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Theorem itg2cnlem1 25796
Description: Lemma for itgcn 25880. (Contributed by Mario Carneiro, 30-Aug-2014.)
Hypotheses
Ref Expression
itg2cn.1 (𝜑𝐹:ℝ⟶(0[,)+∞))
itg2cn.2 (𝜑𝐹 ∈ MblFn)
itg2cn.3 (𝜑 → (∫2𝐹) ∈ ℝ)
Assertion
Ref Expression
itg2cnlem1 (𝜑 → sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)))), ℝ*, < ) = (∫2𝐹))
Distinct variable groups:   𝑥,𝑛,𝐹   𝜑,𝑛,𝑥

Proof of Theorem itg2cnlem1
Dummy variables 𝑚 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fvex 6919 . . . . . . . . . 10 (𝐹𝑥) ∈ V
2 c0ex 11255 . . . . . . . . . 10 0 ∈ V
31, 2ifex 4576 . . . . . . . . 9 if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0) ∈ V
4 eqid 2737 . . . . . . . . . 10 (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)) = (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0))
54fvmpt2 7027 . . . . . . . . 9 ((𝑥 ∈ ℝ ∧ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0) ∈ V) → ((𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0))‘𝑥) = if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0))
63, 5mpan2 691 . . . . . . . 8 (𝑥 ∈ ℝ → ((𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0))‘𝑥) = if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0))
76mpteq2dv 5244 . . . . . . 7 (𝑥 ∈ ℝ → (𝑛 ∈ ℕ ↦ ((𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0))‘𝑥)) = (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)))
87rneqd 5949 . . . . . 6 (𝑥 ∈ ℝ → ran (𝑛 ∈ ℕ ↦ ((𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0))‘𝑥)) = ran (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)))
98supeq1d 9486 . . . . 5 (𝑥 ∈ ℝ → sup(ran (𝑛 ∈ ℕ ↦ ((𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0))‘𝑥)), ℝ, < ) = sup(ran (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)), ℝ, < ))
109mpteq2ia 5245 . . . 4 (𝑥 ∈ ℝ ↦ sup(ran (𝑛 ∈ ℕ ↦ ((𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0))‘𝑥)), ℝ, < )) = (𝑥 ∈ ℝ ↦ sup(ran (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)), ℝ, < ))
11 nfcv 2905 . . . . 5 𝑦sup(ran (𝑛 ∈ ℕ ↦ ((𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0))‘𝑥)), ℝ, < )
12 nfcv 2905 . . . . . . . 8 𝑥
13 nfmpt1 5250 . . . . . . . . . . 11 𝑥(𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0))
1412, 13nfmpt 5249 . . . . . . . . . 10 𝑥(𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)))
15 nfcv 2905 . . . . . . . . . 10 𝑥𝑚
1614, 15nffv 6916 . . . . . . . . 9 𝑥((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)))‘𝑚)
17 nfcv 2905 . . . . . . . . 9 𝑥𝑦
1816, 17nffv 6916 . . . . . . . 8 𝑥(((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)))‘𝑚)‘𝑦)
1912, 18nfmpt 5249 . . . . . . 7 𝑥(𝑚 ∈ ℕ ↦ (((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)))‘𝑚)‘𝑦))
2019nfrn 5963 . . . . . 6 𝑥ran (𝑚 ∈ ℕ ↦ (((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)))‘𝑚)‘𝑦))
21 nfcv 2905 . . . . . 6 𝑥
22 nfcv 2905 . . . . . 6 𝑥 <
2320, 21, 22nfsup 9491 . . . . 5 𝑥sup(ran (𝑚 ∈ ℕ ↦ (((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)))‘𝑚)‘𝑦)), ℝ, < )
24 fveq2 6906 . . . . . . . . 9 (𝑥 = 𝑦 → ((𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0))‘𝑥) = ((𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0))‘𝑦))
2524mpteq2dv 5244 . . . . . . . 8 (𝑥 = 𝑦 → (𝑛 ∈ ℕ ↦ ((𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0))‘𝑥)) = (𝑛 ∈ ℕ ↦ ((𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0))‘𝑦)))
26 breq2 5147 . . . . . . . . . . . . 13 (𝑛 = 𝑚 → ((𝐹𝑥) ≤ 𝑛 ↔ (𝐹𝑥) ≤ 𝑚))
2726ifbid 4549 . . . . . . . . . . . 12 (𝑛 = 𝑚 → if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0) = if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0))
2827mpteq2dv 5244 . . . . . . . . . . 11 (𝑛 = 𝑚 → (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)) = (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0)))
2928fveq1d 6908 . . . . . . . . . 10 (𝑛 = 𝑚 → ((𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0))‘𝑦) = ((𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0))‘𝑦))
3029cbvmptv 5255 . . . . . . . . 9 (𝑛 ∈ ℕ ↦ ((𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0))‘𝑦)) = (𝑚 ∈ ℕ ↦ ((𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0))‘𝑦))
31 eqid 2737 . . . . . . . . . . . 12 (𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0))) = (𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)))
