ps-2b - Mathbox for Norm Megill

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Theorem ps-2b 33041

Description: Variation of projective geometry axiom ps-2 33037. (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 1037	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$
2		simp12 1038	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$
3		simp13 1039	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$
4		simp21 1040	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$
5	2, 3, 4	3jca 1187	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$
6		simp22 1041	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$
7		simp23 1042	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$
8	6, 7	jca 535	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$
9		simp31 1043	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$
10		simp32 1044	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$
11	9, 10	jca 535	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $/\$
12		simp33 1045	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $/\$ $.\/$
13		ps-2b.l	. . . 4
14		ps-2b.j	. . . 4 $.\/$
15		ps-2b.a	. . . 4
16	13, 14, 15	ps-2 33037	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $/\$ $.\/$
17	1, 5, 8, 11, 12, 16	syl32anc 1275	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $/\$ $.\/$
18		simp111 1136	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$
19		hlatl 32920	. . . . 5
20	18, 19	syl 17	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$
21		hllat 32923	. . . . . 6
22	18, 21	syl 17	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$
23		simp112 1137	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$
24		simp121 1139	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$
25		eqid 2450	. . . . . . 7
26	25, 14, 15	hlatjcl 32926	. . . . . 6 $/\$ $/\$ $.\/$
27	18, 23, 24, 26	syl3anc 1267	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$
28		simp122 1140	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$
29		simp123 1141	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$
30	25, 14, 15	hlatjcl 32926	. . . . . 6 $/\$ $/\$ $.\/$
31	18, 28, 29, 30	syl3anc 1267	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$
32		ps-2b.m	. . . . . 6 $./\$
33	25, 32	latmcl 16291	. . . . 5 $/\$ $.\/$ $/\$ $.\/$ $.\/$ $./\$ $.\/$
34	22, 27, 31, 33	syl3anc 1267	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $./\$ $.\/$
35		simp2 1008	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$
36		simp3 1009	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $/\$ $.\/$
37	25, 15	atbase 32849	. . . . . . 7
38	35, 37	syl 17	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$
39	25, 13, 32	latlem12 16317	. . . . . 6 $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$ $./\$ $.\/$
40	22, 38, 27, 31, 39	syl13anc 1269	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$ $./\$ $.\/$
41	36, 40	mpbid 214	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $./\$ $.\/$
42		ps-2b.z	. . . . 5
43	25, 13, 42, 15	atlen0 32870	. . . 4 $/\$ $.\/$ $./\$ $.\/$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$
44	20, 34, 35, 41, 43	syl31anc 1270	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $./\$ $.\/$
45	44	rexlimdv3a 2880	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$ $./\$ $.\/$
46	17, 45	mpd 15	1 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $./\$ $.\/$

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4

wb 188 $/\$ wa 371 $/\$ w3a 984

wceq 1443

wcel 1886

wne 2621

wrex 2737 class class class wbr 4401

cfv 5581 (class class class)co 6288

cbs 15114

cple 15190

cjn 16182

cmee 16183

cp0 16276

clat 16284

catm 32823

cal 32824

chlt 32910

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1668 ax-4 1681 ax-5 1757 ax-6 1804 ax-7 1850 ax-8 1888 ax-9 1895 ax-10 1914 ax-11 1919 ax-12 1932 ax-13 2090 ax-ext 2430 ax-rep 4514 ax-sep 4524 ax-nul 4533 ax-pow 4580 ax-pr 4638 ax-un 6580

This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3an 986 df-tru 1446 df-ex 1663 df-nf 1667 df-sb 1797 df-eu 2302 df-mo 2303 df-clab 2437 df-cleq 2443 df-clel 2446 df-nfc 2580 df-ne 2623 df-ral 2741 df-rex 2742 df-reu 2743 df-rab 2745 df-v 3046 df-sbc 3267 df-csb 3363 df-dif 3406 df-un 3408 df-in 3410 df-ss 3417 df-nul 3731 df-if 3881 df-pw 3952 df-sn 3968 df-pr 3970 df-op 3974 df-uni 4198 df-iun 4279 df-br 4402 df-opab 4461 df-mpt 4462 df-id 4748 df-xp 4839 df-rel 4840 df-cnv 4841 df-co 4842 df-dm 4843 df-rn 4844 df-res 4845 df-ima 4846 df-iota 5545 df-fun 5583 df-fn 5584 df-f 5585 df-f1 5586 df-fo 5587 df-f1o 5588 df-fv 5589 df-riota 6250 df-ov 6291 df-oprab 6292 df-preset 16166 df-poset 16184 df-plt 16197 df-lub 16213 df-glb 16214 df-join 16215 df-meet 16216 df-p0 16278 df-lat 16285 df-clat 16347 df-oposet 32736 df-ol 32738 df-oml 32739 df-covers 32826 df-ats 32827 df-atl 32858 df-cvlat 32882 df-hlat 32911

This theorem is referenced by: ps-2c 33087

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