cdlemg16 - Mathbox for Norm Megill

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Theorem cdlemg16 36780

Description: Part of proof of Lemma G of [Crawley] p. 116; 2nd line p. 117, which says that (our) cdlemg10 36764 "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 36007, in order to make this inference. This final step eliminates the

condition from cdlemg12 36773. 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 997	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
2		simpl21 1072	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $/\$
3		simpl22 1073	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $/\$
4		simpr 459	. . 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 36779	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$
13	1, 2, 3, 4, 12	syl121anc 1231	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $./\$ $.\/$ $./\$
14		simpl1 997	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
15		simpl2 998	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $/\$ $/\$ $/\$
16		simpl31 1075	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $/\$ $.\/$
17		simpl32 1076	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $/\$ $.\/$
18	16, 17	jca 530	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $/\$ $.\/$
19		simpr 459	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $/\$
20		simpl33 1077	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$
21	5, 6, 7, 8, 9, 10, 11	cdlemg12 36773	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$ $./\$
22	14, 15, 18, 19, 20, 21	syl113anc 1238	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $./\$ $.\/$ $./\$
23	13, 22	pm2.61dane 2772	1 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$ $./\$ $.\/$ $./\$

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4 $/\$ wa 367 $/\$ w3a 971

wceq 1398

wcel 1823

wne 2649 class class class wbr 4439

cfv 5570 (class class class)co 6270

cple 14791

cjn 15772

cmee 15773

catm 35385

chlt 35472

clh 36105

cltrn 36222

ctrl 36280

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1623 ax-4 1636 ax-5 1709 ax-6 1752 ax-7 1795 ax-8 1825 ax-9 1827 ax-10 1842 ax-11 1847 ax-12 1859 ax-13 2004 ax-ext 2432 ax-rep 4550 ax-sep 4560 ax-nul 4568 ax-pow 4615 ax-pr 4676 ax-un 6565 ax-riotaBAD 35081

This theorem depends on definitions: df-bi 185 df-or 368 df-an 369 df-3or 972 df-3an 973 df-tru 1401 df-ex 1618 df-nf 1622 df-sb 1745 df-eu 2288 df-mo 2289 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2604 df-ne 2651 df-nel 2652 df-ral 2809 df-rex 2810 df-reu 2811 df-rmo 2812 df-rab 2813 df-v 3108 df-sbc 3325 df-csb 3421 df-dif 3464 df-un 3466 df-in 3468 df-ss 3475 df-nul 3784 df-if 3930 df-pw 4001 df-sn 4017 df-pr 4019 df-op 4023 df-uni 4236 df-iun 4317 df-iin 4318 df-br 4440 df-opab 4498 df-mpt 4499 df-id 4784 df-xp 4994 df-rel 4995 df-cnv 4996 df-co 4997 df-dm 4998 df-rn 4999 df-res 5000 df-ima 5001 df-iota 5534 df-fun 5572 df-fn 5573 df-f 5574 df-f1 5575 df-fo 5576 df-f1o 5577 df-fv 5578 df-riota 6232 df-ov 6273 df-oprab 6274 df-mpt2 6275 df-1st 6773 df-2nd 6774 df-undef 6994 df-map 7414 df-preset 15756 df-poset 15774 df-plt 15787 df-lub 15803 df-glb 15804 df-join 15805 df-meet 15806 df-p0 15868 df-p1 15869 df-lat 15875 df-clat 15937 df-oposet 35298 df-ol 35300 df-oml 35301 df-covers 35388 df-ats 35389 df-atl 35420 df-cvlat 35444 df-hlat 35473 df-llines 35619 df-lplanes 35620 df-lvols 35621 df-lines 35622 df-psubsp 35624 df-pmap 35625 df-padd 35917 df-lhyp 36109 df-laut 36110 df-ldil 36225 df-ltrn 36226 df-trl 36281

This theorem is referenced by: cdlemg16z 36782

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