32 reex 11246 . . . . . . . . . . . . 13 ℝ ∈ V
3332mptex 7243 . . . . . . . . . . . 12 (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0)) ∈ V
3428, 31, 33fvmpt 7016 . . . . . . . . . . 11 (𝑚 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)))‘𝑚) = (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0)))
3534fveq1d 6908 . . . . . . . . . 10 (𝑚 ∈ ℕ → (((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)))‘𝑚)‘𝑦) = ((𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0))‘𝑦))
3635mpteq2ia 5245 . . . . . . . . 9 (𝑚 ∈ ℕ ↦ (((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)))‘𝑚)‘𝑦)) = (𝑚 ∈ ℕ ↦ ((𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0))‘𝑦))
3730, 36eqtr4i 2768 . . . . . . . 8 (𝑛 ∈ ℕ ↦ ((𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0))‘𝑦)) = (𝑚 ∈ ℕ ↦ (((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)))‘𝑚)‘𝑦))
3825, 37eqtrdi 2793 . . . . . . 7 (𝑥 = 𝑦 → (𝑛 ∈ ℕ ↦ ((𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0))‘𝑥)) = (𝑚 ∈ ℕ ↦ (((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)))‘𝑚)‘𝑦)))
3938rneqd 5949 . . . . . 6 (𝑥 = 𝑦 → ran (𝑛 ∈ ℕ ↦ ((𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0))‘𝑥)) = ran (𝑚 ∈ ℕ ↦ (((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)))‘𝑚)‘𝑦)))
4039supeq1d 9486 . . . . 5 (𝑥 = 𝑦 → sup(ran (𝑛 ∈ ℕ ↦ ((𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0))‘𝑥)), ℝ, < ) = sup(ran (𝑚 ∈ ℕ ↦ (((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)))‘𝑚)‘𝑦)), ℝ, < ))
4111, 23, 40cbvmpt 5253 . . . 4 (𝑥 ∈ ℝ ↦ sup(ran (𝑛 ∈ ℕ ↦ ((𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0))‘𝑥)), ℝ, < )) = (𝑦 ∈ ℝ ↦ sup(ran (𝑚 ∈ ℕ ↦ (((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)))‘𝑚)‘𝑦)), ℝ, < ))
4210, 41eqtr3i 2767 . . 3 (𝑥 ∈ ℝ ↦ sup(ran (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)), ℝ, < )) = (𝑦 ∈ ℝ ↦ sup(ran (𝑚 ∈ ℕ ↦ (((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)))‘𝑚)‘𝑦)), ℝ, < ))
43 fveq2 6906 . . . . . . . 8 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
4443breq1d 5153 . . . . . . 7 (𝑥 = 𝑦 → ((𝐹𝑥) ≤ 𝑚 ↔ (𝐹𝑦) ≤ 𝑚))
4544, 43ifbieq1d 4550 . . . . . 6 (𝑥 = 𝑦 → if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0) = if((𝐹𝑦) ≤ 𝑚, (𝐹𝑦), 0))
4645cbvmptv 5255 . . . . 5 (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0)) = (𝑦 ∈ ℝ ↦ if((𝐹𝑦) ≤ 𝑚, (𝐹𝑦), 0))
4734adantl 481 . . . . 5 ((𝜑𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)))‘𝑚) = (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0)))
48 nnre 12273 . . . . . . . . . . . . 13 (𝑚 ∈ ℕ → 𝑚 ∈ ℝ)
4948ad2antlr 727 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → 𝑚 ∈ ℝ)
5049rexrd 11311 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → 𝑚 ∈ ℝ*)
51 elioopnf 13483 . . . . . . . . . . 11 (𝑚 ∈ ℝ* → ((𝐹𝑦) ∈ (𝑚(,)+∞) ↔ ((𝐹𝑦) ∈ ℝ ∧ 𝑚 < (𝐹𝑦))))
5250, 51syl 17 . . . . . . . . . 10 (((𝜑𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → ((𝐹𝑦) ∈ (𝑚(,)+∞) ↔ ((𝐹𝑦) ∈ ℝ ∧ 𝑚 < (𝐹𝑦))))
53 simpr 484 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℝ)
54 itg2cn.1 . . . . . . . . . . . . . 14 (𝜑𝐹:ℝ⟶(0[,)+∞))
5554ffnd 6737 . . . . . . . . . . . . 13 (𝜑𝐹 Fn ℝ)
5655ad2antrr 726 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → 𝐹 Fn ℝ)
57 elpreima 7078 . . . . . . . . . . . 12 (𝐹 Fn ℝ → (𝑦 ∈ (𝐹 “ (𝑚(,)+∞)) ↔ (𝑦 ∈ ℝ ∧ (𝐹𝑦) ∈ (𝑚(,)+∞))))
5856, 57syl 17 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → (𝑦 ∈ (𝐹 “ (𝑚(,)+∞)) ↔ (𝑦 ∈ ℝ ∧ (𝐹𝑦) ∈ (𝑚(,)+∞))))
5953, 58mpbirand 707 . . . . . . . . . 10 (((𝜑𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → (𝑦 ∈ (𝐹 “ (𝑚(,)+∞)) ↔ (𝐹𝑦) ∈ (𝑚(,)+∞)))
60 rge0ssre 13496 . . . . . . . . . . . . . 14 (0[,)+∞) ⊆ ℝ
61 fss 6752 . . . . . . . . . . . . . 14 ((𝐹:ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ ℝ) → 𝐹:ℝ⟶ℝ)
6254, 60, 61sylancl 586 . . . . . . . . . . . . 13 (𝜑𝐹:ℝ⟶ℝ)
6362adantr 480 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → 𝐹:ℝ⟶ℝ)
6463ffvelcdmda 7104 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → (𝐹𝑦) ∈ ℝ)
6564biantrurd 532 . . . . . . . . . 10 (((𝜑𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → (𝑚 < (𝐹𝑦) ↔ ((𝐹𝑦) ∈ ℝ ∧ 𝑚 < (𝐹𝑦))))
6652, 59, 653bitr4d 311 . . . . . . . . 9 (((𝜑𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → (𝑦 ∈ (𝐹 “ (𝑚(,)+∞)) ↔ 𝑚 < (𝐹𝑦)))
