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Theorem i1fadd 25731
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 11239 . . . 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 25712 . . . 4 (𝐹 ∈ dom ∫1𝐹:ℝ⟶ℝ)
53, 4syl 17 . . 3 (𝜑𝐹:ℝ⟶ℝ)
6 i1fadd.2 . . . 4 (𝜑𝐺 ∈ dom ∫1)
7 i1ff 25712 . . . 4 (𝐺 ∈ dom ∫1𝐺:ℝ⟶ℝ)
86, 7syl 17 . . 3 (𝜑𝐺:ℝ⟶ℝ)
9 reex 11247 . . . 4 ℝ ∈ V
109a1i 11 . . 3 (𝜑 → ℝ ∈ V)
11 inidm 4226 . . 3 (ℝ ∩ ℝ) = ℝ
122, 5, 8, 10, 10, 11off 7716 . 2 (𝜑 → (𝐹f + 𝐺):ℝ⟶ℝ)
13 i1frn 25713 . . . . . 6 (𝐹 ∈ dom ∫1 → ran 𝐹 ∈ Fin)
143, 13syl 17 . . . . 5 (𝜑 → ran 𝐹 ∈ Fin)
15 i1frn 25713 . . . . . 6 (𝐺 ∈ dom ∫1 → ran 𝐺 ∈ Fin)
166, 15syl 17 . . . . 5 (𝜑 → ran 𝐺 ∈ Fin)
17 xpfi 9359 . . . . 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 7465 . . . . . 6 (𝑢 + 𝑣) ∈ V
2119, 20fnmpoi 8096 . . . . 5 (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 + 𝑣)) Fn (ran 𝐹 × ran 𝐺)
22 dffn4 6825 . . . . 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 9352 . . . 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 7481 . . . . . . . . 9 ((𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐺 ∧ (𝑥 + 𝑦) = (𝑥 + 𝑦)) → ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺(𝑥 + 𝑦) = (𝑢 + 𝑣))
2826, 27mp3an3 1451 . . . . . . . 8 ((𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐺) → ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺(𝑥 + 𝑦) = (𝑢 + 𝑣))
29 ovex 7465 . . . . . . . . 9 (𝑥 + 𝑦) ∈ V
30 eqeq1 2740 . . . . . . . . . 10 (𝑤 = (𝑥 + 𝑦) → (𝑤 = (𝑢 + 𝑣) ↔ (𝑥 + 𝑦) = (𝑢 + 𝑣)))
31302rexbidv 3221 . . . . . . . . 9 (𝑤 = (𝑥 + 𝑦) → (∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺 𝑤 = (𝑢 + 𝑣) ↔ ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺(𝑥 + 𝑦) = (𝑢 + 𝑣)))
3229, 31elab 3678 . . . . . . . 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 6736 . . . . . . 7 (𝜑𝐹 Fn ℝ)
36 dffn3 6747 . . . . . . 7 (𝐹 Fn ℝ ↔ 𝐹:ℝ⟶ran 𝐹)
3735, 36sylib 218 . . . . . 6 (𝜑𝐹:ℝ⟶ran 𝐹)
388ffnd 6736 . . . . . . 7 (𝜑𝐺 Fn ℝ)
39 dffn3 6747 . . . . . . 7 (𝐺 Fn ℝ ↔ 𝐺:ℝ⟶ran 𝐺)
4038, 39sylib 218 . . . . . 6 (𝜑𝐺:ℝ⟶ran 𝐺)
4134, 37, 40, 10, 10, 11off 7716 . . . . 5 (𝜑 → (𝐹f + 𝐺):ℝ⟶{𝑤 ∣ ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺 𝑤 = (𝑢 + 𝑣)})
4241frnd 6743 . . . 4 (𝜑 → ran (𝐹f + 𝐺) ⊆ {𝑤 ∣ ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺 𝑤 = (𝑢 + 𝑣)})
4319rnmpo 7567 . . . 4 ran (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 + 𝑣)) = {𝑤 ∣ ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺 𝑤 = (𝑢 + 𝑣)}
4442, 43sseqtrrdi 4024 . . 3 (𝜑 → ran (𝐹f + 𝐺) ⊆ ran (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 + 𝑣)))
4525, 44ssfid 9302 . 2 (𝜑 → ran (𝐹f + 𝐺) ∈ Fin)
4612frnd 6743 . . . . . . 7 (𝜑 → ran (𝐹f + 𝐺) ⊆ ℝ)
4746ssdifssd 4146 . . . . . 6 (𝜑 → (ran (𝐹f + 𝐺) ∖ {0}) ⊆ ℝ)
4847sselda 3982 . . . . 5 ((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) → 𝑦 ∈ ℝ)
4948recnd 11290 . . . 4 ((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) → 𝑦 ∈ ℂ)
503, 6i1faddlem 25729 . . . 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 480 . . . 4 ((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) → ran 𝐺 ∈ Fin)
533ad2antrr 726 . . . . . . . 8 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝐹 ∈ dom ∫1)
