jm3.1lem1 - Mathbox for Stefan O'Rear

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Theorem jm3.1lem1 40451

Description: Lemma for jm3.1 40454. (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 12347	. . . 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 12048	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
6	3, 5	reexpcld 13631	. 2 ⊢ (𝜑 → (𝐾↑𝑁) ∈ ℝ)
7		2z 12107	. . . . . . 7 ⊢ 2 ∈ ℤ
8		uzid 12351	. . . . . . 7 ⊢ (2 ∈ ℤ → 2 ∈ (ℤ_≥‘2))
9	7, 8	ax-mp 5	. . . . . 6 ⊢ 2 ∈ (ℤ_≥‘2)
10		uz2mulcl 12420	. . . . . 6 ⊢ ((2 ∈ (ℤ_≥‘2) ∧ 𝐾 ∈ (ℤ_≥‘2)) → (2 · 𝐾) ∈ (ℤ_≥‘2))
11	9, 1, 10	sylancr 590	. . . . 5 ⊢ (𝜑 → (2 · 𝐾) ∈ (ℤ_≥‘2))
12		uz2m1nn 12417	. . . . 5 ⊢ ((2 · 𝐾) ∈ (ℤ_≥‘2) → ((2 · 𝐾) − 1) ∈ ℕ)
13	11, 12	syl 17	. . . 4 ⊢ (𝜑 → ((2 · 𝐾) − 1) ∈ ℕ)
14	13	nnred 11743	. . 3 ⊢ (𝜑 → ((2 · 𝐾) − 1) ∈ ℝ)
15	14, 5	reexpcld 13631	. 2 ⊢ (𝜑 → (((2 · 𝐾) − 1)↑𝑁) ∈ ℝ)
16		jm3.1.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ (ℤ_≥‘2))
17		eluzelre 12347	. . 3 ⊢ (𝐴 ∈ (ℤ_≥‘2) → 𝐴 ∈ ℝ)
18	16, 17	syl 17	. 2 ⊢ (𝜑 → 𝐴 ∈ ℝ)
19		uz2m1nn 12417	. . . . . . 7 ⊢ (𝐾 ∈ (ℤ_≥‘2) → (𝐾 − 1) ∈ ℕ)
20	1, 19	syl 17	. . . . . 6 ⊢ (𝜑 → (𝐾 − 1) ∈ ℕ)
21	20	nngt0d 11777	. . . . 5 ⊢ (𝜑 → 0 < (𝐾 − 1))
22		2cn 11803	. . . . . . . 8 ⊢ 2 ∈ ℂ
23	3	recnd 10759	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ ℂ)
24		mulcl 10711	. . . . . . . 8 ⊢ ((2 ∈ ℂ ∧ 𝐾 ∈ ℂ) → (2 · 𝐾) ∈ ℂ)
25	22, 23, 24	sylancr 590	. . . . . . 7 ⊢ (𝜑 → (2 · 𝐾) ∈ ℂ)
26		1cnd 10726	. . . . . . 7 ⊢ (𝜑 → 1 ∈ ℂ)
27	25, 26, 23	sub32d 11119	. . . . . 6 ⊢ (𝜑 → (((2 · 𝐾) − 1) − 𝐾) = (((2 · 𝐾) − 𝐾) − 1))
28	23	2timesd 11971	. . . . . . . 8 ⊢ (𝜑 → (2 · 𝐾) = (𝐾 + 𝐾))
29	23, 23, 28	mvrladdd 11143	. . . . . . 7 ⊢ (𝜑 → ((2 · 𝐾) − 𝐾) = 𝐾)
30	29	oveq1d 7197	. . . . . 6 ⊢ (𝜑 → (((2 · 𝐾) − 𝐾) − 1) = (𝐾 − 1))
