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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > sitgf | Structured version Visualization version GIF version |
Description: The integral for simple functions is itself a function. (Contributed by Thierry Arnoux, 13-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) |
sitgf.1 | β’ ((π β§ π β dom (πsitgπ)) β ((πsitgπ)βπ) β π΅) |
Ref | Expression |
---|---|
sitgf | β’ (π β (πsitgπ):dom (πsitgπ)βΆπ΅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funmpt 6544 | . . . 4 β’ Fun (π β {π β (dom πMblFnMπ) β£ (ran π β Fin β§ βπ₯ β (ran π β { 0 })(πβ(β‘π β {π₯})) β (0[,)+β))} β¦ (π Ξ£g (π₯ β (ran π β { 0 }) β¦ ((π»β(πβ(β‘π β {π₯}))) Β· π₯)))) | |
2 | sitgval.b | . . . . . 6 β’ π΅ = (Baseβπ) | |
3 | sitgval.j | . . . . . 6 β’ π½ = (TopOpenβπ) | |
4 | sitgval.s | . . . . . 6 β’ π = (sigaGenβπ½) | |
5 | sitgval.0 | . . . . . 6 β’ 0 = (0gβπ) | |
6 | sitgval.x | . . . . . 6 β’ Β· = ( Β·π βπ) | |
7 | sitgval.h | . . . . . 6 β’ π» = (βHomβ(Scalarβπ)) | |
8 | sitgval.1 | . . . . . 6 β’ (π β π β π) | |
9 | sitgval.2 | . . . . . 6 β’ (π β π β βͺ ran measures) | |
10 | 2, 3, 4, 5, 6, 7, 8, 9 | sitgval 32972 | . . . . 5 β’ (π β (πsitgπ) = (π β {π β (dom πMblFnMπ) β£ (ran π β Fin β§ βπ₯ β (ran π β { 0 })(πβ(β‘π β {π₯})) β (0[,)+β))} β¦ (π Ξ£g (π₯ β (ran π β { 0 }) β¦ ((π»β(πβ(β‘π β {π₯}))) Β· π₯))))) |
11 | 10 | funeqd 6528 | . . . 4 β’ (π β (Fun (πsitgπ) β Fun (π β {π β (dom πMblFnMπ) β£ (ran π β Fin β§ βπ₯ β (ran π β { 0 })(πβ(β‘π β {π₯})) β (0[,)+β))} β¦ (π Ξ£g (π₯ β (ran π β { 0 }) β¦ ((π»β(πβ(β‘π β {π₯}))) Β· π₯)))))) |
12 | 1, 11 | mpbiri 258 | . . 3 β’ (π β Fun (πsitgπ)) |
13 | 12 | funfnd 6537 | . 2 β’ (π β (πsitgπ) Fn dom (πsitgπ)) |
14 | sitgf.1 | . . . 4 β’ ((π β§ π β dom (πsitgπ)) β ((πsitgπ)βπ) β π΅) | |
15 | 14 | ralrimiva 3144 | . . 3 β’ (π β βπ β dom (πsitgπ)((πsitgπ)βπ) β π΅) |
16 | fnfvrnss 7073 | . . 3 β’ (((πsitgπ) Fn dom (πsitgπ) β§ βπ β dom (πsitgπ)((πsitgπ)βπ) β π΅) β ran (πsitgπ) β π΅) | |
17 | 13, 15, 16 | syl2anc 585 | . 2 β’ (π β ran (πsitgπ) β π΅) |
18 | df-f 6505 | . 2 β’ ((πsitgπ):dom (πsitgπ)βΆπ΅ β ((πsitgπ) Fn dom (πsitgπ) β§ ran (πsitgπ) β π΅)) | |
19 | 13, 17, 18 | sylanbrc 584 | 1 β’ (π β (πsitgπ):dom (πsitgπ)βΆπ΅) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 βwral 3065 {crab 3410 β cdif 3912 β wss 3915 {csn 4591 βͺ cuni 4870 β¦ cmpt 5193 β‘ccnv 5637 dom cdm 5638 ran crn 5639 β cima 5641 Fun wfun 6495 Fn wfn 6496 βΆwf 6497 βcfv 6501 (class class class)co 7362 Fincfn 8890 0cc0 11058 +βcpnf 11193 [,)cico 13273 Basecbs 17090 Scalarcsca 17143 Β·π cvsca 17144 TopOpenctopn 17310 0gc0g 17328 Ξ£g cgsu 17329 βHomcrrh 32614 sigaGencsigagen 32777 measurescmeas 32834 MblFnMcmbfm 32888 sitgcsitg 32969 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-ov 7365 df-oprab 7366 df-mpo 7367 df-sitg 32970 |
This theorem is referenced by: (None) |
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