54 i1fmbf 25711 . . . . . . . 8 (𝐹 ∈ dom ∫1𝐹 ∈ MblFn)
5553, 54syl 17 . . . . . . 7 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝐹 ∈ MblFn)
565ad2antrr 726 . . . . . . 7 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝐹:ℝ⟶ℝ)
5712ad2antrr 726 . . . . . . . . . 10 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (𝐹f + 𝐺):ℝ⟶ℝ)
5857frnd 6743 . . . . . . . . 9 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → ran (𝐹f + 𝐺) ⊆ ℝ)
59 eldifi 4130 . . . . . . . . . 10 (𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0}) → 𝑦 ∈ ran (𝐹f + 𝐺))
6059ad2antlr 727 . . . . . . . . 9 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝑦 ∈ ran (𝐹f + 𝐺))
6158, 60sseldd 3983 . . . . . . . 8 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝑦 ∈ ℝ)
628adantr 480 . . . . . . . . . 10 ((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) → 𝐺:ℝ⟶ℝ)
6362frnd 6743 . . . . . . . . 9 ((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) → ran 𝐺 ⊆ ℝ)
6463sselda 3982 . . . . . . . 8 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝑧 ∈ ℝ)
6561, 64resubcld 11692 . . . . . . 7 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (𝑦𝑧) ∈ ℝ)
66 mbfimasn 25668 . . . . . . 7 ((𝐹 ∈ MblFn ∧ 𝐹:ℝ⟶ℝ ∧ (𝑦𝑧) ∈ ℝ) → (𝐹 “ {(𝑦𝑧)}) ∈ dom vol)
6755, 56, 65, 66syl3anc 1372 . . . . . 6 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (𝐹 “ {(𝑦𝑧)}) ∈ dom vol)
686ad2antrr 726 . . . . . . . 8 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝐺 ∈ dom ∫1)
69 i1fmbf 25711 . . . . . . . 8 (𝐺 ∈ dom ∫1𝐺 ∈ MblFn)
7068, 69syl 17 . . . . . . 7 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝐺 ∈ MblFn)
718ad2antrr 726 . . . . . . 7 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝐺:ℝ⟶ℝ)
72 mbfimasn 25668 . . . . . . 7 ((𝐺 ∈ MblFn ∧ 𝐺:ℝ⟶ℝ ∧ 𝑧 ∈ ℝ) → (𝐺 “ {𝑧}) ∈ dom vol)
7370, 71, 64, 72syl3anc 1372 . . . . . 6 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (𝐺 “ {𝑧}) ∈ dom vol)
74 inmbl 25578 . . . . . 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 3145 . . . 4 ((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) → ∀𝑧 ∈ ran 𝐺((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧})) ∈ dom vol)
77 finiunmbl 25580 . . . 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 2840 . 2 ((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) → ((𝐹f + 𝐺) “ {𝑦}) ∈ dom vol)
80 mblvol 25566 . . . 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 25567 . . . . 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 4236 . . . . . . . 8 ((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧})) ⊆ (𝐹 “ {(𝑦𝑧)})
8567adantrr 717 . . . . . . . . 9 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺𝑧 = 0)) → (𝐹 “ {(𝑦𝑧)}) ∈ dom vol)
86 mblss 25567 . . . . . . . . 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 25566 . . . . . . . . . 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 772 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺𝑧 = 0)) → 𝑧 = 0)
9190oveq2d 7448 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺𝑧 = 0)) → (𝑦𝑧) = (𝑦 − 0))
9249adantr 480 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺𝑧 = 0)) → 𝑦 ∈ ℂ)
