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Theorem i1fadd 23363
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 9964 . . . 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 23344 . . . 4 (𝐹 ∈ dom ∫1𝐹:ℝ⟶ℝ)
53, 4syl 17 . . 3 (𝜑𝐹:ℝ⟶ℝ)
6 i1fadd.2 . . . 4 (𝜑𝐺 ∈ dom ∫1)
7 i1ff 23344 . . . 4 (𝐺 ∈ dom ∫1𝐺:ℝ⟶ℝ)
86, 7syl 17 . . 3 (𝜑𝐺:ℝ⟶ℝ)
9 reex 9972 . . . 4 ℝ ∈ V
109a1i 11 . . 3 (𝜑 → ℝ ∈ V)
11 inidm 3805 . . 3 (ℝ ∩ ℝ) = ℝ
122, 5, 8, 10, 10, 11off 6866 . 2 (𝜑 → (𝐹𝑓 + 𝐺):ℝ⟶ℝ)
13 i1frn 23345 . . . . . 6 (𝐹 ∈ dom ∫1 → ran 𝐹 ∈ Fin)
143, 13syl 17 . . . . 5 (𝜑 → ran 𝐹 ∈ Fin)
15 i1frn 23345 . . . . . 6 (𝐺 ∈ dom ∫1 → ran 𝐺 ∈ Fin)
166, 15syl 17 . . . . 5 (𝜑 → ran 𝐺 ∈ Fin)
17 xpfi 8176 . . . . 5 ((ran 𝐹 ∈ Fin ∧ ran 𝐺 ∈ Fin) → (ran 𝐹 × ran 𝐺) ∈ Fin)
1814, 16, 17syl2anc 692 . . . 4 (𝜑 → (ran 𝐹 × ran 𝐺) ∈ Fin)
19 eqid 2626 . . . . . 6 (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 + 𝑣)) = (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 + 𝑣))
20 ovex 6633 . . . . . 6 (𝑢 + 𝑣) ∈ V
2119, 20fnmpt2i 7185 . . . . 5 (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 + 𝑣)) Fn (ran 𝐹 × ran 𝐺)
22 dffn4 6080 . . . . 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 220 . . . 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 8197 . . . 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 693 . . 3 (𝜑 → ran (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 + 𝑣)) ∈ Fin)
26 eqid 2626 . . . . . . . . 9 (𝑥 + 𝑦) = (𝑥 + 𝑦)
27 rspceov 6646 . . . . . . . . 9 ((𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐺 ∧ (𝑥 + 𝑦) = (𝑥 + 𝑦)) → ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺(𝑥 + 𝑦) = (𝑢 + 𝑣))
2826, 27mp3an3 1410 . . . . . . . 8 ((𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐺) → ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺(𝑥 + 𝑦) = (𝑢 + 𝑣))
29 ovex 6633 . . . . . . . . 9 (𝑥 + 𝑦) ∈ V
30 eqeq1 2630 . . . . . . . . . 10 (𝑤 = (𝑥 + 𝑦) → (𝑤 = (𝑢 + 𝑣) ↔ (𝑥 + 𝑦) = (𝑢 + 𝑣)))
31302rexbidv 3055 . . . . . . . . 9 (𝑤 = (𝑥 + 𝑦) → (∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺 𝑤 = (𝑢 + 𝑣) ↔ ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺(𝑥 + 𝑦) = (𝑢 + 𝑣)))
3229, 31elab 3338 . . . . . . . 8 ((𝑥 + 𝑦) ∈ {𝑤 ∣ ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺 𝑤 = (𝑢 + 𝑣)} ↔ ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺(𝑥 + 𝑦) = (𝑢 + 𝑣))
3328, 32sylibr 224 . . . . . . 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)})
35 ffn 6004 . . . . . . . 8 (𝐹:ℝ⟶ℝ → 𝐹 Fn ℝ)
365, 35syl 17 . . . . . . 7 (𝜑𝐹 Fn ℝ)
37 dffn3 6013 . . . . . . 7 (𝐹 Fn ℝ ↔ 𝐹:ℝ⟶ran 𝐹)
3836, 37sylib 208 . . . . . 6 (𝜑𝐹:ℝ⟶ran 𝐹)
39 ffn 6004 . . . . . . . 8 (𝐺:ℝ⟶ℝ → 𝐺 Fn ℝ)
408, 39syl 17 . . . . . . 7 (𝜑𝐺 Fn ℝ)
41 dffn3 6013 . . . . . . 7 (𝐺 Fn ℝ ↔ 𝐺:ℝ⟶ran 𝐺)
4240, 41sylib 208 . . . . . 6 (𝜑𝐺:ℝ⟶ran 𝐺)
4334, 38, 42, 10, 10, 11off 6866 . . . . 5 (𝜑 → (𝐹𝑓 + 𝐺):ℝ⟶{𝑤 ∣ ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺 𝑤 = (𝑢 + 𝑣)})
