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Theorem mbfmul 25107
Description: The product of two measurable functions is measurable. (Contributed by Mario Carneiro, 7-Sep-2014.)
Hypotheses
Ref Expression
mbfmul.1 (πœ‘ β†’ 𝐹 ∈ MblFn)
mbfmul.2 (πœ‘ β†’ 𝐺 ∈ MblFn)
Assertion
Ref Expression
mbfmul (πœ‘ β†’ (𝐹 ∘f Β· 𝐺) ∈ MblFn)

Proof of Theorem mbfmul
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 mbfmul.1 . . . . 5 (πœ‘ β†’ 𝐹 ∈ MblFn)
2 mbff 25005 . . . . 5 (𝐹 ∈ MblFn β†’ 𝐹:dom πΉβŸΆβ„‚)
31, 2syl 17 . . . 4 (πœ‘ β†’ 𝐹:dom πΉβŸΆβ„‚)
43ffnd 6670 . . 3 (πœ‘ β†’ 𝐹 Fn dom 𝐹)
5 mbfmul.2 . . . . 5 (πœ‘ β†’ 𝐺 ∈ MblFn)
6 mbff 25005 . . . . 5 (𝐺 ∈ MblFn β†’ 𝐺:dom πΊβŸΆβ„‚)
75, 6syl 17 . . . 4 (πœ‘ β†’ 𝐺:dom πΊβŸΆβ„‚)
87ffnd 6670 . . 3 (πœ‘ β†’ 𝐺 Fn dom 𝐺)
9 mbfdm 25006 . . . 4 (𝐹 ∈ MblFn β†’ dom 𝐹 ∈ dom vol)
101, 9syl 17 . . 3 (πœ‘ β†’ dom 𝐹 ∈ dom vol)
11 mbfdm 25006 . . . 4 (𝐺 ∈ MblFn β†’ dom 𝐺 ∈ dom vol)
125, 11syl 17 . . 3 (πœ‘ β†’ dom 𝐺 ∈ dom vol)
13 eqid 2733 . . 3 (dom 𝐹 ∩ dom 𝐺) = (dom 𝐹 ∩ dom 𝐺)
14 eqidd 2734 . . 3 ((πœ‘ ∧ π‘₯ ∈ dom 𝐹) β†’ (πΉβ€˜π‘₯) = (πΉβ€˜π‘₯))
15 eqidd 2734 . . 3 ((πœ‘ ∧ π‘₯ ∈ dom 𝐺) β†’ (πΊβ€˜π‘₯) = (πΊβ€˜π‘₯))
164, 8, 10, 12, 13, 14, 15offval 7627 . 2 (πœ‘ β†’ (𝐹 ∘f Β· 𝐺) = (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((πΉβ€˜π‘₯) Β· (πΊβ€˜π‘₯))))
17 elinel1 4156 . . . . . . . 8 (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) β†’ π‘₯ ∈ dom 𝐹)
18 ffvelcdm 7033 . . . . . . . 8 ((𝐹:dom πΉβŸΆβ„‚ ∧ π‘₯ ∈ dom 𝐹) β†’ (πΉβ€˜π‘₯) ∈ β„‚)
193, 17, 18syl2an 597 . . . . . . 7 ((πœ‘ ∧ π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺)) β†’ (πΉβ€˜π‘₯) ∈ β„‚)
20 elinel2 4157 . . . . . . . 8 (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) β†’ π‘₯ ∈ dom 𝐺)
21 ffvelcdm 7033 . . . . . . . 8 ((𝐺:dom πΊβŸΆβ„‚ ∧ π‘₯ ∈ dom 𝐺) β†’ (πΊβ€˜π‘₯) ∈ β„‚)
227, 20, 21syl2an 597 . . . . . . 7 ((πœ‘ ∧ π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺)) β†’ (πΊβ€˜π‘₯) ∈ β„‚)
2319, 22remuld 15109 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺)) β†’ (β„œβ€˜((πΉβ€˜π‘₯) Β· (πΊβ€˜π‘₯))) = (((β„œβ€˜(πΉβ€˜π‘₯)) Β· (β„œβ€˜(πΊβ€˜π‘₯))) βˆ’ ((β„‘β€˜(πΉβ€˜π‘₯)) Β· (β„‘β€˜(πΊβ€˜π‘₯)))))
2423mpteq2dva 5206 . . . . 5 (πœ‘ β†’ (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (β„œβ€˜((πΉβ€˜π‘₯) Β· (πΊβ€˜π‘₯)))) = (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (((β„œβ€˜(πΉβ€˜π‘₯)) Β· (β„œβ€˜(πΊβ€˜π‘₯))) βˆ’ ((β„‘β€˜(πΉβ€˜π‘₯)) Β· (β„‘β€˜(πΊβ€˜π‘₯))))))
25 inmbl 24922 . . . . . . 7 ((dom 𝐹 ∈ dom vol ∧ dom 𝐺 ∈ dom vol) β†’ (dom 𝐹 ∩ dom 𝐺) ∈ dom vol)
2610, 12, 25syl2anc 585 . . . . . 6 (πœ‘ β†’ (dom 𝐹 ∩ dom 𝐺) ∈ dom vol)
27 ovexd 7393 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺)) β†’ ((β„œβ€˜(πΉβ€˜π‘₯)) Β· (β„œβ€˜(πΊβ€˜π‘₯))) ∈ V)
28 ovexd 7393 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺)) β†’ ((β„‘β€˜(πΉβ€˜π‘₯)) Β· (β„‘β€˜(πΊβ€˜π‘₯))) ∈ V)