9392subid1d 11610 . . . . . . . . . . . . . 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 4637 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺𝑧 = 0)) → {(𝑦𝑧)} = {𝑦})
9695imaeq2d 6077 . . . . . . . . . . 11 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺𝑧 = 0)) → (𝐹 “ {(𝑦𝑧)}) = (𝐹 “ {𝑦}))
9796fveq2d 6909 . . . . . . . . . 10 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺𝑧 = 0)) → (vol‘(𝐹 “ {(𝑦𝑧)})) = (vol‘(𝐹 “ {𝑦})))
98 i1fima2sn 25716 . . . . . . . . . . . 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 480 . . . . . . . . . 10 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺𝑧 = 0)) → (vol‘(𝐹 “ {𝑦})) ∈ ℝ)
10197, 100eqeltrd 2840 . . . . . . . . 9 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺𝑧 = 0)) → (vol‘(𝐹 “ {(𝑦𝑧)})) ∈ ℝ)
10289, 101eqeltrrd 2841 . . . . . . . 8 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺𝑧 = 0)) → (vol*‘(𝐹 “ {(𝑦𝑧)})) ∈ ℝ)
103 ovolsscl 25522 . . . . . . . 8 ((((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧})) ⊆ (𝐹 “ {(𝑦𝑧)}) ∧ (𝐹 “ {(𝑦𝑧)}) ⊆ ℝ ∧ (vol*‘(𝐹 “ {(𝑦𝑧)})) ∈ ℝ) → (vol*‘((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ)
10484, 87, 102, 103mp3an2i 1467 . . . . . . 7 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺𝑧 = 0)) → (vol*‘((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ)
105104expr 456 . . . . . 6 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (𝑧 = 0 → (vol*‘((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ))
106 eldifsn 4785 . . . . . . . 8 (𝑧 ∈ (ran 𝐺 ∖ {0}) ↔ (𝑧 ∈ ran 𝐺𝑧 ≠ 0))
107 inss2 4237 . . . . . . . . 9 ((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧})) ⊆ (𝐺 “ {𝑧})
108 eldifi 4130 . . . . . . . . . 10 (𝑧 ∈ (ran 𝐺 ∖ {0}) → 𝑧 ∈ ran 𝐺)
109 mblss 25567 . . . . . . . . . . 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 25714 . . . . . . . . . . . . 13 (𝐺 ∈ dom ∫1 → (𝐺 “ {𝑧}) ∈ dom vol)
1136, 112syl 17 . . . . . . . . . . . 12 (𝜑 → (𝐺 “ {𝑧}) ∈ dom vol)
114113ad2antrr 726 . . . . . . . . . . 11 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (𝐺 “ {𝑧}) ∈ dom vol)
115 mblvol 25566 . . . . . . . . . . 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 480 . . . . . . . . . . 11 ((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) → 𝐺 ∈ dom ∫1)
118 i1fima2sn 25716 . . . . . . . . . . 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 2841 . . . . . . . . 9 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol*‘(𝐺 “ {𝑧})) ∈ ℝ)
121 ovolsscl 25522 . . . . . . . . 9 ((((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧})) ⊆ (𝐺 “ {𝑧}) ∧ (𝐺 “ {𝑧}) ⊆ ℝ ∧ (vol*‘(𝐺 “ {𝑧})) ∈ ℝ) → (vol*‘((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ)
122107, 111, 120, 121mp3an2i 1467 . . . . . . . 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 456 . . . . . 6 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (𝑧 ≠ 0 → (vol*‘((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ))
125105, 124pm2.61dne 3027 . . . . 5 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (vol*‘((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ)
12652, 125fsumrecl 15771 . . . 4 ((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) → Σ𝑧 ∈ ran 𝐺(vol*‘((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ)
