jm3.1lem1 - Mathbox for Stefan O'Rear

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Theorem jm3.1lem1 39621

Description: Lemma for jm3.1 39624. (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 12257	. . . 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 11958	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
6	3, 5	reexpcld 13530	. 2 ⊢ (𝜑 → (𝐾↑𝑁) ∈ ℝ)
7		2z 12017	. . . . . . 7 ⊢ 2 ∈ ℤ
8		uzid 12261	. . . . . . 7 ⊢ (2 ∈ ℤ → 2 ∈ (ℤ_≥‘2))
9	7, 8	ax-mp 5	. . . . . 6 ⊢ 2 ∈ (ℤ_≥‘2)
10		uz2mulcl 12329	. . . . . 6 ⊢ ((2 ∈ (ℤ_≥‘2) ∧ 𝐾 ∈ (ℤ_≥‘2)) → (2 · 𝐾) ∈ (ℤ_≥‘2))
11	9, 1, 10	sylancr 589	. . . . 5 ⊢ (𝜑 → (2 · 𝐾) ∈ (ℤ_≥‘2))
12		uz2m1nn 12326	. . . . 5 ⊢ ((2 · 𝐾) ∈ (ℤ_≥‘2) → ((2 · 𝐾) − 1) ∈ ℕ)
13	11, 12	syl 17	. . . 4 ⊢ (𝜑 → ((2 · 𝐾) − 1) ∈ ℕ)
14	13	nnred 11655	. . 3 ⊢ (𝜑 → ((2 · 𝐾) − 1) ∈ ℝ)
15	14, 5	reexpcld 13530	. 2 ⊢ (𝜑 → (((2 · 𝐾) − 1)↑𝑁) ∈ ℝ)
16		jm3.1.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ (ℤ_≥‘2))
17		eluzelre 12257	. . 3 ⊢ (𝐴 ∈ (ℤ_≥‘2) → 𝐴 ∈ ℝ)
18	16, 17	syl 17	. 2 ⊢ (𝜑 → 𝐴 ∈ ℝ)
19		uz2m1nn 12326	. . . . . . 7 ⊢ (𝐾 ∈ (ℤ_≥‘2) → (𝐾 − 1) ∈ ℕ)
20	1, 19	syl 17	. . . . . 6 ⊢ (𝜑 → (𝐾 − 1) ∈ ℕ)
21	20	nngt0d 11689	. . . . 5 ⊢ (𝜑 → 0 < (𝐾 − 1))
22		2cn 11715	. . . . . . . 8 ⊢ 2 ∈ ℂ
23	3	recnd 10671	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ ℂ)
24		mulcl 10623	. . . . . . . 8 ⊢ ((2 ∈ ℂ ∧ 𝐾 ∈ ℂ) → (2 · 𝐾) ∈ ℂ)
25	22, 23, 24	sylancr 589	. . . . . . 7 ⊢ (𝜑 → (2 · 𝐾) ∈ ℂ)
26		1cnd 10638	. . . . . . 7 ⊢ (𝜑 → 1 ∈ ℂ)
27	25, 26, 23	sub32d 11031	. . . . . 6 ⊢ (𝜑 → (((2 · 𝐾) − 1) − 𝐾) = (((2 · 𝐾) − 𝐾) − 1))
28	23	2timesd 11883	. . . . . . . 8 ⊢ (𝜑 → (2 · 𝐾) = (𝐾 + 𝐾))
29	23, 23, 28	mvrladdd 11055	. . . . . . 7 ⊢ (𝜑 → ((2 · 𝐾) − 𝐾) = 𝐾)
30	29	oveq1d 7173	. . . . . 6 ⊢ (𝜑 → (((2 · 𝐾) − 𝐾) − 1) = (𝐾 − 1))
31	27, 30	eqtrd 2858	. . . . 5 ⊢ (𝜑 → (((2 · 𝐾) − 1) − 𝐾) = (𝐾 − 1))
32	21, 31	breqtrrd 5096	. . . 4 ⊢ (𝜑 → 0 < (((2 · 𝐾) − 1) − 𝐾))
33	3, 14	posdifd 11229	. . . 4 ⊢ (𝜑 → (𝐾 < ((2 · 𝐾) − 1) ↔ 0 < (((2 · 𝐾) − 1) − 𝐾)))
34	32, 33	mpbird 259	. . 3 ⊢ (𝜑 → 𝐾 < ((2 · 𝐾) − 1))
35		eluz2nn 12287	. . . . . 6 ⊢ (𝐾 ∈ (ℤ_≥‘2) → 𝐾 ∈ ℕ)
36	1, 35	syl 17	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ ℕ)
37	36	nnrpd 12432	. . . 4 ⊢ (𝜑 → 𝐾 ∈ ℝ⁺)
38	13	nnrpd 12432	. . . 4 ⊢ (𝜑 → ((2 · 𝐾) − 1) ∈ ℝ⁺)
39		rpexpmord 13535	. . . 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 1367	. . 3 ⊢ (𝜑 → (𝐾 < ((2 · 𝐾) − 1) ↔ (𝐾↑𝑁) < (((2 · 𝐾) − 1)↑𝑁)))
41	34, 40	mpbid 234	. 2 ⊢ (𝜑 → (𝐾↑𝑁) < (((2 · 𝐾) − 1)↑𝑁))
42	4	nnzd 12089	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℤ)
43	42	peano2zd 12093	. . . . 5 ⊢ (𝜑 → (𝑁 + 1) ∈ ℤ)
44		frmy 39518	. . . . . 6 ⊢ Y_rm :((ℤ_≥‘2) × ℤ)⟶ℤ
45	44	fovcl 7281	. . . . 5 ⊢ ((𝐾 ∈ (ℤ_≥‘2) ∧ (𝑁 + 1) ∈ ℤ) → (𝐾 Y_rm (𝑁 + 1)) ∈ ℤ)
46	1, 43, 45	syl2anc 586	. . . 4 ⊢ (𝜑 → (𝐾 Y_rm (𝑁 + 1)) ∈ ℤ)
47	46	zred 12090	. . 3 ⊢ (𝜑 → (𝐾 Y_rm (𝑁 + 1)) ∈ ℝ)
48		jm2.17a 39564	. . . 4 ⊢ ((𝐾 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℕ₀) → (((2 · 𝐾) − 1)↑𝑁) ≤ (𝐾 Y_rm (𝑁 + 1)))
49	1, 5, 48	syl2anc 586	. . 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 10799	. 2 ⊢ (𝜑 → (((2 · 𝐾) − 1)↑𝑁) ≤ 𝐴)
52	6, 15, 18, 41, 51	ltletrd 10802	1 ⊢ (𝜑 → (𝐾↑𝑁) < 𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∈ wcel 2114 class class class wbr 5068 ‘cfv 6357 (class class class)co 7158 ℂcc 10537 ℝcr 10538 0cc0 10539 1c1 10540 + caddc 10542 · cmul 10544 < clt 10677 ≤ cle 10678 − cmin 10872 ℕcn 11640 2c2 11695 ℕ₀cn0 11900 ℤcz 11984 ℤ_≥cuz 12246 ℝ⁺crp 12392 ↑cexp 13432 Y_rm crmy 39505