6766notbid 318 . . . . . . . 8 (((𝜑𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → (¬ 𝑦 ∈ (𝐹 “ (𝑚(,)+∞)) ↔ ¬ 𝑚 < (𝐹𝑦)))
68 eldif 3961 . . . . . . . . . 10 (𝑦 ∈ (ℝ ∖ (𝐹 “ (𝑚(,)+∞))) ↔ (𝑦 ∈ ℝ ∧ ¬ 𝑦 ∈ (𝐹 “ (𝑚(,)+∞))))
6968baib 535 . . . . . . . . 9 (𝑦 ∈ ℝ → (𝑦 ∈ (ℝ ∖ (𝐹 “ (𝑚(,)+∞))) ↔ ¬ 𝑦 ∈ (𝐹 “ (𝑚(,)+∞))))
7069adantl 481 . . . . . . . 8 (((𝜑𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → (𝑦 ∈ (ℝ ∖ (𝐹 “ (𝑚(,)+∞))) ↔ ¬ 𝑦 ∈ (𝐹 “ (𝑚(,)+∞))))
7164, 49lenltd 11407 . . . . . . . 8 (((𝜑𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → ((𝐹𝑦) ≤ 𝑚 ↔ ¬ 𝑚 < (𝐹𝑦)))
7267, 70, 713bitr4d 311 . . . . . . 7 (((𝜑𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → (𝑦 ∈ (ℝ ∖ (𝐹 “ (𝑚(,)+∞))) ↔ (𝐹𝑦) ≤ 𝑚))
7372ifbid 4549 . . . . . 6 (((𝜑𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → if(𝑦 ∈ (ℝ ∖ (𝐹 “ (𝑚(,)+∞))), (𝐹𝑦), 0) = if((𝐹𝑦) ≤ 𝑚, (𝐹𝑦), 0))
7473mpteq2dva 5242 . . . . 5 ((𝜑𝑚 ∈ ℕ) → (𝑦 ∈ ℝ ↦ if(𝑦 ∈ (ℝ ∖ (𝐹 “ (𝑚(,)+∞))), (𝐹𝑦), 0)) = (𝑦 ∈ ℝ ↦ if((𝐹𝑦) ≤ 𝑚, (𝐹𝑦), 0)))
7546, 47, 743eqtr4a 2803 . . . 4 ((𝜑𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)))‘𝑚) = (𝑦 ∈ ℝ ↦ if(𝑦 ∈ (ℝ ∖ (𝐹 “ (𝑚(,)+∞))), (𝐹𝑦), 0)))
76 difss 4136 . . . . . 6 (ℝ ∖ (𝐹 “ (𝑚(,)+∞))) ⊆ ℝ
7776a1i 11 . . . . 5 ((𝜑𝑚 ∈ ℕ) → (ℝ ∖ (𝐹 “ (𝑚(,)+∞))) ⊆ ℝ)
78 rembl 25575 . . . . . 6 ℝ ∈ dom vol
7978a1i 11 . . . . 5 ((𝜑𝑚 ∈ ℕ) → ℝ ∈ dom vol)
80 fvex 6919 . . . . . . 7 (𝐹𝑦) ∈ V
8180, 2ifex 4576 . . . . . 6 if(𝑦 ∈ (ℝ ∖ (𝐹 “ (𝑚(,)+∞))), (𝐹𝑦), 0) ∈ V
8281a1i 11 . . . . 5 (((𝜑𝑚 ∈ ℕ) ∧ 𝑦 ∈ (ℝ ∖ (𝐹 “ (𝑚(,)+∞)))) → if(𝑦 ∈ (ℝ ∖ (𝐹 “ (𝑚(,)+∞))), (𝐹𝑦), 0) ∈ V)
83 eldifn 4132 . . . . . . 7 (𝑦 ∈ (ℝ ∖ (ℝ ∖ (𝐹 “ (𝑚(,)+∞)))) → ¬ 𝑦 ∈ (ℝ ∖ (𝐹 “ (𝑚(,)+∞))))
8483adantl 481 . . . . . 6 (((𝜑𝑚 ∈ ℕ) ∧ 𝑦 ∈ (ℝ ∖ (ℝ ∖ (𝐹 “ (𝑚(,)+∞))))) → ¬ 𝑦 ∈ (ℝ ∖ (𝐹 “ (𝑚(,)+∞))))
8584iffalsed 4536 . . . . 5 (((𝜑𝑚 ∈ ℕ) ∧ 𝑦 ∈ (ℝ ∖ (ℝ ∖ (𝐹 “ (𝑚(,)+∞))))) → if(𝑦 ∈ (ℝ ∖ (𝐹 “ (𝑚(,)+∞))), (𝐹𝑦), 0) = 0)
86 iftrue 4531 . . . . . . . . 9 (𝑦 ∈ (ℝ ∖ (𝐹 “ (𝑚(,)+∞))) → if(𝑦 ∈ (ℝ ∖ (𝐹 “ (𝑚(,)+∞))), (𝐹𝑦), 0) = (𝐹𝑦))
8786mpteq2ia 5245 . . . . . . . 8 (𝑦 ∈ (ℝ ∖ (𝐹 “ (𝑚(,)+∞))) ↦ if(𝑦 ∈ (ℝ ∖ (𝐹 “ (𝑚(,)+∞))), (𝐹𝑦), 0)) = (𝑦 ∈ (ℝ ∖ (𝐹 “ (𝑚(,)+∞))) ↦ (𝐹𝑦))
88 resmpt 6055 . . . . . . . . 9 ((ℝ ∖ (𝐹 “ (𝑚(,)+∞))) ⊆ ℝ → ((𝑦 ∈ ℝ ↦ (𝐹𝑦)) ↾ (ℝ ∖ (𝐹 “ (𝑚(,)+∞)))) = (𝑦 ∈ (ℝ ∖ (𝐹 “ (𝑚(,)+∞))) ↦ (𝐹𝑦)))
8976, 88ax-mp 5 . . . . . . . 8 ((𝑦 ∈ ℝ ↦ (𝐹𝑦)) ↾ (ℝ ∖ (𝐹 “ (𝑚(,)+∞)))) = (𝑦 ∈ (ℝ ∖ (𝐹 “ (𝑚(,)+∞))) ↦ (𝐹𝑦))
9087, 89eqtr4i 2768 . . . . . . 7 (𝑦 ∈ (ℝ ∖ (𝐹 “ (𝑚(,)+∞))) ↦ if(𝑦 ∈ (ℝ ∖ (𝐹 “ (𝑚(,)+∞))), (𝐹𝑦), 0)) = ((𝑦 ∈ ℝ ↦ (𝐹𝑦)) ↾ (ℝ ∖ (𝐹 “ (𝑚(,)+∞))))
9154feqmptd 6977 . . . . . . . . 9 (𝜑𝐹 = (𝑦 ∈ ℝ ↦ (𝐹𝑦)))
92 itg2cn.2 . . . . . . . . 9 (𝜑𝐹 ∈ MblFn)
9391, 92eqeltrrd 2842 . . . . . . . 8 (𝜑 → (𝑦 ∈ ℝ ↦ (𝐹𝑦)) ∈ MblFn)
94 mbfima 25665 . . . . . . . . . 10 ((𝐹 ∈ MblFn ∧ 𝐹:ℝ⟶ℝ) → (𝐹 “ (𝑚(,)+∞)) ∈ dom vol)
9592, 62, 94syl2anc 584 . . . . . . . . 9 (𝜑 → (𝐹 “ (𝑚(,)+∞)) ∈ dom vol)
96 cmmbl 25569 . . . . . . . . 9 ((𝐹 “ (𝑚(,)+∞)) ∈ dom vol → (ℝ ∖ (𝐹 “ (𝑚(,)+∞))) ∈ dom vol)
9795, 96syl 17 . . . . . . . 8 (𝜑 → (ℝ ∖ (𝐹 “ (𝑚(,)+∞))) ∈ dom vol)
98 mbfres 25679 . . . . . . . 8 (((𝑦 ∈ ℝ ↦ (𝐹𝑦)) ∈ MblFn ∧ (ℝ ∖ (𝐹 “ (𝑚(,)+∞))) ∈ dom vol) → ((𝑦 ∈ ℝ ↦ (𝐹𝑦)) ↾ (ℝ ∖ (𝐹 “ (𝑚(,)+∞)))) ∈ MblFn)
9993, 97, 98syl2anc 584 . . . . . . 7 (𝜑 → ((𝑦 ∈ ℝ ↦ (𝐹𝑦)) ↾ (ℝ ∖ (𝐹 “ (𝑚(,)+∞)))) ∈ MblFn)
10090, 99eqeltrid 2845 . . . . . 6 (𝜑 → (𝑦 ∈ (ℝ ∖ (𝐹 “ (𝑚(,)+∞))) ↦ if(𝑦 ∈ (ℝ ∖ (𝐹 “ (𝑚(,)+∞))), (𝐹𝑦), 0)) ∈ MblFn)
101100adantr 480 . . . . 5 ((𝜑𝑚 ∈ ℕ) → (𝑦 ∈ (ℝ ∖ (𝐹 “ (𝑚(,)+∞))) ↦ if(𝑦 ∈ (ℝ ∖ (𝐹 “ (𝑚(,)+∞))), (𝐹𝑦), 0)) ∈ MblFn)
10277, 79, 82, 85, 101mbfss 25681 . . . 4 ((𝜑𝑚 ∈ ℕ) → (𝑦 ∈ ℝ ↦ if(𝑦 ∈ (ℝ ∖ (𝐹 “ (𝑚(,)+∞))), (𝐹𝑦), 0)) ∈ MblFn)
10375, 102eqeltrd 2841 . . 3 ((𝜑𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)))‘𝑚) ∈ MblFn)
10454ffvelcdmda 7104 . . . . . 6 ((𝜑𝑥 ∈ ℝ) → (𝐹𝑥) ∈ (0[,)+∞))
105 0e0icopnf 13498 . . . . . 6 0 ∈ (0[,)+∞)
106 ifcl 4571 . . . . . 6 (((𝐹𝑥) ∈ (0[,)+∞) ∧ 0 ∈ (0[,)+∞)) → if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0) ∈ (0[,)+∞))
107104, 105, 106sylancl 586 . . . . 5 ((𝜑𝑥 ∈ ℝ) → if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0) ∈ (0[,)+∞))
108107adantlr 715 . . . 4 (((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0) ∈ (0[,)+∞))
