jm3.1lem1 - Mathbox for Stefan O'Rear

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Theorem jm3.1lem1 40839

Description: Lemma for jm3.1 40842. (Contributed by Stefan O'Rear, 16-Oct-2014.)

**Hypotheses**
Ref	Expression
jm3.1.a	⊢ (𝜑 → 𝐴 ∈ (ℤ_≥‘2))
jm3.1.b	⊢ (𝜑 → 𝐾 ∈ (ℤ_≥‘2))
jm3.1.c	⊢ (𝜑 → 𝑁 ∈ ℕ)
jm3.1.d	⊢ (𝜑 → (𝐾 Y_rm (𝑁 + 1)) ≤ 𝐴)

**Assertion**
Ref	Expression
jm3.1lem1	⊢ (𝜑 → (𝐾↑𝑁) < 𝐴)

**Proof of Theorem jm3.1lem1**
Step	Hyp	Ref	Expression
1		jm3.1.b	. . . 4 ⊢ (𝜑 → 𝐾 ∈ (ℤ_≥‘2))
2		eluzelre 12593	. . . 4 ⊢ (𝐾 ∈ (ℤ_≥‘2) → 𝐾 ∈ ℝ)
3	1, 2	syl 17	. . 3 ⊢ (𝜑 → 𝐾 ∈ ℝ)
4		jm3.1.c	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ)
5	4	nnnn0d 12293	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
6	3, 5	reexpcld 13881	. 2 ⊢ (𝜑 → (𝐾↑𝑁) ∈ ℝ)
7		2z 12352	. . . . . . 7 ⊢ 2 ∈ ℤ
8		uzid 12597	. . . . . . 7 ⊢ (2 ∈ ℤ → 2 ∈ (ℤ_≥‘2))
9	7, 8	ax-mp 5	. . . . . 6 ⊢ 2 ∈ (ℤ_≥‘2)
10		uz2mulcl 12666	. . . . . 6 ⊢ ((2 ∈ (ℤ_≥‘2) ∧ 𝐾 ∈ (ℤ_≥‘2)) → (2 · 𝐾) ∈ (ℤ_≥‘2))
11	9, 1, 10	sylancr 587	. . . . 5 ⊢ (𝜑 → (2 · 𝐾) ∈ (ℤ_≥‘2))
12		uz2m1nn 12663	. . . . 5 ⊢ ((2 · 𝐾) ∈ (ℤ_≥‘2) → ((2 · 𝐾) − 1) ∈ ℕ)
13	11, 12	syl 17	. . . 4 ⊢ (𝜑 → ((2 · 𝐾) − 1) ∈ ℕ)
14	13	nnred 11988	. . 3 ⊢ (𝜑 → ((2 · 𝐾) − 1) ∈ ℝ)
15	14, 5	reexpcld 13881	. 2 ⊢ (𝜑 → (((2 · 𝐾) − 1)↑𝑁) ∈ ℝ)
16		jm3.1.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ (ℤ_≥‘2))
17		eluzelre 12593	. . 3 ⊢ (𝐴 ∈ (ℤ_≥‘2) → 𝐴 ∈ ℝ)
18	16, 17	syl 17	. 2 ⊢ (𝜑 → 𝐴 ∈ ℝ)
19		uz2m1nn 12663	. . . . . . 7 ⊢ (𝐾 ∈ (ℤ_≥‘2) → (𝐾 − 1) ∈ ℕ)
20	1, 19	syl 17	. . . . . 6 ⊢ (𝜑 → (𝐾 − 1) ∈ ℕ)
21	20	nngt0d 12022	. . . . 5 ⊢ (𝜑 → 0 < (𝐾 − 1))
22		2cn 12048	. . . . . . . 8 ⊢ 2 ∈ ℂ
23	3	recnd 11003	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ ℂ)
24		mulcl 10955	. . . . . . . 8 ⊢ ((2 ∈ ℂ ∧ 𝐾 ∈ ℂ) → (2 · 𝐾) ∈ ℂ)
25	22, 23, 24	sylancr 587	. . . . . . 7 ⊢ (𝜑 → (2 · 𝐾) ∈ ℂ)
26		1cnd 10970	. . . . . . 7 ⊢ (𝜑 → 1 ∈ ℂ)
27	25, 26, 23	sub32d 11364	. . . . . 6 ⊢ (𝜑 → (((2 · 𝐾) − 1) − 𝐾) = (((2 · 𝐾) − 𝐾) − 1))
28	23	2timesd 12216	. . . . . . . 8 ⊢ (𝜑 → (2 · 𝐾) = (𝐾 + 𝐾))
