llnexchb2 - Mathbox for Norm Megill

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Theorem llnexchb2 35155

Description: Line exchange property (compare cvlatexchb2 34622 for atoms). (Contributed by NM, 17-Nov-2012.)

**Hypotheses**
Ref	Expression
llnexch.l
llnexch.j	$.\/$
llnexch.m	$./\$
llnexch.a
llnexch.n

**Assertion**
Ref	Expression
llnexchb2	$/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $./\$ $./\$ $./\$

Proof of Theorem llnexchb2 Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp23 1096	. . 3 $/\$ $/\$ $/\$ $/\$ $./\$ $/\$
2		simp1 1061	. . . 4 $/\$ $/\$ $/\$ $/\$ $./\$ $/\$
3		eqid 2622	. . . . . 6
4		llnexch.n	. . . . . 6
5	3, 4	llnbase 34795	. . . . 5
6	1, 5	syl 17	. . . 4 $/\$ $/\$ $/\$ $/\$ $./\$ $/\$
7		llnexch.j	. . . . 5 $.\/$
8		llnexch.a	. . . . 5
9	3, 7, 8, 4	islln3 34796	. . . 4 $/\$ $/\$ $.\/$
10	2, 6, 9	syl2anc 693	. . 3 $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $/\$ $.\/$
11	1, 10	mpbid 222	. 2 $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $/\$ $.\/$
12		simp3r 1090	. . 3 $/\$ $/\$ $/\$ $/\$ $./\$ $/\$
13	12	necomd 2849	. 2 $/\$ $/\$ $/\$ $/\$ $./\$ $/\$
14		simp11 1091	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $/\$ $/\$ $/\$
15		hllat 34650	. . . . . . . . . . . 12
16	14, 15	syl 17	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $/\$ $/\$ $/\$
17		simp2l 1087	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $/\$ $/\$ $/\$
18	3, 8	atbase 34576	. . . . . . . . . . . 12
19	17, 18	syl 17	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $/\$ $/\$ $/\$
20		simp2r 1088	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $/\$ $/\$ $/\$
21	3, 8	atbase 34576	. . . . . . . . . . . 12
22	20, 21	syl 17	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $/\$ $/\$ $/\$
23		simp121 1193	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $/\$ $/\$ $/\$
24	3, 4	llnbase 34795	. . . . . . . . . . . 12
25	23, 24	syl 17	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $/\$ $/\$ $/\$
26		llnexch.l	. . . . . . . . . . . 12
27	3, 26, 7	latjle12 17062	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $.\/$
28	16, 19, 22, 25, 27	syl13anc 1328	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
29		simp3 1063	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $/\$ $/\$ $/\$
30	7, 8, 4	llni2 34798	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $.\/$
31	14, 17, 20, 29, 30	syl31anc 1329	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $/\$ $/\$ $/\$ $.\/$
32	26, 4	llncmp 34808	. . . . . . . . . . 11 $/\$ $.\/$ $/\$ $.\/$ $.\/$
33	14, 31, 23, 32	syl3anc 1326	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$
34	28, 33	bitr2d 269	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$
35	34	necon3abid 2830	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$
36		ianor 509	. . . . . . . 8 $/\$ $\/$
37	35, 36	syl6bb 276	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $\/$
38		simpl11 1136	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$
39	23	adantr 481	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$
40		simp122 1194	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $/\$ $/\$ $/\$
41	40	adantr 481	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$
42		simpl2l 1114	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$
43		simpl2r 1115	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$
44		simpr 477	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$
45		simp13l 1176	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $/\$ $/\$ $/\$ $./\$
46	45	adantr 481	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$
47		llnexch.m	. . . . . . . . . . 11 $./\$
48	26, 7, 47, 8, 4	llnexchb2lem 35154	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $./\$ $.\/$ $./\$ $./\$ $.\/$
49	38, 39, 41, 42, 43, 44, 46, 48	syl331anc 1351	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $.\/$ $./\$ $./\$ $.\/$
