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| Mirrors > Home > MPE Home > Th. List > Mathboxes > knoppndvlem4 | Structured version Visualization version GIF version | ||
| Description: Lemma for knoppndv 37155. (Contributed by Asger C. Ipsen, 15-Jun-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.) |
| Ref | Expression |
|---|---|
| knoppndvlem4.t | ⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) |
| knoppndvlem4.f | ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦))))) |
| knoppndvlem4.w | ⊢ 𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹‘𝑤)‘𝑖)) |
| knoppndvlem4.a | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
| knoppndvlem4.c | ⊢ (𝜑 → 𝐶 ∈ (-1(,)1)) |
| knoppndvlem4.n | ⊢ (𝜑 → 𝑁 ∈ ℕ) |
| Ref | Expression |
|---|---|
| knoppndvlem4 | ⊢ (𝜑 → seq0( + , (𝐹‘𝐴)) ⇝ (𝑊‘𝐴)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | nn0uz 12910 | . 2 ⊢ ℕ0 = (ℤ≥‘0) | |
| 2 | 0zd 12613 | . 2 ⊢ (𝜑 → 0 ∈ ℤ) | |
| 3 | knoppndvlem4.t | . . 3 ⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) | |
| 4 | knoppndvlem4.f | . . 3 ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦))))) | |
| 5 | knoppndvlem4.n | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
| 6 | knoppndvlem4.c | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ (-1(,)1)) | |
| 7 | 6 | knoppndvlem3 37135 | . . . 4 ⊢ (𝜑 → (𝐶 ∈ ℝ ∧ (abs‘𝐶) < 1)) |
| 8 | 7 | simpld 500 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ) |
| 9 | 3, 4, 5, 8 | knoppcnlem8 37121 | . 2 ⊢ (𝜑 → seq0( ∘f + , (𝑚 ∈ ℕ0 ↦ (𝑧 ∈ ℝ ↦ ((𝐹‘𝑧)‘𝑚)))):ℕ0⟶(ℂ ↑m ℝ)) |
| 10 | knoppndvlem4.a | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
| 11 | seqex 14050 | . . 3 ⊢ seq0( + , (𝐹‘𝐴)) ∈ V | |
| 12 | 11 | a1i 11 | . 2 ⊢ (𝜑 → seq0( + , (𝐹‘𝐴)) ∈ V) |
| 13 | 5 | adantr 486 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝑁 ∈ ℕ) |
| 14 | 8 | adantr 486 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝐶 ∈ ℝ) |
| 15 | simpr 490 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0) | |
| 16 | 3, 4, 13, 14, 15 | knoppcnlem7 37120 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (seq0( ∘f + , (𝑚 ∈ ℕ0 ↦ (𝑧 ∈ ℝ ↦ ((𝐹‘𝑧)‘𝑚))))‘𝑘) = (𝑣 ∈ ℝ ↦ (seq0( + , (𝐹‘𝑣))‘𝑘))) |
| 17 | 16 | fveq1d 6887 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((seq0( ∘f + , (𝑚 ∈ ℕ0 ↦ (𝑧 ∈ ℝ ↦ ((𝐹‘𝑧)‘𝑚))))‘𝑘)‘𝐴) = ((𝑣 ∈ ℝ ↦ (seq0( + , (𝐹‘𝑣))‘𝑘))‘𝐴)) |
| 18 | eqid 2766 | . . . . 5 ⊢ (𝑣 ∈ ℝ ↦ (seq0( + , (𝐹‘𝑣))‘𝑘)) = (𝑣 ∈ ℝ ↦ (seq0( + , (𝐹‘𝑣))‘𝑘)) | |
| 19 | fveq2 6885 | . . . . . . 7 ⊢ (𝑣 = 𝐴 → (𝐹‘𝑣) = (𝐹‘𝐴)) | |
| 20 | 19 | seqeq3d 14056 | . . . . . 6 ⊢ (𝑣 = 𝐴 → seq0( + , (𝐹‘𝑣)) = seq0( + , (𝐹‘𝐴))) |
| 21 | 20 | fveq1d 6887 | . . . . 5 ⊢ (𝑣 = 𝐴 → (seq0( + , (𝐹‘𝑣))‘𝑘) = (seq0( + , (𝐹‘𝐴))‘𝑘)) |
| 22 | fvexd 6900 | . . . . 5 ⊢ (𝜑 → (seq0( + , (𝐹‘𝐴))‘𝑘) ∈ V) | |
