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Theorem i1fmul 23382
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 9972 . . . 4 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 · 𝑦) ∈ ℝ)
21adantl 482 . . 3 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (𝑥 · 𝑦) ∈ ℝ)
3 i1fadd.1 . . . 4 (𝜑𝐹 ∈ dom ∫1)
4 i1ff 23362 . . . 4 (𝐹 ∈ dom ∫1𝐹:ℝ⟶ℝ)
53, 4syl 17 . . 3 (𝜑𝐹:ℝ⟶ℝ)
6 i1fadd.2 . . . 4 (𝜑𝐺 ∈ dom ∫1)
7 i1ff 23362 . . . 4 (𝐺 ∈ dom ∫1𝐺:ℝ⟶ℝ)
86, 7syl 17 . . 3 (𝜑𝐺:ℝ⟶ℝ)
9 reex 9978 . . . 4 ℝ ∈ V
109a1i 11 . . 3 (𝜑 → ℝ ∈ V)
11 inidm 3805 . . 3 (ℝ ∩ ℝ) = ℝ
122, 5, 8, 10, 10, 11off 6872 . 2 (𝜑 → (𝐹𝑓 · 𝐺):ℝ⟶ℝ)
13 i1frn 23363 . . . . . 6 (𝐹 ∈ dom ∫1 → ran 𝐹 ∈ Fin)
143, 13syl 17 . . . . 5 (𝜑 → ran 𝐹 ∈ Fin)
15 i1frn 23363 . . . . . 6 (𝐺 ∈ dom ∫1 → ran 𝐺 ∈ Fin)
166, 15syl 17 . . . . 5 (𝜑 → ran 𝐺 ∈ Fin)
17 xpfi 8182 . . . . 5 ((ran 𝐹 ∈ Fin ∧ ran 𝐺 ∈ Fin) → (ran 𝐹 × ran 𝐺) ∈ Fin)
1814, 16, 17syl2anc 692 . . . 4 (𝜑 → (ran 𝐹 × ran 𝐺) ∈ Fin)
19 eqid 2621 . . . . . 6 (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 · 𝑣)) = (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 · 𝑣))
20 ovex 6638 . . . . . 6 (𝑢 · 𝑣) ∈ V
2119, 20fnmpt2i 7191 . . . . 5 (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 · 𝑣)) Fn (ran 𝐹 × ran 𝐺)
22 dffn4 6083 . . . . 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 220 . . . 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 8203 . . . 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 693 . . 3 (𝜑 → ran (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 · 𝑣)) ∈ Fin)
26 eqid 2621 . . . . . . . . 9 (𝑥 · 𝑦) = (𝑥 · 𝑦)
27 rspceov 6652 . . . . . . . . 9 ((𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐺 ∧ (𝑥 · 𝑦) = (𝑥 · 𝑦)) → ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺(𝑥 · 𝑦) = (𝑢 · 𝑣))
2826, 27mp3an3 1410 . . . . . . . 8 ((𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐺) → ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺(𝑥 · 𝑦) = (𝑢 · 𝑣))
29 ovex 6638 . . . . . . . . 9 (𝑥 · 𝑦) ∈ V
30 eqeq1 2625 . . . . . . . . . 10 (𝑤 = (𝑥 · 𝑦) → (𝑤 = (𝑢 · 𝑣) ↔ (𝑥 · 𝑦) = (𝑢 · 𝑣)))
31302rexbidv 3051 . . . . . . . . 9 (𝑤 = (𝑥 · 𝑦) → (∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺 𝑤 = (𝑢 · 𝑣) ↔ ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺(𝑥 · 𝑦) = (𝑢 · 𝑣)))