50	49	ex 450	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $/\$ $/\$ $/\$ $./\$ $.\/$ $./\$ $./\$ $.\/$
51		simpl11 1136	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$
52	23	adantr 481	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$
53	40	adantr 481	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$
54		simpl2r 1115	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$
55		simpl2l 1114	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$
56		simpr 477	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$
57	45	adantr 481	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$
58	26, 7, 47, 8, 4	llnexchb2lem 35154	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $./\$ $.\/$ $./\$ $./\$ $.\/$
59	51, 52, 53, 54, 55, 56, 57, 58	syl331anc 1351	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $.\/$ $./\$ $./\$ $.\/$
60	7, 8	hlatjcom 34654	. . . . . . . . . . . 12 $/\$ $/\$ $.\/$ $.\/$
61	51, 55, 54, 60	syl3anc 1326	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$
62	61	breq2d 4665	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $.\/$ $./\$ $.\/$
63	61	oveq2d 6666	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $.\/$ $./\$ $.\/$
64	63	eqeq2d 2632	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $./\$ $.\/$ $./\$ $./\$ $.\/$
65	59, 62, 64	3bitr4d 300	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $.\/$ $./\$ $./\$ $.\/$
66	65	ex 450	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $/\$ $/\$ $/\$ $./\$ $.\/$ $./\$ $./\$ $.\/$
67	50, 66	jaod 395	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $/\$ $/\$ $/\$ $\/$ $./\$ $.\/$ $./\$ $./\$ $.\/$
68	37, 67	sylbid 230	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$ $./\$ $.\/$
69		neeq1 2856	. . . . . . 7 $.\/$ $.\/$
70		breq2 4657	. . . . . . . 8 $.\/$ $./\$ $./\$ $.\/$
71		oveq2 6658	. . . . . . . . 9 $.\/$ $./\$ $./\$ $.\/$
72	71	eqeq2d 2632	. . . . . . . 8 $.\/$ $./\$ $./\$ $./\$ $./\$ $.\/$
73	70, 72	bibi12d 335	. . . . . . 7 $.\/$ $./\$ $./\$ $./\$ $./\$ $.\/$ $./\$ $./\$ $.\/$
74	69, 73	imbi12d 334	. . . . . 6 $.\/$ $./\$ $./\$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $./\$ $.\/$
75	68, 74	syl5ibrcom 237	. . . . 5 $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $./\$ $./\$
76	75	3exp 1264	. . . 4 $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $/\$ $.\/$ $./\$ $./\$ $./\$
77	76	imp4a 614	. . 3 $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $./\$ $./\$
78	77	rexlimdvv 3037	. 2 $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $/\$ $.\/$ $./\$ $./\$ $./\$
79	11, 13, 78	mp2d 49	1 $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $./\$ $./\$ $./\$

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4

wb 196 $\/$ wo 383 $/\$ wa 384 $/\$ w3a 1037

wceq 1483

wcel 1990

wne 2794

wrex 2913 class class class wbr 4653

cfv 5888 (class class class)co 6650

cbs 15857

cple 15948

cjn 16944

cmee 16945

clat 17045

catm 34550

chlt 34637

clln 34777

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-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 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-ral 2917 df-rex 2918 df-reu 2919 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-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 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-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-1st 7168 df-2nd 7169 df-preset 16928 df-poset 16946 df-plt 16958 df-lub 16974 df-glb 16975 df-join 16976 df-meet 16977 df-p0 17039 df-lat 17046 df-clat 17108 df-oposet 34463 df-ol 34465 df-oml 34466 df-covers 34553 df-ats 34554 df-atl 34585 df-cvlat 34609 df-hlat 34638 df-llines 34784 df-psubsp 34789 df-pmap 34790 df-padd 35082

This theorem is referenced by: llnexch2N 35156 cdleme20l 35610

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