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Theorem i1fadd 25744
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 11236 . . . 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 25725 . . . 4 (𝐹 ∈ dom ∫1𝐹:ℝ⟶ℝ)
53, 4syl 17 . . 3 (𝜑𝐹:ℝ⟶ℝ)
6 i1fadd.2 . . . 4 (𝜑𝐺 ∈ dom ∫1)
7 i1ff 25725 . . . 4 (𝐺 ∈ dom ∫1𝐺:ℝ⟶ℝ)
86, 7syl 17 . . 3 (𝜑𝐺:ℝ⟶ℝ)
9 reex 11244 . . . 4 ℝ ∈ V
109a1i 11 . . 3 (𝜑 → ℝ ∈ V)
11 inidm 4235 . . 3 (ℝ ∩ ℝ) = ℝ
122, 5, 8, 10, 10, 11off 7715 . 2 (𝜑 → (𝐹f + 𝐺):ℝ⟶ℝ)
13 i1frn 25726 . . . . . 6 (𝐹 ∈ dom ∫1 → ran 𝐹 ∈ Fin)
143, 13syl 17 . . . . 5 (𝜑 → ran 𝐹 ∈ Fin)
15 i1frn 25726 . . . . . 6 (𝐺 ∈ dom ∫1 → ran 𝐺 ∈ Fin)
166, 15syl 17 . . . . 5 (𝜑 → ran 𝐺 ∈ Fin)
17 xpfi 9356 . . . . 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 2735 . . . . . 6 (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 + 𝑣)) = (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 + 𝑣))
20 ovex 7464 . . . . . 6 (𝑢 + 𝑣) ∈ V
2119, 20fnmpoi 8094 . . . . 5 (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 + 𝑣)) Fn (ran 𝐹 × ran 𝐺)
22 dffn4 6827 . . . . 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 9349 . . . 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 2735 . . . . . . . . 9 (𝑥 + 𝑦) = (𝑥 + 𝑦)
27 rspceov 7480 . . . . . . . . 9 ((𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐺 ∧ (𝑥 + 𝑦) = (𝑥 + 𝑦)) → ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺(𝑥 + 𝑦) = (𝑢 + 𝑣))
2826, 27mp3an3 1449 . . . . . . . 8 ((𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐺) → ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺(𝑥 + 𝑦) = (𝑢 + 𝑣))
29 ovex 7464 . . . . . . . . 9 (𝑥 + 𝑦) ∈ V
30 eqeq1 2739 . . . . . . . . . 10 (𝑤 = (𝑥 + 𝑦) → (𝑤 = (𝑢 + 𝑣) ↔ (𝑥 + 𝑦) = (𝑢 + 𝑣)))
31302rexbidv 3220 . . . . . . . . 9 (𝑤 = (𝑥 + 𝑦) → (∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺 𝑤 = (𝑢 + 𝑣) ↔ ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺(𝑥 + 𝑦) = (𝑢 + 𝑣)))
3229, 31elab 3681 . . . . . . . 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 6738 . . . . . . 7 (𝜑𝐹 Fn ℝ)
36 dffn3 6749 . . . . . . 7 (𝐹 Fn ℝ ↔ 𝐹:ℝ⟶ran 𝐹)
3735, 36sylib 218 . . . . . 6 (𝜑𝐹:ℝ⟶ran 𝐹)
388ffnd 6738 . . . . . . 7 (𝜑𝐺 Fn ℝ)
39 dffn3 6749 . . . . . . 7 (𝐺 Fn ℝ ↔ 𝐺:ℝ⟶ran 𝐺)
4038, 39sylib 218 . . . . . 6 (𝜑𝐺:ℝ⟶ran 𝐺)
4134, 37, 40, 10, 10, 11off 7715 . . . . 5 (𝜑 → (𝐹f + 𝐺):ℝ⟶{𝑤 ∣ ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺 𝑤 = (𝑢 + 𝑣)})
4241frnd 6745 . . . 4 (𝜑 → ran (𝐹f + 𝐺) ⊆ {𝑤 ∣ ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺 𝑤 = (𝑢 + 𝑣)})
4319rnmpo 7566 . . . 4 ran (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 + 𝑣)) = {𝑤 ∣ ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺 𝑤 = (𝑢 + 𝑣)}
4442, 43sseqtrrdi 4047 . . 3 (𝜑 → ran (𝐹f + 𝐺) ⊆ ran (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 + 𝑣)))
4525, 44ssfid 9299 . 2 (𝜑 → ran (𝐹f + 𝐺) ∈ Fin)
4612frnd 6745 . . . . . . 7 (𝜑 → ran (𝐹f + 𝐺) ⊆ ℝ)
4746ssdifssd 4157 . . . . . 6 (𝜑 → (ran (𝐹f + 𝐺) ∖ {0}) ⊆ ℝ)
4847sselda 3995 . . . . 5 ((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) → 𝑦 ∈ ℝ)
4948recnd 11287 . . . 4 ((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) → 𝑦 ∈ ℂ)
503, 6i1faddlem 25742 . . . 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 25724 . . . . . . . 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 6745 . . . . . . . . 9 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → ran (𝐹f + 𝐺) ⊆ ℝ)
59 eldifi 4141 . . . . . . . . . 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 3996 . . . . . . . 8 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝑦 ∈ ℝ)
628adantr 480 . . . . . . . . . 10 ((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) → 𝐺:ℝ⟶ℝ)
6362frnd 6745 . . . . . . . . 9 ((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) → ran 𝐺 ⊆ ℝ)
6463sselda 3995 . . . . . . . 8 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝑧 ∈ ℝ)
6561, 64resubcld 11689 . . . . . . 7 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (𝑦𝑧) ∈ ℝ)
66 mbfimasn 25681 . . . . . . 7 ((𝐹 ∈ MblFn ∧ 𝐹:ℝ⟶ℝ ∧ (𝑦𝑧) ∈ ℝ) → (𝐹 “ {(𝑦𝑧)}) ∈ dom vol)
6755, 56, 65, 66syl3anc 1370 . . . . . 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 25724 . . . . . . . 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 25681 . . . . . . 7 ((𝐺 ∈ MblFn ∧ 𝐺:ℝ⟶ℝ ∧ 𝑧 ∈ ℝ) → (𝐺 “ {𝑧}) ∈ dom vol)
7370, 71, 64, 72syl3anc 1370 . . . . . 6 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (𝐺 “ {𝑧}) ∈ dom vol)
74 inmbl 25591 . . . . . 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 3144 . . . 4 ((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) → ∀𝑧 ∈ ran 𝐺((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧})) ∈ dom vol)
77 finiunmbl 25593 . . . 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 2839 . 2 ((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) → ((𝐹f + 𝐺) “ {𝑦}) ∈ dom vol)
80 mblvol 25579 . . . 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 25580 . . . . 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 4245 . . . . . . . 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 25580 . . . . . . . . 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 25579 . . . . . . . . . 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 773 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺𝑧 = 0)) → 𝑧 = 0)
9190oveq2d 7447 . . . . . . . . . . . . . 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 11607 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺𝑧 = 0)) → (𝑦 − 0) = 𝑦)
9491, 93eqtrd 2775 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺𝑧 = 0)) → (𝑦𝑧) = 𝑦)
9594sneqd 4643 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺𝑧 = 0)) → {(𝑦𝑧)} = {𝑦})
9695imaeq2d 6080 . . . . . . . . . . 11 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺𝑧 = 0)) → (𝐹 “ {(𝑦𝑧)}) = (𝐹 “ {𝑦}))
9796fveq2d 6911 . . . . . . . . . 10 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺𝑧 = 0)) → (vol‘(𝐹 “ {(𝑦𝑧)})) = (vol‘(𝐹 “ {𝑦})))
98 i1fima2sn 25729 . . . . . . . . . . . 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 2839 . . . . . . . . 9 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺𝑧 = 0)) → (vol‘(𝐹 “ {(𝑦𝑧)})) ∈ ℝ)
10289, 101eqeltrrd 2840 . . . . . . . 8 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺𝑧 = 0)) → (vol*‘(𝐹 “ {(𝑦𝑧)})) ∈ ℝ)
103 ovolsscl 25535 . . . . . . . 8 ((((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧})) ⊆ (𝐹 “ {(𝑦𝑧)}) ∧ (𝐹 “ {(𝑦𝑧)}) ⊆ ℝ ∧ (vol*‘(𝐹 “ {(𝑦𝑧)})) ∈ ℝ) → (vol*‘((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ)
10484, 87, 102, 103mp3an2i 1465 . . . . . . 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 4791 . . . . . . . 8 (𝑧 ∈ (ran 𝐺 ∖ {0}) ↔ (𝑧 ∈ ran 𝐺𝑧 ≠ 0))
107 inss2 4246 . . . . . . . . 9 ((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧})) ⊆ (𝐺 “ {𝑧})
108 eldifi 4141 . . . . . . . . . 10 (𝑧 ∈ (ran 𝐺 ∖ {0}) → 𝑧 ∈ ran 𝐺)
109 mblss 25580 . . . . . . . . . . 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 25727 . . . . . . . . . . . . 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 25579 . . . . . . . . . . 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 25729 . . . . . . . . . . 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 2840 . . . . . . . . 9 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol*‘(𝐺 “ {𝑧})) ∈ ℝ)
121 ovolsscl 25535 . . . . . . . . 9 ((((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧})) ⊆ (𝐺 “ {𝑧}) ∧ (𝐺 “ {𝑧}) ⊆ ℝ ∧ (vol*‘(𝐺 “ {𝑧})) ∈ ℝ) → (vol*‘((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ)
122107, 111, 120, 121mp3an2i 1465 . . . . . . . 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 3026 . . . . 5 (((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (vol*‘((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ)
12652, 125fsumrecl 15767 . . . 4 ((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) → Σ𝑧 ∈ ran 𝐺(vol*‘((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ)
12751fveq2d 6911 . . . . 5 ((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) → (vol*‘((𝐹f + 𝐺) “ {𝑦})) = (vol*‘ 𝑧 ∈ ran 𝐺((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))))
128107, 110sstrid 4007 . . . . . . . 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 3144 . . . . . 6 ((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) → ∀𝑧 ∈ ran 𝐺(((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧})) ⊆ ℝ ∧ (vol*‘((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ))
131 ovolfiniun 25550 . . . . . 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 5170 . . . 4 ((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) → (vol*‘((𝐹f + 𝐺) “ {𝑦})) ≤ Σ𝑧 ∈ ran 𝐺(vol*‘((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))))
134 ovollecl 25532 . . . 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 1370 . . 3 ((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) → (vol*‘((𝐹f + 𝐺) “ {𝑦})) ∈ ℝ)
13681, 135eqeltrd 2839 . 2 ((𝜑𝑦 ∈ (ran (𝐹f + 𝐺) ∖ {0})) → (vol‘((𝐹f + 𝐺) “ {𝑦})) ∈ ℝ)
13712, 45, 79, 136i1fd 25730 1 (𝜑 → (𝐹f + 𝐺) ∈ dom ∫1)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  {cab 2712  wne 2938  wral 3059  wrex 3068  Vcvv 3478  cdif 3960  cin 3962  wss 3963  {csn 4631   ciun 4996   class class class wbr 5148   × cxp 5687  ccnv 5688  dom cdm 5689  ran crn 5690  cima 5692   Fn wfn 6558  wf 6559  ontowfo 6561  cfv 6563  (class class class)co 7431  cmpo 7433  f cof 7695  Fincfn 8984  cc 11151  cr 11152  0cc0 11153   + caddc 11156  cle 11294  cmin 11490  Σcsu 15719  vol*covol 25511  volcvol 25512  MblFncmbf 25663  1citg1 25664
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-er 8744  df-map 8867  df-pm 8868  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-sup 9480  df-inf 9481  df-oi 9548  df-dju 9939  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-2 12327  df-3 12328  df-n0 12525  df-z 12612  df-uz 12877  df-q 12989  df-rp 13033  df-xadd 13153  df-ioo 13388  df-ico 13390  df-icc 13391  df-fz 13545  df-fzo 13692  df-fl 13829  df-seq 14040  df-exp 14100  df-hash 14367  df-cj 15135  df-re 15136  df-im 15137  df-sqrt 15271  df-abs 15272  df-clim 15521  df-sum 15720  df-xmet 21375  df-met 21376  df-ovol 25513  df-vol 25514  df-mbf 25668  df-itg1 25669
This theorem is referenced by:  itg1addlem4  25748  itg1addlem4OLD  25749  i1fsub  25758  itg2splitlem  25798  itg2split  25799  itg2addlem  25808  itg2addnc  37661  ftc1anclem3  37682  ftc1anclem5  37684  ftc1anclem8  37687
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