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Theorem sitgfval 33340
Description: Value of the Bochner integral for a simple function 𝐹. (Contributed by Thierry Arnoux, 30-Jan-2018.)
Hypotheses
Ref Expression
sitgval.b 𝐡 = (Baseβ€˜π‘Š)
sitgval.j 𝐽 = (TopOpenβ€˜π‘Š)
sitgval.s 𝑆 = (sigaGenβ€˜π½)
sitgval.0 0 = (0gβ€˜π‘Š)
sitgval.x Β· = ( ·𝑠 β€˜π‘Š)
sitgval.h 𝐻 = (ℝHomβ€˜(Scalarβ€˜π‘Š))
sitgval.1 (πœ‘ β†’ π‘Š ∈ 𝑉)
sitgval.2 (πœ‘ β†’ 𝑀 ∈ βˆͺ ran measures)
sibfmbl.1 (πœ‘ β†’ 𝐹 ∈ dom (π‘Šsitg𝑀))
Assertion
Ref Expression
sitgfval (πœ‘ β†’ ((π‘Šsitg𝑀)β€˜πΉ) = (π‘Š Ξ£g (π‘₯ ∈ (ran 𝐹 βˆ– { 0 }) ↦ ((π»β€˜(π‘€β€˜(◑𝐹 β€œ {π‘₯}))) Β· π‘₯))))
Distinct variable groups:   π‘₯,𝐹   π‘₯,𝑀   π‘₯,π‘Š   π‘₯, 0   πœ‘,π‘₯
Allowed substitution hints:   𝐡(π‘₯)   𝑆(π‘₯)   Β· (π‘₯)   𝐻(π‘₯)   𝐽(π‘₯)   𝑉(π‘₯)

Proof of Theorem sitgfval
Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sitgval.b . . 3 𝐡 = (Baseβ€˜π‘Š)
2 sitgval.j . . 3 𝐽 = (TopOpenβ€˜π‘Š)
3 sitgval.s . . 3 𝑆 = (sigaGenβ€˜π½)
4 sitgval.0 . . 3 0 = (0gβ€˜π‘Š)
5 sitgval.x . . 3 Β· = ( ·𝑠 β€˜π‘Š)
6 sitgval.h . . 3 𝐻 = (ℝHomβ€˜(Scalarβ€˜π‘Š))
7 sitgval.1 . . 3 (πœ‘ β†’ π‘Š ∈ 𝑉)
8 sitgval.2 . . 3 (πœ‘ β†’ 𝑀 ∈ βˆͺ ran measures)
91, 2, 3, 4, 5, 6, 7, 8sitgval 33331 . 2 (πœ‘ β†’ (π‘Šsitg𝑀) = (𝑓 ∈ {𝑔 ∈ (dom 𝑀MblFnM𝑆) ∣ (ran 𝑔 ∈ Fin ∧ βˆ€π‘₯ ∈ (ran 𝑔 βˆ– { 0 })(π‘€β€˜(◑𝑔 β€œ {π‘₯})) ∈ (0[,)+∞))} ↦ (π‘Š Ξ£g (π‘₯ ∈ (ran 𝑓 βˆ– { 0 }) ↦ ((π»β€˜(π‘€β€˜(◑𝑓 β€œ {π‘₯}))) Β· π‘₯)))))
10 simpr 486 . . . . . 6 ((πœ‘ ∧ 𝑓 = 𝐹) β†’ 𝑓 = 𝐹)
1110rneqd 5938 . . . . 5 ((πœ‘ ∧ 𝑓 = 𝐹) β†’ ran 𝑓 = ran 𝐹)
1211difeq1d 4122 . . . 4 ((πœ‘ ∧ 𝑓 = 𝐹) β†’ (ran 𝑓 βˆ– { 0 }) = (ran 𝐹 βˆ– { 0 }))
1310cnveqd 5876 . . . . . . . 8 ((πœ‘ ∧ 𝑓 = 𝐹) β†’ ◑𝑓 = ◑𝐹)
