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Mirrors > Home > MPE Home > Th. List > Mathboxes > cnndvlem1 | Structured version Visualization version GIF version |
Description: Lemma for cnndv 36522. (Contributed by Asger C. Ipsen, 25-Aug-2021.) |
Ref | Expression |
---|---|
cnndvlem1.t | ⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) |
cnndvlem1.f | ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ (((1 / 2)↑𝑛) · (𝑇‘(((2 · 3)↑𝑛) · 𝑦))))) |
cnndvlem1.w | ⊢ 𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹‘𝑤)‘𝑖)) |
Ref | Expression |
---|---|
cnndvlem1 | ⊢ (𝑊 ∈ (ℝ–cn→ℝ) ∧ dom (ℝ D 𝑊) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnndvlem1.t | . . . 4 ⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) | |
2 | cnndvlem1.f | . . . 4 ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ (((1 / 2)↑𝑛) · (𝑇‘(((2 · 3)↑𝑛) · 𝑦))))) | |
3 | cnndvlem1.w | . . . 4 ⊢ 𝑊 = (𝑤 ∈ ℝ ↦ Σ𝑖 ∈ ℕ0 ((𝐹‘𝑤)‘𝑖)) | |
4 | 3nn 12343 | . . . . 5 ⊢ 3 ∈ ℕ | |
5 | 4 | a1i 11 | . . . 4 ⊢ (⊤ → 3 ∈ ℕ) |
6 | neg1rr 12379 | . . . . . . . . 9 ⊢ -1 ∈ ℝ | |
7 | 6 | rexri 11317 | . . . . . . . 8 ⊢ -1 ∈ ℝ* |
8 | 1re 11259 | . . . . . . . . 9 ⊢ 1 ∈ ℝ | |
9 | 8 | rexri 11317 | . . . . . . . 8 ⊢ 1 ∈ ℝ* |
10 | halfre 12478 | . . . . . . . . 9 ⊢ (1 / 2) ∈ ℝ | |
11 | 10 | rexri 11317 | . . . . . . . 8 ⊢ (1 / 2) ∈ ℝ* |
12 | 7, 9, 11 | 3pm3.2i 1338 | . . . . . . 7 ⊢ (-1 ∈ ℝ* ∧ 1 ∈ ℝ* ∧ (1 / 2) ∈ ℝ*) |
13 | neg1lt0 12381 | . . . . . . . . . 10 ⊢ -1 < 0 | |
14 | halfgt0 12480 | . . . . . . . . . 10 ⊢ 0 < (1 / 2) | |
15 | 13, 14 | pm3.2i 470 | . . . . . . . . 9 ⊢ (-1 < 0 ∧ 0 < (1 / 2)) |
16 | 0re 11261 | . . . . . . . . . 10 ⊢ 0 ∈ ℝ | |
17 | 6, 16, 10 | lttri 11385 | . . . . . . . . 9 ⊢ ((-1 < 0 ∧ 0 < (1 / 2)) → -1 < (1 / 2)) |
18 | 15, 17 | ax-mp 5 | . . . . . . . 8 ⊢ -1 < (1 / 2) |
19 | halflt1 12482 | . . . . . . . 8 ⊢ (1 / 2) < 1 | |
20 | 18, 19 | pm3.2i 470 | . . . . . . 7 ⊢ (-1 < (1 / 2) ∧ (1 / 2) < 1) |
21 | 12, 20 | pm3.2i 470 | . . . . . 6 ⊢ ((-1 ∈ ℝ* ∧ 1 ∈ ℝ* ∧ (1 / 2) ∈ ℝ*) ∧ (-1 < (1 / 2) ∧ (1 / 2) < 1)) |
22 | elioo3g 13413 | . . . . . 6 ⊢ ((1 / 2) ∈ (-1(,)1) ↔ ((-1 ∈ ℝ* ∧ 1 ∈ ℝ* ∧ (1 / 2) ∈ ℝ*) ∧ (-1 < (1 / 2) ∧ (1 / 2) < 1))) | |
23 | 21, 22 | mpbir 231 | . . . . 5 ⊢ (1 / 2) ∈ (-1(,)1) |
24 | 23 | a1i 11 | . . . 4 ⊢ (⊤ → (1 / 2) ∈ (-1(,)1)) |
25 | 1, 2, 3, 5, 24 | knoppcn2 36519 | . . 3 ⊢ (⊤ → 𝑊 ∈ (ℝ–cn→ℝ)) |
26 | 25 | mptru 1544 | . 2 ⊢ 𝑊 ∈ (ℝ–cn→ℝ) |
27 | 2cn 12339 | . . . . . . . . 9 ⊢ 2 ∈ ℂ | |
28 | 27 | mullidi 11264 | . . . . . . . 8 ⊢ (1 · 2) = 2 |
29 | 2lt3 12436 | . . . . . . . 8 ⊢ 2 < 3 | |
30 | 28, 29 | eqbrtri 5169 | . . . . . . 7 ⊢ (1 · 2) < 3 |
31 | 2pos 12367 | . . . . . . . 8 ⊢ 0 < 2 | |
32 | 4 | nnrei 12273 | . . . . . . . . 9 ⊢ 3 ∈ ℝ |
33 | 2re 12338 | . . . . . . . . 9 ⊢ 2 ∈ ℝ | |
