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Theorem iblsplit 44668
Description: The union of two integrable functions is integrable. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypotheses
Ref Expression
iblsplit.1 (πœ‘ β†’ (vol*β€˜(𝐴 ∩ 𝐡)) = 0)
iblsplit.2 (πœ‘ β†’ π‘ˆ = (𝐴 βˆͺ 𝐡))
iblsplit.3 ((πœ‘ ∧ π‘₯ ∈ π‘ˆ) β†’ 𝐢 ∈ β„‚)
iblsplit.4 (πœ‘ β†’ (π‘₯ ∈ 𝐴 ↦ 𝐢) ∈ 𝐿1)
iblsplit.5 (πœ‘ β†’ (π‘₯ ∈ 𝐡 ↦ 𝐢) ∈ 𝐿1)
Assertion
Ref Expression
iblsplit (πœ‘ β†’ (π‘₯ ∈ π‘ˆ ↦ 𝐢) ∈ 𝐿1)
Distinct variable groups:   π‘₯,𝐴   π‘₯,𝐡   π‘₯,π‘ˆ   πœ‘,π‘₯
Allowed substitution hint:   𝐢(π‘₯)
Proof of Theorem iblsplit
Dummy variables π‘˜ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 iblsplit.3 . . . 4 ((πœ‘ ∧ π‘₯ ∈ π‘ˆ) β†’ 𝐢 ∈ β„‚)
21fmpttd 7111 . . 3 (πœ‘ β†’ (π‘₯ ∈ π‘ˆ ↦ 𝐢):π‘ˆβŸΆβ„‚)
3 ssun1 4171 . . . . . 6 𝐴 βŠ† (𝐴 βˆͺ 𝐡)
4 iblsplit.2 . . . . . 6 (πœ‘ β†’ π‘ˆ = (𝐴 βˆͺ 𝐡))
53, 4sseqtrrid 4034 . . . . 5 (πœ‘ β†’ 𝐴 βŠ† π‘ˆ)
65resmptd 6038 . . . 4 (πœ‘ β†’ ((π‘₯ ∈ π‘ˆ ↦ 𝐢) β†Ύ 𝐴) = (π‘₯ ∈ 𝐴 ↦ 𝐢))
7 iblsplit.4 . . . . . 6 (πœ‘ β†’ (π‘₯ ∈ 𝐴 ↦ 𝐢) ∈ 𝐿1)
8 eqidd 2733 . . . . . . 7 (πœ‘ β†’ (π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ (β„œβ€˜(𝐢 / (i↑𝑦)))), (β„œβ€˜(𝐢 / (i↑𝑦))), 0)) = (π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ (β„œβ€˜(𝐢 / (i↑𝑦)))), (β„œβ€˜(𝐢 / (i↑𝑦))), 0)))
9 eqidd 2733 . . . . . . 7 ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ (β„œβ€˜(𝐢 / (i↑𝑦))) = (β„œβ€˜(𝐢 / (i↑𝑦))))
105sseld 3980 . . . . . . . . 9 (πœ‘ β†’ (π‘₯ ∈ 𝐴 β†’ π‘₯ ∈ π‘ˆ))
1110imdistani 569 . . . . . . . 8 ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ (πœ‘ ∧ π‘₯ ∈ π‘ˆ))
1211, 1syl 17 . . . . . . 7 ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ 𝐢 ∈ β„‚)
138, 9, 12isibl2 25275 . . . . . 6 (πœ‘ β†’ ((π‘₯ ∈ 𝐴 ↦ 𝐢) ∈ 𝐿1 ↔ ((π‘₯ ∈ 𝐴 ↦ 𝐢) ∈ MblFn ∧ βˆ€π‘¦ ∈ (0...3)(∫2β€˜(π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ (β„œβ€˜(𝐢 / (i↑𝑦)))), (β„œβ€˜(𝐢 / (i↑𝑦))), 0))) ∈ ℝ)))
