cdlemd6 - Mathbox for Norm Megill

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Theorem cdlemd6 34999

Description: Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 31-May-2012.)

**Hypotheses**
Ref	Expression
cdlemd4.l
cdlemd4.j	$.\/$
cdlemd4.a
cdlemd4.h
cdlemd4.t

**Assertion**
Ref	Expression
cdlemd6	$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$

**Proof of Theorem cdlemd6**
Step	Hyp	Ref	Expression
1		simp3 998	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$
2	1	oveq2d 6298	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$
3	2	oveq1d 6297	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$
4		simp1l 1020	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$
5		simp1rl 1061	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$
6		simp21 1029	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$
7		cdlemd4.l	. . . . . . 7
8		cdlemd4.j	. . . . . . 7 $.\/$
9		eqid 2467	. . . . . . 7
10		cdlemd4.a	. . . . . . 7
11		cdlemd4.h	. . . . . . 7
12		cdlemd4.t	. . . . . . 7
13		eqid 2467	. . . . . . 7
14	7, 8, 9, 10, 11, 12, 13	trlval2 34959	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $.\/$
15	4, 5, 6, 14	syl3anc 1228	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
16		simp1rr 1062	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$
17	7, 8, 9, 10, 11, 12, 13	trlval2 34959	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $.\/$
18	4, 16, 6, 17	syl3anc 1228	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
19	3, 15, 18	3eqtr4d 2518	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$
20	19	oveq2d 6298	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$
21	1	oveq1d 6297	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$
22	20, 21	oveq12d 6300	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$
23		simp22 1030	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$
24		simp23 1031	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
25	7, 8, 9, 10, 11, 12, 13	cdlemc 34993	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$
26	4, 5, 6, 23, 24, 25	syl131anc 1241	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$
27		oveq2 6290	. . . . . . 7 $.\/$ $.\/$
28	27	breq2d 4459	. . . . . 6 $.\/$ $.\/$
29	28	notbid 294	. . . . 5 $.\/$ $.\/$
30	29	biimpd 207	. . . 4 $.\/$ $.\/$
31	1, 24, 30	sylc 60	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
32	7, 8, 9, 10, 11, 12, 13	cdlemc 34993	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$
33	4, 16, 6, 23, 31, 32	syl131anc 1241	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$
34	22, 26, 33	3eqtr4d 2518	1 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4 $/\$ wa 369 $/\$ w3a 973

wceq 1379

wcel 1767 class class class wbr 4447

cfv 5586 (class class class)co 6282

cple 14555

cjn 15424

cmee 15425

catm 34060

chlt 34147

clh 34780

cltrn 34897

ctrl 34954

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-iin 4328 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-mpt2 6287 df-1st 6781 df-2nd 6782 df-map 7419 df-poset 15426 df-plt 15438 df-lub 15454 df-glb 15455 df-join 15456 df-meet 15457 df-p0 15519 df-p1 15520 df-lat 15526 df-clat 15588 df-oposet 33973 df-ol 33975 df-oml 33976 df-covers 34063 df-ats 34064 df-atl 34095 df-cvlat 34119 df-hlat 34148 df-llines 34294 df-psubsp 34299 df-pmap 34300 df-padd 34592 df-lhyp 34784 df-laut 34785 df-ldil 34900 df-ltrn 34901 df-trl 34955

This theorem is referenced by: cdlemd7 35000

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