44 frn 6012 . . . . 5 ((𝐹𝑓 + 𝐺):ℝ⟶{𝑤 ∣ ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺 𝑤 = (𝑢 + 𝑣)} → ran (𝐹𝑓 + 𝐺) ⊆ {𝑤 ∣ ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺 𝑤 = (𝑢 + 𝑣)})
4543, 44syl 17 . . . 4 (𝜑 → ran (𝐹𝑓 + 𝐺) ⊆ {𝑤 ∣ ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺 𝑤 = (𝑢 + 𝑣)})
4619rnmpt2 6724 . . . 4 ran (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 + 𝑣)) = {𝑤 ∣ ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺 𝑤 = (𝑢 + 𝑣)}
4745, 46syl6sseqr 3636 . . 3 (𝜑 → ran (𝐹𝑓 + 𝐺) ⊆ ran (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 + 𝑣)))
48 ssfi 8125 . . 3 ((ran (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 + 𝑣)) ∈ Fin ∧ ran (𝐹𝑓 + 𝐺) ⊆ ran (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 + 𝑣))) → ran (𝐹𝑓 + 𝐺) ∈ Fin)
4925, 47, 48syl2anc 692 . 2 (𝜑 → ran (𝐹𝑓 + 𝐺) ∈ Fin)
50 frn 6012 . . . . . . . 8 ((𝐹𝑓 + 𝐺):ℝ⟶ℝ → ran (𝐹𝑓 + 𝐺) ⊆ ℝ)
5112, 50syl 17 . . . . . . 7 (𝜑 → ran (𝐹𝑓 + 𝐺) ⊆ ℝ)
5251ssdifssd 3731 . . . . . 6 (𝜑 → (ran (𝐹𝑓 + 𝐺) ∖ {0}) ⊆ ℝ)
5352sselda 3588 . . . . 5 ((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) → 𝑦 ∈ ℝ)
5453recnd 10013 . . . 4 ((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) → 𝑦 ∈ ℂ)
553, 6i1faddlem 23361 . . . 4 ((𝜑𝑦 ∈ ℂ) → ((𝐹𝑓 + 𝐺) “ {𝑦}) = 𝑧 ∈ ran 𝐺((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧})))
5654, 55syldan 487 . . 3 ((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) → ((𝐹𝑓 + 𝐺) “ {𝑦}) = 𝑧 ∈ ran 𝐺((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧})))
5716adantr 481 . . . 4 ((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) → ran 𝐺 ∈ Fin)
583ad2antrr 761 . . . . . . . 8 (((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝐹 ∈ dom ∫1)
59 i1fmbf 23343 . . . . . . . 8 (𝐹 ∈ dom ∫1𝐹 ∈ MblFn)
6058, 59syl 17 . . . . . . 7 (((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝐹 ∈ MblFn)
615ad2antrr 761 . . . . . . 7 (((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝐹:ℝ⟶ℝ)
6212ad2antrr 761 . . . . . . . . . 10 (((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (𝐹𝑓 + 𝐺):ℝ⟶ℝ)
6362, 50syl 17 . . . . . . . . 9 (((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → ran (𝐹𝑓 + 𝐺) ⊆ ℝ)
64 eldifi 3715 . . . . . . . . . 10 (𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0}) → 𝑦 ∈ ran (𝐹𝑓 + 𝐺))
6564ad2antlr 762 . . . . . . . . 9 (((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝑦 ∈ ran (𝐹𝑓 + 𝐺))
6663, 65sseldd 3589 . . . . . . . 8 (((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝑦 ∈ ℝ)
678adantr 481 . . . . . . . . . 10 ((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) → 𝐺:ℝ⟶ℝ)
68 frn 6012 . . . . . . . . . 10 (𝐺:ℝ⟶ℝ → ran 𝐺 ⊆ ℝ)
6967, 68syl 17 . . . . . . . . 9 ((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) → ran 𝐺 ⊆ ℝ)