147, 13mpbid 231 . . . . 5 (πœ‘ β†’ ((π‘₯ ∈ 𝐴 ↦ 𝐢) ∈ MblFn ∧ βˆ€π‘¦ ∈ (0...3)(∫2β€˜(π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ (β„œβ€˜(𝐢 / (i↑𝑦)))), (β„œβ€˜(𝐢 / (i↑𝑦))), 0))) ∈ ℝ))
1514simpld 495 . . . 4 (πœ‘ β†’ (π‘₯ ∈ 𝐴 ↦ 𝐢) ∈ MblFn)
166, 15eqeltrd 2833 . . 3 (πœ‘ β†’ ((π‘₯ ∈ π‘ˆ ↦ 𝐢) β†Ύ 𝐴) ∈ MblFn)
17 ssun2 4172 . . . . . 6 𝐡 βŠ† (𝐴 βˆͺ 𝐡)
1817, 4sseqtrrid 4034 . . . . 5 (πœ‘ β†’ 𝐡 βŠ† π‘ˆ)
1918resmptd 6038 . . . 4 (πœ‘ β†’ ((π‘₯ ∈ π‘ˆ ↦ 𝐢) β†Ύ 𝐡) = (π‘₯ ∈ 𝐡 ↦ 𝐢))
20 iblsplit.5 . . . . . 6 (πœ‘ β†’ (π‘₯ ∈ 𝐡 ↦ 𝐢) ∈ 𝐿1)
21 eqidd 2733 . . . . . . 7 (πœ‘ β†’ (π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐡 ∧ 0 ≀ (β„œβ€˜(𝐢 / (i↑𝑦)))), (β„œβ€˜(𝐢 / (i↑𝑦))), 0)) = (π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐡 ∧ 0 ≀ (β„œβ€˜(𝐢 / (i↑𝑦)))), (β„œβ€˜(𝐢 / (i↑𝑦))), 0)))
22 eqidd 2733 . . . . . . 7 ((πœ‘ ∧ π‘₯ ∈ 𝐡) β†’ (β„œβ€˜(𝐢 / (i↑𝑦))) = (β„œβ€˜(𝐢 / (i↑𝑦))))
2318sseld 3980 . . . . . . . . 9 (πœ‘ β†’ (π‘₯ ∈ 𝐡 β†’ π‘₯ ∈ π‘ˆ))
2423imdistani 569 . . . . . . . 8 ((πœ‘ ∧ π‘₯ ∈ 𝐡) β†’ (πœ‘ ∧ π‘₯ ∈ π‘ˆ))
2524, 1syl 17 . . . . . . 7 ((πœ‘ ∧ π‘₯ ∈ 𝐡) β†’ 𝐢 ∈ β„‚)
2621, 22, 25isibl2 25275 . . . . . 6 (πœ‘ β†’ ((π‘₯ ∈ 𝐡 ↦ 𝐢) ∈ 𝐿1 ↔ ((π‘₯ ∈ 𝐡 ↦ 𝐢) ∈ MblFn ∧ βˆ€π‘¦ ∈ (0...3)(∫2β€˜(π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐡 ∧ 0 ≀ (β„œβ€˜(𝐢 / (i↑𝑦)))), (β„œβ€˜(𝐢 / (i↑𝑦))), 0))) ∈ ℝ)))
2720, 26mpbid 231 . . . . 5 (πœ‘ β†’ ((π‘₯ ∈ 𝐡 ↦ 𝐢) ∈ MblFn ∧ βˆ€π‘¦ ∈ (0...3)(∫2β€˜(π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐡 ∧ 0 ≀ (β„œβ€˜(𝐢 / (i↑𝑦)))), (β„œβ€˜(𝐢 / (i↑𝑦))), 0))) ∈ ℝ))
2827simpld 495 . . . 4 (πœ‘ β†’ (π‘₯ ∈ 𝐡 ↦ 𝐢) ∈ MblFn)
2919, 28eqeltrd 2833 . . 3 (πœ‘ β†’ ((π‘₯ ∈ π‘ˆ ↦ 𝐢) β†Ύ 𝐡) ∈ MblFn)
304eqcomd 2738 . . 3 (πœ‘ β†’ (𝐴 βˆͺ 𝐡) = π‘ˆ)
312, 16, 29, 30mbfres2cn 44660 . 2 (πœ‘ β†’ (π‘₯ ∈ π‘ˆ ↦ 𝐢) ∈ MblFn)
3215, 12mbfdm2 25145 . . . . . 6 (πœ‘ β†’ 𝐴 ∈ dom vol)
3332adantr 481 . . . . 5 ((πœ‘ ∧ π‘˜ ∈ (0...3)) β†’ 𝐴 ∈ dom vol)
3428, 25mbfdm2 25145 . . . . . 6 (πœ‘ β†’ 𝐡 ∈ dom vol)
3534adantr 481 . . . . 5 ((πœ‘ ∧ π‘˜ ∈ (0...3)) β†’ 𝐡 ∈ dom vol)
