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Theorem i1fadd 23507
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 10057 . . . 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 23488 . . . 4 (𝐹 ∈ dom ∫1𝐹:ℝ⟶ℝ)
53, 4syl 17 . . 3 (𝜑𝐹:ℝ⟶ℝ)
6 i1fadd.2 . . . 4 (𝜑𝐺 ∈ dom ∫1)
7 i1ff 23488 . . . 4 (𝐺 ∈ dom ∫1𝐺:ℝ⟶ℝ)
86, 7syl 17 . . 3 (𝜑𝐺:ℝ⟶ℝ)
9 reex 10065 . . . 4 ℝ ∈ V
109a1i 11 . . 3 (𝜑 → ℝ ∈ V)
11 inidm 3855 . . 3 (ℝ ∩ ℝ) = ℝ
122, 5, 8, 10, 10, 11off 6954 . 2 (𝜑 → (𝐹𝑓 + 𝐺):ℝ⟶ℝ)
13 i1frn 23489 . . . . . 6 (𝐹 ∈ dom ∫1 → ran 𝐹 ∈ Fin)
143, 13syl 17 . . . . 5 (𝜑 → ran 𝐹 ∈ Fin)
15 i1frn 23489 . . . . . 6 (𝐺 ∈ dom ∫1 → ran 𝐺 ∈ Fin)
166, 15syl 17 . . . . 5 (𝜑 → ran 𝐺 ∈ Fin)
17 xpfi 8272 . . . . 5 ((ran 𝐹 ∈ Fin ∧ ran 𝐺 ∈ Fin) → (ran 𝐹 × ran 𝐺) ∈ Fin)
1814, 16, 17syl2anc 694 . . . 4 (𝜑 → (ran 𝐹 × ran 𝐺) ∈ Fin)
19 eqid 2651 . . . . . 6 (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 + 𝑣)) = (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 + 𝑣))
20 ovex 6718 . . . . . 6 (𝑢 + 𝑣) ∈ V
2119, 20fnmpt2i 7284 . . . . 5 (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 + 𝑣)) Fn (ran 𝐹 × ran 𝐺)
22 dffn4 6159 . . . . 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 8293 . . . 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 695 . . 3 (𝜑 → ran (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 + 𝑣)) ∈ Fin)
26 eqid 2651 . . . . . . . . 9 (𝑥 + 𝑦) = (𝑥 + 𝑦)
27 rspceov 6732 . . . . . . . . 9 ((𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐺 ∧ (𝑥 + 𝑦) = (𝑥 + 𝑦)) → ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺(𝑥 + 𝑦) = (𝑢 + 𝑣))
2826, 27mp3an3 1453 . . . . . . . 8 ((𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐺) → ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺(𝑥 + 𝑦) = (𝑢 + 𝑣))
29 ovex 6718 . . . . . . . . 9 (𝑥 + 𝑦) ∈ V
30 eqeq1 2655 . . . . . . . . . 10 (𝑤 = (𝑥 + 𝑦) → (𝑤 = (𝑢 + 𝑣) ↔ (𝑥 + 𝑦) = (𝑢 + 𝑣)))
31302rexbidv 3086 . . . . . . . . 9 (𝑤 = (𝑥 + 𝑦) → (∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺 𝑤 = (𝑢 + 𝑣) ↔ ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺(𝑥 + 𝑦) = (𝑢 + 𝑣)))
3229, 31elab 3382 . . . . . . . 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 481 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ ran 𝐹𝑦 ∈ ran 𝐺)) → (𝑥 + 𝑦) ∈ {𝑤 ∣ ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺 𝑤 = (𝑢 + 𝑣)})
35 ffn 6083 . . . . . . . 8 (𝐹:ℝ⟶ℝ → 𝐹 Fn ℝ)
365, 35syl 17 . . . . . . 7 (𝜑𝐹 Fn ℝ)
37 dffn3 6092 . . . . . . 7 (𝐹 Fn ℝ ↔ 𝐹:ℝ⟶ran 𝐹)
3836, 37sylib 208 . . . . . 6 (𝜑𝐹:ℝ⟶ran 𝐹)
39 ffn 6083 . . . . . . . 8 (𝐺:ℝ⟶ℝ → 𝐺 Fn ℝ)
408, 39syl 17 . . . . . . 7 (𝜑𝐺 Fn ℝ)
41 dffn3 6092 . . . . . . 7 (𝐺 Fn ℝ ↔ 𝐺:ℝ⟶ran 𝐺)
4240, 41sylib 208 . . . . . 6 (𝜑𝐺:ℝ⟶ran 𝐺)
4334, 38, 42, 10, 10, 11off 6954 . . . . 5 (𝜑 → (𝐹𝑓 + 𝐺):ℝ⟶{𝑤 ∣ ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺 𝑤 = (𝑢 + 𝑣)})
44 frn 6091 . . . . 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 6812 . . . 4 ran (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 + 𝑣)) = {𝑤 ∣ ∃𝑢 ∈ ran 𝐹𝑣 ∈ ran 𝐺 𝑤 = (𝑢 + 𝑣)}
4745, 46syl6sseqr 3685 . . 3 (𝜑 → ran (𝐹𝑓 + 𝐺) ⊆ ran (𝑢 ∈ ran 𝐹, 𝑣 ∈ ran 𝐺 ↦ (𝑢 + 𝑣)))
48 ssfi 8221 . . 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 694 . 2 (𝜑 → ran (𝐹𝑓 + 𝐺) ∈ Fin)
50 frn 6091 . . . . . . . 8 ((𝐹𝑓 + 𝐺):ℝ⟶ℝ → ran (𝐹𝑓 + 𝐺) ⊆ ℝ)
5112, 50syl 17 . . . . . . 7 (𝜑 → ran (𝐹𝑓 + 𝐺) ⊆ ℝ)
5251ssdifssd 3781 . . . . . 6 (𝜑 → (ran (𝐹𝑓 + 𝐺) ∖ {0}) ⊆ ℝ)
5352sselda 3636 . . . . 5 ((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) → 𝑦 ∈ ℝ)
5453recnd 10106 . . . 4 ((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) → 𝑦 ∈ ℂ)
