jm3.1lem1 - Mathbox for Stefan O'Rear

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Theorem jm3.1lem1 43041

Description: Lemma for jm3.1 43044. (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 12863	. . . 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 12562	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
6	3, 5	reexpcld 14181	. 2 ⊢ (𝜑 → (𝐾↑𝑁) ∈ ℝ)
7		2z 12624	. . . . . . 7 ⊢ 2 ∈ ℤ
8		uzid 12867	. . . . . . 7 ⊢ (2 ∈ ℤ → 2 ∈ (ℤ_≥‘2))
9	7, 8	ax-mp 5	. . . . . 6 ⊢ 2 ∈ (ℤ_≥‘2)
10		uz2mulcl 12942	. . . . . 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 12939	. . . . 5 ⊢ ((2 · 𝐾) ∈ (ℤ_≥‘2) → ((2 · 𝐾) − 1) ∈ ℕ)
13	11, 12	syl 17	. . . 4 ⊢ (𝜑 → ((2 · 𝐾) − 1) ∈ ℕ)
14	13	nnred 12255	. . 3 ⊢ (𝜑 → ((2 · 𝐾) − 1) ∈ ℝ)
15	14, 5	reexpcld 14181	. 2 ⊢ (𝜑 → (((2 · 𝐾) − 1)↑𝑁) ∈ ℝ)
16		jm3.1.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ (ℤ_≥‘2))
17		eluzelre 12863	. . 3 ⊢ (𝐴 ∈ (ℤ_≥‘2) → 𝐴 ∈ ℝ)
18	16, 17	syl 17	. 2 ⊢ (𝜑 → 𝐴 ∈ ℝ)
19		uz2m1nn 12939	. . . . . . 7 ⊢ (𝐾 ∈ (ℤ_≥‘2) → (𝐾 − 1) ∈ ℕ)
20	1, 19	syl 17	. . . . . 6 ⊢ (𝜑 → (𝐾 − 1) ∈ ℕ)
21	20	nngt0d 12289	. . . . 5 ⊢ (𝜑 → 0 < (𝐾 − 1))
22		2cn 12315	. . . . . . . 8 ⊢ 2 ∈ ℂ
23	3	recnd 11263	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ ℂ)
24		mulcl 11213	. . . . . . . 8 ⊢ ((2 ∈ ℂ ∧ 𝐾 ∈ ℂ) → (2 · 𝐾) ∈ ℂ)
25	22, 23, 24	sylancr 587	. . . . . . 7 ⊢ (𝜑 → (2 · 𝐾) ∈ ℂ)
26		1cnd 11230	. . . . . . 7 ⊢ (𝜑 → 1 ∈ ℂ)
27	25, 26, 23	sub32d 11626	. . . . . 6 ⊢ (𝜑 → (((2 · 𝐾) − 1) − 𝐾) = (((2 · 𝐾) − 𝐾) − 1))
28	23	2timesd 12484	. . . . . . . 8 ⊢ (𝜑 → (2 · 𝐾) = (𝐾 + 𝐾))
29	23, 23, 28	mvrladdd 11650	. . . . . . 7 ⊢ (𝜑 → ((2 · 𝐾) − 𝐾) = 𝐾)
30	29	oveq1d 7420	. . . . . 6 ⊢ (𝜑 → (((2 · 𝐾) − 𝐾) − 1) = (𝐾 − 1))
31	27, 30	eqtrd 2770	. . . . 5 ⊢ (𝜑 → (((2 · 𝐾) − 1) − 𝐾) = (𝐾 − 1))
32	21, 31	breqtrrd 5147	. . . 4 ⊢ (𝜑 → 0 < (((2 · 𝐾) − 1) − 𝐾))
33	3, 14	posdifd 11824	. . . 4 ⊢ (𝜑 → (𝐾 < ((2 · 𝐾) − 1) ↔ 0 < (((2 · 𝐾) − 1) − 𝐾)))
34	32, 33	mpbird 257	. . 3 ⊢ (𝜑 → 𝐾 < ((2 · 𝐾) − 1))
35		eluz2nn 12898	. . . . . 6 ⊢ (𝐾 ∈ (ℤ_≥‘2) → 𝐾 ∈ ℕ)
36	1, 35	syl 17	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ ℕ)
37	36	nnrpd 13049	. . . 4 ⊢ (𝜑 → 𝐾 ∈ ℝ⁺)
38	13	nnrpd 13049	. . . 4 ⊢ (𝜑 → ((2 · 𝐾) − 1) ∈ ℝ⁺)
39		rpexpmord 14186	. . . 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 1373	. . 3 ⊢ (𝜑 → (𝐾 < ((2 · 𝐾) − 1) ↔ (𝐾↑𝑁) < (((2 · 𝐾) − 1)↑𝑁)))
41	34, 40	mpbid 232	. 2 ⊢ (𝜑 → (𝐾↑𝑁) < (((2 · 𝐾) − 1)↑𝑁))
42	4	nnzd 12615	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℤ)
43	42	peano2zd 12700	. . . . 5 ⊢ (𝜑 → (𝑁 + 1) ∈ ℤ)
44		frmy 42938	. . . . . 6 ⊢ Y_rm :((ℤ_≥‘2) × ℤ)⟶ℤ
45	44	fovcl 7535	. . . . 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 12697	. . 3 ⊢ (𝜑 → (𝐾 Y_rm (𝑁 + 1)) ∈ ℝ)
48		jm2.17a 42984	. . . 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 11392	. 2 ⊢ (𝜑 → (((2 · 𝐾) − 1)↑𝑁) ≤ 𝐴)
52	6, 15, 18, 41, 51	ltletrd 11395	1 ⊢ (𝜑 → (𝐾↑𝑁) < 𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∈ wcel 2108 class class class wbr 5119 ‘cfv 6531 (class class class)co 7405 ℂcc 11127 ℝcr 11128 0cc0 11129 1c1 11130 + caddc 11132 · cmul 11134 < clt 11269 ≤ cle 11270 − cmin 11466 ℕcn 12240 2c2 12295 ℕ₀cn0 12501 ℤcz 12588 ℤ_≥cuz 12852 ℝ⁺crp 13008 ↑cexp 14079 Y_rm crmy 42924

