Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
||
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 | ⊢ 𝐵 = (Base‘𝑊) |
sitgval.j | ⊢ 𝐽 = (TopOpen‘𝑊) |
sitgval.s | ⊢ 𝑆 = (sigaGen‘𝐽) |
sitgval.0 | ⊢ 0 = (0g‘𝑊) |
sitgval.x | ⊢ · = ( ·𝑠 ‘𝑊) |
sitgval.h | ⊢ 𝐻 = (ℝHom‘(Scalar‘𝑊)) |
sitgval.1 | ⊢ (𝜑 → 𝑊 ∈ 𝑉) |
sitgval.2 | ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures) |
sitgf.1 | ⊢ ((𝜑 ∧ 𝑓 ∈ dom (𝑊sitg𝑀)) → ((𝑊sitg𝑀)‘𝑓) ∈ 𝐵) |
Ref | Expression |
---|---|
sitgf | ⊢ (𝜑 → (𝑊sitg𝑀):dom (𝑊sitg𝑀)⟶𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funmpt 6386 | . . . 4 ⊢ Fun (𝑓 ∈ {𝑔 ∈ (dom 𝑀MblFnM𝑆) ∣ (ran 𝑔 ∈ Fin ∧ ∀𝑥 ∈ (ran 𝑔 ∖ { 0 })(𝑀‘(◡𝑔 “ {𝑥})) ∈ (0[,)+∞))} ↦ (𝑊 Σg (𝑥 ∈ (ran 𝑓 ∖ { 0 }) ↦ ((𝐻‘(𝑀‘(◡𝑓 “ {𝑥}))) · 𝑥)))) | |
2 | sitgval.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑊) | |
3 | sitgval.j | . . . . . 6 ⊢ 𝐽 = (TopOpen‘𝑊) | |
4 | sitgval.s | . . . . . 6 ⊢ 𝑆 = (sigaGen‘𝐽) | |
5 | sitgval.0 | . . . . . 6 ⊢ 0 = (0g‘𝑊) | |
6 | sitgval.x | . . . . . 6 ⊢ · = ( ·𝑠 ‘𝑊) | |
7 | sitgval.h | . . . . . 6 ⊢ 𝐻 = (ℝHom‘(Scalar‘𝑊)) | |
8 | sitgval.1 | . . . . . 6 ⊢ (𝜑 → 𝑊 ∈ 𝑉) | |
9 | sitgval.2 | . . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ∪ ran measures) | |
10 | 2, 3, 4, 5, 6, 7, 8, 9 | sitgval 31611 | . . . . 5 ⊢ (𝜑 → (𝑊sitg𝑀) = (𝑓 ∈ {𝑔 ∈ (dom 𝑀MblFnM𝑆) ∣ (ran 𝑔 ∈ Fin ∧ ∀𝑥 ∈ (ran 𝑔 ∖ { 0 })(𝑀‘(◡𝑔 “ {𝑥})) ∈ (0[,)+∞))} ↦ (𝑊 Σg (𝑥 ∈ (ran 𝑓 ∖ { 0 }) ↦ ((𝐻‘(𝑀‘(◡𝑓 “ {𝑥}))) · 𝑥))))) |
11 | 10 | funeqd 6370 | . . . 4 ⊢ (𝜑 → (Fun (𝑊sitg𝑀) ↔ Fun (𝑓 ∈ {𝑔 ∈ (dom 𝑀MblFnM𝑆) ∣ (ran 𝑔 ∈ Fin ∧ ∀𝑥 ∈ (ran 𝑔 ∖ { 0 })(𝑀‘(◡𝑔 “ {𝑥})) ∈ (0[,)+∞))} ↦ (𝑊 Σg (𝑥 ∈ (ran 𝑓 ∖ { 0 }) ↦ ((𝐻‘(𝑀‘(◡𝑓 “ {𝑥}))) · 𝑥)))))) |
12 | 1, 11 | mpbiri 260 | . . 3 ⊢ (𝜑 → Fun (𝑊sitg𝑀)) |
13 | 12 | funfnd 6379 | . 2 ⊢ (𝜑 → (𝑊sitg𝑀) Fn dom (𝑊sitg𝑀)) |
14 | sitgf.1 | . . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ dom (𝑊sitg𝑀)) → ((𝑊sitg𝑀)‘𝑓) ∈ 𝐵) | |
15 | 14 | ralrimiva 3181 | . . 3 ⊢ (𝜑 → ∀𝑓 ∈ dom (𝑊sitg𝑀)((𝑊sitg𝑀)‘𝑓) ∈ 𝐵) |
16 | fnfvrnss 6877 | . . 3 ⊢ (((𝑊sitg𝑀) Fn dom (𝑊sitg𝑀) ∧ ∀𝑓 ∈ dom (𝑊sitg𝑀)((𝑊sitg𝑀)‘𝑓) ∈ 𝐵) → ran (𝑊sitg𝑀) ⊆ 𝐵) | |
17 | 13, 15, 16 | syl2anc 586 | . 2 ⊢ (𝜑 → ran (𝑊sitg𝑀) ⊆ 𝐵) |
18 | df-f 6352 | . 2 ⊢ ((𝑊sitg𝑀):dom (𝑊sitg𝑀)⟶𝐵 ↔ ((𝑊sitg𝑀) Fn dom (𝑊sitg𝑀) ∧ ran (𝑊sitg𝑀) ⊆ 𝐵)) | |
19 | 13, 17, 18 | sylanbrc 585 | 1 ⊢ (𝜑 → (𝑊sitg𝑀):dom (𝑊sitg𝑀)⟶𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ∀wral 3137 {crab 3141 ∖ cdif 3926 ⊆ wss 3929 {csn 4560 ∪ cuni 4831 ↦ cmpt 5139 ◡ccnv 5547 dom cdm 5548 ran crn 5549 “ cima 5551 Fun wfun 6342 Fn wfn 6343 ⟶wf 6344 ‘cfv 6348 (class class class)co 7149 Fincfn 8502 0cc0 10530 +∞cpnf 10665 [,)cico 12734 Basecbs 16478 Scalarcsca 16563 ·𝑠 cvsca 16564 TopOpenctopn 16690 0gc0g 16708 Σg cgsu 16709 ℝHomcrrh 31255 sigaGencsigagen 31418 measurescmeas 31475 MblFnMcmbfm 31529 sitgcsitg 31608 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pr 5323 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-ral 3142 df-rex 3143 df-reu 3144 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7152 df-oprab 7153 df-mpo 7154 df-sitg 31609 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |