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Theorem i1fmul 25549
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 11192 . . . 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 25529 . . . 4 (𝐹 ∈ dom ∫1𝐹:ℝ⟶ℝ)
53, 4syl 17 . . 3 (𝜑𝐹:ℝ⟶ℝ)
6 i1fadd.2 . . . 4 (𝜑𝐺 ∈ dom ∫1)
7 i1ff 25529 . . . 4 (𝐺 ∈ dom ∫1𝐺:ℝ⟶ℝ)
86, 7syl 17 . . 3 (𝜑𝐺:ℝ⟶ℝ)
9 reex 11198 . . . 4 ℝ ∈ V
109a1i 11 . . 3 (𝜑 → ℝ ∈ V)
11 inidm 4211 . . 3 (ℝ ∩ ℝ) = ℝ
122, 5, 8, 10, 10, 11off 7682 . 2 (𝜑 → (𝐹f · 𝐺):ℝ⟶ℝ)
13 i1frn 25530 . . . . . 6 (𝐹 ∈ dom ∫1 → ran 𝐹 ∈ Fin)
143, 13syl 17 . . . . 5 (𝜑 → ran 𝐹 ∈ Fin)
15 i1frn 25530 . . . . . 6 (𝐺 ∈ dom ∫1 → ran 𝐺 ∈ Fin)
166, 15syl 17 . . . . 5 (𝜑 → ran 𝐺 ∈ Fin)
17 xpfi 9314 . . . . 5 ((ran 𝐹 ∈ Fin ∧ ran 𝐺 ∈ Fin) → (ran 𝐹 × ran 𝐺) ∈ Fin)
1814, 16, 17syl2anc 583 . . . 4 (𝜑 → (ran 𝐹 × ran 𝐺) ∈ Fin)
19 eqid 2724 . . . . . 6 (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 · 𝑣)) = (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 · 𝑣))
20 ovex 7435 . . . . . 6 (𝑢 · 𝑣) ∈ V
2119, 20fnmpoi 8050 . . . . 5 (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 · 𝑣)) Fn (ran 𝐹 × ran 𝐺)
22 dffn4 6802 . . . . 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 229 . . . 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 9335 . . . 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 585 . . 3 (𝜑 → ran (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 · 𝑣)) ∈ Fin)
26 eqid 2724 . . . . . . . . 9 (𝑥 · 𝑦) = (𝑥 · 𝑦)
27 rspceov 7449 . . . . . . . . 9 ((𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐺 ∧ (𝑥 · 𝑦) = (𝑥 · 𝑦)) → ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺(𝑥 · 𝑦) = (𝑢 · 𝑣))
2826, 27mp3an3 1446 . . . . . . . 8 ((𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐺) → ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺(𝑥 · 𝑦) = (𝑢 · 𝑣))
29 ovex 7435 . . . . . . . . 9 (𝑥 · 𝑦) ∈ V
30 eqeq1 2728 . . . . . . . . . 10 (𝑤 = (𝑥 · 𝑦) → (𝑤 = (𝑢 · 𝑣) ↔ (𝑥 · 𝑦) = (𝑢 · 𝑣)))
31302rexbidv 3211 . . . . . . . . 9 (𝑤 = (𝑥 · 𝑦) → (∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺 𝑤 = (𝑢 · 𝑣) ↔ ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺(𝑥 · 𝑦) = (𝑢 · 𝑣)))
3229, 31elab 3661 . . . . . . . 8 ((𝑥 · 𝑦) ∈ {𝑤 ∣ ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺 𝑤 = (𝑢 · 𝑣)} ↔ ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺(𝑥 · 𝑦) = (𝑢 · 𝑣))
3328, 32sylibr 233 . . . . . . 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 6709 . . . . . . 7 (𝜑𝐹 Fn ℝ)
36 dffn3 6721 . . . . . . 7 (𝐹 Fn ℝ ↔ 𝐹:ℝ⟶ran 𝐹)
3735, 36sylib 217 . . . . . 6 (𝜑𝐹:ℝ⟶ran 𝐹)
388ffnd 6709 . . . . . . 7 (𝜑𝐺 Fn ℝ)
39 dffn3 6721 . . . . . . 7 (𝐺 Fn ℝ ↔ 𝐺:ℝ⟶ran 𝐺)
4038, 39sylib 217 . . . . . 6 (𝜑𝐺:ℝ⟶ran 𝐺)
4134, 37, 40, 10, 10, 11off 7682 . . . . 5 (𝜑 → (𝐹f · 𝐺):ℝ⟶{𝑤 ∣ ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺 𝑤 = (𝑢 · 𝑣)})
4241frnd 6716 . . . 4 (𝜑 → ran (𝐹f · 𝐺) ⊆ {𝑤 ∣ ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺 𝑤 = (𝑢 · 𝑣)})
4319rnmpo 7535 . . . 4 ran (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 · 𝑣)) = {𝑤 ∣ ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺 𝑤 = (𝑢 · 𝑣)}
4442, 43sseqtrrdi 4026 . . 3 (𝜑 → ran (𝐹f · 𝐺) ⊆ ran (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 · 𝑣)))