2919recld 15085 . . . . . . 7 ((πœ‘ ∧ π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺)) β†’ (β„œβ€˜(πΉβ€˜π‘₯)) ∈ ℝ)
3022recld 15085 . . . . . . 7 ((πœ‘ ∧ π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺)) β†’ (β„œβ€˜(πΊβ€˜π‘₯)) ∈ ℝ)
31 eqidd 2734 . . . . . . 7 (πœ‘ β†’ (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (β„œβ€˜(πΉβ€˜π‘₯))) = (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (β„œβ€˜(πΉβ€˜π‘₯))))
32 eqidd 2734 . . . . . . 7 (πœ‘ β†’ (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (β„œβ€˜(πΊβ€˜π‘₯))) = (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (β„œβ€˜(πΊβ€˜π‘₯))))
3326, 29, 30, 31, 32offval2 7638 . . . . . 6 (πœ‘ β†’ ((π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (β„œβ€˜(πΉβ€˜π‘₯))) ∘f Β· (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (β„œβ€˜(πΊβ€˜π‘₯)))) = (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((β„œβ€˜(πΉβ€˜π‘₯)) Β· (β„œβ€˜(πΊβ€˜π‘₯)))))
3419imcld 15086 . . . . . . 7 ((πœ‘ ∧ π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺)) β†’ (β„‘β€˜(πΉβ€˜π‘₯)) ∈ ℝ)
3522imcld 15086 . . . . . . 7 ((πœ‘ ∧ π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺)) β†’ (β„‘β€˜(πΊβ€˜π‘₯)) ∈ ℝ)
36 eqidd 2734 . . . . . . 7 (πœ‘ β†’ (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (β„‘β€˜(πΉβ€˜π‘₯))) = (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (β„‘β€˜(πΉβ€˜π‘₯))))
37 eqidd 2734 . . . . . . 7 (πœ‘ β†’ (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (β„‘β€˜(πΊβ€˜π‘₯))) = (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (β„‘β€˜(πΊβ€˜π‘₯))))
3826, 34, 35, 36, 37offval2 7638 . . . . . 6 (πœ‘ β†’ ((π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (β„‘β€˜(πΉβ€˜π‘₯))) ∘f Β· (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (β„‘β€˜(πΊβ€˜π‘₯)))) = (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((β„‘β€˜(πΉβ€˜π‘₯)) Β· (β„‘β€˜(πΊβ€˜π‘₯)))))
3926, 27, 28, 33, 38offval2 7638 . . . . 5 (πœ‘ β†’ (((π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (β„œβ€˜(πΉβ€˜π‘₯))) ∘f Β· (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (β„œβ€˜(πΊβ€˜π‘₯)))) ∘f βˆ’ ((π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (β„‘β€˜(πΉβ€˜π‘₯))) ∘f Β· (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (β„‘β€˜(πΊβ€˜π‘₯))))) = (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (((β„œβ€˜(πΉβ€˜π‘₯)) Β· (β„œβ€˜(πΊβ€˜π‘₯))) βˆ’ ((β„‘β€˜(πΉβ€˜π‘₯)) Β· (β„‘β€˜(πΊβ€˜π‘₯))))))
4024, 39eqtr4d 2776 . . . 4 (πœ‘ β†’ (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (β„œβ€˜((πΉβ€˜π‘₯) Β· (πΊβ€˜π‘₯)))) = (((π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (β„œβ€˜(πΉβ€˜π‘₯))) ∘f Β· (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (β„œβ€˜(πΊβ€˜π‘₯)))) ∘f βˆ’ ((π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (β„‘β€˜(πΉβ€˜π‘₯))) ∘f Β· (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (β„‘β€˜(πΊβ€˜π‘₯))))))
