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Theorem mbfi1fseqlem4 23291
 Description: Lemma for mbfi1fseq 23294. This lemma is not as interesting as it is long - it is simply checking that 𝐺 is in fact a sequence of simple functions, by verifying that its range is in (0...𝑛2↑𝑛) / (2↑𝑛) (which is to say, the numbers from 0 to 𝑛 in increments of 1 / (2↑𝑛)), and also that the preimage of each point 𝑘 is measurable, because it is equal to (-𝑛[,]𝑛) ∩ (◡𝐹 “ (𝑘[,)𝑘 + 1 / (2↑𝑛))) for 𝑘 < 𝑛 and (-𝑛[,]𝑛) ∩ (◡𝐹 “ (𝑘[,)+∞)) for 𝑘 = 𝑛. (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
mbfi1fseqlem4 (𝜑𝐺:ℕ⟶dom ∫1)
Distinct variable groups:   𝑥,𝑚,𝑦,𝐹   𝑥,𝐺   𝑚,𝐽   𝜑,𝑚,𝑥,𝑦
Allowed substitution hints:   𝐺(𝑦,𝑚)   𝐽(𝑥,𝑦)

Proof of Theorem mbfi1fseqlem4
Dummy variables 𝑘 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 reex 9906 . . . . 5 ℝ ∈ V
21mptex 6390 . . . 4 (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝑚[,]𝑚), if((𝑚𝐽𝑥) ≤ 𝑚, (𝑚𝐽𝑥), 𝑚), 0)) ∈ V
3 mbfi1fseq.4 . . . 4 𝐺 = (𝑚 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝑚[,]𝑚), if((𝑚𝐽𝑥) ≤ 𝑚, (𝑚𝐽𝑥), 𝑚), 0)))
42, 3fnmpti 5935 . . 3 𝐺 Fn ℕ
54a1i 11 . 2 (𝜑𝐺 Fn ℕ)
6 mbfi1fseq.1 . . . . . 6 (𝜑𝐹 ∈ MblFn)
7 mbfi1fseq.2 . . . . . 6 (𝜑𝐹:ℝ⟶(0[,)+∞))
8 mbfi1fseq.3 . . . . . 6 𝐽 = (𝑚 ∈ ℕ, 𝑦 ∈ ℝ ↦ ((⌊‘((𝐹𝑦) · (2↑𝑚))) / (2↑𝑚)))
96, 7, 8, 3mbfi1fseqlem3 23290 . . . . 5 ((𝜑𝑛 ∈ ℕ) → (𝐺𝑛):ℝ⟶ran (𝑚 ∈ (0...(𝑛 · (2↑𝑛))) ↦ (𝑚 / (2↑𝑛))))
10 elfznn0 12302 . . . . . . . . 9 (𝑚 ∈ (0...(𝑛 · (2↑𝑛))) → 𝑚 ∈ ℕ0)
1110nn0red 11229 . . . . . . . 8 (𝑚 ∈ (0...(𝑛 · (2↑𝑛))) → 𝑚 ∈ ℝ)
12 2nn 11062 . . . . . . . . . 10 2 ∈ ℕ
13 nnnn0 11176 . . . . . . . . . 10 (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0)
14 nnexpcl 12735 . . . . . . . . . 10 ((2 ∈ ℕ ∧ 𝑛 ∈ ℕ0) → (2↑𝑛) ∈ ℕ)
1512, 13, 14sylancr 694 . . . . . . . . 9 (𝑛 ∈ ℕ → (2↑𝑛) ∈ ℕ)
1615adantl 481 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (2↑𝑛) ∈ ℕ)
17 nndivre 10933 . . . . . . . 8 ((𝑚 ∈ ℝ ∧ (2↑𝑛) ∈ ℕ) → (𝑚 / (2↑𝑛)) ∈ ℝ)
1811, 16, 17syl2anr 494 . . . . . . 7 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝑛 · (2↑𝑛)))) → (𝑚 / (2↑𝑛)) ∈ ℝ)
19 eqid 2610 . . . . . . 7 (𝑚 ∈ (0...(𝑛 · (2↑𝑛))) ↦ (𝑚 / (2↑𝑛))) = (𝑚 ∈ (0...(𝑛 · (2↑𝑛))) ↦ (𝑚 / (2↑𝑛)))
2018, 19fmptd 6292 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (𝑚 ∈ (0...(𝑛 · (2↑𝑛))) ↦ (𝑚 / (2↑𝑛))):(0...(𝑛 · (2↑𝑛)))⟶ℝ)
21 frn 5966 . . . . . 6 ((𝑚 ∈ (0...(𝑛 · (2↑𝑛))) ↦ (𝑚 / (2↑𝑛))):(0...(𝑛 · (2↑𝑛)))⟶ℝ → ran (𝑚 ∈ (0...(𝑛 · (2↑𝑛))) ↦ (𝑚 / (2↑𝑛))) ⊆ ℝ)
2220, 21syl 17 . . . . 5 ((𝜑𝑛 ∈ ℕ) → ran (𝑚 ∈ (0...(𝑛 · (2↑𝑛))) ↦ (𝑚 / (2↑𝑛))) ⊆ ℝ)
239, 22fssd 5970 . . . 4 ((𝜑𝑛 ∈ ℕ) → (𝐺𝑛):ℝ⟶ℝ)
24 fzfid 12634 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (0...(𝑛 · (2↑𝑛))) ∈ Fin)
25 ffn 5958 . . . . . . . 8 ((𝑚 ∈ (0...(𝑛 · (2↑𝑛))) ↦ (𝑚 / (2↑𝑛))):(0...(𝑛 · (2↑𝑛)))⟶ℝ → (𝑚 ∈ (0...(𝑛 · (2↑𝑛))) ↦ (𝑚 / (2↑𝑛))) Fn (0...(𝑛 · (2↑𝑛))))
2620, 25syl 17 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → (𝑚 ∈ (0...(𝑛 · (2↑𝑛))) ↦ (𝑚 / (2↑𝑛))) Fn (0...(𝑛 · (2↑𝑛))))
27 dffn4 6034 . . . . . . 7 ((𝑚 ∈ (0...(𝑛 · (2↑𝑛))) ↦ (𝑚 / (2↑𝑛))) Fn (0...(𝑛 · (2↑𝑛))) ↔ (𝑚 ∈ (0...(𝑛 · (2↑𝑛))) ↦ (𝑚 / (2↑𝑛))):(0...(𝑛 · (2↑𝑛)))–onto→ran (𝑚 ∈ (0...(𝑛 · (2↑𝑛))) ↦ (𝑚 / (2↑𝑛))))
2826, 27sylib 207 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (𝑚 ∈ (0...(𝑛 · (2↑𝑛))) ↦ (𝑚 / (2↑𝑛))):(0...(𝑛 · (2↑𝑛)))–onto→ran (𝑚 ∈ (0...(𝑛 · (2↑𝑛))) ↦ (𝑚 / (2↑𝑛))))
29 fofi 8135 . . . . . 6 (((0...(𝑛 · (2↑𝑛))) ∈ Fin ∧ (𝑚 ∈ (0...(𝑛 · (2↑𝑛))) ↦ (𝑚 / (2↑𝑛))):(0...(𝑛 · (2↑𝑛)))–onto→ran (𝑚 ∈ (0...(𝑛 · (2↑𝑛))) ↦ (𝑚 / (2↑𝑛)))) → ran (𝑚 ∈ (0...(𝑛 · (2↑𝑛))) ↦ (𝑚 / (2↑𝑛))) ∈ Fin)
3024, 28, 29syl2anc 691 . . . . 5 ((𝜑𝑛 ∈ ℕ) → ran (𝑚 ∈ (0...(𝑛 · (2↑𝑛))) ↦ (𝑚 / (2↑𝑛))) ∈ Fin)
31 frn 5966 . . . . . 6 ((𝐺𝑛):ℝ⟶ran (𝑚 ∈ (0...(𝑛 · (2↑𝑛))) ↦ (𝑚 / (2↑𝑛))) → ran (𝐺𝑛) ⊆ ran (𝑚 ∈ (0...(𝑛 · (2↑𝑛))) ↦ (𝑚 / (2↑𝑛))))
329, 31syl 17 . . . . 5 ((𝜑𝑛 ∈ ℕ) → ran (𝐺𝑛) ⊆ ran (𝑚 ∈ (0...(𝑛 · (2↑𝑛))) ↦ (𝑚 / (2↑𝑛))))
33 ssfi 8065 . . . . 5 ((ran (𝑚 ∈ (0...(𝑛 · (2↑𝑛))) ↦ (𝑚 / (2↑𝑛))) ∈ Fin ∧ ran (𝐺𝑛) ⊆ ran (𝑚 ∈ (0...(𝑛 · (2↑𝑛))) ↦ (𝑚 / (2↑𝑛)))) → ran (𝐺𝑛) ∈ Fin)
3430, 32, 33syl2anc 691 . . . 4 ((𝜑𝑛 ∈ ℕ) → ran (𝐺𝑛) ∈ Fin)
356, 7, 8, 3mbfi1fseqlem2 23289 . . . . . . . . . . . . . 14 (𝑛 ∈ ℕ → (𝐺𝑛) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝑛[,]𝑛), if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛), 0)))
3635fveq1d 6105 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ → ((𝐺𝑛)‘𝑥) = ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝑛[,]𝑛), if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛), 0))‘𝑥))
3736ad2antlr 759 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐺𝑛)‘𝑥) = ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝑛[,]𝑛), if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛), 0))‘𝑥))
38 simpr 476 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℝ)
39 ovex 6577 . . . . . . . . . . . . . . 15 (𝑛𝐽𝑥) ∈ V
40 vex 3176 . . . . . . . . . . . . . . 15 𝑛 ∈ V
4139, 40ifex 4106 . . . . . . . . . . . . . 14 if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) ∈ V
42 c0ex 9913 . . . . . . . . . . . . . 14 0 ∈ V
4341, 42ifex 4106 . . . . . . . . . . . . 13 if(𝑥 ∈ (-𝑛[,]𝑛), if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛), 0) ∈ V
44 eqid 2610 . . . . . . . . . . . . . 14 (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝑛[,]𝑛), if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛), 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝑛[,]𝑛), if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛), 0))
4544fvmpt2 6200 . . . . . . . . . . . . 13 ((𝑥 ∈ ℝ ∧ if(𝑥 ∈ (-𝑛[,]𝑛), if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛), 0) ∈ V) → ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝑛[,]𝑛), if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛), 0))‘𝑥) = if(𝑥 ∈ (-𝑛[,]𝑛), if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛), 0))
4638, 43, 45sylancl 693 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝑛[,]𝑛), if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛), 0))‘𝑥) = if(𝑥 ∈ (-𝑛[,]𝑛), if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛), 0))