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 ax-addf 10618 ax-mulf 10619

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-oadd 8108 df-omul 8109 df-er 8291 df-map 8410 df-pm 8411 df-ixp 8464 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fsupp 8836 df-fi 8877 df-sup 8908 df-inf 8909 df-oi 8976 df-card 9370 df-acn 9373 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-xnn0 11971 df-z 11985 df-dec 12102 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-ioo 12745 df-ioc 12746 df-ico 12747 df-icc 12748 df-fz 12896 df-fzo 13037 df-fl 13165 df-mod 13241 df-seq 13373 df-exp 13433 df-fac 13637 df-bc 13666 df-hash 13694 df-shft 14428 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-limsup 14830 df-clim 14847 df-rlim 14848 df-sum 15045 df-ef 15423 df-sin 15425 df-cos 15426 df-pi 15428 df-dvds 15610 df-gcd 15846 df-numer 16077 df-denom 16078 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-starv 16582 df-sca 16583 df-vsca 16584 df-ip 16585 df-tset 16586 df-ple 16587 df-ds 16589 df-unif 16590 df-hom 16591 df-cco 16592 df-rest 16698 df-topn 16699 df-0g 16717 df-gsum 16718 df-topgen 16719 df-pt 16720 df-prds 16723 df-xrs 16777 df-qtop 16782 df-imas 16783 df-xps 16785 df-mre 16859 df-mrc 16860 df-acs 16862 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-submnd 17959 df-mulg 18227 df-cntz 18449 df-cmn 18910 df-psmet 20539 df-xmet 20540 df-met 20541 df-bl 20542 df-mopn 20543 df-fbas 20544 df-fg 20545 df-cnfld 20548 df-top 21504 df-topon 21521 df-topsp 21543 df-bases 21556 df-cld 21629 df-ntr 21630 df-cls 21631 df-nei 21708 df-lp 21746 df-perf 21747 df-cn 21837 df-cnp 21838 df-haus 21925 df-tx 22172 df-hmeo 22365 df-fil 22456 df-fm 22548 df-flim 22549 df-flf 22550 df-xms 22932 df-ms 22933 df-tms 22934 df-cncf 23488 df-limc 24466 df-dv 24467 df-log 25142 df-squarenn 39445 df-pell1qr 39446 df-pell14qr 39447 df-pell1234qr 39448 df-pellfund 39449 df-rmx 39506 df-rmy 39507

This theorem is referenced by: jm3.1lem2 39622

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