lcmineqlem17 - Mathbox for metakunt

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Theorem lcmineqlem17 40053

Description: Inequality of 2^{2n}. (Contributed by metakunt, 29-Apr-2024.)

**Hypothesis**
Ref	Expression
lcmineqlem17.1	⊢ (𝜑 → 𝑁 ∈ ℕ₀)

**Assertion**
Ref	Expression
lcmineqlem17	⊢ (𝜑 → (2↑(2 · 𝑁)) ≤ (((2 · 𝑁) + 1) · ((2 · 𝑁)C𝑁)))

Proof of Theorem lcmineqlem17 Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		2nn0 12250	. . . . . 6 ⊢ 2 ∈ ℕ₀
2	1	a1i 11	. . . . 5 ⊢ (𝜑 → 2 ∈ ℕ₀)
3		lcmineqlem17.1	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
4	2, 3	nn0mulcld 12298	. . . 4 ⊢ (𝜑 → (2 · 𝑁) ∈ ℕ₀)
5		binom11 15544	. . . 4 ⊢ ((2 · 𝑁) ∈ ℕ₀ → (2↑(2 · 𝑁)) = Σ𝑘 ∈ (0...(2 · 𝑁))((2 · 𝑁)C𝑘))
6	4, 5	syl 17	. . 3 ⊢ (𝜑 → (2↑(2 · 𝑁)) = Σ𝑘 ∈ (0...(2 · 𝑁))((2 · 𝑁)C𝑘))
7		fzfid 13693	. . . 4 ⊢ (𝜑 → (0...(2 · 𝑁)) ∈ Fin)
8	4	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(2 · 𝑁))) → (2 · 𝑁) ∈ ℕ₀)
9		elfzelz 13256	. . . . . . . 8 ⊢ (𝑘 ∈ (0...(2 · 𝑁)) → 𝑘 ∈ ℤ)
10	9	adantl 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(2 · 𝑁))) → 𝑘 ∈ ℤ)
11	8, 10	jca 512	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(2 · 𝑁))) → ((2 · 𝑁) ∈ ℕ₀ ∧ 𝑘 ∈ ℤ))
12		bccl 14036	. . . . . 6 ⊢ (((2 · 𝑁) ∈ ℕ₀ ∧ 𝑘 ∈ ℤ) → ((2 · 𝑁)C𝑘) ∈ ℕ₀)
13	11, 12	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(2 · 𝑁))) → ((2 · 𝑁)C𝑘) ∈ ℕ₀)
14	13	nn0red 12294	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(2 · 𝑁))) → ((2 · 𝑁)C𝑘) ∈ ℝ)
15	3	nn0zd 12424	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℤ)
16		bccl 14036	. . . . . . 7 ⊢ (((2 · 𝑁) ∈ ℕ₀ ∧ 𝑁 ∈ ℤ) → ((2 · 𝑁)C𝑁) ∈ ℕ₀)
17	4, 15, 16	syl2anc 584	. . . . . 6 ⊢ (𝜑 → ((2 · 𝑁)C𝑁) ∈ ℕ₀)
18	17	nn0red 12294	. . . . 5 ⊢ (𝜑 → ((2 · 𝑁)C𝑁) ∈ ℝ)
19	18	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(2 · 𝑁))) → ((2 · 𝑁)C𝑁) ∈ ℝ)
20		bcmax 26426	. . . . 5 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑘 ∈ ℤ) → ((2 · 𝑁)C𝑘) ≤ ((2 · 𝑁)C𝑁))
21	3, 9, 20	syl2an 596	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...(2 · 𝑁))) → ((2 · 𝑁)C𝑘) ≤ ((2 · 𝑁)C𝑁))
22	7, 14, 19, 21	fsumle 15511	. . 3 ⊢ (𝜑 → Σ𝑘 ∈ (0...(2 · 𝑁))((2 · 𝑁)C𝑘) ≤ Σ𝑘 ∈ (0...(2 · 𝑁))((2 · 𝑁)C𝑁))
23	6, 22	eqbrtrd 5096	. 2 ⊢ (𝜑 → (2↑(2 · 𝑁)) ≤ Σ𝑘 ∈ (0...(2 · 𝑁))((2 · 𝑁)C𝑁))
24	17	nn0cnd 12295	. . . 4 ⊢ (𝜑 → ((2 · 𝑁)C𝑁) ∈ ℂ)
25		fsumconst 15502	. . . 4 ⊢ (((0...(2 · 𝑁)) ∈ Fin ∧ ((2 · 𝑁)C𝑁) ∈ ℂ) → Σ𝑘 ∈ (0...(2 · 𝑁))((2 · 𝑁)C𝑁) = ((♯‘(0...(2 · 𝑁))) · ((2 · 𝑁)C𝑁)))
26	7, 24, 25	syl2anc 584	. . 3 ⊢ (𝜑 → Σ𝑘 ∈ (0...(2 · 𝑁))((2 · 𝑁)C𝑁) = ((♯‘(0...(2 · 𝑁))) · ((2 · 𝑁)C𝑁)))
27		hashfz0 14147	. . . . 5 ⊢ ((2 · 𝑁) ∈ ℕ₀ → (♯‘(0...(2 · 𝑁))) = ((2 · 𝑁) + 1))
28	4, 27	syl 17	. . . 4 ⊢ (𝜑 → (♯‘(0...(2 · 𝑁))) = ((2 · 𝑁) + 1))
29	28	oveq1d 7290	. . 3 ⊢ (𝜑 → ((♯‘(0...(2 · 𝑁))) · ((2 · 𝑁)C𝑁)) = (((2 · 𝑁) + 1) · ((2 · 𝑁)C𝑁)))
30	26, 29	eqtrd 2778	. 2 ⊢ (𝜑 → Σ𝑘 ∈ (0...(2 · 𝑁))((2 · 𝑁)C𝑁) = (((2 · 𝑁) + 1) · ((2 · 𝑁)C𝑁)))
31	23, 30	breqtrd 5100	1 ⊢ (𝜑 → (2↑(2 · 𝑁)) ≤ (((2 · 𝑁) + 1) · ((2 · 𝑁)C𝑁)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 class class class wbr 5074 ‘cfv 6433 (class class class)co 7275 Fincfn 8733 ℂcc 10869 ℝcr 10870 0cc0 10871 1c1 10872 + caddc 10874 · cmul 10876 ≤ cle 11010 2c2 12028 ℕ₀cn0 12233 ℤcz 12319 ...cfz 13239 ↑cexp 13782 Ccbc 14016 ♯chash 14044 Σcsu 15397

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-sup 9201 df-oi 9269 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-n0 12234 df-z 12320 df-uz 12583 df-rp 12731 df-ico 13085 df-fz 13240 df-fzo 13383 df-seq 13722 df-exp 13783 df-fac 13988 df-bc 14017 df-hash 14045 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-clim 15197 df-sum 15398

This theorem is referenced by: lcmineqlem20 40056

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