36 iblsplit.1 . . . . . 6 (πœ‘ β†’ (vol*β€˜(𝐴 ∩ 𝐡)) = 0)
3736adantr 481 . . . . 5 ((πœ‘ ∧ π‘˜ ∈ (0...3)) β†’ (vol*β€˜(𝐴 ∩ 𝐡)) = 0)
384adantr 481 . . . . 5 ((πœ‘ ∧ π‘˜ ∈ (0...3)) β†’ π‘ˆ = (𝐴 βˆͺ 𝐡))
391adantlr 713 . . . . . . . . . . 11 (((πœ‘ ∧ π‘˜ ∈ (0...3)) ∧ π‘₯ ∈ π‘ˆ) β†’ 𝐢 ∈ β„‚)
40 ax-icn 11165 . . . . . . . . . . . . . 14 i ∈ β„‚
4140a1i 11 . . . . . . . . . . . . 13 (π‘˜ ∈ (0...3) β†’ i ∈ β„‚)
42 elfznn0 13590 . . . . . . . . . . . . 13 (π‘˜ ∈ (0...3) β†’ π‘˜ ∈ β„•0)
4341, 42expcld 14107 . . . . . . . . . . . 12 (π‘˜ ∈ (0...3) β†’ (iβ†‘π‘˜) ∈ β„‚)
4443ad2antlr 725 . . . . . . . . . . 11 (((πœ‘ ∧ π‘˜ ∈ (0...3)) ∧ π‘₯ ∈ π‘ˆ) β†’ (iβ†‘π‘˜) ∈ β„‚)
4540a1i 11 . . . . . . . . . . . 12 (((πœ‘ ∧ π‘˜ ∈ (0...3)) ∧ π‘₯ ∈ π‘ˆ) β†’ i ∈ β„‚)
46 ine0 11645 . . . . . . . . . . . . 13 i β‰  0
4746a1i 11 . . . . . . . . . . . 12 (((πœ‘ ∧ π‘˜ ∈ (0...3)) ∧ π‘₯ ∈ π‘ˆ) β†’ i β‰  0)
48 elfzelz 13497 . . . . . . . . . . . . 13 (π‘˜ ∈ (0...3) β†’ π‘˜ ∈ β„€)
4948ad2antlr 725 . . . . . . . . . . . 12 (((πœ‘ ∧ π‘˜ ∈ (0...3)) ∧ π‘₯ ∈ π‘ˆ) β†’ π‘˜ ∈ β„€)
5045, 47, 49expne0d 14113 . . . . . . . . . . 11 (((πœ‘ ∧ π‘˜ ∈ (0...3)) ∧ π‘₯ ∈ π‘ˆ) β†’ (iβ†‘π‘˜) β‰  0)
5139, 44, 50divcld 11986 . . . . . . . . . 10 (((πœ‘ ∧ π‘˜ ∈ (0...3)) ∧ π‘₯ ∈ π‘ˆ) β†’ (𝐢 / (iβ†‘π‘˜)) ∈ β„‚)
5251recld 15137 . . . . . . . . 9 (((πœ‘ ∧ π‘˜ ∈ (0...3)) ∧ π‘₯ ∈ π‘ˆ) β†’ (β„œβ€˜(𝐢 / (iβ†‘π‘˜))) ∈ ℝ)
5352rexrd 11260 . . . . . . . 8 (((πœ‘ ∧ π‘˜ ∈ (0...3)) ∧ π‘₯ ∈ π‘ˆ) β†’ (β„œβ€˜(𝐢 / (iβ†‘π‘˜))) ∈ ℝ*)
5453adantr 481 . . . . . . 7 ((((πœ‘ ∧ π‘˜ ∈ (0...3)) ∧ π‘₯ ∈ π‘ˆ) ∧ 0 ≀ (β„œβ€˜(𝐢 / (iβ†‘π‘˜)))) β†’ (β„œβ€˜(𝐢 / (iβ†‘π‘˜))) ∈ ℝ*)
55 simpr 485 . . . . . . 7 ((((πœ‘ ∧ π‘˜ ∈ (0...3)) ∧ π‘₯ ∈ π‘ˆ) ∧ 0 ≀ (β„œβ€˜(𝐢 / (iβ†‘π‘˜)))) β†’ 0 ≀ (β„œβ€˜(𝐢 / (iβ†‘π‘˜))))
56 pnfge 13106 . . . . . . . 8 ((β„œβ€˜(𝐢 / (iβ†‘π‘˜))) ∈ ℝ* β†’ (β„œβ€˜(𝐢 / (iβ†‘π‘˜))) ≀ +∞)
5754, 56syl 17 . . . . . . 7 ((((πœ‘ ∧ π‘˜ ∈ (0...3)) ∧ π‘₯ ∈ π‘ˆ) ∧ 0 ≀ (β„œβ€˜(𝐢 / (iβ†‘π‘˜)))) β†’ (β„œβ€˜(𝐢 / (iβ†‘π‘˜))) ≀ +∞)
58 0xr 11257 . . . . . . . 8 0 ∈ ℝ*