553, 6i1faddlem 23505 . . . 4 ((𝜑𝑦 ∈ ℂ) → ((𝐹𝑓 + 𝐺) “ {𝑦}) = 𝑧 ∈ ran 𝐺((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧})))
5654, 55syldan 486 . . 3 ((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) → ((𝐹𝑓 + 𝐺) “ {𝑦}) = 𝑧 ∈ ran 𝐺((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧})))
5716adantr 480 . . . 4 ((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) → ran 𝐺 ∈ Fin)
583ad2antrr 762 . . . . . . . 8 (((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝐹 ∈ dom ∫1)
59 i1fmbf 23487 . . . . . . . 8 (𝐹 ∈ dom ∫1𝐹 ∈ MblFn)
6058, 59syl 17 . . . . . . 7 (((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝐹 ∈ MblFn)
615ad2antrr 762 . . . . . . 7 (((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝐹:ℝ⟶ℝ)
6212ad2antrr 762 . . . . . . . . . 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 3765 . . . . . . . . . 10 (𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0}) → 𝑦 ∈ ran (𝐹𝑓 + 𝐺))
6564ad2antlr 763 . . . . . . . . 9 (((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝑦 ∈ ran (𝐹𝑓 + 𝐺))
6663, 65sseldd 3637 . . . . . . . 8 (((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝑦 ∈ ℝ)
678adantr 480 . . . . . . . . . 10 ((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) → 𝐺:ℝ⟶ℝ)
68 frn 6091 . . . . . . . . . 10 (𝐺:ℝ⟶ℝ → ran 𝐺 ⊆ ℝ)
6967, 68syl 17 . . . . . . . . 9 ((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) → ran 𝐺 ⊆ ℝ)
7069sselda 3636 . . . . . . . 8 (((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝑧 ∈ ℝ)
7166, 70resubcld 10496 . . . . . . 7 (((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (𝑦𝑧) ∈ ℝ)
72 mbfimasn 23446 . . . . . . 7 ((𝐹 ∈ MblFn ∧ 𝐹:ℝ⟶ℝ ∧ (𝑦𝑧) ∈ ℝ) → (𝐹 “ {(𝑦𝑧)}) ∈ dom vol)
7360, 61, 71, 72syl3anc 1366 . . . . . 6 (((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (𝐹 “ {(𝑦𝑧)}) ∈ dom vol)
746ad2antrr 762 . . . . . . . 8 (((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝐺 ∈ dom ∫1)
75 i1fmbf 23487 . . . . . . . 8 (𝐺 ∈ dom ∫1𝐺 ∈ MblFn)
7674, 75syl 17 . . . . . . 7 (((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝐺 ∈ MblFn)
778ad2antrr 762 . . . . . . 7 (((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → 𝐺:ℝ⟶ℝ)
78 mbfimasn 23446 . . . . . . 7 ((𝐺 ∈ MblFn ∧ 𝐺:ℝ⟶ℝ ∧ 𝑧 ∈ ℝ) → (𝐺 “ {𝑧}) ∈ dom vol)
7976, 77, 70, 78syl3anc 1366 . . . . . 6 (((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (𝐺 “ {𝑧}) ∈ dom vol)
80 inmbl 23356 . . . . . 6 (((𝐹 “ {(𝑦𝑧)}) ∈ dom vol ∧ (𝐺 “ {𝑧}) ∈ dom vol) → ((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧})) ∈ dom vol)
8173, 79, 80syl2anc 694 . . . . 5 (((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → ((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧})) ∈ dom vol)
8281ralrimiva 2995 . . . 4 ((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) → ∀𝑧 ∈ ran 𝐺((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧})) ∈ dom vol)
83 finiunmbl 23358 . . . 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 694 . . 3 ((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) → 𝑧 ∈ ran 𝐺((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧})) ∈ dom vol)
8556, 84eqeltrd 2730 . 2 ((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) → ((𝐹𝑓 + 𝐺) “ {𝑦}) ∈ dom vol)
86 mblvol 23344 . . . 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 23345 . . . . 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 3866 . . . . . . . . 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 753 . . . . . . . . 9 (((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺𝑧 = 0)) → (𝐹 “ {(𝑦𝑧)}) ∈ dom vol)
