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Theorem mbfi1fseqlem3 23385
Description: Lemma for mbfi1fseq 23389. (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 12219 . . . . . . . . . . . . . . . . . . 19 (0[,)+∞) ⊆ ℝ
2 mbfi1fseq.2 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐹:ℝ⟶(0[,)+∞))
3 simpr 477 . . . . . . . . . . . . . . . . . . . 20 ((𝑚 ∈ ℕ ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℝ)
4 ffvelrn 6314 . . . . . . . . . . . . . . . . . . . 20 ((𝐹:ℝ⟶(0[,)+∞) ∧ 𝑦 ∈ ℝ) → (𝐹𝑦) ∈ (0[,)+∞))
52, 3, 4syl2an 494 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑦 ∈ ℝ)) → (𝐹𝑦) ∈ (0[,)+∞))
61, 5sseldi 3586 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑦 ∈ ℝ)) → (𝐹𝑦) ∈ ℝ)
7 2nn 11130 . . . . . . . . . . . . . . . . . . . . 21 2 ∈ ℕ
8 nnnn0 11244 . . . . . . . . . . . . . . . . . . . . 21 (𝑚 ∈ ℕ → 𝑚 ∈ ℕ0)
9 nnexpcl 12810 . . . . . . . . . . . . . . . . . . . . 21 ((2 ∈ ℕ ∧ 𝑚 ∈ ℕ0) → (2↑𝑚) ∈ ℕ)
107, 8, 9sylancr 694 . . . . . . . . . . . . . . . . . . . 20 (𝑚 ∈ ℕ → (2↑𝑚) ∈ ℕ)
1110ad2antrl 763 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑦 ∈ ℝ)) → (2↑𝑚) ∈ ℕ)
1211nnred 10980 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑦 ∈ ℝ)) → (2↑𝑚) ∈ ℝ)
136, 12remulcld 10015 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑦 ∈ ℝ)) → ((𝐹𝑦) · (2↑𝑚)) ∈ ℝ)
14 reflcl 12534 . . . . . . . . . . . . . . . . 17 (((𝐹𝑦) · (2↑𝑚)) ∈ ℝ → (⌊‘((𝐹𝑦) · (2↑𝑚))) ∈ ℝ)
1513, 14syl 17 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑦 ∈ ℝ)) → (⌊‘((𝐹𝑦) · (2↑𝑚))) ∈ ℝ)
1615, 11nndivred 11014 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑦 ∈ ℝ)) → ((⌊‘((𝐹𝑦) · (2↑𝑚))) / (2↑𝑚)) ∈ ℝ)
1716ralrimivva 2970 . . . . . . . . . . . . . 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 7183 . . . . . . . . . . . . . 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 6758 . . . . . . . . . . . . 13 ((𝐽:(ℕ × ℝ)⟶ℝ ∧ 𝐴 ∈ ℕ ∧ 𝑥 ∈ ℝ) → (𝐴𝐽𝑥) ∈ ℝ)
2220, 21syl3an1 1356 . . . . . . . . . . . 12 ((𝜑𝐴 ∈ ℕ ∧ 𝑥 ∈ ℝ) → (𝐴𝐽𝑥) ∈ ℝ)
23223expa 1262 . . . . . . . . . . 11 (((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐴𝐽𝑥) ∈ ℝ)
24 nnre 10972 . . . . . . . . . . . 12 (𝐴 ∈ ℕ → 𝐴 ∈ ℝ)
2524ad2antlr 762 . . . . . . . . . . 11 (((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 𝐴 ∈ ℝ)
26 nnnn0 11244 . . . . . . . . . . . . . 14 (𝐴 ∈ ℕ → 𝐴 ∈ ℕ0)
27 nnexpcl 12810 . . . . . . . . . . . . . 14 ((2 ∈ ℕ ∧ 𝐴 ∈ ℕ0) → (2↑𝐴) ∈ ℕ)
287, 26, 27sylancr 694 . . . . . . . . . . . . 13 (𝐴 ∈ ℕ → (2↑𝐴) ∈ ℕ)
2928ad2antlr 762 . . . . . . . . . . . 12 (((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (2↑𝐴) ∈ ℕ)
30 nnre 10972 . . . . . . . . . . . . 13 ((2↑𝐴) ∈ ℕ → (2↑𝐴) ∈ ℝ)
31 nngt0 10994 . . . . . . . . . . . . 13 ((2↑𝐴) ∈ ℕ → 0 < (2↑𝐴))
3230, 31jca 554 . . . . . . . . . . . 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 10820 . . . . . . . . . . 11 (((𝐴𝐽𝑥) ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ ((2↑𝐴) ∈ ℝ ∧ 0 < (2↑𝐴))) → ((𝐴𝐽𝑥) ≤ 𝐴 ↔ ((𝐴𝐽𝑥) · (2↑𝐴)) ≤ (𝐴 · (2↑𝐴))))