12751fveq2d 6909 . . . . 5 ((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) → (vol*‘((𝐹f + 𝐺) “ {𝑦})) = (vol*‘ 𝑧 ∈ ran 𝐺((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))))
128107, 110sstrid 3994 . . . . . . . 8 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → ((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧})) ⊆ ℝ)
129128, 125jca 511 . . . . . . 7 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧})) ⊆ ℝ ∧ (vol*‘((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ))
130129ralrimiva 3145 . . . . . 6 ((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) → ∀𝑧 ∈ ran 𝐺(((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧})) ⊆ ℝ ∧ (vol*‘((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ))
131 ovolfiniun 25537 . . . . . 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 5164 . . . 4 ((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) → (vol*‘((𝐹f + 𝐺) “ {𝑦})) ≤ Σ𝑧 ∈ ran 𝐺(vol*‘((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))))
134 ovollecl 25519 . . . 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 1372 . . 3 ((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) → (vol*‘((𝐹f + 𝐺) “ {𝑦})) ∈ ℝ)
13681, 135eqeltrd 2840 . 2 ((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) → (vol‘((𝐹f + 𝐺) “ {𝑦})) ∈ ℝ)
13712, 45, 79, 136i1fd 25717 1 (𝜑 → (𝐹f + 𝐺) ∈ dom ∫1)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1539  wcel 2107  {cab 2713  wne 2939  wral 3060  wrex 3069  Vcvv 3479  cdif 3947  cin 3949  wss 3950  {csn 4625   ciun 4990   class class class wbr 5142   × cxp 5682  ccnv 5683  dom cdm 5684  ran crn 5685  cima 5687   Fn wfn 6555  wf 6556  ontowfo 6558  cfv 6560  (class class class)co 7432  cmpo 7434  f cof 7696  Fincfn 8986  cc 11154  cr 11155  0cc0 11156   + caddc 11159  cle 11297  cmin 11493  Σcsu 15723  vol*covol 25498  volcvol 25499  MblFncmbf 25650  1citg1 25651
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-inf2 9682  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233  ax-pre-sup 11234
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-se 5637  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-isom 6569  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-of 7698  df-om 7889  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-2o 8508  df-er 8746  df-map 8869  df-pm 8870  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-sup 9483  df-inf 9484  df-oi 9551  df-dju 9942  df-card 9980  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-div 11922  df-nn 12268  df-2 12330  df-3 12331  df-n0 12529  df-z 12616  df-uz 12880  df-q 12992  df-rp 13036  df-xadd 13156  df-ioo 13392  df-ico 13394  df-icc 13395  df-fz 13549  df-fzo 13696  df-fl 13833  df-seq 14044  df-exp 14104  df-hash 14371  df-cj 15139  df-re 15140  df-im 15141  df-sqrt 15275  df-abs 15276  df-clim 15525  df-sum 15724  df-xmet 21358  df-met 21359  df-ovol 25500  df-vol 25501  df-mbf 25655  df-itg1 25656
This theorem is referenced by:  itg1addlem4  25735  i1fsub  25744  itg2splitlem  25784  itg2split  25785  itg2addlem  25794  itg2addnc  37682  ftc1anclem3  37703  ftc1anclem5  37705  ftc1anclem8  37708
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