llnmlplnN - Mathbox for Norm Megill

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Theorem llnmlplnN 33104

Description: The intersection of a line with a plane not containing it is an atom. (Contributed by NM, 29-Jun-2012.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
llnmlpln.l
llnmlpln.m	$./\$
llnmlpln.z
llnmlpln.a
llnmlpln.n
llnmlpln.p

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

Proof of Theorem llnmlplnN Dummy variable

is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simprl 764	. 2 $/\$ $/\$ $/\$ $/\$ $./\$
2		simp11 1038	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $./\$
3		hllat 32929	. . . . . . . 8
4	2, 3	syl 17	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $./\$
5		simp12 1039	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $./\$
6		eqid 2451	. . . . . . . . 9
7		llnmlpln.n	. . . . . . . . 9
8	6, 7	llnbase 33074	. . . . . . . 8
9	5, 8	syl 17	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $./\$
10		simp13 1040	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $./\$
11		llnmlpln.p	. . . . . . . . 9
12	6, 11	lplnbase 33099	. . . . . . . 8
13	10, 12	syl 17	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $./\$
14		llnmlpln.m	. . . . . . . 8 $./\$
15	6, 14	latmcl 16298	. . . . . . 7 $/\$ $/\$ $./\$
16	4, 9, 13, 15	syl3anc 1268	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $./\$ $./\$
17		simp2r 1035	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $./\$ $./\$
18		simp3 1010	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $./\$ $./\$
19		llnmlpln.l	. . . . . . 7
20		llnmlpln.z	. . . . . . 7
21		llnmlpln.a	. . . . . . 7
22	6, 19, 20, 21, 7	llnle 33083	. . . . . 6 $/\$ $./\$ $/\$ $./\$ $/\$ $./\$ $./\$
23	2, 16, 17, 18, 22	syl22anc 1269	. . . . 5 $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $./\$ $./\$
24	4	adantr 467	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $./\$ $/\$ $/\$ $./\$
25	16	adantr 467	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $./\$ $/\$ $/\$ $./\$ $./\$
26	9	adantr 467	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $./\$ $/\$ $/\$ $./\$
27	6, 19, 14	latmle1 16322	. . . . . . . 8 $/\$ $/\$ $./\$
28	4, 9, 13, 27	syl3anc 1268	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $./\$ $./\$
29	28	adantr 467	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $./\$ $/\$ $/\$ $./\$ $./\$
30	6, 7	llnbase 33074	. . . . . . . . . 10
31	30	ad2antrl 734	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $./\$ $/\$ $/\$ $./\$
32		simprr 766	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $./\$ $/\$ $/\$ $./\$ $./\$
33	6, 19, 24, 31, 25, 26, 32, 29	lattrd 16304	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $./\$ $/\$ $/\$ $./\$
34		simpl11 1083	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $./\$ $/\$ $/\$ $./\$
35		simprl 764	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $./\$ $/\$ $/\$ $./\$
36		simpl12 1084	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $./\$ $/\$ $/\$ $./\$
37	19, 7	llncmp 33087	. . . . . . . . 9 $/\$ $/\$
38	34, 35, 36, 37	syl3anc 1268	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $./\$ $/\$ $/\$ $./\$
39	33, 38	mpbid 214	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $./\$ $/\$ $/\$ $./\$
40	39, 32	eqbrtrrd 4425	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $./\$ $/\$ $/\$ $./\$ $./\$
41	6, 19, 24, 25, 26, 29, 40	latasymd 16303	. . . . 5 $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $./\$ $/\$ $/\$ $./\$ $./\$
42	23, 41	rexlimddv 2883	. . . 4 $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $./\$ $./\$
43	6, 19, 14	latleeqm1 16325	. . . . 5 $/\$ $/\$ $./\$
44	4, 9, 13, 43	syl3anc 1268	. . . 4 $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $./\$ $./\$
45	42, 44	mpbird 236	. . 3 $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $./\$
46	45	3expia 1210	. 2 $/\$ $/\$ $/\$ $/\$ $./\$ $./\$
47	1, 46	mt3d 129	1 $/\$ $/\$ $/\$ $/\$ $./\$ $./\$

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4

wb 188 $/\$ wa 371 $/\$ w3a 985

wceq 1444

wcel 1887

wne 2622

wrex 2738 class class class wbr 4402

cfv 5582 (class class class)co 6290

cbs 15121

cple 15197

cmee 16190

cp0 16283

clat 16291

catm 32829

chlt 32916

clln 33056

clpl 33057

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1669 ax-4 1682 ax-5 1758 ax-6 1805 ax-7 1851 ax-8 1889 ax-9 1896 ax-10 1915 ax-11 1920 ax-12 1933 ax-13 2091 ax-ext 2431 ax-rep 4515 ax-sep 4525 ax-nul 4534 ax-pow 4581 ax-pr 4639 ax-un 6583

This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3an 987 df-tru 1447 df-ex 1664 df-nf 1668 df-sb 1798 df-eu 2303 df-mo 2304 df-clab 2438 df-cleq 2444 df-clel 2447 df-nfc 2581 df-ne 2624 df-ral 2742 df-rex 2743 df-reu 2744 df-rab 2746 df-v 3047 df-sbc 3268 df-csb 3364 df-dif 3407 df-un 3409 df-in 3411 df-ss 3418 df-nul 3732 df-if 3882 df-pw 3953 df-sn 3969 df-pr 3971 df-op 3975 df-uni 4199 df-iun 4280 df-br 4403 df-opab 4462 df-mpt 4463 df-id 4749 df-xp 4840 df-rel 4841 df-cnv 4842 df-co 4843 df-dm 4844 df-rn 4845 df-res 4846 df-ima 4847 df-iota 5546 df-fun 5584 df-fn 5585 df-f 5586 df-f1 5587 df-fo 5588 df-f1o 5589 df-fv 5590 df-riota 6252 df-ov 6293 df-oprab 6294 df-preset 16173 df-poset 16191 df-plt 16204 df-lub 16220 df-glb 16221 df-join 16222 df-meet 16223 df-p0 16285 df-lat 16292 df-clat 16354 df-oposet 32742 df-ol 32744 df-oml 32745 df-covers 32832 df-ats 32833 df-atl 32864 df-cvlat 32888 df-hlat 32917 df-llines 33063 df-lplanes 33064

This theorem is referenced by: (None)

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