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Theorem isi1f 25054
Description: The predicate "𝐹 is a simple function". A simple function is a finite nonnegative linear combination of indicator functions for finitely measurable sets. We use the idiom 𝐹 ∈ dom ∫1 to represent this concept because ∫1 is the first preparation function for our final definition ∫ (see df-itg 25003); unlike that operator, which can integrate any function, this operator can only integrate simple functions. (Contributed by Mario Carneiro, 18-Jun-2014.)
Assertion
Ref Expression
isi1f (𝐹 ∈ dom ∫1 ↔ (𝐹 ∈ MblFn ∧ (𝐹:β„βŸΆβ„ ∧ ran 𝐹 ∈ Fin ∧ (volβ€˜(◑𝐹 β€œ (ℝ βˆ– {0}))) ∈ ℝ)))

Proof of Theorem isi1f
Dummy variables 𝑓 𝑔 π‘₯ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 feq1 6654 . . 3 (𝑔 = 𝐹 β†’ (𝑔:β„βŸΆβ„ ↔ 𝐹:β„βŸΆβ„))
2 rneq 5896 . . . 4 (𝑔 = 𝐹 β†’ ran 𝑔 = ran 𝐹)
32eleq1d 2823 . . 3 (𝑔 = 𝐹 β†’ (ran 𝑔 ∈ Fin ↔ ran 𝐹 ∈ Fin))
4 cnveq 5834 . . . . . 6 (𝑔 = 𝐹 β†’ ◑𝑔 = ◑𝐹)
54imaeq1d 6017 . . . . 5 (𝑔 = 𝐹 β†’ (◑𝑔 β€œ (ℝ βˆ– {0})) = (◑𝐹 β€œ (ℝ βˆ– {0})))
65fveq2d 6851 . . . 4 (𝑔 = 𝐹 β†’ (volβ€˜(◑𝑔 β€œ (ℝ βˆ– {0}))) = (volβ€˜(◑𝐹 β€œ (ℝ βˆ– {0}))))
76eleq1d 2823 . . 3 (𝑔 = 𝐹 β†’ ((volβ€˜(◑𝑔 β€œ (ℝ βˆ– {0}))) ∈ ℝ ↔ (volβ€˜(◑𝐹 β€œ (ℝ βˆ– {0}))) ∈ ℝ))
81, 3, 73anbi123d 1437 . 2 (𝑔 = 𝐹 β†’ ((𝑔:β„βŸΆβ„ ∧ ran 𝑔 ∈ Fin ∧ (volβ€˜(◑𝑔 β€œ (ℝ βˆ– {0}))) ∈ ℝ) ↔ (𝐹:β„βŸΆβ„ ∧ ran 𝐹 ∈ Fin ∧ (volβ€˜(◑𝐹 β€œ (ℝ βˆ– {0}))) ∈ ℝ)))
9 sumex 15579 . . 3 Ξ£π‘₯ ∈ (ran 𝑓 βˆ– {0})(π‘₯ Β· (volβ€˜(◑𝑓 β€œ {π‘₯}))) ∈ V
10 df-itg1 25000 . . 3 ∫1 = (𝑓 ∈ {𝑔 ∈ MblFn ∣ (𝑔:β„βŸΆβ„ ∧ ran 𝑔 ∈ Fin ∧ (volβ€˜(◑𝑔 β€œ (ℝ βˆ– {0}))) ∈ ℝ)} ↦ Ξ£π‘₯ ∈ (ran 𝑓 βˆ– {0})(π‘₯ Β· (volβ€˜(◑𝑓 β€œ {π‘₯}))))
119, 10dmmpti 6650 . 2 dom ∫1 = {𝑔 ∈ MblFn ∣ (𝑔:β„βŸΆβ„ ∧ ran 𝑔 ∈ Fin ∧ (volβ€˜(◑𝑔 β€œ (ℝ βˆ– {0}))) ∈ ℝ)}
128, 11elrab2 3653 1 (𝐹 ∈ dom ∫1 ↔ (𝐹 ∈ MblFn ∧ (𝐹:β„βŸΆβ„ ∧ ran 𝐹 ∈ Fin ∧ (volβ€˜(◑𝐹 β€œ (ℝ βˆ– {0}))) ∈ ℝ)))
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  {crab 3410   βˆ– cdif 3912  {csn 4591  β—‘ccnv 5637  dom cdm 5638  ran crn 5639   β€œ cima 5641  βŸΆwf 6497  β€˜cfv 6501  (class class class)co 7362  Fincfn 8890  β„cr 11057  0cc0 11058   Β· cmul 11063  Ξ£csu 15577  volcvol 24843  MblFncmbf 24994  βˆ«1citg1 24995
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-sep 5261  ax-nul 5268  ax-pr 5389
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-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  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-fv 6509  df-sum 15578  df-itg1 25000
This theorem is referenced by:  i1fmbf  25055  i1ff  25056  i1frn  25057  i1fima2  25059  i1fd  25061
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