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Theorem i1fadd 24302
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 10618 . . . 4 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 + 𝑦) ∈ ℝ)
21adantl 485 . . 3 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (𝑥 + 𝑦) ∈ ℝ)
3 i1fadd.1 . . . 4 (𝜑𝐹 ∈ dom ∫1)
4 i1ff 24283 . . . 4 (𝐹 ∈ dom ∫1𝐹:ℝ⟶ℝ)
53, 4syl 17 . . 3 (𝜑𝐹:ℝ⟶ℝ)
6 i1fadd.2 . . . 4 (𝜑𝐺 ∈ dom ∫1)
7 i1ff 24283 . . . 4 (𝐺 ∈ dom ∫1𝐺:ℝ⟶ℝ)
86, 7syl 17 . . 3 (𝜑𝐺:ℝ⟶ℝ)
9 reex 10626 . . . 4 ℝ ∈ V
109a1i 11 . . 3 (𝜑 → ℝ ∈ V)
11 inidm 4180 . . 3 (ℝ ∩ ℝ) = ℝ
122, 5, 8, 10, 10, 11off 7418 . 2 (𝜑 → (𝐹f + 𝐺):ℝ⟶ℝ)
13 i1frn 24284 . . . . . 6 (𝐹 ∈ dom ∫1 → ran 𝐹 ∈ Fin)
143, 13syl 17 . . . . 5 (𝜑 → ran 𝐹 ∈ Fin)
15 i1frn 24284 . . . . . 6 (𝐺 ∈ dom ∫1 → ran 𝐺 ∈ Fin)
166, 15syl 17 . . . . 5 (𝜑 → ran 𝐺 ∈ Fin)
17 xpfi 8786 . . . . 5 ((ran 𝐹 ∈ Fin ∧ ran 𝐺 ∈ Fin) → (ran 𝐹 × ran 𝐺) ∈ Fin)
1814, 16, 17syl2anc 587 . . . 4 (𝜑 → (ran 𝐹 × ran 𝐺) ∈ Fin)
19 eqid 2824 . . . . . 6 (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 + 𝑣)) = (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 + 𝑣))
20 ovex 7182 . . . . . 6 (𝑢 + 𝑣) ∈ V
2119, 20fnmpoi 7763 . . . . 5 (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 + 𝑣)) Fn (ran 𝐹 × ran 𝐺)
22 dffn4 6587 . . . . 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 233 . . . 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 8807 . . . 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 589 . . 3 (𝜑 → ran (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 + 𝑣)) ∈ Fin)
26 eqid 2824 . . . . . . . . 9 (𝑥 + 𝑦) = (𝑥 + 𝑦)
27 rspceov 7196 . . . . . . . . 9 ((𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐺 ∧ (𝑥 + 𝑦) = (𝑥 + 𝑦)) → ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺(𝑥 + 𝑦) = (𝑢 + 𝑣))
2826, 27mp3an3 1447 . . . . . . . 8 ((𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐺) → ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺(𝑥 + 𝑦) = (𝑢 + 𝑣))
29 ovex 7182 . . . . . . . . 9 (𝑥 + 𝑦) ∈ V
30 eqeq1 2828 . . . . . . . . . 10 (𝑤 = (𝑥 + 𝑦) → (𝑤 = (𝑢 + 𝑣) ↔ (𝑥 + 𝑦) = (𝑢 + 𝑣)))
31302rexbidv 3292 . . . . . . . . 9 (𝑤 = (𝑥 + 𝑦) → (∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺 𝑤 = (𝑢 + 𝑣) ↔ ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺(𝑥 + 𝑦) = (𝑢 + 𝑣)))
3229, 31elab 3653 . . . . . . . 8 ((𝑥 + 𝑦) ∈ {𝑤 ∣ ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺 𝑤 = (𝑢 + 𝑣)} ↔ ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺(𝑥 + 𝑦) = (𝑢 + 𝑣))
3328, 32sylibr 237 . . . . . . 7 ((𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐺) → (𝑥 + 𝑦) ∈ {𝑤 ∣ ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺 𝑤 = (𝑢 + 𝑣)})
3433adantl 485 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐺)) → (𝑥 + 𝑦) ∈ {𝑤 ∣ ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺 𝑤 = (𝑢 + 𝑣)})
355ffnd 6504 . . . . . . 7 (𝜑𝐹 Fn ℝ)
36 dffn3 6515 . . . . . . 7 (𝐹 Fn ℝ ↔ 𝐹:ℝ⟶ran 𝐹)
3735, 36sylib 221 . . . . . 6 (𝜑𝐹:ℝ⟶ran 𝐹)
388ffnd 6504 . . . . . . 7 (𝜑𝐺 Fn ℝ)
39 dffn3 6515 . . . . . . 7 (𝐺 Fn ℝ ↔ 𝐺:ℝ⟶ran 𝐺)
4038, 39sylib 221 . . . . . 6 (𝜑𝐺:ℝ⟶ran 𝐺)
4134, 37, 40, 10, 10, 11off 7418 . . . . 5 (𝜑 → (𝐹f + 𝐺):ℝ⟶{𝑤 ∣ ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺 𝑤 = (𝑢 + 𝑣)})
4241frnd 6510 . . . 4 (𝜑 → ran (𝐹f + 𝐺) ⊆ {𝑤 ∣ ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺 𝑤 = (𝑢 + 𝑣)})
4319rnmpo 7277 . . . 4 ran (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 + 𝑣)) = {𝑤 ∣ ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺 𝑤 = (𝑢 + 𝑣)}
4442, 43sseqtrrdi 4004 . . 3 (𝜑 → ran (𝐹f + 𝐺) ⊆ ran (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 + 𝑣)))
4525, 44ssfid 8738 . 2 (𝜑 → ran (𝐹f + 𝐺) ∈ Fin)
4612frnd 6510 . . . . . . 7 (𝜑 → ran (𝐹f + 𝐺) ⊆ ℝ)
4746ssdifssd 4105 . . . . . 6 (𝜑 → (ran (𝐹f + 𝐺) ∖ {0}) ⊆ ℝ)
4847sselda 3953 . . . . 5 ((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) → 𝑦 ∈ ℝ)
