dyaddisjlem - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > dyaddisjlem		Structured version Visualization version Unicode version

Theorem dyaddisjlem 23363

Description: Lemma for dyaddisj 23364. (Contributed by Mario Carneiro, 26-Mar-2015.)

**Hypothesis**
Ref	Expression
dyadmbl.1

**Assertion**
Ref	Expression
dyaddisjlem	$/\$ $/\$ $/\$ $/\$ $\/$ $\/$

Distinct variable groups:

**Proof of Theorem dyaddisjlem**
Step	Hyp	Ref	Expression
1		simplll 798	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
2		simplrl 800	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
3		dyadmbl.1	. . . . . . . . . . 11
4	3	dyadval 23360	. . . . . . . . . 10 $/\$
5	1, 2, 4	syl2anc 693	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
6	5	fveq2d 6195	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
7		df-ov 6653	. . . . . . . 8
8	6, 7	syl6eqr 2674	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
9		simpllr 799	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
10		simplrr 801	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
11	3	dyadval 23360	. . . . . . . . . 10 $/\$
12	9, 10, 11	syl2anc 693	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
13	12	fveq2d 6195	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
14		df-ov 6653	. . . . . . . 8
15	13, 14	syl6eqr 2674	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
16	8, 15	ineq12d 3815	. . . . . 6 $/\$ $/\$ $/\$ $/\$
17		incom 3805	. . . . . 6
18	16, 17	syl6eq 2672	. . . . 5 $/\$ $/\$ $/\$ $/\$
19	18	adantr 481	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
20	1	zred 11482	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
21	20	recnd 10068	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
22		2nn 11185	. . . . . . . . . . . 12
23		nnexpcl 12873	. . . . . . . . . . . 12 $/\$
24	22, 2, 23	sylancr 695	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
25	24	nncnd 11036	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
26		nnexpcl 12873	. . . . . . . . . . . 12 $/\$
27	22, 10, 26	sylancr 695	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
28	27	nncnd 11036	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
29	24	nnne0d 11065	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
30	21, 25, 28, 29	div13d 10825	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
31		2cnd 11093	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
32		2ne0 11113	. . . . . . . . . . . . 13
33	32	a1i 11	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
34	2	nn0zd 11480	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
35	10	nn0zd 11480	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
36	31, 33, 34, 35	expsubd 13019	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
37		2z 11409	. . . . . . . . . . . 12
38		simpr 477	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
39		znn0sub 11424	. . . . . . . . . . . . . 14 $/\$
40	34, 35, 39	syl2anc 693	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
41	38, 40	mpbid 222	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
42		zexpcl 12875	. . . . . . . . . . . 12 $/\$
43	37, 41, 42	sylancr 695	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
44	36, 43	eqeltrrd 2702	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
45	44, 1	zmulcld 11488	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
46	30, 45	eqeltrd 2701	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
47		zltp1le 11427	. . . . . . . 8 $/\$
48	9, 46, 47	syl2anc 693	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
49	9	zred 11482	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
50	20, 24	nndivred 11069	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
51	27	nnred 11035	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
52	27	nngt0d 11064	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
53		ltdivmul2 10900	. . . . . . . 8 $/\$ $/\$ $/\$
54	49, 50, 51, 52, 53	syl112anc 1330	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
55		peano2re 10209	. . . . . . . . 9
56	49, 55	syl 17	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
57		ledivmul2 10902	. . . . . . . 8 $/\$ $/\$ $/\$
58	56, 50, 51, 52, 57	syl112anc 1330	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
59	48, 54, 58	3bitr4d 300	. . . . . 6 $/\$ $/\$ $/\$ $/\$
60	49, 27	nndivred 11069	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
61	60	rexrd 10089	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
62	56, 27	nndivred 11069	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
63	62	rexrd 10089	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
64	50	rexrd 10089	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
65		peano2re 10209	. . . . . . . . . 10
66	20, 65	syl 17	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
67	66, 24	nndivred 11069	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
68	67	rexrd 10089	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
69		ioodisj 12302	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
70	69	ex 450	. . . . . . 7 $/\$ $/\$ $/\$
71	61, 63, 64, 68, 70	syl22anc 1327	. . . . . 6 $/\$ $/\$ $/\$ $/\$
72	59, 71	sylbid 230	. . . . 5 $/\$ $/\$ $/\$ $/\$
73	72	imp 445	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
74	19, 73	eqtrd 2656	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
75	74	3mix3d 1238	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $\/$
76	50	adantr 481	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
77	67	adantr 481	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
78		simprl 794	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
79	66	recnd 10068	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
80	79, 25, 28, 29	div13d 10825	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
81	1	peano2zd 11485	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
82	44, 81	zmulcld 11488	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
83	80, 82	eqeltrd 2701	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
84		zltp1le 11427	. . . . . . . . . . 11 $/\$
85	9, 83, 84	syl2anc 693	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
86		ltdivmul2 10900	. . . . . . . . . . 11 $/\$ $/\$ $/\$
87	49, 67, 51, 52, 86	syl112anc 1330	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
88		ledivmul2 10902	. . . . . . . . . . 11 $/\$ $/\$ $/\$
89	56, 67, 51, 52, 88	syl112anc 1330	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
90	85, 87, 89	3bitr4d 300	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
91	90	biimpa 501	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
92	91	adantrl 752	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
93		iccss 12241	. . . . . . 7 $/\$ $/\$ $/\$
94	76, 77, 78, 92, 93	syl22anc 1327	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
95	12	fveq2d 6195	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
96		df-ov 6653	. . . . . . . 8
97	95, 96	syl6eqr 2674	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
98	97	adantr 481	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
99	5	fveq2d 6195	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
100		df-ov 6653	. . . . . . . 8
101	99, 100	syl6eqr 2674	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
102	101	adantr 481	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
103	94, 98, 102	3sstr4d 3648	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
104	103	3mix2d 1237	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $\/$
105	104	anassrs 680	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $\/$
106	16	adantr 481	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
107		ioodisj 12302	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
108	107	ex 450	. . . . . . . 8 $/\$ $/\$ $/\$
109	64, 68, 61, 63, 108	syl22anc 1327	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
110	109	imp 445	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
111	106, 110	eqtrd 2656	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
112	111	3mix3d 1238	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $\/$
113	112	adantlr 751	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $\/$
114	60	adantr 481	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
115	67	adantr 481	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
116	105, 113, 114, 115	ltlecasei 10145	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $\/$ $\/$
117	75, 116, 60, 50	ltlecasei 10145	1 $/\$ $/\$ $/\$ $/\$ $\/$ $\/$

Colors of variables: wff setvar class

Syntax hints:

wi 4

wb 196 $/\$ wa 384 $\/$ w3o 1036

wceq 1483

wcel 1990

wne 2794

cin 3573

wss 3574

c0 3915

cop 4183 class class class wbr 4653

cfv 5888 (class class class)co 6650

cmpt2 6652

cr 9935

cc0 9936

c1 9937

caddc 9939

cmul 9941

cxr 10073

clt 10074

cle 10075

cmin 10266

cdiv 10684

cn 11020

c2 11070

cn0 11292

cz 11377

cioo 12175

cicc 12178

cexp 12860

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

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-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-ioo 12179 df-icc 12182 df-seq 12802 df-exp 12861

This theorem is referenced by: dyaddisj 23364

W3C validator