7069sselda 3588 . . . . . . . 8 (((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝑧 ∈ ℝ)
7166, 70resubcld 10403 . . . . . . 7 (((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (𝑦𝑧) ∈ ℝ)
72 mbfimasn 23302 . . . . . . 7 ((𝐹 ∈ MblFn ∧ 𝐹:ℝ⟶ℝ ∧ (𝑦𝑧) ∈ ℝ) → (𝐹 “ {(𝑦𝑧)}) ∈ dom vol)
7360, 61, 71, 72syl3anc 1323 . . . . . 6 (((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (𝐹 “ {(𝑦𝑧)}) ∈ dom vol)
746ad2antrr 761 . . . . . . . 8 (((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝐺 ∈ dom ∫1)
75 i1fmbf 23343 . . . . . . . 8 (𝐺 ∈ dom ∫1𝐺 ∈ MblFn)
7674, 75syl 17 . . . . . . 7 (((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝐺 ∈ MblFn)
778ad2antrr 761 . . . . . . 7 (((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝐺:ℝ⟶ℝ)
78 mbfimasn 23302 . . . . . . 7 ((𝐺 ∈ MblFn ∧ 𝐺:ℝ⟶ℝ ∧ 𝑧 ∈ ℝ) → (𝐺 “ {𝑧}) ∈ dom vol)
7976, 77, 70, 78syl3anc 1323 . . . . . 6 (((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (𝐺 “ {𝑧}) ∈ dom vol)
80 inmbl 23212 . . . . . 6 (((𝐹 “ {(𝑦𝑧)}) ∈ dom vol ∧ (𝐺 “ {𝑧}) ∈ dom vol) → ((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧})) ∈ dom vol)
8173, 79, 80syl2anc 692 . . . . 5 (((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → ((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧})) ∈ dom vol)
8281ralrimiva 2965 . . . 4 ((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) → ∀𝑧 ∈ ran 𝐺((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧})) ∈ dom vol)
83 finiunmbl 23214 . . . 4 ((ran 𝐺 ∈ Fin ∧ ∀𝑧 ∈ ran 𝐺((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧})) ∈ dom vol) → 𝑧 ∈ ran 𝐺((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧})) ∈ dom vol)
8457, 82, 83syl2anc 692 . . 3 ((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) → 𝑧 ∈ ran 𝐺((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧})) ∈ dom vol)
8556, 84eqeltrd 2704 . 2 ((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) → ((𝐹𝑓 + 𝐺) “ {𝑦}) ∈ dom vol)
86 mblvol 23200 . . . 4 (((𝐹𝑓 + 𝐺) “ {𝑦}) ∈ dom vol → (vol‘((𝐹𝑓 + 𝐺) “ {𝑦})) = (vol*‘((𝐹𝑓 + 𝐺) “ {𝑦})))
8785, 86syl 17 . . 3 ((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) → (vol‘((𝐹𝑓 + 𝐺) “ {𝑦})) = (vol*‘((𝐹𝑓 + 𝐺) “ {𝑦})))
88 mblss 23201 . . . . 5 (((𝐹𝑓 + 𝐺) “ {𝑦}) ∈ dom vol → ((𝐹𝑓 + 𝐺) “ {𝑦}) ⊆ ℝ)
8985, 88syl 17 . . . 4 ((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) → ((𝐹𝑓 + 𝐺) “ {𝑦}) ⊆ ℝ)
90 inss1 3816 . . . . . . . . 9 ((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧})) ⊆ (𝐹 “ {(𝑦𝑧)})
9190a1i 11 . . . . . . . 8 (((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺𝑧 = 0)) → ((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧})) ⊆ (𝐹 “ {(𝑦𝑧)}))
9273adantrr 752 . . . . . . . . 9 (((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺𝑧 = 0)) → (𝐹 “ {(𝑦𝑧)}) ∈ dom vol)
93 mblss 23201 . . . . . . . . 9 ((𝐹 “ {(𝑦𝑧)}) ∈ dom vol → (𝐹 “ {(𝑦𝑧)}) ⊆ ℝ)
9492, 93syl 17 . . . . . . . 8 (((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺𝑧 = 0)) → (𝐹 “ {(𝑦𝑧)}) ⊆ ℝ)