4948recnd 10667 . . . 4 ((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) → 𝑦 ∈ ℂ)
503, 6i1faddlem 24300 . . . 4 ((𝜑𝑦 ∈ ℂ) → ((𝐹f + 𝐺) “ {𝑦}) = 𝑧 ∈ ran 𝐺((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧})))
5149, 50syldan 594 . . 3 ((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) → ((𝐹f + 𝐺) “ {𝑦}) = 𝑧 ∈ ran 𝐺((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧})))
5216adantr 484 . . . 4 ((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) → ran 𝐺 ∈ Fin)
533ad2antrr 725 . . . . . . . 8 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝐹 ∈ dom ∫1)
54 i1fmbf 24282 . . . . . . . 8 (𝐹 ∈ dom ∫1𝐹 ∈ MblFn)
5553, 54syl 17 . . . . . . 7 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝐹 ∈ MblFn)
565ad2antrr 725 . . . . . . 7 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝐹:ℝ⟶ℝ)
5712ad2antrr 725 . . . . . . . . . 10 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (𝐹f + 𝐺):ℝ⟶ℝ)
5857frnd 6510 . . . . . . . . 9 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → ran (𝐹f + 𝐺) ⊆ ℝ)
59 eldifi 4089 . . . . . . . . . 10 (𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0}) → 𝑦 ∈ ran (𝐹f + 𝐺))
6059ad2antlr 726 . . . . . . . . 9 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝑦 ∈ ran (𝐹f + 𝐺))
6158, 60sseldd 3954 . . . . . . . 8 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝑦 ∈ ℝ)
628adantr 484 . . . . . . . . . 10 ((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) → 𝐺:ℝ⟶ℝ)
6362frnd 6510 . . . . . . . . 9 ((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) → ran 𝐺 ⊆ ℝ)
6463sselda 3953 . . . . . . . 8 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝑧 ∈ ℝ)
6561, 64resubcld 11066 . . . . . . 7 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (𝑦𝑧) ∈ ℝ)
66 mbfimasn 24239 . . . . . . 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 725 . . . . . . . 8 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝐺 ∈ dom ∫1)
69 i1fmbf 24282 . . . . . . . 8 (𝐺 ∈ dom ∫1𝐺 ∈ MblFn)
7068, 69syl 17 . . . . . . 7 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝐺 ∈ MblFn)
718ad2antrr 725 . . . . . . 7 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝐺:ℝ⟶ℝ)
72 mbfimasn 24239 . . . . . . 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 24149 . . . . . 6 (((𝐹 “ {(𝑦𝑧)}) ∈ dom vol ∧ (𝐺 “ {𝑧}) ∈ dom vol) → ((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧})) ∈ dom vol)
7567, 73, 74syl2anc 587 . . . . 5 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → ((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧})) ∈ dom vol)
7675ralrimiva 3177 . . . 4 ((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) → ∀𝑧 ∈ ran 𝐺((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧})) ∈ dom vol)
77 finiunmbl 24151 . . . 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 587 . . 3 ((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) → 𝑧 ∈ ran 𝐺((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧})) ∈ dom vol)
7951, 78eqeltrd 2916 . 2 ((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) → ((𝐹f + 𝐺) “ {𝑦}) ∈ dom vol)
80 mblvol 24137 . . . 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 24138 . . . . 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 4190 . . . . . . . 8 ((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧})) ⊆ (𝐹 “ {(𝑦𝑧)})
8567adantrr 716 . . . . . . . . 9 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺𝑧 = 0)) → (𝐹 “ {(𝑦𝑧)}) ∈ dom vol)
86 mblss 24138 . . . . . . . . 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 24137 . . . . . . . . . 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 7165 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺𝑧 = 0)) → (𝑦𝑧) = (𝑦 − 0))
9249adantr 484 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺𝑧 = 0)) → 𝑦 ∈ ℂ)
9392subid1d 10984 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺𝑧 = 0)) → (𝑦 − 0) = 𝑦)
9491, 93eqtrd 2859 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺𝑧 = 0)) → (𝑦𝑧) = 𝑦)
9594sneqd 4562 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺𝑧 = 0)) → {(𝑦𝑧)} = {𝑦})
9695imaeq2d 5916 . . . . . . . . . . 11 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺𝑧 = 0)) → (𝐹 “ {(𝑦𝑧)}) = (𝐹 “ {𝑦}))
9796fveq2d 6665 . . . . . . . . . 10 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺𝑧 = 0)) → (vol‘(𝐹 “ {(𝑦𝑧)})) = (vol‘(𝐹 “ {𝑦})))
98 i1fima2sn 24287 . . . . . . . . . . . 12 ((𝐹 ∈ dom ∫1𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) → (vol‘(𝐹 “ {𝑦})) ∈ ℝ)
993, 98sylan 583 . . . . . . . . . . 11 ((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) → (vol‘(𝐹 “ {𝑦})) ∈ ℝ)
10099adantr 484 . . . . . . . . . 10 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺𝑧 = 0)) → (vol‘(𝐹 “ {𝑦})) ∈ ℝ)
10197, 100eqeltrd 2916 . . . . . . . . 9 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺𝑧 = 0)) → (vol‘(𝐹 “ {(𝑦𝑧)})) ∈ ℝ)
10289, 101eqeltrrd 2917 . . . . . . . 8 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺𝑧 = 0)) → (vol*‘(𝐹 “ {(𝑦𝑧)})) ∈ ℝ)
103 ovolsscl 24093 . . . . . . . 8 ((((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧})) ⊆ (𝐹 “ {(𝑦𝑧)}) ∧ (𝐹 “ {(𝑦𝑧)}) ⊆ ℝ ∧ (vol*‘(𝐹 “ {(𝑦𝑧)})) ∈ ℝ) → (vol*‘((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ)
10484, 87, 102, 103mp3an2i 1463 . . . . . . 7 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺𝑧 = 0)) → (vol*‘((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ)
105104expr 460 . . . . . 6 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (𝑧 = 0 → (vol*‘((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ))
106 eldifsn 4704 . . . . . . . 8 (𝑧 ∈ (ran 𝐺 ∖ {0}) ↔ (𝑧 ∈ ran 𝐺𝑧 ≠ 0))
107 inss2 4191 . . . . . . . . 9 ((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧})) ⊆ (𝐺 “ {𝑧})
108 eldifi 4089 . . . . . . . . . 10 (𝑧 ∈ (ran 𝐺 ∖ {0}) → 𝑧 ∈ ran 𝐺)
109 mblss 24138 . . . . . . . . . . 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 595 . . . . . . . . 9 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (𝐺 “ {𝑧}) ⊆ ℝ)
112 i1fima 24285 . . . . . . . . . . . . 13 (𝐺 ∈ dom ∫1 → (𝐺 “ {𝑧}) ∈ dom vol)
1136, 112syl 17 . . . . . . . . . . . 12 (𝜑 → (𝐺 “ {𝑧}) ∈ dom vol)
114113ad2antrr 725 . . . . . . . . . . 11 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (𝐺 “ {𝑧}) ∈ dom vol)
115 mblvol 24137 . . . . . . . . . . 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 484 . . . . . . . . . . 11 ((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) → 𝐺 ∈ dom ∫1)
118 i1fima2sn 24287 . . . . . . . . . . 11 ((𝐺 ∈ dom ∫1𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol‘(𝐺 “ {𝑧})) ∈ ℝ)
119117, 118sylan 583 . . . . . . . . . 10 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol‘(𝐺 “ {𝑧})) ∈ ℝ)
120116, 119eqeltrrd 2917 . . . . . . . . 9 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol*‘(𝐺 “ {𝑧})) ∈ ℝ)
121 ovolsscl 24093 . . . . . . . . 9 ((((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧})) ⊆ (𝐺 “ {𝑧}) ∧ (𝐺 “ {𝑧}) ⊆ ℝ ∧ (vol*‘(𝐺 “ {𝑧})) ∈ ℝ) → (vol*‘((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ)
122107, 111, 120, 121mp3an2i 1463 . . . . . . . 8 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol*‘((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ)
123106, 122sylan2br 597 . . . . . . 7 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺𝑧 ≠ 0)) → (vol*‘((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ)
124123expr 460 . . . . . 6 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (𝑧 ≠ 0 → (vol*‘((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ))
125105, 124pm2.61dne 3100 . . . . 5 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (vol*‘((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ)
