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Theorem mbfmul 25607
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 25505 . . . . 5 (𝐹 ∈ MblFn β†’ 𝐹:dom πΉβŸΆβ„‚)
31, 2syl 17 . . . 4 (πœ‘ β†’ 𝐹:dom πΉβŸΆβ„‚)
43ffnd 6711 . . 3 (πœ‘ β†’ 𝐹 Fn dom 𝐹)
5 mbfmul.2 . . . . 5 (πœ‘ β†’ 𝐺 ∈ MblFn)
6 mbff 25505 . . . . 5 (𝐺 ∈ MblFn β†’ 𝐺:dom πΊβŸΆβ„‚)
75, 6syl 17 . . . 4 (πœ‘ β†’ 𝐺:dom πΊβŸΆβ„‚)
87ffnd 6711 . . 3 (πœ‘ β†’ 𝐺 Fn dom 𝐺)
9 mbfdm 25506 . . . 4 (𝐹 ∈ MblFn β†’ dom 𝐹 ∈ dom vol)
101, 9syl 17 . . 3 (πœ‘ β†’ dom 𝐹 ∈ dom vol)
11 mbfdm 25506 . . . 4 (𝐺 ∈ MblFn β†’ dom 𝐺 ∈ dom vol)
125, 11syl 17 . . 3 (πœ‘ β†’ dom 𝐺 ∈ dom vol)
13 eqid 2726 . . 3 (dom 𝐹 ∩ dom 𝐺) = (dom 𝐹 ∩ dom 𝐺)
14 eqidd 2727 . . 3 ((πœ‘ ∧ π‘₯ ∈ dom 𝐹) β†’ (πΉβ€˜π‘₯) = (πΉβ€˜π‘₯))
15 eqidd 2727 . . 3 ((πœ‘ ∧ π‘₯ ∈ dom 𝐺) β†’ (πΊβ€˜π‘₯) = (πΊβ€˜π‘₯))
164, 8, 10, 12, 13, 14, 15offval 7675 . 2 (πœ‘ β†’ (𝐹 ∘f Β· 𝐺) = (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((πΉβ€˜π‘₯) Β· (πΊβ€˜π‘₯))))
17 elinel1 4190 . . . . . . . 8 (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) β†’ π‘₯ ∈ dom 𝐹)
18 ffvelcdm 7076 . . . . . . . 8 ((𝐹:dom πΉβŸΆβ„‚ ∧ π‘₯ ∈ dom 𝐹) β†’ (πΉβ€˜π‘₯) ∈ β„‚)
193, 17, 18syl2an 595 . . . . . . 7 ((πœ‘ ∧ π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺)) β†’ (πΉβ€˜π‘₯) ∈ β„‚)
20 elinel2 4191 . . . . . . . 8 (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) β†’ π‘₯ ∈ dom 𝐺)
21 ffvelcdm 7076 . . . . . . . 8 ((𝐺:dom πΊβŸΆβ„‚ ∧ π‘₯ ∈ dom 𝐺) β†’ (πΊβ€˜π‘₯) ∈ β„‚)
227, 20, 21syl2an 595 . . . . . . 7 ((πœ‘ ∧ π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺)) β†’ (πΊβ€˜π‘₯) ∈ β„‚)
2319, 22remuld 15169 . . . . . 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 5241 . . . . 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 25422 . . . . . . 7 ((dom 𝐹 ∈ dom vol ∧ dom 𝐺 ∈ dom vol) β†’ (dom 𝐹 ∩ dom 𝐺) ∈ dom vol)
2610, 12, 25syl2anc 583 . . . . . 6 (πœ‘ β†’ (dom 𝐹 ∩ dom 𝐺) ∈ dom vol)
27 ovexd 7439 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺)) β†’ ((β„œβ€˜(πΉβ€˜π‘₯)) Β· (β„œβ€˜(πΊβ€˜π‘₯))) ∈ V)
28 ovexd 7439 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺)) β†’ ((β„‘β€˜(πΉβ€˜π‘₯)) Β· (β„‘β€˜(πΊβ€˜π‘₯))) ∈ V)
2919recld 15145 . . . . . . 7 ((πœ‘ ∧ π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺)) β†’ (β„œβ€˜(πΉβ€˜π‘₯)) ∈ ℝ)
3022recld 15145 . . . . . . 7 ((πœ‘ ∧ π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺)) β†’ (β„œβ€˜(πΊβ€˜π‘₯)) ∈ ℝ)
31 eqidd 2727 . . . . . . 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 2727 . . . . . . 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 7686 . . . . . 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 15146 . . . . . . 7 ((πœ‘ ∧ π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺)) β†’ (β„‘β€˜(πΉβ€˜π‘₯)) ∈ ℝ)
3522imcld 15146 . . . . . . 7 ((πœ‘ ∧ π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺)) β†’ (β„‘β€˜(πΊβ€˜π‘₯)) ∈ ℝ)