93 mblss 23345 . . . . . . . . 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 23344 . . . . . . . . . 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 811 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺𝑧 = 0)) → 𝑧 = 0)
9897oveq2d 6706 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺𝑧 = 0)) → (𝑦𝑧) = (𝑦 − 0))
9954adantr 480 . . . . . . . . . . . . . . 15 (((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺𝑧 = 0)) → 𝑦 ∈ ℂ)
10099subid1d 10419 . . . . . . . . . . . . . 14 (((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺𝑧 = 0)) → (𝑦 − 0) = 𝑦)
10198, 100eqtrd 2685 . . . . . . . . . . . . 13 (((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺𝑧 = 0)) → (𝑦𝑧) = 𝑦)
102101sneqd 4222 . . . . . . . . . . . 12 (((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺𝑧 = 0)) → {(𝑦𝑧)} = {𝑦})
103102imaeq2d 5501 . . . . . . . . . . 11 (((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺𝑧 = 0)) → (𝐹 “ {(𝑦𝑧)}) = (𝐹 “ {𝑦}))
104103fveq2d 6233 . . . . . . . . . 10 (((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺𝑧 = 0)) → (vol‘(𝐹 “ {(𝑦𝑧)})) = (vol‘(𝐹 “ {𝑦})))
105 i1fima2sn 23492 . . . . . . . . . . . 12 ((𝐹 ∈ dom ∫1𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) → (vol‘(𝐹 “ {𝑦})) ∈ ℝ)
1063, 105sylan 487 . . . . . . . . . . 11 ((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) → (vol‘(𝐹 “ {𝑦})) ∈ ℝ)
107106adantr 480 . . . . . . . . . 10 (((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺𝑧 = 0)) → (vol‘(𝐹 “ {𝑦})) ∈ ℝ)
108104, 107eqeltrd 2730 . . . . . . . . 9 (((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺𝑧 = 0)) → (vol‘(𝐹 “ {(𝑦𝑧)})) ∈ ℝ)
10996, 108eqeltrrd 2731 . . . . . . . 8 (((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) ∧ (𝑧 ∈ ran 𝐺𝑧 = 0)) → (vol*‘(𝐹 “ {(𝑦𝑧)})) ∈ ℝ)
110 ovolsscl 23300 . . . . . . . 8 ((((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧})) ⊆ (𝐹 “ {(𝑦𝑧)}) ∧ (𝐹 “ {(𝑦𝑧)}) ⊆ ℝ ∧ (vol*‘(𝐹 “ {(𝑦𝑧)})) ∈ ℝ) → (vol*‘((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ)
11191, 94, 109, 110syl3anc 1366 . . . . . . 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 4350 . . . . . . . 8 (𝑧 ∈ (ran 𝐺 ∖ {0}) ↔ (𝑧 ∈ ran 𝐺𝑧 ≠ 0))
114 inss2 3867 . . . . . . . . . 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 3765 . . . . . . . . . 10 (𝑧 ∈ (ran 𝐺 ∖ {0}) → 𝑧 ∈ ran 𝐺)
117 mblss 23345 . . . . . . . . . . 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 490 . . . . . . . . 9 (((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (𝐺 “ {𝑧}) ⊆ ℝ)
120 i1fima 23490 . . . . . . . . . . . . 13 (𝐺 ∈ dom ∫1 → (𝐺 “ {𝑧}) ∈ dom vol)
1216, 120syl 17 . . . . . . . . . . . 12 (𝜑 → (𝐺 “ {𝑧}) ∈ dom vol)
122121ad2antrr 762 . . . . . . . . . . 11 (((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (𝐺 “ {𝑧}) ∈ dom vol)
123 mblvol 23344 . . . . . . . . . . 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 480 . . . . . . . . . . 11 ((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) → 𝐺 ∈ dom ∫1)
126 i1fima2sn 23492 . . . . . . . . . . 11 ((𝐺 ∈ dom ∫1𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol‘(𝐺 “ {𝑧})) ∈ ℝ)
127125, 126sylan 487 . . . . . . . . . 10 (((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol‘(𝐺 “ {𝑧})) ∈ ℝ)
128124, 127eqeltrrd 2731 . . . . . . . . 9 (((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol*‘(𝐺 “ {𝑧})) ∈ ℝ)
129 ovolsscl 23300 . . . . . . . . 9 ((((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧})) ⊆ (𝐺 “ {𝑧}) ∧ (𝐺 “ {𝑧}) ⊆ ℝ ∧ (vol*‘(𝐺 “ {𝑧})) ∈ ℝ) → (vol*‘((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ)