41 inss1 4189 . . . . . . . . . 10 (dom 𝐹 ∩ dom 𝐺) βŠ† dom 𝐹
42 resmpt 5992 . . . . . . . . . 10 ((dom 𝐹 ∩ dom 𝐺) βŠ† dom 𝐹 β†’ ((π‘₯ ∈ dom 𝐹 ↦ (πΉβ€˜π‘₯)) β†Ύ (dom 𝐹 ∩ dom 𝐺)) = (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (πΉβ€˜π‘₯)))
4341, 42ax-mp 5 . . . . . . . . 9 ((π‘₯ ∈ dom 𝐹 ↦ (πΉβ€˜π‘₯)) β†Ύ (dom 𝐹 ∩ dom 𝐺)) = (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (πΉβ€˜π‘₯))
443feqmptd 6911 . . . . . . . . . . 11 (πœ‘ β†’ 𝐹 = (π‘₯ ∈ dom 𝐹 ↦ (πΉβ€˜π‘₯)))
4544, 1eqeltrrd 2835 . . . . . . . . . 10 (πœ‘ β†’ (π‘₯ ∈ dom 𝐹 ↦ (πΉβ€˜π‘₯)) ∈ MblFn)
46 mbfres 25024 . . . . . . . . . 10 (((π‘₯ ∈ dom 𝐹 ↦ (πΉβ€˜π‘₯)) ∈ MblFn ∧ (dom 𝐹 ∩ dom 𝐺) ∈ dom vol) β†’ ((π‘₯ ∈ dom 𝐹 ↦ (πΉβ€˜π‘₯)) β†Ύ (dom 𝐹 ∩ dom 𝐺)) ∈ MblFn)
4745, 26, 46syl2anc 585 . . . . . . . . 9 (πœ‘ β†’ ((π‘₯ ∈ dom 𝐹 ↦ (πΉβ€˜π‘₯)) β†Ύ (dom 𝐹 ∩ dom 𝐺)) ∈ MblFn)
4843, 47eqeltrrid 2839 . . . . . . . 8 (πœ‘ β†’ (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (πΉβ€˜π‘₯)) ∈ MblFn)
4919ismbfcn2 25018 . . . . . . . 8 (πœ‘ β†’ ((π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (πΉβ€˜π‘₯)) ∈ MblFn ↔ ((π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (β„œβ€˜(πΉβ€˜π‘₯))) ∈ MblFn ∧ (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (β„‘β€˜(πΉβ€˜π‘₯))) ∈ MblFn)))
5048, 49mpbid 231 . . . . . . 7 (πœ‘ β†’ ((π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (β„œβ€˜(πΉβ€˜π‘₯))) ∈ MblFn ∧ (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (β„‘β€˜(πΉβ€˜π‘₯))) ∈ MblFn))
5150simpld 496 . . . . . 6 (πœ‘ β†’ (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (β„œβ€˜(πΉβ€˜π‘₯))) ∈ MblFn)
52 inss2 4190 . . . . . . . . . 10 (dom 𝐹 ∩ dom 𝐺) βŠ† dom 𝐺
53 resmpt 5992 . . . . . . . . . 10 ((dom 𝐹 ∩ dom 𝐺) βŠ† dom 𝐺 β†’ ((π‘₯ ∈ dom 𝐺 ↦ (πΊβ€˜π‘₯)) β†Ύ (dom 𝐹 ∩ dom 𝐺)) = (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (πΊβ€˜π‘₯)))
5452, 53ax-mp 5 . . . . . . . . 9 ((π‘₯ ∈ dom 𝐺 ↦ (πΊβ€˜π‘₯)) β†Ύ (dom 𝐹 ∩ dom 𝐺)) = (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (πΊβ€˜π‘₯))
557feqmptd 6911 . . . . . . . . . . 11 (πœ‘ β†’ 𝐺 = (π‘₯ ∈ dom 𝐺 ↦ (πΊβ€˜π‘₯)))
5655, 5eqeltrrd 2835 . . . . . . . . . 10 (πœ‘ β†’ (π‘₯ ∈ dom 𝐺 ↦ (πΊβ€˜π‘₯)) ∈ MblFn)
57 mbfres 25024 . . . . . . . . . 10 (((π‘₯ ∈ dom 𝐺 ↦ (πΊβ€˜π‘₯)) ∈ MblFn ∧ (dom 𝐹 ∩ dom 𝐺) ∈ dom vol) β†’ ((π‘₯ ∈ dom 𝐺 ↦ (πΊβ€˜π‘₯)) β†Ύ (dom 𝐹 ∩ dom 𝐺)) ∈ MblFn)
5856, 26, 57syl2anc 585 . . . . . . . . 9 (πœ‘ β†’ ((π‘₯ ∈ dom 𝐺 ↦ (πΊβ€˜π‘₯)) β†Ύ (dom 𝐹 ∩ dom 𝐺)) ∈ MblFn)
5954, 58eqeltrrid 2839 . . . . . . . 8 (πœ‘ β†’ (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (πΊβ€˜π‘₯)) ∈ MblFn)
6022ismbfcn2 25018 . . . . . . . 8 (πœ‘ β†’ ((π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (πΊβ€˜π‘₯)) ∈ MblFn ↔ ((π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (β„œβ€˜(πΊβ€˜π‘₯))) ∈ MblFn ∧ (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (β„‘β€˜(πΊβ€˜π‘₯))) ∈ MblFn)))