4737, 46eqtrd 2644 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐺𝑛)‘𝑥) = if(𝑥 ∈ (-𝑛[,]𝑛), if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛), 0))
4847adantlr 747 . . . . . . . . . 10 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → ((𝐺𝑛)‘𝑥) = if(𝑥 ∈ (-𝑛[,]𝑛), if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛), 0))
4948eqeq1d 2612 . . . . . . . . 9 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → (((𝐺𝑛)‘𝑥) = 𝑘 ↔ if(𝑥 ∈ (-𝑛[,]𝑛), if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛), 0) = 𝑘))
50 eldifsni 4261 . . . . . . . . . . . . 13 (𝑘 ∈ (ran (𝐺𝑛) ∖ {0}) → 𝑘 ≠ 0)
5150ad2antlr 759 . . . . . . . . . . . 12 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → 𝑘 ≠ 0)
52 neeq1 2844 . . . . . . . . . . . 12 (if(𝑥 ∈ (-𝑛[,]𝑛), if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛), 0) = 𝑘 → (if(𝑥 ∈ (-𝑛[,]𝑛), if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛), 0) ≠ 0 ↔ 𝑘 ≠ 0))
5351, 52syl5ibrcom 236 . . . . . . . . . . 11 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → (if(𝑥 ∈ (-𝑛[,]𝑛), if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛), 0) = 𝑘 → if(𝑥 ∈ (-𝑛[,]𝑛), if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛), 0) ≠ 0))
54 iffalse 4045 . . . . . . . . . . . 12 𝑥 ∈ (-𝑛[,]𝑛) → if(𝑥 ∈ (-𝑛[,]𝑛), if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛), 0) = 0)
5554necon1ai 2809 . . . . . . . . . . 11 (if(𝑥 ∈ (-𝑛[,]𝑛), if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛), 0) ≠ 0 → 𝑥 ∈ (-𝑛[,]𝑛))
5653, 55syl6 34 . . . . . . . . . 10 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → (if(𝑥 ∈ (-𝑛[,]𝑛), if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛), 0) = 𝑘𝑥 ∈ (-𝑛[,]𝑛)))
5756pm4.71rd 665 . . . . . . . . 9 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → (if(𝑥 ∈ (-𝑛[,]𝑛), if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛), 0) = 𝑘 ↔ (𝑥 ∈ (-𝑛[,]𝑛) ∧ if(𝑥 ∈ (-𝑛[,]𝑛), if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛), 0) = 𝑘)))
58 iftrue 4042 . . . . . . . . . . . 12 (𝑥 ∈ (-𝑛[,]𝑛) → if(𝑥 ∈ (-𝑛[,]𝑛), if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛), 0) = if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛))
5958eqeq1d 2612 . . . . . . . . . . 11 (𝑥 ∈ (-𝑛[,]𝑛) → (if(𝑥 ∈ (-𝑛[,]𝑛), if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛), 0) = 𝑘 ↔ if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) = 𝑘))
60 simpllr 795 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → 𝑛 ∈ ℕ)
6160nnred 10912 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → 𝑛 ∈ ℝ)
6261adantr 480 . . . . . . . . . . . . . . . 16 (((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) ∧ 𝑘 = 𝑛) → 𝑛 ∈ ℝ)
63 rge0ssre 12151 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (0[,)+∞) ⊆ ℝ
64 simpr 476 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑚 ∈ ℕ ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℝ)
65 ffvelrn 6265 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝐹:ℝ⟶(0[,)+∞) ∧ 𝑦 ∈ ℝ) → (𝐹𝑦) ∈ (0[,)+∞))
667, 64, 65syl2an 493 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑦 ∈ ℝ)) → (𝐹𝑦) ∈ (0[,)+∞))
6763, 66sseldi 3566 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑦 ∈ ℝ)) → (𝐹𝑦) ∈ ℝ)
68 nnnn0 11176 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑚 ∈ ℕ → 𝑚 ∈ ℕ0)
69 nnexpcl 12735 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((2 ∈ ℕ ∧ 𝑚 ∈ ℕ0) → (2↑𝑚) ∈ ℕ)
7012, 68, 69sylancr 694 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑚 ∈ ℕ → (2↑𝑚) ∈ ℕ)
7170ad2antrl 760 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑦 ∈ ℝ)) → (2↑𝑚) ∈ ℕ)
7271nnred 10912 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑦 ∈ ℝ)) → (2↑𝑚) ∈ ℝ)
7367, 72remulcld 9949 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑦 ∈ ℝ)) → ((𝐹𝑦) · (2↑𝑚)) ∈ ℝ)
74 reflcl 12459 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝐹𝑦) · (2↑𝑚)) ∈ ℝ → (⌊‘((𝐹𝑦) · (2↑𝑚))) ∈ ℝ)
7573, 74syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑦 ∈ ℝ)) → (⌊‘((𝐹𝑦) · (2↑𝑚))) ∈ ℝ)
7675, 71nndivred 10946 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑦 ∈ ℝ)) → ((⌊‘((𝐹𝑦) · (2↑𝑚))) / (2↑𝑚)) ∈ ℝ)
7776ralrimivva 2954 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → ∀𝑚 ∈ ℕ ∀𝑦 ∈ ℝ ((⌊‘((𝐹𝑦) · (2↑𝑚))) / (2↑𝑚)) ∈ ℝ)
788fmpt2 7126 . . . . . . . . . . . . . . . . . . . . 21 (∀𝑚 ∈ ℕ ∀𝑦 ∈ ℝ ((⌊‘((𝐹𝑦) · (2↑𝑚))) / (2↑𝑚)) ∈ ℝ ↔ 𝐽:(ℕ × ℝ)⟶ℝ)
7977, 78sylib 207 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐽:(ℕ × ℝ)⟶ℝ)
80 fovrn 6702 . . . . . . . . . . . . . . . . . . . 20 ((𝐽:(ℕ × ℝ)⟶ℝ ∧ 𝑛 ∈ ℕ ∧ 𝑥 ∈ ℝ) → (𝑛𝐽𝑥) ∈ ℝ)
8179, 80syl3an1 1351 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛 ∈ ℕ ∧ 𝑥 ∈ ℝ) → (𝑛𝐽𝑥) ∈ ℝ)
82813expa 1257 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝑛𝐽𝑥) ∈ ℝ)
8382adantlr 747 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → (𝑛𝐽𝑥) ∈ ℝ)
8483adantr 480 . . . . . . . . . . . . . . . 16 (((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) ∧ 𝑘 = 𝑛) → (𝑛𝐽𝑥) ∈ ℝ)
85 lemin 11897 . . . . . . . . . . . . . . . 16 ((𝑛 ∈ ℝ ∧ (𝑛𝐽𝑥) ∈ ℝ ∧ 𝑛 ∈ ℝ) → (𝑛 ≤ if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) ↔ (𝑛 ≤ (𝑛𝐽𝑥) ∧ 𝑛𝑛)))
8662, 84, 62, 85syl3anc 1318 . . . . . . . . . . . . . . 15 (((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) ∧ 𝑘 = 𝑛) → (𝑛 ≤ if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) ↔ (𝑛 ≤ (𝑛𝐽𝑥) ∧ 𝑛𝑛)))
8784, 62ifcld 4081 . . . . . . . . . . . . . . . . 17 (((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) ∧ 𝑘 = 𝑛) → if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) ∈ ℝ)
8887, 62letri3d 10058 . . . . . . . . . . . . . . . 16 (((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) ∧ 𝑘 = 𝑛) → (if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) = 𝑛 ↔ (if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) ≤ 𝑛𝑛 ≤ if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛))))
89 simpr 476 . . . . . . . . . . . . . . . . 17 (((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) ∧ 𝑘 = 𝑛) → 𝑘 = 𝑛)
9089eqeq2d 2620 . . . . . . . . . . . . . . . 16 (((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) ∧ 𝑘 = 𝑛) → (if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) = 𝑘 ↔ if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) = 𝑛))
