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Theorem mbfmul 25669
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 25567 . . . . 5 (𝐹 ∈ MblFn β†’ 𝐹:dom πΉβŸΆβ„‚)
31, 2syl 17 . . . 4 (πœ‘ β†’ 𝐹:dom πΉβŸΆβ„‚)
43ffnd 6723 . . 3 (πœ‘ β†’ 𝐹 Fn dom 𝐹)
5 mbfmul.2 . . . . 5 (πœ‘ β†’ 𝐺 ∈ MblFn)
6 mbff 25567 . . . . 5 (𝐺 ∈ MblFn β†’ 𝐺:dom πΊβŸΆβ„‚)
75, 6syl 17 . . . 4 (πœ‘ β†’ 𝐺:dom πΊβŸΆβ„‚)
87ffnd 6723 . . 3 (πœ‘ β†’ 𝐺 Fn dom 𝐺)
9 mbfdm 25568 . . . 4 (𝐹 ∈ MblFn β†’ dom 𝐹 ∈ dom vol)
101, 9syl 17 . . 3 (πœ‘ β†’ dom 𝐹 ∈ dom vol)
11 mbfdm 25568 . . . 4 (𝐺 ∈ MblFn β†’ dom 𝐺 ∈ dom vol)
125, 11syl 17 . . 3 (πœ‘ β†’ dom 𝐺 ∈ dom vol)
13 eqid 2728 . . 3 (dom 𝐹 ∩ dom 𝐺) = (dom 𝐹 ∩ dom 𝐺)
14 eqidd 2729 . . 3 ((πœ‘ ∧ π‘₯ ∈ dom 𝐹) β†’ (πΉβ€˜π‘₯) = (πΉβ€˜π‘₯))
15 eqidd 2729 . . 3 ((πœ‘ ∧ π‘₯ ∈ dom 𝐺) β†’ (πΊβ€˜π‘₯) = (πΊβ€˜π‘₯))
164, 8, 10, 12, 13, 14, 15offval 7694 . 2 (πœ‘ β†’ (𝐹 ∘f Β· 𝐺) = (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((πΉβ€˜π‘₯) Β· (πΊβ€˜π‘₯))))
17 elinel1 4195 . . . . . . . 8 (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) β†’ π‘₯ ∈ dom 𝐹)
18 ffvelcdm 7091 . . . . . . . 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 4196 . . . . . . . 8 (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) β†’ π‘₯ ∈ dom 𝐺)
21 ffvelcdm 7091 . . . . . . . 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 15198 . . . . . 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 5248 . . . . 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 25484 . . . . . . 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 7455 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺)) β†’ ((β„œβ€˜(πΉβ€˜π‘₯)) Β· (β„œβ€˜(πΊβ€˜π‘₯))) ∈ V)
28 ovexd 7455 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺)) β†’ ((β„‘β€˜(πΉβ€˜π‘₯)) Β· (β„‘β€˜(πΊβ€˜π‘₯))) ∈ V)
2919recld 15174 . . . . . . 7 ((πœ‘ ∧ π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺)) β†’ (β„œβ€˜(πΉβ€˜π‘₯)) ∈ ℝ)
3022recld 15174 . . . . . . 7 ((πœ‘ ∧ π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺)) β†’ (β„œβ€˜(πΊβ€˜π‘₯)) ∈ ℝ)
31 eqidd 2729 . . . . . . 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 2729 . . . . . . 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 7705 . . . . . 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 15175 . . . . . . 7 ((πœ‘ ∧ π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺)) β†’ (β„‘β€˜(πΉβ€˜π‘₯)) ∈ ℝ)
3522imcld 15175 . . . . . . 7 ((πœ‘ ∧ π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺)) β†’ (β„‘β€˜(πΊβ€˜π‘₯)) ∈ ℝ)
36 eqidd 2729 . . . . . . 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 2729 . . . . . . 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 7705 . . . . . 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 7705 . . . . 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 2771 . . . 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 4229 . . . . . . . . . 10 (dom 𝐹 ∩ dom 𝐺) βŠ† dom 𝐹
42 resmpt 6041 . . . . . . . . . 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 6967 . . . . . . . . . . 11 (πœ‘ β†’ 𝐹 = (π‘₯ ∈ dom 𝐹 ↦ (πΉβ€˜π‘₯)))
4544, 1eqeltrrd 2830 . . . . . . . . . 10 (πœ‘ β†’ (π‘₯ ∈ dom 𝐹 ↦ (πΉβ€˜π‘₯)) ∈ MblFn)
46 mbfres 25586 . . . . . . . . . 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 2834 . . . . . . . 8 (πœ‘ β†’ (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (πΉβ€˜π‘₯)) ∈ MblFn)
4919ismbfcn2 25580 . . . . . . . 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 4230 . . . . . . . . . 10 (dom 𝐹 ∩ dom 𝐺) βŠ† dom 𝐺
53 resmpt 6041 . . . . . . . . . 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 6967 . . . . . . . . . . 11 (πœ‘ β†’ 𝐺 = (π‘₯ ∈ dom 𝐺 ↦ (πΊβ€˜π‘₯)))
5655, 5eqeltrrd 2830 . . . . . . . . . 10 (πœ‘ β†’ (π‘₯ ∈ dom 𝐺 ↦ (πΊβ€˜π‘₯)) ∈ MblFn)
57 mbfres 25586 . . . . . . . . . 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 2834 . . . . . . . 8 (πœ‘ β†’ (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (πΊβ€˜π‘₯)) ∈ MblFn)
6022ismbfcn2 25580 . . . . . . . 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 7125 . . . . . 6 (πœ‘ β†’ (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (β„œβ€˜(πΉβ€˜π‘₯))):(dom 𝐹 ∩ dom 𝐺)βŸΆβ„)
6430fmpttd 7125 . . . . . 6 (πœ‘ β†’ (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (β„œβ€˜(πΊβ€˜π‘₯))):(dom 𝐹 ∩ dom 𝐺)βŸΆβ„)
