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Mirrors > Home > MPE Home > Th. List > mbfi1fseqlem2 | Structured version Visualization version GIF version |
Description: Lemma for mbfi1fseq 23294. (Contributed by Mario Carneiro, 16-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.) |
Ref | Expression |
---|---|
mbfi1fseq.1 | ⊢ (𝜑 → 𝐹 ∈ MblFn) |
mbfi1fseq.2 | ⊢ (𝜑 → 𝐹:ℝ⟶(0[,)+∞)) |
mbfi1fseq.3 | ⊢ 𝐽 = (𝑚 ∈ ℕ, 𝑦 ∈ ℝ ↦ ((⌊‘((𝐹‘𝑦) · (2↑𝑚))) / (2↑𝑚))) |
mbfi1fseq.4 | ⊢ 𝐺 = (𝑚 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝑚[,]𝑚), if((𝑚𝐽𝑥) ≤ 𝑚, (𝑚𝐽𝑥), 𝑚), 0))) |
Ref | Expression |
---|---|
mbfi1fseqlem2 | ⊢ (𝐴 ∈ ℕ → (𝐺‘𝐴) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝐴[,]𝐴), if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴), 0))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | negeq 10152 | . . . . . 6 ⊢ (𝑚 = 𝐴 → -𝑚 = -𝐴) | |
2 | id 22 | . . . . . 6 ⊢ (𝑚 = 𝐴 → 𝑚 = 𝐴) | |
3 | 1, 2 | oveq12d 6567 | . . . . 5 ⊢ (𝑚 = 𝐴 → (-𝑚[,]𝑚) = (-𝐴[,]𝐴)) |
4 | 3 | eleq2d 2673 | . . . 4 ⊢ (𝑚 = 𝐴 → (𝑥 ∈ (-𝑚[,]𝑚) ↔ 𝑥 ∈ (-𝐴[,]𝐴))) |
5 | oveq1 6556 | . . . . . 6 ⊢ (𝑚 = 𝐴 → (𝑚𝐽𝑥) = (𝐴𝐽𝑥)) | |
6 | 5, 2 | breq12d 4596 | . . . . 5 ⊢ (𝑚 = 𝐴 → ((𝑚𝐽𝑥) ≤ 𝑚 ↔ (𝐴𝐽𝑥) ≤ 𝐴)) |
7 | 6, 5, 2 | ifbieq12d 4063 | . . . 4 ⊢ (𝑚 = 𝐴 → if((𝑚𝐽𝑥) ≤ 𝑚, (𝑚𝐽𝑥), 𝑚) = if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴)) |
8 | 4, 7 | ifbieq1d 4059 | . . 3 ⊢ (𝑚 = 𝐴 → if(𝑥 ∈ (-𝑚[,]𝑚), if((𝑚𝐽𝑥) ≤ 𝑚, (𝑚𝐽𝑥), 𝑚), 0) = if(𝑥 ∈ (-𝐴[,]𝐴), if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴), 0)) |
9 | 8 | mpteq2dv 4673 | . 2 ⊢ (𝑚 = 𝐴 → (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝑚[,]𝑚), if((𝑚𝐽𝑥) ≤ 𝑚, (𝑚𝐽𝑥), 𝑚), 0)) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝐴[,]𝐴), if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴), 0))) |
10 | mbfi1fseq.4 | . 2 ⊢ 𝐺 = (𝑚 ∈ ℕ ↦ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝑚[,]𝑚), if((𝑚𝐽𝑥) ≤ 𝑚, (𝑚𝐽𝑥), 𝑚), 0))) | |
11 | reex 9906 | . . 3 ⊢ ℝ ∈ V | |
12 | 11 | mptex 6390 | . 2 ⊢ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝐴[,]𝐴), if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴), 0)) ∈ V |
13 | 9, 10, 12 | fvmpt 6191 | 1 ⊢ (𝐴 ∈ ℕ → (𝐺‘𝐴) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝐴[,]𝐴), if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴), 0))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ifcif 4036 class class class wbr 4583 ↦ cmpt 4643 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 ℝcr 9814 0cc0 9815 · cmul 9820 +∞cpnf 9950 ≤ cle 9954 -cneg 10146 / cdiv 10563 ℕcn 10897 2c2 10947 [,)cico 12048 [,]cicc 12049 ⌊cfl 12453 ↑cexp 12722 MblFncmbf 23189 |
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-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-pr 4833 ax-cnex 9871 ax-resscn 9872 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 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-ral 2901 df-rex 2902 df-reu 2903 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-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 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-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-neg 10148 |
This theorem is referenced by: mbfi1fseqlem3 23290 mbfi1fseqlem4 23291 mbfi1fseqlem5 23292 mbfi1fseqlem6 23293 |
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