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Theorem mbfi1fseqlem3 23529
Description: Lemma for mbfi1fseq 23533. (Contributed by Mario Carneiro, 16-Aug-2014.)
Hypotheses
Ref Expression
mbfi1fseq.1 (𝜑𝐹 ∈ MblFn)
mbfi1fseq.2 (𝜑𝐹:ℝ⟶(0[,)+∞))
mbfi1fseq.3 𝐽 = (𝑚 ∈ ℕ, 𝑦 ∈ ℝ ↦ ((⌊‘((𝐹𝑦) · (2↑𝑚))) / (2↑𝑚)))
mbfi1fseq.4 𝐺 = (𝑚 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝑚[,]𝑚), if((𝑚𝐽𝑥) ≤ 𝑚, (𝑚𝐽𝑥), 𝑚), 0)))
Assertion
Ref Expression
mbfi1fseqlem3 ((𝜑𝐴 ∈ ℕ) → (𝐺𝐴):ℝ⟶ran (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴))))
Distinct variable groups:   𝑥,𝑚,𝑦,𝐹   𝑥,𝐺   𝑚,𝐽   𝜑,𝑚,𝑥,𝑦   𝐴,𝑚,𝑥,𝑦
Allowed substitution hints:   𝐺(𝑦,𝑚)   𝐽(𝑥,𝑦)

Proof of Theorem mbfi1fseqlem3
StepHypRef Expression
1 rge0ssre 12318 . . . . . . . . . . . . . . . . . . 19 (0[,)+∞) ⊆ ℝ
2 mbfi1fseq.2 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐹:ℝ⟶(0[,)+∞))
3 simpr 476 . . . . . . . . . . . . . . . . . . . 20 ((𝑚 ∈ ℕ ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℝ)
4 ffvelrn 6397 . . . . . . . . . . . . . . . . . . . 20 ((𝐹:ℝ⟶(0[,)+∞) ∧ 𝑦 ∈ ℝ) → (𝐹𝑦) ∈ (0[,)+∞))
52, 3, 4syl2an 493 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑦 ∈ ℝ)) → (𝐹𝑦) ∈ (0[,)+∞))
61, 5sseldi 3634 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑦 ∈ ℝ)) → (𝐹𝑦) ∈ ℝ)
7 2nn 11223 . . . . . . . . . . . . . . . . . . . . 21 2 ∈ ℕ
8 nnnn0 11337 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 ∈ ℕ → 𝑚 ∈ ℕ0)
9 nnexpcl 12913 . . . . . . . . . . . . . . . . . . . . 21 ((2 ∈ ℕ ∧ 𝑚 ∈ ℕ0) → (2↑𝑚) ∈ ℕ)
107, 8, 9sylancr 696 . . . . . . . . . . . . . . . . . . . 20 (𝑚 ∈ ℕ → (2↑𝑚) ∈ ℕ)
1110ad2antrl 764 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑦 ∈ ℝ)) → (2↑𝑚) ∈ ℕ)
1211nnred 11073 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑦 ∈ ℝ)) → (2↑𝑚) ∈ ℝ)
136, 12remulcld 10108 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑦 ∈ ℝ)) → ((𝐹𝑦) · (2↑𝑚)) ∈ ℝ)
14 reflcl 12637 . . . . . . . . . . . . . . . . 17 (((𝐹𝑦) · (2↑𝑚)) ∈ ℝ → (⌊‘((𝐹𝑦) · (2↑𝑚))) ∈ ℝ)
1513, 14syl 17 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑦 ∈ ℝ)) → (⌊‘((𝐹𝑦) · (2↑𝑚))) ∈ ℝ)
1615, 11nndivred 11107 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑦 ∈ ℝ)) → ((⌊‘((𝐹𝑦) · (2↑𝑚))) / (2↑𝑚)) ∈ ℝ)
1716ralrimivva 3000 . . . . . . . . . . . . . 14 (𝜑 → ∀𝑚 ∈ ℕ ∀𝑦 ∈ ℝ ((⌊‘((𝐹𝑦) · (2↑𝑚))) / (2↑𝑚)) ∈ ℝ)
18 mbfi1fseq.3 . . . . . . . . . . . . . . 15 𝐽 = (𝑚 ∈ ℕ, 𝑦 ∈ ℝ ↦ ((⌊‘((𝐹𝑦) · (2↑𝑚))) / (2↑𝑚)))
1918fmpt2 7282 . . . . . . . . . . . . . 14 (∀𝑚 ∈ ℕ ∀𝑦 ∈ ℝ ((⌊‘((𝐹𝑦) · (2↑𝑚))) / (2↑𝑚)) ∈ ℝ ↔ 𝐽:(ℕ × ℝ)⟶ℝ)
2017, 19sylib 208 . . . . . . . . . . . . 13 (𝜑𝐽:(ℕ × ℝ)⟶ℝ)
21 fovrn 6846 . . . . . . . . . . . . 13 ((𝐽:(ℕ × ℝ)⟶ℝ ∧ 𝐴 ∈ ℕ ∧ 𝑥 ∈ ℝ) → (𝐴𝐽𝑥) ∈ ℝ)
2220, 21syl3an1 1399 . . . . . . . . . . . 12 ((𝜑𝐴 ∈ ℕ ∧ 𝑥 ∈ ℝ) → (𝐴𝐽𝑥) ∈ ℝ)