| 23 | 18, 21, 10, 22 | fvmptd3 7017 | . . . 4 ⊢ (𝜑 → ((𝑣 ∈ ℝ ↦ (seq0( + , (𝐹‘𝑣))‘𝑘))‘𝐴) = (seq0( + , (𝐹‘𝐴))‘𝑘)) |
| 24 | 23 | adantr 486 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝑣 ∈ ℝ ↦ (seq0( + , (𝐹‘𝑣))‘𝑘))‘𝐴) = (seq0( + , (𝐹‘𝐴))‘𝑘)) |
| 25 | 17, 24 | eqtrd 2801 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((seq0( ∘f + , (𝑚 ∈ ℕ0 ↦ (𝑧 ∈ ℝ ↦ ((𝐹‘𝑧)‘𝑚))))‘𝑘)‘𝐴) = (seq0( + , (𝐹‘𝐴))‘𝑘)) |
| 26 | knoppndvlem4.w | . . 3 ⊢ 𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹‘𝑤)‘𝑖)) | |
| 27 | 7 | simprd 501 | . . 3 ⊢ (𝜑 → (abs‘𝐶) < 1) |
| 28 | 3, 4, 26, 5, 8, 27 | knoppcnlem9 37122 | . 2 ⊢ (𝜑 → seq0( ∘f + , (𝑚 ∈ ℕ0 ↦ (𝑧 ∈ ℝ ↦ ((𝐹‘𝑧)‘𝑚))))(⇝𝑢‘ℝ)𝑊) |
| 29 | 1, 2, 9, 10, 12, 25, 28 | ulmclm 26565 | 1 ⊢ (𝜑 → seq0( + , (𝐹‘𝐴)) ⇝ (𝑊‘𝐴)) |
| Colors of variables: wff setvar class |
| This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2146 Vcvv 3458 class class class wbr 5112 ↦ cmpt 5195 ‘cfv 6540 (class class class)co 7416 ∘f cof 7678 ℝcr 11109 0cc0 11110 1c1 11111 + caddc 11113 · cmul 11115 < clt 11253 − cmin 11451 -cneg 11452 / cdiv 11881 ℕcn 12243 2c2 12305 ℕ0cn0 12514 (,)cioo 13382 ⌊cfl 13834 seqcseq 14048 ↑cexp 14108 abscabs 15296 ⇝ cli 15546 Σcsu 15748 |
| 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 5241 ax-sep 5260 ax-nul 5272 ax-pow 5339 ax-pr 5407 ax-un 7738 ax-inf2 9612 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188 |
| This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 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-nel 3068 df-ral 3083 df-rex 3093 df-rmo 3372 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-pss 3928 df-nul 4290 df-if 4491 df-pw 4567 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4876 df-int 4916 df-iun 4961 df-br 5113 df-opab 5177 df-mpt 5196 df-tr 5222 df-id 5559 df-eprel 5564 df-po 5572 df-so 5573 df-fr 5617 df-se 5618 df-we 5619 df-xp 5670 df-rel 5671 df-cnv 5672 df-co 5673 df-dm 5674 df-rn 5675 df-res 5676 df-ima 5677 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7680 df-om 7865 df-1st 7988 df-2nd 7989 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-1o 8455 df-er 8696 df-map 8828 df-pm 8829 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-sup 9404 df-inf 9405 df-oi 9474 df-card 9936 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11453 df-neg 11454 df-div 11882 df-nn 12244 df-2 12313 df-3 12314 df-n0 12515 df-z 12602 df-uz 12873 df-rp 13027 df-ioo 13386 df-ico 13388 df-fz 13546 df-fzo 13694 df-fl 13836 df-seq 14049 df-exp 14109 df-hash 14378 df-cj 15161 df-re 15162 df-im 15163 df-sqrt 15297 df-abs 15298 df-limsup 15533 df-clim 15550 df-rlim 15551 df-sum 15749 df-ulm 26555 |
| This theorem is used by: knoppndvlem6 37138 knoppf 37156 |
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