3229, 31elab 3337 . . . . . . . 8 ((𝑥 · 𝑦) ∈ {𝑤 ∣ ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺 𝑤 = (𝑢 · 𝑣)} ↔ ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺(𝑥 · 𝑦) = (𝑢 · 𝑣))
3328, 32sylibr 224 . . . . . . 7 ((𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐺) → (𝑥 · 𝑦) ∈ {𝑤 ∣ ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺 𝑤 = (𝑢 · 𝑣)})
3433adantl 482 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐺)) → (𝑥 · 𝑦) ∈ {𝑤 ∣ ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺 𝑤 = (𝑢 · 𝑣)})
35 ffn 6007 . . . . . . . 8 (𝐹:ℝ⟶ℝ → 𝐹 Fn ℝ)
365, 35syl 17 . . . . . . 7 (𝜑𝐹 Fn ℝ)
37 dffn3 6016 . . . . . . 7 (𝐹 Fn ℝ ↔ 𝐹:ℝ⟶ran 𝐹)
3836, 37sylib 208 . . . . . 6 (𝜑𝐹:ℝ⟶ran 𝐹)
39 ffn 6007 . . . . . . . 8 (𝐺:ℝ⟶ℝ → 𝐺 Fn ℝ)
408, 39syl 17 . . . . . . 7 (𝜑𝐺 Fn ℝ)
41 dffn3 6016 . . . . . . 7 (𝐺 Fn ℝ ↔ 𝐺:ℝ⟶ran 𝐺)
4240, 41sylib 208 . . . . . 6 (𝜑𝐺:ℝ⟶ran 𝐺)
4334, 38, 42, 10, 10, 11off 6872 . . . . 5 (𝜑 → (𝐹𝑓 · 𝐺):ℝ⟶{𝑤 ∣ ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺 𝑤 = (𝑢 · 𝑣)})
44 frn 6015 . . . . 5 ((𝐹𝑓 · 𝐺):ℝ⟶{𝑤 ∣ ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺 𝑤 = (𝑢 · 𝑣)} → ran (𝐹𝑓 · 𝐺) ⊆ {𝑤 ∣ ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺 𝑤 = (𝑢 · 𝑣)})
4543, 44syl 17 . . . 4 (𝜑 → ran (𝐹𝑓 · 𝐺) ⊆ {𝑤 ∣ ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺 𝑤 = (𝑢 · 𝑣)})
4619rnmpt2 6730 . . . 4 ran (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 · 𝑣)) = {𝑤 ∣ ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺 𝑤 = (𝑢 · 𝑣)}
4745, 46syl6sseqr 3636 . . 3 (𝜑 → ran (𝐹𝑓 · 𝐺) ⊆ ran (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 · 𝑣)))
48 ssfi 8131 . . 3 ((ran (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 · 𝑣)) ∈ Fin ∧ ran (𝐹𝑓 · 𝐺) ⊆ ran (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 · 𝑣))) → ran (𝐹𝑓 · 𝐺) ∈ Fin)
4925, 47, 48syl2anc 692 . 2 (𝜑 → ran (𝐹𝑓 · 𝐺) ∈ Fin)
50 frn 6015 . . . . . . . 8 ((𝐹𝑓 · 𝐺):ℝ⟶ℝ → ran (𝐹𝑓 · 𝐺) ⊆ ℝ)
5112, 50syl 17 . . . . . . 7 (𝜑 → ran (𝐹𝑓 · 𝐺) ⊆ ℝ)
52 ax-resscn 9944 . . . . . . 7 ℝ ⊆ ℂ
5351, 52syl6ss 3599 . . . . . 6 (𝜑 → ran (𝐹𝑓 · 𝐺) ⊆ ℂ)
5453ssdifd 3729 . . . . 5 (𝜑 → (ran (𝐹𝑓 · 𝐺) ∖ {0}) ⊆ (ℂ ∖ {0}))
5554sselda 3587 . . . 4 ((𝜑𝑦 ∈ (ran (𝐹𝑓 · 𝐺) ∖ {0})) → 𝑦 ∈ (ℂ ∖ {0}))
563, 6i1fmullem 23380 . . . 4 ((𝜑𝑦 ∈ (ℂ ∖ {0})) → ((𝐹𝑓 · 𝐺) “ {𝑦}) = 𝑧 ∈ (ran 𝐺 ∖ {0})((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧})))
5755, 56syldan 487 . . 3 ((𝜑𝑦 ∈ (ran (𝐹𝑓 · 𝐺) ∖ {0})) → ((𝐹𝑓 · 𝐺) “ {𝑦}) = 𝑧 ∈ (ran 𝐺 ∖ {0})((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧})))
