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Theorem itg2cnlem1 23468
Description: Lemma for itgcn 23549. (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 6168 . . . . . . . . . 10 (𝐹𝑥) ∈ V
2 c0ex 9994 . . . . . . . . . 10 0 ∈ V
31, 2ifex 4134 . . . . . . . . 9 if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0) ∈ V
4 eqid 2621 . . . . . . . . . 10 (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)) = (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0))
54fvmpt2 6258 . . . . . . . . 9 ((𝑥 ∈ ℝ ∧ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0) ∈ V) → ((𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0))‘𝑥) = if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0))
63, 5mpan2 706 . . . . . . . 8 (𝑥 ∈ ℝ → ((𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0))‘𝑥) = if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0))
76mpteq2dv 4715 . . . . . . 7 (𝑥 ∈ ℝ → (𝑛 ∈ ℕ ↦ ((𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0))‘𝑥)) = (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)))
87rneqd 5323 . . . . . 6 (𝑥 ∈ ℝ → ran (𝑛 ∈ ℕ ↦ ((𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0))‘𝑥)) = ran (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)))
98supeq1d 8312 . . . . 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 4710 . . . 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 2761 . . . . 5 𝑦sup(ran (𝑛 ∈ ℕ ↦ ((𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0))‘𝑥)), ℝ, < )
12 nfcv 2761 . . . . . . . 8 𝑥
13 nfmpt1 4717 . . . . . . . . . . 11 𝑥(𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0))
1412, 13nfmpt 4716 . . . . . . . . . 10 𝑥(𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)))
15 nfcv 2761 . . . . . . . . . 10 𝑥𝑚
1614, 15nffv 6165 . . . . . . . . 9 𝑥((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)))‘𝑚)
17 nfcv 2761 . . . . . . . . 9 𝑥𝑦
1816, 17nffv 6165 . . . . . . . 8 𝑥(((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)))‘𝑚)‘𝑦)
1912, 18nfmpt 4716 . . . . . . 7 𝑥(𝑚 ∈ ℕ ↦ (((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)))‘𝑚)‘𝑦))
2019nfrn 5338 . . . . . 6 𝑥ran (𝑚 ∈ ℕ ↦ (((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)))‘𝑚)‘𝑦))
21 nfcv 2761 . . . . . 6 𝑥
22 nfcv 2761 . . . . . 6 𝑥 <
2320, 21, 22nfsup 8317 . . . . 5 𝑥sup(ran (𝑚 ∈ ℕ ↦ (((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)))‘𝑚)‘𝑦)), ℝ, < )
24 fveq2 6158 . . . . . . . . 9 (𝑥 = 𝑦 → ((𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0))‘𝑥) = ((𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0))‘𝑦))
2524mpteq2dv 4715 . . . . . . . 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 4627 . . . . . . . . . . . . 13 (𝑛 = 𝑚 → ((𝐹𝑥) ≤ 𝑛 ↔ (𝐹𝑥) ≤ 𝑚))
2726ifbid 4086 . . . . . . . . . . . 12 (𝑛 = 𝑚 → if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0) = if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0))
2827mpteq2dv 4715 . . . . . . . . . . 11 (𝑛 = 𝑚 → (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)) = (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0)))
2928fveq1d 6160 . . . . . . . . . 10 (𝑛 = 𝑚 → ((𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0))‘𝑦) = ((𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0))‘𝑦))
3029cbvmptv 4720 . . . . . . . . 9 (𝑛 ∈ ℕ ↦ ((𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0))‘𝑦)) = (𝑚 ∈ ℕ ↦ ((𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0))‘𝑦))
31 eqid 2621 . . . . . . . . . . . 12 (𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0))) = (𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)))
32 reex 9987 . . . . . . . . . . . . 13 ℝ ∈ V
3332mptex 6451 . . . . . . . . . . . 12 (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0)) ∈ V
3428, 31, 33fvmpt 6249 . . . . . . . . . . 11 (𝑚 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)))‘𝑚) = (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0)))
3534fveq1d 6160 . . . . . . . . . 10 (𝑚 ∈ ℕ → (((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)))‘𝑚)‘𝑦) = ((𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0))‘𝑦))