34 | 8, 32, 33 | ltmuldivi 12186 | . . . . . . . 8 ⊢ (0 < 2 → ((1 · 2) < 3 ↔ 1 < (3 / 2))) |
35 | 31, 34 | ax-mp 5 | . . . . . . 7 ⊢ ((1 · 2) < 3 ↔ 1 < (3 / 2)) |
36 | 30, 35 | mpbi 230 | . . . . . 6 ⊢ 1 < (3 / 2) |
37 | 16, 10, 14 | ltleii 11382 | . . . . . . . . 9 ⊢ 0 ≤ (1 / 2) |
38 | 10 | absidi 15413 | . . . . . . . . 9 ⊢ (0 ≤ (1 / 2) → (abs‘(1 / 2)) = (1 / 2)) |
39 | 37, 38 | ax-mp 5 | . . . . . . . 8 ⊢ (abs‘(1 / 2)) = (1 / 2) |
40 | 39 | oveq2i 7442 | . . . . . . 7 ⊢ (3 · (abs‘(1 / 2))) = (3 · (1 / 2)) |
41 | 4 | nncni 12274 | . . . . . . . . 9 ⊢ 3 ∈ ℂ |
42 | 2ne0 12368 | . . . . . . . . 9 ⊢ 2 ≠ 0 | |
43 | 41, 27, 42 | divreci 12010 | . . . . . . . 8 ⊢ (3 / 2) = (3 · (1 / 2)) |
44 | 43 | eqcomi 2744 | . . . . . . 7 ⊢ (3 · (1 / 2)) = (3 / 2) |
45 | 40, 44 | eqtri 2763 | . . . . . 6 ⊢ (3 · (abs‘(1 / 2))) = (3 / 2) |
46 | 36, 45 | breqtrri 5175 | . . . . 5 ⊢ 1 < (3 · (abs‘(1 / 2))) |
47 | 46 | a1i 11 | . . . 4 ⊢ (⊤ → 1 < (3 · (abs‘(1 / 2)))) |
48 | 1, 2, 3, 24, 5, 47 | knoppndv 36517 | . . 3 ⊢ (⊤ → dom (ℝ D 𝑊) = ∅) |
49 | 48 | mptru 1544 | . 2 ⊢ dom (ℝ D 𝑊) = ∅ |
50 | 26, 49 | pm3.2i 470 | 1 ⊢ (𝑊 ∈ (ℝ–cn→ℝ) ∧ dom (ℝ D 𝑊) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ⊤wtru 1538 ∈ wcel 2106 ∅c0 4339 class class class wbr 5148 ↦ cmpt 5231 dom cdm 5689 ‘cfv 6563 (class class class)co 7431 ℝcr 11152 0cc0 11153 1c1 11154 + caddc 11156 · cmul 11158 ℝ*cxr 11292 < clt 11293 ≤ cle 11294 − cmin 11490 -cneg 11491 / cdiv 11918 ℕcn 12264 2c2 12319 3c3 12320 ℕ0cn0 12524 (,)cioo 13384 ⌊cfl 13827 ↑cexp 14099 abscabs 15270 Σcsu 15719 –cn→ccncf 24916 D cdv 25913 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-pm 8868 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-fi 9449 df-sup 9480 df-inf 9481 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-q 12989 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-ioo 13388 df-ico 13390 df-icc 13391 df-fz 13545 df-fzo 13692 df-fl 13829 df-seq 14040 df-exp 14100 df-hash 14367 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-limsup 15504 df-clim 15521 df-rlim 15522 df-sum 15720 df-dvds 16288 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-rest 17469 df-topn 17470 df-0g 17488 df-gsum 17489 df-topgen 17490 df-pt 17491 df-prds 17494 df-xrs 17549 df-qtop 17554 df-imas 17555 df-xps 17557 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-mulg 19099 df-cntz 19348 df-cmn 19815 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-cnfld 21383 df-top 22916 df-topon 22933 df-topsp 22955 df-bases 22969 df-ntr 23044 df-cn 23251 df-cnp 23252 df-tx 23586 df-hmeo 23779 df-xms 24346 df-ms 24347 df-tms 24348 df-cncf 24918 df-limc 25916 df-dv 25917 df-ulm 26435 |
This theorem is referenced by: cnndvlem2 36521 |
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