95 mblvol 23200 . . . . . . . . . 10 ((𝐹 “ {(𝑦𝑧)}) ∈ dom vol → (vol‘(𝐹 “ {(𝑦𝑧)})) = (vol*‘(𝐹 “ {(𝑦𝑧)})))
9692, 95syl 17 . . . . . . . . 9 (((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺𝑧 = 0)) → (vol‘(𝐹 “ {(𝑦𝑧)})) = (vol*‘(𝐹 “ {(𝑦𝑧)})))
97 simprr 795 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺𝑧 = 0)) → 𝑧 = 0)
9897oveq2d 6621 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺𝑧 = 0)) → (𝑦𝑧) = (𝑦 − 0))
9954adantr 481 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺𝑧 = 0)) → 𝑦 ∈ ℂ)
10099subid1d 10326 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺𝑧 = 0)) → (𝑦 − 0) = 𝑦)
10198, 100eqtrd 2660 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺𝑧 = 0)) → (𝑦𝑧) = 𝑦)
102101sneqd 4165 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺𝑧 = 0)) → {(𝑦𝑧)} = {𝑦})
103102imaeq2d 5429 . . . . . . . . . . 11 (((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺𝑧 = 0)) → (𝐹 “ {(𝑦𝑧)}) = (𝐹 “ {𝑦}))
104103fveq2d 6154 . . . . . . . . . 10 (((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺𝑧 = 0)) → (vol‘(𝐹 “ {(𝑦𝑧)})) = (vol‘(𝐹 “ {𝑦})))
105 i1fima2sn 23348 . . . . . . . . . . . 12 ((𝐹 ∈ dom ∫1𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) → (vol‘(𝐹 “ {𝑦})) ∈ ℝ)
1063, 105sylan 488 . . . . . . . . . . 11 ((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) → (vol‘(𝐹 “ {𝑦})) ∈ ℝ)
107106adantr 481 . . . . . . . . . 10 (((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺𝑧 = 0)) → (vol‘(𝐹 “ {𝑦})) ∈ ℝ)
108104, 107eqeltrd 2704 . . . . . . . . 9 (((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺𝑧 = 0)) → (vol‘(𝐹 “ {(𝑦𝑧)})) ∈ ℝ)
10996, 108eqeltrrd 2705 . . . . . . . 8 (((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺𝑧 = 0)) → (vol*‘(𝐹 “ {(𝑦𝑧)})) ∈ ℝ)
110 ovolsscl 23156 . . . . . . . 8 ((((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧})) ⊆ (𝐹 “ {(𝑦𝑧)}) ∧ (𝐹 “ {(𝑦𝑧)}) ⊆ ℝ ∧ (vol*‘(𝐹 “ {(𝑦𝑧)})) ∈ ℝ) → (vol*‘((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ)
11191, 94, 109, 110syl3anc 1323 . . . . . . 7 (((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺𝑧 = 0)) → (vol*‘((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ)
112111expr 642 . . . . . 6 (((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (𝑧 = 0 → (vol*‘((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ))
113 eldifsn 4292 . . . . . . . 8 (𝑧 ∈ (ran 𝐺 ∖ {0}) ↔ (𝑧 ∈ ran 𝐺𝑧 ≠ 0))
114 inss2 3817 . . . . . . . . . 10 ((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧})) ⊆ (𝐺 “ {𝑧})
115114a1i 11 . . . . . . . . 9 (((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → ((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧})) ⊆ (𝐺 “ {𝑧}))
116 eldifi 3715 . . . . . . . . . 10 (𝑧 ∈ (ran 𝐺 ∖ {0}) → 𝑧 ∈ ran 𝐺)
117 mblss 23201 . . . . . . . . . . 11 ((𝐺 “ {𝑧}) ∈ dom vol → (𝐺 “ {𝑧}) ⊆ ℝ)