3635mpteq2ia 4710 . . . . . . . . 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 2646 . . . . . . . 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, 37syl6eq 2671 . . . . . . 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 5323 . . . . . 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 8312 . . . . 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 4719 . . . 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 2645 . . 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 6158 . . . . . . . 8 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
4443breq1d 4633 . . . . . . 7 (𝑥 = 𝑦 → ((𝐹𝑥) ≤ 𝑚 ↔ (𝐹𝑦) ≤ 𝑚))
4544, 43ifbieq1d 4087 . . . . . 6 (𝑥 = 𝑦 → if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0) = if((𝐹𝑦) ≤ 𝑚, (𝐹𝑦), 0))
4645cbvmptv 4720 . . . . 5 (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0)) = (𝑦 ∈ ℝ ↦ if((𝐹𝑦) ≤ 𝑚, (𝐹𝑦), 0))
4734adantl 482 . . . . 5 ((𝜑𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)))‘𝑚) = (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0)))
48 nnre 10987 . . . . . . . . . . . . 13 (𝑚 ∈ ℕ → 𝑚 ∈ ℝ)
4948ad2antlr 762 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → 𝑚 ∈ ℝ)
5049rexrd 10049 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → 𝑚 ∈ ℝ*)
51 elioopnf 12225 . . . . . . . . . . 11 (𝑚 ∈ ℝ* → ((𝐹𝑦) ∈ (𝑚(,)+∞) ↔ ((𝐹𝑦) ∈ ℝ ∧ 𝑚 < (𝐹𝑦))))
5250, 51syl 17 . . . . . . . . . 10 (((𝜑𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → ((𝐹𝑦) ∈ (𝑚(,)+∞) ↔ ((𝐹𝑦) ∈ ℝ ∧ 𝑚 < (𝐹𝑦))))
53 itg2cn.1 . . . . . . . . . . . . . 14 (𝜑𝐹:ℝ⟶(0[,)+∞))
54 ffn 6012 . . . . . . . . . . . . . 14 (𝐹:ℝ⟶(0[,)+∞) → 𝐹 Fn ℝ)
5553, 54syl 17 . . . . . . . . . . . . 13 (𝜑𝐹 Fn ℝ)
5655ad2antrr 761 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → 𝐹 Fn ℝ)
57 elpreima 6303 . . . . . . . . . . . 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(,)+∞))))
59 simpr 477 . . . . . . . . . . . 12 (((𝜑𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℝ)
6059biantrurd 529 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → ((𝐹𝑦) ∈ (𝑚(,)+∞) ↔ (𝑦 ∈ ℝ ∧ (𝐹𝑦) ∈ (𝑚(,)+∞))))
6158, 60bitr4d 271 . . . . . . . . . 10 (((𝜑𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → (𝑦 ∈ (𝐹 “ (𝑚(,)+∞)) ↔ (𝐹𝑦) ∈ (𝑚(,)+∞)))
62 rge0ssre 12238 . . . . . . . . . . . . . 14 (0[,)+∞) ⊆ ℝ
63 fss 6023 . . . . . . . . . . . . . 14 ((𝐹:ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ ℝ) → 𝐹:ℝ⟶ℝ)
6453, 62, 63sylancl 693 . . . . . . . . . . . . 13 (𝜑𝐹:ℝ⟶ℝ)
6564adantr 481 . . . . . . . . . . . 12 ((𝜑𝑚 ∈ ℕ) → 𝐹:ℝ⟶ℝ)
6665ffvelrnda 6325 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → (𝐹𝑦) ∈ ℝ)
6766biantrurd 529 . . . . . . . . . 10 (((𝜑𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → (𝑚 < (𝐹𝑦) ↔ ((𝐹𝑦) ∈ ℝ ∧ 𝑚 < (𝐹𝑦))))
6852, 61, 673bitr4d 300 . . . . . . . . 9 (((𝜑𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → (𝑦 ∈ (𝐹 “ (𝑚(,)+∞)) ↔ 𝑚 < (𝐹𝑦)))
6968notbid 308 . . . . . . . 8 (((𝜑𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → (¬ 𝑦 ∈ (𝐹 “ (𝑚(,)+∞)) ↔ ¬ 𝑚 < (𝐹𝑦)))
70 eldif 3570 . . . . . . . . . 10 (𝑦 ∈ (ℝ ∖ (𝐹 “ (𝑚(,)+∞))) ↔ (𝑦 ∈ ℝ ∧ ¬ 𝑦 ∈ (𝐹 “ (𝑚(,)+∞))))
7170baib 943 . . . . . . . . 9 (𝑦 ∈ ℝ → (𝑦 ∈ (ℝ ∖ (𝐹 “ (𝑚(,)+∞))) ↔ ¬ 𝑦 ∈ (𝐹 “ (𝑚(,)+∞))))
7271adantl 482 . . . . . . . 8 (((𝜑𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → (𝑦 ∈ (ℝ ∖ (𝐹 “ (𝑚(,)+∞))) ↔ ¬ 𝑦 ∈ (𝐹 “ (𝑚(,)+∞))))
7366, 49lenltd 10143 . . . . . . . 8 (((𝜑𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → ((𝐹𝑦) ≤ 𝑚 ↔ ¬ 𝑚 < (𝐹𝑦)))
7469, 72, 733bitr4d 300 . . . . . . 7 (((𝜑𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → (𝑦 ∈ (ℝ ∖ (𝐹 “ (𝑚(,)+∞))) ↔ (𝐹𝑦) ≤ 𝑚))
7574ifbid 4086 . . . . . 6 (((𝜑𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → if(𝑦 ∈ (ℝ ∖ (𝐹 “ (𝑚(,)+∞))), (𝐹𝑦), 0) = if((𝐹𝑦) ≤ 𝑚, (𝐹𝑦), 0))
7675mpteq2dva 4714 . . . . 5 ((𝜑𝑚 ∈ ℕ) → (𝑦 ∈ ℝ ↦ if(𝑦 ∈ (ℝ ∖ (𝐹 “ (𝑚(,)+∞))), (𝐹𝑦), 0)) = (𝑦 ∈ ℝ ↦ if((𝐹𝑦) ≤ 𝑚, (𝐹𝑦), 0)))
7746, 47, 763eqtr4a 2681 . . . 4 ((𝜑𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)))‘𝑚) = (𝑦 ∈ ℝ ↦ if(𝑦 ∈ (ℝ ∖ (𝐹 “ (𝑚(,)+∞))), (𝐹𝑦), 0)))
78 difss 3721 . . . . . 6 (ℝ ∖ (𝐹 “ (𝑚(,)+∞))) ⊆ ℝ
