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 | ⊢ 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 5840 | . . . 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 29721 | . . . . 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 5825 | . . . 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 247 | . . 3 ⊢ (𝜑 → Fun (𝑊sitg𝑀)) |
13 | funfn 5833 | . . 3 ⊢ (Fun (𝑊sitg𝑀) ↔ (𝑊sitg𝑀) Fn dom (𝑊sitg𝑀)) | |
14 | 12, 13 | sylib 207 | . 2 ⊢ (𝜑 → (𝑊sitg𝑀) Fn dom (𝑊sitg𝑀)) |
15 | sitgf.1 | . . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ dom (𝑊sitg𝑀)) → ((𝑊sitg𝑀)‘𝑓) ∈ 𝐵) | |
16 | 15 | ralrimiva 2949 | . . 3 ⊢ (𝜑 → ∀𝑓 ∈ dom (𝑊sitg𝑀)((𝑊sitg𝑀)‘𝑓) ∈ 𝐵) |
17 | fnfvrnss 6297 | . . 3 ⊢ (((𝑊sitg𝑀) Fn dom (𝑊sitg𝑀) ∧ ∀𝑓 ∈ dom (𝑊sitg𝑀)((𝑊sitg𝑀)‘𝑓) ∈ 𝐵) → ran (𝑊sitg𝑀) ⊆ 𝐵) | |
18 | 14, 16, 17 | syl2anc 691 | . 2 ⊢ (𝜑 → ran (𝑊sitg𝑀) ⊆ 𝐵) |
19 | df-f 5808 | . 2 ⊢ ((𝑊sitg𝑀):dom (𝑊sitg𝑀)⟶𝐵 ↔ ((𝑊sitg𝑀) Fn dom (𝑊sitg𝑀) ∧ ran (𝑊sitg𝑀) ⊆ 𝐵)) | |
20 | 14, 18, 19 | sylanbrc 695 | 1 ⊢ (𝜑 → (𝑊sitg𝑀):dom (𝑊sitg𝑀)⟶𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 {crab 2900 ∖ cdif 3537 ⊆ wss 3540 {csn 4125 ∪ cuni 4372 ↦ cmpt 4643 ◡ccnv 5037 dom cdm 5038 ran crn 5039 “ cima 5041 Fun wfun 5798 Fn wfn 5799 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 Fincfn 7841 0cc0 9815 +∞cpnf 9950 [,)cico 12048 Basecbs 15695 Scalarcsca 15771 ·𝑠 cvsca 15772 TopOpenctopn 15905 0gc0g 15923 Σg cgsu 15924 ℝHomcrrh 29365 sigaGencsigagen 29528 measurescmeas 29585 MblFnMcmbfm 29639 sitgcsitg 29718 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-sitg 29719 |
This theorem is referenced by: (None) |
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