91 min2 11895 . . . . . . . . . . . . . . . . . 18 (((𝑛𝐽𝑥) ∈ ℝ ∧ 𝑛 ∈ ℝ) → if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) ≤ 𝑛)
9284, 62, 91syl2anc 691 . . . . . . . . . . . . . . . . 17 (((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) ∧ 𝑘 = 𝑛) → if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) ≤ 𝑛)
9392biantrurd 528 . . . . . . . . . . . . . . . 16 (((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) ∧ 𝑘 = 𝑛) → (𝑛 ≤ if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) ↔ (if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) ≤ 𝑛𝑛 ≤ if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛))))
9488, 90, 933bitr4d 299 . . . . . . . . . . . . . . 15 (((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) ∧ 𝑘 = 𝑛) → (if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) = 𝑘𝑛 ≤ if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛)))
9562leidd 10473 . . . . . . . . . . . . . . . 16 (((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) ∧ 𝑘 = 𝑛) → 𝑛𝑛)
9695biantrud 527 . . . . . . . . . . . . . . 15 (((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) ∧ 𝑘 = 𝑛) → (𝑛 ≤ (𝑛𝐽𝑥) ↔ (𝑛 ≤ (𝑛𝐽𝑥) ∧ 𝑛𝑛)))
9786, 94, 963bitr4d 299 . . . . . . . . . . . . . 14 (((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) ∧ 𝑘 = 𝑛) → (if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) = 𝑘𝑛 ≤ (𝑛𝐽𝑥)))
98 breq1 4586 . . . . . . . . . . . . . . 15 (𝑘 = 𝑛 → (𝑘 ≤ (𝐹𝑥) ↔ 𝑛 ≤ (𝐹𝑥)))
997adantr 480 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑛 ∈ ℕ) → 𝐹:ℝ⟶(0[,)+∞))
10099ffvelrnda 6267 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐹𝑥) ∈ (0[,)+∞))
101 elrege0 12149 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐹𝑥) ∈ (0[,)+∞) ↔ ((𝐹𝑥) ∈ ℝ ∧ 0 ≤ (𝐹𝑥)))
102100, 101sylib 207 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐹𝑥) ∈ ℝ ∧ 0 ≤ (𝐹𝑥)))
103102simpld 474 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → (𝐹𝑥) ∈ ℝ)
104103adantlr 747 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → (𝐹𝑥) ∈ ℝ)
10560, 15syl 17 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → (2↑𝑛) ∈ ℕ)
106105nnred 10912 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → (2↑𝑛) ∈ ℝ)
107104, 106remulcld 9949 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → ((𝐹𝑥) · (2↑𝑛)) ∈ ℝ)
108 reflcl 12459 . . . . . . . . . . . . . . . . . 18 (((𝐹𝑥) · (2↑𝑛)) ∈ ℝ → (⌊‘((𝐹𝑥) · (2↑𝑛))) ∈ ℝ)
109107, 108syl 17 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → (⌊‘((𝐹𝑥) · (2↑𝑛))) ∈ ℝ)
110105nngt0d 10941 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → 0 < (2↑𝑛))
111 lemuldiv 10782 . . . . . . . . . . . . . . . . 17 ((𝑛 ∈ ℝ ∧ (⌊‘((𝐹𝑥) · (2↑𝑛))) ∈ ℝ ∧ ((2↑𝑛) ∈ ℝ ∧ 0 < (2↑𝑛))) → ((𝑛 · (2↑𝑛)) ≤ (⌊‘((𝐹𝑥) · (2↑𝑛))) ↔ 𝑛 ≤ ((⌊‘((𝐹𝑥) · (2↑𝑛))) / (2↑𝑛))))
11261, 109, 106, 110, 111syl112anc 1322 . . . . . . . . . . . . . . . 16 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → ((𝑛 · (2↑𝑛)) ≤ (⌊‘((𝐹𝑥) · (2↑𝑛))) ↔ 𝑛 ≤ ((⌊‘((𝐹𝑥) · (2↑𝑛))) / (2↑𝑛))))
113 lemul1 10754 . . . . . . . . . . . . . . . . . 18 ((𝑛 ∈ ℝ ∧ (𝐹𝑥) ∈ ℝ ∧ ((2↑𝑛) ∈ ℝ ∧ 0 < (2↑𝑛))) → (𝑛 ≤ (𝐹𝑥) ↔ (𝑛 · (2↑𝑛)) ≤ ((𝐹𝑥) · (2↑𝑛))))
11461, 104, 106, 110, 113syl112anc 1322 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → (𝑛 ≤ (𝐹𝑥) ↔ (𝑛 · (2↑𝑛)) ≤ ((𝐹𝑥) · (2↑𝑛))))
115 nnmulcl 10920 . . . . . . . . . . . . . . . . . . . . 21 ((𝑛 ∈ ℕ ∧ (2↑𝑛) ∈ ℕ) → (𝑛 · (2↑𝑛)) ∈ ℕ)
11615, 115mpdan 699 . . . . . . . . . . . . . . . . . . . 20 (𝑛 ∈ ℕ → (𝑛 · (2↑𝑛)) ∈ ℕ)
11760, 116syl 17 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → (𝑛 · (2↑𝑛)) ∈ ℕ)
118117nnzd 11357 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → (𝑛 · (2↑𝑛)) ∈ ℤ)
119 flge 12468 . . . . . . . . . . . . . . . . . 18 ((((𝐹𝑥) · (2↑𝑛)) ∈ ℝ ∧ (𝑛 · (2↑𝑛)) ∈ ℤ) → ((𝑛 · (2↑𝑛)) ≤ ((𝐹𝑥) · (2↑𝑛)) ↔ (𝑛 · (2↑𝑛)) ≤ (⌊‘((𝐹𝑥) · (2↑𝑛)))))
120107, 118, 119syl2anc 691 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → ((𝑛 · (2↑𝑛)) ≤ ((𝐹𝑥) · (2↑𝑛)) ↔ (𝑛 · (2↑𝑛)) ≤ (⌊‘((𝐹𝑥) · (2↑𝑛)))))
121114, 120bitrd 267 . . . . . . . . . . . . . . . 16 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → (𝑛 ≤ (𝐹𝑥) ↔ (𝑛 · (2↑𝑛)) ≤ (⌊‘((𝐹𝑥) · (2↑𝑛)))))
122 simpr 476 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℝ)
123 simpr 476 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑚 = 𝑛𝑦 = 𝑥) → 𝑦 = 𝑥)
124123fveq2d 6107 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑚 = 𝑛𝑦 = 𝑥) → (𝐹𝑦) = (𝐹𝑥))
125 simpl 472 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑚 = 𝑛𝑦 = 𝑥) → 𝑚 = 𝑛)
126125oveq2d 6565 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑚 = 𝑛𝑦 = 𝑥) → (2↑𝑚) = (2↑𝑛))
127124, 126oveq12d 6567 . . . . . . . . . . . . . . . . . . . . 21 ((𝑚 = 𝑛𝑦 = 𝑥) → ((𝐹𝑦) · (2↑𝑚)) = ((𝐹𝑥) · (2↑𝑛)))
128127fveq2d 6107 . . . . . . . . . . . . . . . . . . . 20 ((𝑚 = 𝑛𝑦 = 𝑥) → (⌊‘((𝐹𝑦) · (2↑𝑚))) = (⌊‘((𝐹𝑥) · (2↑𝑛))))
129128, 126oveq12d 6567 . . . . . . . . . . . . . . . . . . 19 ((𝑚 = 𝑛𝑦 = 𝑥) → ((⌊‘((𝐹𝑦) · (2↑𝑚))) / (2↑𝑚)) = ((⌊‘((𝐹𝑥) · (2↑𝑛))) / (2↑𝑛)))
130 ovex 6577 . . . . . . . . . . . . . . . . . . 19 ((⌊‘((𝐹𝑥) · (2↑𝑛))) / (2↑𝑛)) ∈ V
131129, 8, 130ovmpt2a 6689 . . . . . . . . . . . . . . . . . 18 ((𝑛 ∈ ℕ ∧ 𝑥 ∈ ℝ) → (𝑛𝐽𝑥) = ((⌊‘((𝐹𝑥) · (2↑𝑛))) / (2↑𝑛)))
13260, 122, 131syl2anc 691 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → (𝑛𝐽𝑥) = ((⌊‘((𝐹𝑥) · (2↑𝑛))) / (2↑𝑛)))
133132breq2d 4595 . . . . . . . . . . . . . . . 16 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → (𝑛 ≤ (𝑛𝐽𝑥) ↔ 𝑛 ≤ ((⌊‘((𝐹𝑥) · (2↑𝑛))) / (2↑𝑛))))
134112, 121, 1333bitr4d 299 . . . . . . . . . . . . . . 15 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → (𝑛 ≤ (𝐹𝑥) ↔ 𝑛 ≤ (𝑛𝐽𝑥)))
13598, 134sylan9bbr 733 . . . . . . . . . . . . . 14 (((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) ∧ 𝑘 = 𝑛) → (𝑘 ≤ (𝐹𝑥) ↔ 𝑛 ≤ (𝑛𝐽𝑥)))
136122adantr 480 . . . . . . . . . . . . . . . 16 (((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) ∧ 𝑘 = 𝑛) → 𝑥 ∈ ℝ)
137 iftrue 4042 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑛 → if(𝑘 = 𝑛, ℝ, (𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛)))))) = ℝ)