7978a1i 11 . . . . 5 ((𝜑𝑚 ∈ ℕ) → (ℝ ∖ (𝐹 “ (𝑚(,)+∞))) ⊆ ℝ)
80 rembl 23248 . . . . . 6 ℝ ∈ dom vol
8180a1i 11 . . . . 5 ((𝜑𝑚 ∈ ℕ) → ℝ ∈ dom vol)
82 fvex 6168 . . . . . . 7 (𝐹𝑦) ∈ V
8382, 2ifex 4134 . . . . . 6 if(𝑦 ∈ (ℝ ∖ (𝐹 “ (𝑚(,)+∞))), (𝐹𝑦), 0) ∈ V
8483a1i 11 . . . . 5 (((𝜑𝑚 ∈ ℕ) ∧ 𝑦 ∈ (ℝ ∖ (𝐹 “ (𝑚(,)+∞)))) → if(𝑦 ∈ (ℝ ∖ (𝐹 “ (𝑚(,)+∞))), (𝐹𝑦), 0) ∈ V)
85 eldifn 3717 . . . . . . 7 (𝑦 ∈ (ℝ ∖ (ℝ ∖ (𝐹 “ (𝑚(,)+∞)))) → ¬ 𝑦 ∈ (ℝ ∖ (𝐹 “ (𝑚(,)+∞))))
8685adantl 482 . . . . . 6 (((𝜑𝑚 ∈ ℕ) ∧ 𝑦 ∈ (ℝ ∖ (ℝ ∖ (𝐹 “ (𝑚(,)+∞))))) → ¬ 𝑦 ∈ (ℝ ∖ (𝐹 “ (𝑚(,)+∞))))
8786iffalsed 4075 . . . . 5 (((𝜑𝑚 ∈ ℕ) ∧ 𝑦 ∈ (ℝ ∖ (ℝ ∖ (𝐹 “ (𝑚(,)+∞))))) → if(𝑦 ∈ (ℝ ∖ (𝐹 “ (𝑚(,)+∞))), (𝐹𝑦), 0) = 0)
88 iftrue 4070 . . . . . . . . 9 (𝑦 ∈ (ℝ ∖ (𝐹 “ (𝑚(,)+∞))) → if(𝑦 ∈ (ℝ ∖ (𝐹 “ (𝑚(,)+∞))), (𝐹𝑦), 0) = (𝐹𝑦))
8988mpteq2ia 4710 . . . . . . . 8 (𝑦 ∈ (ℝ ∖ (𝐹 “ (𝑚(,)+∞))) ↦ if(𝑦 ∈ (ℝ ∖ (𝐹 “ (𝑚(,)+∞))), (𝐹𝑦), 0)) = (𝑦 ∈ (ℝ ∖ (𝐹 “ (𝑚(,)+∞))) ↦ (𝐹𝑦))
90 resmpt 5418 . . . . . . . . 9 ((ℝ ∖ (𝐹 “ (𝑚(,)+∞))) ⊆ ℝ → ((𝑦 ∈ ℝ ↦ (𝐹𝑦)) ↾ (ℝ ∖ (𝐹 “ (𝑚(,)+∞)))) = (𝑦 ∈ (ℝ ∖ (𝐹 “ (𝑚(,)+∞))) ↦ (𝐹𝑦)))
9178, 90ax-mp 5 . . . . . . . 8 ((𝑦 ∈ ℝ ↦ (𝐹𝑦)) ↾ (ℝ ∖ (𝐹 “ (𝑚(,)+∞)))) = (𝑦 ∈ (ℝ ∖ (𝐹 “ (𝑚(,)+∞))) ↦ (𝐹𝑦))
9289, 91eqtr4i 2646 . . . . . . 7 (𝑦 ∈ (ℝ ∖ (𝐹 “ (𝑚(,)+∞))) ↦ if(𝑦 ∈ (ℝ ∖ (𝐹 “ (𝑚(,)+∞))), (𝐹𝑦), 0)) = ((𝑦 ∈ ℝ ↦ (𝐹𝑦)) ↾ (ℝ ∖ (𝐹 “ (𝑚(,)+∞))))
9353feqmptd 6216 . . . . . . . . 9 (𝜑𝐹 = (𝑦 ∈ ℝ ↦ (𝐹𝑦)))
94 itg2cn.2 . . . . . . . . 9 (𝜑𝐹 ∈ MblFn)
9593, 94eqeltrrd 2699 . . . . . . . 8 (𝜑 → (𝑦 ∈ ℝ ↦ (𝐹𝑦)) ∈ MblFn)
96 mbfima 23339 . . . . . . . . . 10 ((𝐹 ∈ MblFn ∧ 𝐹:ℝ⟶ℝ) → (𝐹 “ (𝑚(,)+∞)) ∈ dom vol)
9794, 64, 96syl2anc 692 . . . . . . . . 9 (𝜑 → (𝐹 “ (𝑚(,)+∞)) ∈ dom vol)
98 cmmbl 23242 . . . . . . . . 9 ((𝐹 “ (𝑚(,)+∞)) ∈ dom vol → (ℝ ∖ (𝐹 “ (𝑚(,)+∞))) ∈ dom vol)
9997, 98syl 17 . . . . . . . 8 (𝜑 → (ℝ ∖ (𝐹 “ (𝑚(,)+∞))) ∈ dom vol)
100 mbfres 23351 . . . . . . . 8 (((𝑦 ∈ ℝ ↦ (𝐹𝑦)) ∈ MblFn ∧ (ℝ ∖ (𝐹 “ (𝑚(,)+∞))) ∈ dom vol) → ((𝑦 ∈ ℝ ↦ (𝐹𝑦)) ↾ (ℝ ∖ (𝐹 “ (𝑚(,)+∞)))) ∈ MblFn)
10195, 99, 100syl2anc 692 . . . . . . 7 (𝜑 → ((𝑦 ∈ ℝ ↦ (𝐹𝑦)) ↾ (ℝ ∖ (𝐹 “ (𝑚(,)+∞)))) ∈ MblFn)
10292, 101syl5eqel 2702 . . . . . 6 (𝜑 → (𝑦 ∈ (ℝ ∖ (𝐹 “ (𝑚(,)+∞))) ↦ if(𝑦 ∈ (ℝ ∖ (𝐹 “ (𝑚(,)+∞))), (𝐹𝑦), 0)) ∈ MblFn)
103102adantr 481 . . . . 5 ((𝜑𝑚 ∈ ℕ) → (𝑦 ∈ (ℝ ∖ (𝐹 “ (𝑚(,)+∞))) ↦ if(𝑦 ∈ (ℝ ∖ (𝐹 “ (𝑚(,)+∞))), (𝐹𝑦), 0)) ∈ MblFn)
10479, 81, 84, 87, 103mbfss 23353 . . . 4 ((𝜑𝑚 ∈ ℕ) → (𝑦 ∈ ℝ ↦ if(𝑦 ∈ (ℝ ∖ (𝐹 “ (𝑚(,)+∞))), (𝐹𝑦), 0)) ∈ MblFn)
10577, 104eqeltrd 2698 . . 3 ((𝜑𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)))‘𝑚) ∈ MblFn)
10653ffvelrnda 6325 . . . . . . 7 ((𝜑𝑥 ∈ ℝ) → (𝐹𝑥) ∈ (0[,)+∞))
107 0e0icopnf 12240 . . . . . . 7 0 ∈ (0[,)+∞)
108 ifcl 4108 . . . . . . 7 (((𝐹𝑥) ∈ (0[,)+∞) ∧ 0 ∈ (0[,)+∞)) → if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0) ∈ (0[,)+∞))
109106, 107, 108sylancl 693 . . . . . 6 ((𝜑𝑥 ∈ ℝ) → if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0) ∈ (0[,)+∞))
110109adantlr 750 . . . . 5 (((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0) ∈ (0[,)+∞))
111 eqid 2621 . . . . 5 (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0)) = (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0))
112110, 111fmptd 6351 . . . 4 ((𝜑𝑚 ∈ ℕ) → (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0)):ℝ⟶(0[,)+∞))
11347feq1d 5997 . . . 4 ((𝜑𝑚 ∈ ℕ) → (((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)))‘𝑚):ℝ⟶(0[,)+∞) ↔ (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0)):ℝ⟶(0[,)+∞)))
114112, 113mpbird 247 . . 3 ((𝜑𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)))‘𝑚):ℝ⟶(0[,)+∞))
115 elrege0 12236 . . . . . . . . . . . . 13 ((𝐹𝑥) ∈ (0[,)+∞) ↔ ((𝐹𝑥) ∈ ℝ ∧ 0 ≤ (𝐹𝑥)))
116106, 115sylib 208 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ ℝ) → ((𝐹𝑥) ∈ ℝ ∧ 0 ≤ (𝐹𝑥)))
117116simpld 475 . . . . . . . . . . 11 ((𝜑𝑥 ∈ ℝ) → (𝐹𝑥) ∈ ℝ)
118117adantlr 750 . . . . . . . . . 10 (((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐹𝑥) ∈ ℝ)
119118adantr 481 . . . . . . . . 9 ((((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐹𝑥) ≤ 𝑚) → (𝐹𝑥) ∈ ℝ)
120119leidd 10554 . . . . . . . 8 ((((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐹𝑥) ≤ 𝑚) → (𝐹𝑥) ≤ (𝐹𝑥))