31	27, 30	eqtrd 2774	. . . . 5 ⊢ (𝜑 → (((2 · 𝐾) − 1) − 𝐾) = (𝐾 − 1))
32	21, 31	breqtrrd 5068	. . . 4 ⊢ (𝜑 → 0 < (((2 · 𝐾) − 1) − 𝐾))
33	3, 14	posdifd 11317	. . . 4 ⊢ (𝜑 → (𝐾 < ((2 · 𝐾) − 1) ↔ 0 < (((2 · 𝐾) − 1) − 𝐾)))
34	32, 33	mpbird 260	. . 3 ⊢ (𝜑 → 𝐾 < ((2 · 𝐾) − 1))
35		eluz2nn 12378	. . . . . 6 ⊢ (𝐾 ∈ (ℤ_≥‘2) → 𝐾 ∈ ℕ)
36	1, 35	syl 17	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ ℕ)
37	36	nnrpd 12524	. . . 4 ⊢ (𝜑 → 𝐾 ∈ ℝ⁺)
38	13	nnrpd 12524	. . . 4 ⊢ (𝜑 → ((2 · 𝐾) − 1) ∈ ℝ⁺)
39		rpexpmord 13636	. . . 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 1372	. . 3 ⊢ (𝜑 → (𝐾 < ((2 · 𝐾) − 1) ↔ (𝐾↑𝑁) < (((2 · 𝐾) − 1)↑𝑁)))
41	34, 40	mpbid 235	. 2 ⊢ (𝜑 → (𝐾↑𝑁) < (((2 · 𝐾) − 1)↑𝑁))
42	4	nnzd 12179	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℤ)
43	42	peano2zd 12183	. . . . 5 ⊢ (𝜑 → (𝑁 + 1) ∈ ℤ)
44		frmy 40348	. . . . . 6 ⊢ Y_rm :((ℤ_≥‘2) × ℤ)⟶ℤ
45	44	fovcl 7306	. . . . 5 ⊢ ((𝐾 ∈ (ℤ_≥‘2) ∧ (𝑁 + 1) ∈ ℤ) → (𝐾 Y_rm (𝑁 + 1)) ∈ ℤ)
46	1, 43, 45	syl2anc 587	. . . 4 ⊢ (𝜑 → (𝐾 Y_rm (𝑁 + 1)) ∈ ℤ)
47	46	zred 12180	. . 3 ⊢ (𝜑 → (𝐾 Y_rm (𝑁 + 1)) ∈ ℝ)
48		jm2.17a 40394	. . . 4 ⊢ ((𝐾 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℕ₀) → (((2 · 𝐾) − 1)↑𝑁) ≤ (𝐾 Y_rm (𝑁 + 1)))
49	1, 5, 48	syl2anc 587	. . 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 10887	. 2 ⊢ (𝜑 → (((2 · 𝐾) − 1)↑𝑁) ≤ 𝐴)
52	6, 15, 18, 41, 51	ltletrd 10890	1 ⊢ (𝜑 → (𝐾↑𝑁) < 𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∈ wcel 2114 class class class wbr 5040 ‘cfv 6349 (class class class)co 7182 ℂcc 10625 ℝcr 10626 0cc0 10627 1c1 10628 + caddc 10630 · cmul 10632 < clt 10765 ≤ cle 10766 − cmin 10960 ℕcn 11728 2c2 11783 ℕ₀cn0 11988 ℤcz 12074 ℤ_≥cuz 12336 ℝ⁺crp 12484 ↑cexp 13533 Y_rm crmy 40335