11879, 117syl 17 . . . . . . . . . 10 (((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (𝐺 “ {𝑧}) ⊆ ℝ)
119116, 118sylan2 491 . . . . . . . . 9 (((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (𝐺 “ {𝑧}) ⊆ ℝ)
120 i1fima 23346 . . . . . . . . . . . . 13 (𝐺 ∈ dom ∫1 → (𝐺 “ {𝑧}) ∈ dom vol)
1216, 120syl 17 . . . . . . . . . . . 12 (𝜑 → (𝐺 “ {𝑧}) ∈ dom vol)
122121ad2antrr 761 . . . . . . . . . . 11 (((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (𝐺 “ {𝑧}) ∈ dom vol)
123 mblvol 23200 . . . . . . . . . . 11 ((𝐺 “ {𝑧}) ∈ dom vol → (vol‘(𝐺 “ {𝑧})) = (vol*‘(𝐺 “ {𝑧})))
124122, 123syl 17 . . . . . . . . . 10 (((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol‘(𝐺 “ {𝑧})) = (vol*‘(𝐺 “ {𝑧})))
1256adantr 481 . . . . . . . . . . 11 ((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) → 𝐺 ∈ dom ∫1)
126 i1fima2sn 23348 . . . . . . . . . . 11 ((𝐺 ∈ dom ∫1𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol‘(𝐺 “ {𝑧})) ∈ ℝ)
127125, 126sylan 488 . . . . . . . . . 10 (((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol‘(𝐺 “ {𝑧})) ∈ ℝ)
128124, 127eqeltrrd 2705 . . . . . . . . 9 (((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol*‘(𝐺 “ {𝑧})) ∈ ℝ)
129 ovolsscl 23156 . . . . . . . . 9 ((((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧})) ⊆ (𝐺 “ {𝑧}) ∧ (𝐺 “ {𝑧}) ⊆ ℝ ∧ (vol*‘(𝐺 “ {𝑧})) ∈ ℝ) → (vol*‘((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ)
130115, 119, 128, 129syl3anc 1323 . . . . . . . 8 (((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol*‘((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ)
131113, 130sylan2br 493 . . . . . . 7 (((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺𝑧 ≠ 0)) → (vol*‘((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ)
132131expr 642 . . . . . 6 (((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (𝑧 ≠ 0 → (vol*‘((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ))
133112, 132pm2.61dne 2882 . . . . 5 (((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (vol*‘((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ)
13457, 133fsumrecl 14393 . . . 4 ((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) → Σ𝑧 ∈ ran 𝐺(vol*‘((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ)
13556fveq2d 6154 . . . . 5 ((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) → (vol*‘((𝐹𝑓 + 𝐺) “ {𝑦})) = (vol*‘ 𝑧 ∈ ran 𝐺((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))))
136114, 118syl5ss 3599 . . . . . . . 8 (((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → ((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧})) ⊆ ℝ)
137136, 133jca 554 . . . . . . 7 (((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧})) ⊆ ℝ ∧ (vol*‘((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ))
138137ralrimiva 2965 . . . . . 6 ((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) → ∀𝑧 ∈ ran 𝐺(((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧})) ⊆ ℝ ∧ (vol*‘((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ))
