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| Mirrors > Home > MPE Home > Th. List > mbfi1fseqlem2 | Structured version Visualization version GIF version | ||
| Description: Lemma for mbfi1fseq 25757. (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 11501 | . . . . . 6 ⊢ (𝑚 = 𝐴 → -𝑚 = -𝐴) | |
| 2 | id 22 | . . . . . 6 ⊢ (𝑚 = 𝐴 → 𝑚 = 𝐴) | |
| 3 | 1, 2 | oveq12d 7450 | . . . . 5 ⊢ (𝑚 = 𝐴 → (-𝑚[,]𝑚) = (-𝐴[,]𝐴)) | 
| 4 | 3 | eleq2d 2826 | . . . 4 ⊢ (𝑚 = 𝐴 → (𝑥 ∈ (-𝑚[,]𝑚) ↔ 𝑥 ∈ (-𝐴[,]𝐴))) | 
| 5 | oveq1 7439 | . . . . . 6 ⊢ (𝑚 = 𝐴 → (𝑚𝐽𝑥) = (𝐴𝐽𝑥)) | |
| 6 | 5, 2 | breq12d 5155 | . . . . 5 ⊢ (𝑚 = 𝐴 → ((𝑚𝐽𝑥) ≤ 𝑚 ↔ (𝐴𝐽𝑥) ≤ 𝐴)) | 
| 7 | 6, 5, 2 | ifbieq12d 4553 | . . . 4 ⊢ (𝑚 = 𝐴 → if((𝑚𝐽𝑥) ≤ 𝑚, (𝑚𝐽𝑥), 𝑚) = if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴)) | 
| 8 | 4, 7 | ifbieq1d 4549 | . . 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 5243 | . 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 11247 | . . 3 ⊢ ℝ ∈ V | |
| 12 | 11 | mptex 7244 | . 2 ⊢ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝐴[,]𝐴), if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴), 0)) ∈ V | 
| 13 | 9, 10, 12 | fvmpt 7015 | 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 1539 ∈ wcel 2107 ifcif 4524 class class class wbr 5142 ↦ cmpt 5224 ⟶wf 6556 ‘cfv 6560 (class class class)co 7432 ∈ cmpo 7434 ℝcr 11155 0cc0 11156 · cmul 11161 +∞cpnf 11293 ≤ cle 11297 -cneg 11494 / cdiv 11921 ℕcn 12267 2c2 12322 [,)cico 13390 [,]cicc 13391 ⌊cfl 13831 ↑cexp 14103 MblFncmbf 25650 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pr 5431 ax-cnex 11212 ax-resscn 11213 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-neg 11496 | 
| This theorem is referenced by: mbfi1fseqlem3 25753 mbfi1fseqlem4 25754 mbfi1fseqlem5 25755 mbfi1fseqlem6 25756 | 
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