36 eqidd 2727 . . . . . . 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 2727 . . . . . . 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 7686 . . . . . 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 7686 . . . . 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 2769 . . . 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 4223 . . . . . . . . . 10 (dom 𝐹 ∩ dom 𝐺) βŠ† dom 𝐹
42 resmpt 6030 . . . . . . . . . 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 6953 . . . . . . . . . . 11 (πœ‘ β†’ 𝐹 = (π‘₯ ∈ dom 𝐹 ↦ (πΉβ€˜π‘₯)))
4544, 1eqeltrrd 2828 . . . . . . . . . 10 (πœ‘ β†’ (π‘₯ ∈ dom 𝐹 ↦ (πΉβ€˜π‘₯)) ∈ MblFn)
46 mbfres 25524 . . . . . . . . . 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 583 . . . . . . . . 9 (πœ‘ β†’ ((π‘₯ ∈ dom 𝐹 ↦ (πΉβ€˜π‘₯)) β†Ύ (dom 𝐹 ∩ dom 𝐺)) ∈ MblFn)
4843, 47eqeltrrid 2832 . . . . . . . 8 (πœ‘ β†’ (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (πΉβ€˜π‘₯)) ∈ MblFn)
4919ismbfcn2 25518 . . . . . . . 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 494 . . . . . 6 (πœ‘ β†’ (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (β„œβ€˜(πΉβ€˜π‘₯))) ∈ MblFn)
52 inss2 4224 . . . . . . . . . 10 (dom 𝐹 ∩ dom 𝐺) βŠ† dom 𝐺
53 resmpt 6030 . . . . . . . . . 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 6953 . . . . . . . . . . 11 (πœ‘ β†’ 𝐺 = (π‘₯ ∈ dom 𝐺 ↦ (πΊβ€˜π‘₯)))
5655, 5eqeltrrd 2828 . . . . . . . . . 10 (πœ‘ β†’ (π‘₯ ∈ dom 𝐺 ↦ (πΊβ€˜π‘₯)) ∈ MblFn)
57 mbfres 25524 . . . . . . . . . 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 583 . . . . . . . . 9 (πœ‘ β†’ ((π‘₯ ∈ dom 𝐺 ↦ (πΊβ€˜π‘₯)) β†Ύ (dom 𝐹 ∩ dom 𝐺)) ∈ MblFn)
5954, 58eqeltrrid 2832 . . . . . . . 8 (πœ‘ β†’ (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (πΊβ€˜π‘₯)) ∈ MblFn)
6022ismbfcn2 25518 . . . . . . . 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 494 . . . . . 6 (πœ‘ β†’ (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (β„œβ€˜(πΊβ€˜π‘₯))) ∈ MblFn)
6329fmpttd 7109 . . . . . 6 (πœ‘ β†’ (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (β„œβ€˜(πΉβ€˜π‘₯))):(dom 𝐹 ∩ dom 𝐺)βŸΆβ„)
6430fmpttd 7109 . . . . . 6 (πœ‘ β†’ (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (β„œβ€˜(πΊβ€˜π‘₯))):(dom 𝐹 ∩ dom 𝐺)βŸΆβ„)
6551, 62, 63, 64mbfmullem 25606 . . . . 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 495 . . . . . 6 (πœ‘ β†’ (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (β„‘β€˜(πΉβ€˜π‘₯))) ∈ MblFn)
6761simprd 495 . . . . . 6 (πœ‘ β†’ (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (β„‘β€˜(πΊβ€˜π‘₯))) ∈ MblFn)
6834fmpttd 7109 . . . . . 6 (πœ‘ β†’ (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (β„‘β€˜(πΉβ€˜π‘₯))):(dom 𝐹 ∩ dom 𝐺)βŸΆβ„)
6935fmpttd 7109 . . . . . 6 (πœ‘ β†’ (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (β„‘β€˜(πΊβ€˜π‘₯))):(dom 𝐹 ∩ dom 𝐺)βŸΆβ„)
7066, 67, 68, 69mbfmullem 25606 . . . . 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 25542 . . . 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 2827 . . 3 (πœ‘ β†’ (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (β„œβ€˜((πΉβ€˜π‘₯) Β· (πΊβ€˜π‘₯)))) ∈ MblFn)
7319, 22immuld 15170 . . . . . 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 5241 . . . . 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 7439 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺)) β†’ ((β„œβ€˜(πΉβ€˜π‘₯)) Β· (β„‘β€˜(πΊβ€˜π‘₯))) ∈ V)