139 ovolfiniun 23171 . . . . . 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}))))
14057, 138, 139syl2anc 692 . . . . 5 ((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) → (vol*‘ 𝑧 ∈ ran 𝐺((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))) ≤ Σ𝑧 ∈ ran 𝐺(vol*‘((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))))
141135, 140eqbrtrd 4640 . . . 4 ((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) → (vol*‘((𝐹𝑓 + 𝐺) “ {𝑦})) ≤ Σ𝑧 ∈ ran 𝐺(vol*‘((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))))
142 ovollecl 23153 . . . 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)
14389, 134, 141, 142syl3anc 1323 . . 3 ((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) → (vol*‘((𝐹𝑓 + 𝐺) “ {𝑦})) ∈ ℝ)
14487, 143eqeltrd 2704 . 2 ((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) → (vol‘((𝐹𝑓 + 𝐺) “ {𝑦})) ∈ ℝ)
14512, 49, 85, 144i1fd 23349 1 (𝜑 → (𝐹𝑓 + 𝐺) ∈ dom ∫1)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1992  {cab 2612  wne 2796  wral 2912  wrex 2913  Vcvv 3191  cdif 3557  cin 3559  wss 3560  {csn 4153   ciun 4490   class class class wbr 4618   × cxp 5077  ccnv 5078  dom cdm 5079  ran crn 5080  cima 5082   Fn wfn 5845  wf 5846  ontowfo 5848  cfv 5850  (class class class)co 6605  cmpt2 6607  𝑓 cof 6849  Fincfn 7900  cc 9879  cr 9880  0cc0 9881   + caddc 9884  cle 10020  cmin 10211  Σcsu 14345  vol*covol 23133  volcvol 23134  MblFncmbf 23284  1citg1 23285
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-inf2 8483  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957  ax-pre-mulgt0 9958  ax-pre-sup 9959
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-isom 5859  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-of 6851  df-om 7014  df-1st 7116  df-2nd 7117  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-1o 7506  df-2o 7507  df-oadd 7510  df-er 7688  df-map 7805  df-pm 7806  df-en 7901  df-dom 7902  df-sdom 7903  df-fin 7904  df-sup 8293  df-inf 8294  df-oi 8360  df-card 8710  df-cda 8935  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-sub 10213  df-neg 10214  df-div 10630  df-nn 10966  df-2 11024  df-3 11025  df-n0 11238  df-z 11323  df-uz 11632  df-q 11733  df-rp 11777  df-xadd 11891  df-ioo 12118  df-ico 12120  df-icc 12121  df-fz 12266  df-fzo 12404  df-fl 12530  df-seq 12739  df-exp 12798  df-hash 13055  df-cj 13768  df-re 13769  df-im 13770  df-sqrt 13904  df-abs 13905  df-clim 14148  df-sum 14346  df-xmet 19653  df-met 19654  df-ovol 23135  df-vol 23136  df-mbf 23289  df-itg1 23290
This theorem is referenced by:  itg1addlem4  23367  i1fsub  23376  itg2splitlem  23416  itg2split  23417  itg2addlem  23426  itg2addnc  33082  ftc1anclem3  33105  ftc1anclem5  33107  ftc1anclem8  33110
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