4525, 44ssfid 9264 . 2 (𝜑 → ran (𝐹f · 𝐺) ∈ Fin)
4612frnd 6716 . . . . . . 7 (𝜑 → ran (𝐹f · 𝐺) ⊆ ℝ)
47 ax-resscn 11164 . . . . . . 7 ℝ ⊆ ℂ
4846, 47sstrdi 3987 . . . . . 6 (𝜑 → ran (𝐹f · 𝐺) ⊆ ℂ)
4948ssdifd 4133 . . . . 5 (𝜑 → (ran (𝐹f · 𝐺) ∖ {0}) ⊆ (ℂ ∖ {0}))
5049sselda 3975 . . . 4 ((𝜑𝑦 ∈ (ran (𝐹f · 𝐺) ∖ {0})) → 𝑦 ∈ (ℂ ∖ {0}))
513, 6i1fmullem 25547 . . . 4 ((𝜑𝑦 ∈ (ℂ ∖ {0})) → ((𝐹f · 𝐺) “ {𝑦}) = 𝑧 ∈ (ran 𝐺 ∖ {0})((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧})))
5250, 51syldan 590 . . 3 ((𝜑𝑦 ∈ (ran (𝐹f · 𝐺) ∖ {0})) → ((𝐹f · 𝐺) “ {𝑦}) = 𝑧 ∈ (ran 𝐺 ∖ {0})((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧})))
53 difss 4124 . . . . . 6 (ran 𝐺 ∖ {0}) ⊆ ran 𝐺
54 ssfi 9170 . . . . . 6 ((ran 𝐺 ∈ Fin ∧ (ran 𝐺 ∖ {0}) ⊆ ran 𝐺) → (ran 𝐺 ∖ {0}) ∈ Fin)
5516, 53, 54sylancl 585 . . . . 5 (𝜑 → (ran 𝐺 ∖ {0}) ∈ Fin)
56 i1fima 25531 . . . . . . . 8 (𝐹 ∈ dom ∫1 → (𝐹 “ {(𝑦 / 𝑧)}) ∈ dom vol)
573, 56syl 17 . . . . . . 7 (𝜑 → (𝐹 “ {(𝑦 / 𝑧)}) ∈ dom vol)
58 i1fima 25531 . . . . . . . 8 (𝐺 ∈ dom ∫1 → (𝐺 “ {𝑧}) ∈ dom vol)
596, 58syl 17 . . . . . . 7 (𝜑 → (𝐺 “ {𝑧}) ∈ dom vol)
60 inmbl 25395 . . . . . . 7 (((𝐹 “ {(𝑦 / 𝑧)}) ∈ dom vol ∧ (𝐺 “ {𝑧}) ∈ dom vol) → ((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧})) ∈ dom vol)
6157, 59, 60syl2anc 583 . . . . . 6 (𝜑 → ((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧})) ∈ dom vol)
6261ralrimivw 3142 . . . . 5 (𝜑 → ∀𝑧 ∈ (ran 𝐺 ∖ {0})((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧})) ∈ dom vol)
63 finiunmbl 25397 . . . . 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 583 . . . 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 2825 . 2 ((𝜑𝑦 ∈ (ran (𝐹f · 𝐺) ∖ {0})) → ((𝐹f · 𝐺) “ {𝑦}) ∈ dom vol)
67 mblvol 25383 . . . 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 25384 . . . . 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 4222 . . . . . . 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 723 . . . . . . 7 (((𝜑𝑦 ∈ (ran (𝐹f · 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (𝐺 “ {𝑧}) ∈ dom vol)
75 mblss 25384 . . . . . . 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 25383 . . . . . . . 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 25533 . . . . . . . 8 ((𝐺 ∈ dom ∫1𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol‘(𝐺 “ {𝑧})) ∈ ℝ)
8179, 80sylan 579 . . . . . . 7 (((𝜑𝑦 ∈ (ran (𝐹f · 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol‘(𝐺 “ {𝑧})) ∈ ℝ)
8278, 81eqeltrrd 2826 . . . . . 6 (((𝜑𝑦 ∈ (ran (𝐹f · 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol*‘(𝐺 “ {𝑧})) ∈ ℝ)
83 ovolsscl 25339 . . . . . 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 1368 . . . . 5 (((𝜑𝑦 ∈ (ran (𝐹f · 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol*‘((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ)
8571, 84fsumrecl 15678 . . . 4 ((𝜑𝑦 ∈ (ran (𝐹f · 𝐺) ∖ {0})) → Σ𝑧 ∈ (ran 𝐺 ∖ {0})(vol*‘((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ)
8652fveq2d 6886 . . . . 5 ((𝜑𝑦 ∈ (ran (𝐹f · 𝐺) ∖ {0})) → (vol*‘((𝐹f · 𝐺) “ {𝑦})) = (vol*‘ 𝑧 ∈ (ran 𝐺 ∖ {0})((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧}))))