121 iftrue 4070 . . . . . . . . 9 ((𝐹𝑥) ≤ 𝑚 → if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0) = (𝐹𝑥))
122121adantl 482 . . . . . . . 8 ((((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐹𝑥) ≤ 𝑚) → if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0) = (𝐹𝑥))
12348ad3antlr 766 . . . . . . . . . 10 ((((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐹𝑥) ≤ 𝑚) → 𝑚 ∈ ℝ)
124 peano2re 10169 . . . . . . . . . . 11 (𝑚 ∈ ℝ → (𝑚 + 1) ∈ ℝ)
125123, 124syl 17 . . . . . . . . . 10 ((((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐹𝑥) ≤ 𝑚) → (𝑚 + 1) ∈ ℝ)
126 simpr 477 . . . . . . . . . 10 ((((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐹𝑥) ≤ 𝑚) → (𝐹𝑥) ≤ 𝑚)
127123lep1d 10915 . . . . . . . . . 10 ((((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐹𝑥) ≤ 𝑚) → 𝑚 ≤ (𝑚 + 1))
128119, 123, 125, 126, 127letrd 10154 . . . . . . . . 9 ((((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐹𝑥) ≤ 𝑚) → (𝐹𝑥) ≤ (𝑚 + 1))
129128iftrued 4072 . . . . . . . 8 ((((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐹𝑥) ≤ 𝑚) → if((𝐹𝑥) ≤ (𝑚 + 1), (𝐹𝑥), 0) = (𝐹𝑥))
130120, 122, 1293brtr4d 4655 . . . . . . 7 ((((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐹𝑥) ≤ 𝑚) → if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0) ≤ if((𝐹𝑥) ≤ (𝑚 + 1), (𝐹𝑥), 0))
131 iffalse 4073 . . . . . . . . 9 (¬ (𝐹𝑥) ≤ 𝑚 → if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0) = 0)
132131adantl 482 . . . . . . . 8 ((((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ ¬ (𝐹𝑥) ≤ 𝑚) → if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0) = 0)
133116simprd 479 . . . . . . . . . . 11 ((𝜑𝑥 ∈ ℝ) → 0 ≤ (𝐹𝑥))
134 0le0 11070 . . . . . . . . . . 11 0 ≤ 0
135 breq2 4627 . . . . . . . . . . . 12 ((𝐹𝑥) = if((𝐹𝑥) ≤ (𝑚 + 1), (𝐹𝑥), 0) → (0 ≤ (𝐹𝑥) ↔ 0 ≤ if((𝐹𝑥) ≤ (𝑚 + 1), (𝐹𝑥), 0)))
136 breq2 4627 . . . . . . . . . . . 12 (0 = if((𝐹𝑥) ≤ (𝑚 + 1), (𝐹𝑥), 0) → (0 ≤ 0 ↔ 0 ≤ if((𝐹𝑥) ≤ (𝑚 + 1), (𝐹𝑥), 0)))
137135, 136ifboth 4102 . . . . . . . . . . 11 ((0 ≤ (𝐹𝑥) ∧ 0 ≤ 0) → 0 ≤ if((𝐹𝑥) ≤ (𝑚 + 1), (𝐹𝑥), 0))
138133, 134, 137sylancl 693 . . . . . . . . . 10 ((𝜑𝑥 ∈ ℝ) → 0 ≤ if((𝐹𝑥) ≤ (𝑚 + 1), (𝐹𝑥), 0))
139138adantlr 750 . . . . . . . . 9 (((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 0 ≤ if((𝐹𝑥) ≤ (𝑚 + 1), (𝐹𝑥), 0))
140139adantr 481 . . . . . . . 8 ((((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ ¬ (𝐹𝑥) ≤ 𝑚) → 0 ≤ if((𝐹𝑥) ≤ (𝑚 + 1), (𝐹𝑥), 0))
141132, 140eqbrtrd 4645 . . . . . . 7 ((((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ ¬ (𝐹𝑥) ≤ 𝑚) → if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0) ≤ if((𝐹𝑥) ≤ (𝑚 + 1), (𝐹𝑥), 0))
142130, 141pm2.61dan 831 . . . . . 6 (((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0) ≤ if((𝐹𝑥) ≤ (𝑚 + 1), (𝐹𝑥), 0))
143142ralrimiva 2962 . . . . 5 ((𝜑𝑚 ∈ ℕ) → ∀𝑥 ∈ ℝ if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0) ≤ if((𝐹𝑥) ≤ (𝑚 + 1), (𝐹𝑥), 0))
1441, 2ifex 4134 . . . . . . 7 if((𝐹𝑥) ≤ (𝑚 + 1), (𝐹𝑥), 0) ∈ V
145144a1i 11 . . . . . 6 (((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → if((𝐹𝑥) ≤ (𝑚 + 1), (𝐹𝑥), 0) ∈ V)
146 eqidd 2622 . . . . . 6 ((𝜑𝑚 ∈ ℕ) → (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0)) = (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0)))
147 eqidd 2622 . . . . . 6 ((𝜑𝑚 ∈ ℕ) → (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ (𝑚 + 1), (𝐹𝑥), 0)) = (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ (𝑚 + 1), (𝐹𝑥), 0)))
14881, 110, 145, 146, 147ofrfval2 6880 . . . . 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)))
149143, 148mpbird 247 . . . 4 ((𝜑𝑚 ∈ ℕ) → (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0)) ∘𝑟 ≤ (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ (𝑚 + 1), (𝐹𝑥), 0)))
150 peano2nn 10992 . . . . . 6 (𝑚 ∈ ℕ → (𝑚 + 1) ∈ ℕ)
151150adantl 482 . . . . 5 ((𝜑𝑚 ∈ ℕ) → (𝑚 + 1) ∈ ℕ)
152 breq2 4627 . . . . . . . 8 (𝑛 = (𝑚 + 1) → ((𝐹𝑥) ≤ 𝑛 ↔ (𝐹𝑥) ≤ (𝑚 + 1)))
153152ifbid 4086 . . . . . . 7 (𝑛 = (𝑚 + 1) → if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0) = if((𝐹𝑥) ≤ (𝑚 + 1), (𝐹𝑥), 0))
154153mpteq2dv 4715 . . . . . 6 (𝑛 = (𝑚 + 1) → (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)) = (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ (𝑚 + 1), (𝐹𝑥), 0)))