3523, 25, 33, 34syl3anc 1323 . . . . . . . . . 10 (((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐴𝐽𝑥) ≤ 𝐴 ↔ ((𝐴𝐽𝑥) · (2↑𝐴)) ≤ (𝐴 · (2↑𝐴))))
3635biimpa 501 . . . . . . . . 9 ((((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → ((𝐴𝐽𝑥) · (2↑𝐴)) ≤ (𝐴 · (2↑𝐴)))
37 simplr 791 . . . . . . . . . . . . . . . 16 (((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 𝐴 ∈ ℕ)
3837adantr 481 . . . . . . . . . . . . . . 15 ((((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → 𝐴 ∈ ℕ)
39 simplr 791 . . . . . . . . . . . . . . 15 ((((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → 𝑥 ∈ ℝ)
40 simpr 477 . . . . . . . . . . . . . . . . . . . 20 ((𝑚 = 𝐴𝑦 = 𝑥) → 𝑦 = 𝑥)
4140fveq2d 6154 . . . . . . . . . . . . . . . . . . 19 ((𝑚 = 𝐴𝑦 = 𝑥) → (𝐹𝑦) = (𝐹𝑥))
42 simpl 473 . . . . . . . . . . . . . . . . . . . 20 ((𝑚 = 𝐴𝑦 = 𝑥) → 𝑚 = 𝐴)
4342oveq2d 6621 . . . . . . . . . . . . . . . . . . 19 ((𝑚 = 𝐴𝑦 = 𝑥) → (2↑𝑚) = (2↑𝐴))
4441, 43oveq12d 6623 . . . . . . . . . . . . . . . . . 18 ((𝑚 = 𝐴𝑦 = 𝑥) → ((𝐹𝑦) · (2↑𝑚)) = ((𝐹𝑥) · (2↑𝐴)))
4544fveq2d 6154 . . . . . . . . . . . . . . . . 17 ((𝑚 = 𝐴𝑦 = 𝑥) → (⌊‘((𝐹𝑦) · (2↑𝑚))) = (⌊‘((𝐹𝑥) · (2↑𝐴))))
4645, 43oveq12d 6623 . . . . . . . . . . . . . . . 16 ((𝑚 = 𝐴𝑦 = 𝑥) → ((⌊‘((𝐹𝑦) · (2↑𝑚))) / (2↑𝑚)) = ((⌊‘((𝐹𝑥) · (2↑𝐴))) / (2↑𝐴)))
47 ovex 6633 . . . . . . . . . . . . . . . 16 ((⌊‘((𝐹𝑥) · (2↑𝐴))) / (2↑𝐴)) ∈ V
4846, 18, 47ovmpt2a 6745 . . . . . . . . . . . . . . 15 ((𝐴 ∈ ℕ ∧ 𝑥 ∈ ℝ) → (𝐴𝐽𝑥) = ((⌊‘((𝐹𝑥) · (2↑𝐴))) / (2↑𝐴)))
4938, 39, 48syl2anc 692 . . . . . . . . . . . . . 14 ((((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → (𝐴𝐽𝑥) = ((⌊‘((𝐹𝑥) · (2↑𝐴))) / (2↑𝐴)))
5049oveq1d 6620 . . . . . . . . . . . . 13 ((((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → ((𝐴𝐽𝑥) · (2↑𝐴)) = (((⌊‘((𝐹𝑥) · (2↑𝐴))) / (2↑𝐴)) · (2↑𝐴)))
512adantr 481 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝐴 ∈ ℕ) → 𝐹:ℝ⟶(0[,)+∞))
5251ffvelrnda 6316 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐹𝑥) ∈ (0[,)+∞))
53 elrege0 12217 . . . . . . . . . . . . . . . . . . . 20 ((𝐹𝑥) ∈ (0[,)+∞) ↔ ((𝐹𝑥) ∈ ℝ ∧ 0 ≤ (𝐹𝑥)))
5452, 53sylib 208 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹𝑥) ∈ ℝ ∧ 0 ≤ (𝐹𝑥)))
5554simpld 475 . . . . . . . . . . . . . . . . . 18 (((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐹𝑥) ∈ ℝ)
5629nnred 10980 . . . . . . . . . . . . . . . . . 18 (((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (2↑𝐴) ∈ ℝ)
5755, 56remulcld 10015 . . . . . . . . . . . . . . . . 17 (((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹𝑥) · (2↑𝐴)) ∈ ℝ)
5829nnnn0d 11296 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (2↑𝐴) ∈ ℕ0)
5958nn0ge0d 11299 . . . . . . . . . . . . . . . . . 18 (((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 0 ≤ (2↑𝐴))
60 mulge0 10491 . . . . . . . . . . . . . . . . . 18 ((((𝐹𝑥) ∈ ℝ ∧ 0 ≤ (𝐹𝑥)) ∧ ((2↑𝐴) ∈ ℝ ∧ 0 ≤ (2↑𝐴))) → 0 ≤ ((𝐹𝑥) · (2↑𝐴)))
