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Theorem i1fadd 25668
Description: The sum of two simple functions is a simple function. (Contributed by Mario Carneiro, 18-Jun-2014.)
Hypotheses
Ref Expression
i1fadd.1 (𝜑𝐹 ∈ dom ∫1)
i1fadd.2 (𝜑𝐺 ∈ dom ∫1)
Assertion
Ref Expression
i1fadd (𝜑 → (𝐹f + 𝐺) ∈ dom ∫1)

Proof of Theorem i1fadd
Dummy variables 𝑦 𝑧 𝑤 𝑣 𝑥 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 readdcl 11223 . . . 4 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 + 𝑦) ∈ ℝ)
21adantl 480 . . 3 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (𝑥 + 𝑦) ∈ ℝ)
3 i1fadd.1 . . . 4 (𝜑𝐹 ∈ dom ∫1)
4 i1ff 25649 . . . 4 (𝐹 ∈ dom ∫1𝐹:ℝ⟶ℝ)
53, 4syl 17 . . 3 (𝜑𝐹:ℝ⟶ℝ)
6 i1fadd.2 . . . 4 (𝜑𝐺 ∈ dom ∫1)
7 i1ff 25649 . . . 4 (𝐺 ∈ dom ∫1𝐺:ℝ⟶ℝ)
86, 7syl 17 . . 3 (𝜑𝐺:ℝ⟶ℝ)
9 reex 11231 . . . 4 ℝ ∈ V
109a1i 11 . . 3 (𝜑 → ℝ ∈ V)
11 inidm 4217 . . 3 (ℝ ∩ ℝ) = ℝ
122, 5, 8, 10, 10, 11off 7703 . 2 (𝜑 → (𝐹f + 𝐺):ℝ⟶ℝ)
13 i1frn 25650 . . . . . 6 (𝐹 ∈ dom ∫1 → ran 𝐹 ∈ Fin)
143, 13syl 17 . . . . 5 (𝜑 → ran 𝐹 ∈ Fin)
15 i1frn 25650 . . . . . 6 (𝐺 ∈ dom ∫1 → ran 𝐺 ∈ Fin)
166, 15syl 17 . . . . 5 (𝜑 → ran 𝐺 ∈ Fin)
17 xpfi 9343 . . . . 5 ((ran 𝐹 ∈ Fin ∧ ran 𝐺 ∈ Fin) → (ran 𝐹 × ran 𝐺) ∈ Fin)
1814, 16, 17syl2anc 582 . . . 4 (𝜑 → (ran 𝐹 × ran 𝐺) ∈ Fin)
19 eqid 2725 . . . . . 6 (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 + 𝑣)) = (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 + 𝑣))
20 ovex 7452 . . . . . 6 (𝑢 + 𝑣) ∈ V
2119, 20fnmpoi 8075 . . . . 5 (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 + 𝑣)) Fn (ran 𝐹 × ran 𝐺)
22 dffn4 6816 . . . . 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 9364 . . . 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 584 . . 3 (𝜑 → ran (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 + 𝑣)) ∈ Fin)
26 eqid 2725 . . . . . . . . 9 (𝑥 + 𝑦) = (𝑥 + 𝑦)
27 rspceov 7467 . . . . . . . . 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 7452 . . . . . . . . 9 (𝑥 + 𝑦) ∈ V
30 eqeq1 2729 . . . . . . . . . 10 (𝑤 = (𝑥 + 𝑦) → (𝑤 = (𝑢 + 𝑣) ↔ (𝑥 + 𝑦) = (𝑢 + 𝑣)))
31302rexbidv 3209 . . . . . . . . 9 (𝑤 = (𝑥 + 𝑦) → (∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺 𝑤 = (𝑢 + 𝑣) ↔ ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺(𝑥 + 𝑦) = (𝑢 + 𝑣)))
3229, 31elab 3664 . . . . . . . 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 480 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐺)) → (𝑥 + 𝑦) ∈ {𝑤 ∣ ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺 𝑤 = (𝑢 + 𝑣)})
355ffnd 6724 . . . . . . 7 (𝜑𝐹 Fn ℝ)
36 dffn3 6735 . . . . . . 7 (𝐹 Fn ℝ ↔ 𝐹:ℝ⟶ran 𝐹)
3735, 36sylib 217 . . . . . 6 (𝜑𝐹:ℝ⟶ran 𝐹)
388ffnd 6724 . . . . . . 7 (𝜑𝐺 Fn ℝ)
39 dffn3 6735 . . . . . . 7 (𝐺 Fn ℝ ↔ 𝐺:ℝ⟶ran 𝐺)
4038, 39sylib 217 . . . . . 6 (𝜑𝐺:ℝ⟶ran 𝐺)
4134, 37, 40, 10, 10, 11off 7703 . . . . 5 (𝜑 → (𝐹f + 𝐺):ℝ⟶{𝑤 ∣ ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺 𝑤 = (𝑢 + 𝑣)})
4241frnd 6731 . . . 4 (𝜑 → ran (𝐹f + 𝐺) ⊆ {𝑤 ∣ ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺 𝑤 = (𝑢 + 𝑣)})
4319rnmpo 7554 . . . 4 ran (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 + 𝑣)) = {𝑤 ∣ ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺 𝑤 = (𝑢 + 𝑣)}
4442, 43sseqtrrdi 4028 . . 3 (𝜑 → ran (𝐹f + 𝐺) ⊆ ran (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 + 𝑣)))
4525, 44ssfid 9292 . 2 (𝜑 → ran (𝐹f + 𝐺) ∈ Fin)
4612frnd 6731 . . . . . . 7 (𝜑 → ran (𝐹f + 𝐺) ⊆ ℝ)
4746ssdifssd 4139 . . . . . 6 (𝜑 → (ran (𝐹f + 𝐺) ∖ {0}) ⊆ ℝ)
4847sselda 3976 . . . . 5 ((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) → 𝑦 ∈ ℝ)
4948recnd 11274 . . . 4 ((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) → 𝑦 ∈ ℂ)
503, 6i1faddlem 25666 . . . 4 ((𝜑𝑦 ∈ ℂ) → ((𝐹f + 𝐺) “ {𝑦}) = 𝑧 ∈ ran 𝐺((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧})))
5149, 50syldan 589 . . 3 ((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) → ((𝐹f + 𝐺) “ {𝑦}) = 𝑧 ∈ ran 𝐺((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧})))
5216adantr 479 . . . 4 ((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) → ran 𝐺 ∈ Fin)
533ad2antrr 724 . . . . . . . 8 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝐹 ∈ dom ∫1)
54 i1fmbf 25648 . . . . . . . 8 (𝐹 ∈ dom ∫1𝐹 ∈ MblFn)
5553, 54syl 17 . . . . . . 7 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝐹 ∈ MblFn)
565ad2antrr 724 . . . . . . 7 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝐹:ℝ⟶ℝ)
5712ad2antrr 724 . . . . . . . . . 10 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (𝐹f + 𝐺):ℝ⟶ℝ)
5857frnd 6731 . . . . . . . . 9 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → ran (𝐹f + 𝐺) ⊆ ℝ)
59 eldifi 4123 . . . . . . . . . 10 (𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0}) → 𝑦 ∈ ran (𝐹f + 𝐺))
6059ad2antlr 725 . . . . . . . . 9 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝑦 ∈ ran (𝐹f + 𝐺))
6158, 60sseldd 3977 . . . . . . . 8 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝑦 ∈ ℝ)
628adantr 479 . . . . . . . . . 10 ((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) → 𝐺:ℝ⟶ℝ)
6362frnd 6731 . . . . . . . . 9 ((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) → ran 𝐺 ⊆ ℝ)
6463sselda 3976 . . . . . . . 8 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝑧 ∈ ℝ)
6561, 64resubcld 11674 . . . . . . 7 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (𝑦𝑧) ∈ ℝ)
66 mbfimasn 25605 . . . . . . 7 ((𝐹 ∈ MblFn ∧ 𝐹:ℝ⟶ℝ ∧ (𝑦𝑧) ∈ ℝ) → (𝐹 “ {(𝑦𝑧)}) ∈ dom vol)
6755, 56, 65, 66syl3anc 1368 . . . . . 6 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (𝐹 “ {(𝑦𝑧)}) ∈ dom vol)
686ad2antrr 724 . . . . . . . 8 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝐺 ∈ dom ∫1)
69 i1fmbf 25648 . . . . . . . 8 (𝐺 ∈ dom ∫1𝐺 ∈ MblFn)
7068, 69syl 17 . . . . . . 7 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝐺 ∈ MblFn)
718ad2antrr 724 . . . . . . 7 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝐺:ℝ⟶ℝ)
72 mbfimasn 25605 . . . . . . 7 ((𝐺 ∈ MblFn ∧ 𝐺:ℝ⟶ℝ ∧ 𝑧 ∈ ℝ) → (𝐺 “ {𝑧}) ∈ dom vol)
7370, 71, 64, 72syl3anc 1368 . . . . . 6 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (𝐺 “ {𝑧}) ∈ dom vol)
74 inmbl 25515 . . . . . 6 (((𝐹 “ {(𝑦𝑧)}) ∈ dom vol ∧ (𝐺 “ {𝑧}) ∈ dom vol) → ((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧})) ∈ dom vol)
7567, 73, 74syl2anc 582 . . . . 5 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → ((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧})) ∈ dom vol)
7675ralrimiva 3135 . . . 4 ((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) → ∀𝑧 ∈ ran 𝐺((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧})) ∈ dom vol)
77 finiunmbl 25517 . . . 4 ((ran 𝐺 ∈ Fin ∧ ∀𝑧 ∈ ran 𝐺((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧})) ∈ dom vol) → 𝑧 ∈ ran 𝐺((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧})) ∈ dom vol)
7852, 76, 77syl2anc 582 . . 3 ((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) → 𝑧 ∈ ran 𝐺((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧})) ∈ dom vol)
7951, 78eqeltrd 2825 . 2 ((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) → ((𝐹f + 𝐺) “ {𝑦}) ∈ dom vol)
80 mblvol 25503 . . . 4 (((𝐹f + 𝐺) “ {𝑦}) ∈ dom vol → (vol‘((𝐹f + 𝐺) “ {𝑦})) = (vol*‘((𝐹f + 𝐺) “ {𝑦})))
8179, 80syl 17 . . 3 ((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) → (vol‘((𝐹f + 𝐺) “ {𝑦})) = (vol*‘((𝐹f + 𝐺) “ {𝑦})))
82 mblss 25504 . . . . 5 (((𝐹f + 𝐺) “ {𝑦}) ∈ dom vol → ((𝐹f + 𝐺) “ {𝑦}) ⊆ ℝ)
8379, 82syl 17 . . . 4 ((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) → ((𝐹f + 𝐺) “ {𝑦}) ⊆ ℝ)
84 inss1 4227 . . . . . . . 8 ((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧})) ⊆ (𝐹 “ {(𝑦𝑧)})
8567adantrr 715 . . . . . . . . 9 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺𝑧 = 0)) → (𝐹 “ {(𝑦𝑧)}) ∈ dom vol)
86 mblss 25504 . . . . . . . . 9 ((𝐹 “ {(𝑦𝑧)}) ∈ dom vol → (𝐹 “ {(𝑦𝑧)}) ⊆ ℝ)
8785, 86syl 17 . . . . . . . 8 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺𝑧 = 0)) → (𝐹 “ {(𝑦𝑧)}) ⊆ ℝ)
88 mblvol 25503 . . . . . . . . . 10 ((𝐹 “ {(𝑦𝑧)}) ∈ dom vol → (vol‘(𝐹 “ {(𝑦𝑧)})) = (vol*‘(𝐹 “ {(𝑦𝑧)})))
8985, 88syl 17 . . . . . . . . 9 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺𝑧 = 0)) → (vol‘(𝐹 “ {(𝑦𝑧)})) = (vol*‘(𝐹 “ {(𝑦𝑧)})))
90 simprr 771 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺𝑧 = 0)) → 𝑧 = 0)
9190oveq2d 7435 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺𝑧 = 0)) → (𝑦𝑧) = (𝑦 − 0))
9249adantr 479 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺𝑧 = 0)) → 𝑦 ∈ ℂ)
9392subid1d 11592 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺𝑧 = 0)) → (𝑦 − 0) = 𝑦)
9491, 93eqtrd 2765 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺𝑧 = 0)) → (𝑦𝑧) = 𝑦)
9594sneqd 4642 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺𝑧 = 0)) → {(𝑦𝑧)} = {𝑦})
9695imaeq2d 6064 . . . . . . . . . . 11 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺𝑧 = 0)) → (𝐹 “ {(𝑦𝑧)}) = (𝐹 “ {𝑦}))
9796fveq2d 6900 . . . . . . . . . 10 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺𝑧 = 0)) → (vol‘(𝐹 “ {(𝑦𝑧)})) = (vol‘(𝐹 “ {𝑦})))
98 i1fima2sn 25653 . . . . . . . . . . . 12 ((𝐹 ∈ dom ∫1𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) → (vol‘(𝐹 “ {𝑦})) ∈ ℝ)
993, 98sylan 578 . . . . . . . . . . 11 ((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) → (vol‘(𝐹 “ {𝑦})) ∈ ℝ)
10099adantr 479 . . . . . . . . . 10 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺𝑧 = 0)) → (vol‘(𝐹 “ {𝑦})) ∈ ℝ)
10197, 100eqeltrd 2825 . . . . . . . . 9 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺𝑧 = 0)) → (vol‘(𝐹 “ {(𝑦𝑧)})) ∈ ℝ)
10289, 101eqeltrrd 2826 . . . . . . . 8 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺𝑧 = 0)) → (vol*‘(𝐹 “ {(𝑦𝑧)})) ∈ ℝ)
103 ovolsscl 25459 . . . . . . . 8 ((((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧})) ⊆ (𝐹 “ {(𝑦𝑧)}) ∧ (𝐹 “ {(𝑦𝑧)}) ⊆ ℝ ∧ (vol*‘(𝐹 “ {(𝑦𝑧)})) ∈ ℝ) → (vol*‘((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ)
10484, 87, 102, 103mp3an2i 1462 . . . . . . 7 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺𝑧 = 0)) → (vol*‘((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ)
105104expr 455 . . . . . 6 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (𝑧 = 0 → (vol*‘((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ))
106 eldifsn 4792 . . . . . . . 8 (𝑧 ∈ (ran 𝐺 ∖ {0}) ↔ (𝑧 ∈ ran 𝐺𝑧 ≠ 0))
107 inss2 4228 . . . . . . . . 9 ((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧})) ⊆ (𝐺 “ {𝑧})