10947, 108fmpt3d 7136 . . 3 ((𝜑𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)))‘𝑚):ℝ⟶(0[,)+∞))
110 elrege0 13494 . . . . . . . . . . . . 13 ((𝐹𝑥) ∈ (0[,)+∞) ↔ ((𝐹𝑥) ∈ ℝ ∧ 0 ≤ (𝐹𝑥)))
111104, 110sylib 218 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ ℝ) → ((𝐹𝑥) ∈ ℝ ∧ 0 ≤ (𝐹𝑥)))
112111simpld 494 . . . . . . . . . . 11 ((𝜑𝑥 ∈ ℝ) → (𝐹𝑥) ∈ ℝ)
113112adantlr 715 . . . . . . . . . 10 (((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐹𝑥) ∈ ℝ)
114113adantr 480 . . . . . . . . 9 ((((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐹𝑥) ≤ 𝑚) → (𝐹𝑥) ∈ ℝ)
115114leidd 11829 . . . . . . . 8 ((((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐹𝑥) ≤ 𝑚) → (𝐹𝑥) ≤ (𝐹𝑥))
116 iftrue 4531 . . . . . . . . 9 ((𝐹𝑥) ≤ 𝑚 → if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0) = (𝐹𝑥))
117116adantl 481 . . . . . . . 8 ((((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐹𝑥) ≤ 𝑚) → if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0) = (𝐹𝑥))
11848ad3antlr 731 . . . . . . . . . 10 ((((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐹𝑥) ≤ 𝑚) → 𝑚 ∈ ℝ)
119 peano2re 11434 . . . . . . . . . . 11 (𝑚 ∈ ℝ → (𝑚 + 1) ∈ ℝ)
120118, 119syl 17 . . . . . . . . . 10 ((((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐹𝑥) ≤ 𝑚) → (𝑚 + 1) ∈ ℝ)
121 simpr 484 . . . . . . . . . 10 ((((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐹𝑥) ≤ 𝑚) → (𝐹𝑥) ≤ 𝑚)
122118lep1d 12199 . . . . . . . . . 10 ((((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐹𝑥) ≤ 𝑚) → 𝑚 ≤ (𝑚 + 1))
123114, 118, 120, 121, 122letrd 11418 . . . . . . . . 9 ((((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐹𝑥) ≤ 𝑚) → (𝐹𝑥) ≤ (𝑚 + 1))
124123iftrued 4533 . . . . . . . 8 ((((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐹𝑥) ≤ 𝑚) → if((𝐹𝑥) ≤ (𝑚 + 1), (𝐹𝑥), 0) = (𝐹𝑥))
125115, 117, 1243brtr4d 5175 . . . . . . 7 ((((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐹𝑥) ≤ 𝑚) → if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0) ≤ if((𝐹𝑥) ≤ (𝑚 + 1), (𝐹𝑥), 0))
126 iffalse 4534 . . . . . . . . 9 (¬ (𝐹𝑥) ≤ 𝑚 → if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0) = 0)
127126adantl 481 . . . . . . . 8 ((((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ ¬ (𝐹𝑥) ≤ 𝑚) → if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0) = 0)
128111simprd 495 . . . . . . . . . . 11 ((𝜑𝑥 ∈ ℝ) → 0 ≤ (𝐹𝑥))
129 0le0 12367 . . . . . . . . . . 11 0 ≤ 0
130 breq2 5147 . . . . . . . . . . . 12 ((𝐹𝑥) = if((𝐹𝑥) ≤ (𝑚 + 1), (𝐹𝑥), 0) → (0 ≤ (𝐹𝑥) ↔ 0 ≤ if((𝐹𝑥) ≤ (𝑚 + 1), (𝐹𝑥), 0)))
131 breq2 5147 . . . . . . . . . . . 12 (0 = if((𝐹𝑥) ≤ (𝑚 + 1), (𝐹𝑥), 0) → (0 ≤ 0 ↔ 0 ≤ if((𝐹𝑥) ≤ (𝑚 + 1), (𝐹𝑥), 0)))
132130, 131ifboth 4565 . . . . . . . . . . 11 ((0 ≤ (𝐹𝑥) ∧ 0 ≤ 0) → 0 ≤ if((𝐹𝑥) ≤ (𝑚 + 1), (𝐹𝑥), 0))
133128, 129, 132sylancl 586 . . . . . . . . . 10 ((𝜑𝑥 ∈ ℝ) → 0 ≤ if((𝐹𝑥) ≤ (𝑚 + 1), (𝐹𝑥), 0))
134133adantlr 715 . . . . . . . . 9 (((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 0 ≤ if((𝐹𝑥) ≤ (𝑚 + 1), (𝐹𝑥), 0))
135134adantr 480 . . . . . . . 8 ((((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ ¬ (𝐹𝑥) ≤ 𝑚) → 0 ≤ if((𝐹𝑥) ≤ (𝑚 + 1), (𝐹𝑥), 0))
136127, 135eqbrtrd 5165 . . . . . . 7 ((((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ ¬ (𝐹𝑥) ≤ 𝑚) → if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0) ≤ if((𝐹𝑥) ≤ (𝑚 + 1), (𝐹𝑥), 0))
137125, 136pm2.61dan 813 . . . . . 6 (((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0) ≤ if((𝐹𝑥) ≤ (𝑚 + 1), (𝐹𝑥), 0))
138137ralrimiva 3146 . . . . 5 ((𝜑𝑚 ∈ ℕ) → ∀𝑥 ∈ ℝ if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0) ≤ if((𝐹𝑥) ≤ (𝑚 + 1), (𝐹𝑥), 0))
1391, 2ifex 4576 . . . . . . 7 if((𝐹𝑥) ≤ (𝑚 + 1), (𝐹𝑥), 0) ∈ V
140139a1i 11 . . . . . 6 (((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → if((𝐹𝑥) ≤ (𝑚 + 1), (𝐹𝑥), 0) ∈ V)
141 eqidd 2738 . . . . . 6 ((𝜑𝑚 ∈ ℕ) → (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0)) = (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0)))
142 eqidd 2738 . . . . . 6 ((𝜑𝑚 ∈ ℕ) → (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ (𝑚 + 1), (𝐹𝑥), 0)) = (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ (𝑚 + 1), (𝐹𝑥), 0)))
14379, 108, 140, 141, 142ofrfval2 7718 . . . . 5 ((𝜑𝑚 ∈ ℕ) → ((𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0)) ∘r ≤ (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ (𝑚 + 1), (𝐹𝑥), 0)) ↔ ∀𝑥 ∈ ℝ if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0) ≤ if((𝐹𝑥) ≤ (𝑚 + 1), (𝐹𝑥), 0)))