87 mblss 25384 . . . . . . . . . 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 723 . . . . . . . 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 3138 . . . . . 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 25354 . . . . . 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 583 . . . . 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 5161 . . . 4 ((𝜑𝑦 ∈ (ran (𝐹f · 𝐺) ∖ {0})) → (vol*‘((𝐹f · 𝐺) “ {𝑦})) ≤ Σ𝑧 ∈ (ran 𝐺 ∖ {0})(vol*‘((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧}))))
95 ovollecl 25336 . . . 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 1368 . . 3 ((𝜑𝑦 ∈ (ran (𝐹f · 𝐺) ∖ {0})) → (vol*‘((𝐹f · 𝐺) “ {𝑦})) ∈ ℝ)
9768, 96eqeltrd 2825 . 2 ((𝜑𝑦 ∈ (ran (𝐹f · 𝐺) ∖ {0})) → (vol‘((𝐹f · 𝐺) “ {𝑦})) ∈ ℝ)
9812, 45, 66, 97i1fd 25534 1 (𝜑 → (𝐹f · 𝐺) ∈ dom ∫1)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  {cab 2701  wral 3053  wrex 3062  Vcvv 3466  cdif 3938  cin 3940  wss 3941  {csn 4621   ciun 4988   class class class wbr 5139   × cxp 5665  ccnv 5666  dom cdm 5667  ran crn 5668  cima 5670   Fn wfn 6529  wf 6530  ontowfo 6532  cfv 6534  (class class class)co 7402  cmpo 7404  f cof 7662  Fincfn 8936  cc 11105  cr 11106  0cc0 11107   · cmul 11112  cle 11247   / cdiv 11869  Σcsu 15630  vol*covol 25315  volcvol 25316  1citg1 25468
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 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719  ax-inf2 9633  ax-cnex 11163  ax-resscn 11164  ax-1cn 11165  ax-icn 11166  ax-addcl 11167  ax-addrcl 11168  ax-mulcl 11169  ax-mulrcl 11170  ax-mulcom 11171  ax-addass 11172  ax-mulass 11173  ax-distr 11174  ax-i2m1 11175  ax-1ne0 11176  ax-1rid 11177  ax-rnegex 11178  ax-rrecex 11179  ax-cnre 11180  ax-pre-lttri 11181  ax-pre-lttrn 11182  ax-pre-ltadd 11183  ax-pre-mulgt0 11184  ax-pre-sup 11185
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 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3960  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-int 4942  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-tr 5257  df-id 5565  df-eprel 5571  df-po 5579  df-so 5580  df-fr 5622  df-se 5623  df-we 5624  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-pred 6291  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-isom 6543  df-riota 7358  df-ov 7405  df-oprab 7406  df-mpo 7407  df-of 7664  df-om 7850  df-1st 7969  df-2nd 7970  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-2o 8463  df-er 8700  df-map 8819  df-pm 8820  df-en 8937  df-dom 8938  df-sdom 8939  df-fin 8940  df-sup 9434  df-inf 9435  df-oi 9502  df-dju 9893  df-card 9931  df-pnf 11248  df-mnf 11249  df-xr 11250  df-ltxr 11251  df-le 11252  df-sub 11444  df-neg 11445  df-div 11870  df-nn 12211  df-2 12273  df-3 12274  df-n0 12471  df-z 12557  df-uz 12821  df-q 12931  df-rp 12973  df-xadd 13091  df-ioo 13326  df-ico 13328  df-icc 13329  df-fz 13483  df-fzo 13626  df-fl 13755  df-seq 13965  df-exp 14026  df-hash 14289  df-cj 15044  df-re 15045  df-im 15046  df-sqrt 15180  df-abs 15181  df-clim 15430  df-sum 15631  df-xmet 21223  df-met 21224  df-ovol 25317  df-vol 25318  df-mbf 25472  df-itg1 25473
This theorem is referenced by:  mbfmullem2  25578  ftc1anclem3  37057
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