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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 6587 | . . . 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 33331 | . . . . 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 6571 | . . . 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 6580 | . 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 3147 | . . 3 β’ (π β βπ β dom (πsitgπ)((πsitgπ)βπ) β π΅) |
16 | fnfvrnss 7120 | . . 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 6548 | . 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 3062 {crab 3433 β cdif 3946 β wss 3949 {csn 4629 βͺ cuni 4909 β¦ cmpt 5232 β‘ccnv 5676 dom cdm 5677 ran crn 5678 β cima 5680 Fun wfun 6538 Fn wfn 6539 βΆwf 6540 βcfv 6544 (class class class)co 7409 Fincfn 8939 0cc0 11110 +βcpnf 11245 [,)cico 13326 Basecbs 17144 Scalarcsca 17200 Β·π cvsca 17201 TopOpenctopn 17367 0gc0g 17385 Ξ£g cgsu 17386 βHomcrrh 32973 sigaGencsigagen 33136 measurescmeas 33193 MblFnMcmbfm 33247 sitgcsitg 33328 |
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 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7412 df-oprab 7413 df-mpo 7414 df-sitg 33329 |
This theorem is referenced by: (None) |
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