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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > sibff | Structured version Visualization version GIF version |
Description: A simple function is a function. (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 |
---|---|
sibff | β’ (π β πΉ:βͺ dom πβΆβͺ π½) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sitgval.2 | . . . 4 β’ (π β π β βͺ ran measures) | |
2 | dmmeas 32840 | . . . 4 β’ (π β βͺ ran measures β dom π β βͺ ran sigAlgebra) | |
3 | 1, 2 | syl 17 | . . 3 β’ (π β dom π β βͺ ran sigAlgebra) |
4 | sitgval.s | . . . 4 β’ π = (sigaGenβπ½) | |
5 | sitgval.j | . . . . . 6 β’ π½ = (TopOpenβπ) | |
6 | fvexd 6862 | . . . . . 6 β’ (π β (TopOpenβπ) β V) | |
7 | 5, 6 | eqeltrid 2842 | . . . . 5 β’ (π β π½ β V) |
8 | 7 | sgsiga 32781 | . . . 4 β’ (π β (sigaGenβπ½) β βͺ ran sigAlgebra) |
9 | 4, 8 | eqeltrid 2842 | . . 3 β’ (π β π β βͺ ran sigAlgebra) |
10 | sitgval.b | . . . 4 β’ π΅ = (Baseβπ) | |
11 | sitgval.0 | . . . 4 β’ 0 = (0gβπ) | |
12 | sitgval.x | . . . 4 β’ Β· = ( Β·π βπ) | |
13 | sitgval.h | . . . 4 β’ π» = (βHomβ(Scalarβπ)) | |
14 | sitgval.1 | . . . 4 β’ (π β π β π) | |
15 | sibfmbl.1 | . . . 4 β’ (π β πΉ β dom (πsitgπ)) | |
16 | 10, 5, 4, 11, 12, 13, 14, 1, 15 | sibfmbl 32975 | . . 3 β’ (π β πΉ β (dom πMblFnMπ)) |
17 | 3, 9, 16 | mbfmf 32893 | . 2 β’ (π β πΉ:βͺ dom πβΆβͺ π) |
18 | 4 | unieqi 4883 | . . . 4 β’ βͺ π = βͺ (sigaGenβπ½) |
19 | unisg 32782 | . . . . 5 β’ (π½ β V β βͺ (sigaGenβπ½) = βͺ π½) | |
20 | 7, 19 | syl 17 | . . . 4 β’ (π β βͺ (sigaGenβπ½) = βͺ π½) |
21 | 18, 20 | eqtrid 2789 | . . 3 β’ (π β βͺ π = βͺ π½) |
22 | 21 | feq3d 6660 | . 2 β’ (π β (πΉ:βͺ dom πβΆβͺ π β πΉ:βͺ dom πβΆβͺ π½)) |
23 | 17, 22 | mpbid 231 | 1 β’ (π β πΉ:βͺ dom πβΆβͺ π½) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 Vcvv 3448 βͺ cuni 4870 dom cdm 5638 ran crn 5639 βΆwf 6497 βcfv 6501 (class class class)co 7362 Basecbs 17090 Scalarcsca 17143 Β·π cvsca 17144 TopOpenctopn 17310 0gc0g 17328 βHomcrrh 32614 sigAlgebracsiga 32747 sigaGencsigagen 32777 measurescmeas 32834 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-pow 5325 ax-pr 5389 ax-un 7677 |
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-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-int 4913 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-1st 7926 df-2nd 7927 df-map 8774 df-esum 32667 df-siga 32748 df-sigagen 32778 df-meas 32835 df-mbfm 32889 df-sitg 32970 |
This theorem is referenced by: sibfinima 32979 sibfof 32980 sitgaddlemb 32988 sitmcl 32991 |
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