29	23, 23, 28	mvrladdd 11388	. . . . . . 7 ⊢ (𝜑 → ((2 · 𝐾) − 𝐾) = 𝐾)
30	29	oveq1d 7290	. . . . . 6 ⊢ (𝜑 → (((2 · 𝐾) − 𝐾) − 1) = (𝐾 − 1))
31	27, 30	eqtrd 2778	. . . . 5 ⊢ (𝜑 → (((2 · 𝐾) − 1) − 𝐾) = (𝐾 − 1))
32	21, 31	breqtrrd 5102	. . . 4 ⊢ (𝜑 → 0 < (((2 · 𝐾) − 1) − 𝐾))
33	3, 14	posdifd 11562	. . . 4 ⊢ (𝜑 → (𝐾 < ((2 · 𝐾) − 1) ↔ 0 < (((2 · 𝐾) − 1) − 𝐾)))
34	32, 33	mpbird 256	. . 3 ⊢ (𝜑 → 𝐾 < ((2 · 𝐾) − 1))
35		eluz2nn 12624	. . . . . 6 ⊢ (𝐾 ∈ (ℤ_≥‘2) → 𝐾 ∈ ℕ)
36	1, 35	syl 17	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ ℕ)
37	36	nnrpd 12770	. . . 4 ⊢ (𝜑 → 𝐾 ∈ ℝ⁺)
38	13	nnrpd 12770	. . . 4 ⊢ (𝜑 → ((2 · 𝐾) − 1) ∈ ℝ⁺)
39		rpexpmord 13886	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ ℝ⁺ ∧ ((2 · 𝐾) − 1) ∈ ℝ⁺) → (𝐾 < ((2 · 𝐾) − 1) ↔ (𝐾↑𝑁) < (((2 · 𝐾) − 1)↑𝑁)))
40	4, 37, 38, 39	syl3anc 1370	. . 3 ⊢ (𝜑 → (𝐾 < ((2 · 𝐾) − 1) ↔ (𝐾↑𝑁) < (((2 · 𝐾) − 1)↑𝑁)))
41	34, 40	mpbid 231	. 2 ⊢ (𝜑 → (𝐾↑𝑁) < (((2 · 𝐾) − 1)↑𝑁))
42	4	nnzd 12425	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℤ)
43	42	peano2zd 12429	. . . . 5 ⊢ (𝜑 → (𝑁 + 1) ∈ ℤ)
44		frmy 40736	. . . . . 6 ⊢ Y_rm :((ℤ_≥‘2) × ℤ)⟶ℤ
45	44	fovcl 7402	. . . . 5 ⊢ ((𝐾 ∈ (ℤ_≥‘2) ∧ (𝑁 + 1) ∈ ℤ) → (𝐾 Y_rm (𝑁 + 1)) ∈ ℤ)
46	1, 43, 45	syl2anc 584	. . . 4 ⊢ (𝜑 → (𝐾 Y_rm (𝑁 + 1)) ∈ ℤ)
47	46	zred 12426	. . 3 ⊢ (𝜑 → (𝐾 Y_rm (𝑁 + 1)) ∈ ℝ)
48		jm2.17a 40782	. . . 4 ⊢ ((𝐾 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℕ₀) → (((2 · 𝐾) − 1)↑𝑁) ≤ (𝐾 Y_rm (𝑁 + 1)))
49	1, 5, 48	syl2anc 584	. . 3 ⊢ (𝜑 → (((2 · 𝐾) − 1)↑𝑁) ≤ (𝐾 Y_rm (𝑁 + 1)))
50		jm3.1.d	. . 3 ⊢ (𝜑 → (𝐾 Y_rm (𝑁 + 1)) ≤ 𝐴)
51	15, 47, 18, 49, 50	letrd 11132	. 2 ⊢ (𝜑 → (((2 · 𝐾) − 1)↑𝑁) ≤ 𝐴)
52	6, 15, 18, 41, 51	ltletrd 11135	1 ⊢ (𝜑 → (𝐾↑𝑁) < 𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∈ wcel 2106 class class class wbr 5074 ‘cfv 6433 (class class class)co 7275 ℂcc 10869 ℝcr 10870 0cc0 10871 1c1 10872 + caddc 10874 · cmul 10876 < clt 11009 ≤ cle 11010 − cmin 11205 ℕcn 11973 2c2 12028 ℕ₀cn0 12233 ℤcz 12319 ℤ_≥cuz 12582 ℝ⁺crp 12730 ↑cexp 13782 Y_rm crmy 40723