144138, 143mpbird 257 . . . 4 ((𝜑𝑚 ∈ ℕ) → (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0)) ∘r ≤ (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ (𝑚 + 1), (𝐹𝑥), 0)))
145 peano2nn 12278 . . . . . 6 (𝑚 ∈ ℕ → (𝑚 + 1) ∈ ℕ)
146145adantl 481 . . . . 5 ((𝜑𝑚 ∈ ℕ) → (𝑚 + 1) ∈ ℕ)
147 breq2 5147 . . . . . . . 8 (𝑛 = (𝑚 + 1) → ((𝐹𝑥) ≤ 𝑛 ↔ (𝐹𝑥) ≤ (𝑚 + 1)))
148147ifbid 4549 . . . . . . 7 (𝑛 = (𝑚 + 1) → if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0) = if((𝐹𝑥) ≤ (𝑚 + 1), (𝐹𝑥), 0))
149148mpteq2dv 5244 . . . . . 6 (𝑛 = (𝑚 + 1) → (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)) = (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ (𝑚 + 1), (𝐹𝑥), 0)))
15032mptex 7243 . . . . . 6 (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ (𝑚 + 1), (𝐹𝑥), 0)) ∈ V
151149, 31, 150fvmpt 7016 . . . . 5 ((𝑚 + 1) ∈ ℕ → ((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)))‘(𝑚 + 1)) = (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ (𝑚 + 1), (𝐹𝑥), 0)))
152146, 151syl 17 . . . 4 ((𝜑𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)))‘(𝑚 + 1)) = (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ (𝑚 + 1), (𝐹𝑥), 0)))
153144, 47, 1523brtr4d 5175 . . 3 ((𝜑𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)))‘𝑚) ∘r ≤ ((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)))‘(𝑚 + 1)))
15462ffvelcdmda 7104 . . . 4 ((𝜑𝑦 ∈ ℝ) → (𝐹𝑦) ∈ ℝ)
15534adantl 481 . . . . . . 7 (((𝜑𝑦 ∈ ℝ) ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)))‘𝑚) = (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0)))
156155fveq1d 6908 . . . . . 6 (((𝜑𝑦 ∈ ℝ) ∧ 𝑚 ∈ ℕ) → (((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)))‘𝑚)‘𝑦) = ((𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0))‘𝑦))
157112leidd 11829 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ ℝ) → (𝐹𝑥) ≤ (𝐹𝑥))
158 breq1 5146 . . . . . . . . . . . . . 14 ((𝐹𝑥) = if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0) → ((𝐹𝑥) ≤ (𝐹𝑥) ↔ if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0) ≤ (𝐹𝑥)))
159 breq1 5146 . . . . . . . . . . . . . 14 (0 = if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0) → (0 ≤ (𝐹𝑥) ↔ if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0) ≤ (𝐹𝑥)))
160158, 159ifboth 4565 . . . . . . . . . . . . 13 (((𝐹𝑥) ≤ (𝐹𝑥) ∧ 0 ≤ (𝐹𝑥)) → if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0) ≤ (𝐹𝑥))
161157, 128, 160syl2anc 584 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ ℝ) → if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0) ≤ (𝐹𝑥))
162161adantlr 715 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0) ≤ (𝐹𝑥))
163162ralrimiva 3146 . . . . . . . . . 10 ((𝜑𝑚 ∈ ℕ) → ∀𝑥 ∈ ℝ if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0) ≤ (𝐹𝑥))
16432a1i 11 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ) → ℝ ∈ V)
1651, 2ifex 4576 . . . . . . . . . . . 12 if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0) ∈ V
166165a1i 11 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0) ∈ V)
16754feqmptd 6977 . . . . . . . . . . . 12 (𝜑𝐹 = (𝑥 ∈ ℝ ↦ (𝐹𝑥)))
168167adantr 480 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ) → 𝐹 = (𝑥 ∈ ℝ ↦ (𝐹𝑥)))
169164, 166, 113, 141, 168ofrfval2 7718 . . . . . . . . . 10 ((𝜑𝑚 ∈ ℕ) → ((𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0)) ∘r𝐹 ↔ ∀𝑥 ∈ ℝ if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0) ≤ (𝐹𝑥)))
170163, 169mpbird 257 . . . . . . . . 9 ((𝜑𝑚 ∈ ℕ) → (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0)) ∘r𝐹)
171166fmpttd 7135 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ) → (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0)):ℝ⟶V)
172171ffnd 6737 . . . . . . . . . 10 ((𝜑𝑚 ∈ ℕ) → (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0)) Fn ℝ)
17355adantr 480 . . . . . . . . . 10 ((𝜑𝑚 ∈ ℕ) → 𝐹 Fn ℝ)
174 inidm 4227 . . . . . . . . . 10 (ℝ ∩ ℝ) = ℝ
175 eqidd 2738 . . . . . . . . . 10 (((𝜑𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → ((𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0))‘𝑦) = ((𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0))‘𝑦))
176 eqidd 2738 . . . . . . . . . 10 (((𝜑𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → (𝐹𝑦) = (𝐹𝑦))