1413imaeq1d 6059 . . . . . . 7 ((πœ‘ ∧ 𝑓 = 𝐹) β†’ (◑𝑓 β€œ {π‘₯}) = (◑𝐹 β€œ {π‘₯}))
1514fveq2d 6896 . . . . . 6 ((πœ‘ ∧ 𝑓 = 𝐹) β†’ (π‘€β€˜(◑𝑓 β€œ {π‘₯})) = (π‘€β€˜(◑𝐹 β€œ {π‘₯})))
1615fveq2d 6896 . . . . 5 ((πœ‘ ∧ 𝑓 = 𝐹) β†’ (π»β€˜(π‘€β€˜(◑𝑓 β€œ {π‘₯}))) = (π»β€˜(π‘€β€˜(◑𝐹 β€œ {π‘₯}))))
1716oveq1d 7424 . . . 4 ((πœ‘ ∧ 𝑓 = 𝐹) β†’ ((π»β€˜(π‘€β€˜(◑𝑓 β€œ {π‘₯}))) Β· π‘₯) = ((π»β€˜(π‘€β€˜(◑𝐹 β€œ {π‘₯}))) Β· π‘₯))
1812, 17mpteq12dv 5240 . . 3 ((πœ‘ ∧ 𝑓 = 𝐹) β†’ (π‘₯ ∈ (ran 𝑓 βˆ– { 0 }) ↦ ((π»β€˜(π‘€β€˜(◑𝑓 β€œ {π‘₯}))) Β· π‘₯)) = (π‘₯ ∈ (ran 𝐹 βˆ– { 0 }) ↦ ((π»β€˜(π‘€β€˜(◑𝐹 β€œ {π‘₯}))) Β· π‘₯)))
1918oveq2d 7425 . 2 ((πœ‘ ∧ 𝑓 = 𝐹) β†’ (π‘Š Ξ£g (π‘₯ ∈ (ran 𝑓 βˆ– { 0 }) ↦ ((π»β€˜(π‘€β€˜(◑𝑓 β€œ {π‘₯}))) Β· π‘₯))) = (π‘Š Ξ£g (π‘₯ ∈ (ran 𝐹 βˆ– { 0 }) ↦ ((π»β€˜(π‘€β€˜(◑𝐹 β€œ {π‘₯}))) Β· π‘₯))))
20 sibfmbl.1 . . . . 5 (πœ‘ β†’ 𝐹 ∈ dom (π‘Šsitg𝑀))
211, 2, 3, 4, 5, 6, 7, 8, 20sibfmbl 33334 . . . 4 (πœ‘ β†’ 𝐹 ∈ (dom 𝑀MblFnM𝑆))
221, 2, 3, 4, 5, 6, 7, 8, 20sibfrn 33336 . . . 4 (πœ‘ β†’ ran 𝐹 ∈ Fin)
231, 2, 3, 4, 5, 6, 7, 8, 20sibfima 33337 . . . . 5 ((πœ‘ ∧ π‘₯ ∈ (ran 𝐹 βˆ– { 0 })) β†’ (π‘€β€˜(◑𝐹 β€œ {π‘₯})) ∈ (0[,)+∞))
2423ralrimiva 3147 . . . 4 (πœ‘ β†’ βˆ€π‘₯ ∈ (ran 𝐹 βˆ– { 0 })(π‘€β€˜(◑𝐹 β€œ {π‘₯})) ∈ (0[,)+∞))
2521, 22, 24jca32 517 . . 3 (πœ‘ β†’ (𝐹 ∈ (dom 𝑀MblFnM𝑆) ∧ (ran 𝐹 ∈ Fin ∧ βˆ€π‘₯ ∈ (ran 𝐹 βˆ– { 0 })(π‘€β€˜(◑𝐹 β€œ {π‘₯})) ∈ (0[,)+∞))))
26 rneq 5936 . . . . . 6 (𝑔 = 𝐹 β†’ ran 𝑔 = ran 𝐹)
2726eleq1d 2819 . . . . 5 (𝑔 = 𝐹 β†’ (ran 𝑔 ∈ Fin ↔ ran 𝐹 ∈ Fin))
2826difeq1d 4122 . . . . . 6 (𝑔 = 𝐹 β†’ (ran 𝑔 βˆ– { 0 }) = (ran 𝐹 βˆ– { 0 }))
29 cnveq 5874 . . . . . . . . 9 (𝑔 = 𝐹 β†’ ◑𝑔 = ◑𝐹)
3029imaeq1d 6059 . . . . . . . 8 (𝑔 = 𝐹 β†’ (◑𝑔 β€œ {π‘₯}) = (◑𝐹 β€œ {π‘₯}))