138137adantl 481 . . . . . . . . . . . . . . . 16 (((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) ∧ 𝑘 = 𝑛) → if(𝑘 = 𝑛, ℝ, (𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛)))))) = ℝ)
139136, 138eleqtrrd 2691 . . . . . . . . . . . . . . 15 (((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) ∧ 𝑘 = 𝑛) → 𝑥 ∈ if(𝑘 = 𝑛, ℝ, (𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛)))))))
140139biantrurd 528 . . . . . . . . . . . . . 14 (((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) ∧ 𝑘 = 𝑛) → (𝑘 ≤ (𝐹𝑥) ↔ (𝑥 ∈ if(𝑘 = 𝑛, ℝ, (𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛)))))) ∧ 𝑘 ≤ (𝐹𝑥))))
14197, 135, 1403bitr2d 295 . . . . . . . . . . . . 13 (((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) ∧ 𝑘 = 𝑛) → (if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) = 𝑘 ↔ (𝑥 ∈ if(𝑘 = 𝑛, ℝ, (𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛)))))) ∧ 𝑘 ≤ (𝐹𝑥))))
14232ssdifssd 3710 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛 ∈ ℕ) → (ran (𝐺𝑛) ∖ {0}) ⊆ ran (𝑚 ∈ (0...(𝑛 · (2↑𝑛))) ↦ (𝑚 / (2↑𝑛))))
143142sselda 3568 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) → 𝑘 ∈ ran (𝑚 ∈ (0...(𝑛 · (2↑𝑛))) ↦ (𝑚 / (2↑𝑛))))
14419rnmpt 5292 . . . . . . . . . . . . . . . . . . . . 21 ran (𝑚 ∈ (0...(𝑛 · (2↑𝑛))) ↦ (𝑚 / (2↑𝑛))) = {𝑘 ∣ ∃𝑚 ∈ (0...(𝑛 · (2↑𝑛)))𝑘 = (𝑚 / (2↑𝑛))}
145144abeq2i 2722 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∈ ran (𝑚 ∈ (0...(𝑛 · (2↑𝑛))) ↦ (𝑚 / (2↑𝑛))) ↔ ∃𝑚 ∈ (0...(𝑛 · (2↑𝑛)))𝑘 = (𝑚 / (2↑𝑛)))
146 elfzelz 12213 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑚 ∈ (0...(𝑛 · (2↑𝑛))) → 𝑚 ∈ ℤ)
147146adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝑛 · (2↑𝑛)))) → 𝑚 ∈ ℤ)
148147zcnd 11359 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝑛 · (2↑𝑛)))) → 𝑚 ∈ ℂ)
14915ad2antlr 759 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝑛 · (2↑𝑛)))) → (2↑𝑛) ∈ ℕ)
150149nncnd 10913 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝑛 · (2↑𝑛)))) → (2↑𝑛) ∈ ℂ)
151149nnne0d 10942 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝑛 · (2↑𝑛)))) → (2↑𝑛) ≠ 0)
152148, 150, 151divcan1d 10681 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝑛 · (2↑𝑛)))) → ((𝑚 / (2↑𝑛)) · (2↑𝑛)) = 𝑚)
153152, 147eqeltrd 2688 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝑛 · (2↑𝑛)))) → ((𝑚 / (2↑𝑛)) · (2↑𝑛)) ∈ ℤ)
154 oveq1 6556 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 = (𝑚 / (2↑𝑛)) → (𝑘 · (2↑𝑛)) = ((𝑚 / (2↑𝑛)) · (2↑𝑛)))
155154eleq1d 2672 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = (𝑚 / (2↑𝑛)) → ((𝑘 · (2↑𝑛)) ∈ ℤ ↔ ((𝑚 / (2↑𝑛)) · (2↑𝑛)) ∈ ℤ))
156153, 155syl5ibrcom 236 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑛 ∈ ℕ) ∧ 𝑚 ∈ (0...(𝑛 · (2↑𝑛)))) → (𝑘 = (𝑚 / (2↑𝑛)) → (𝑘 · (2↑𝑛)) ∈ ℤ))
157156rexlimdva 3013 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑛 ∈ ℕ) → (∃𝑚 ∈ (0...(𝑛 · (2↑𝑛)))𝑘 = (𝑚 / (2↑𝑛)) → (𝑘 · (2↑𝑛)) ∈ ℤ))
158145, 157syl5bi 231 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛 ∈ ℕ) → (𝑘 ∈ ran (𝑚 ∈ (0...(𝑛 · (2↑𝑛))) ↦ (𝑚 / (2↑𝑛))) → (𝑘 · (2↑𝑛)) ∈ ℤ))
159158imp 444 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ ran (𝑚 ∈ (0...(𝑛 · (2↑𝑛))) ↦ (𝑚 / (2↑𝑛)))) → (𝑘 · (2↑𝑛)) ∈ ℤ)
160143, 159syldan 486 . . . . . . . . . . . . . . . . 17 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) → (𝑘 · (2↑𝑛)) ∈ ℤ)
161160adantr 480 . . . . . . . . . . . . . . . 16 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → (𝑘 · (2↑𝑛)) ∈ ℤ)
162 flbi 12479 . . . . . . . . . . . . . . . 16 ((((𝐹𝑥) · (2↑𝑛)) ∈ ℝ ∧ (𝑘 · (2↑𝑛)) ∈ ℤ) → ((⌊‘((𝐹𝑥) · (2↑𝑛))) = (𝑘 · (2↑𝑛)) ↔ ((𝑘 · (2↑𝑛)) ≤ ((𝐹𝑥) · (2↑𝑛)) ∧ ((𝐹𝑥) · (2↑𝑛)) < ((𝑘 · (2↑𝑛)) + 1))))
163107, 161, 162syl2anc 691 . . . . . . . . . . . . . . 15 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → ((⌊‘((𝐹𝑥) · (2↑𝑛))) = (𝑘 · (2↑𝑛)) ↔ ((𝑘 · (2↑𝑛)) ≤ ((𝐹𝑥) · (2↑𝑛)) ∧ ((𝐹𝑥) · (2↑𝑛)) < ((𝑘 · (2↑𝑛)) + 1))))
164163adantr 480 . . . . . . . . . . . . . 14 (((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) ∧ 𝑘𝑛) → ((⌊‘((𝐹𝑥) · (2↑𝑛))) = (𝑘 · (2↑𝑛)) ↔ ((𝑘 · (2↑𝑛)) ≤ ((𝐹𝑥) · (2↑𝑛)) ∧ ((𝐹𝑥) · (2↑𝑛)) < ((𝑘 · (2↑𝑛)) + 1))))
165 neeq1 2844 . . . . . . . . . . . . . . . . . . . . . . . 24 (if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) = 𝑘 → (if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) ≠ 𝑛𝑘𝑛))
166165biimparc 503 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑘𝑛 ∧ if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) = 𝑘) → if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) ≠ 𝑛)
167 iffalse 4045 . . . . . . . . . . . . . . . . . . . . . . . 24 (¬ (𝑛𝐽𝑥) ≤ 𝑛 → if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) = 𝑛)
168167necon1ai 2809 . . . . . . . . . . . . . . . . . . . . . . 23 (if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) ≠ 𝑛 → (𝑛𝐽𝑥) ≤ 𝑛)
169166, 168syl 17 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑘𝑛 ∧ if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) = 𝑘) → (𝑛𝐽𝑥) ≤ 𝑛)
170169iftrued 4044 . . . . . . . . . . . . . . . . . . . . 21 ((𝑘𝑛 ∧ if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) = 𝑘) → if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) = (𝑛𝐽𝑥))
171 simpr 476 . . . . . . . . . . . . . . . . . . . . 21 ((𝑘𝑛 ∧ if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) = 𝑘) → if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) = 𝑘)
172170, 171eqtr3d 2646 . . . . . . . . . . . . . . . . . . . 20 ((𝑘𝑛 ∧ if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) = 𝑘) → (𝑛𝐽𝑥) = 𝑘)
173172, 169eqbrtrrd 4607 . . . . . . . . . . . . . . . . . . 19 ((𝑘𝑛 ∧ if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) = 𝑘) → 𝑘𝑛)
174173, 172jca 553 . . . . . . . . . . . . . . . . . 18 ((𝑘𝑛 ∧ if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) = 𝑘) → (𝑘𝑛 ∧ (𝑛𝐽𝑥) = 𝑘))
175174ex 449 . . . . . . . . . . . . . . . . 17 (𝑘𝑛 → (if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) = 𝑘 → (𝑘𝑛 ∧ (𝑛𝐽𝑥) = 𝑘)))
176 breq1 4586 . . . . . . . . . . . . . . . . . . . 20 ((𝑛𝐽𝑥) = 𝑘 → ((𝑛𝐽𝑥) ≤ 𝑛𝑘𝑛))
177176biimparc 503 . . . . . . . . . . . . . . . . . . 19 ((𝑘𝑛 ∧ (𝑛𝐽𝑥) = 𝑘) → (𝑛𝐽𝑥) ≤ 𝑛)
178177iftrued 4044 . . . . . . . . . . . . . . . . . 18 ((𝑘𝑛 ∧ (𝑛𝐽𝑥) = 𝑘) → if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) = (𝑛𝐽𝑥))