6154, 56, 59, 60syl12anc 1321 . . . . . . . . . . . . . . . . 17 (((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 0 ≤ ((𝐹𝑥) · (2↑𝐴)))
62 flge0nn0 12558 . . . . . . . . . . . . . . . . 17 ((((𝐹𝑥) · (2↑𝐴)) ∈ ℝ ∧ 0 ≤ ((𝐹𝑥) · (2↑𝐴))) → (⌊‘((𝐹𝑥) · (2↑𝐴))) ∈ ℕ0)
6357, 61, 62syl2anc 692 . . . . . . . . . . . . . . . 16 (((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (⌊‘((𝐹𝑥) · (2↑𝐴))) ∈ ℕ0)
6463adantr 481 . . . . . . . . . . . . . . 15 ((((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → (⌊‘((𝐹𝑥) · (2↑𝐴))) ∈ ℕ0)
6564nn0cnd 11298 . . . . . . . . . . . . . 14 ((((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → (⌊‘((𝐹𝑥) · (2↑𝐴))) ∈ ℂ)
6629adantr 481 . . . . . . . . . . . . . . 15 ((((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → (2↑𝐴) ∈ ℕ)
6766nncnd 10981 . . . . . . . . . . . . . 14 ((((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → (2↑𝐴) ∈ ℂ)
6866nnne0d 11010 . . . . . . . . . . . . . 14 ((((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → (2↑𝐴) ≠ 0)
6965, 67, 68divcan1d 10747 . . . . . . . . . . . . 13 ((((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → (((⌊‘((𝐹𝑥) · (2↑𝐴))) / (2↑𝐴)) · (2↑𝐴)) = (⌊‘((𝐹𝑥) · (2↑𝐴))))
7050, 69eqtrd 2660 . . . . . . . . . . . 12 ((((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → ((𝐴𝐽𝑥) · (2↑𝐴)) = (⌊‘((𝐹𝑥) · (2↑𝐴))))
7170, 64eqeltrd 2704 . . . . . . . . . . 11 ((((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → ((𝐴𝐽𝑥) · (2↑𝐴)) ∈ ℕ0)
72 nn0uz 11666 . . . . . . . . . . 11 0 = (ℤ‘0)
7371, 72syl6eleq 2714 . . . . . . . . . 10 ((((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → ((𝐴𝐽𝑥) · (2↑𝐴)) ∈ (ℤ‘0))
74 nnmulcl 10988 . . . . . . . . . . . . . 14 ((𝐴 ∈ ℕ ∧ (2↑𝐴) ∈ ℕ) → (𝐴 · (2↑𝐴)) ∈ ℕ)
7528, 74mpdan 701 . . . . . . . . . . . . 13 (𝐴 ∈ ℕ → (𝐴 · (2↑𝐴)) ∈ ℕ)
7675ad2antlr 762 . . . . . . . . . . . 12 (((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐴 · (2↑𝐴)) ∈ ℕ)
7776adantr 481 . . . . . . . . . . 11 ((((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → (𝐴 · (2↑𝐴)) ∈ ℕ)
7877nnzd 11425 . . . . . . . . . 10 ((((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → (𝐴 · (2↑𝐴)) ∈ ℤ)
79 elfz5 12273 . . . . . . . . . 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 692 . . . . . . . . 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 6612 . . . . . . . . 9 (𝑚 = ((𝐴𝐽𝑥) · (2↑𝐴)) → (𝑚 / (2↑𝐴)) = (((𝐴𝐽𝑥) · (2↑𝐴)) / (2↑𝐴)))
83 eqid 2626 . . . . . . . . 9 (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴))) = (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴)))
84 ovex 6633 . . . . . . . . 9 (((𝐴𝐽𝑥) · (2↑𝐴)) / (2↑𝐴)) ∈ V
8582, 83, 84fvmpt 6240 . . . . . . . 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 481 . . . . . . . . 9 ((((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → (𝐴𝐽𝑥) ∈ ℝ)
8887recnd 10013 . . . . . . . 8 ((((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → (𝐴𝐽𝑥) ∈ ℂ)