3130fveq2d 6896 . . . . . . 7 (𝑔 = 𝐹 β†’ (π‘€β€˜(◑𝑔 β€œ {π‘₯})) = (π‘€β€˜(◑𝐹 β€œ {π‘₯})))
3231eleq1d 2819 . . . . . 6 (𝑔 = 𝐹 β†’ ((π‘€β€˜(◑𝑔 β€œ {π‘₯})) ∈ (0[,)+∞) ↔ (π‘€β€˜(◑𝐹 β€œ {π‘₯})) ∈ (0[,)+∞)))
3328, 32raleqbidv 3343 . . . . 5 (𝑔 = 𝐹 β†’ (βˆ€π‘₯ ∈ (ran 𝑔 βˆ– { 0 })(π‘€β€˜(◑𝑔 β€œ {π‘₯})) ∈ (0[,)+∞) ↔ βˆ€π‘₯ ∈ (ran 𝐹 βˆ– { 0 })(π‘€β€˜(◑𝐹 β€œ {π‘₯})) ∈ (0[,)+∞)))
3427, 33anbi12d 632 . . . 4 (𝑔 = 𝐹 β†’ ((ran 𝑔 ∈ Fin ∧ βˆ€π‘₯ ∈ (ran 𝑔 βˆ– { 0 })(π‘€β€˜(◑𝑔 β€œ {π‘₯})) ∈ (0[,)+∞)) ↔ (ran 𝐹 ∈ Fin ∧ βˆ€π‘₯ ∈ (ran 𝐹 βˆ– { 0 })(π‘€β€˜(◑𝐹 β€œ {π‘₯})) ∈ (0[,)+∞))))
3534elrab 3684 . . 3 (𝐹 ∈ {𝑔 ∈ (dom 𝑀MblFnM𝑆) ∣ (ran 𝑔 ∈ Fin ∧ βˆ€π‘₯ ∈ (ran 𝑔 βˆ– { 0 })(π‘€β€˜(◑𝑔 β€œ {π‘₯})) ∈ (0[,)+∞))} ↔ (𝐹 ∈ (dom 𝑀MblFnM𝑆) ∧ (ran 𝐹 ∈ Fin ∧ βˆ€π‘₯ ∈ (ran 𝐹 βˆ– { 0 })(π‘€β€˜(◑𝐹 β€œ {π‘₯})) ∈ (0[,)+∞))))
3625, 35sylibr 233 . 2 (πœ‘ β†’ 𝐹 ∈ {𝑔 ∈ (dom 𝑀MblFnM𝑆) ∣ (ran 𝑔 ∈ Fin ∧ βˆ€π‘₯ ∈ (ran 𝑔 βˆ– { 0 })(π‘€β€˜(◑𝑔 β€œ {π‘₯})) ∈ (0[,)+∞))})
37 ovexd 7444 . 2 (πœ‘ β†’ (π‘Š Ξ£g (π‘₯ ∈ (ran 𝐹 βˆ– { 0 }) ↦ ((π»β€˜(π‘€β€˜(◑𝐹 β€œ {π‘₯}))) Β· π‘₯))) ∈ V)
389, 19, 36, 37fvmptd 7006 1 (πœ‘ β†’ ((π‘Šsitg𝑀)β€˜πΉ) = (π‘Š Ξ£g (π‘₯ ∈ (ran 𝐹 βˆ– { 0 }) ↦ ((π»β€˜(π‘€β€˜(◑𝐹 β€œ {π‘₯}))) Β· π‘₯))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062  {crab 3433  Vcvv 3475   βˆ– cdif 3946  {csn 4629  βˆͺ cuni 4909   ↦ cmpt 5232  β—‘ccnv 5676  dom cdm 5677  ran crn 5678   β€œ cima 5680  β€˜cfv 6544  (class class class)co 7409  Fincfn 8939  0cc0 11110  +∞cpnf 11245  [,)cico 13326  Basecbs 17144  Scalarcsca 17200   ·𝑠 cvsca 17201  TopOpenctopn 17367  0gc0g 17385   Ξ£g cgsu 17386  β„Homcrrh 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:  sitgclg  33341  sitg0  33345
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