177172, 173, 164, 164, 174, 175, 176ofrfval 7707 . . . . . . . . 9 ((𝜑𝑚 ∈ ℕ) → ((𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0)) ∘r𝐹 ↔ ∀𝑦 ∈ ℝ ((𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0))‘𝑦) ≤ (𝐹𝑦)))
178170, 177mpbid 232 . . . . . . . 8 ((𝜑𝑚 ∈ ℕ) → ∀𝑦 ∈ ℝ ((𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0))‘𝑦) ≤ (𝐹𝑦))
179178r19.21bi 3251 . . . . . . 7 (((𝜑𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → ((𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0))‘𝑦) ≤ (𝐹𝑦))
180179an32s 652 . . . . . 6 (((𝜑𝑦 ∈ ℝ) ∧ 𝑚 ∈ ℕ) → ((𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0))‘𝑦) ≤ (𝐹𝑦))
181156, 180eqbrtrd 5165 . . . . 5 (((𝜑𝑦 ∈ ℝ) ∧ 𝑚 ∈ ℕ) → (((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)))‘𝑚)‘𝑦) ≤ (𝐹𝑦))
182181ralrimiva 3146 . . . 4 ((𝜑𝑦 ∈ ℝ) → ∀𝑚 ∈ ℕ (((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)))‘𝑚)‘𝑦) ≤ (𝐹𝑦))
183 brralrspcev 5203 . . . 4 (((𝐹𝑦) ∈ ℝ ∧ ∀𝑚 ∈ ℕ (((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)))‘𝑚)‘𝑦) ≤ (𝐹𝑦)) → ∃𝑧 ∈ ℝ ∀𝑚 ∈ ℕ (((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)))‘𝑚)‘𝑦) ≤ 𝑧)
184154, 182, 183syl2anc 584 . . 3 ((𝜑𝑦 ∈ ℝ) → ∃𝑧 ∈ ℝ ∀𝑚 ∈ ℕ (((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)))‘𝑚)‘𝑦) ≤ 𝑧)
18528fveq2d 6910 . . . . . . 7 (𝑛 = 𝑚 → (∫2‘(𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0))) = (∫2‘(𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0))))
186185cbvmptv 5255 . . . . . 6 (𝑛 ∈ ℕ ↦ (∫2‘(𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)))) = (𝑚 ∈ ℕ ↦ (∫2‘(𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0))))
18734fveq2d 6910 . . . . . . 7 (𝑚 ∈ ℕ → (∫2‘((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)))‘𝑚)) = (∫2‘(𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0))))
188187mpteq2ia 5245 . . . . . 6 (𝑚 ∈ ℕ ↦ (∫2‘((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)))‘𝑚))) = (𝑚 ∈ ℕ ↦ (∫2‘(𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0))))
189186, 188eqtr4i 2768 . . . . 5 (𝑛 ∈ ℕ ↦ (∫2‘(𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)))) = (𝑚 ∈ ℕ ↦ (∫2‘((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)))‘𝑚)))
190189rneqi 5948 . . . 4 ran (𝑛 ∈ ℕ ↦ (∫2‘(𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)))) = ran (𝑚 ∈ ℕ ↦ (∫2‘((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)))‘𝑚)))
191190supeq1i 9487 . . 3 sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)))), ℝ*, < ) = sup(ran (𝑚 ∈ ℕ ↦ (∫2‘((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)))‘𝑚))), ℝ*, < )
19242, 103, 109, 153, 184, 191itg2mono 25788 . 2 (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ sup(ran (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)), ℝ, < ))) = sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)))), ℝ*, < ))
193 eqid 2737 . . . . . . . . . . . 12 (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)) = (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0))
19427, 193, 165fvmpt 7016 . . . . . . . . . . 11 (𝑚 ∈ ℕ → ((𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0))‘𝑚) = if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0))
195194adantl 481 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ) ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0))‘𝑚) = if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0))
196161adantr 480 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ) ∧ 𝑚 ∈ ℕ) → if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0) ≤ (𝐹𝑥))
197195, 196eqbrtrd 5165 . . . . . . . . 9 (((𝜑𝑥 ∈ ℝ) ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0))‘𝑚) ≤ (𝐹𝑥))
198197ralrimiva 3146 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ) → ∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0))‘𝑚) ≤ (𝐹𝑥))
1993a1i 11 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℕ) → if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0) ∈ V)
200199fmpttd 7135 . . . . . . . . . 10 ((𝜑𝑥 ∈ ℝ) → (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)):ℕ⟶V)
201200ffnd 6737 . . . . . . . . 9 ((𝜑𝑥 ∈ ℝ) → (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)) Fn ℕ)
202 breq1 5146 . . . . . . . . . 10 (𝑤 = ((𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0))‘𝑚) → (𝑤 ≤ (𝐹𝑥) ↔ ((𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0))‘𝑚) ≤ (𝐹𝑥)))