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-inf2 9655 ax-cnex 11185 ax-resscn 11186 ax-1cn 11187 ax-icn 11188 ax-addcl 11189 ax-addrcl 11190 ax-mulcl 11191 ax-mulrcl 11192 ax-mulcom 11193 ax-addass 11194 ax-mulass 11195 ax-distr 11196 ax-i2m1 11197 ax-1ne0 11198 ax-1rid 11199 ax-rnegex 11200 ax-rrecex 11201 ax-cnre 11202 ax-pre-lttri 11203 ax-pre-lttrn 11204 ax-pre-ltadd 11205 ax-pre-mulgt0 11206 ax-pre-sup 11207 ax-addf 11208

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-tp 4606 df-op 4608 df-uni 4884 df-int 4923 df-iun 4969 df-iin 4970 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-se 5607 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-isom 6540 df-riota 7362 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7671 df-om 7862 df-1st 7988 df-2nd 7989 df-supp 8160 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-oadd 8484 df-omul 8485 df-er 8719 df-map 8842 df-pm 8843 df-ixp 8912 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-fsupp 9374 df-fi 9423 df-sup 9454 df-inf 9455 df-oi 9524 df-card 9953 df-acn 9956 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275 df-sub 11468 df-neg 11469 df-div 11895 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12502 df-xnn0 12575 df-z 12589 df-dec 12709 df-uz 12853 df-q 12965 df-rp 13009 df-xneg 13128 df-xadd 13129 df-xmul 13130 df-ioo 13366 df-ioc 13367 df-ico 13368 df-icc 13369 df-fz 13525 df-fzo 13672 df-fl 13809 df-mod 13887 df-seq 14020 df-exp 14080 df-fac 14292 df-bc 14321 df-hash 14349 df-shft 15086 df-cj 15118 df-re 15119 df-im 15120 df-sqrt 15254 df-abs 15255 df-limsup 15487 df-clim 15504 df-rlim 15505 df-sum 15703 df-ef 16083 df-sin 16085 df-cos 16086 df-pi 16088 df-dvds 16273 df-gcd 16514 df-numer 16754 df-denom 16755 df-struct 17166 df-sets 17183 df-slot 17201 df-ndx 17213 df-base 17229 df-ress 17252 df-plusg 17284 df-mulr 17285 df-starv 17286 df-sca 17287 df-vsca 17288 df-ip 17289 df-tset 17290 df-ple 17291 df-ds 17293 df-unif 17294 df-hom 17295 df-cco 17296 df-rest 17436 df-topn 17437 df-0g 17455 df-gsum 17456 df-topgen 17457 df-pt 17458 df-prds 17461 df-xrs 17516 df-qtop 17521 df-imas 17522 df-xps 17524 df-mre 17598 df-mrc 17599 df-acs 17601 df-mgm 18618 df-sgrp 18697 df-mnd 18713 df-submnd 18762 df-mulg 19051 df-cntz 19300 df-cmn 19763 df-psmet 21307 df-xmet 21308 df-met 21309 df-bl 21310 df-mopn 21311 df-fbas 21312 df-fg 21313 df-cnfld 21316 df-top 22832 df-topon 22849 df-topsp 22871 df-bases 22884 df-cld 22957 df-ntr 22958 df-cls 22959 df-nei 23036 df-lp 23074 df-perf 23075 df-cn 23165 df-cnp 23166 df-haus 23253 df-tx 23500 df-hmeo 23693 df-fil 23784 df-fm 23876 df-flim 23877 df-flf 23878 df-xms 24259 df-ms 24260 df-tms 24261 df-cncf 24822 df-limc 25819 df-dv 25820 df-log 26517 df-squarenn 42864 df-pell1qr 42865 df-pell14qr 42866 df-pell1234qr 42867 df-pellfund 42868 df-rmx 42925 df-rmy 42926

This theorem is referenced by: jm3.1lem2 43042

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