6159, 60mpbid 231 . . . . . . 7 (πœ‘ β†’ ((π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (β„œβ€˜(πΊβ€˜π‘₯))) ∈ MblFn ∧ (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (β„‘β€˜(πΊβ€˜π‘₯))) ∈ MblFn))
6261simpld 496 . . . . . 6 (πœ‘ β†’ (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (β„œβ€˜(πΊβ€˜π‘₯))) ∈ MblFn)
6329fmpttd 7064 . . . . . 6 (πœ‘ β†’ (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (β„œβ€˜(πΉβ€˜π‘₯))):(dom 𝐹 ∩ dom 𝐺)βŸΆβ„)
6430fmpttd 7064 . . . . . 6 (πœ‘ β†’ (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (β„œβ€˜(πΊβ€˜π‘₯))):(dom 𝐹 ∩ dom 𝐺)βŸΆβ„)
6551, 62, 63, 64mbfmullem 25106 . . . . 5 (πœ‘ β†’ ((π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (β„œβ€˜(πΉβ€˜π‘₯))) ∘f Β· (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (β„œβ€˜(πΊβ€˜π‘₯)))) ∈ MblFn)
6650simprd 497 . . . . . 6 (πœ‘ β†’ (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (β„‘β€˜(πΉβ€˜π‘₯))) ∈ MblFn)
6761simprd 497 . . . . . 6 (πœ‘ β†’ (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (β„‘β€˜(πΊβ€˜π‘₯))) ∈ MblFn)
6834fmpttd 7064 . . . . . 6 (πœ‘ β†’ (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (β„‘β€˜(πΉβ€˜π‘₯))):(dom 𝐹 ∩ dom 𝐺)βŸΆβ„)
6935fmpttd 7064 . . . . . 6 (πœ‘ β†’ (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (β„‘β€˜(πΊβ€˜π‘₯))):(dom 𝐹 ∩ dom 𝐺)βŸΆβ„)
7066, 67, 68, 69mbfmullem 25106 . . . . 5 (πœ‘ β†’ ((π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (β„‘β€˜(πΉβ€˜π‘₯))) ∘f Β· (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (β„‘β€˜(πΊβ€˜π‘₯)))) ∈ MblFn)
7165, 70mbfsub 25042 . . . 4 (πœ‘ β†’ (((π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (β„œβ€˜(πΉβ€˜π‘₯))) ∘f Β· (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (β„œβ€˜(πΊβ€˜π‘₯)))) ∘f βˆ’ ((π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (β„‘β€˜(πΉβ€˜π‘₯))) ∘f Β· (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (β„‘β€˜(πΊβ€˜π‘₯))))) ∈ MblFn)
7240, 71eqeltrd 2834 . . 3 (πœ‘ β†’ (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (β„œβ€˜((πΉβ€˜π‘₯) Β· (πΊβ€˜π‘₯)))) ∈ MblFn)
7319, 22immuld 15110 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺)) β†’ (β„‘β€˜((πΉβ€˜π‘₯) Β· (πΊβ€˜π‘₯))) = (((β„œβ€˜(πΉβ€˜π‘₯)) Β· (β„‘β€˜(πΊβ€˜π‘₯))) + ((β„‘β€˜(πΉβ€˜π‘₯)) Β· (β„œβ€˜(πΊβ€˜π‘₯)))))
7473mpteq2dva 5206 . . . . 5 (πœ‘ β†’ (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (β„‘β€˜((πΉβ€˜π‘₯) Β· (πΊβ€˜π‘₯)))) = (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (((β„œβ€˜(πΉβ€˜π‘₯)) Β· (β„‘β€˜(πΊβ€˜π‘₯))) + ((β„‘β€˜(πΉβ€˜π‘₯)) Β· (β„œβ€˜(πΊβ€˜π‘₯))))))
75 ovexd 7393 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺)) β†’ ((β„œβ€˜(πΉβ€˜π‘₯)) Β· (β„‘β€˜(πΊβ€˜π‘₯))) ∈ V)
76 ovexd 7393 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺)) β†’ ((β„‘β€˜(πΉβ€˜π‘₯)) Β· (β„œβ€˜(πΊβ€˜π‘₯))) ∈ V)