8988, 67, 68divcan4d 10752 . . . . . . 7 ((((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → (((𝐴𝐽𝑥) · (2↑𝐴)) / (2↑𝐴)) = (𝐴𝐽𝑥))
9086, 89eqtrd 2660 . . . . . 6 ((((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → ((𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴)))‘((𝐴𝐽𝑥) · (2↑𝐴))) = (𝐴𝐽𝑥))
91 elfznn0 12371 . . . . . . . . . . . . 13 (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) → 𝑚 ∈ ℕ0)
9291nn0red 11297 . . . . . . . . . . . 12 (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) → 𝑚 ∈ ℝ)
9328adantl 482 . . . . . . . . . . . 12 ((𝜑𝐴 ∈ ℕ) → (2↑𝐴) ∈ ℕ)
94 nndivre 11001 . . . . . . . . . . . 12 ((𝑚 ∈ ℝ ∧ (2↑𝐴) ∈ ℕ) → (𝑚 / (2↑𝐴)) ∈ ℝ)
9592, 93, 94syl2anr 495 . . . . . . . . . . 11 (((𝜑𝐴 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝐴 · (2↑𝐴)))) → (𝑚 / (2↑𝐴)) ∈ ℝ)
9695, 83fmptd 6341 . . . . . . . . . 10 ((𝜑𝐴 ∈ ℕ) → (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴))):(0...(𝐴 · (2↑𝐴)))⟶ℝ)
97 ffn 6004 . . . . . . . . . 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 481 . . . . . . . 8 (((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴))) Fn (0...(𝐴 · (2↑𝐴))))
10099adantr 481 . . . . . . 7 ((((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴))) Fn (0...(𝐴 · (2↑𝐴))))
101 fnfvelrn 6313 . . . . . . 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 692 . . . . . 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 2705 . . . . 5 ((((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ (𝐴𝐽𝑥) ≤ 𝐴) → (𝐴𝐽𝑥) ∈ ran (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴))))
10476nnnn0d 11296 . . . . . . . . . . 11 (((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐴 · (2↑𝐴)) ∈ ℕ0)
105104, 72syl6eleq 2714 . . . . . . . . . 10 (((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐴 · (2↑𝐴)) ∈ (ℤ‘0))
106 eluzfz2 12288 . . . . . . . . . 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 6612 . . . . . . . . . 10 (𝑚 = (𝐴 · (2↑𝐴)) → (𝑚 / (2↑𝐴)) = ((𝐴 · (2↑𝐴)) / (2↑𝐴)))
109 ovex 6633 . . . . . . . . . 10 ((𝐴 · (2↑𝐴)) / (2↑𝐴)) ∈ V
110108, 83, 109fvmpt 6240 . . . . . . . . 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 10013 . . . . . . . . 9 (((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 𝐴 ∈ ℂ)
11329nncnd 10981 . . . . . . . . 9 (((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (2↑𝐴) ∈ ℂ)
11429nnne0d 11010 . . . . . . . . 9 (((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (2↑𝐴) ≠ 0)
115112, 113, 114divcan4d 10752 . . . . . . . 8 (((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐴 · (2↑𝐴)) / (2↑𝐴)) = 𝐴)
116111, 115eqtrd 2660 . . . . . . 7 (((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴)))‘(𝐴 · (2↑𝐴))) = 𝐴)
117 fnfvelrn 6313 . . . . . . . 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 692 . . . . . . 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 2705 . . . . . 6 (((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 𝐴 ∈ ran (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴))))
