cdlemg16 - Mathbox for Norm Megill

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Theorem cdlemg16 34610

Description: Part of proof of Lemma G of [Crawley] p. 116; 2nd line p. 117, which says that (our) cdlemg10 34594 "implies (2)" (of p. 116). No details are provided by the authors, so there may be a shorter proof; but ours requires the 14 lemmas, one using Desargues' law dalaw 33839, in order to make this inference. This final step eliminates the

condition from cdlemg12 34603. TODO: FIX COMMENT TODO: should we also eliminate

here (or earlier)? Do it if we don't need to add it in for something else later. (Contributed by NM, 6-May-2013.)

**Hypotheses**
Ref	Expression
cdlemg12.l
cdlemg12.j	$.\/$
cdlemg12.m	$./\$
cdlemg12.a
cdlemg12.h
cdlemg12.t
cdlemg12b.r

**Assertion**
Ref	Expression
cdlemg16	$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$ $./\$

**Proof of Theorem cdlemg16**
Step	Hyp	Ref	Expression
1		simpl1 991	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
2		simpl21 1066	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $/\$
3		simpl22 1067	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $/\$
4		simpr 461	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $/\$
5		cdlemg12.l	. . . 4
6		cdlemg12.j	. . . 4 $.\/$
7		cdlemg12.m	. . . 4 $./\$
8		cdlemg12.a	. . . 4
9		cdlemg12.h	. . . 4
10		cdlemg12.t	. . . 4
11		cdlemg12b.r	. . . 4
12	5, 6, 7, 8, 9, 10, 11	cdlemg15 34609	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$
13	1, 2, 3, 4, 12	syl121anc 1224	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $./\$ $.\/$ $./\$
14		simpl1 991	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
15		simpl2 992	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $/\$
16		simpl31 1069	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $/\$ $.\/$
17		simpl32 1070	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $/\$ $.\/$
18	16, 17	jca 532	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $/\$ $.\/$
19		simpr 461	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $/\$
20		simpl33 1071	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$
21	5, 6, 7, 8, 9, 10, 11	cdlemg12 34603	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$ $./\$
22	14, 15, 18, 19, 20, 21	syl113anc 1231	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $./\$ $.\/$ $./\$
23	13, 22	pm2.61dane 2766	1 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$ $./\$

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4 $/\$ wa 369 $/\$ w3a 965

wceq 1370

wcel 1758

wne 2644 class class class wbr 4393

cfv 5519 (class class class)co 6193

cple 14356

cjn 15225

cmee 15226

catm 33217

chlt 33304

clh 33937

cltrn 34054

ctrl 34111

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 ax-riotaBAD 32913

This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 966 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-nel 2647 df-ral 2800 df-rex 2801 df-reu 2802 df-rmo 2803 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-iin 4275 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-mpt2 6198 df-1st 6680 df-2nd 6681 df-undef 6895 df-map 7319 df-poset 15227 df-plt 15239 df-lub 15255 df-glb 15256 df-join 15257 df-meet 15258 df-p0 15320 df-p1 15321 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 df-llines 33451 df-lplanes 33452 df-lvols 33453 df-lines 33454 df-psubsp 33456 df-pmap 33457 df-padd 33749 df-lhyp 33941 df-laut 33942 df-ldil 34057 df-ltrn 34058 df-trl 34112

This theorem is referenced by: cdlemg16z 34612

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