12652, 125fsumrecl 15091 . . . 4 ((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) → Σ𝑧 ∈ ran 𝐺(vol*‘((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ)
12751fveq2d 6665 . . . . 5 ((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) → (vol*‘((𝐹f + 𝐺) “ {𝑦})) = (vol*‘ 𝑧 ∈ ran 𝐺((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))))
128107, 110sstrid 3964 . . . . . . . 8 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → ((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧})) ⊆ ℝ)
129128, 125jca 515 . . . . . . 7 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧})) ⊆ ℝ ∧ (vol*‘((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ))
130129ralrimiva 3177 . . . . . 6 ((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) → ∀𝑧 ∈ ran 𝐺(((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧})) ⊆ ℝ ∧ (vol*‘((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ))
131 ovolfiniun 24108 . . . . . 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 587 . . . . 5 ((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) → (vol*‘ 𝑧 ∈ ran 𝐺((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))) ≤ Σ𝑧 ∈ ran 𝐺(vol*‘((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))))
133127, 132eqbrtrd 5074 . . . 4 ((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) → (vol*‘((𝐹f + 𝐺) “ {𝑦})) ≤ Σ𝑧 ∈ ran 𝐺(vol*‘((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))))
134 ovollecl 24090 . . . 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 2916 . 2 ((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) → (vol‘((𝐹f + 𝐺) “ {𝑦})) ∈ ℝ)
13712, 45, 79, 136i1fd 24288 1 (𝜑 → (𝐹f + 𝐺) ∈ dom ∫1)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  {cab 2802  wne 3014  wral 3133  wrex 3134  Vcvv 3480  cdif 3916  cin 3918  wss 3919  {csn 4550   ciun 4905   class class class wbr 5052   × cxp 5540  ccnv 5541  dom cdm 5542  ran crn 5543  cima 5545   Fn wfn 6338  wf 6339  ontowfo 6341  cfv 6343  (class class class)co 7149  cmpo 7151  f cof 7401  Fincfn 8505  cc 10533  cr 10534  0cc0 10535   + caddc 10538  cle 10674  cmin 10868  Σcsu 15042  vol*covol 24069  volcvol 24070  MblFncmbf 24221  1citg1 24222
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455  ax-inf2 9101  ax-cnex 10591  ax-resscn 10592  ax-1cn 10593  ax-icn 10594  ax-addcl 10595  ax-addrcl 10596  ax-mulcl 10597  ax-mulrcl 10598  ax-mulcom 10599  ax-addass 10600  ax-mulass 10601  ax-distr 10602  ax-i2m1 10603  ax-1ne0 10604  ax-1rid 10605  ax-rnegex 10606  ax-rrecex 10607  ax-cnre 10608  ax-pre-lttri 10609  ax-pre-lttrn 10610  ax-pre-ltadd 10611  ax-pre-mulgt0 10612  ax-pre-sup 10613
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-se 5502  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-isom 6352  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-of 7403  df-om 7575  df-1st 7684  df-2nd 7685  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-2o 8099  df-oadd 8102  df-er 8285  df-map 8404  df-pm 8405  df-en 8506  df-dom 8507  df-sdom 8508  df-fin 8509  df-sup 8903  df-inf 8904  df-oi 8971  df-dju 9327  df-card 9365  df-pnf 10675  df-mnf 10676  df-xr 10677  df-ltxr 10678  df-le 10679  df-sub 10870  df-neg 10871  df-div 11296  df-nn 11635  df-2 11697  df-3 11698  df-n0 11895  df-z 11979  df-uz 12241  df-q 12346  df-rp 12387  df-xadd 12505  df-ioo 12739  df-ico 12741  df-icc 12742  df-fz 12895  df-fzo 13038  df-fl 13166  df-seq 13374  df-exp 13435  df-hash 13696  df-cj 14458  df-re 14459  df-im 14460  df-sqrt 14594  df-abs 14595  df-clim 14845  df-sum 15043  df-xmet 20538  df-met 20539  df-ovol 24071  df-vol 24072  df-mbf 24226  df-itg1 24227
This theorem is referenced by:  itg1addlem4  24306  i1fsub  24315  itg2splitlem  24355  itg2split  24356  itg2addlem  24365  itg2addnc  35056  ftc1anclem3  35077  ftc1anclem5  35079  ftc1anclem8  35082
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