		simpl 473	. . . . . . 7 |- ( ( a e. RR /\ 0 < a ) -> a e. RR )
33		simpr 477	. . . . . . 7 |- ( ( a e. RR /\ 0 < a ) -> 0 < a )
34	32, 33	elrpd 11869	. . . . . 6 |- ( ( a e. RR /\ 0 < a ) -> a e. RR+ )
35	34	adantll 750	. . . . 5 |- ( ( ( ph /\ a e. RR ) /\ 0 < a ) -> a e. RR+ )
36	29, 30, 31, 35	pimrecltpos 40919	. . . 4 |- ( ( ( ph /\ a e. RR ) /\ 0 < a ) -> { x e. C | ( 1 / B ) < a } = ( { x e. C | ( 1 / a ) < B } u. { x e. C | B < 0 } ) )
37		smfrec.a	. . . . . . . 8 |- ( ph -> A e. V )
38	4, 37	rabexd 4814	. . . . . . 7 |- ( ph -> C e. _V )
39		eqid 2622	. . . . . . 7 |- ( S ↾t C ) = ( S ↾t C )
40	3, 38, 39	subsalsal 40577	. . . . . 6 |- ( ph -> ( S ↾t C ) e. SAlg )
41	40	ad2antrr 762	. . . . 5 |- ( ( ( ph /\ a e. RR ) /\ 0 < a ) -> ( S ↾t C ) e. SAlg )
42	3	adantr 481	. . . . . . 7 |- ( ( ph /\ a e. RR ) -> S e. SAlg )
43	42	adantr 481	. . . . . 6 |- ( ( ( ph /\ a e. RR ) /\ 0 < a ) -> S e. SAlg )
44	6	a1i 11	. . . . . . . . 9 |- ( ph -> C C_ A )
45	3, 11, 44	sssmfmpt 40959	. . . . . . . 8 |- ( ph -> ( x e. C |-> B ) e. (SMblFn ` S ) )
46	45	adantr 481	. . . . . . 7 |- ( ( ph /\ a e. RR ) -> ( x e. C |-> B ) e. (SMblFn ` S ) )
47	46	adantr 481	. . . . . 6 |- ( ( ( ph /\ a e. RR ) /\ 0 < a ) -> ( x e. C |-> B ) e. (SMblFn ` S ) )
48	34	rprecred 11883	. . . . . . 7 |- ( ( a e. RR /\ 0 < a ) -> ( 1 / a ) e. RR )
49	48	adantll 750	. . . . . 6 |- ( ( ( ph /\ a e. RR ) /\ 0 < a ) -> ( 1 / a ) e. RR )
50	29, 43, 30, 47, 49	smfpimgtmpt 40989	. . . . 5 |- ( ( ( ph /\ a e. RR ) /\ 0 < a ) -> { x e. C | ( 1 / a ) < B } e. ( S ↾t C ) )
51		0red 10041	. . . . . . 7 |- ( ph -> 0 e. RR )
52	1, 3, 19, 45, 51	smfpimltmpt 40955	. . . . . 6 |- ( ph -> { x e. C | B < 0 } e. ( S ↾t C ) )
53	52	ad2antrr 762	. . . . 5 |- ( ( ( ph /\ a e. RR ) /\ 0 < a ) -> { x e. C | B < 0 } e. ( S ↾t C ) )
54	41, 50, 53	saluncld 40566	. . . 4 |- ( ( ( ph /\ a e. RR ) /\ 0 < a ) -> ( { x e. C | ( 1 / a ) < B } u. { x e. C | B < 0 } ) e. ( S ↾t C ) )
55	36, 54	eqeltrd 2701	. . 3 |- ( ( ( ph /\ a e. RR ) /\ 0 < a ) -> { x e. C | ( 1 / B ) < a } e. ( S ↾t C ) )
56		nfv 1843	. . . . . . . 8 |- F/ x a = 0
57	1, 56	nfan 1828	. . . . . . 7 |- F/ x ( ph /\ a = 0 )
58		breq2 4657	. . . . . . . . 9 |- ( a = 0 -> ( ( 1 / B ) < a <-> ( 1 / B ) < 0 ) )
59	58	ad2antlr 763	. . . . . . . 8 |- ( ( ( ph /\ a = 0 ) /\ x e. C ) -> ( ( 1 / B ) < a <-> ( 1 / B ) < 0 ) )
60	19, 24	reclt0 39614	. . . . . . . . . 10 |- ( ( ph /\ x e. C ) -> ( B < 0 <-> ( 1 / B ) < 0 ) )