58 difss 3720 . . . . . 6 (ran 𝐺 ∖ {0}) ⊆ ran 𝐺
59 ssfi 8131 . . . . . 6 ((ran 𝐺 ∈ Fin ∧ (ran 𝐺 ∖ {0}) ⊆ ran 𝐺) → (ran 𝐺 ∖ {0}) ∈ Fin)
6016, 58, 59sylancl 693 . . . . 5 (𝜑 → (ran 𝐺 ∖ {0}) ∈ Fin)
61 i1fima 23364 . . . . . . . 8 (𝐹 ∈ dom ∫1 → (𝐹 “ {(𝑦 / 𝑧)}) ∈ dom vol)
623, 61syl 17 . . . . . . 7 (𝜑 → (𝐹 “ {(𝑦 / 𝑧)}) ∈ dom vol)
63 i1fima 23364 . . . . . . . 8 (𝐺 ∈ dom ∫1 → (𝐺 “ {𝑧}) ∈ dom vol)
646, 63syl 17 . . . . . . 7 (𝜑 → (𝐺 “ {𝑧}) ∈ dom vol)
65 inmbl 23229 . . . . . . 7 (((𝐹 “ {(𝑦 / 𝑧)}) ∈ dom vol ∧ (𝐺 “ {𝑧}) ∈ dom vol) → ((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧})) ∈ dom vol)
6662, 64, 65syl2anc 692 . . . . . 6 (𝜑 → ((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧})) ∈ dom vol)
6766ralrimivw 2962 . . . . 5 (𝜑 → ∀𝑧 ∈ (ran 𝐺 ∖ {0})((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧})) ∈ dom vol)
68 finiunmbl 23231 . . . . 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)
6960, 67, 68syl2anc 692 . . . 4 (𝜑 𝑧 ∈ (ran 𝐺 ∖ {0})((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧})) ∈ dom vol)
7069adantr 481 . . 3 ((𝜑𝑦 ∈ (ran (𝐹𝑓 · 𝐺) ∖ {0})) → 𝑧 ∈ (ran 𝐺 ∖ {0})((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧})) ∈ dom vol)
7157, 70eqeltrd 2698 . 2 ((𝜑𝑦 ∈ (ran (𝐹𝑓 · 𝐺) ∖ {0})) → ((𝐹𝑓 · 𝐺) “ {𝑦}) ∈ dom vol)
72 mblvol 23217 . . . 4 (((𝐹𝑓 · 𝐺) “ {𝑦}) ∈ dom vol → (vol‘((𝐹𝑓 · 𝐺) “ {𝑦})) = (vol*‘((𝐹𝑓 · 𝐺) “ {𝑦})))
7371, 72syl 17 . . 3 ((𝜑𝑦 ∈ (ran (𝐹𝑓 · 𝐺) ∖ {0})) → (vol‘((𝐹𝑓 · 𝐺) “ {𝑦})) = (vol*‘((𝐹𝑓 · 𝐺) “ {𝑦})))
74 mblss 23218 . . . . 5 (((𝐹𝑓 · 𝐺) “ {𝑦}) ∈ dom vol → ((𝐹𝑓 · 𝐺) “ {𝑦}) ⊆ ℝ)
7571, 74syl 17 . . . 4 ((𝜑𝑦 ∈ (ran (𝐹𝑓 · 𝐺) ∖ {0})) → ((𝐹𝑓 · 𝐺) “ {𝑦}) ⊆ ℝ)
7660adantr 481 . . . . 5 ((𝜑𝑦 ∈ (ran (𝐹𝑓 · 𝐺) ∖ {0})) → (ran 𝐺 ∖ {0}) ∈ Fin)
77 inss2 3817 . . . . . . 7 ((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧})) ⊆ (𝐺 “ {𝑧})
7877a1i 11 . . . . . 6 (((𝜑𝑦 ∈ (ran (𝐹𝑓 · 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → ((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧})) ⊆ (𝐺 “ {𝑧}))
7964ad2antrr 761 . . . . . . 7 (((𝜑𝑦 ∈ (ran (𝐹𝑓 · 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (𝐺 “ {𝑧}) ∈ dom vol)
80 mblss 23218 . . . . . . 7 ((𝐺 “ {𝑧}) ∈ dom vol → (𝐺 “ {𝑧}) ⊆ ℝ)
8179, 80syl 17 . . . . . 6 (((𝜑𝑦 ∈ (ran (𝐹𝑓 · 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (𝐺 “ {𝑧}) ⊆ ℝ)