23223expa 1284 . . . . . . . . . . 11 (((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐴𝐽𝑥) ∈ ℝ)
24 nnre 11065 . . . . . . . . . . . 12 (𝐴 ∈ ℕ → 𝐴 ∈ ℝ)
2524ad2antlr 763 . . . . . . . . . . 11 (((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 𝐴 ∈ ℝ)
26 nnnn0 11337 . . . . . . . . . . . . . 14 (𝐴 ∈ ℕ → 𝐴 ∈ ℕ0)
27 nnexpcl 12913 . . . . . . . . . . . . . 14 ((2 ∈ ℕ ∧ 𝐴 ∈ ℕ0) → (2↑𝐴) ∈ ℕ)
287, 26, 27sylancr 696 . . . . . . . . . . . . 13 (𝐴 ∈ ℕ → (2↑𝐴) ∈ ℕ)
2928ad2antlr 763 . . . . . . . . . . . 12 (((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (2↑𝐴) ∈ ℕ)
30 nnre 11065 . . . . . . . . . . . . 13 ((2↑𝐴) ∈ ℕ → (2↑𝐴) ∈ ℝ)
31 nngt0 11087 . . . . . . . . . . . . 13 ((2↑𝐴) ∈ ℕ → 0 < (2↑𝐴))
3230, 31jca 553 . . . . . . . . . . . 12 ((2↑𝐴) ∈ ℕ → ((2↑𝐴) ∈ ℝ ∧ 0 < (2↑𝐴)))
3329, 32syl 17 . . . . . . . . . . 11 (((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((2↑𝐴) ∈ ℝ ∧ 0 < (2↑𝐴)))
34 lemul1 10913 . . . . . . . . . . 11 (((𝐴𝐽𝑥) ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ ((2↑𝐴) ∈ ℝ ∧ 0 < (2↑𝐴))) → ((𝐴𝐽𝑥) ≤ 𝐴 ↔ ((𝐴𝐽𝑥) · (2↑𝐴)) ≤ (𝐴 · (2↑𝐴))))
3523, 25, 33, 34syl3anc 1366 . . . . . . . . . 10 (((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐴𝐽𝑥) ≤ 𝐴 ↔ ((𝐴𝐽𝑥) · (2↑𝐴)) ≤ (𝐴 · (2↑𝐴))))
3635biimpa 500 . . . . . . . . 9 ((((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → ((𝐴𝐽𝑥) · (2↑𝐴)) ≤ (𝐴 · (2↑𝐴)))
37 simplr 807 . . . . . . . . . . . . . . . 16 (((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 𝐴 ∈ ℕ)
3837adantr 480 . . . . . . . . . . . . . . 15 ((((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → 𝐴 ∈ ℕ)
39 simplr 807 . . . . . . . . . . . . . . 15 ((((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → 𝑥 ∈ ℝ)
40 simpr 476 . . . . . . . . . . . . . . . . . . . 20 ((𝑚 = 𝐴𝑦 = 𝑥) → 𝑦 = 𝑥)
4140fveq2d 6233 . . . . . . . . . . . . . . . . . . 19 ((𝑚 = 𝐴𝑦 = 𝑥) → (𝐹𝑦) = (𝐹𝑥))
42 simpl 472 . . . . . . . . . . . . . . . . . . . 20 ((𝑚 = 𝐴𝑦 = 𝑥) → 𝑚 = 𝐴)
4342oveq2d 6706 . . . . . . . . . . . . . . . . . . 19 ((𝑚 = 𝐴𝑦 = 𝑥) → (2↑𝑚) = (2↑𝐴))
4441, 43oveq12d 6708 . . . . . . . . . . . . . . . . . 18 ((𝑚 = 𝐴𝑦 = 𝑥) → ((𝐹𝑦) · (2↑𝑚)) = ((𝐹𝑥) · (2↑𝐴)))
4544fveq2d 6233 . . . . . . . . . . . . . . . . 17 ((𝑚 = 𝐴𝑦 = 𝑥) → (⌊‘((𝐹𝑦) · (2↑𝑚))) = (⌊‘((𝐹𝑥) · (2↑𝐴))))
4645, 43oveq12d 6708 . . . . . . . . . . . . . . . 16 ((𝑚 = 𝐴𝑦 = 𝑥) → ((⌊‘((𝐹𝑦) · (2↑𝑚))) / (2↑𝑚)) = ((⌊‘((𝐹𝑥) · (2↑𝐴))) / (2↑𝐴)))
47 ovex 6718 . . . . . . . . . . . . . . . 16 ((⌊‘((𝐹𝑥) · (2↑𝐴))) / (2↑𝐴)) ∈ V
4846, 18, 47ovmpt2a 6833 . . . . . . . . . . . . . . 15 ((𝐴 ∈ ℕ ∧ 𝑥 ∈ ℝ) → (𝐴𝐽𝑥) = ((⌊‘((𝐹𝑥) · (2↑𝐴))) / (2↑𝐴)))
4938, 39, 48syl2anc 694 . . . . . . . . . . . . . 14 ((((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → (𝐴𝐽𝑥) = ((⌊‘((𝐹𝑥) · (2↑𝐴))) / (2↑𝐴)))