59 pnfxr 11264 . . . . . . . 8 +∞ ∈ ℝ*
60 elicc1 13364 . . . . . . . 8 ((0 ∈ ℝ* ∧ +∞ ∈ ℝ*) β†’ ((β„œβ€˜(𝐢 / (iβ†‘π‘˜))) ∈ (0[,]+∞) ↔ ((β„œβ€˜(𝐢 / (iβ†‘π‘˜))) ∈ ℝ* ∧ 0 ≀ (β„œβ€˜(𝐢 / (iβ†‘π‘˜))) ∧ (β„œβ€˜(𝐢 / (iβ†‘π‘˜))) ≀ +∞)))
6158, 59, 60mp2an 690 . . . . . . 7 ((β„œβ€˜(𝐢 / (iβ†‘π‘˜))) ∈ (0[,]+∞) ↔ ((β„œβ€˜(𝐢 / (iβ†‘π‘˜))) ∈ ℝ* ∧ 0 ≀ (β„œβ€˜(𝐢 / (iβ†‘π‘˜))) ∧ (β„œβ€˜(𝐢 / (iβ†‘π‘˜))) ≀ +∞))
6254, 55, 57, 61syl3anbrc 1343 . . . . . 6 ((((πœ‘ ∧ π‘˜ ∈ (0...3)) ∧ π‘₯ ∈ π‘ˆ) ∧ 0 ≀ (β„œβ€˜(𝐢 / (iβ†‘π‘˜)))) β†’ (β„œβ€˜(𝐢 / (iβ†‘π‘˜))) ∈ (0[,]+∞))
63 0e0iccpnf 13432 . . . . . . 7 0 ∈ (0[,]+∞)
6463a1i 11 . . . . . 6 ((((πœ‘ ∧ π‘˜ ∈ (0...3)) ∧ π‘₯ ∈ π‘ˆ) ∧ Β¬ 0 ≀ (β„œβ€˜(𝐢 / (iβ†‘π‘˜)))) β†’ 0 ∈ (0[,]+∞))
6562, 64ifclda 4562 . . . . 5 (((πœ‘ ∧ π‘˜ ∈ (0...3)) ∧ π‘₯ ∈ π‘ˆ) β†’ if(0 ≀ (β„œβ€˜(𝐢 / (iβ†‘π‘˜))), (β„œβ€˜(𝐢 / (iβ†‘π‘˜))), 0) ∈ (0[,]+∞))
66 eqid 2732 . . . . 5 (π‘₯ ∈ ℝ ↦ if(π‘₯ ∈ 𝐴, if(0 ≀ (β„œβ€˜(𝐢 / (iβ†‘π‘˜))), (β„œβ€˜(𝐢 / (iβ†‘π‘˜))), 0), 0)) = (π‘₯ ∈ ℝ ↦ if(π‘₯ ∈ 𝐴, if(0 ≀ (β„œβ€˜(𝐢 / (iβ†‘π‘˜))), (β„œβ€˜(𝐢 / (iβ†‘π‘˜))), 0), 0))
67 eqid 2732 . . . . 5 (π‘₯ ∈ ℝ ↦ if(π‘₯ ∈ 𝐡, if(0 ≀ (β„œβ€˜(𝐢 / (iβ†‘π‘˜))), (β„œβ€˜(𝐢 / (iβ†‘π‘˜))), 0), 0)) = (π‘₯ ∈ ℝ ↦ if(π‘₯ ∈ 𝐡, if(0 ≀ (β„œβ€˜(𝐢 / (iβ†‘π‘˜))), (β„œβ€˜(𝐢 / (iβ†‘π‘˜))), 0), 0))
68 ifan 4580 . . . . . 6 if((π‘₯ ∈ π‘ˆ ∧ 0 ≀ (β„œβ€˜(𝐢 / (iβ†‘π‘˜)))), (β„œβ€˜(𝐢 / (iβ†‘π‘˜))), 0) = if(π‘₯ ∈ π‘ˆ, if(0 ≀ (β„œβ€˜(𝐢 / (iβ†‘π‘˜))), (β„œβ€˜(𝐢 / (iβ†‘π‘˜))), 0), 0)
6968mpteq2i 5252 . . . . 5 (π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ π‘ˆ ∧ 0 ≀ (β„œβ€˜(𝐢 / (iβ†‘π‘˜)))), (β„œβ€˜(𝐢 / (iβ†‘π‘˜))), 0)) = (π‘₯ ∈ ℝ ↦ if(π‘₯ ∈ π‘ˆ, if(0 ≀ (β„œβ€˜(𝐢 / (iβ†‘π‘˜))), (β„œβ€˜(𝐢 / (iβ†‘π‘˜))), 0), 0))
70 ifan 4580 . . . . . . . . . 10 if((π‘₯ ∈ 𝐴 ∧ 0 ≀ (β„œβ€˜(𝐢 / (iβ†‘π‘˜)))), (β„œβ€˜(𝐢 / (iβ†‘π‘˜))), 0) = if(π‘₯ ∈ 𝐴, if(0 ≀ (β„œβ€˜(𝐢 / (iβ†‘π‘˜))), (β„œβ€˜(𝐢 / (iβ†‘π‘˜))), 0), 0)
7170eqcomi 2741 . . . . . . . . 9 if(π‘₯ ∈ 𝐴, if(0 ≀ (β„œβ€˜(𝐢 / (iβ†‘π‘˜))), (β„œβ€˜(𝐢 / (iβ†‘π‘˜))), 0), 0) = if((π‘₯ ∈ 𝐴 ∧ 0 ≀ (β„œβ€˜(𝐢 / (iβ†‘π‘˜)))), (β„œβ€˜(𝐢 / (iβ†‘π‘˜))), 0)
7271mpteq2i 5252 . . . . . . . 8 (π‘₯ ∈ ℝ ↦ if(π‘₯ ∈ 𝐴, if(0 ≀ (β„œβ€˜(𝐢 / (iβ†‘π‘˜))), (β„œβ€˜(𝐢 / (iβ†‘π‘˜))), 0), 0)) = (π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ (β„œβ€˜(𝐢 / (iβ†‘π‘˜)))), (β„œβ€˜(𝐢 / (iβ†‘π‘˜))), 0))
7372a1i 11 . . . . . . 7 ((πœ‘ ∧ π‘˜ ∈ (0...3)) β†’ (π‘₯ ∈ ℝ ↦ if(π‘₯ ∈ 𝐴, if(0 ≀ (β„œβ€˜(𝐢 / (iβ†‘π‘˜))), (β„œβ€˜(𝐢 / (iβ†‘π‘˜))), 0), 0)) = (π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ (β„œβ€˜(𝐢 / (iβ†‘π‘˜)))), (β„œβ€˜(𝐢 / (iβ†‘π‘˜))), 0)))
7473fveq2d 6892 . . . . . 6 ((πœ‘ ∧ π‘˜ ∈ (0...3)) β†’ (∫2β€˜(π‘₯ ∈ ℝ ↦ if(π‘₯ ∈ 𝐴, if(0 ≀ (β„œβ€˜(𝐢 / (iβ†‘π‘˜))), (β„œβ€˜(𝐢 / (iβ†‘π‘˜))), 0), 0))) = (∫2β€˜(π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ (β„œβ€˜(𝐢 / (iβ†‘π‘˜)))), (β„œβ€˜(𝐢 / (iβ†‘π‘˜))), 0))))
75 eqidd 2733 . . . . . . . . . 10 (πœ‘ β†’ (π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ (β„œβ€˜(𝐢 / (iβ†‘π‘˜)))), (β„œβ€˜(𝐢 / (iβ†‘π‘˜))), 0)) = (π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ (β„œβ€˜(𝐢 / (iβ†‘π‘˜)))), (β„œβ€˜(𝐢 / (iβ†‘π‘˜))), 0)))
76 eqidd 2733 . . . . . . . . . 10 ((πœ‘ ∧ π‘₯ ∈ 𝐴) β†’ (β„œβ€˜(𝐢 / (iβ†‘π‘˜))) = (β„œβ€˜(𝐢 / (iβ†‘π‘˜))))
7775, 76, 12isibl2 25275 . . . . . . . . 9 (πœ‘ β†’ ((π‘₯ ∈ 𝐴 ↦ 𝐢) ∈ 𝐿1 ↔ ((π‘₯ ∈ 𝐴 ↦ 𝐢) ∈ MblFn ∧ βˆ€π‘˜ ∈ (0...3)(∫2β€˜(π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ (β„œβ€˜(𝐢 / (iβ†‘π‘˜)))), (β„œβ€˜(𝐢 / (iβ†‘π‘˜))), 0))) ∈ ℝ)))
787, 77mpbid 231 . . . . . . . 8 (πœ‘ β†’ ((π‘₯ ∈ 𝐴 ↦ 𝐢) ∈ MblFn ∧ βˆ€π‘˜ ∈ (0...3)(∫2β€˜(π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ (β„œβ€˜(𝐢 / (iβ†‘π‘˜)))), (β„œβ€˜(𝐢 / (iβ†‘π‘˜))), 0))) ∈ ℝ))
7978simprd 496 . . . . . . 7 (πœ‘ β†’ βˆ€π‘˜ ∈ (0...3)(∫2β€˜(π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ (β„œβ€˜(𝐢 / (iβ†‘π‘˜)))), (β„œβ€˜(𝐢 / (iβ†‘π‘˜))), 0))) ∈ ℝ)
8079r19.21bi 3248 . . . . . 6 ((πœ‘ ∧ π‘˜ ∈ (0...3)) β†’ (∫2β€˜(π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐴 ∧ 0 ≀ (β„œβ€˜(𝐢 / (iβ†‘π‘˜)))), (β„œβ€˜(𝐢 / (iβ†‘π‘˜))), 0))) ∈ ℝ)
8174, 80eqeltrd 2833 . . . . 5 ((πœ‘ ∧ π‘˜ ∈ (0...3)) β†’ (∫2β€˜(π‘₯ ∈ ℝ ↦ if(π‘₯ ∈ 𝐴, if(0 ≀ (β„œβ€˜(𝐢 / (iβ†‘π‘˜))), (β„œβ€˜(𝐢 / (iβ†‘π‘˜))), 0), 0))) ∈ ℝ)
82 ifan 4580 . . . . . . . . 9 if((π‘₯ ∈ 𝐡 ∧ 0 ≀ (β„œβ€˜(𝐢 / (iβ†‘π‘˜)))), (β„œβ€˜(𝐢 / (iβ†‘π‘˜))), 0) = if(π‘₯ ∈ 𝐡, if(0 ≀ (β„œβ€˜(𝐢 / (iβ†‘π‘˜))), (β„œβ€˜(𝐢 / (iβ†‘π‘˜))), 0), 0)
8382eqcomi 2741 . . . . . . . 8 if(π‘₯ ∈ 𝐡, if(0 ≀ (β„œβ€˜(𝐢 / (iβ†‘π‘˜))), (β„œβ€˜(𝐢 / (iβ†‘π‘˜))), 0), 0) = if((π‘₯ ∈ 𝐡 ∧ 0 ≀ (β„œβ€˜(𝐢 / (iβ†‘π‘˜)))), (β„œβ€˜(𝐢 / (iβ†‘π‘˜))), 0)
8483mpteq2i 5252 . . . . . . 7 (π‘₯ ∈ ℝ ↦ if(π‘₯ ∈ 𝐡, if(0 ≀ (β„œβ€˜(𝐢 / (iβ†‘π‘˜))), (β„œβ€˜(𝐢 / (iβ†‘π‘˜))), 0), 0)) = (π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐡 ∧ 0 ≀ (β„œβ€˜(𝐢 / (iβ†‘π‘˜)))), (β„œβ€˜(𝐢 / (iβ†‘π‘˜))), 0))
8584fveq2i 6891 . . . . . 6 (∫2β€˜(π‘₯ ∈ ℝ ↦ if(π‘₯ ∈ 𝐡, if(0 ≀ (β„œβ€˜(𝐢 / (iβ†‘π‘˜))), (β„œβ€˜(𝐢 / (iβ†‘π‘˜))), 0), 0))) = (∫2β€˜(π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐡 ∧ 0 ≀ (β„œβ€˜(𝐢 / (iβ†‘π‘˜)))), (β„œβ€˜(𝐢 / (iβ†‘π‘˜))), 0)))
86 eqidd 2733 . . . . . . . . . 10 (πœ‘ β†’ (π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐡 ∧ 0 ≀ (β„œβ€˜(𝐢 / (iβ†‘π‘˜)))), (β„œβ€˜(𝐢 / (iβ†‘π‘˜))), 0)) = (π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐡 ∧ 0 ≀ (β„œβ€˜(𝐢 / (iβ†‘π‘˜)))), (β„œβ€˜(𝐢 / (iβ†‘π‘˜))), 0)))
87 eqidd 2733 . . . . . . . . . 10 ((πœ‘ ∧ π‘₯ ∈ 𝐡) β†’ (β„œβ€˜(𝐢 / (iβ†‘π‘˜))) = (β„œβ€˜(𝐢 / (iβ†‘π‘˜))))
8886, 87, 25isibl2 25275 . . . . . . . . 9 (πœ‘ β†’ ((π‘₯ ∈ 𝐡 ↦ 𝐢) ∈ 𝐿1 ↔ ((π‘₯ ∈ 𝐡 ↦ 𝐢) ∈ MblFn ∧ βˆ€π‘˜ ∈ (0...3)(∫2β€˜(π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐡 ∧ 0 ≀ (β„œβ€˜(𝐢 / (iβ†‘π‘˜)))), (β„œβ€˜(𝐢 / (iβ†‘π‘˜))), 0))) ∈ ℝ)))
8920, 88mpbid 231 . . . . . . . 8 (πœ‘ β†’ ((π‘₯ ∈ 𝐡 ↦ 𝐢) ∈ MblFn ∧ βˆ€π‘˜ ∈ (0...3)(∫2β€˜(π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐡 ∧ 0 ≀ (β„œβ€˜(𝐢 / (iβ†‘π‘˜)))), (β„œβ€˜(𝐢 / (iβ†‘π‘˜))), 0))) ∈ ℝ))
9089simprd 496 . . . . . . 7 (πœ‘ β†’ βˆ€π‘˜ ∈ (0...3)(∫2β€˜(π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐡 ∧ 0 ≀ (β„œβ€˜(𝐢 / (iβ†‘π‘˜)))), (β„œβ€˜(𝐢 / (iβ†‘π‘˜))), 0))) ∈ ℝ)
9190r19.21bi 3248 . . . . . 6 ((πœ‘ ∧ π‘˜ ∈ (0...3)) β†’ (∫2β€˜(π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ 𝐡 ∧ 0 ≀ (β„œβ€˜(𝐢 / (iβ†‘π‘˜)))), (β„œβ€˜(𝐢 / (iβ†‘π‘˜))), 0))) ∈ ℝ)
9285, 91eqeltrid 2837 . . . . 5 ((πœ‘ ∧ π‘˜ ∈ (0...3)) β†’ (∫2β€˜(π‘₯ ∈ ℝ ↦ if(π‘₯ ∈ 𝐡, if(0 ≀ (β„œβ€˜(𝐢 / (iβ†‘π‘˜))), (β„œβ€˜(𝐢 / (iβ†‘π‘˜))), 0), 0))) ∈ ℝ)
9333, 35, 37, 38, 65, 66, 67, 69, 81, 92itg2split 25258 . . . 4 ((πœ‘ ∧ π‘˜ ∈ (0...3)) β†’ (∫2β€˜(π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ π‘ˆ ∧ 0 ≀ (β„œβ€˜(𝐢 / (iβ†‘π‘˜)))), (β„œβ€˜(𝐢 / (iβ†‘π‘˜))), 0))) = ((∫2β€˜(π‘₯ ∈ ℝ ↦ if(π‘₯ ∈ 𝐴, if(0 ≀ (β„œβ€˜(𝐢 / (iβ†‘π‘˜))), (β„œβ€˜(𝐢 / (iβ†‘π‘˜))), 0), 0))) + (∫2β€˜(π‘₯ ∈ ℝ ↦ if(π‘₯ ∈ 𝐡, if(0 ≀ (β„œβ€˜(𝐢 / (iβ†‘π‘˜))), (β„œβ€˜(𝐢 / (iβ†‘π‘˜))), 0), 0)))))
9481, 92readdcld 11239 . . . 4 ((πœ‘ ∧ π‘˜ ∈ (0...3)) β†’ ((∫2β€˜(π‘₯ ∈ ℝ ↦ if(π‘₯ ∈ 𝐴, if(0 ≀ (β„œβ€˜(𝐢 / (iβ†‘π‘˜))), (β„œβ€˜(𝐢 / (iβ†‘π‘˜))), 0), 0))) + (∫2β€˜(π‘₯ ∈ ℝ ↦ if(π‘₯ ∈ 𝐡, if(0 ≀ (β„œβ€˜(𝐢 / (iβ†‘π‘˜))), (β„œβ€˜(𝐢 / (iβ†‘π‘˜))), 0), 0)))) ∈ ℝ)
9593, 94eqeltrd 2833 . . 3 ((πœ‘ ∧ π‘˜ ∈ (0...3)) β†’ (∫2β€˜(π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ π‘ˆ ∧ 0 ≀ (β„œβ€˜(𝐢 / (iβ†‘π‘˜)))), (β„œβ€˜(𝐢 / (iβ†‘π‘˜))), 0))) ∈ ℝ)
9695ralrimiva 3146 . 2 (πœ‘ β†’ βˆ€π‘˜ ∈ (0...3)(∫2β€˜(π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ π‘ˆ ∧ 0 ≀ (β„œβ€˜(𝐢 / (iβ†‘π‘˜)))), (β„œβ€˜(𝐢 / (iβ†‘π‘˜))), 0))) ∈ ℝ)
97 eqidd 2733 . . 3 (πœ‘ β†’ (π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ π‘ˆ ∧ 0 ≀ (β„œβ€˜(𝐢 / (iβ†‘π‘˜)))), (β„œβ€˜(𝐢 / (iβ†‘π‘˜))), 0)) = (π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ π‘ˆ ∧ 0 ≀ (β„œβ€˜(𝐢 / (iβ†‘π‘˜)))), (β„œβ€˜(𝐢 / (iβ†‘π‘˜))), 0)))
98 eqidd 2733 . . 3 ((πœ‘ ∧ π‘₯ ∈ π‘ˆ) β†’ (β„œβ€˜(𝐢 / (iβ†‘π‘˜))) = (β„œβ€˜(𝐢 / (iβ†‘π‘˜))))
9997, 98, 1isibl2 25275 . 2 (πœ‘ β†’ ((π‘₯ ∈ π‘ˆ ↦ 𝐢) ∈ 𝐿1 ↔ ((π‘₯ ∈ π‘ˆ ↦ 𝐢) ∈ MblFn ∧ βˆ€π‘˜ ∈ (0...3)(∫2β€˜(π‘₯ ∈ ℝ ↦ if((π‘₯ ∈ π‘ˆ ∧ 0 ≀ (β„œβ€˜(𝐢 / (iβ†‘π‘˜)))), (β„œβ€˜(𝐢 / (iβ†‘π‘˜))), 0))) ∈ ℝ)))
10031, 96, 99mpbir2and 711 1 (πœ‘ β†’ (π‘₯ ∈ π‘ˆ ↦ 𝐢) ∈ 𝐿1)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   β‰  wne 2940  βˆ€wral 3061   βˆͺ cun 3945   ∩ cin 3946  ifcif 4527   class class class wbr 5147   ↦ cmpt 5230  dom cdm 5675   β†Ύ cres 5677  β€˜cfv 6540  (class class class)co 7405  β„‚cc 11104  β„cr 11105  0cc0 11106  ici 11108   + caddc 11109  +∞cpnf 11241  β„*cxr 11243   ≀ cle 11245   / cdiv 11867  3c3 12264  β„€cz 12554  [,]cicc 13323  ...cfz 13480  β†‘cexp 14023  β„œcre 15040  vol*covol 24970  volcvol 24971  MblFncmbf 25122  βˆ«2citg2 25124  πΏ1cibl 25125
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-inf2 9632  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183  ax-pre-sup 11184  ax-addf 11185
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-disj 5113  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-of 7666  df-ofr 7667  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-2o 8463  df-er 8699  df-map 8818  df-pm 8819  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-fi 9402  df-sup 9433  df-inf 9434  df-oi 9501  df-dju 9892  df-card 9930  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-div 11868  df-nn 12209  df-2 12271  df-3 12272  df-n0 12469  df-z 12555  df-uz 12819  df-q 12929  df-rp 12971  df-xneg 13088  df-xadd 13089  df-xmul 13090  df-ioo 13324  df-ico 13326  df-icc 13327  df-fz 13481  df-fzo 13624  df-fl 13753  df-seq 13963  df-exp 14024  df-hash 14287  df-cj 15042  df-re 15043  df-im 15044  df-sqrt 15178  df-abs 15179  df-clim 15428  df-sum 15629  df-rest 17364  df-topgen 17385  df-psmet 20928  df-xmet 20929  df-met 20930  df-bl 20931  df-mopn 20932  df-top 22387  df-topon 22404  df-bases 22440  df-cmp 22882  df-ovol 24972  df-vol 24973  df-mbf 25127  df-itg1 25128  df-itg2 25129  df-ibl 25130
This theorem is referenced by:  iblsplitf  44672
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