61	60	bicomd 213	. . . . . . . . 9 |- ( ( ph /\ x e. C ) -> ( ( 1 / B ) < 0 <-> B < 0 ) )
62	61	adantlr 751	. . . . . . . 8 |- ( ( ( ph /\ a = 0 ) /\ x e. C ) -> ( ( 1 / B ) < 0 <-> B < 0 ) )
63	59, 62	bitrd 268	. . . . . . 7 |- ( ( ( ph /\ a = 0 ) /\ x e. C ) -> ( ( 1 / B ) < a <-> B < 0 ) )
64	57, 63	rabbida 39274	. . . . . 6 |- ( ( ph /\ a = 0 ) -> { x e. C | ( 1 / B ) < a } = { x e. C | B < 0 } )
65	52	adantr 481	. . . . . 6 |- ( ( ph /\ a = 0 ) -> { x e. C | B < 0 } e. ( S ↾t C ) )
66	64, 65	eqeltrd 2701	. . . . 5 |- ( ( ph /\ a = 0 ) -> { x e. C | ( 1 / B ) < a } e. ( S ↾t C ) )
67	66	ad4ant14 1293	. . . 4 |- ( ( ( ( ph /\ a e. RR ) /\ -. 0 < a ) /\ a = 0 ) -> { x e. C | ( 1 / B ) < a } e. ( S ↾t C ) )
68		simpll 790	. . . . 5 |- ( ( ( ( ph /\ a e. RR ) /\ -. 0 < a ) /\ -. a = 0 ) -> ( ph /\ a e. RR ) )
69		simpll 790	. . . . . . 7 |- ( ( ( a e. RR /\ -. 0 < a ) /\ -. a = 0 ) -> a e. RR )
70		0red 10041	. . . . . . 7 |- ( ( ( a e. RR /\ -. 0 < a ) /\ -. a = 0 ) -> 0 e. RR )
71		neqne 2802	. . . . . . . 8 |- ( -. a = 0 -> a =/= 0 )
72	71	adantl 482	. . . . . . 7 |- ( ( ( a e. RR /\ -. 0 < a ) /\ -. a = 0 ) -> a =/= 0 )
73		simplr 792	. . . . . . 7 |- ( ( ( a e. RR /\ -. 0 < a ) /\ -. a = 0 ) -> -. 0 < a )
74	69, 70, 72, 73	lttri5d 39513	. . . . . 6 |- ( ( ( a e. RR /\ -. 0 < a ) /\ -. a = 0 ) -> a < 0 )
75	74	adantlll 754	. . . . 5 |- ( ( ( ( ph /\ a e. RR ) /\ -. 0 < a ) /\ -. a = 0 ) -> a < 0 )
76		nfv 1843	. . . . . . . 8 |- F/ x a < 0
77	27, 76	nfan 1828	. . . . . . 7 |- F/ x ( ( ph /\ a e. RR ) /\ a < 0 )
78	8	adantlr 751	. . . . . . . . 9 |- ( ( ( ph /\ a e. RR ) /\ x e. A ) -> B e. RR )
79	17, 78	sylan2 491	. . . . . . . 8 |- ( ( ( ph /\ a e. RR ) /\ x e. C ) -> B e. RR )
80	79	adantlr 751	. . . . . . 7 |- ( ( ( ( ph /\ a e. RR ) /\ a < 0 ) /\ x e. C ) -> B e. RR )
81	23	adantl 482	. . . . . . 7 |- ( ( ( ( ph /\ a e. RR ) /\ a < 0 ) /\ x e. C ) -> B =/= 0 )
82		simpr 477	. . . . . . . 8 |- ( ( ph /\ a e. RR ) -> a e. RR )
83	82	adantr 481	. . . . . . 7 |- ( ( ( ph /\ a e. RR ) /\ a < 0 ) -> a e. RR )
84		simpr 477	. . . . . . 7 |- ( ( ( ph /\ a e. RR ) /\ a < 0 ) -> a < 0 )
85	77, 80, 81, 83, 84	pimrecltneg 40933	. . . . . 6 |- ( ( ( ph /\ a e. RR ) /\ a < 0 ) -> { x e. C | ( 1 / B ) < a } = { x e. C | B e. ( ( 1 / a ) (,) 0 ) } )
86	42	adantr 481	. . . . . . 7 |- ( ( ( ph /\ a e. RR ) /\ a < 0 ) -> S e. SAlg )
87	38	ad2antrr 762	. . . . . . 7 |- ( ( ( ph /\ a e. RR ) /\ a < 0 ) -> C e. _V )