7726, 29, 35, 31, 37offval2 7638 . . . . . 6 (πœ‘ β†’ ((π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (β„œβ€˜(πΉβ€˜π‘₯))) ∘f Β· (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (β„‘β€˜(πΊβ€˜π‘₯)))) = (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((β„œβ€˜(πΉβ€˜π‘₯)) Β· (β„‘β€˜(πΊβ€˜π‘₯)))))
7826, 34, 30, 36, 32offval2 7638 . . . . . 6 (πœ‘ β†’ ((π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (β„‘β€˜(πΉβ€˜π‘₯))) ∘f Β· (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (β„œβ€˜(πΊβ€˜π‘₯)))) = (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((β„‘β€˜(πΉβ€˜π‘₯)) Β· (β„œβ€˜(πΊβ€˜π‘₯)))))
7926, 75, 76, 77, 78offval2 7638 . . . . 5 (πœ‘ β†’ (((π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (β„œβ€˜(πΉβ€˜π‘₯))) ∘f Β· (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (β„‘β€˜(πΊβ€˜π‘₯)))) ∘f + ((π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (β„‘β€˜(πΉβ€˜π‘₯))) ∘f Β· (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (β„œβ€˜(πΊβ€˜π‘₯))))) = (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (((β„œβ€˜(πΉβ€˜π‘₯)) Β· (β„‘β€˜(πΊβ€˜π‘₯))) + ((β„‘β€˜(πΉβ€˜π‘₯)) Β· (β„œβ€˜(πΊβ€˜π‘₯))))))
8074, 79eqtr4d 2776 . . . 4 (πœ‘ β†’ (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (β„‘β€˜((πΉβ€˜π‘₯) Β· (πΊβ€˜π‘₯)))) = (((π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (β„œβ€˜(πΉβ€˜π‘₯))) ∘f Β· (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (β„‘β€˜(πΊβ€˜π‘₯)))) ∘f + ((π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (β„‘β€˜(πΉβ€˜π‘₯))) ∘f Β· (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (β„œβ€˜(πΊβ€˜π‘₯))))))
8151, 67, 63, 69mbfmullem 25106 . . . . 5 (πœ‘ β†’ ((π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (β„œβ€˜(πΉβ€˜π‘₯))) ∘f Β· (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (β„‘β€˜(πΊβ€˜π‘₯)))) ∈ MblFn)
8266, 62, 68, 64mbfmullem 25106 . . . . 5 (πœ‘ β†’ ((π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (β„‘β€˜(πΉβ€˜π‘₯))) ∘f Β· (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (β„œβ€˜(πΊβ€˜π‘₯)))) ∈ MblFn)
8381, 82mbfadd 25041 . . . 4 (πœ‘ β†’ (((π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (β„œβ€˜(πΉβ€˜π‘₯))) ∘f Β· (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (β„‘β€˜(πΊβ€˜π‘₯)))) ∘f + ((π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (β„‘β€˜(πΉβ€˜π‘₯))) ∘f Β· (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (β„œβ€˜(πΊβ€˜π‘₯))))) ∈ MblFn)
8480, 83eqeltrd 2834 . . 3 (πœ‘ β†’ (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (β„‘β€˜((πΉβ€˜π‘₯) Β· (πΊβ€˜π‘₯)))) ∈ MblFn)
8519, 22mulcld 11180 . . . 4 ((πœ‘ ∧ π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺)) β†’ ((πΉβ€˜π‘₯) Β· (πΊβ€˜π‘₯)) ∈ β„‚)
