![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > sitg0 | Structured version Visualization version GIF version |
Description: The integral of the constant zero function is zero. (Contributed by Thierry Arnoux, 13-Mar-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) |
sitg0.1 | β’ (π β π β TopSp) |
sitg0.2 | β’ (π β π β Mnd) |
Ref | Expression |
---|---|
sitg0 | β’ (π β ((πsitgπ)β(βͺ dom π Γ { 0 })) = 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sitgval.b | . . 3 β’ π΅ = (Baseβπ) | |
2 | sitgval.j | . . 3 β’ π½ = (TopOpenβπ) | |
3 | sitgval.s | . . 3 β’ π = (sigaGenβπ½) | |
4 | sitgval.0 | . . 3 β’ 0 = (0gβπ) | |
5 | sitgval.x | . . 3 β’ Β· = ( Β·π βπ) | |
6 | sitgval.h | . . 3 β’ π» = (βHomβ(Scalarβπ)) | |
7 | sitgval.1 | . . 3 β’ (π β π β π) | |
8 | sitgval.2 | . . 3 β’ (π β π β βͺ ran measures) | |
9 | sitg0.1 | . . . 4 β’ (π β π β TopSp) | |
10 | sitg0.2 | . . . 4 β’ (π β π β Mnd) | |
11 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | sibf0 33333 | . . 3 β’ (π β (βͺ dom π Γ { 0 }) β dom (πsitgπ)) |
12 | 1, 2, 3, 4, 5, 6, 7, 8, 11 | sitgfval 33340 | . 2 β’ (π β ((πsitgπ)β(βͺ dom π Γ { 0 })) = (π Ξ£g (π₯ β (ran (βͺ dom π Γ { 0 }) β { 0 }) β¦ ((π»β(πβ(β‘(βͺ dom π Γ { 0 }) β {π₯}))) Β· π₯)))) |
13 | rnxpss 6172 | . . . . . . 7 β’ ran (βͺ dom π Γ { 0 }) β { 0 } | |
14 | ssdif0 4364 | . . . . . . 7 β’ (ran (βͺ dom π Γ { 0 }) β { 0 } β (ran (βͺ dom π Γ { 0 }) β { 0 }) = β ) | |
15 | 13, 14 | mpbi 229 | . . . . . 6 β’ (ran (βͺ dom π Γ { 0 }) β { 0 }) = β |
16 | mpteq1 5242 | . . . . . 6 β’ ((ran (βͺ dom π Γ { 0 }) β { 0 }) = β β (π₯ β (ran (βͺ dom π Γ { 0 }) β { 0 }) β¦ ((π»β(πβ(β‘(βͺ dom π Γ { 0 }) β {π₯}))) Β· π₯)) = (π₯ β β β¦ ((π»β(πβ(β‘(βͺ dom π Γ { 0 }) β {π₯}))) Β· π₯))) | |
17 | 15, 16 | ax-mp 5 | . . . . 5 β’ (π₯ β (ran (βͺ dom π Γ { 0 }) β { 0 }) β¦ ((π»β(πβ(β‘(βͺ dom π Γ { 0 }) β {π₯}))) Β· π₯)) = (π₯ β β β¦ ((π»β(πβ(β‘(βͺ dom π Γ { 0 }) β {π₯}))) Β· π₯)) |
18 | mpt0 6693 | . . . . 5 β’ (π₯ β β β¦ ((π»β(πβ(β‘(βͺ dom π Γ { 0 }) β {π₯}))) Β· π₯)) = β | |
19 | 17, 18 | eqtri 2761 | . . . 4 β’ (π₯ β (ran (βͺ dom π Γ { 0 }) β { 0 }) β¦ ((π»β(πβ(β‘(βͺ dom π Γ { 0 }) β {π₯}))) Β· π₯)) = β |
20 | 19 | oveq2i 7420 | . . 3 β’ (π Ξ£g (π₯ β (ran (βͺ dom π Γ { 0 }) β { 0 }) β¦ ((π»β(πβ(β‘(βͺ dom π Γ { 0 }) β {π₯}))) Β· π₯))) = (π Ξ£g β ) |
21 | 4 | gsum0 18603 | . . 3 β’ (π Ξ£g β ) = 0 |
22 | 20, 21 | eqtri 2761 | . 2 β’ (π Ξ£g (π₯ β (ran (βͺ dom π Γ { 0 }) β { 0 }) β¦ ((π»β(πβ(β‘(βͺ dom π Γ { 0 }) β {π₯}))) Β· π₯))) = 0 |
23 | 12, 22 | eqtrdi 2789 | 1 β’ (π β ((πsitgπ)β(βͺ dom π Γ { 0 })) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 β cdif 3946 β wss 3949 β c0 4323 {csn 4629 βͺ cuni 4909 β¦ cmpt 5232 Γ cxp 5675 β‘ccnv 5676 dom cdm 5677 ran crn 5678 β cima 5680 βcfv 6544 (class class class)co 7409 Basecbs 17144 Scalarcsca 17200 Β·π cvsca 17201 TopOpenctopn 17367 0gc0g 17385 Ξ£g cgsu 17386 Mndcmnd 18625 TopSpctps 22434 βHomcrrh 32973 sigaGencsigagen 33136 measurescmeas 33193 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-pow 5364 ax-pr 5428 ax-un 7725 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 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-rmo 3377 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-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 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-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 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-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-map 8822 df-en 8940 df-fin 8943 df-seq 13967 df-0g 17387 df-gsum 17388 df-mgm 18561 df-sgrp 18610 df-mnd 18626 df-top 22396 df-topon 22413 df-topsp 22435 df-esum 33026 df-siga 33107 df-sigagen 33137 df-meas 33194 df-mbfm 33248 df-sitg 33329 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |