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Theorem i1fadd 25059
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 11134 . . . 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 25040 . . . 4 (𝐹 ∈ dom ∫1𝐹:ℝ⟶ℝ)
53, 4syl 17 . . 3 (𝜑𝐹:ℝ⟶ℝ)
6 i1fadd.2 . . . 4 (𝜑𝐺 ∈ dom ∫1)
7 i1ff 25040 . . . 4 (𝐺 ∈ dom ∫1𝐺:ℝ⟶ℝ)
86, 7syl 17 . . 3 (𝜑𝐺:ℝ⟶ℝ)
9 reex 11142 . . . 4 ℝ ∈ V
109a1i 11 . . 3 (𝜑 → ℝ ∈ V)
11 inidm 4178 . . 3 (ℝ ∩ ℝ) = ℝ
122, 5, 8, 10, 10, 11off 7635 . 2 (𝜑 → (𝐹f + 𝐺):ℝ⟶ℝ)
13 i1frn 25041 . . . . . 6 (𝐹 ∈ dom ∫1 → ran 𝐹 ∈ Fin)
143, 13syl 17 . . . . 5 (𝜑 → ran 𝐹 ∈ Fin)
15 i1frn 25041 . . . . . 6 (𝐺 ∈ dom ∫1 → ran 𝐺 ∈ Fin)
166, 15syl 17 . . . . 5 (𝜑 → ran 𝐺 ∈ Fin)
17 xpfi 9261 . . . . 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 2736 . . . . . 6 (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 + 𝑣)) = (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 + 𝑣))
20 ovex 7390 . . . . . 6 (𝑢 + 𝑣) ∈ V
2119, 20fnmpoi 8002 . . . . 5 (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 + 𝑣)) Fn (ran 𝐹 × ran 𝐺)
22 dffn4 6762 . . . . 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 9282 . . . 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 2736 . . . . . . . . 9 (𝑥 + 𝑦) = (𝑥 + 𝑦)
27 rspceov 7404 . . . . . . . . 9 ((𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐺 ∧ (𝑥 + 𝑦) = (𝑥 + 𝑦)) → ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺(𝑥 + 𝑦) = (𝑢 + 𝑣))
2826, 27mp3an3 1450 . . . . . . . 8 ((𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐺) → ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺(𝑥 + 𝑦) = (𝑢 + 𝑣))
29 ovex 7390 . . . . . . . . 9 (𝑥 + 𝑦) ∈ V
30 eqeq1 2740 . . . . . . . . . 10 (𝑤 = (𝑥 + 𝑦) → (𝑤 = (𝑢 + 𝑣) ↔ (𝑥 + 𝑦) = (𝑢 + 𝑣)))
31302rexbidv 3213 . . . . . . . . 9 (𝑤 = (𝑥 + 𝑦) → (∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺 𝑤 = (𝑢 + 𝑣) ↔ ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺(𝑥 + 𝑦) = (𝑢 + 𝑣)))
3229, 31elab 3630 . . . . . . . 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 482 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐺)) → (𝑥 + 𝑦) ∈ {𝑤 ∣ ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺 𝑤 = (𝑢 + 𝑣)})
355ffnd 6669 . . . . . . 7 (𝜑𝐹 Fn ℝ)
36 dffn3 6681 . . . . . . 7 (𝐹 Fn ℝ ↔ 𝐹:ℝ⟶ran 𝐹)
3735, 36sylib 217 . . . . . 6 (𝜑𝐹:ℝ⟶ran 𝐹)
388ffnd 6669 . . . . . . 7 (𝜑𝐺 Fn ℝ)
39 dffn3 6681 . . . . . . 7 (𝐺 Fn ℝ ↔ 𝐺:ℝ⟶ran 𝐺)
4038, 39sylib 217 . . . . . 6 (𝜑𝐺:ℝ⟶ran 𝐺)
4134, 37, 40, 10, 10, 11off 7635 . . . . 5 (𝜑 → (𝐹f + 𝐺):ℝ⟶{𝑤 ∣ ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺 𝑤 = (𝑢 + 𝑣)})
4241frnd 6676 . . . 4 (𝜑 → ran (𝐹f + 𝐺) ⊆ {𝑤 ∣ ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺 𝑤 = (𝑢 + 𝑣)})
4319rnmpo 7489 . . . 4 ran (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 + 𝑣)) = {𝑤 ∣ ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺 𝑤 = (𝑢 + 𝑣)}
4442, 43sseqtrrdi 3995 . . 3 (𝜑 → ran (𝐹f + 𝐺) ⊆ ran (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 + 𝑣)))
4525, 44ssfid 9211 . 2 (𝜑 → ran (𝐹f + 𝐺) ∈ Fin)
4612frnd 6676 . . . . . . 7 (𝜑 → ran (𝐹f + 𝐺) ⊆ ℝ)
4746ssdifssd 4102 . . . . . 6 (𝜑 → (ran (𝐹f + 𝐺) ∖ {0}) ⊆ ℝ)
4847sselda 3944 . . . . 5 ((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) → 𝑦 ∈ ℝ)
4948recnd 11183 . . . 4 ((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) → 𝑦 ∈ ℂ)
503, 6i1faddlem 25057 . . . 4 ((𝜑𝑦 ∈ ℂ) → ((𝐹f + 𝐺) “ {𝑦}) = 𝑧 ∈ ran 𝐺((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧})))
5149, 50syldan 591 . . 3 ((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) → ((𝐹f + 𝐺) “ {𝑦}) = 𝑧 ∈ ran 𝐺((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧})))
5216adantr 481 . . . 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 25039 . . . . . . . 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 6676 . . . . . . . . 9 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → ran (𝐹f + 𝐺) ⊆ ℝ)
59 eldifi 4086 . . . . . . . . . 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 3945 . . . . . . . 8 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝑦 ∈ ℝ)
628adantr 481 . . . . . . . . . 10 ((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) → 𝐺:ℝ⟶ℝ)
6362frnd 6676 . . . . . . . . 9 ((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) → ran 𝐺 ⊆ ℝ)
6463sselda 3944 . . . . . . . 8 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝑧 ∈ ℝ)
6561, 64resubcld 11583 . . . . . . 7 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (𝑦𝑧) ∈ ℝ)
66 mbfimasn 24996 . . . . . . 7 ((𝐹 ∈ MblFn ∧ 𝐹:ℝ⟶ℝ ∧ (𝑦𝑧) ∈ ℝ) → (𝐹 “ {(𝑦𝑧)}) ∈ dom vol)
6755, 56, 65, 66syl3anc 1371 . . . . . 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 25039 . . . . . . . 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 24996 . . . . . . 7 ((𝐺 ∈ MblFn ∧ 𝐺:ℝ⟶ℝ ∧ 𝑧 ∈ ℝ) → (𝐺 “ {𝑧}) ∈ dom vol)
7370, 71, 64, 72syl3anc 1371 . . . . . 6 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (𝐺 “ {𝑧}) ∈ dom vol)
74 inmbl 24906 . . . . . 6 (((𝐹 “ {(𝑦𝑧)}) ∈ dom vol ∧ (𝐺 “ {𝑧}) ∈ dom vol) → ((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧})) ∈ dom vol)
7567, 73, 74syl2anc 584 . . . . 5 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → ((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧})) ∈ dom vol)
7675ralrimiva 3143 . . . 4 ((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) → ∀𝑧 ∈ ran 𝐺((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧})) ∈ dom vol)
77 finiunmbl 24908 . . . 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 584 . . 3 ((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) → 𝑧 ∈ ran 𝐺((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧})) ∈ dom vol)
7951, 78eqeltrd 2838 . 2 ((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) → ((𝐹f + 𝐺) “ {𝑦}) ∈ dom vol)
80 mblvol 24894 . . . 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 24895 . . . . 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 4188 . . . . . . . 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 24895 . . . . . . . . 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 24894 . . . . . . . . . 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 7373 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺𝑧 = 0)) → (𝑦𝑧) = (𝑦 − 0))
9249adantr 481 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺𝑧 = 0)) → 𝑦 ∈ ℂ)
9392subid1d 11501 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺𝑧 = 0)) → (𝑦 − 0) = 𝑦)
9491, 93eqtrd 2776 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺𝑧 = 0)) → (𝑦𝑧) = 𝑦)
9594sneqd 4598 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺𝑧 = 0)) → {(𝑦𝑧)} = {𝑦})
9695imaeq2d 6013 . . . . . . . . . . 11 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺𝑧 = 0)) → (𝐹 “ {(𝑦𝑧)}) = (𝐹 “ {𝑦}))
9796fveq2d 6846 . . . . . . . . . 10 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺𝑧 = 0)) → (vol‘(𝐹 “ {(𝑦𝑧)})) = (vol‘(𝐹 “ {𝑦})))
98 i1fima2sn 25044 . . . . . . . . . . . 12 ((𝐹 ∈ dom ∫1𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) → (vol‘(𝐹 “ {𝑦})) ∈ ℝ)
993, 98sylan 580 . . . . . . . . . . 11 ((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) → (vol‘(𝐹 “ {𝑦})) ∈ ℝ)
10099adantr 481 . . . . . . . . . 10 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺𝑧 = 0)) → (vol‘(𝐹 “ {𝑦})) ∈ ℝ)
10197, 100eqeltrd 2838 . . . . . . . . 9 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺𝑧 = 0)) → (vol‘(𝐹 “ {(𝑦𝑧)})) ∈ ℝ)
10289, 101eqeltrrd 2839 . . . . . . . 8 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺𝑧 = 0)) → (vol*‘(𝐹 “ {(𝑦𝑧)})) ∈ ℝ)
103 ovolsscl 24850 . . . . . . . 8 ((((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧})) ⊆ (𝐹 “ {(𝑦𝑧)}) ∧ (𝐹 “ {(𝑦𝑧)}) ⊆ ℝ ∧ (vol*‘(𝐹 “ {(𝑦𝑧)})) ∈ ℝ) → (vol*‘((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ)
10484, 87, 102, 103mp3an2i 1466 . . . . . . 7 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺𝑧 = 0)) → (vol*‘((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ)
105104expr 457 . . . . . 6 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (𝑧 = 0 → (vol*‘((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ))
106 eldifsn 4747 . . . . . . . 8 (𝑧 ∈ (ran 𝐺 ∖ {0}) ↔ (𝑧 ∈ ran 𝐺𝑧 ≠ 0))
107 inss2 4189 . . . . . . . . 9 ((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧})) ⊆ (𝐺 “ {𝑧})
108 eldifi 4086 . . . . . . . . . 10 (𝑧 ∈ (ran 𝐺 ∖ {0}) → 𝑧 ∈ ran 𝐺)
109 mblss 24895 . . . . . . . . . . 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 593 . . . . . . . . 9 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (𝐺 “ {𝑧}) ⊆ ℝ)
112 i1fima 25042 . . . . . . . . . . . . 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 24894 . . . . . . . . . . 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 481 . . . . . . . . . . 11 ((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) → 𝐺 ∈ dom ∫1)
118 i1fima2sn 25044 . . . . . . . . . . 11 ((𝐺 ∈ dom ∫1𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol‘(𝐺 “ {𝑧})) ∈ ℝ)
119117, 118sylan 580 . . . . . . . . . 10 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol‘(𝐺 “ {𝑧})) ∈ ℝ)
120116, 119eqeltrrd 2839 . . . . . . . . 9 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol*‘(𝐺 “ {𝑧})) ∈ ℝ)
121 ovolsscl 24850 . . . . . . . . 9 ((((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧})) ⊆ (𝐺 “ {𝑧}) ∧ (𝐺 “ {𝑧}) ⊆ ℝ ∧ (vol*‘(𝐺 “ {𝑧})) ∈ ℝ) → (vol*‘((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ)
122107, 111, 120, 121mp3an2i 1466 . . . . . . . 8 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol*‘((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ)
123106, 122sylan2br 595 . . . . . . 7 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺𝑧 ≠ 0)) → (vol*‘((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ)
124123expr 457 . . . . . 6 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (𝑧 ≠ 0 → (vol*‘((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ))