179 simpr 476 . . . . . . . . . . . . . . . . . 18 ((𝑘𝑛 ∧ (𝑛𝐽𝑥) = 𝑘) → (𝑛𝐽𝑥) = 𝑘)
180178, 179eqtrd 2644 . . . . . . . . . . . . . . . . 17 ((𝑘𝑛 ∧ (𝑛𝐽𝑥) = 𝑘) → if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) = 𝑘)
181175, 180impbid1 214 . . . . . . . . . . . . . . . 16 (𝑘𝑛 → (if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) = 𝑘 ↔ (𝑘𝑛 ∧ (𝑛𝐽𝑥) = 𝑘)))
182181adantl 481 . . . . . . . . . . . . . . 15 (((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) ∧ 𝑘𝑛) → (if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) = 𝑘 ↔ (𝑘𝑛 ∧ (𝑛𝐽𝑥) = 𝑘)))
183 eldifi 3694 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ (ran (𝐺𝑛) ∖ {0}) → 𝑘 ∈ ran (𝐺𝑛))
184 nnre 10904 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑛 ∈ ℕ → 𝑛 ∈ ℝ)
185184ad2antlr 759 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 𝑛 ∈ ℝ)
18682, 185, 91syl2anc 691 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) ≤ 𝑛)
18713ad2antlr 759 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 𝑛 ∈ ℕ0)
188187nn0ge0d 11231 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → 0 ≤ 𝑛)
189 breq1 4586 . . . . . . . . . . . . . . . . . . . . . . . 24 (if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) = if(𝑥 ∈ (-𝑛[,]𝑛), if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛), 0) → (if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) ≤ 𝑛 ↔ if(𝑥 ∈ (-𝑛[,]𝑛), if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛), 0) ≤ 𝑛))
190 breq1 4586 . . . . . . . . . . . . . . . . . . . . . . . 24 (0 = if(𝑥 ∈ (-𝑛[,]𝑛), if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛), 0) → (0 ≤ 𝑛 ↔ if(𝑥 ∈ (-𝑛[,]𝑛), if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛), 0) ≤ 𝑛))
191189, 190ifboth 4074 . . . . . . . . . . . . . . . . . . . . . . 23 ((if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) ≤ 𝑛 ∧ 0 ≤ 𝑛) → if(𝑥 ∈ (-𝑛[,]𝑛), if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛), 0) ≤ 𝑛)
192186, 188, 191syl2anc 691 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → if(𝑥 ∈ (-𝑛[,]𝑛), if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛), 0) ≤ 𝑛)
19347, 192eqbrtrd 4605 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐺𝑛)‘𝑥) ≤ 𝑛)
194193ralrimiva 2949 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑛 ∈ ℕ) → ∀𝑥 ∈ ℝ ((𝐺𝑛)‘𝑥) ≤ 𝑛)
195 ffn 5958 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐺𝑛):ℝ⟶ℝ → (𝐺𝑛) Fn ℝ)
19623, 195syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑛 ∈ ℕ) → (𝐺𝑛) Fn ℝ)
197 breq1 4586 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 = ((𝐺𝑛)‘𝑥) → (𝑘𝑛 ↔ ((𝐺𝑛)‘𝑥) ≤ 𝑛))
198197ralrn 6270 . . . . . . . . . . . . . . . . . . . . 21 ((𝐺𝑛) Fn ℝ → (∀𝑘 ∈ ran (𝐺𝑛)𝑘𝑛 ↔ ∀𝑥 ∈ ℝ ((𝐺𝑛)‘𝑥) ≤ 𝑛))
199196, 198syl 17 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑛 ∈ ℕ) → (∀𝑘 ∈ ran (𝐺𝑛)𝑘𝑛 ↔ ∀𝑥 ∈ ℝ ((𝐺𝑛)‘𝑥) ≤ 𝑛))
200194, 199mpbird 246 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑛 ∈ ℕ) → ∀𝑘 ∈ ran (𝐺𝑛)𝑘𝑛)
201200r19.21bi 2916 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ ran (𝐺𝑛)) → 𝑘𝑛)
202183, 201sylan2 490 . . . . . . . . . . . . . . . . 17 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) → 𝑘𝑛)
203202ad2antrr 758 . . . . . . . . . . . . . . . 16 (((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) ∧ 𝑘𝑛) → 𝑘𝑛)
204203biantrurd 528 . . . . . . . . . . . . . . 15 (((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) ∧ 𝑘𝑛) → ((𝑛𝐽𝑥) = 𝑘 ↔ (𝑘𝑛 ∧ (𝑛𝐽𝑥) = 𝑘)))
205132eqeq1d 2612 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → ((𝑛𝐽𝑥) = 𝑘 ↔ ((⌊‘((𝐹𝑥) · (2↑𝑛))) / (2↑𝑛)) = 𝑘))
206109recnd 9947 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → (⌊‘((𝐹𝑥) · (2↑𝑛))) ∈ ℂ)
20732, 22sstrd 3578 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑛 ∈ ℕ) → ran (𝐺𝑛) ⊆ ℝ)
208207ssdifssd 3710 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑛 ∈ ℕ) → (ran (𝐺𝑛) ∖ {0}) ⊆ ℝ)
209208sselda 3568 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) → 𝑘 ∈ ℝ)
210209adantr 480 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → 𝑘 ∈ ℝ)
211210recnd 9947 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → 𝑘 ∈ ℂ)
212105nncnd 10913 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → (2↑𝑛) ∈ ℂ)
213105nnne0d 10942 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → (2↑𝑛) ≠ 0)
214206, 211, 212, 213divmul3d 10714 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → (((⌊‘((𝐹𝑥) · (2↑𝑛))) / (2↑𝑛)) = 𝑘 ↔ (⌊‘((𝐹𝑥) · (2↑𝑛))) = (𝑘 · (2↑𝑛))))
215205, 214bitrd 267 . . . . . . . . . . . . . . . 16 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → ((𝑛𝐽𝑥) = 𝑘 ↔ (⌊‘((𝐹𝑥) · (2↑𝑛))) = (𝑘 · (2↑𝑛))))
216215adantr 480 . . . . . . . . . . . . . . 15 (((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) ∧ 𝑘𝑛) → ((𝑛𝐽𝑥) = 𝑘 ↔ (⌊‘((𝐹𝑥) · (2↑𝑛))) = (𝑘 · (2↑𝑛))))
217182, 204, 2163bitr2d 295 . . . . . . . . . . . . . 14 (((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) ∧ 𝑘𝑛) → (if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) = 𝑘 ↔ (⌊‘((𝐹𝑥) · (2↑𝑛))) = (𝑘 · (2↑𝑛))))
218 ifnefalse 4048 . . . . . . . . . . . . . . . . . 18 (𝑘𝑛 → if(𝑘 = 𝑛, ℝ, (𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛)))))) = (𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛))))))
219218eleq2d 2673 . . . . . . . . . . . . . . . . 17 (𝑘𝑛 → (𝑥 ∈ if(𝑘 = 𝑛, ℝ, (𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛)))))) ↔ 𝑥 ∈ (𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛)))))))
220105nnrecred 10943 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → (1 / (2↑𝑛)) ∈ ℝ)
221210, 220readdcld 9948 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → (𝑘 + (1 / (2↑𝑛))) ∈ ℝ)
222221rexrd 9968 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → (𝑘 + (1 / (2↑𝑛))) ∈ ℝ*)
223 elioomnf 12139 . . . . . . . . . . . . . . . . . . . 20 ((𝑘 + (1 / (2↑𝑛))) ∈ ℝ* → ((𝐹𝑥) ∈ (-∞(,)(𝑘 + (1 / (2↑𝑛)))) ↔ ((𝐹𝑥) ∈ ℝ ∧ (𝐹𝑥) < (𝑘 + (1 / (2↑𝑛))))))
224222, 223syl 17 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → ((𝐹𝑥) ∈ (-∞(,)(𝑘 + (1 / (2↑𝑛)))) ↔ ((𝐹𝑥) ∈ ℝ ∧ (𝐹𝑥) < (𝑘 + (1 / (2↑𝑛))))))
22599ad2antrr 758 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → 𝐹:ℝ⟶(0[,)+∞))
226 ffn 5958 . . . . . . . . . . . . . . . . . . . . . 22 (𝐹:ℝ⟶(0[,)+∞) → 𝐹 Fn ℝ)
