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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > sibfmbl | Structured version Visualization version GIF version |
Description: A simple function is measurable. (Contributed by Thierry Arnoux, 19-Feb-2018.) |
Ref | Expression |
---|---|
sitgval.b | β’ π΅ = (Baseβπ) |
sitgval.j | β’ π½ = (TopOpenβπ) |
sitgval.s | β’ π = (sigaGenβπ½) |
sitgval.0 | β’ 0 = (0gβπ) |
sitgval.x | β’ Β· = ( Β·π βπ) |
sitgval.h | β’ π» = (βHomβ(Scalarβπ)) |
sitgval.1 | β’ (π β π β π) |
sitgval.2 | β’ (π β π β βͺ ran measures) |
sibfmbl.1 | β’ (π β πΉ β dom (πsitgπ)) |
Ref | Expression |
---|---|
sibfmbl | β’ (π β πΉ β (dom πMblFnMπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sibfmbl.1 | . . 3 β’ (π β πΉ β dom (πsitgπ)) | |
2 | sitgval.b | . . . 4 β’ π΅ = (Baseβπ) | |
3 | sitgval.j | . . . 4 β’ π½ = (TopOpenβπ) | |
4 | sitgval.s | . . . 4 β’ π = (sigaGenβπ½) | |
5 | sitgval.0 | . . . 4 β’ 0 = (0gβπ) | |
6 | sitgval.x | . . . 4 β’ Β· = ( Β·π βπ) | |
7 | sitgval.h | . . . 4 β’ π» = (βHomβ(Scalarβπ)) | |
8 | sitgval.1 | . . . 4 β’ (π β π β π) | |
9 | sitgval.2 | . . . 4 β’ (π β π β βͺ ran measures) | |
10 | 2, 3, 4, 5, 6, 7, 8, 9 | issibf 33986 | . . 3 β’ (π β (πΉ β dom (πsitgπ) β (πΉ β (dom πMblFnMπ) β§ ran πΉ β Fin β§ βπ₯ β (ran πΉ β { 0 })(πβ(β‘πΉ β {π₯})) β (0[,)+β)))) |
11 | 1, 10 | mpbid 231 | . 2 β’ (π β (πΉ β (dom πMblFnMπ) β§ ran πΉ β Fin β§ βπ₯ β (ran πΉ β { 0 })(πβ(β‘πΉ β {π₯})) β (0[,)+β))) |
12 | 11 | simp1d 1139 | 1 β’ (π β πΉ β (dom πMblFnMπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ w3a 1084 = wceq 1533 β wcel 2098 βwral 3058 β cdif 3946 {csn 4632 βͺ cuni 4912 β‘ccnv 5681 dom cdm 5682 ran crn 5683 β cima 5685 βcfv 6553 (class class class)co 7426 Fincfn 8970 0cc0 11146 +βcpnf 11283 [,)cico 13366 Basecbs 17187 Scalarcsca 17243 Β·π cvsca 17244 TopOpenctopn 17410 0gc0g 17428 βHomcrrh 33627 sigaGencsigagen 33790 measurescmeas 33847 MblFnMcmbfm 33901 sitgcsitg 33982 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-ov 7429 df-oprab 7430 df-mpo 7431 df-sitg 33983 |
This theorem is referenced by: sibff 33989 sibfinima 33992 sibfof 33993 sitgfval 33994 sitgclg 33995 |
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