cdlemd6 - Mathbox for Norm Megill

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Theorem cdlemd6 33769

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 1010	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$
2	1	oveq2d 6306	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$
3	2	oveq1d 6305	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$
4		simp1l 1032	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$
5		simp1rl 1073	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$
6		simp21 1041	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$
7		cdlemd4.l	. . . . . . 7
8		cdlemd4.j	. . . . . . 7 $.\/$
9		eqid 2451	. . . . . . 7
10		cdlemd4.a	. . . . . . 7
11		cdlemd4.h	. . . . . . 7
12		cdlemd4.t	. . . . . . 7
13		eqid 2451	. . . . . . 7
14	7, 8, 9, 10, 11, 12, 13	trlval2 33729	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $.\/$
15	4, 5, 6, 14	syl3anc 1268	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
16		simp1rr 1074	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$
17	7, 8, 9, 10, 11, 12, 13	trlval2 33729	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $.\/$
18	4, 16, 6, 17	syl3anc 1268	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
19	3, 15, 18	3eqtr4d 2495	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$
20	19	oveq2d 6306	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$
21	1	oveq1d 6305	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$
22	20, 21	oveq12d 6308	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$
23		simp22 1042	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$
24		simp23 1043	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
25	7, 8, 9, 10, 11, 12, 13	cdlemc 33763	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$
26	4, 5, 6, 23, 24, 25	syl131anc 1281	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$
27		oveq2 6298	. . . . . . 7 $.\/$ $.\/$
28	27	breq2d 4414	. . . . . 6 $.\/$ $.\/$
29	28	notbid 296	. . . . 5 $.\/$ $.\/$
30	29	biimpd 211	. . . 4 $.\/$ $.\/$
31	1, 24, 30	sylc 62	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
32	7, 8, 9, 10, 11, 12, 13	cdlemc 33763	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$
33	4, 16, 6, 23, 31, 32	syl131anc 1281	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$
34	22, 26, 33	3eqtr4d 2495	1 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4 $/\$ wa 371 $/\$ w3a 985

wceq 1444

wcel 1887 class class class wbr 4402

cfv 5582 (class class class)co 6290

cple 15197

cjn 16189

cmee 16190

catm 32829

chlt 32916

clh 33549

cltrn 33666

ctrl 33724

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1669 ax-4 1682 ax-5 1758 ax-6 1805 ax-7 1851 ax-8 1889 ax-9 1896 ax-10 1915 ax-11 1920 ax-12 1933 ax-13 2091 ax-ext 2431 ax-rep 4515 ax-sep 4525 ax-nul 4534 ax-pow 4581 ax-pr 4639 ax-un 6583

This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3an 987 df-tru 1447 df-ex 1664 df-nf 1668 df-sb 1798 df-eu 2303 df-mo 2304 df-clab 2438 df-cleq 2444 df-clel 2447 df-nfc 2581 df-ne 2624 df-ral 2742 df-rex 2743 df-reu 2744 df-rab 2746 df-v 3047 df-sbc 3268 df-csb 3364 df-dif 3407 df-un 3409 df-in 3411 df-ss 3418 df-nul 3732 df-if 3882 df-pw 3953 df-sn 3969 df-pr 3971 df-op 3975 df-uni 4199 df-iun 4280 df-iin 4281 df-br 4403 df-opab 4462 df-mpt 4463 df-id 4749 df-xp 4840 df-rel 4841 df-cnv 4842 df-co 4843 df-dm 4844 df-rn 4845 df-res 4846 df-ima 4847 df-iota 5546 df-fun 5584 df-fn 5585 df-f 5586 df-f1 5587 df-fo 5588 df-f1o 5589 df-fv 5590 df-riota 6252 df-ov 6293 df-oprab 6294 df-mpt2 6295 df-1st 6793 df-2nd 6794 df-map 7474 df-preset 16173 df-poset 16191 df-plt 16204 df-lub 16220 df-glb 16221 df-join 16222 df-meet 16223 df-p0 16285 df-p1 16286 df-lat 16292 df-clat 16354 df-oposet 32742 df-ol 32744 df-oml 32745 df-covers 32832 df-ats 32833 df-atl 32864 df-cvlat 32888 df-hlat 32917 df-llines 33063 df-psubsp 33068 df-pmap 33069 df-padd 33361 df-lhyp 33553 df-laut 33554 df-ldil 33669 df-ltrn 33670 df-trl 33725

This theorem is referenced by: cdlemd7 33770

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