15532mptex 6451 . . . . . 6 (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ (𝑚 + 1), (𝐹𝑥), 0)) ∈ V
156154, 31, 155fvmpt 6249 . . . . 5 ((𝑚 + 1) ∈ ℕ → ((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)))‘(𝑚 + 1)) = (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ (𝑚 + 1), (𝐹𝑥), 0)))
157151, 156syl 17 . . . 4 ((𝜑𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)))‘(𝑚 + 1)) = (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ (𝑚 + 1), (𝐹𝑥), 0)))
158149, 47, 1573brtr4d 4655 . . 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)))
15964ffvelrnda 6325 . . . 4 ((𝜑𝑦 ∈ ℝ) → (𝐹𝑦) ∈ ℝ)
16034adantl 482 . . . . . . 7 (((𝜑𝑦 ∈ ℝ) ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)))‘𝑚) = (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0)))
161160fveq1d 6160 . . . . . 6 (((𝜑𝑦 ∈ ℝ) ∧ 𝑚 ∈ ℕ) → (((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)))‘𝑚)‘𝑦) = ((𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0))‘𝑦))
162117leidd 10554 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ ℝ) → (𝐹𝑥) ≤ (𝐹𝑥))
163 breq1 4626 . . . . . . . . . . . . . 14 ((𝐹𝑥) = if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0) → ((𝐹𝑥) ≤ (𝐹𝑥) ↔ if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0) ≤ (𝐹𝑥)))
164 breq1 4626 . . . . . . . . . . . . . 14 (0 = if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0) → (0 ≤ (𝐹𝑥) ↔ if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0) ≤ (𝐹𝑥)))
165163, 164ifboth 4102 . . . . . . . . . . . . 13 (((𝐹𝑥) ≤ (𝐹𝑥) ∧ 0 ≤ (𝐹𝑥)) → if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0) ≤ (𝐹𝑥))
166162, 133, 165syl2anc 692 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ ℝ) → if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0) ≤ (𝐹𝑥))
167166adantlr 750 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0) ≤ (𝐹𝑥))
168167ralrimiva 2962 . . . . . . . . . 10 ((𝜑𝑚 ∈ ℕ) → ∀𝑥 ∈ ℝ if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0) ≤ (𝐹𝑥))
16932a1i 11 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ) → ℝ ∈ V)
1701, 2ifex 4134 . . . . . . . . . . . 12 if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0) ∈ V
171170a1i 11 . . . . . . . . . . 11 (((𝜑𝑚 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0) ∈ V)
17253feqmptd 6216 . . . . . . . . . . . 12 (𝜑𝐹 = (𝑥 ∈ ℝ ↦ (𝐹𝑥)))
173172adantr 481 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ) → 𝐹 = (𝑥 ∈ ℝ ↦ (𝐹𝑥)))
174169, 171, 118, 146, 173ofrfval2 6880 . . . . . . . . . 10 ((𝜑𝑚 ∈ ℕ) → ((𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0)) ∘𝑟𝐹 ↔ ∀𝑥 ∈ ℝ if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0) ≤ (𝐹𝑥)))
175168, 174mpbird 247 . . . . . . . . 9 ((𝜑𝑚 ∈ ℕ) → (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0)) ∘𝑟𝐹)
176171, 111fmptd 6351 . . . . . . . . . . 11 ((𝜑𝑚 ∈ ℕ) → (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0)):ℝ⟶V)
177 ffn 6012 . . . . . . . . . . 11 ((𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0)):ℝ⟶V → (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0)) Fn ℝ)
178176, 177syl 17 . . . . . . . . . 10 ((𝜑𝑚 ∈ ℕ) → (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0)) Fn ℝ)
17955adantr 481 . . . . . . . . . 10 ((𝜑𝑚 ∈ ℕ) → 𝐹 Fn ℝ)
180 inidm 3806 . . . . . . . . . 10 (ℝ ∩ ℝ) = ℝ
181 eqidd 2622 . . . . . . . . . 10 (((𝜑𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → ((𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0))‘𝑦) = ((𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0))‘𝑦))
182 eqidd 2622 . . . . . . . . . 10 (((𝜑𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → (𝐹𝑦) = (𝐹𝑦))
183178, 179, 169, 169, 180, 181, 182ofrfval 6870 . . . . . . . . 9 ((𝜑𝑚 ∈ ℕ) → ((𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0)) ∘𝑟𝐹 ↔ ∀𝑦 ∈ ℝ ((𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0))‘𝑦) ≤ (𝐹𝑦)))
184175, 183mpbid 222 . . . . . . . 8 ((𝜑𝑚 ∈ ℕ) → ∀𝑦 ∈ ℝ ((𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0))‘𝑦) ≤ (𝐹𝑦))
185184r19.21bi 2928 . . . . . . 7 (((𝜑𝑚 ∈ ℕ) ∧ 𝑦 ∈ ℝ) → ((𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0))‘𝑦) ≤ (𝐹𝑦))
