2llnjN - Mathbox for Norm Megill

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Theorem 2llnjN 34853

Description: The join of two different lattice lines in a lattice plane equals the plane. (Contributed by NM, 4-Jul-2012.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
2llnj.l
2llnj.j	$.\/$
2llnj.n
2llnj.p

**Assertion**
Ref	Expression
2llnjN	$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$

Proof of Theorem 2llnjN Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . . . . . . 8
2		2llnj.j	. . . . . . . 8 $.\/$
3		eqid 2622	. . . . . . . 8
4		2llnj.n	. . . . . . . 8
5	1, 2, 3, 4	islln2 34797	. . . . . . 7 $/\$ $/\$ $.\/$
6		simpr 477	. . . . . . 7 $/\$ $/\$ $.\/$ $/\$ $.\/$
7	5, 6	syl6bi 243	. . . . . 6 $/\$ $.\/$
8	1, 2, 3, 4	islln2 34797	. . . . . . 7 $/\$ $/\$ $.\/$
9		simpr 477	. . . . . . 7 $/\$ $/\$ $.\/$ $/\$ $.\/$
10	8, 9	syl6bi 243	. . . . . 6 $/\$ $.\/$
11	7, 10	anim12d 586	. . . . 5 $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$
12	11	imp 445	. . . 4 $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$
13	12	3adantr3 1222	. . 3 $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$
14	13	3adant3 1081	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$
15		simp2rr 1131	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$
16		simp3rr 1135	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$
17	15, 16	oveq12d 6668	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$
18		simp13 1093	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$
19		breq1 4656	. . . . . . . . . . . . . . 15 $.\/$ $.\/$
20		neeq1 2856	. . . . . . . . . . . . . . 15 $.\/$ $.\/$
21	19, 20	3anbi13d 1401	. . . . . . . . . . . . . 14 $.\/$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$
22		breq1 4656	. . . . . . . . . . . . . . 15 $.\/$ $.\/$
23		neeq2 2857	. . . . . . . . . . . . . . 15 $.\/$ $.\/$ $.\/$ $.\/$
24	22, 23	3anbi23d 1402	. . . . . . . . . . . . . 14 $.\/$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$
25	21, 24	sylan9bb 736	. . . . . . . . . . . . 13 $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$
26	15, 16, 25	syl2anc 693	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$
27	18, 26	mpbid 222	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$
28		simp11 1091	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$
29		simp123 1195	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$
30		simp2ll 1128	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$
31		simp2lr 1129	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$
32		simp2rl 1130	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$
33		simp3ll 1132	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$
34		simp3lr 1133	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$
35		simp3rl 1134	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$
36		2llnj.l	. . . . . . . . . . . . . 14
37		2llnj.p	. . . . . . . . . . . . . 14
38	36, 2, 3, 4, 37	2llnjaN 34852	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$
39	38	ex 450	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$
40	28, 29, 30, 31, 32, 33, 34, 35, 39	syl233anc 1355	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$
41	27, 40	mpd 15	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$
42	17, 41	eqtrd 2656	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$
43	42	3exp 1264	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $.\/$ $.\/$
44	43	3impib 1262	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $.\/$ $.\/$
45	44	expd 452	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$
46	45	rexlimdvv 3037	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$
47	46	3exp 1264	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$
48	47	rexlimdvv 3037	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$
49	48	impd 447	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$
50	14, 49	mpd 15	1 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$

Colors of variables: wff setvar class

Syntax hints:

wi 4

wb 196 $/\$ 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

catm 34550

chlt 34637

clln 34777

clpl 34778

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-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-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-lplanes 34785

This theorem is referenced by: 2llnm2N 34854

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