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 ax-addf 10950 ax-mulf 10951

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-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 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-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7978 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-2o 8298 df-oadd 8301 df-omul 8302 df-er 8498 df-map 8617 df-pm 8618 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fsupp 9129 df-fi 9170 df-sup 9201 df-inf 9202 df-oi 9269 df-card 9697 df-acn 9700 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-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-xnn0 12306 df-z 12320 df-dec 12438 df-uz 12583 df-q 12689 df-rp 12731 df-xneg 12848 df-xadd 12849 df-xmul 12850 df-ioo 13083 df-ioc 13084 df-ico 13085 df-icc 13086 df-fz 13240 df-fzo 13383 df-fl 13512 df-mod 13590 df-seq 13722 df-exp 13783 df-fac 13988 df-bc 14017 df-hash 14045 df-shft 14778 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-limsup 15180 df-clim 15197 df-rlim 15198 df-sum 15398 df-ef 15777 df-sin 15779 df-cos 15780 df-pi 15782 df-dvds 15964 df-gcd 16202 df-numer 16439 df-denom 16440 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-starv 16977 df-sca 16978 df-vsca 16979 df-ip 16980 df-tset 16981 df-ple 16982 df-ds 16984 df-unif 16985 df-hom 16986 df-cco 16987 df-rest 17133 df-topn 17134 df-0g 17152 df-gsum 17153 df-topgen 17154 df-pt 17155 df-prds 17158 df-xrs 17213 df-qtop 17218 df-imas 17219 df-xps 17221 df-mre 17295 df-mrc 17296 df-acs 17298 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-submnd 18431 df-mulg 18701 df-cntz 18923 df-cmn 19388 df-psmet 20589 df-xmet 20590 df-met 20591 df-bl 20592 df-mopn 20593 df-fbas 20594 df-fg 20595 df-cnfld 20598 df-top 22043 df-topon 22060 df-topsp 22082 df-bases 22096 df-cld 22170 df-ntr 22171 df-cls 22172 df-nei 22249 df-lp 22287 df-perf 22288 df-cn 22378 df-cnp 22379 df-haus 22466 df-tx 22713 df-hmeo 22906 df-fil 22997 df-fm 23089 df-flim 23090 df-flf 23091 df-xms 23473 df-ms 23474 df-tms 23475 df-cncf 24041 df-limc 25030 df-dv 25031 df-log 25712 df-squarenn 40663 df-pell1qr 40664 df-pell14qr 40665 df-pell1234qr 40666 df-pellfund 40667 df-rmx 40724 df-rmy 40725

This theorem is referenced by: jm3.1lem2 40840

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