125105, 124pm2.61dne 3031 . . . . 5 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (vol*‘((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ)
12652, 125fsumrecl 15619 . . . 4 ((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) → Σ𝑧 ∈ ran 𝐺(vol*‘((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ)
12751fveq2d 6846 . . . . 5 ((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) → (vol*‘((𝐹f + 𝐺) “ {𝑦})) = (vol*‘ 𝑧 ∈ ran 𝐺((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))))
128107, 110sstrid 3955 . . . . . . . 8 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → ((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧})) ⊆ ℝ)
129128, 125jca 512 . . . . . . 7 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧})) ⊆ ℝ ∧ (vol*‘((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ))
130129ralrimiva 3143 . . . . . 6 ((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) → ∀𝑧 ∈ ran 𝐺(((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧})) ⊆ ℝ ∧ (vol*‘((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ))
131 ovolfiniun 24865 . . . . . 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 584 . . . . 5 ((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) → (vol*‘ 𝑧 ∈ ran 𝐺((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))) ≤ Σ𝑧 ∈ ran 𝐺(vol*‘((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))))
133127, 132eqbrtrd 5127 . . . 4 ((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) → (vol*‘((𝐹f + 𝐺) “ {𝑦})) ≤ Σ𝑧 ∈ ran 𝐺(vol*‘((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))))
134 ovollecl 24847 . . . 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 1371 . . 3 ((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) → (vol*‘((𝐹f + 𝐺) “ {𝑦})) ∈ ℝ)
13681, 135eqeltrd 2838 . 2 ((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) → (vol‘((𝐹f + 𝐺) “ {𝑦})) ∈ ℝ)
13712, 45, 79, 136i1fd 25045 1 (𝜑 → (𝐹f + 𝐺) ∈ dom ∫1)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  {cab 2713  wne 2943  wral 3064  wrex 3073  Vcvv 3445  cdif 3907  cin 3909  wss 3910  {csn 4586   ciun 4954   class class class wbr 5105   × cxp 5631  ccnv 5632  dom cdm 5633  ran crn 5634  cima 5636   Fn wfn 6491  wf 6492  ontowfo 6494  cfv 6496  (class class class)co 7357  cmpo 7359  f cof 7615  Fincfn 8883  cc 11049  cr 11050  0cc0 11051   + caddc 11054  cle 11190  cmin 11385  Σcsu 15570  vol*covol 24826  volcvol 24827  MblFncmbf 24978  1citg1 24979
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-er 8648  df-map 8767  df-pm 8768  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9378  df-inf 9379  df-oi 9446  df-dju 9837  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-z 12500  df-uz 12764  df-q 12874  df-rp 12916  df-xadd 13034  df-ioo 13268  df-ico 13270  df-icc 13271  df-fz 13425  df-fzo 13568  df-fl 13697  df-seq 13907  df-exp 13968  df-hash 14231  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-clim 15370  df-sum 15571  df-xmet 20789  df-met 20790  df-ovol 24828  df-vol 24829  df-mbf 24983  df-itg1 24984
This theorem is referenced by:  itg1addlem4  25063  itg1addlem4OLD  25064  i1fsub  25073  itg2splitlem  25113  itg2split  25114  itg2addlem  25123  itg2addnc  36132  ftc1anclem3  36153  ftc1anclem5  36155  ftc1anclem8  36158
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