cdlemd3 - Mathbox for Norm Megill

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Theorem cdlemd3 34150

Description: Part of proof of Lemma D in [Crawley] p. 113. The

requirement is not mentioned in their proof. (Contributed by NM, 29-May-2012.)

**Hypotheses**
Ref	Expression
cdlemd3.l
cdlemd3.j	$.\/$
cdlemd3.a
cdlemd3.h

**Assertion**
Ref	Expression
cdlemd3	$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$

**Proof of Theorem cdlemd3**
Step	Hyp	Ref	Expression
1		simp33 1026	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$
2		simp1l 1012	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$
3		simp31 1024	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$
4		simp32 1025	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$
5		simp21l 1105	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$
6		simp233 1134	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$
7		cdlemd3.l	. . . . 5
8		cdlemd3.j	. . . . 5 $.\/$
9		cdlemd3.a	. . . . 5
10	7, 8, 9	hlatexch1 33345	. . . 4 $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$
11	2, 3, 4, 5, 6, 10	syl131anc 1232	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$
12		simp22l 1107	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$
13	7, 8, 9	hlatlej1 33325	. . . . . 6 $/\$ $/\$ $.\/$
14	2, 5, 12, 13	syl3anc 1219	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$
15		simp232 1133	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$
16		hllat 33314	. . . . . . 7
17	2, 16	syl 16	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$
18		eqid 2451	. . . . . . . 8
19	18, 9	atbase 33240	. . . . . . 7
20	5, 19	syl 16	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$
21	18, 9	atbase 33240	. . . . . . 7
22	3, 21	syl 16	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$
23	18, 9	atbase 33240	. . . . . . . 8
24	12, 23	syl 16	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$
25	18, 8	latjcl 15323	. . . . . . 7 $/\$ $/\$ $.\/$
26	17, 20, 24, 25	syl3anc 1219	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$
27	18, 7, 8	latjle12 15334	. . . . . 6 $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$
28	17, 20, 22, 26, 27	syl13anc 1221	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$
29	14, 15, 28	mpbi2and 912	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$
30	18, 9	atbase 33240	. . . . . 6
31	4, 30	syl 16	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$
32	18, 8	latjcl 15323	. . . . . 6 $/\$ $/\$ $.\/$
33	17, 20, 22, 32	syl3anc 1219	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$
34	18, 7	lattr 15328	. . . . 5 $/\$ $/\$ $.\/$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$
35	17, 31, 33, 26, 34	syl13anc 1221	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$
36	29, 35	mpan2d 674	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$
37	11, 36	syld 44	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$
38	1, 37	mtod 177	1 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4

wb 184 $/\$ wa 369 $/\$ w3a 965

wceq 1370

wcel 1758

wne 2644 class class class wbr 4390

cfv 5516 (class class class)co 6190

cbs 14276

cple 14347

cjn 15216

clat 15317

catm 33214

chlt 33301

clh 33934

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 4501 ax-sep 4511 ax-nul 4519 ax-pow 4568 ax-pr 4629 ax-un 6472

This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 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-ral 2800 df-rex 2801 df-reu 2802 df-rab 2804 df-v 3070 df-sbc 3285 df-csb 3387 df-dif 3429 df-un 3431 df-in 3433 df-ss 3440 df-nul 3736 df-if 3890 df-pw 3960 df-sn 3976 df-pr 3978 df-op 3982 df-uni 4190 df-iun 4271 df-br 4391 df-opab 4449 df-mpt 4450 df-id 4734 df-xp 4944 df-rel 4945 df-cnv 4946 df-co 4947 df-dm 4948 df-rn 4949 df-res 4950 df-ima 4951 df-iota 5479 df-fun 5518 df-fn 5519 df-f 5520 df-f1 5521 df-fo 5522 df-f1o 5523 df-fv 5524 df-riota 6151 df-ov 6193 df-oprab 6194 df-poset 15218 df-plt 15230 df-lub 15246 df-glb 15247 df-join 15248 df-meet 15249 df-p0 15311 df-lat 15318 df-covers 33217 df-ats 33218 df-atl 33249 df-cvlat 33273 df-hlat 33302

This theorem is referenced by: cdlemd4 34151

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