120119adantr 481 . . . . 5 ((((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) ∧ ¬ (𝐴𝐽𝑥) ≤ 𝐴) → 𝐴 ∈ ran (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴))))
121103, 120ifclda 4097 . . . 4 (((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴) ∈ ran (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴))))
122 eluzfz1 12287 . . . . . . . 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 6612 . . . . . . . 8 (𝑚 = 0 → (𝑚 / (2↑𝐴)) = (0 / (2↑𝐴)))
125 ovex 6633 . . . . . . . 8 (0 / (2↑𝐴)) ∈ V
126124, 83, 125fvmpt 6240 . . . . . . 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 10973 . . . . . . . 8 ((2↑𝐴) ∈ ℕ → (2↑𝐴) ∈ ℂ)
129 nnne0 10998 . . . . . . . 8 ((2↑𝐴) ∈ ℕ → (2↑𝐴) ≠ 0)
130128, 129div0d 10745 . . . . . . 7 ((2↑𝐴) ∈ ℕ → (0 / (2↑𝐴)) = 0)
13129, 130syl 17 . . . . . 6 (((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (0 / (2↑𝐴)) = 0)
132127, 131eqtrd 2660 . . . . 5 (((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴)))‘0) = 0)
133 fnfvelrn 6313 . . . . . 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 692 . . . . 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 2705 . . . 4 (((𝜑𝐴 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 0 ∈ ran (𝑚 ∈ (0...(𝐴 · (2↑𝐴))) ↦ (𝑚 / (2↑𝐴))))
136121, 135ifcld 4108 . . 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 2626 . . 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 6341 . 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 23384 . . . 4 (𝐴 ∈ ℕ → (𝐺𝐴) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝐴[,]𝐴), if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴), 0)))
142141adantl 482 . . 3 ((𝜑𝐴 ∈ ℕ) → (𝐺𝐴) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝐴[,]𝐴), if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴), 0)))
143142feq1d 5989 . 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 384   = wceq 1480  wcel 1992  wral 2912  ifcif 4063   class class class wbr 4618  cmpt 4678   × cxp 5077  ran crn 5080   Fn wfn 5845  wf 5846  cfv 5850  (class class class)co 6605  cmpt2 6607  cr 9880  0cc0 9881   · cmul 9886  +∞cpnf 10016   < clt 10019  cle 10020  -cneg 10212   / cdiv 10629  cn 10965  2c2 11015  0cn0 11237  cz 11322  cuz 11631  [,)cico 12116  [,]cicc 12117  ...cfz 12265  cfl 12528  cexp 12797  MblFncmbf 23284
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957  ax-pre-mulgt0 9958  ax-pre-sup 9959
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-1st 7116  df-2nd 7117  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-er 7688  df-en 7901  df-dom 7902  df-sdom 7903  df-sup 8293  df-inf 8294  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-sub 10213  df-neg 10214  df-div 10630  df-nn 10966  df-2 11024  df-n0 11238  df-z 11323  df-uz 11632  df-ico 12120  df-fz 12266  df-fl 12530  df-seq 12739  df-exp 12798
This theorem is referenced by:  mbfi1fseqlem4  23386
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