2lgslem1a1 - Metamath Proof Explorer

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Theorem 2lgslem1a1 25114

Description: Lemma 1 for 2lgslem1a 25116. (Contributed by AV, 16-Jun-2021.)

**Assertion**
Ref	Expression
2lgslem1a1	$/\$

Distinct variable group:

**Proof of Theorem 2lgslem1a1**
Step	Hyp	Ref	Expression
1		nnrp 11842	. . . . . 6
2	1	adantr 481	. . . . 5 $/\$
3		elfzelz 12342	. . . . . 6
4		zre 11381	. . . . . . 7
5		2re 11090	. . . . . . . 8
6	5	a1i 11	. . . . . . 7
7	4, 6	remulcld 10070	. . . . . 6
8	3, 7	syl 17	. . . . 5
9	2, 8	anim12ci 591	. . . 4 $/\$ $/\$ $/\$
10		elfznn 12370	. . . . . 6
11		nnre 11027	. . . . . . 7
12		nnnn0 11299	. . . . . . . 8
13	12	nn0ge0d 11354	. . . . . . 7
14		0le2 11111	. . . . . . . . 9
15	5, 14	pm3.2i 471	. . . . . . . 8 $/\$
16	15	a1i 11	. . . . . . 7 $/\$
17		mulge0 10546	. . . . . . 7 $/\$ $/\$ $/\$
18	11, 13, 16, 17	syl21anc 1325	. . . . . 6
19	10, 18	syl 17	. . . . 5
20	19	adantl 482	. . . 4 $/\$ $/\$
21		elfz2 12333	. . . . . 6 $/\$ $/\$ $/\$ $/\$
22	4	3ad2ant3 1084	. . . . . . . . . 10 $/\$ $/\$
23		zre 11381	. . . . . . . . . . 11
24	23	3ad2ant2 1083	. . . . . . . . . 10 $/\$ $/\$
25		2pos 11112	. . . . . . . . . . . 12
26	5, 25	pm3.2i 471	. . . . . . . . . . 11 $/\$
27	26	a1i 11	. . . . . . . . . 10 $/\$ $/\$ $/\$
28		lemul1 10875	. . . . . . . . . 10 $/\$ $/\$ $/\$
29	22, 24, 27, 28	syl3anc 1326	. . . . . . . . 9 $/\$ $/\$
30		nncn 11028	. . . . . . . . . . . . . . . . 17
31		peano2cnm 10347	. . . . . . . . . . . . . . . . 17
32	30, 31	syl 17	. . . . . . . . . . . . . . . 16
33		2cnd 11093	. . . . . . . . . . . . . . . 16
34		2ne0 11113	. . . . . . . . . . . . . . . . 17
35	34	a1i 11	. . . . . . . . . . . . . . . 16
36	32, 33, 35	divcan1d 10802	. . . . . . . . . . . . . . 15
37	36	adantr 481	. . . . . . . . . . . . . 14 $/\$
38	37	adantl 482	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
39	38	breq2d 4665	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
40		id 22	. . . . . . . . . . . . . . . 16
41		2z 11409	. . . . . . . . . . . . . . . . 17
42	41	a1i 11	. . . . . . . . . . . . . . . 16
43	40, 42	zmulcld 11488	. . . . . . . . . . . . . . 15
44	43	3ad2ant3 1084	. . . . . . . . . . . . . 14 $/\$ $/\$
45		nnz 11399	. . . . . . . . . . . . . . 15
46	45	adantr 481	. . . . . . . . . . . . . 14 $/\$
47		zltlem1 11430	. . . . . . . . . . . . . 14 $/\$
48	44, 46, 47	syl2an 494	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
49	48	biimprd 238	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
50	39, 49	sylbid 230	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
51	50	ex 450	. . . . . . . . . 10 $/\$ $/\$ $/\$
52	51	com23 86	. . . . . . . . 9 $/\$ $/\$ $/\$
53	29, 52	sylbid 230	. . . . . . . 8 $/\$ $/\$ $/\$
54	53	a1d 25	. . . . . . 7 $/\$ $/\$ $/\$
55	54	imp32 449	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
56	21, 55	sylbi 207	. . . . 5 $/\$
57	56	impcom 446	. . . 4 $/\$ $/\$
58		modid 12695	. . . 4 $/\$ $/\$ $/\$
59	9, 20, 57, 58	syl12anc 1324	. . 3 $/\$ $/\$
60	59	eqcomd 2628	. 2 $/\$ $/\$
61	60	ralrimiva 2966	1 $/\$

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4

wb 196 $/\$ wa 384 $/\$ w3a 1037

wceq 1483

wcel 1990

wne 2794

wral 2912 class class class wbr 4653 (class class class)co 6650

cc 9934

cr 9935

cc0 9936

c1 9937

cmul 9941

clt 10074

cle 10075

cmin 10266

cdiv 10684

cn 11020

c2 11070

cz 11377

crp 11832

cfz 12326

cmo 12668

cdvds 14983

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-inf 8349 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-fz 12327 df-fl 12593 df-mod 12669

This theorem is referenced by: 2lgslem1a 25116

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