82 mblvol 23217 . . . . . . . 8 ((𝐺 “ {𝑧}) ∈ dom vol → (vol‘(𝐺 “ {𝑧})) = (vol*‘(𝐺 “ {𝑧})))
8379, 82syl 17 . . . . . . 7 (((𝜑𝑦 ∈ (ran (𝐹𝑓 · 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol‘(𝐺 “ {𝑧})) = (vol*‘(𝐺 “ {𝑧})))
846adantr 481 . . . . . . . 8 ((𝜑𝑦 ∈ (ran (𝐹𝑓 · 𝐺) ∖ {0})) → 𝐺 ∈ dom ∫1)
85 i1fima2sn 23366 . . . . . . . 8 ((𝐺 ∈ dom ∫1𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol‘(𝐺 “ {𝑧})) ∈ ℝ)
8684, 85sylan 488 . . . . . . 7 (((𝜑𝑦 ∈ (ran (𝐹𝑓 · 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol‘(𝐺 “ {𝑧})) ∈ ℝ)
8783, 86eqeltrrd 2699 . . . . . 6 (((𝜑𝑦 ∈ (ran (𝐹𝑓 · 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol*‘(𝐺 “ {𝑧})) ∈ ℝ)
88 ovolsscl 23173 . . . . . 6 ((((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧})) ⊆ (𝐺 “ {𝑧}) ∧ (𝐺 “ {𝑧}) ⊆ ℝ ∧ (vol*‘(𝐺 “ {𝑧})) ∈ ℝ) → (vol*‘((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ)
8978, 81, 87, 88syl3anc 1323 . . . . 5 (((𝜑𝑦 ∈ (ran (𝐹𝑓 · 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol*‘((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ)
9076, 89fsumrecl 14405 . . . 4 ((𝜑𝑦 ∈ (ran (𝐹𝑓 · 𝐺) ∖ {0})) → Σ𝑧 ∈ (ran 𝐺 ∖ {0})(vol*‘((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ)
9157fveq2d 6157 . . . . 5 ((𝜑𝑦 ∈ (ran (𝐹𝑓 · 𝐺) ∖ {0})) → (vol*‘((𝐹𝑓 · 𝐺) “ {𝑦})) = (vol*‘ 𝑧 ∈ (ran 𝐺 ∖ {0})((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧}))))
92 mblss 23218 . . . . . . . . . 10 (((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧})) ∈ dom vol → ((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧})) ⊆ ℝ)
9366, 92syl 17 . . . . . . . . 9 (𝜑 → ((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧})) ⊆ ℝ)
9493ad2antrr 761 . . . . . . . 8 (((𝜑𝑦 ∈ (ran (𝐹𝑓 · 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → ((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧})) ⊆ ℝ)
9594, 89jca 554 . . . . . . 7 (((𝜑𝑦 ∈ (ran (𝐹𝑓 · 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧})) ⊆ ℝ ∧ (vol*‘((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ))
9695ralrimiva 2961 . . . . . 6 ((𝜑𝑦 ∈ (ran (𝐹𝑓 · 𝐺) ∖ {0})) → ∀𝑧 ∈ (ran 𝐺 ∖ {0})(((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧})) ⊆ ℝ ∧ (vol*‘((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ))
97 ovolfiniun 23188 . . . . . 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}))))
9876, 96, 97syl2anc 692 . . . . 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}))))