108 eldifi 4123 . . . . . . . . . 10 (𝑧 ∈ (ran 𝐺 ∖ {0}) → 𝑧 ∈ ran 𝐺)
109 mblss 25504 . . . . . . . . . . 11 ((𝐺 “ {𝑧}) ∈ dom vol → (𝐺 “ {𝑧}) ⊆ ℝ)
11073, 109syl 17 . . . . . . . . . 10 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (𝐺 “ {𝑧}) ⊆ ℝ)
111108, 110sylan2 591 . . . . . . . . 9 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (𝐺 “ {𝑧}) ⊆ ℝ)
112 i1fima 25651 . . . . . . . . . . . . 13 (𝐺 ∈ dom ∫1 → (𝐺 “ {𝑧}) ∈ dom vol)
1136, 112syl 17 . . . . . . . . . . . 12 (𝜑 → (𝐺 “ {𝑧}) ∈ dom vol)
114113ad2antrr 724 . . . . . . . . . . 11 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (𝐺 “ {𝑧}) ∈ dom vol)
115 mblvol 25503 . . . . . . . . . . 11 ((𝐺 “ {𝑧}) ∈ dom vol → (vol‘(𝐺 “ {𝑧})) = (vol*‘(𝐺 “ {𝑧})))
116114, 115syl 17 . . . . . . . . . 10 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol‘(𝐺 “ {𝑧})) = (vol*‘(𝐺 “ {𝑧})))
1176adantr 479 . . . . . . . . . . 11 ((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) → 𝐺 ∈ dom ∫1)
118 i1fima2sn 25653 . . . . . . . . . . 11 ((𝐺 ∈ dom ∫1𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol‘(𝐺 “ {𝑧})) ∈ ℝ)
119117, 118sylan 578 . . . . . . . . . 10 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol‘(𝐺 “ {𝑧})) ∈ ℝ)
120116, 119eqeltrrd 2826 . . . . . . . . 9 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol*‘(𝐺 “ {𝑧})) ∈ ℝ)
121 ovolsscl 25459 . . . . . . . . 9 ((((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧})) ⊆ (𝐺 “ {𝑧}) ∧ (𝐺 “ {𝑧}) ⊆ ℝ ∧ (vol*‘(𝐺 “ {𝑧})) ∈ ℝ) → (vol*‘((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ)
122107, 111, 120, 121mp3an2i 1462 . . . . . . . 8 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol*‘((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ)
123106, 122sylan2br 593 . . . . . . 7 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺𝑧 ≠ 0)) → (vol*‘((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ)
124123expr 455 . . . . . 6 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (𝑧 ≠ 0 → (vol*‘((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ))
125105, 124pm2.61dne 3017 . . . . 5 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (vol*‘((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ)
12652, 125fsumrecl 15716 . . . 4 ((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) → Σ𝑧 ∈ ran 𝐺(vol*‘((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ)
12751fveq2d 6900 . . . . 5 ((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) → (vol*‘((𝐹f + 𝐺) “ {𝑦})) = (vol*‘ 𝑧 ∈ ran 𝐺((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))))
128107, 110sstrid 3988 . . . . . . . 8 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → ((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧})) ⊆ ℝ)
129128, 125jca 510 . . . . . . 7 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧})) ⊆ ℝ ∧ (vol*‘((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ))
130129ralrimiva 3135 . . . . . 6 ((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) → ∀𝑧 ∈ ran 𝐺(((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧})) ⊆ ℝ ∧ (vol*‘((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ))
131 ovolfiniun 25474 . . . . . 6 ((ran 𝐺 ∈ Fin ∧ ∀𝑧 ∈ ran 𝐺(((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧})) ⊆ ℝ ∧ (vol*‘((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ)) → (vol*‘ 𝑧 ∈ ran 𝐺((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))) ≤ Σ𝑧 ∈ ran 𝐺(vol*‘((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))))