88	46	adantr 481	. . . . . . 7 |- ( ( ( ph /\ a e. RR ) /\ a < 0 ) -> ( x e. C |-> B ) e. (SMblFn ` S ) )
89		1red 10055	. . . . . . . . . 10 |- ( ( a e. RR /\ a < 0 ) -> 1 e. RR )
90		simpl 473	. . . . . . . . . 10 |- ( ( a e. RR /\ a < 0 ) -> a e. RR )
91		lt0ne0 10494	. . . . . . . . . 10 |- ( ( a e. RR /\ a < 0 ) -> a =/= 0 )
92	89, 90, 91	redivcld 10853	. . . . . . . . 9 |- ( ( a e. RR /\ a < 0 ) -> ( 1 / a ) e. RR )
93	92	adantll 750	. . . . . . . 8 |- ( ( ( ph /\ a e. RR ) /\ a < 0 ) -> ( 1 / a ) e. RR )
94	93	rexrd 10089	. . . . . . 7 |- ( ( ( ph /\ a e. RR ) /\ a < 0 ) -> ( 1 / a ) e. RR* )
95	51	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ a e. RR ) /\ a < 0 ) -> 0 e. RR )
96	95	rexrd 10089	. . . . . . 7 |- ( ( ( ph /\ a e. RR ) /\ a < 0 ) -> 0 e. RR* )
97	77, 86, 87, 80, 88, 94, 96	smfpimioompt 40993	. . . . . 6 |- ( ( ( ph /\ a e. RR ) /\ a < 0 ) -> { x e. C | B e. ( ( 1 / a ) (,) 0 ) } e. ( S ↾t C ) )
98	85, 97	eqeltrd 2701	. . . . 5 |- ( ( ( ph /\ a e. RR ) /\ a < 0 ) -> { x e. C | ( 1 / B ) < a } e. ( S ↾t C ) )
99	68, 75, 98	syl2anc 693	. . . 4 |- ( ( ( ( ph /\ a e. RR ) /\ -. 0 < a ) /\ -. a = 0 ) -> { x e. C | ( 1 / B ) < a } e. ( S ↾t C ) )
100	67, 99	pm2.61dan 832	. . 3 |- ( ( ( ph /\ a e. RR ) /\ -. 0 < a ) -> { x e. C | ( 1 / B ) < a } e. ( S ↾t C ) )
101	55, 100	pm2.61dan 832	. 2 |- ( ( ph /\ a e. RR ) -> { x e. C | ( 1 / B ) < a } e. ( S ↾t C ) )
102	1, 2, 3, 15, 25, 101	issmfdmpt 40957	1 |- ( ph -> ( x e. C |-> ( 1 / B ) ) e. (SMblFn ` S ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   F/wnf 1708   e. wcel 1990   =/= wne 2794   {crab 2916   _Vcvv 3200   u. cun 3572   C_ wss 3574   U.cuni 4436   class class class wbr 4653   |-> cmpt 4729   dom cdm 5114   ` cfv 5888  (class class class)co 6650   RRcr 9935   0cc0 9936   1c1 9937   < clt 10074   / cdiv 10684   RR+crp 11832   (,)cioo 12175   ↾_t crest 16081  SAlgcsalg 40528  SMblFncsmblfn 40909

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

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-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-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-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  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-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-ioo 12179  df-ico 12181  df-fl 12593  df-rest 16083  df-salg 40529  df-smblfn 40910

This theorem is referenced by:  smfdiv  41004