MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  i1fmul Structured version   Visualization version   GIF version

Theorem i1fmul 25731
Description: The pointwise product of two simple functions is a simple function. (Contributed by Mario Carneiro, 5-Sep-2014.)
Hypotheses
Ref Expression
i1fadd.1 (𝜑𝐹 ∈ dom ∫1)
i1fadd.2 (𝜑𝐺 ∈ dom ∫1)
Assertion
Ref Expression
i1fmul (𝜑 → (𝐹f · 𝐺) ∈ dom ∫1)

Proof of Theorem i1fmul
Dummy variables 𝑦 𝑧 𝑤 𝑣 𝑥 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 remulcl 11240 . . . 4 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 · 𝑦) ∈ ℝ)
21adantl 481 . . 3 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (𝑥 · 𝑦) ∈ ℝ)
3 i1fadd.1 . . . 4 (𝜑𝐹 ∈ dom ∫1)
4 i1ff 25711 . . . 4 (𝐹 ∈ dom ∫1𝐹:ℝ⟶ℝ)
53, 4syl 17 . . 3 (𝜑𝐹:ℝ⟶ℝ)
6 i1fadd.2 . . . 4 (𝜑𝐺 ∈ dom ∫1)
7 i1ff 25711 . . . 4 (𝐺 ∈ dom ∫1𝐺:ℝ⟶ℝ)
86, 7syl 17 . . 3 (𝜑𝐺:ℝ⟶ℝ)
9 reex 11246 . . . 4 ℝ ∈ V
109a1i 11 . . 3 (𝜑 → ℝ ∈ V)
11 inidm 4227 . . 3 (ℝ ∩ ℝ) = ℝ
122, 5, 8, 10, 10, 11off 7715 . 2 (𝜑 → (𝐹f · 𝐺):ℝ⟶ℝ)
13 i1frn 25712 . . . . . 6 (𝐹 ∈ dom ∫1 → ran 𝐹 ∈ Fin)
143, 13syl 17 . . . . 5 (𝜑 → ran 𝐹 ∈ Fin)
15 i1frn 25712 . . . . . 6 (𝐺 ∈ dom ∫1 → ran 𝐺 ∈ Fin)
166, 15syl 17 . . . . 5 (𝜑 → ran 𝐺 ∈ Fin)
17 xpfi 9358 . . . . 5 ((ran 𝐹 ∈ Fin ∧ ran 𝐺 ∈ Fin) → (ran 𝐹 × ran 𝐺) ∈ Fin)
1814, 16, 17syl2anc 584 . . . 4 (𝜑 → (ran 𝐹 × ran 𝐺) ∈ Fin)
19 eqid 2737 . . . . . 6 (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 · 𝑣)) = (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 · 𝑣))
20 ovex 7464 . . . . . 6 (𝑢 · 𝑣) ∈ V
2119, 20fnmpoi 8095 . . . . 5 (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 · 𝑣)) Fn (ran 𝐹 × ran 𝐺)
22 dffn4 6826 . . . . 5 ((𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 · 𝑣)) Fn (ran 𝐹 × ran 𝐺) ↔ (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 · 𝑣)):(ran 𝐹 × ran 𝐺)–onto→ran (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 · 𝑣)))
2321, 22mpbi 230 . . . 4 (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 · 𝑣)):(ran 𝐹 × ran 𝐺)–onto→ran (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 · 𝑣))
24 fofi 9351 . . . 4 (((ran 𝐹 × ran 𝐺) ∈ Fin ∧ (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 · 𝑣)):(ran 𝐹 × ran 𝐺)–onto→ran (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 · 𝑣))) → ran (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 · 𝑣)) ∈ Fin)
2518, 23, 24sylancl 586 . . 3 (𝜑 → ran (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 · 𝑣)) ∈ Fin)