186185an32s 845 . . . . . 6 (((𝜑𝑦 ∈ ℝ) ∧ 𝑚 ∈ ℕ) → ((𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0))‘𝑦) ≤ (𝐹𝑦))
187161, 186eqbrtrd 4645 . . . . 5 (((𝜑𝑦 ∈ ℝ) ∧ 𝑚 ∈ ℕ) → (((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)))‘𝑚)‘𝑦) ≤ (𝐹𝑦))
188187ralrimiva 2962 . . . 4 ((𝜑𝑦 ∈ ℝ) → ∀𝑚 ∈ ℕ (((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)))‘𝑚)‘𝑦) ≤ (𝐹𝑦))
189 breq2 4627 . . . . . 6 (𝑧 = (𝐹𝑦) → ((((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)))‘𝑚)‘𝑦) ≤ 𝑧 ↔ (((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)))‘𝑚)‘𝑦) ≤ (𝐹𝑦)))
190189ralbidv 2982 . . . . 5 (𝑧 = (𝐹𝑦) → (∀𝑚 ∈ ℕ (((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)))‘𝑚)‘𝑦) ≤ 𝑧 ↔ ∀𝑚 ∈ ℕ (((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)))‘𝑚)‘𝑦) ≤ (𝐹𝑦)))
191190rspcev 3299 . . . 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)
192159, 188, 191syl2anc 692 . . 3 ((𝜑𝑦 ∈ ℝ) → ∃𝑧 ∈ ℝ ∀𝑚 ∈ ℕ (((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)))‘𝑚)‘𝑦) ≤ 𝑧)
19328fveq2d 6162 . . . . . . 7 (𝑛 = 𝑚 → (∫2‘(𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0))) = (∫2‘(𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0))))
194193cbvmptv 4720 . . . . . 6 (𝑛 ∈ ℕ ↦ (∫2‘(𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)))) = (𝑚 ∈ ℕ ↦ (∫2‘(𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0))))
19534fveq2d 6162 . . . . . . 7 (𝑚 ∈ ℕ → (∫2‘((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)))‘𝑚)) = (∫2‘(𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0))))
196195mpteq2ia 4710 . . . . . 6 (𝑚 ∈ ℕ ↦ (∫2‘((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)))‘𝑚))) = (𝑚 ∈ ℕ ↦ (∫2‘(𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0))))
197194, 196eqtr4i 2646 . . . . 5 (𝑛 ∈ ℕ ↦ (∫2‘(𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)))) = (𝑚 ∈ ℕ ↦ (∫2‘((𝑛 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)))‘𝑚)))
198197rneqi 5322 . . . 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)))
199198supeq1i 8313 . . 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*, < )
20042, 105, 114, 158, 192, 199itg2mono 23460 . 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*, < ))
201 eqid 2621 . . . . . . . . . . . 12 (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)) = (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0))
20227, 201, 170fvmpt 6249 . . . . . . . . . . 11 (𝑚 ∈ ℕ → ((𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0))‘𝑚) = if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0))
203202adantl 482 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ) ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0))‘𝑚) = if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0))
204166adantr 481 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ) ∧ 𝑚 ∈ ℕ) → if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0) ≤ (𝐹𝑥))
205203, 204eqbrtrd 4645 . . . . . . . . 9 (((𝜑𝑥 ∈ ℝ) ∧ 𝑚 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0))‘𝑚) ≤ (𝐹𝑥))
206205ralrimiva 2962 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ) → ∀𝑚 ∈ ℕ ((𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0))‘𝑚) ≤ (𝐹𝑥))
2073a1i 11 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℕ) → if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0) ∈ V)
208207, 201fmptd 6351 . . . . . . . . . 10 ((𝜑𝑥 ∈ ℝ) → (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)):ℕ⟶V)
209 ffn 6012 . . . . . . . . . 10 ((𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)):ℕ⟶V → (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)) Fn ℕ)
210208, 209syl 17 . . . . . . . . 9 ((𝜑𝑥 ∈ ℝ) → (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)) Fn ℕ)
211 breq1 4626 . . . . . . . . . 10 (𝑤 = ((𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0))‘𝑚) → (𝑤 ≤ (𝐹𝑥) ↔ ((𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0))‘𝑚) ≤ (𝐹𝑥)))
212211ralrn 6328 . . . . . . . . 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)))
213210, 212syl 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)))