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7491 ax-inf2 9189 ax-cnex 10683 ax-resscn 10684 ax-1cn 10685 ax-icn 10686 ax-addcl 10687 ax-addrcl 10688 ax-mulcl 10689 ax-mulrcl 10690 ax-mulcom 10691 ax-addass 10692 ax-mulass 10693 ax-distr 10694 ax-i2m1 10695 ax-1ne0 10696 ax-1rid 10697 ax-rnegex 10698 ax-rrecex 10699 ax-cnre 10700 ax-pre-lttri 10701 ax-pre-lttrn 10702 ax-pre-ltadd 10703 ax-pre-mulgt0 10704 ax-pre-sup 10705 ax-addf 10706 ax-mulf 10707

This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-tp 4531 df-op 4533 df-uni 4807 df-int 4847 df-iun 4893 df-iin 4894 df-br 5041 df-opab 5103 df-mpt 5121 df-tr 5147 df-id 5439 df-eprel 5444 df-po 5452 df-so 5453 df-fr 5493 df-se 5494 df-we 5495 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-pred 6139 df-ord 6185 df-on 6186 df-lim 6187 df-suc 6188 df-iota 6307 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7139 df-ov 7185 df-oprab 7186 df-mpo 7187 df-of 7437 df-om 7612 df-1st 7726 df-2nd 7727 df-supp 7869 df-wrecs 7988 df-recs 8049 df-rdg 8087 df-1o 8143 df-2o 8144 df-oadd 8147 df-omul 8148 df-er 8332 df-map 8451 df-pm 8452 df-ixp 8520 df-en 8568 df-dom 8569 df-sdom 8570 df-fin 8571 df-fsupp 8919 df-fi 8960 df-sup 8991 df-inf 8992 df-oi 9059 df-card 9453 df-acn 9456 df-pnf 10767 df-mnf 10768 df-xr 10769 df-ltxr 10770 df-le 10771 df-sub 10962 df-neg 10963 df-div 11388 df-nn 11729 df-2 11791 df-3 11792 df-4 11793 df-5 11794 df-6 11795 df-7 11796 df-8 11797 df-9 11798 df-n0 11989 df-xnn0 12061 df-z 12075 df-dec 12192 df-uz 12337 df-q 12443 df-rp 12485 df-xneg 12602 df-xadd 12603 df-xmul 12604 df-ioo 12837 df-ioc 12838 df-ico 12839 df-icc 12840 df-fz 12994 df-fzo 13137 df-fl 13265 df-mod 13341 df-seq 13473 df-exp 13534 df-fac 13738 df-bc 13767 df-hash 13795 df-shft 14528 df-cj 14560 df-re 14561 df-im 14562 df-sqrt 14696 df-abs 14697 df-limsup 14930 df-clim 14947 df-rlim 14948 df-sum 15148 df-ef 15525 df-sin 15527 df-cos 15528 df-pi 15530 df-dvds 15712 df-gcd 15950 df-numer 16187 df-denom 16188 df-struct 16600 df-ndx 16601 df-slot 16602 df-base 16604 df-sets 16605 df-ress 16606 df-plusg 16693 df-mulr 16694 df-starv 16695 df-sca 16696 df-vsca 16697 df-ip 16698 df-tset 16699 df-ple 16700 df-ds 16702 df-unif 16703 df-hom 16704 df-cco 16705 df-rest 16811 df-topn 16812 df-0g 16830 df-gsum 16831 df-topgen 16832 df-pt 16833 df-prds 16836 df-xrs 16890 df-qtop 16895 df-imas 16896 df-xps 16898 df-mre 16972 df-mrc 16973 df-acs 16975 df-mgm 17980 df-sgrp 18029 df-mnd 18040 df-submnd 18085 df-mulg 18355 df-cntz 18577 df-cmn 19038 df-psmet 20221 df-xmet 20222 df-met 20223 df-bl 20224 df-mopn 20225 df-fbas 20226 df-fg 20227 df-cnfld 20230 df-top 21657 df-topon 21674 df-topsp 21696 df-bases 21709 df-cld 21782 df-ntr 21783 df-cls 21784 df-nei 21861 df-lp 21899 df-perf 21900 df-cn 21990 df-cnp 21991 df-haus 22078 df-tx 22325 df-hmeo 22518 df-fil 22609 df-fm 22701 df-flim 22702 df-flf 22703 df-xms 23085 df-ms 23086 df-tms 23087 df-cncf 23642 df-limc 24630 df-dv 24631 df-log 25312 df-squarenn 40275 df-pell1qr 40276 df-pell14qr 40277 df-pell1234qr 40278 df-pellfund 40279 df-rmx 40336 df-rmy 40337

This theorem is referenced by: jm3.1lem2 40452

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