jm3.1lem1 - Mathbox for Stefan O'Rear

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Theorem jm3.1lem1 43558

Description: Lemma for jm3.1 43561. (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 12847	. . . 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 12539	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
6	3, 5	reexpcld 14173	. 2 ⊢ (𝜑 → (𝐾↑𝑁) ∈ ℝ)
7		2z 12600	. . . . . . 7 ⊢ 2 ∈ ℤ
8		uzid 12851	. . . . . . 7 ⊢ (2 ∈ ℤ → 2 ∈ (ℤ_≥‘2))
9	7, 8	ax-mp 5	. . . . . 6 ⊢ 2 ∈ (ℤ_≥‘2)
10		uz2mulcl 12924	. . . . . 6 ⊢ ((2 ∈ (ℤ_≥‘2) ∧ 𝐾 ∈ (ℤ_≥‘2)) → (2 · 𝐾) ∈ (ℤ_≥‘2))
11	9, 1, 10	sylancr 596	. . . . 5 ⊢ (𝜑 → (2 · 𝐾) ∈ (ℤ_≥‘2))
12		uz2m1nn 12921	. . . . 5 ⊢ ((2 · 𝐾) ∈ (ℤ_≥‘2) → ((2 · 𝐾) − 1) ∈ ℕ)
13	11, 12	syl 17	. . . 4 ⊢ (𝜑 → ((2 · 𝐾) − 1) ∈ ℕ)
14	13	nnred 12222	. . 3 ⊢ (𝜑 → ((2 · 𝐾) − 1) ∈ ℝ)
15	14, 5	reexpcld 14173	. 2 ⊢ (𝜑 → (((2 · 𝐾) − 1)↑𝑁) ∈ ℝ)
16		jm3.1.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ (ℤ_≥‘2))
17		eluzelre 12847	. . 3 ⊢ (𝐴 ∈ (ℤ_≥‘2) → 𝐴 ∈ ℝ)
18	16, 17	syl 17	. 2 ⊢ (𝜑 → 𝐴 ∈ ℝ)
19		uz2m1nn 12921	. . . . . . 7 ⊢ (𝐾 ∈ (ℤ_≥‘2) → (𝐾 − 1) ∈ ℕ)
20	1, 19	syl 17	. . . . . 6 ⊢ (𝜑 → (𝐾 − 1) ∈ ℕ)
21	20	nngt0d 12259	. . . . 5 ⊢ (𝜑 → 0 < (𝐾 − 1))
22		2cn 12290	. . . . . . . 8 ⊢ 2 ∈ ℂ
23	3	recnd 11207	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ ℂ)
24		mulcl 11154	. . . . . . . 8 ⊢ ((2 ∈ ℂ ∧ 𝐾 ∈ ℂ) → (2 · 𝐾) ∈ ℂ)
25	22, 23, 24	sylancr 596	. . . . . . 7 ⊢ (𝜑 → (2 · 𝐾) ∈ ℂ)
26		1cnd 11172	. . . . . . 7 ⊢ (𝜑 → 1 ∈ ℂ)
27	25, 26, 23	sub32d 11571	. . . . . 6 ⊢ (𝜑 → (((2 · 𝐾) − 1) − 𝐾) = (((2 · 𝐾) − 𝐾) − 1))
28	23	2timesd 12461	. . . . . . . 8 ⊢ (𝜑 → (2 · 𝐾) = (𝐾 + 𝐾))
29	23, 23, 28	mvrladdd 11597	. . . . . . 7 ⊢ (𝜑 → ((2 · 𝐾) − 𝐾) = 𝐾)