203202ralrn 7108 . . . . . . . . 9 ((𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)) Fn ℕ → (∀𝑤 ∈ ran (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0))𝑤 ≤ (𝐹𝑥) ↔ ∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0))‘𝑚) ≤ (𝐹𝑥)))
204201, 203syl 17 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ) → (∀𝑤 ∈ ran (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0))𝑤 ≤ (𝐹𝑥) ↔ ∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0))‘𝑚) ≤ (𝐹𝑥)))
205198, 204mpbird 257 . . . . . . 7 ((𝜑𝑥 ∈ ℝ) → ∀𝑤 ∈ ran (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0))𝑤 ≤ (𝐹𝑥))
206112adantr 480 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (𝐹𝑥) ∈ ℝ)
207 0re 11263 . . . . . . . . . . 11 0 ∈ ℝ
208 ifcl 4571 . . . . . . . . . . 11 (((𝐹𝑥) ∈ ℝ ∧ 0 ∈ ℝ) → if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0) ∈ ℝ)
209206, 207, 208sylancl 586 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℕ) → if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0) ∈ ℝ)
210209fmpttd 7135 . . . . . . . . 9 ((𝜑𝑥 ∈ ℝ) → (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)):ℕ⟶ℝ)
211210frnd 6744 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ) → ran (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)) ⊆ ℝ)
212 1nn 12277 . . . . . . . . . 10 1 ∈ ℕ
213193, 209dmmptd 6713 . . . . . . . . . 10 ((𝜑𝑥 ∈ ℝ) → dom (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)) = ℕ)
214212, 213eleqtrrid 2848 . . . . . . . . 9 ((𝜑𝑥 ∈ ℝ) → 1 ∈ dom (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)))
215 n0i 4340 . . . . . . . . . 10 (1 ∈ dom (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)) → ¬ dom (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)) = ∅)
216 dm0rn0 5935 . . . . . . . . . . 11 (dom (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)) = ∅ ↔ ran (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)) = ∅)
217216necon3bbii 2988 . . . . . . . . . 10 (¬ dom (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)) = ∅ ↔ ran (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)) ≠ ∅)
218215, 217sylib 218 . . . . . . . . 9 (1 ∈ dom (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)) → ran (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)) ≠ ∅)
219214, 218syl 17 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ) → ran (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)) ≠ ∅)
220 brralrspcev 5203 . . . . . . . . 9 (((𝐹𝑥) ∈ ℝ ∧ ∀𝑤 ∈ ran (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0))𝑤 ≤ (𝐹𝑥)) → ∃𝑧 ∈ ℝ ∀𝑤 ∈ ran (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0))𝑤𝑧)
221112, 205, 220syl2anc 584 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ) → ∃𝑧 ∈ ℝ ∀𝑤 ∈ ran (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0))𝑤𝑧)
222 suprleub 12234 . . . . . . . 8 (((ran (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)) ⊆ ℝ ∧ ran (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)) ≠ ∅ ∧ ∃𝑧 ∈ ℝ ∀𝑤 ∈ ran (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0))𝑤𝑧) ∧ (𝐹𝑥) ∈ ℝ) → (sup(ran (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)), ℝ, < ) ≤ (𝐹𝑥) ↔ ∀𝑤 ∈ ran (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0))𝑤 ≤ (𝐹𝑥)))
223211, 219, 221, 112, 222syl31anc 1375 . . . . . . 7 ((𝜑𝑥 ∈ ℝ) → (sup(ran (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)), ℝ, < ) ≤ (𝐹𝑥) ↔ ∀𝑤 ∈ ran (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0))𝑤 ≤ (𝐹𝑥)))
224205, 223mpbird 257 . . . . . 6 ((𝜑𝑥 ∈ ℝ) → sup(ran (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)), ℝ, < ) ≤ (𝐹𝑥))
225 arch 12523 . . . . . . . . 9 ((𝐹𝑥) ∈ ℝ → ∃𝑚 ∈ ℕ (𝐹𝑥) < 𝑚)
226112, 225syl 17 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ) → ∃𝑚 ∈ ℕ (𝐹𝑥) < 𝑚)
227194ad2antrl 728 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ) ∧ (𝑚 ∈ ℕ ∧ (𝐹𝑥) < 𝑚)) → ((𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0))‘𝑚) = if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0))
228 ltle 11349 . . . . . . . . . . . . 13 (((𝐹𝑥) ∈ ℝ ∧ 𝑚 ∈ ℝ) → ((𝐹𝑥) < 𝑚 → (𝐹𝑥) ≤ 𝑚))
229112, 48, 228syl2an 596 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℝ) ∧ 𝑚 ∈ ℕ) → ((𝐹𝑥) < 𝑚 → (𝐹𝑥) ≤ 𝑚))
230229impr 454 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ) ∧ (𝑚 ∈ ℕ ∧ (𝐹𝑥) < 𝑚)) → (𝐹𝑥) ≤ 𝑚)