76 ovexd 7439 . . . . . 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 7686 . . . . . 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 7686 . . . . . 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 7686 . . . . 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 2769 . . . 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 25606 . . . . 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 25606 . . . . 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 25541 . . . 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 2827 . . 3 (πœ‘ β†’ (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (β„‘β€˜((πΉβ€˜π‘₯) Β· (πΊβ€˜π‘₯)))) ∈ MblFn)
8519, 22mulcld 11235 . . . 4 ((πœ‘ ∧ π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺)) β†’ ((πΉβ€˜π‘₯) Β· (πΊβ€˜π‘₯)) ∈ β„‚)
8685ismbfcn2 25518 . . 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 710 . 2 (πœ‘ β†’ (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((πΉβ€˜π‘₯) Β· (πΊβ€˜π‘₯))) ∈ MblFn)
8816, 87eqeltrd 2827 1 (πœ‘ β†’ (𝐹 ∘f Β· 𝐺) ∈ MblFn)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  Vcvv 3468   ∩ cin 3942   βŠ† wss 3943   ↦ cmpt 5224  dom cdm 5669   β†Ύ cres 5671  βŸΆwf 6532  β€˜cfv 6536  (class class class)co 7404   ∘f cof 7664  β„‚cc 11107  β„cr 11108   + caddc 11112   Β· cmul 11114   βˆ’ cmin 11445  β„œcre 15048  β„‘cim 15049  volcvol 25343  MblFncmbf 25494
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721  ax-inf2 9635  ax-cc 10429  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186  ax-pre-sup 11187
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-disj 5107  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-se 5625  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6293  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-of 7666  df-ofr 7667  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8264  df-wrecs 8295  df-recs 8369  df-rdg 8408  df-1o 8464  df-2o 8465  df-oadd 8468  df-omul 8469  df-er 8702  df-map 8821  df-pm 8822  df-en 8939  df-dom 8940  df-sdom 8941  df-fin 8942  df-fi 9405  df-sup 9436  df-inf 9437  df-oi 9504  df-dju 9895  df-card 9933  df-acn 9936  df-pnf 11251  df-mnf 11252  df-xr 11253  df-ltxr 11254  df-le 11255  df-sub 11447  df-neg 11448  df-div 11873  df-nn 12214  df-2 12276  df-3 12277  df-n0 12474  df-z 12560  df-uz 12824  df-q 12934  df-rp 12978  df-xneg 13095  df-xadd 13096  df-xmul 13097  df-ioo 13331  df-ioc 13332  df-ico 13333  df-icc 13334  df-fz 13488  df-fzo 13631  df-fl 13760  df-seq 13970  df-exp 14031  df-hash 14294  df-cj 15050  df-re 15051  df-im 15052  df-sqrt 15186  df-abs 15187  df-limsup 15419  df-clim 15436  df-rlim 15437  df-sum 15637  df-rest 17375  df-topgen 17396  df-psmet 21228  df-xmet 21229  df-met 21230  df-bl 21231  df-mopn 21232  df-top 22747  df-topon 22764  df-bases 22800  df-cmp 23242  df-ovol 25344  df-vol 25345  df-mbf 25499  df-itg1 25500  df-0p 25550
This theorem is referenced by:  bddmulibl  25719
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