13252, 130, 131syl2anc 582 . . . . 5 ((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) → (vol*‘ 𝑧 ∈ ran 𝐺((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))) ≤ Σ𝑧 ∈ ran 𝐺(vol*‘((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))))
133127, 132eqbrtrd 5171 . . . 4 ((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) → (vol*‘((𝐹f + 𝐺) “ {𝑦})) ≤ Σ𝑧 ∈ ran 𝐺(vol*‘((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))))
134 ovollecl 25456 . . . 4 ((((𝐹f + 𝐺) “ {𝑦}) ⊆ ℝ ∧ Σ𝑧 ∈ ran 𝐺(vol*‘((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ ∧ (vol*‘((𝐹f + 𝐺) “ {𝑦})) ≤ Σ𝑧 ∈ ran 𝐺(vol*‘((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧})))) → (vol*‘((𝐹f + 𝐺) “ {𝑦})) ∈ ℝ)
13583, 126, 133, 134syl3anc 1368 . . 3 ((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) → (vol*‘((𝐹f + 𝐺) “ {𝑦})) ∈ ℝ)
13681, 135eqeltrd 2825 . 2 ((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) → (vol‘((𝐹f + 𝐺) “ {𝑦})) ∈ ℝ)
13712, 45, 79, 136i1fd 25654 1 (𝜑 → (𝐹f + 𝐺) ∈ dom ∫1)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  {cab 2702  wne 2929  wral 3050  wrex 3059  Vcvv 3461  cdif 3941  cin 3943  wss 3944  {csn 4630   ciun 4997   class class class wbr 5149   × cxp 5676  ccnv 5677  dom cdm 5678  ran crn 5679  cima 5681   Fn wfn 6544  wf 6545  ontowfo 6547  cfv 6549  (class class class)co 7419  cmpo 7421  f cof 7683  Fincfn 8964  cc 11138  cr 11139  0cc0 11140   + caddc 11143  cle 11281  cmin 11476  Σcsu 15668  vol*covol 25435  volcvol 25436  MblFncmbf 25587  1citg1 25588
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 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-inf2 9666  ax-cnex 11196  ax-resscn 11197  ax-1cn 11198  ax-icn 11199  ax-addcl 11200  ax-addrcl 11201  ax-mulcl 11202  ax-mulrcl 11203  ax-mulcom 11204  ax-addass 11205  ax-mulass 11206  ax-distr 11207  ax-i2m1 11208  ax-1ne0 11209  ax-1rid 11210  ax-rnegex 11211  ax-rrecex 11212  ax-cnre 11213  ax-pre-lttri 11214  ax-pre-lttrn 11215  ax-pre-ltadd 11216  ax-pre-mulgt0 11217  ax-pre-sup 11218
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  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 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  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 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-isom 6558  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-of 7685  df-om 7872  df-1st 7994  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-2o 8488  df-er 8725  df-map 8847  df-pm 8848  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-sup 9467  df-inf 9468  df-oi 9535  df-dju 9926  df-card 9964  df-pnf 11282  df-mnf 11283  df-xr 11284  df-ltxr 11285  df-le 11286  df-sub 11478  df-neg 11479  df-div 11904  df-nn 12246  df-2 12308  df-3 12309  df-n0 12506  df-z 12592  df-uz 12856  df-q 12966  df-rp 13010  df-xadd 13128  df-ioo 13363  df-ico 13365  df-icc 13366  df-fz 13520  df-fzo 13663  df-fl 13793  df-seq 14003  df-exp 14063  df-hash 14326  df-cj 15082  df-re 15083  df-im 15084  df-sqrt 15218  df-abs 15219  df-clim 15468  df-sum 15669  df-xmet 21289  df-met 21290  df-ovol 25437  df-vol 25438  df-mbf 25592  df-itg1 25593
This theorem is referenced by:  itg1addlem4  25672  itg1addlem4OLD  25673  i1fsub  25682  itg2splitlem  25722  itg2split  25723  itg2addlem  25732  itg2addnc  37278  ftc1anclem3  37299  ftc1anclem5  37301  ftc1anclem8  37304
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