26 eqid 2737 . . . . . . . . 9 (𝑥 · 𝑦) = (𝑥 · 𝑦)
27 rspceov 7480 . . . . . . . . 9 ((𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐺 ∧ (𝑥 · 𝑦) = (𝑥 · 𝑦)) → ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺(𝑥 · 𝑦) = (𝑢 · 𝑣))
2826, 27mp3an3 1452 . . . . . . . 8 ((𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐺) → ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺(𝑥 · 𝑦) = (𝑢 · 𝑣))
29 ovex 7464 . . . . . . . . 9 (𝑥 · 𝑦) ∈ V
30 eqeq1 2741 . . . . . . . . . 10 (𝑤 = (𝑥 · 𝑦) → (𝑤 = (𝑢 · 𝑣) ↔ (𝑥 · 𝑦) = (𝑢 · 𝑣)))
31302rexbidv 3222 . . . . . . . . 9 (𝑤 = (𝑥 · 𝑦) → (∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺 𝑤 = (𝑢 · 𝑣) ↔ ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺(𝑥 · 𝑦) = (𝑢 · 𝑣)))
3229, 31elab 3679 . . . . . . . 8 ((𝑥 · 𝑦) ∈ {𝑤 ∣ ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺 𝑤 = (𝑢 · 𝑣)} ↔ ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺(𝑥 · 𝑦) = (𝑢 · 𝑣))
3328, 32sylibr 234 . . . . . . 7 ((𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐺) → (𝑥 · 𝑦) ∈ {𝑤 ∣ ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺 𝑤 = (𝑢 · 𝑣)})
3433adantl 481 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐺)) → (𝑥 · 𝑦) ∈ {𝑤 ∣ ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺 𝑤 = (𝑢 · 𝑣)})
355ffnd 6737 . . . . . . 7 (𝜑𝐹 Fn ℝ)
36 dffn3 6748 . . . . . . 7 (𝐹 Fn ℝ ↔ 𝐹:ℝ⟶ran 𝐹)
3735, 36sylib 218 . . . . . 6 (𝜑𝐹:ℝ⟶ran 𝐹)
388ffnd 6737 . . . . . . 7 (𝜑𝐺 Fn ℝ)
39 dffn3 6748 . . . . . . 7 (𝐺 Fn ℝ ↔ 𝐺:ℝ⟶ran 𝐺)
4038, 39sylib 218 . . . . . 6 (𝜑𝐺:ℝ⟶ran 𝐺)
4134, 37, 40, 10, 10, 11off 7715 . . . . 5 (𝜑 → (𝐹f · 𝐺):ℝ⟶{𝑤 ∣ ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺 𝑤 = (𝑢 · 𝑣)})
4241frnd 6744 . . . 4 (𝜑 → ran (𝐹f · 𝐺) ⊆ {𝑤 ∣ ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺 𝑤 = (𝑢 · 𝑣)})
4319rnmpo 7566 . . . 4 ran (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 · 𝑣)) = {𝑤 ∣ ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺 𝑤 = (𝑢 · 𝑣)}
4442, 43sseqtrrdi 4025 . . 3 (𝜑 → ran (𝐹f · 𝐺) ⊆ ran (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 · 𝑣)))
4525, 44ssfid 9301 . 2 (𝜑 → ran (𝐹f · 𝐺) ∈ Fin)
4612frnd 6744 . . . . . . 7 (𝜑 → ran (𝐹f · 𝐺) ⊆ ℝ)
47 ax-resscn 11212 . . . . . . 7 ℝ ⊆ ℂ
4846, 47sstrdi 3996 . . . . . 6 (𝜑 → ran (𝐹f · 𝐺) ⊆ ℂ)
4948ssdifd 4145 . . . . 5 (𝜑 → (ran (𝐹f · 𝐺) ∖ {0}) ⊆ (ℂ ∖ {0}))
5049sselda 3983 . . . 4 ((𝜑𝑦 ∈ (ran (𝐹f · 𝐺) ∖ {0})) → 𝑦 ∈ (ℂ ∖ {0}))
513, 6i1fmullem 25729 . . . 4 ((𝜑𝑦 ∈ (ℂ ∖ {0})) → ((𝐹f · 𝐺) “ {𝑦}) = 𝑧 ∈ (ran 𝐺 ∖ {0})((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧})))