6551, 62, 63, 64mbfmullem 25668 . . . . 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 7125 . . . . . 6 (πœ‘ β†’ (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (β„‘β€˜(πΉβ€˜π‘₯))):(dom 𝐹 ∩ dom 𝐺)βŸΆβ„)
6935fmpttd 7125 . . . . . 6 (πœ‘ β†’ (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (β„‘β€˜(πΊβ€˜π‘₯))):(dom 𝐹 ∩ dom 𝐺)βŸΆβ„)
7066, 67, 68, 69mbfmullem 25668 . . . . 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 25604 . . . 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 2829 . . 3 (πœ‘ β†’ (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (β„œβ€˜((πΉβ€˜π‘₯) Β· (πΊβ€˜π‘₯)))) ∈ MblFn)
7319, 22immuld 15199 . . . . . 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 5248 . . . . 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 7455 . . . . . 6 ((πœ‘ ∧ π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺)) β†’ ((β„œβ€˜(πΉβ€˜π‘₯)) Β· (β„‘β€˜(πΊβ€˜π‘₯))) ∈ V)
76 ovexd 7455 . . . . . 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 7705 . . . . . 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 7705 . . . . . 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 7705 . . . . 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 2771 . . . 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 25668 . . . . 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 25668 . . . . 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 25603 . . . 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 2829 . . 3 (πœ‘ β†’ (π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺) ↦ (β„‘β€˜((πΉβ€˜π‘₯) Β· (πΊβ€˜π‘₯)))) ∈ MblFn)
8519, 22mulcld 11265 . . . 4 ((πœ‘ ∧ π‘₯ ∈ (dom 𝐹 ∩ dom 𝐺)) β†’ ((πΉβ€˜π‘₯) Β· (πΊβ€˜π‘₯)) ∈ β„‚)
8685ismbfcn2 25580 . . 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 2829 1 (πœ‘ β†’ (𝐹 ∘f Β· 𝐺) ∈ MblFn)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1534   ∈ wcel 2099  Vcvv 3471   ∩ cin 3946   βŠ† wss 3947   ↦ cmpt 5231  dom cdm 5678   β†Ύ cres 5680  βŸΆwf 6544  β€˜cfv 6548  (class class class)co 7420   ∘f cof 7683  β„‚cc 11137  β„cr 11138   + caddc 11142   Β· cmul 11144   βˆ’ cmin 11475  β„œcre 15077  β„‘cim 15078  volcvol 25405  MblFncmbf 25556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740  ax-inf2 9665  ax-cc 10459  ax-cnex 11195  ax-resscn 11196  ax-1cn 11197  ax-icn 11198  ax-addcl 11199  ax-addrcl 11200  ax-mulcl 11201  ax-mulrcl 11202  ax-mulcom 11203  ax-addass 11204  ax-mulass 11205  ax-distr 11206  ax-i2m1 11207  ax-1ne0 11208  ax-1rid 11209  ax-rnegex 11210  ax-rrecex 11211  ax-cnre 11212  ax-pre-lttri 11213  ax-pre-lttrn 11214  ax-pre-ltadd 11215  ax-pre-mulgt0 11216  ax-pre-sup 11217
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3373  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-int 4950  df-iun 4998  df-disj 5114  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-se 5634  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6305  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-isom 6557  df-riota 7376  df-ov 7423  df-oprab 7424  df-mpo 7425  df-of 7685  df-ofr 7686  df-om 7871  df-1st 7993  df-2nd 7994  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-2o 8488  df-oadd 8491  df-omul 8492  df-er 8725  df-map 8847  df-pm 8848  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-fi 9435  df-sup 9466  df-inf 9467  df-oi 9534  df-dju 9925  df-card 9963  df-acn 9966  df-pnf 11281  df-mnf 11282  df-xr 11283  df-ltxr 11284  df-le 11285  df-sub 11477  df-neg 11478  df-div 11903  df-nn 12244  df-2 12306  df-3 12307  df-n0 12504  df-z 12590  df-uz 12854  df-q 12964  df-rp 13008  df-xneg 13125  df-xadd 13126  df-xmul 13127  df-ioo 13361  df-ioc 13362  df-ico 13363  df-icc 13364  df-fz 13518  df-fzo 13661  df-fl 13790  df-seq 14000  df-exp 14060  df-hash 14323  df-cj 15079  df-re 15080  df-im 15081  df-sqrt 15215  df-abs 15216  df-limsup 15448  df-clim 15465  df-rlim 15466  df-sum 15666  df-rest 17404  df-topgen 17425  df-psmet 21271  df-xmet 21272  df-met 21273  df-bl 21274  df-mopn 21275  df-top 22809  df-topon 22826  df-bases 22862  df-cmp 23304  df-ovol 25406  df-vol 25407  df-mbf 25561  df-itg1 25562  df-0p 25612
This theorem is referenced by:  bddmulibl  25781
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