231230iftrued 4533 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ) ∧ (𝑚 ∈ ℕ ∧ (𝐹𝑥) < 𝑚)) → if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0) = (𝐹𝑥))
232227, 231eqtrd 2777 . . . . . . . . 9 (((𝜑𝑥 ∈ ℝ) ∧ (𝑚 ∈ ℕ ∧ (𝐹𝑥) < 𝑚)) → ((𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0))‘𝑚) = (𝐹𝑥))
233201adantr 480 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ) ∧ (𝑚 ∈ ℕ ∧ (𝐹𝑥) < 𝑚)) → (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)) Fn ℕ)
234 simprl 771 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ) ∧ (𝑚 ∈ ℕ ∧ (𝐹𝑥) < 𝑚)) → 𝑚 ∈ ℕ)
235 fnfvelrn 7100 . . . . . . . . . 10 (((𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)) Fn ℕ ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0))‘𝑚) ∈ ran (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)))
236233, 234, 235syl2anc 584 . . . . . . . . 9 (((𝜑𝑥 ∈ ℝ) ∧ (𝑚 ∈ ℕ ∧ (𝐹𝑥) < 𝑚)) → ((𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0))‘𝑚) ∈ ran (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)))
237232, 236eqeltrrd 2842 . . . . . . . 8 (((𝜑𝑥 ∈ ℝ) ∧ (𝑚 ∈ ℕ ∧ (𝐹𝑥) < 𝑚)) → (𝐹𝑥) ∈ ran (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)))
238226, 237rexlimddv 3161 . . . . . . 7 ((𝜑𝑥 ∈ ℝ) → (𝐹𝑥) ∈ ran (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)))
239211, 219, 221, 238suprubd 12230 . . . . . 6 ((𝜑𝑥 ∈ ℝ) → (𝐹𝑥) ≤ sup(ran (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)), ℝ, < ))
240211, 219, 221suprcld 12231 . . . . . . 7 ((𝜑𝑥 ∈ ℝ) → sup(ran (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)), ℝ, < ) ∈ ℝ)
241240, 112letri3d 11403 . . . . . 6 ((𝜑𝑥 ∈ ℝ) → (sup(ran (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)), ℝ, < ) = (𝐹𝑥) ↔ (sup(ran (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)), ℝ, < ) ≤ (𝐹𝑥) ∧ (𝐹𝑥) ≤ sup(ran (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)), ℝ, < ))))
242224, 239, 241mpbir2and 713 . . . . 5 ((𝜑𝑥 ∈ ℝ) → sup(ran (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)), ℝ, < ) = (𝐹𝑥))
243242mpteq2dva 5242 . . . 4 (𝜑 → (𝑥 ∈ ℝ ↦ sup(ran (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)), ℝ, < )) = (𝑥 ∈ ℝ ↦ (𝐹𝑥)))
244243, 167eqtr4d 2780 . . 3 (𝜑 → (𝑥 ∈ ℝ ↦ sup(ran (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)), ℝ, < )) = 𝐹)
245244fveq2d 6910 . 2 (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ sup(ran (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)), ℝ, < ))) = (∫2𝐹))
246192, 245eqtr3d 2779 1 (𝜑 → sup(ran (𝑛 ∈ ℕ ↦ (∫2‘(𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)))), ℝ*, < ) = (∫2𝐹))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wne 2940  wral 3061  wrex 3070  Vcvv 3480  cdif 3948  wss 3951  c0 4333  ifcif 4525   class class class wbr 5143  cmpt 5225  ccnv 5684  dom cdm 5685  ran crn 5686  cres 5687  cima 5688   Fn wfn 6556  wf 6557  cfv 6561  (class class class)co 7431  r cofr 7696  supcsup 9480  cr 11154  0cc0 11155  1c1 11156   + caddc 11158  +∞cpnf 11292  *cxr 11294   < clt 11295  cle 11296  cn 12266  (,)cioo 13387  [,)cico 13389  volcvol 25498  MblFncmbf 25649  2citg2 25651
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cc 10475  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-disj 5111  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-ofr 7698  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-oadd 8510  df-omul 8511  df-er 8745  df-map 8868  df-pm 8869  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fi 9451  df-sup 9482  df-inf 9483  df-oi 9550  df-dju 9941  df-card 9979  df-acn 9982  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-z 12614  df-uz 12879  df-q 12991  df-rp 13035  df-xneg 13154  df-xadd 13155  df-xmul 13156  df-ioo 13391  df-ioc 13392  df-ico 13393  df-icc 13394  df-fz 13548  df-fzo 13695  df-fl 13832  df-seq 14043  df-exp 14103  df-hash 14370  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-clim 15524  df-rlim 15525  df-sum 15723  df-rest 17467  df-topgen 17488  df-psmet 21356  df-xmet 21357  df-met 21358  df-bl 21359  df-mopn 21360  df-top 22900  df-topon 22917  df-bases 22953  df-cmp 23395  df-ovol 25499  df-vol 25500  df-mbf 25654  df-itg1 25655  df-itg2 25656
This theorem is referenced by:  itg2cn  25798
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