8685ismbfcn2 25018 . . 3 (πœ‘ β†’ ((π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((πΉβ€˜π‘₯) Β· (πΊβ€˜π‘₯))) ∈ MblFn ↔ ((π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (β„œβ€˜((πΉβ€˜π‘₯) Β· (πΊβ€˜π‘₯)))) ∈ MblFn ∧ (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (β„‘β€˜((πΉβ€˜π‘₯) Β· (πΊβ€˜π‘₯)))) ∈ MblFn)))
8772, 84, 86mpbir2and 712 . 2 (πœ‘ β†’ (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((πΉβ€˜π‘₯) Β· (πΊβ€˜π‘₯))) ∈ MblFn)
8816, 87eqeltrd 2834 1 (πœ‘ β†’ (𝐹 ∘f Β· 𝐺) ∈ MblFn)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  Vcvv 3444   ∩ cin 3910   βŠ† wss 3911   ↦ cmpt 5189  dom cdm 5634   β†Ύ cres 5636  βŸΆwf 6493  β€˜cfv 6497  (class class class)co 7358   ∘f cof 7616  β„‚cc 11054  β„cr 11055   + caddc 11059   Β· cmul 11061   βˆ’ cmin 11390  β„œcre 14988  β„‘cim 14989  volcvol 24843  MblFncmbf 24994
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 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-inf2 9582  ax-cc 10376  ax-cnex 11112  ax-resscn 11113  ax-1cn 11114  ax-icn 11115  ax-addcl 11116  ax-addrcl 11117  ax-mulcl 11118  ax-mulrcl 11119  ax-mulcom 11120  ax-addass 11121  ax-mulass 11122  ax-distr 11123  ax-i2m1 11124  ax-1ne0 11125  ax-1rid 11126  ax-rnegex 11127  ax-rrecex 11128  ax-cnre 11129  ax-pre-lttri 11130  ax-pre-lttrn 11131  ax-pre-ltadd 11132  ax-pre-mulgt0 11133  ax-pre-sup 11134
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  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 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-disj 5072  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-se 5590  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-isom 6506  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-of 7618  df-ofr 7619  df-om 7804  df-1st 7922  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-1o 8413  df-2o 8414  df-oadd 8417  df-omul 8418  df-er 8651  df-map 8770  df-pm 8771  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-fi 9352  df-sup 9383  df-inf 9384  df-oi 9451  df-dju 9842  df-card 9880  df-acn 9883  df-pnf 11196  df-mnf 11197  df-xr 11198  df-ltxr 11199  df-le 11200  df-sub 11392  df-neg 11393  df-div 11818  df-nn 12159  df-2 12221  df-3 12222  df-n0 12419  df-z 12505  df-uz 12769  df-q 12879  df-rp 12921  df-xneg 13038  df-xadd 13039  df-xmul 13040  df-ioo 13274  df-ioc 13275  df-ico 13276  df-icc 13277  df-fz 13431  df-fzo 13574  df-fl 13703  df-seq 13913  df-exp 13974  df-hash 14237  df-cj 14990  df-re 14991  df-im 14992  df-sqrt 15126  df-abs 15127  df-limsup 15359  df-clim 15376  df-rlim 15377  df-sum 15577  df-rest 17309  df-topgen 17330  df-psmet 20804  df-xmet 20805  df-met 20806  df-bl 20807  df-mopn 20808  df-top 22259  df-topon 22276  df-bases 22312  df-cmp 22754  df-ovol 24844  df-vol 24845  df-mbf 24999  df-itg1 25000  df-0p 25050
This theorem is referenced by:  bddmulibl  25219
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