5250, 51syldan 591 . . 3 ((𝜑𝑦 ∈ (ran (𝐹f · 𝐺) ∖ {0})) → ((𝐹f · 𝐺) “ {𝑦}) = 𝑧 ∈ (ran 𝐺 ∖ {0})((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧})))
53 difss 4136 . . . . . 6 (ran 𝐺 ∖ {0}) ⊆ ran 𝐺
54 ssfi 9213 . . . . . 6 ((ran 𝐺 ∈ Fin ∧ (ran 𝐺 ∖ {0}) ⊆ ran 𝐺) → (ran 𝐺 ∖ {0}) ∈ Fin)
5516, 53, 54sylancl 586 . . . . 5 (𝜑 → (ran 𝐺 ∖ {0}) ∈ Fin)
56 i1fima 25713 . . . . . . . 8 (𝐹 ∈ dom ∫1 → (𝐹 “ {(𝑦 / 𝑧)}) ∈ dom vol)
573, 56syl 17 . . . . . . 7 (𝜑 → (𝐹 “ {(𝑦 / 𝑧)}) ∈ dom vol)
58 i1fima 25713 . . . . . . . 8 (𝐺 ∈ dom ∫1 → (𝐺 “ {𝑧}) ∈ dom vol)
596, 58syl 17 . . . . . . 7 (𝜑 → (𝐺 “ {𝑧}) ∈ dom vol)
60 inmbl 25577 . . . . . . 7 (((𝐹 “ {(𝑦 / 𝑧)}) ∈ dom vol ∧ (𝐺 “ {𝑧}) ∈ dom vol) → ((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧})) ∈ dom vol)
6157, 59, 60syl2anc 584 . . . . . 6 (𝜑 → ((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧})) ∈ dom vol)
6261ralrimivw 3150 . . . . 5 (𝜑 → ∀𝑧 ∈ (ran 𝐺 ∖ {0})((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧})) ∈ dom vol)
63 finiunmbl 25579 . . . . 5 (((ran 𝐺 ∖ {0}) ∈ Fin ∧ ∀𝑧 ∈ (ran 𝐺 ∖ {0})((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧})) ∈ dom vol) → 𝑧 ∈ (ran 𝐺 ∖ {0})((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧})) ∈ dom vol)
6455, 62, 63syl2anc 584 . . . 4 (𝜑 𝑧 ∈ (ran 𝐺 ∖ {0})((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧})) ∈ dom vol)
6564adantr 480 . . 3 ((𝜑𝑦 ∈ (ran (𝐹f · 𝐺) ∖ {0})) → 𝑧 ∈ (ran 𝐺 ∖ {0})((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧})) ∈ dom vol)
6652, 65eqeltrd 2841 . 2 ((𝜑𝑦 ∈ (ran (𝐹f · 𝐺) ∖ {0})) → ((𝐹f · 𝐺) “ {𝑦}) ∈ dom vol)
67 mblvol 25565 . . . 4 (((𝐹f · 𝐺) “ {𝑦}) ∈ dom vol → (vol‘((𝐹f · 𝐺) “ {𝑦})) = (vol*‘((𝐹f · 𝐺) “ {𝑦})))
6866, 67syl 17 . . 3 ((𝜑𝑦 ∈ (ran (𝐹f · 𝐺) ∖ {0})) → (vol‘((𝐹f · 𝐺) “ {𝑦})) = (vol*‘((𝐹f · 𝐺) “ {𝑦})))
69 mblss 25566 . . . . 5 (((𝐹f · 𝐺) “ {𝑦}) ∈ dom vol → ((𝐹f · 𝐺) “ {𝑦}) ⊆ ℝ)
7066, 69syl 17 . . . 4 ((𝜑𝑦 ∈ (ran (𝐹f · 𝐺) ∖ {0})) → ((𝐹f · 𝐺) “ {𝑦}) ⊆ ℝ)
7155adantr 480 . . . . 5 ((𝜑𝑦 ∈ (ran (𝐹f · 𝐺) ∖ {0})) → (ran 𝐺 ∖ {0}) ∈ Fin)
72 inss2 4238 . . . . . . 7 ((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧})) ⊆ (𝐺 “ {𝑧})
7372a1i 11 . . . . . 6 (((𝜑𝑦 ∈ (ran (𝐹f · 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → ((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧})) ⊆ (𝐺 “ {𝑧}))
7459ad2antrr 726 . . . . . . 7 (((𝜑𝑦 ∈ (ran (𝐹f · 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (𝐺 “ {𝑧}) ∈ dom vol)