5049oveq1d 6705 . . . . . . . . . . . . 13 ((((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → ((𝐴𝐽𝑥) · (2↑𝐴)) = (((⌊‘((𝐹𝑥) · (2↑𝐴))) / (2↑𝐴)) · (2↑𝐴)))
512adantr 480 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝐴 ∈ ℕ) → 𝐹:ℝ⟶(0[,)+∞))
5251ffvelrnda 6399 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐹𝑥) ∈ (0[,)+∞))
53 elrege0 12316 . . . . . . . . . . . . . . . . . . . 20 ((𝐹𝑥) ∈ (0[,)+∞) ↔ ((𝐹𝑥) ∈ ℝ ∧ 0 ≤ (𝐹𝑥)))
5452, 53sylib 208 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹𝑥) ∈ ℝ ∧ 0 ≤ (𝐹𝑥)))
5554simpld 474 . . . . . . . . . . . . . . . . . 18 (((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐹𝑥) ∈ ℝ)
5629nnred 11073 . . . . . . . . . . . . . . . . . 18 (((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (2↑𝐴) ∈ ℝ)
5755, 56remulcld 10108 . . . . . . . . . . . . . . . . 17 (((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹𝑥) · (2↑𝐴)) ∈ ℝ)
5829nnnn0d 11389 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (2↑𝐴) ∈ ℕ0)
5958nn0ge0d 11392 . . . . . . . . . . . . . . . . . 18 (((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 0 ≤ (2↑𝐴))
60 mulge0 10584 . . . . . . . . . . . . . . . . . 18 ((((𝐹𝑥) ∈ ℝ ∧ 0 ≤ (𝐹𝑥)) ∧ ((2↑𝐴) ∈ ℝ ∧ 0 ≤ (2↑𝐴))) → 0 ≤ ((𝐹𝑥) · (2↑𝐴)))
6154, 56, 59, 60syl12anc 1364 . . . . . . . . . . . . . . . . 17 (((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 0 ≤ ((𝐹𝑥) · (2↑𝐴)))
62 flge0nn0 12661 . . . . . . . . . . . . . . . . 17 ((((𝐹𝑥) · (2↑𝐴)) ∈ ℝ ∧ 0 ≤ ((𝐹𝑥) · (2↑𝐴))) → (⌊‘((𝐹𝑥) · (2↑𝐴))) ∈ ℕ0)
6357, 61, 62syl2anc 694 . . . . . . . . . . . . . . . 16 (((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (⌊‘((𝐹𝑥) · (2↑𝐴))) ∈ ℕ0)
6463adantr 480 . . . . . . . . . . . . . . 15 ((((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → (⌊‘((𝐹𝑥) · (2↑𝐴))) ∈ ℕ0)
6564nn0cnd 11391 . . . . . . . . . . . . . 14 ((((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → (⌊‘((𝐹𝑥) · (2↑𝐴))) ∈ ℂ)
6629adantr 480 . . . . . . . . . . . . . . 15 ((((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → (2↑𝐴) ∈ ℕ)
6766nncnd 11074 . . . . . . . . . . . . . 14 ((((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → (2↑𝐴) ∈ ℂ)
6866nnne0d 11103 . . . . . . . . . . . . . 14 ((((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → (2↑𝐴) ≠ 0)
6965, 67, 68divcan1d 10840 . . . . . . . . . . . . 13 ((((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → (((⌊‘((𝐹𝑥) · (2↑𝐴))) / (2↑𝐴)) · (2↑𝐴)) = (⌊‘((𝐹𝑥) · (2↑𝐴))))
7050, 69eqtrd 2685 . . . . . . . . . . . 12 ((((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → ((𝐴𝐽𝑥) · (2↑𝐴)) = (⌊‘((𝐹𝑥) · (2↑𝐴))))
7170, 64eqeltrd 2730 . . . . . . . . . . 11 ((((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → ((𝐴𝐽𝑥) · (2↑𝐴)) ∈ ℕ0)