130115, 119, 128, 129syl3anc 1366 . . . . . . . 8 (((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) ∧ 𝑧 ∈ (ran 𝐺 ∖ {0})) → (vol*‘((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ)
131113, 130sylan2br 492 . . . . . . 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 2909 . . . . 5 (((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (vol*‘((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ)
13457, 133fsumrecl 14509 . . . 4 ((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) → Σ𝑧 ∈ ran 𝐺(vol*‘((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ)
13556fveq2d 6233 . . . . 5 ((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) → (vol*‘((𝐹𝑓 + 𝐺) “ {𝑦})) = (vol*‘ 𝑧 ∈ ran 𝐺((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))))
136114, 118syl5ss 3647 . . . . . . . 8 (((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → ((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧})) ⊆ ℝ)
137136, 133jca 553 . . . . . . 7 (((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) ∧ 𝑧 ∈ ran 𝐺) → (((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧})) ⊆ ℝ ∧ (vol*‘((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ))
138137ralrimiva 2995 . . . . . 6 ((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) → ∀𝑧 ∈ ran 𝐺(((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧})) ⊆ ℝ ∧ (vol*‘((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))) ∈ ℝ))
139 ovolfiniun 23315 . . . . . 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 694 . . . . 5 ((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) → (vol*‘ 𝑧 ∈ ran 𝐺((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))) ≤ Σ𝑧 ∈ ran 𝐺(vol*‘((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))))
141135, 140eqbrtrd 4707 . . . 4 ((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) → (vol*‘((𝐹𝑓 + 𝐺) “ {𝑦})) ≤ Σ𝑧 ∈ ran 𝐺(vol*‘((𝐹 “ {(𝑦𝑧)}) ∩ (𝐺 “ {𝑧}))))
142 ovollecl 23297 . . . 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 1366 . . 3 ((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) → (vol*‘((𝐹𝑓 + 𝐺) “ {𝑦})) ∈ ℝ)
14487, 143eqeltrd 2730 . 2 ((𝜑𝑦 ∈ (ran (𝐹𝑓 + 𝐺) ∖ {0})) → (vol‘((𝐹𝑓 + 𝐺) “ {𝑦})) ∈ ℝ)
14512, 49, 85, 144i1fd 23493 1 (𝜑 → (𝐹𝑓 + 𝐺) ∈ dom ∫1)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  {cab 2637  wne 2823  wral 2941  wrex 2942  Vcvv 3231  cdif 3604  cin 3606  wss 3607  {csn 4210   ciun 4552   class class class wbr 4685   × cxp 5141  ccnv 5142  dom cdm 5143  ran crn 5144  cima 5146   Fn wfn 5921  wf 5922  ontowfo 5924  cfv 5926  (class class class)co 6690  cmpt2 6692  𝑓 cof 6937  Fincfn 7997  cc 9972  cr 9973  0cc0 9974   + caddc 9977  cle 10113  cmin 10304  Σcsu 14460  vol*covol 23277  volcvol 23278  MblFncmbf 23428  1citg1 23429
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-sup 8389  df-inf 8390  df-oi 8456  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-z 11416  df-uz 11726  df-q 11827  df-rp 11871  df-xadd 11985  df-ioo 12217  df-ico 12219  df-icc 12220  df-fz 12365  df-fzo 12505  df-fl 12633  df-seq 12842  df-exp 12901  df-hash 13158  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-clim 14263  df-sum 14461  df-xmet 19787  df-met 19788  df-ovol 23279  df-vol 23280  df-mbf 23433  df-itg1 23434
This theorem is referenced by:  itg1addlem4  23511  i1fsub  23520  itg2splitlem  23560  itg2split  23561  itg2addlem  23570  itg2addnc  33594  ftc1anclem3  33617  ftc1anclem5  33619  ftc1anclem8  33622
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