75 mblss 25566 . . . . . . 7 ((𝐺 “ {𝑧}) ∈ dom vol → (𝐺 “ {𝑧}) ⊆ ℝ)
7674, 75syl 17 . . . . . 6 (((𝜑𝑦 ∈ (ran (𝐹f · 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (𝐺 “ {𝑧}) ⊆ ℝ)
77 mblvol 25565 . . . . . . . 8 ((𝐺 “ {𝑧}) ∈ dom vol → (vol‘(𝐺 “ {𝑧})) = (vol*‘(𝐺 “ {𝑧})))
7874, 77syl 17 . . . . . . 7 (((𝜑𝑦 ∈ (ran (𝐹f · 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol‘(𝐺 “ {𝑧})) = (vol*‘(𝐺 “ {𝑧})))
796adantr 480 . . . . . . . 8 ((𝜑𝑦 ∈ (ran (𝐹f · 𝐺) ∖ {0})) → 𝐺 ∈ dom ∫1)
80 i1fima2sn 25715 . . . . . . . 8 ((𝐺 ∈ dom ∫1𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol‘(𝐺 “ {𝑧})) ∈ ℝ)
8179, 80sylan 580 . . . . . . 7 (((𝜑𝑦 ∈ (ran (𝐹f · 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol‘(𝐺 “ {𝑧})) ∈ ℝ)
8278, 81eqeltrrd 2842 . . . . . 6 (((𝜑𝑦 ∈ (ran (𝐹f · 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol*‘(𝐺 “ {𝑧})) ∈ ℝ)
83 ovolsscl 25521 . . . . . 6 ((((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧})) ⊆ (𝐺 “ {𝑧}) ∧ (𝐺 “ {𝑧}) ⊆ ℝ ∧ (vol*‘(𝐺 “ {𝑧})) ∈ ℝ) → (vol*‘((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ)
8473, 76, 82, 83syl3anc 1373 . . . . 5 (((𝜑𝑦 ∈ (ran (𝐹f · 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol*‘((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ)
8571, 84fsumrecl 15770 . . . 4 ((𝜑𝑦 ∈ (ran (𝐹f · 𝐺) ∖ {0})) → Σ𝑧 ∈ (ran 𝐺 ∖ {0})(vol*‘((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ)
8652fveq2d 6910 . . . . 5 ((𝜑𝑦 ∈ (ran (𝐹f · 𝐺) ∖ {0})) → (vol*‘((𝐹f · 𝐺) “ {𝑦})) = (vol*‘ 𝑧 ∈ (ran 𝐺 ∖ {0})((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧}))))
87 mblss 25566 . . . . . . . . . 10 (((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧})) ∈ dom vol → ((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧})) ⊆ ℝ)
8861, 87syl 17 . . . . . . . . 9 (𝜑 → ((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧})) ⊆ ℝ)
8988ad2antrr 726 . . . . . . . 8 (((𝜑𝑦 ∈ (ran (𝐹f · 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → ((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧})) ⊆ ℝ)
9089, 84jca 511 . . . . . . 7 (((𝜑𝑦 ∈ (ran (𝐹f · 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧})) ⊆ ℝ ∧ (vol*‘((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ))
9190ralrimiva 3146 . . . . . 6 ((𝜑𝑦 ∈ (ran (𝐹f · 𝐺) ∖ {0})) → ∀𝑧 ∈ (ran 𝐺 ∖ {0})(((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧})) ⊆ ℝ ∧ (vol*‘((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ))
