ps-2b - Mathbox for Norm Megill

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Theorem ps-2b 33435

Description: Variation of projective geometry axiom ps-2 33431. (Contributed by NM, 3-Jul-2012.)

**Hypotheses**
Ref	Expression
ps-2b.l
ps-2b.j	$.\/$
ps-2b.m	$./\$
ps-2b.z
ps-2b.a

**Assertion**
Ref	Expression
ps-2b	$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $./\$ $.\/$

Proof of Theorem ps-2b Dummy variable

is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp11 1018	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$
2		simp12 1019	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$
3		simp13 1020	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$
4		simp21 1021	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$
5	2, 3, 4	3jca 1168	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$
6		simp22 1022	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$
7		simp23 1023	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$
8	6, 7	jca 532	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$
9		simp31 1024	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$
10		simp32 1025	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$
11	9, 10	jca 532	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $/\$
12		simp33 1026	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $/\$ $.\/$
13		ps-2b.l	. . . 4
14		ps-2b.j	. . . 4 $.\/$
15		ps-2b.a	. . . 4
16	13, 14, 15	ps-2 33431	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $/\$ $.\/$
17	1, 5, 8, 11, 12, 16	syl32anc 1227	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $/\$ $.\/$
18		simp111 1117	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$
19		hlatl 33314	. . . . 5
20	18, 19	syl 16	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$
21		hllat 33317	. . . . . 6
22	18, 21	syl 16	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$
23		simp112 1118	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$
24		simp121 1120	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$
25		eqid 2451	. . . . . . 7
26	25, 14, 15	hlatjcl 33320	. . . . . 6 $/\$ $/\$ $.\/$
27	18, 23, 24, 26	syl3anc 1219	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$
28		simp122 1121	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$
29		simp123 1122	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$
30	25, 14, 15	hlatjcl 33320	. . . . . 6 $/\$ $/\$ $.\/$
31	18, 28, 29, 30	syl3anc 1219	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$
32		ps-2b.m	. . . . . 6 $./\$
33	25, 32	latmcl 15333	. . . . 5 $/\$ $.\/$ $/\$ $.\/$ $.\/$ $./\$ $.\/$
34	22, 27, 31, 33	syl3anc 1219	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $./\$ $.\/$
35		simp2 989	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$
36		simp3 990	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $/\$ $.\/$
37	25, 15	atbase 33243	. . . . . . 7
38	35, 37	syl 16	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$
39	25, 13, 32	latlem12 15359	. . . . . 6 $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$ $./\$ $.\/$
40	22, 38, 27, 31, 39	syl13anc 1221	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$ $./\$ $.\/$
41	36, 40	mpbid 210	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $./\$ $.\/$
42		ps-2b.z	. . . . 5
43	25, 13, 42, 15	atlen0 33264	. . . 4 $/\$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$
44	20, 34, 35, 41, 43	syl31anc 1222	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $./\$ $.\/$
45	44	rexlimdv3a 2942	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$ $./\$ $.\/$
46	17, 45	mpd 15	1 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $./\$ $.\/$

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4

wb 184 $/\$ wa 369 $/\$ w3a 965

wceq 1370

wcel 1758

wne 2644

wrex 2796 class class class wbr 4393

cfv 5519 (class class class)co 6193

cbs 14285

cple 14356

cjn 15225

cmee 15226

cp0 15318

clat 15326

catm 33217

cal 33218

chlt 33304

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-8 1760 ax-9 1762 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1952 ax-ext 2430 ax-rep 4504 ax-sep 4514 ax-nul 4522 ax-pow 4571 ax-pr 4632 ax-un 6475

This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3an 967 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1703 df-eu 2264 df-mo 2265 df-clab 2437 df-cleq 2443 df-clel 2446 df-nfc 2601 df-ne 2646 df-ral 2800 df-rex 2801 df-reu 2802 df-rab 2804 df-v 3073 df-sbc 3288 df-csb 3390 df-dif 3432 df-un 3434 df-in 3436 df-ss 3443 df-nul 3739 df-if 3893 df-pw 3963 df-sn 3979 df-pr 3981 df-op 3985 df-uni 4193 df-iun 4274 df-br 4394 df-opab 4452 df-mpt 4453 df-id 4737 df-xp 4947 df-rel 4948 df-cnv 4949 df-co 4950 df-dm 4951 df-rn 4952 df-res 4953 df-ima 4954 df-iota 5482 df-fun 5521 df-fn 5522 df-f 5523 df-f1 5524 df-fo 5525 df-f1o 5526 df-fv 5527 df-riota 6154 df-ov 6196 df-oprab 6197 df-poset 15227 df-plt 15239 df-lub 15255 df-glb 15256 df-join 15257 df-meet 15258 df-p0 15320 df-lat 15327 df-clat 15389 df-oposet 33130 df-ol 33132 df-oml 33133 df-covers 33220 df-ats 33221 df-atl 33252 df-cvlat 33276 df-hlat 33305

This theorem is referenced by: ps-2c 33481

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