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Mirrors > Home > MPE Home > Th. List > Mathboxes > knoppndvlem4 | Structured version Visualization version GIF version |
Description: Lemma for knoppndv 32525. (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 11722 | . 2 ⊢ ℕ0 = (ℤ≥‘0) | |
2 | 0zd 11389 | . 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 32505 | . . . 4 ⊢ (𝜑 → (𝐶 ∈ ℝ ∧ (abs‘𝐶) < 1)) |
8 | 7 | simpld 475 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ) |
9 | 3, 4, 5, 8 | knoppcnlem8 32490 | . 2 ⊢ (𝜑 → seq0( ∘𝑓 + , (𝑚 ∈ ℕ0 ↦ (𝑧 ∈ ℝ ↦ ((𝐹‘𝑧)‘𝑚)))):ℕ0⟶(ℂ ↑𝑚 ℝ)) |
10 | knoppndvlem4.a | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
11 | seqex 12803 | . . 3 ⊢ seq0( + , (𝐹‘𝐴)) ∈ V | |
12 | 11 | a1i 11 | . 2 ⊢ (𝜑 → seq0( + , (𝐹‘𝐴)) ∈ V) |
13 | 5 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝑁 ∈ ℕ) |
14 | 8 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝐶 ∈ ℝ) |
15 | simpr 477 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0) | |
16 | 3, 4, 13, 14, 15 | knoppcnlem7 32489 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (seq0( ∘𝑓 + , (𝑚 ∈ ℕ0 ↦ (𝑧 ∈ ℝ ↦ ((𝐹‘𝑧)‘𝑚))))‘𝑘) = (𝑣 ∈ ℝ ↦ (seq0( + , (𝐹‘𝑣))‘𝑘))) |
17 | 16 | fveq1d 6193 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((seq0( ∘𝑓 + , (𝑚 ∈ ℕ0 ↦ (𝑧 ∈ ℝ ↦ ((𝐹‘𝑧)‘𝑚))))‘𝑘)‘𝐴) = ((𝑣 ∈ ℝ ↦ (seq0( + , (𝐹‘𝑣))‘𝑘))‘𝐴)) |
18 | eqid 2622 | . . . . . 6 ⊢ (𝑣 ∈ ℝ ↦ (seq0( + , (𝐹‘𝑣))‘𝑘)) = (𝑣 ∈ ℝ ↦ (seq0( + , (𝐹‘𝑣))‘𝑘)) | |
19 | 18 | a1i 11 | . . . . 5 ⊢ (𝜑 → (𝑣 ∈ ℝ ↦ (seq0( + , (𝐹‘𝑣))‘𝑘)) = (𝑣 ∈ ℝ ↦ (seq0( + , (𝐹‘𝑣))‘𝑘))) |
20 | fveq2 6191 | . . . . . . . 8 ⊢ (𝑣 = 𝐴 → (𝐹‘𝑣) = (𝐹‘𝐴)) | |
21 | 20 | seqeq3d 12809 | . . . . . . 7 ⊢ (𝑣 = 𝐴 → seq0( + , (𝐹‘𝑣)) = seq0( + , (𝐹‘𝐴))) |
22 | 21 | fveq1d 6193 | . . . . . 6 ⊢ (𝑣 = 𝐴 → (seq0( + , (𝐹‘𝑣))‘𝑘) = (seq0( + , (𝐹‘𝐴))‘𝑘)) |
23 | 22 | adantl 482 | . . . . 5 ⊢ ((𝜑 ∧ 𝑣 = 𝐴) → (seq0( + , (𝐹‘𝑣))‘𝑘) = (seq0( + , (𝐹‘𝐴))‘𝑘)) |
24 | fvexd 6203 | . . . . 5 ⊢ (𝜑 → (seq0( + , (𝐹‘𝐴))‘𝑘) ∈ V) | |
25 | 19, 23, 10, 24 | fvmptd 6288 | . . . 4 ⊢ (𝜑 → ((𝑣 ∈ ℝ ↦ (seq0( + , (𝐹‘𝑣))‘𝑘))‘𝐴) = (seq0( + , (𝐹‘𝐴))‘𝑘)) |
26 | 25 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝑣 ∈ ℝ ↦ (seq0( + , (𝐹‘𝑣))‘𝑘))‘𝐴) = (seq0( + , (𝐹‘𝐴))‘𝑘)) |
27 | 17, 26 | eqtrd 2656 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((seq0( ∘𝑓 + , (𝑚 ∈ ℕ0 ↦ (𝑧 ∈ ℝ ↦ ((𝐹‘𝑧)‘𝑚))))‘𝑘)‘𝐴) = (seq0( + , (𝐹‘𝐴))‘𝑘)) |
28 | knoppndvlem4.w | . . 3 ⊢ 𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹‘𝑤)‘𝑖)) | |
29 | 7 | simprd 479 | . . 3 ⊢ (𝜑 → (abs‘𝐶) < 1) |
30 | 3, 4, 28, 5, 8, 29 | knoppcnlem9 32491 | . 2 ⊢ (𝜑 → seq0( ∘𝑓 + , (𝑚 ∈ ℕ0 ↦ (𝑧 ∈ ℝ ↦ ((𝐹‘𝑧)‘𝑚))))(⇝𝑢‘ℝ)𝑊) |
31 | 1, 2, 9, 10, 12, 27, 30 | ulmclm 24141 | 1 ⊢ (𝜑 → seq0( + , (𝐹‘𝐴)) ⇝ (𝑊‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 Vcvv 3200 class class class wbr 4653 ↦ cmpt 4729 ‘cfv 5888 (class class class)co 6650 ∘𝑓 cof 6895 ℝcr 9935 0cc0 9936 1c1 9937 + caddc 9939 · cmul 9941 < clt 10074 − cmin 10266 -cneg 10267 / cdiv 10684 ℕcn 11020 2c2 11070 ℕ0cn0 11292 (,)cioo 12175 ⌊cfl 12591 seqcseq 12801 ↑cexp 12860 abscabs 13974 ⇝ cli 14215 Σcsu 14416 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-ioo 12179 df-ico 12181 df-fz 12327 df-fzo 12466 df-fl 12593 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-limsup 14202 df-clim 14219 df-rlim 14220 df-sum 14417 df-ulm 24131 |
This theorem is referenced by: knoppndvlem6 32508 knoppf 32526 |
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