72 nn0uz 11760 . . . . . . . . . . 11 0 = (ℤ‘0)
7371, 72syl6eleq 2740 . . . . . . . . . 10 ((((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → ((𝐴𝐽𝑥) · (2↑𝐴)) ∈ (ℤ‘0))
74 nnmulcl 11081 . . . . . . . . . . . . . 14 ((𝐴 ∈ ℕ ∧ (2↑𝐴) ∈ ℕ) → (𝐴 · (2↑𝐴)) ∈ ℕ)
7528, 74mpdan 703 . . . . . . . . . . . . 13 (𝐴 ∈ ℕ → (𝐴 · (2↑𝐴)) ∈ ℕ)
7675ad2antlr 763 . . . . . . . . . . . 12 (((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐴 · (2↑𝐴)) ∈ ℕ)
7776adantr 480 . . . . . . . . . . 11 ((((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → (𝐴 · (2↑𝐴)) ∈ ℕ)
7877nnzd 11519 . . . . . . . . . 10 ((((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → (𝐴 · (2↑𝐴)) ∈ ℤ)
79 elfz5 12372 . . . . . . . . . 10 ((((𝐴𝐽𝑥) · (2↑𝐴)) ∈ (ℤ‘0) ∧ (𝐴 · (2↑𝐴)) ∈ ℤ) → (((𝐴𝐽𝑥) · (2↑𝐴)) ∈ (0...(𝐴 · (2↑𝐴))) ↔ ((𝐴𝐽𝑥) · (2↑𝐴)) ≤ (𝐴 · (2↑𝐴))))
8073, 78, 79syl2anc 694 . . . . . . . . 9 ((((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → (((𝐴𝐽𝑥) · (2↑𝐴)) ∈ (0...(𝐴 · (2↑𝐴))) ↔ ((𝐴𝐽𝑥) · (2↑𝐴)) ≤ (𝐴 · (2↑𝐴))))
8136, 80mpbird 247 . . . . . . . 8 ((((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → ((𝐴𝐽𝑥) · (2↑𝐴)) ∈ (0...(𝐴 · (2↑𝐴))))
82 oveq1 6697 . . . . . . . . 9 (𝑚 = ((𝐴𝐽𝑥) · (2↑𝐴)) → (𝑚 / (2↑𝐴)) = (((𝐴𝐽𝑥) · (2↑𝐴)) / (2↑𝐴)))
83 eqid 2651 . . . . . . . . 9 (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴))) = (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴)))
84 ovex 6718 . . . . . . . . 9 (((𝐴𝐽𝑥) · (2↑𝐴)) / (2↑𝐴)) ∈ V
8582, 83, 84fvmpt 6321 . . . . . . . 8 (((𝐴𝐽𝑥) · (2↑𝐴)) ∈ (0...(𝐴 · (2↑𝐴))) → ((𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴)))‘((𝐴𝐽𝑥) · (2↑𝐴))) = (((𝐴𝐽𝑥) · (2↑𝐴)) / (2↑𝐴)))
8681, 85syl 17 . . . . . . 7 ((((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → ((𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴)))‘((𝐴𝐽𝑥) · (2↑𝐴))) = (((𝐴𝐽𝑥) · (2↑𝐴)) / (2↑𝐴)))
8723adantr 480 . . . . . . . . 9 ((((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → (𝐴𝐽𝑥) ∈ ℝ)
8887recnd 10106 . . . . . . . 8 ((((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → (𝐴𝐽𝑥) ∈ ℂ)
8988, 67, 68divcan4d 10845 . . . . . . 7 ((((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → (((𝐴𝐽𝑥) · (2↑𝐴)) / (2↑𝐴)) = (𝐴𝐽𝑥))
9086, 89eqtrd 2685 . . . . . 6 ((((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → ((𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴)))‘((𝐴𝐽𝑥) · (2↑𝐴))) = (𝐴𝐽𝑥))
91 elfznn0 12471 . . . . . . . . . . . . 13 (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) → 𝑚 ∈ ℕ0)
9291nn0red 11390 . . . . . . . . . . . 12 (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) → 𝑚 ∈ ℝ)
9328adantl 481 . . . . . . . . . . . 12 ((𝜑𝐴 ∈ ℕ) → (2↑𝐴) ∈ ℕ)
94 nndivre 11094 . . . . . . . . . . . 12 ((𝑚 ∈ ℝ ∧ (2↑𝐴) ∈ ℕ) → (𝑚 / (2↑𝐴)) ∈ ℝ)
