Theorem mbfi1fseqlem2 25912

Description: Lemma for mbfi1fseq 25917. (Contributed by Mario Carneiro, 16-Aug-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)

Hypotheses

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)))

Assertion

Ref	Expression
mbfi1fseqlem2	⊢ (𝐴 ∈ ℕ → (𝐺‘𝐴) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝐴[,]𝐴), if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴), 0)))

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 mbfi1fseqlem2

Step	Hyp	Ref	Expression
1		negeq 11467	. . . . . 6 ⊢ (𝑚 = 𝐴 → -𝑚 = -𝐴)
2		id 23	. . . . . 6 ⊢ (𝑚 = 𝐴 → 𝑚 = 𝐴)
3	1, 2	oveq12d 7441	. . . . 5 ⊢ (𝑚 = 𝐴 → (-𝑚[,]𝑚) = (-𝐴[,]𝐴))
4	3	eleq2d 2852	. . . 4 ⊢ (𝑚 = 𝐴 → (𝑥 ∈ (-𝑚[,]𝑚) ↔ 𝑥 ∈ (-𝐴[,]𝐴)))
5		oveq1 7430	. . . . . 6 ⊢ (𝑚 = 𝐴 → (𝑚𝐽𝑥) = (𝐴𝐽𝑥))
6	5, 2	breq12d 5127	. . . . 5 ⊢ (𝑚 = 𝐴 → ((𝑚𝐽𝑥) ≤ 𝑚 ↔ (𝐴𝐽𝑥) ≤ 𝐴))
7	6, 5, 2	ifbieq12d 4521	. . . 4 ⊢ (𝑚 = 𝐴 → if((𝑚𝐽𝑥) ≤ 𝑚, (𝑚𝐽𝑥), 𝑚) = if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴))
8	4, 7	ifbieq1d 4517	. . 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 5210	. 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 11209	. . 3 ⊢ ℝ ∈ V
12	11	mptex 7228	. 2 ⊢ (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝐴[,]𝐴), if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴), 0)) ∈ V
13	9, 10, 12	fvmpt 6996	1 ⊢ (𝐴 ∈ ℕ → (𝐺‘𝐴) = (𝑥 ∈ ℝ ↦ if(𝑥 ∈ (-𝐴[,]𝐴), if((𝐴𝐽𝑥) ≤ 𝐴, (𝐴𝐽𝑥), 𝐴), 0)))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2146  ifcif 4492   class class class wbr 5114   ↦ cmpt 5197  ⟶wf 6539  ‘cfv 6543  (class class class)co 7423   ∈ cmpo 7425  ℝcr 11117  0cc0 11118   · cmul 11123  +∞cpnf 11258   ≤ cle 11262  -cneg 11460   / cdiv 11889  ℕcn 12251  2c2 12313  [,)cico 13392  [,]cicc 13393  ⌊cfl 13843  ↑cexp 14117  MblFncmbf 25810

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pr 5409  ax-cnex 11174  ax-resscn 11175

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7426  df-neg 11462

This theorem is used by:  mbfi1fseqlem3  25913  mbfi1fseqlem4  25914  mbfi1fseqlem5  25915  mbfi1fseqlem6  25916