30	29	oveq1d 7407	. . . . . 6 ⊢ (𝜑 → (((2 · 𝐾) − 𝐾) − 1) = (𝐾 − 1))
31	27, 30	eqtrd 2796	. . . . 5 ⊢ (𝜑 → (((2 · 𝐾) − 1) − 𝐾) = (𝐾 − 1))
32	21, 31	breqtrrd 5127	. . . 4 ⊢ (𝜑 → 0 < (((2 · 𝐾) − 1) − 𝐾))
33	3, 14	posdifd 11771	. . . 4 ⊢ (𝜑 → (𝐾 < ((2 · 𝐾) − 1) ↔ 0 < (((2 · 𝐾) − 1) − 𝐾)))
34	32, 33	mpbird 259	. . 3 ⊢ (𝜑 → 𝐾 < ((2 · 𝐾) − 1))
35		eluz2nn 12886	. . . . . 6 ⊢ (𝐾 ∈ (ℤ_≥‘2) → 𝐾 ∈ ℕ)
36	1, 35	syl 17	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ ℕ)
37	36	nnrpd 13032	. . . 4 ⊢ (𝜑 → 𝐾 ∈ ℝ⁺)
38	13	nnrpd 13032	. . . 4 ⊢ (𝜑 → ((2 · 𝐾) − 1) ∈ ℝ⁺)
39		rpexpmord 14178	. . . 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 1389	. . 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 12591	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℤ)
43	42	peano2zd 12677	. . . . 5 ⊢ (𝜑 → (𝑁 + 1) ∈ ℤ)
44		frmy 43455	. . . . . 6 ⊢ Y_rm :((ℤ_≥‘2) × ℤ)⟶ℤ
45	44	fovcl 7520	. . . . 5 ⊢ ((𝐾 ∈ (ℤ_≥‘2) ∧ (𝑁 + 1) ∈ ℤ) → (𝐾 Y_rm (𝑁 + 1)) ∈ ℤ)
46	1, 43, 45	syl2anc 593	. . . 4 ⊢ (𝜑 → (𝐾 Y_rm (𝑁 + 1)) ∈ ℤ)
47	46	zred 12674	. . 3 ⊢ (𝜑 → (𝐾 Y_rm (𝑁 + 1)) ∈ ℝ)
48		jm2.17a 43501	. . . 4 ⊢ ((𝐾 ∈ (ℤ_≥‘2) ∧ 𝑁 ∈ ℕ₀) → (((2 · 𝐾) − 1)↑𝑁) ≤ (𝐾 Y_rm (𝑁 + 1)))
49	1, 5, 48	syl2anc 593	. . 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 11337	. 2 ⊢ (𝜑 → (((2 · 𝐾) − 1)↑𝑁) ≤ 𝐴)
52	6, 15, 18, 41, 51	ltletrd 11340	1 ⊢ (𝜑 → (𝐾↑𝑁) < 𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∈ wcel 2141 class class class wbr 5099 ‘cfv 6517 (class class class)co 7392 ℂcc 11068 ℝcr 11069 0cc0 11070 1c1 11071 + caddc 11073 · cmul 11075 < clt 11213 ≤ cle 11214 − cmin 11411 ℕcn 12207 2c2 12269 ℕ₀cn0 12478 ℤcz 12565 ℤ_≥cuz 12836 ℝ⁺crp 12990 ↑cexp 14071 Y_rm crmy 43442