9592, 93, 94syl2anr 494 . . . . . . . . . . 11 (((𝜑𝐴 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐴 · (2↑𝐴)))) → (𝑚 / (2↑𝐴)) ∈ ℝ)
9695, 83fmptd 6425 . . . . . . . . . 10 ((𝜑𝐴 ∈ ℕ) → (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴))):(0...(𝐴 · (2↑𝐴)))⟶ℝ)
97 ffn 6083 . . . . . . . . . 10 ((𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴))):(0...(𝐴 · (2↑𝐴)))⟶ℝ → (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴))) Fn (0...(𝐴 · (2↑𝐴))))
9896, 97syl 17 . . . . . . . . 9 ((𝜑𝐴 ∈ ℕ) → (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴))) Fn (0...(𝐴 · (2↑𝐴))))
9998adantr 480 . . . . . . . 8 (((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴))) Fn (0...(𝐴 · (2↑𝐴))))
10099adantr 480 . . . . . . 7 ((((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴))) Fn (0...(𝐴 · (2↑𝐴))))
101 fnfvelrn 6396 . . . . . . 7 (((𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴))) Fn (0...(𝐴 · (2↑𝐴))) ∧ ((𝐴𝐽𝑥) · (2↑𝐴)) ∈ (0...(𝐴 · (2↑𝐴)))) → ((𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴)))‘((𝐴𝐽𝑥) · (2↑𝐴))) ∈ ran (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴))))
102100, 81, 101syl2anc 694 . . . . . 6 ((((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → ((𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴)))‘((𝐴𝐽𝑥) · (2↑𝐴))) ∈ ran (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴))))
10390, 102eqeltrrd 2731 . . . . 5 ((((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → (𝐴𝐽𝑥) ∈ ran (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴))))
10476nnnn0d 11389 . . . . . . . . . . 11 (((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐴 · (2↑𝐴)) ∈ ℕ0)
105104, 72syl6eleq 2740 . . . . . . . . . 10 (((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐴 · (2↑𝐴)) ∈ (ℤ‘0))
106 eluzfz2 12387 . . . . . . . . . 10 ((𝐴 · (2↑𝐴)) ∈ (ℤ‘0) → (𝐴 · (2↑𝐴)) ∈ (0...(𝐴 · (2↑𝐴))))
107105, 106syl 17 . . . . . . . . 9 (((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐴 · (2↑𝐴)) ∈ (0...(𝐴 · (2↑𝐴))))
108 oveq1 6697 . . . . . . . . . 10 (𝑚 = (𝐴 · (2↑𝐴)) → (𝑚 / (2↑𝐴)) = ((𝐴 · (2↑𝐴)) / (2↑𝐴)))
109 ovex 6718 . . . . . . . . . 10 ((𝐴 · (2↑𝐴)) / (2↑𝐴)) ∈ V
110108, 83, 109fvmpt 6321 . . . . . . . . 9 ((𝐴 · (2↑𝐴)) ∈ (0...(𝐴 · (2↑𝐴))) → ((𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴)))‘(𝐴 · (2↑𝐴))) = ((𝐴 · (2↑𝐴)) / (2↑𝐴)))
111107, 110syl 17 . . . . . . . 8 (((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴)))‘(𝐴 · (2↑𝐴))) = ((𝐴 · (2↑𝐴)) / (2↑𝐴)))
11225recnd 10106 . . . . . . . . 9 (((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 𝐴 ∈ ℂ)
11329nncnd 11074 . . . . . . . . 9 (((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (2↑𝐴) ∈ ℂ)
11429nnne0d 11103 . . . . . . . . 9 (((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (2↑𝐴) ≠ 0)