227225, 226syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → 𝐹 Fn ℝ)
228 elpreima 6245 . . . . . . . . . . . . . . . . . . . . 21 (𝐹 Fn ℝ → (𝑥 ∈ (𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛))))) ↔ (𝑥 ∈ ℝ ∧ (𝐹𝑥) ∈ (-∞(,)(𝑘 + (1 / (2↑𝑛)))))))
229227, 228syl 17 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → (𝑥 ∈ (𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛))))) ↔ (𝑥 ∈ ℝ ∧ (𝐹𝑥) ∈ (-∞(,)(𝑘 + (1 / (2↑𝑛)))))))
230122biantrurd 528 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → ((𝐹𝑥) ∈ (-∞(,)(𝑘 + (1 / (2↑𝑛)))) ↔ (𝑥 ∈ ℝ ∧ (𝐹𝑥) ∈ (-∞(,)(𝑘 + (1 / (2↑𝑛)))))))
231229, 230bitr4d 270 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → (𝑥 ∈ (𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛))))) ↔ (𝐹𝑥) ∈ (-∞(,)(𝑘 + (1 / (2↑𝑛))))))
232104biantrurd 528 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → ((𝐹𝑥) < (𝑘 + (1 / (2↑𝑛))) ↔ ((𝐹𝑥) ∈ ℝ ∧ (𝐹𝑥) < (𝑘 + (1 / (2↑𝑛))))))
233224, 231, 2323bitr4d 299 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → (𝑥 ∈ (𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛))))) ↔ (𝐹𝑥) < (𝑘 + (1 / (2↑𝑛)))))
234 ltmul1 10752 . . . . . . . . . . . . . . . . . . 19 (((𝐹𝑥) ∈ ℝ ∧ (𝑘 + (1 / (2↑𝑛))) ∈ ℝ ∧ ((2↑𝑛) ∈ ℝ ∧ 0 < (2↑𝑛))) → ((𝐹𝑥) < (𝑘 + (1 / (2↑𝑛))) ↔ ((𝐹𝑥) · (2↑𝑛)) < ((𝑘 + (1 / (2↑𝑛))) · (2↑𝑛))))
235104, 221, 106, 110, 234syl112anc 1322 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → ((𝐹𝑥) < (𝑘 + (1 / (2↑𝑛))) ↔ ((𝐹𝑥) · (2↑𝑛)) < ((𝑘 + (1 / (2↑𝑛))) · (2↑𝑛))))
236220recnd 9947 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → (1 / (2↑𝑛)) ∈ ℂ)
237211, 236, 212adddird 9944 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → ((𝑘 + (1 / (2↑𝑛))) · (2↑𝑛)) = ((𝑘 · (2↑𝑛)) + ((1 / (2↑𝑛)) · (2↑𝑛))))
238212, 213recid2d 10676 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → ((1 / (2↑𝑛)) · (2↑𝑛)) = 1)
239238oveq2d 6565 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → ((𝑘 · (2↑𝑛)) + ((1 / (2↑𝑛)) · (2↑𝑛))) = ((𝑘 · (2↑𝑛)) + 1))
240237, 239eqtrd 2644 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → ((𝑘 + (1 / (2↑𝑛))) · (2↑𝑛)) = ((𝑘 · (2↑𝑛)) + 1))
241240breq2d 4595 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → (((𝐹𝑥) · (2↑𝑛)) < ((𝑘 + (1 / (2↑𝑛))) · (2↑𝑛)) ↔ ((𝐹𝑥) · (2↑𝑛)) < ((𝑘 · (2↑𝑛)) + 1)))
242233, 235, 2413bitrd 293 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → (𝑥 ∈ (𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛))))) ↔ ((𝐹𝑥) · (2↑𝑛)) < ((𝑘 · (2↑𝑛)) + 1)))
243219, 242sylan9bbr 733 . . . . . . . . . . . . . . . 16 (((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) ∧ 𝑘𝑛) → (𝑥 ∈ if(𝑘 = 𝑛, ℝ, (𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛)))))) ↔ ((𝐹𝑥) · (2↑𝑛)) < ((𝑘 · (2↑𝑛)) + 1)))
244 lemul1 10754 . . . . . . . . . . . . . . . . . 18 ((𝑘 ∈ ℝ ∧ (𝐹𝑥) ∈ ℝ ∧ ((2↑𝑛) ∈ ℝ ∧ 0 < (2↑𝑛))) → (𝑘 ≤ (𝐹𝑥) ↔ (𝑘 · (2↑𝑛)) ≤ ((𝐹𝑥) · (2↑𝑛))))
245210, 104, 106, 110, 244syl112anc 1322 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → (𝑘 ≤ (𝐹𝑥) ↔ (𝑘 · (2↑𝑛)) ≤ ((𝐹𝑥) · (2↑𝑛))))
246245adantr 480 . . . . . . . . . . . . . . . 16 (((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) ∧ 𝑘𝑛) → (𝑘 ≤ (𝐹𝑥) ↔ (𝑘 · (2↑𝑛)) ≤ ((𝐹𝑥) · (2↑𝑛))))
247243, 246anbi12d 743 . . . . . . . . . . . . . . 15 (((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) ∧ 𝑘𝑛) → ((𝑥 ∈ if(𝑘 = 𝑛, ℝ, (𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛)))))) ∧ 𝑘 ≤ (𝐹𝑥)) ↔ (((𝐹𝑥) · (2↑𝑛)) < ((𝑘 · (2↑𝑛)) + 1) ∧ (𝑘 · (2↑𝑛)) ≤ ((𝐹𝑥) · (2↑𝑛)))))
248 ancom 465 . . . . . . . . . . . . . . 15 ((((𝐹𝑥) · (2↑𝑛)) < ((𝑘 · (2↑𝑛)) + 1) ∧ (𝑘 · (2↑𝑛)) ≤ ((𝐹𝑥) · (2↑𝑛))) ↔ ((𝑘 · (2↑𝑛)) ≤ ((𝐹𝑥) · (2↑𝑛)) ∧ ((𝐹𝑥) · (2↑𝑛)) < ((𝑘 · (2↑𝑛)) + 1)))
249247, 248syl6bb 275 . . . . . . . . . . . . . 14 (((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) ∧ 𝑘𝑛) → ((𝑥 ∈ if(𝑘 = 𝑛, ℝ, (𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛)))))) ∧ 𝑘 ≤ (𝐹𝑥)) ↔ ((𝑘 · (2↑𝑛)) ≤ ((𝐹𝑥) · (2↑𝑛)) ∧ ((𝐹𝑥) · (2↑𝑛)) < ((𝑘 · (2↑𝑛)) + 1))))
250164, 217, 2493bitr4d 299 . . . . . . . . . . . . 13 (((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) ∧ 𝑘𝑛) → (if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) = 𝑘 ↔ (𝑥 ∈ if(𝑘 = 𝑛, ℝ, (𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛)))))) ∧ 𝑘 ≤ (𝐹𝑥))))
251141, 250pm2.61dane 2869 . . . . . . . . . . . 12 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → (if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) = 𝑘 ↔ (𝑥 ∈ if(𝑘 = 𝑛, ℝ, (𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛)))))) ∧ 𝑘 ≤ (𝐹𝑥))))
252 eldif 3550 . . . . . . . . . . . . 13 (𝑥 ∈ (if(𝑘 = 𝑛, ℝ, (𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛)))))) ∖ (𝐹 “ (-∞(,)𝑘))) ↔ (𝑥 ∈ if(𝑘 = 𝑛, ℝ, (𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛)))))) ∧ ¬ 𝑥 ∈ (𝐹 “ (-∞(,)𝑘))))
253210rexrd 9968 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → 𝑘 ∈ ℝ*)
254 elioomnf 12139 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ ℝ* → ((𝐹𝑥) ∈ (-∞(,)𝑘) ↔ ((𝐹𝑥) ∈ ℝ ∧ (𝐹𝑥) < 𝑘)))
255253, 254syl 17 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → ((𝐹𝑥) ∈ (-∞(,)𝑘) ↔ ((𝐹𝑥) ∈ ℝ ∧ (𝐹𝑥) < 𝑘)))
256 elpreima 6245 . . . . . . . . . . . . . . . . . . 19 (𝐹 Fn ℝ → (𝑥 ∈ (𝐹 “ (-∞(,)𝑘)) ↔ (𝑥 ∈ ℝ ∧ (𝐹𝑥) ∈ (-∞(,)𝑘))))
257227, 256syl 17 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → (𝑥 ∈ (𝐹 “ (-∞(,)𝑘)) ↔ (𝑥 ∈ ℝ ∧ (𝐹𝑥) ∈ (-∞(,)𝑘))))
258122biantrurd 528 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → ((𝐹𝑥) ∈ (-∞(,)𝑘) ↔ (𝑥 ∈ ℝ ∧ (𝐹𝑥) ∈ (-∞(,)𝑘))))
259257, 258bitr4d 270 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → (𝑥 ∈ (𝐹 “ (-∞(,)𝑘)) ↔ (𝐹𝑥) ∈ (-∞(,)𝑘)))
260104biantrurd 528 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → ((𝐹𝑥) < 𝑘 ↔ ((𝐹𝑥) ∈ ℝ ∧ (𝐹𝑥) < 𝑘)))
261255, 259, 2603bitr4d 299 . . . . . . . . . . . . . . . 16 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → (𝑥 ∈ (𝐹 “ (-∞(,)𝑘)) ↔ (𝐹𝑥) < 𝑘))
262261notbid 307 . . . . . . . . . . . . . . 15 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → (¬ 𝑥 ∈ (𝐹 “ (-∞(,)𝑘)) ↔ ¬ (𝐹𝑥) < 𝑘))
263210, 104lenltd 10062 . . . . . . . . . . . . . . 15 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → (𝑘 ≤ (𝐹𝑥) ↔ ¬ (𝐹𝑥) < 𝑘))