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5226 ax-sep 5245 ax-nul 5255 ax-pow 5321 ax-pr 5389 ax-un 7714 ax-inf2 9593 ax-cnex 11126 ax-resscn 11127 ax-1cn 11128 ax-icn 11129 ax-addcl 11130 ax-addrcl 11131 ax-mulcl 11132 ax-mulrcl 11133 ax-mulcom 11134 ax-addass 11135 ax-mulass 11136 ax-distr 11137 ax-i2m1 11138 ax-1ne0 11139 ax-1rid 11140 ax-rnegex 11141 ax-rrecex 11142 ax-cnre 11143 ax-pre-lttri 11144 ax-pre-lttrn 11145 ax-pre-ltadd 11146 ax-pre-mulgt0 11147 ax-pre-sup 11148 ax-addf 11149

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4582 df-pr 4584 df-tp 4586 df-op 4588 df-uni 4865 df-int 4905 df-iun 4950 df-iin 4951 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5540 df-eprel 5545 df-po 5553 df-so 5554 df-fr 5598 df-se 5599 df-we 5600 df-xp 5651 df-rel 5652 df-cnv 5653 df-co 5654 df-dm 5655 df-rn 5656 df-res 5657 df-ima 5658 df-pred 6284 df-ord 6345 df-on 6346 df-lim 6347 df-suc 6348 df-iota 6473 df-fun 6519 df-fn 6520 df-f 6521 df-f1 6522 df-fo 6523 df-f1o 6524 df-fv 6525 df-isom 6526 df-riota 7349 df-ov 7395 df-oprab 7396 df-mpo 7397 df-of 7656 df-om 7843 df-1st 7966 df-2nd 7967 df-supp 8136 df-frecs 8257 df-wrecs 8288 df-recs 8337 df-rdg 8376 df-1o 8432 df-2o 8433 df-oadd 8436 df-omul 8437 df-er 8673 df-map 8805 df-pm 8806 df-ixp 8876 df-en 8924 df-dom 8925 df-sdom 8926 df-fin 8927 df-fsupp 9305 df-fi 9354 df-sup 9385 df-inf 9386 df-oi 9455 df-card 9894 df-acn 9897 df-pnf 11215 df-mnf 11216 df-xr 11217 df-ltxr 11218 df-le 11219 df-sub 11413 df-neg 11414 df-div 11842 df-nn 12208 df-2 12277 df-3 12278 df-4 12279 df-5 12280 df-6 12281 df-7 12282 df-8 12283 df-9 12284 df-n0 12479 df-xnn0 12552 df-z 12566 df-dec 12686 df-uz 12837 df-q 12947 df-rp 12991 df-xneg 13111 df-xadd 13112 df-xmul 13113 df-ioo 13350 df-ioc 13351 df-ico 13352 df-icc 13353 df-fz 13510 df-fzo 13657 df-fl 13799 df-mod 13877 df-seq 14012 df-exp 14072 df-fac 14284 df-bc 14313 df-hash 14341 df-shft 15077 df-cj 15109 df-re 15110 df-im 15111 df-sqrt 15245 df-abs 15246 df-limsup 15481 df-clim 15498 df-rlim 15499 df-sum 15697 df-ef 16080 df-sin 16082 df-cos 16083 df-pi 16085 df-dvds 16270 df-gcd 16512 df-numer 16753 df-denom 16754 df-struct 17166 df-sets 17183 df-slot 17201 df-ndx 17213 df-base 17229 df-ress 17250 df-plusg 17282 df-mulr 17283 df-starv 17284 df-sca 17285 df-vsca 17286 df-ip 17287 df-tset 17288 df-ple 17289 df-ds 17291 df-unif 17292 df-hom 17293 df-cco 17294 df-rest 17434 df-topn 17435 df-0g 17453 df-gsum 17454 df-topgen 17455 df-pt 17456 df-prds 17459 df-xrs 17515 df-qtop 17520 df-imas 17521 df-xps 17523 df-mre 17597 df-mrc 17598 df-acs 17600 df-mgm 18657 df-sgrp 18736 df-mnd 18752 df-submnd 18801 df-mulg 19093 df-cntz 19340 df-cmn 19805 df-psmet 21396 df-xmet 21397 df-met 21398 df-bl 21399 df-mopn 21400 df-fbas 21401 df-fg 21402 df-cnfld 21405 df-top 22934 df-topon 22951 df-topsp 22973 df-bases 22986 df-cld 23059 df-ntr 23060 df-cls 23061 df-nei 23138 df-lp 23176 df-perf 23177 df-cn 23267 df-cnp 23268 df-haus 23355 df-tx 23602 df-hmeo 23795 df-fil 23886 df-fm 23978 df-flim 23979 df-flf 23980 df-xms 24360 df-ms 24361 df-tms 24362 df-cncf 24920 df-limc 25908 df-dv 25909 df-log 26598 df-squarenn 43382 df-pell1qr 43383 df-pell14qr 43384 df-pell1234qr 43385 df-pellfund 43386 df-rmx 43443 df-rmy 43444

This theorem is referenced by: jm3.1lem2 43559

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