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Mirrors > Home > MPE Home > Th. List > Mathboxes > knoppndvlem8 | Structured version Visualization version GIF version |
Description: Lemma for knoppndv 36517. (Contributed by Asger C. Ipsen, 15-Jun-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.) |
Ref | Expression |
---|---|
knoppndvlem8.t | ⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) |
knoppndvlem8.f | ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦))))) |
knoppndvlem8.a | ⊢ 𝐴 = ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀) |
knoppndvlem8.c | ⊢ (𝜑 → 𝐶 ∈ (-1(,)1)) |
knoppndvlem8.j | ⊢ (𝜑 → 𝐽 ∈ ℕ0) |
knoppndvlem8.m | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
knoppndvlem8.n | ⊢ (𝜑 → 𝑁 ∈ ℕ) |
knoppndvlem8.1 | ⊢ (𝜑 → 2 ∥ 𝑀) |
Ref | Expression |
---|---|
knoppndvlem8 | ⊢ (𝜑 → ((𝐹‘𝐴)‘𝐽) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | knoppndvlem8.t | . . 3 ⊢ 𝑇 = (𝑥 ∈ ℝ ↦ (abs‘((⌊‘(𝑥 + (1 / 2))) − 𝑥))) | |
2 | knoppndvlem8.f | . . 3 ⊢ 𝐹 = (𝑦 ∈ ℝ ↦ (𝑛 ∈ ℕ0 ↦ ((𝐶↑𝑛) · (𝑇‘(((2 · 𝑁)↑𝑛) · 𝑦))))) | |
3 | knoppndvlem8.a | . . 3 ⊢ 𝐴 = ((((2 · 𝑁)↑-𝐽) / 2) · 𝑀) | |
4 | knoppndvlem8.j | . . 3 ⊢ (𝜑 → 𝐽 ∈ ℕ0) | |
5 | knoppndvlem8.m | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
6 | knoppndvlem8.n | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
7 | 1, 2, 3, 4, 5, 6 | knoppndvlem7 36501 | . 2 ⊢ (𝜑 → ((𝐹‘𝐴)‘𝐽) = ((𝐶↑𝐽) · (𝑇‘(𝑀 / 2)))) |
8 | knoppndvlem8.1 | . . . . 5 ⊢ (𝜑 → 2 ∥ 𝑀) | |
9 | 2z 12647 | . . . . . . . 8 ⊢ 2 ∈ ℤ | |
10 | 9 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 2 ∈ ℤ) |
11 | 2ne0 12368 | . . . . . . . 8 ⊢ 2 ≠ 0 | |
12 | 11 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 2 ≠ 0) |
13 | 10, 12, 5 | 3jca 1127 | . . . . . 6 ⊢ (𝜑 → (2 ∈ ℤ ∧ 2 ≠ 0 ∧ 𝑀 ∈ ℤ)) |
14 | dvdsval2 16290 | . . . . . 6 ⊢ ((2 ∈ ℤ ∧ 2 ≠ 0 ∧ 𝑀 ∈ ℤ) → (2 ∥ 𝑀 ↔ (𝑀 / 2) ∈ ℤ)) | |
15 | 13, 14 | syl 17 | . . . . 5 ⊢ (𝜑 → (2 ∥ 𝑀 ↔ (𝑀 / 2) ∈ ℤ)) |
16 | 8, 15 | mpbid 232 | . . . 4 ⊢ (𝜑 → (𝑀 / 2) ∈ ℤ) |
17 | 1, 16 | dnizeq0 36458 | . . 3 ⊢ (𝜑 → (𝑇‘(𝑀 / 2)) = 0) |
18 | 17 | oveq2d 7447 | . 2 ⊢ (𝜑 → ((𝐶↑𝐽) · (𝑇‘(𝑀 / 2))) = ((𝐶↑𝐽) · 0)) |
19 | knoppndvlem8.c | . . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ (-1(,)1)) | |
20 | 19 | knoppndvlem3 36497 | . . . . . 6 ⊢ (𝜑 → (𝐶 ∈ ℝ ∧ (abs‘𝐶) < 1)) |
21 | 20 | simpld 494 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℝ) |
22 | 21 | recnd 11287 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ℂ) |
23 | 22, 4 | expcld 14183 | . . 3 ⊢ (𝜑 → (𝐶↑𝐽) ∈ ℂ) |
24 | 23 | mul01d 11458 | . 2 ⊢ (𝜑 → ((𝐶↑𝐽) · 0) = 0) |
25 | 7, 18, 24 | 3eqtrd 2779 | 1 ⊢ (𝜑 → ((𝐹‘𝐴)‘𝐽) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 class class class wbr 5148 ↦ cmpt 5231 ‘cfv 6563 (class class class)co 7431 ℝcr 11152 0cc0 11153 1c1 11154 + caddc 11156 · cmul 11158 < clt 11293 − cmin 11490 -cneg 11491 / cdiv 11918 ℕcn 12264 2c2 12319 ℕ0cn0 12524 ℤcz 12611 (,)cioo 13384 ⌊cfl 13827 ↑cexp 14099 abscabs 15270 ∥ cdvds 16287 |
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-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 |
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-op 4638 df-uni 4913 df-iun 4998 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-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-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-sup 9480 df-inf 9481 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-n0 12525 df-z 12612 df-uz 12877 df-rp 13033 df-ioo 13388 df-ico 13390 df-fl 13829 df-seq 14040 df-exp 14100 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-dvds 16288 |
This theorem is referenced by: knoppndvlem10 36504 |
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