115112, 113, 114divcan4d 10845 . . . . . . . 8 (((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐴 · (2↑𝐴)) / (2↑𝐴)) = 𝐴)
116111, 115eqtrd 2685 . . . . . . 7 (((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴)))‘(𝐴 · (2↑𝐴))) = 𝐴)
117 fnfvelrn 6396 . . . . . . . 8 (((𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴))) Fn (0...(𝐴 · (2↑𝐴))) ∧ (𝐴 · (2↑𝐴)) ∈ (0...(𝐴 · (2↑𝐴)))) → ((𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴)))‘(𝐴 · (2↑𝐴))) ∈ ran (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴))))
11899, 107, 117syl2anc 694 . . . . . . 7 (((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴)))‘(𝐴 · (2↑𝐴))) ∈ ran (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴))))
119116, 118eqeltrrd 2731 . . . . . 6 (((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 𝐴 ∈ ran (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴))))
120119adantr 480 . . . . 5 ((((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ ¬ (𝐴𝐽𝑥) ≤ 𝐴) → 𝐴 ∈ ran (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴))))
121103, 120ifclda 4153 . . . 4 (((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴) ∈ ran (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴))))
122 eluzfz1 12386 . . . . . . . 8 ((𝐴 · (2↑𝐴)) ∈ (ℤ‘0) → 0 ∈ (0...(𝐴 · (2↑𝐴))))
123105, 122syl 17 . . . . . . 7 (((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 0 ∈ (0...(𝐴 · (2↑𝐴))))
124 oveq1 6697 . . . . . . . 8 (𝑚 = 0 → (𝑚 / (2↑𝐴)) = (0 / (2↑𝐴)))
125 ovex 6718 . . . . . . . 8 (0 / (2↑𝐴)) ∈ V
126124, 83, 125fvmpt 6321 . . . . . . 7 (0 ∈ (0...(𝐴 · (2↑𝐴))) → ((𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴)))‘0) = (0 / (2↑𝐴)))
127123, 126syl 17 . . . . . 6 (((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴)))‘0) = (0 / (2↑𝐴)))
128 nncn 11066 . . . . . . . 8 ((2↑𝐴) ∈ ℕ → (2↑𝐴) ∈ ℂ)
129 nnne0 11091 . . . . . . . 8 ((2↑𝐴) ∈ ℕ → (2↑𝐴) ≠ 0)
130128, 129div0d 10838 . . . . . . 7 ((2↑𝐴) ∈ ℕ → (0 / (2↑𝐴)) = 0)
13129, 130syl 17 . . . . . 6 (((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (0 / (2↑𝐴)) = 0)
132127, 131eqtrd 2685 . . . . 5 (((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴)))‘0) = 0)
133 fnfvelrn 6396 . . . . . 6 (((𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴))) Fn (0...(𝐴 · (2↑𝐴))) ∧ 0 ∈ (0...(𝐴 · (2↑𝐴)))) → ((𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴)))‘0) ∈ ran (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴))))
13499, 123, 133syl2anc 694 . . . . 5 (((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴)))‘0) ∈ ran (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴))))
135132, 134eqeltrrd 2731 . . . 4 (((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 0 ∈ ran (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴))))