214206, 213mpbird 247 . . . . . . 7 ((𝜑𝑥 ∈ ℝ) → ∀𝑤 ∈ ran (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0))𝑤 ≤ (𝐹𝑥))
215117adantr 481 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℕ) → (𝐹𝑥) ∈ ℝ)
216 0re 10000 . . . . . . . . . . 11 0 ∈ ℝ
217 ifcl 4108 . . . . . . . . . . 11 (((𝐹𝑥) ∈ ℝ ∧ 0 ∈ ℝ) → if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0) ∈ ℝ)
218215, 216, 217sylancl 693 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ) ∧ 𝑛 ∈ ℕ) → if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0) ∈ ℝ)
219218, 201fmptd 6351 . . . . . . . . 9 ((𝜑𝑥 ∈ ℝ) → (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)):ℕ⟶ℝ)
220 frn 6020 . . . . . . . . 9 ((𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)):ℕ⟶ℝ → ran (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)) ⊆ ℝ)
221219, 220syl 17 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ) → ran (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)) ⊆ ℝ)
222 1nn 10991 . . . . . . . . . 10 1 ∈ ℕ
223201, 218dmmptd 5991 . . . . . . . . . 10 ((𝜑𝑥 ∈ ℝ) → dom (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)) = ℕ)
224222, 223syl5eleqr 2705 . . . . . . . . 9 ((𝜑𝑥 ∈ ℝ) → 1 ∈ dom (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)))
225 n0i 3902 . . . . . . . . . 10 (1 ∈ dom (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)) → ¬ dom (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)) = ∅)
226 dm0rn0 5312 . . . . . . . . . . 11 (dom (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)) = ∅ ↔ ran (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)) = ∅)
227226necon3bbii 2837 . . . . . . . . . 10 (¬ dom (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)) = ∅ ↔ ran (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)) ≠ ∅)
228225, 227sylib 208 . . . . . . . . 9 (1 ∈ dom (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)) → ran (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)) ≠ ∅)
229224, 228syl 17 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ) → ran (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)) ≠ ∅)
230 breq2 4627 . . . . . . . . . . 11 (𝑧 = (𝐹𝑥) → (𝑤𝑧𝑤 ≤ (𝐹𝑥)))
231230ralbidv 2982 . . . . . . . . . 10 (𝑧 = (𝐹𝑥) → (∀𝑤 ∈ ran (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0))𝑤𝑧 ↔ ∀𝑤 ∈ ran (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0))𝑤 ≤ (𝐹𝑥)))
232231rspcev 3299 . . . . . . . . 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)
233117, 214, 232syl2anc 692 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ) → ∃𝑧 ∈ ℝ ∀𝑤 ∈ ran (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0))𝑤𝑧)
234 suprleub 10949 . . . . . . . 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)))
235221, 229, 233, 117, 234syl31anc 1326 . . . . . . 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)))
236214, 235mpbird 247 . . . . . 6 ((𝜑𝑥 ∈ ℝ) → sup(ran (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)), ℝ, < ) ≤ (𝐹𝑥))
237 arch 11249 . . . . . . . . 9 ((𝐹𝑥) ∈ ℝ → ∃𝑚 ∈ ℕ (𝐹𝑥) < 𝑚)
238117, 237syl 17 . . . . . . . 8 ((𝜑𝑥 ∈ ℝ) → ∃𝑚 ∈ ℕ (𝐹𝑥) < 𝑚)
239202ad2antrl 763 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ) ∧ (𝑚 ∈ ℕ ∧ (𝐹𝑥) < 𝑚)) → ((𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0))‘𝑚) = if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0))
240 ltle 10086 . . . . . . . . . . . . 13 (((𝐹𝑥) ∈ ℝ ∧ 𝑚 ∈ ℝ) → ((𝐹𝑥) < 𝑚 → (𝐹𝑥) ≤ 𝑚))
241117, 48, 240syl2an 494 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℝ) ∧ 𝑚 ∈ ℕ) → ((𝐹𝑥) < 𝑚 → (𝐹𝑥) ≤ 𝑚))
242241impr 648 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ) ∧ (𝑚 ∈ ℕ ∧ (𝐹𝑥) < 𝑚)) → (𝐹𝑥) ≤ 𝑚)
243242iftrued 4072 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ) ∧ (𝑚 ∈ ℕ ∧ (𝐹𝑥) < 𝑚)) → if((𝐹𝑥) ≤ 𝑚, (𝐹𝑥), 0) = (𝐹𝑥))
244239, 243eqtrd 2655 . . . . . . . . 9 (((𝜑𝑥 ∈ ℝ) ∧ (𝑚 ∈ ℕ ∧ (𝐹𝑥) < 𝑚)) → ((𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0))‘𝑚) = (𝐹𝑥))
245210adantr 481 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ) ∧ (𝑚 ∈ ℕ ∧ (𝐹𝑥) < 𝑚)) → (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)) Fn ℕ)
246 simprl 793 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ) ∧ (𝑚 ∈ ℕ ∧ (𝐹𝑥) < 𝑚)) → 𝑚 ∈ ℕ)
