cdlemd6 - Mathbox for Norm Megill

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Theorem cdlemd6 33535

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 985	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$
2	1	oveq2d 6106	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$
3	2	oveq1d 6105	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$
4		simp1l 1007	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$
5		simp1rl 1048	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$
6		simp21 1016	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$
7		cdlemd4.l	. . . . . . 7
8		cdlemd4.j	. . . . . . 7 $.\/$
9		eqid 2441	. . . . . . 7
10		cdlemd4.a	. . . . . . 7
11		cdlemd4.h	. . . . . . 7
12		cdlemd4.t	. . . . . . 7
13		eqid 2441	. . . . . . 7
14	7, 8, 9, 10, 11, 12, 13	trlval2 33495	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $.\/$
15	4, 5, 6, 14	syl3anc 1213	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
16		simp1rr 1049	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$
17	7, 8, 9, 10, 11, 12, 13	trlval2 33495	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $.\/$
18	4, 16, 6, 17	syl3anc 1213	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
19	3, 15, 18	3eqtr4d 2483	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$
20	19	oveq2d 6106	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$
21	1	oveq1d 6105	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$
22	20, 21	oveq12d 6108	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$
23		simp22 1017	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$
24		simp23 1018	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$
25	7, 8, 9, 10, 11, 12, 13	cdlemc 33529	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$
26	4, 5, 6, 23, 24, 25	syl131anc 1226	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$
27		oveq2 6098	. . . . . . 7 $.\/$ $.\/$
28	27	breq2d 4301	. . . . . 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 33529	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$
33	4, 16, 6, 23, 31, 32	syl131anc 1226	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$
34	22, 26, 33	3eqtr4d 2483	1 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4 $/\$ wa 369 $/\$ w3a 960

wceq 1364

wcel 1761 class class class wbr 4289

cfv 5415 (class class class)co 6090

cple 14241

cjn 15110

cmee 15111

catm 32596

chlt 32683

clh 33316

cltrn 33433

ctrl 33490

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1596 ax-4 1607 ax-5 1675 ax-6 1713 ax-7 1733 ax-8 1763 ax-9 1765 ax-10 1780 ax-11 1785 ax-12 1797 ax-13 1948 ax-ext 2422 ax-rep 4400 ax-sep 4410 ax-nul 4418 ax-pow 4467 ax-pr 4528 ax-un 6371

This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3an 962 df-tru 1367 df-ex 1592 df-nf 1595 df-sb 1706 df-eu 2263 df-mo 2264 df-clab 2428 df-cleq 2434 df-clel 2437 df-nfc 2566 df-ne 2606 df-ral 2718 df-rex 2719 df-reu 2720 df-rab 2722 df-v 2972 df-sbc 3184 df-csb 3286 df-dif 3328 df-un 3330 df-in 3332 df-ss 3339 df-nul 3635 df-if 3789 df-pw 3859 df-sn 3875 df-pr 3877 df-op 3881 df-uni 4089 df-iun 4170 df-iin 4171 df-br 4290 df-opab 4348 df-mpt 4349 df-id 4632 df-xp 4842 df-rel 4843 df-cnv 4844 df-co 4845 df-dm 4846 df-rn 4847 df-res 4848 df-ima 4849 df-iota 5378 df-fun 5417 df-fn 5418 df-f 5419 df-f1 5420 df-fo 5421 df-f1o 5422 df-fv 5423 df-riota 6049 df-ov 6093 df-oprab 6094 df-mpt2 6095 df-1st 6576 df-2nd 6577 df-map 7212 df-poset 15112 df-plt 15124 df-lub 15140 df-glb 15141 df-join 15142 df-meet 15143 df-p0 15205 df-p1 15206 df-lat 15212 df-clat 15274 df-oposet 32509 df-ol 32511 df-oml 32512 df-covers 32599 df-ats 32600 df-atl 32631 df-cvlat 32655 df-hlat 32684 df-llines 32830 df-psubsp 32835 df-pmap 32836 df-padd 33128 df-lhyp 33320 df-laut 33321 df-ldil 33436 df-ltrn 33437 df-trl 33491

This theorem is referenced by: cdlemd7 33536

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