92 ovolfiniun 25536 . . . . . 6 (((ran 𝐺 ∖ {0}) ∈ Fin ∧ ∀𝑧 ∈ (ran 𝐺 ∖ {0})(((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧})) ⊆ ℝ ∧ (vol*‘((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ)) → (vol*‘ 𝑧 ∈ (ran 𝐺 ∖ {0})((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧}))) ≤ Σ𝑧 ∈ (ran 𝐺 ∖ {0})(vol*‘((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧}))))
9371, 91, 92syl2anc 584 . . . . 5 ((𝜑𝑦 ∈ (ran (𝐹f · 𝐺) ∖ {0})) → (vol*‘ 𝑧 ∈ (ran 𝐺 ∖ {0})((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧}))) ≤ Σ𝑧 ∈ (ran 𝐺 ∖ {0})(vol*‘((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧}))))
9486, 93eqbrtrd 5165 . . . 4 ((𝜑𝑦 ∈ (ran (𝐹f · 𝐺) ∖ {0})) → (vol*‘((𝐹f · 𝐺) “ {𝑦})) ≤ Σ𝑧 ∈ (ran 𝐺 ∖ {0})(vol*‘((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧}))))
95 ovollecl 25518 . . . 4 ((((𝐹f · 𝐺) “ {𝑦}) ⊆ ℝ ∧ Σ𝑧 ∈ (ran 𝐺 ∖ {0})(vol*‘((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ ∧ (vol*‘((𝐹f · 𝐺) “ {𝑦})) ≤ Σ𝑧 ∈ (ran 𝐺 ∖ {0})(vol*‘((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧})))) → (vol*‘((𝐹f · 𝐺) “ {𝑦})) ∈ ℝ)
9670, 85, 94, 95syl3anc 1373 . . 3 ((𝜑𝑦 ∈ (ran (𝐹f · 𝐺) ∖ {0})) → (vol*‘((𝐹f · 𝐺) “ {𝑦})) ∈ ℝ)
9768, 96eqeltrd 2841 . 2 ((𝜑𝑦 ∈ (ran (𝐹f · 𝐺) ∖ {0})) → (vol‘((𝐹f · 𝐺) “ {𝑦})) ∈ ℝ)
9812, 45, 66, 97i1fd 25716 1 (𝜑 → (𝐹f · 𝐺) ∈ dom ∫1)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  {cab 2714  wral 3061  wrex 3070  Vcvv 3480  cdif 3948  cin 3950  wss 3951  {csn 4626   ciun 4991   class class class wbr 5143   × cxp 5683  ccnv 5684  dom cdm 5685  ran crn 5686  cima 5688   Fn wfn 6556  wf 6557  ontowfo 6559  cfv 6561  (class class class)co 7431  cmpo 7433  f cof 7695  Fincfn 8985  cc 11153  cr 11154  0cc0 11155   · cmul 11160  cle 11296   / cdiv 11920  Σcsu 15722  vol*covol 25497  volcvol 25498  1citg1 25650
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-map 8868  df-pm 8869  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-sup 9482  df-inf 9483  df-oi 9550  df-dju 9941  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-z 12614  df-uz 12879  df-q 12991  df-rp 13035  df-xadd 13155  df-ioo 13391  df-ico 13393  df-icc 13394  df-fz 13548  df-fzo 13695  df-fl 13832  df-seq 14043  df-exp 14103  df-hash 14370  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-clim 15524  df-sum 15723  df-xmet 21357  df-met 21358  df-ovol 25499  df-vol 25500  df-mbf 25654  df-itg1 25655
This theorem is referenced by:  mbfmullem2  25759  ftc1anclem3  37702
  Copyright terms: Public domain W3C validator