247 fnfvelrn 6322 . . . . . . . . . 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)))
248245, 246, 247syl2anc 692 . . . . . . . . 9 (((𝜑𝑥 ∈ ℝ) ∧ (𝑚 ∈ ℕ ∧ (𝐹𝑥) < 𝑚)) → ((𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0))‘𝑚) ∈ ran (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)))
249244, 248eqeltrrd 2699 . . . . . . . 8 (((𝜑𝑥 ∈ ℝ) ∧ (𝑚 ∈ ℕ ∧ (𝐹𝑥) < 𝑚)) → (𝐹𝑥) ∈ ran (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)))
250238, 249rexlimddv 3030 . . . . . . 7 ((𝜑𝑥 ∈ ℝ) → (𝐹𝑥) ∈ ran (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)))
251 suprub 10944 . . . . . . 7 (((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) ∈ ran (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0))) → (𝐹𝑥) ≤ sup(ran (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)), ℝ, < ))
252221, 229, 233, 250, 251syl31anc 1326 . . . . . 6 ((𝜑𝑥 ∈ ℝ) → (𝐹𝑥) ≤ sup(ran (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)), ℝ, < ))
253 suprcl 10943 . . . . . . . 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) → sup(ran (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)), ℝ, < ) ∈ ℝ)
254221, 229, 233, 253syl3anc 1323 . . . . . . 7 ((𝜑𝑥 ∈ ℝ) → sup(ran (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)), ℝ, < ) ∈ ℝ)
255254, 117letri3d 10139 . . . . . 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, < ))))
256236, 252, 255mpbir2and 956 . . . . 5 ((𝜑𝑥 ∈ ℝ) → sup(ran (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)), ℝ, < ) = (𝐹𝑥))
257256mpteq2dva 4714 . . . 4 (𝜑 → (𝑥 ∈ ℝ ↦ sup(ran (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)), ℝ, < )) = (𝑥 ∈ ℝ ↦ (𝐹𝑥)))
258257, 172eqtr4d 2658 . . 3 (𝜑 → (𝑥 ∈ ℝ ↦ sup(ran (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)), ℝ, < )) = 𝐹)
259258fveq2d 6162 . 2 (𝜑 → (∫2‘(𝑥 ∈ ℝ ↦ sup(ran (𝑛 ∈ ℕ ↦ if((𝐹𝑥) ≤ 𝑛, (𝐹𝑥), 0)), ℝ, < ))) = (∫2𝐹))
260200, 259eqtr3d 2657 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 196  wa 384   = wceq 1480  wcel 1987  wne 2790  wral 2908  wrex 2909  Vcvv 3190  cdif 3557  wss 3560  c0 3897  ifcif 4064   class class class wbr 4623  cmpt 4683  ccnv 5083  dom cdm 5084  ran crn 5085  cres 5086  cima 5087   Fn wfn 5852  wf 5853  cfv 5857  (class class class)co 6615  𝑟 cofr 6861  supcsup 8306  cr 9895  0cc0 9896  1c1 9897   + caddc 9899  +∞cpnf 10031  *cxr 10033   < clt 10034  cle 10035  cn 10980  (,)cioo 12133  [,)cico 12135  volcvol 23172  MblFncmbf 23323  2citg2 23325
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 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-inf2 8498  ax-cc 9217  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973  ax-pre-sup 9974  ax-addf 9975
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 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-disj 4594  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-se 5044  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-isom 5866  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-of 6862  df-ofr 6863  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-2o 7521  df-oadd 7524  df-omul 7525  df-er 7702  df-map 7819  df-pm 7820  df-en 7916  df-dom 7917  df-sdom 7918  df-fin 7919  df-fi 8277  df-sup 8308  df-inf 8309  df-oi 8375  df-card 8725  df-acn 8728  df-cda 8950  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-div 10645  df-nn 10981  df-2 11039  df-3 11040  df-n0 11253  df-z 11338  df-uz 11648  df-q 11749  df-rp 11793  df-xneg 11906  df-xadd 11907  df-xmul 11908  df-ioo 12137  df-ioc 12138  df-ico 12139  df-icc 12140  df-fz 12285  df-fzo 12423  df-fl 12549  df-seq 12758  df-exp 12817  df-hash 13074  df-cj 13789  df-re 13790  df-im 13791  df-sqrt 13925  df-abs 13926  df-clim 14169  df-rlim 14170  df-sum 14367  df-rest 16023  df-topgen 16044  df-psmet 19678  df-xmet 19679  df-met 19680  df-bl 19681  df-mopn 19682  df-top 20639  df-topon 20656  df-bases 20690  df-cmp 21130  df-ovol 23173  df-vol 23174  df-mbf 23328  df-itg1 23329  df-itg2 23330
This theorem is referenced by:  itg2cn  23470
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