9991, 98eqbrtrd 4640 . . . 4 ((𝜑𝑦 ∈ (ran (𝐹𝑓 · 𝐺) ∖ {0})) → (vol*‘((𝐹𝑓 · 𝐺) “ {𝑦})) ≤ Σ𝑧 ∈ (ran 𝐺 ∖ {0})(vol*‘((𝐹 “ {(𝑦 / 𝑧)}) ∩ (𝐺 “ {𝑧}))))
100 ovollecl 23170 . . . 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)
10175, 90, 99, 100syl3anc 1323 . . 3 ((𝜑𝑦 ∈ (ran (𝐹𝑓 · 𝐺) ∖ {0})) → (vol*‘((𝐹𝑓 · 𝐺) “ {𝑦})) ∈ ℝ)
10273, 101eqeltrd 2698 . 2 ((𝜑𝑦 ∈ (ran (𝐹𝑓 · 𝐺) ∖ {0})) → (vol‘((𝐹𝑓 · 𝐺) “ {𝑦})) ∈ ℝ)
10312, 49, 71, 102i1fd 23367 1 (𝜑 → (𝐹𝑓 · 𝐺) ∈ dom ∫1)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  {cab 2607  wral 2907  wrex 2908  Vcvv 3189  cdif 3556  cin 3558  wss 3559  {csn 4153   ciun 4490   class class class wbr 4618   × cxp 5077  ccnv 5078  dom cdm 5079  ran crn 5080  cima 5082   Fn wfn 5847  wf 5848  ontowfo 5850  cfv 5852  (class class class)co 6610  cmpt2 6612  𝑓 cof 6855  Fincfn 7906  cc 9885  cr 9886  0cc0 9887   · cmul 9892  cle 10026   / cdiv 10635  Σcsu 14357  vol*covol 23150  volcvol 23151  1citg1 23303
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-inf2 8489  ax-cnex 9943  ax-resscn 9944  ax-1cn 9945  ax-icn 9946  ax-addcl 9947  ax-addrcl 9948  ax-mulcl 9949  ax-mulrcl 9950  ax-mulcom 9951  ax-addass 9952  ax-mulass 9953  ax-distr 9954  ax-i2m1 9955  ax-1ne0 9956  ax-1rid 9957  ax-rnegex 9958  ax-rrecex 9959  ax-cnre 9960  ax-pre-lttri 9961  ax-pre-lttrn 9962  ax-pre-ltadd 9963  ax-pre-mulgt0 9964  ax-pre-sup 9965
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-isom 5861  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-of 6857  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-2o 7513  df-oadd 7516  df-er 7694  df-map 7811  df-pm 7812  df-en 7907  df-dom 7908  df-sdom 7909  df-fin 7910  df-sup 8299  df-inf 8300  df-oi 8366  df-card 8716  df-cda 8941  df-pnf 10027  df-mnf 10028  df-xr 10029  df-ltxr 10030  df-le 10031  df-sub 10219  df-neg 10220  df-div 10636  df-nn 10972  df-2 11030  df-3 11031  df-n0 11244  df-z 11329  df-uz 11639  df-q 11740  df-rp 11784  df-xadd 11898  df-ioo 12128  df-ico 12130  df-icc 12131  df-fz 12276  df-fzo 12414  df-fl 12540  df-seq 12749  df-exp 12808  df-hash 13065  df-cj 13780  df-re 13781  df-im 13782  df-sqrt 13916  df-abs 13917  df-clim 14160  df-sum 14358  df-xmet 19667  df-met 19668  df-ovol 23152  df-vol 23153  df-mbf 23307  df-itg1 23308
This theorem is referenced by:  mbfmullem2  23410  ftc1anclem3  33146
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