136121, 135ifcld 4164 . . 3 (((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → if(𝑥 ∈ (-𝐴[,]𝐴), if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴), 0) ∈ ran (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴))))
137 eqid 2651 . . 3 (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝐴[,]𝐴), if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴), 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝐴[,]𝐴), if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴), 0))
138136, 137fmptd 6425 . 2 ((𝜑𝐴 ∈ ℕ) → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝐴[,]𝐴), if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴), 0)):ℝ⟶ran (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴))))
139 mbfi1fseq.1 . . . . 5 (𝜑𝐹 ∈ MblFn)
140 mbfi1fseq.4 . . . . 5 𝐺 = (𝑚 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝑚[,]𝑚), if((𝑚𝐽𝑥) ≤ 𝑚, (𝑚𝐽𝑥), 𝑚), 0)))
141139, 2, 18, 140mbfi1fseqlem2 23528 . . . 4 (𝐴 ∈ ℕ → (𝐺𝐴) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝐴[,]𝐴), if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴), 0)))
142141adantl 481 . . 3 ((𝜑𝐴 ∈ ℕ) → (𝐺𝐴) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝐴[,]𝐴), if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴), 0)))
143142feq1d 6068 . 2 ((𝜑𝐴 ∈ ℕ) → ((𝐺𝐴):ℝ⟶ran (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴))) ↔ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝐴[,]𝐴), if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴), 0)):ℝ⟶ran (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴)))))
144138, 143mpbird 247 1 ((𝜑𝐴 ∈ ℕ) → (𝐺𝐴):ℝ⟶ran (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wral 2941  ifcif 4119   class class class wbr 4685  cmpt 4762   × cxp 5141  ran crn 5144   Fn wfn 5921  wf 5922  cfv 5926  (class class class)co 6690  cmpt2 6692  cr 9973  0cc0 9974   · cmul 9979  +∞cpnf 10109   < clt 10112  cle 10113  -cneg 10305   / cdiv 10722  cn 11058  2c2 11108  0cn0 11330  cz 11415  cuz 11725  [,)cico 12215  [,]cicc 12216  ...cfz 12364  cfl 12631  cexp 12900  MblFncmbf 23428
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-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-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-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-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-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-sup 8389  df-inf 8390  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-n0 11331  df-z 11416  df-uz 11726  df-ico 12219  df-fz 12365  df-fl 12633  df-seq 12842  df-exp 12901
This theorem is referenced by:  mbfi1fseqlem4  23530
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