264262, 263bitr4d 270 . . . . . . . . . . . . . 14 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → (¬ 𝑥 ∈ (𝐹 “ (-∞(,)𝑘)) ↔ 𝑘 ≤ (𝐹𝑥)))
265264anbi2d 736 . . . . . . . . . . . . 13 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → ((𝑥 ∈ if(𝑘 = 𝑛, ℝ, (𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛)))))) ∧ ¬ 𝑥 ∈ (𝐹 “ (-∞(,)𝑘))) ↔ (𝑥 ∈ if(𝑘 = 𝑛, ℝ, (𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛)))))) ∧ 𝑘 ≤ (𝐹𝑥))))
266252, 265syl5bb 271 . . . . . . . . . . . 12 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → (𝑥 ∈ (if(𝑘 = 𝑛, ℝ, (𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛)))))) ∖ (𝐹 “ (-∞(,)𝑘))) ↔ (𝑥 ∈ if(𝑘 = 𝑛, ℝ, (𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛)))))) ∧ 𝑘 ≤ (𝐹𝑥))))
267251, 266bitr4d 270 . . . . . . . . . . 11 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → (if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) = 𝑘𝑥 ∈ (if(𝑘 = 𝑛, ℝ, (𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛)))))) ∖ (𝐹 “ (-∞(,)𝑘)))))
26859, 267sylan9bbr 733 . . . . . . . . . 10 (((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) ∧ 𝑥 ∈ (-𝑛[,]𝑛)) → (if(𝑥 ∈ (-𝑛[,]𝑛), if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛), 0) = 𝑘𝑥 ∈ (if(𝑘 = 𝑛, ℝ, (𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛)))))) ∖ (𝐹 “ (-∞(,)𝑘)))))
269268pm5.32da 671 . . . . . . . . 9 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → ((𝑥 ∈ (-𝑛[,]𝑛) ∧ if(𝑥 ∈ (-𝑛[,]𝑛), if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛), 0) = 𝑘) ↔ (𝑥 ∈ (-𝑛[,]𝑛) ∧ 𝑥 ∈ (if(𝑘 = 𝑛, ℝ, (𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛)))))) ∖ (𝐹 “ (-∞(,)𝑘))))))
27049, 57, 2693bitrd 293 . . . . . . . 8 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → (((𝐺𝑛)‘𝑥) = 𝑘 ↔ (𝑥 ∈ (-𝑛[,]𝑛) ∧ 𝑥 ∈ (if(𝑘 = 𝑛, ℝ, (𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛)))))) ∖ (𝐹 “ (-∞(,)𝑘))))))
271270pm5.32da 671 . . . . . . 7 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) → ((𝑥 ∈ ℝ ∧ ((𝐺𝑛)‘𝑥) = 𝑘) ↔ (𝑥 ∈ ℝ ∧ (𝑥 ∈ (-𝑛[,]𝑛) ∧ 𝑥 ∈ (if(𝑘 = 𝑛, ℝ, (𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛)))))) ∖ (𝐹 “ (-∞(,)𝑘)))))))
27223adantr 480 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) → (𝐺𝑛):ℝ⟶ℝ)
273272, 195syl 17 . . . . . . . 8 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) → (𝐺𝑛) Fn ℝ)
274 fniniseg 6246 . . . . . . . 8 ((𝐺𝑛) Fn ℝ → (𝑥 ∈ ((𝐺𝑛) “ {𝑘}) ↔ (𝑥 ∈ ℝ ∧ ((𝐺𝑛)‘𝑥) = 𝑘)))
275273, 274syl 17 . . . . . . 7 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) → (𝑥 ∈ ((𝐺𝑛) “ {𝑘}) ↔ (𝑥 ∈ ℝ ∧ ((𝐺𝑛)‘𝑥) = 𝑘)))
276 elin 3758 . . . . . . . 8 (𝑥 ∈ ((-𝑛[,]𝑛) ∩ (if(𝑘 = 𝑛, ℝ, (𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛)))))) ∖ (𝐹 “ (-∞(,)𝑘)))) ↔ (𝑥 ∈ (-𝑛[,]𝑛) ∧ 𝑥 ∈ (if(𝑘 = 𝑛, ℝ, (𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛)))))) ∖ (𝐹 “ (-∞(,)𝑘)))))
277184ad2antlr 759 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) → 𝑛 ∈ ℝ)
278277renegcld 10336 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) → -𝑛 ∈ ℝ)
279 iccmbl 23141 . . . . . . . . . . . . 13 ((-𝑛 ∈ ℝ ∧ 𝑛 ∈ ℝ) → (-𝑛[,]𝑛) ∈ dom vol)
280278, 277, 279syl2anc 691 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) → (-𝑛[,]𝑛) ∈ dom vol)
281 mblss 23106 . . . . . . . . . . . 12 ((-𝑛[,]𝑛) ∈ dom vol → (-𝑛[,]𝑛) ⊆ ℝ)
282280, 281syl 17 . . . . . . . . . . 11 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) → (-𝑛[,]𝑛) ⊆ ℝ)
283282sseld 3567 . . . . . . . . . 10 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) → (𝑥 ∈ (-𝑛[,]𝑛) → 𝑥 ∈ ℝ))
284283adantrd 483 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) → ((𝑥 ∈ (-𝑛[,]𝑛) ∧ 𝑥 ∈ (if(𝑘 = 𝑛, ℝ, (𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛)))))) ∖ (𝐹 “ (-∞(,)𝑘)))) → 𝑥 ∈ ℝ))
285284pm4.71rd 665 . . . . . . . 8 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) → ((𝑥 ∈ (-𝑛[,]𝑛) ∧ 𝑥 ∈ (if(𝑘 = 𝑛, ℝ, (𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛)))))) ∖ (𝐹 “ (-∞(,)𝑘)))) ↔ (𝑥 ∈ ℝ ∧ (𝑥 ∈ (-𝑛[,]𝑛) ∧ 𝑥 ∈ (if(𝑘 = 𝑛, ℝ, (𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛)))))) ∖ (𝐹 “ (-∞(,)𝑘)))))))
286276, 285syl5bb 271 . . . . . . 7 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) → (𝑥 ∈ ((-𝑛[,]𝑛) ∩ (if(𝑘 = 𝑛, ℝ, (𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛)))))) ∖ (𝐹 “ (-∞(,)𝑘)))) ↔ (𝑥 ∈ ℝ ∧ (𝑥 ∈ (-𝑛[,]𝑛) ∧ 𝑥 ∈ (if(𝑘 = 𝑛, ℝ, (𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛)))))) ∖ (𝐹 “ (-∞(,)𝑘)))))))
287271, 275, 2863bitr4d 299 . . . . . 6 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) → (𝑥 ∈ ((𝐺𝑛) “ {𝑘}) ↔ 𝑥 ∈ ((-𝑛[,]𝑛) ∩ (if(𝑘 = 𝑛, ℝ, (𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛)))))) ∖ (𝐹 “ (-∞(,)𝑘))))))
288287eqrdv 2608 . . . . 5 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) → ((𝐺𝑛) “ {𝑘}) = ((-𝑛[,]𝑛) ∩ (if(𝑘 = 𝑛, ℝ, (𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛)))))) ∖ (𝐹 “ (-∞(,)𝑘)))))
289 rembl 23115 . . . . . . . . 9 ℝ ∈ dom vol
290 fss 5969 . . . . . . . . . . 11 ((𝐹:ℝ⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ ℝ) → 𝐹:ℝ⟶ℝ)
2917, 63, 290sylancl 693 . . . . . . . . . 10 (𝜑𝐹:ℝ⟶ℝ)
292 mbfima 23205 . . . . . . . . . 10 ((𝐹 ∈ MblFn ∧ 𝐹:ℝ⟶ℝ) → (𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛))))) ∈ dom vol)
2936, 291, 292syl2anc 691 . . . . . . . . 9 (𝜑 → (𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛))))) ∈ dom vol)
294 ifcl 4080 . . . . . . . . 9 ((ℝ ∈ dom vol ∧ (𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛))))) ∈ dom vol) → if(𝑘 = 𝑛, ℝ, (𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛)))))) ∈ dom vol)
295289, 293, 294sylancr 694 . . . . . . . 8 (𝜑 → if(𝑘 = 𝑛, ℝ, (𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛)))))) ∈ dom vol)
296 mbfima 23205 . . . . . . . . 9 ((𝐹 ∈ MblFn ∧ 𝐹:ℝ⟶ℝ) → (𝐹 “ (-∞(,)𝑘)) ∈ dom vol)
2976, 291, 296syl2anc 691 . . . . . . . 8 (𝜑 → (𝐹 “ (-∞(,)𝑘)) ∈ dom vol)
298 difmbl 23118 . . . . . . . 8 ((if(𝑘 = 𝑛, ℝ, (𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛)))))) ∈ dom vol ∧ (𝐹 “ (-∞(,)𝑘)) ∈ dom vol) → (if(𝑘 = 𝑛, ℝ, (𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛)))))) ∖ (𝐹 “ (-∞(,)𝑘))) ∈ dom vol)
299295, 297, 298syl2anc 691 . . . . . . 7 (𝜑 → (if(𝑘 = 𝑛, ℝ, (𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛)))))) ∖ (𝐹 “ (-∞(,)𝑘))) ∈ dom vol)
