ps-2b - Mathbox for Norm Megill

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Theorem ps-2b 34278

Description: Variation of projective geometry axiom ps-2 34274. (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 1026	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$
2		simp12 1027	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$
3		simp13 1028	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$
4		simp21 1029	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$
5	2, 3, 4	3jca 1176	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$
6		simp22 1030	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$
7		simp23 1031	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$
8	6, 7	jca 532	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$
9		simp31 1032	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$
10		simp32 1033	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$
11	9, 10	jca 532	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $/\$
12		simp33 1034	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $/\$ $.\/$
13		ps-2b.l	. . . 4
14		ps-2b.j	. . . 4 $.\/$
15		ps-2b.a	. . . 4
16	13, 14, 15	ps-2 34274	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $/\$ $.\/$
17	1, 5, 8, 11, 12, 16	syl32anc 1236	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $/\$ $.\/$
18		simp111 1125	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$
19		hlatl 34157	. . . . 5
20	18, 19	syl 16	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$
21		hllat 34160	. . . . . 6
22	18, 21	syl 16	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$
23		simp112 1126	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$
24		simp121 1128	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$
25		eqid 2467	. . . . . . 7
26	25, 14, 15	hlatjcl 34163	. . . . . 6 $/\$ $/\$ $.\/$
27	18, 23, 24, 26	syl3anc 1228	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$
28		simp122 1129	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$
29		simp123 1130	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$
30	25, 14, 15	hlatjcl 34163	. . . . . 6 $/\$ $/\$ $.\/$
31	18, 28, 29, 30	syl3anc 1228	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$
32		ps-2b.m	. . . . . 6 $./\$
33	25, 32	latmcl 15535	. . . . 5 $/\$ $.\/$ $/\$ $.\/$ $.\/$ $./\$ $.\/$
34	22, 27, 31, 33	syl3anc 1228	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $./\$ $.\/$
35		simp2 997	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$
36		simp3 998	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $/\$ $.\/$
37	25, 15	atbase 34086	. . . . . . 7
38	35, 37	syl 16	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$
39	25, 13, 32	latlem12 15561	. . . . . 6 $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$ $./\$ $.\/$
40	22, 38, 27, 31, 39	syl13anc 1230	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$ $./\$ $.\/$
41	36, 40	mpbid 210	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $./\$ $.\/$
42		ps-2b.z	. . . . 5
43	25, 13, 42, 15	atlen0 34107	. . . 4 $/\$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$
44	20, 34, 35, 41, 43	syl31anc 1231	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $./\$ $.\/$
45	44	rexlimdv3a 2957	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$ $./\$ $.\/$
46	17, 45	mpd 15	1 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $./\$ $.\/$

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4

wb 184 $/\$ wa 369 $/\$ w3a 973

wceq 1379

wcel 1767

wne 2662

wrex 2815 class class class wbr 4447

cfv 5586 (class class class)co 6282

cbs 14486

cple 14558

cjn 15427

cmee 15428

cp0 15520

clat 15528

catm 34060

cal 34061

chlt 34147

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1601 ax-4 1612 ax-5 1680 ax-6 1719 ax-7 1739 ax-8 1769 ax-9 1771 ax-10 1786 ax-11 1791 ax-12 1803 ax-13 1968 ax-ext 2445 ax-rep 4558 ax-sep 4568 ax-nul 4576 ax-pow 4625 ax-pr 4686 ax-un 6574

This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3an 975 df-tru 1382 df-ex 1597 df-nf 1600 df-sb 1712 df-eu 2279 df-mo 2280 df-clab 2453 df-cleq 2459 df-clel 2462 df-nfc 2617 df-ne 2664 df-ral 2819 df-rex 2820 df-reu 2821 df-rab 2823 df-v 3115 df-sbc 3332 df-csb 3436 df-dif 3479 df-un 3481 df-in 3483 df-ss 3490 df-nul 3786 df-if 3940 df-pw 4012 df-sn 4028 df-pr 4030 df-op 4034 df-uni 4246 df-iun 4327 df-br 4448 df-opab 4506 df-mpt 4507 df-id 4795 df-xp 5005 df-rel 5006 df-cnv 5007 df-co 5008 df-dm 5009 df-rn 5010 df-res 5011 df-ima 5012 df-iota 5549 df-fun 5588 df-fn 5589 df-f 5590 df-f1 5591 df-fo 5592 df-f1o 5593 df-fv 5594 df-riota 6243 df-ov 6285 df-oprab 6286 df-poset 15429 df-plt 15441 df-lub 15457 df-glb 15458 df-join 15459 df-meet 15460 df-p0 15522 df-lat 15529 df-clat 15591 df-oposet 33973 df-ol 33975 df-oml 33976 df-covers 34063 df-ats 34064 df-atl 34095 df-cvlat 34119 df-hlat 34148

This theorem is referenced by: ps-2c 34324

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