300299ad2antrr 758 . . . . . 6 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) → (if(𝑘 = 𝑛, ℝ, (𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛)))))) ∖ (𝐹 “ (-∞(,)𝑘))) ∈ dom vol)
301 inmbl 23117 . . . . . 6 (((-𝑛[,]𝑛) ∈ dom vol ∧ (if(𝑘 = 𝑛, ℝ, (𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛)))))) ∖ (𝐹 “ (-∞(,)𝑘))) ∈ dom vol) → ((-𝑛[,]𝑛) ∩ (if(𝑘 = 𝑛, ℝ, (𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛)))))) ∖ (𝐹 “ (-∞(,)𝑘)))) ∈ dom vol)
302280, 300, 301syl2anc 691 . . . . 5 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) → ((-𝑛[,]𝑛) ∩ (if(𝑘 = 𝑛, ℝ, (𝐹 “ (-∞(,)(𝑘 + (1 / (2↑𝑛)))))) ∖ (𝐹 “ (-∞(,)𝑘)))) ∈ dom vol)
303288, 302eqeltrd 2688 . . . 4 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) → ((𝐺𝑛) “ {𝑘}) ∈ dom vol)
304 mblvol 23105 . . . . . 6 (((𝐺𝑛) “ {𝑘}) ∈ dom vol → (vol‘((𝐺𝑛) “ {𝑘})) = (vol*‘((𝐺𝑛) “ {𝑘})))
305303, 304syl 17 . . . . 5 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) → (vol‘((𝐺𝑛) “ {𝑘})) = (vol*‘((𝐺𝑛) “ {𝑘})))
306196adantr 480 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) → (𝐺𝑛) Fn ℝ)
307306, 274syl 17 . . . . . . . 8 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) → (𝑥 ∈ ((𝐺𝑛) “ {𝑘}) ↔ (𝑥 ∈ ℝ ∧ ((𝐺𝑛)‘𝑥) = 𝑘)))
30882, 185ifcld 4081 . . . . . . . . . . . . . . 15 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) ∈ ℝ)
309 0re 9919 . . . . . . . . . . . . . . 15 0 ∈ ℝ
310 ifcl 4080 . . . . . . . . . . . . . . 15 ((if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛) ∈ ℝ ∧ 0 ∈ ℝ) → if(𝑥 ∈ (-𝑛[,]𝑛), if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛), 0) ∈ ℝ)
311308, 309, 310sylancl 693 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → if(𝑥 ∈ (-𝑛[,]𝑛), if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛), 0) ∈ ℝ)
31244fvmpt2 6200 . . . . . . . . . . . . . 14 ((𝑥 ∈ ℝ ∧ if(𝑥 ∈ (-𝑛[,]𝑛), if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛), 0) ∈ ℝ) → ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝑛[,]𝑛), if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛), 0))‘𝑥) = if(𝑥 ∈ (-𝑛[,]𝑛), if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛), 0))
31338, 311, 312syl2anc 691 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝑛[,]𝑛), if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛), 0))‘𝑥) = if(𝑥 ∈ (-𝑛[,]𝑛), if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛), 0))
31437, 313eqtrd 2644 . . . . . . . . . . . 12 (((𝜑𝑛 ∈ ℕ) ∧ 𝑥 ∈ ℝ) → ((𝐺𝑛)‘𝑥) = if(𝑥 ∈ (-𝑛[,]𝑛), if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛), 0))
315314adantlr 747 . . . . . . . . . . 11 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → ((𝐺𝑛)‘𝑥) = if(𝑥 ∈ (-𝑛[,]𝑛), if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛), 0))
316315eqeq1d 2612 . . . . . . . . . 10 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → (((𝐺𝑛)‘𝑥) = 𝑘 ↔ if(𝑥 ∈ (-𝑛[,]𝑛), if((𝑛𝐽𝑥) ≤ 𝑛, (𝑛𝐽𝑥), 𝑛), 0) = 𝑘))
317316, 56sylbid 229 . . . . . . . . 9 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) ∧ 𝑥 ∈ ℝ) → (((𝐺𝑛)‘𝑥) = 𝑘𝑥 ∈ (-𝑛[,]𝑛)))
318317expimpd 627 . . . . . . . 8 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) → ((𝑥 ∈ ℝ ∧ ((𝐺𝑛)‘𝑥) = 𝑘) → 𝑥 ∈ (-𝑛[,]𝑛)))
319307, 318sylbid 229 . . . . . . 7 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) → (𝑥 ∈ ((𝐺𝑛) “ {𝑘}) → 𝑥 ∈ (-𝑛[,]𝑛)))
320319ssrdv 3574 . . . . . 6 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) → ((𝐺𝑛) “ {𝑘}) ⊆ (-𝑛[,]𝑛))
321 iccssre 12126 . . . . . . 7 ((-𝑛 ∈ ℝ ∧ 𝑛 ∈ ℝ) → (-𝑛[,]𝑛) ⊆ ℝ)
322278, 277, 321syl2anc 691 . . . . . 6 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) → (-𝑛[,]𝑛) ⊆ ℝ)
323 mblvol 23105 . . . . . . . 8 ((-𝑛[,]𝑛) ∈ dom vol → (vol‘(-𝑛[,]𝑛)) = (vol*‘(-𝑛[,]𝑛)))
324280, 323syl 17 . . . . . . 7 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) → (vol‘(-𝑛[,]𝑛)) = (vol*‘(-𝑛[,]𝑛)))
325 iccvolcl 23142 . . . . . . . 8 ((-𝑛 ∈ ℝ ∧ 𝑛 ∈ ℝ) → (vol‘(-𝑛[,]𝑛)) ∈ ℝ)
326278, 277, 325syl2anc 691 . . . . . . 7 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) → (vol‘(-𝑛[,]𝑛)) ∈ ℝ)
327324, 326eqeltrrd 2689 . . . . . 6 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) → (vol*‘(-𝑛[,]𝑛)) ∈ ℝ)
328 ovolsscl 23061 . . . . . 6 ((((𝐺𝑛) “ {𝑘}) ⊆ (-𝑛[,]𝑛) ∧ (-𝑛[,]𝑛) ⊆ ℝ ∧ (vol*‘(-𝑛[,]𝑛)) ∈ ℝ) → (vol*‘((𝐺𝑛) “ {𝑘})) ∈ ℝ)
329320, 322, 327, 328syl3anc 1318 . . . . 5 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) → (vol*‘((𝐺𝑛) “ {𝑘})) ∈ ℝ)
330305, 329eqeltrd 2688 . . . 4 (((𝜑𝑛 ∈ ℕ) ∧ 𝑘 ∈ (ran (𝐺𝑛) ∖ {0})) → (vol‘((𝐺𝑛) “ {𝑘})) ∈ ℝ)
33123, 34, 303, 330i1fd 23254 . . 3 ((𝜑𝑛 ∈ ℕ) → (𝐺𝑛) ∈ dom ∫1)
332331ralrimiva 2949 . 2 (𝜑 → ∀𝑛 ∈ ℕ (𝐺𝑛) ∈ dom ∫1)
333 ffnfv 6295 . 2 (𝐺:ℕ⟶dom ∫1 ↔ (𝐺 Fn ℕ ∧ ∀𝑛 ∈ ℕ (𝐺𝑛) ∈ dom ∫1))
3345, 332, 333sylanbrc 695 1 (𝜑𝐺:ℕ⟶dom ∫1)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ∖ cdif 3537   ∩ cin 3539   ⊆ wss 3540  ifcif 4036  {csn 4125   class class class wbr 4583   ↦ cmpt 4643   × cxp 5036  ◡ccnv 5037  dom cdm 5038  ran crn 5039   “ cima 5041   Fn wfn 5799  ⟶wf 5800  –onto→wfo 5802  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  Fincfn 7841  ℝcr 9814  0cc0 9815  1c1 9816   + caddc 9818   · cmul 9820  +∞cpnf 9950  -∞cmnf 9951  ℝ*cxr 9952   < clt 9953   ≤ cle 9954  -cneg 10146   / cdiv 10563  ℕcn 10897  2c2 10947  ℕ0cn0 11169  ℤcz 11254  (,)cioo 12046  [,)cico 12048  [,]cicc 12049  ...cfz 12197  ⌊cfl 12453  ↑cexp 12722  vol*covol 23038  volcvol 23039  MblFncmbf 23189  ∫1citg1 23190 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fi 8200  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-q 11665  df-rp 11709  df-xneg 11822  df-xadd 11823  df-xmul 11824  df-ioo 12050  df-ico 12052  df-icc 12053  df-fz 12198  df-fzo 12335  df-fl 12455  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-rlim 14068  df-sum 14265  df-rest 15906  df-topgen 15927  df-psmet 19559  df-xmet 19560  df-met 19561  df-bl 19562  df-mopn 19563  df-top 20521  df-bases 20522  df-topon 20523  df-cmp 21000  df-ovol 23040  df-vol 23041  df-mbf 23194  df-itg1 23195 This theorem is referenced by:  mbfi1fseqlem5  23292  mbfi1fseqlem6  23293
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