cdlemg42 - Mathbox for Norm Megill

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Theorem cdlemg42 34679

Description: Part of proof of Lemma G of [Crawley] p. 116, first line of third paragraph on p. 117. (Contributed by NM, 3-Jun-2013.)

**Hypotheses**
Ref	Expression
cdlemg42.l
cdlemg42.j	$.\/$
cdlemg42.a
cdlemg42.h
cdlemg42.t
cdlemg42.r

**Assertion**
Ref	Expression
cdlemg42	$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$

**Proof of Theorem cdlemg42**
Step	Hyp	Ref	Expression
1		simp33 1026	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
2		simpl1l 1039	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
3		simp31l 1111	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
4	3	adantr 465	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
5		simp1 988	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
6		simp2l 1014	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
7		cdlemg42.l	. . . . . . . . . . . 12
8		cdlemg42.a	. . . . . . . . . . . 12
9		cdlemg42.h	. . . . . . . . . . . 12
10		cdlemg42.t	. . . . . . . . . . . 12
11	7, 8, 9, 10	ltrnat 34090	. . . . . . . . . . 11 $/\$ $/\$ $/\$
12	5, 6, 3, 11	syl3anc 1219	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
13	12	adantr 465	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
14		cdlemg42.j	. . . . . . . . . 10 $.\/$
15	7, 14, 8	hlatlej1 33325	. . . . . . . . 9 $/\$ $/\$ $.\/$
16	2, 4, 13, 15	syl3anc 1219	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$
17		simpr 461	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$
18		hllat 33314	. . . . . . . . . 10
19	2, 18	syl 16	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
20		eqid 2451	. . . . . . . . . . 11
21	20, 8	atbase 33240	. . . . . . . . . 10
22	4, 21	syl 16	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
23		simp2r 1015	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
24	7, 8, 9, 10	ltrnat 34090	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
25	5, 23, 3, 24	syl3anc 1219	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
26	25	adantr 465	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
27	20, 8	atbase 33240	. . . . . . . . . 10
28	26, 27	syl 16	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
29	20, 14, 8	hlatjcl 33317	. . . . . . . . . 10 $/\$ $/\$ $.\/$
30	2, 4, 13, 29	syl3anc 1219	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$
31	20, 7, 14	latjle12 15334	. . . . . . . . 9 $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$
32	19, 22, 28, 30, 31	syl13anc 1221	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$
33	16, 17, 32	mpbi2and 912	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$
34		simpl32 1070	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
35	34	necomd 2719	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
36	7, 14, 8	ps-1 33427	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$
37	2, 4, 26, 35, 4, 13, 36	syl132anc 1237	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$
38	33, 37	mpbid 210	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$
39	38	oveq1d 6205	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$
40		simpl1 991	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$
41		simpl2r 1042	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
42		simpl31 1069	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$
43		eqid 2451	. . . . . . 7
44		cdlemg42.r	. . . . . . 7
45	7, 14, 43, 8, 9, 10, 44	trlval2 34113	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $.\/$
46	40, 41, 42, 45	syl3anc 1219	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$
47		simpl2l 1041	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
48	7, 14, 43, 8, 9, 10, 44	trlval2 34113	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $.\/$
49	40, 47, 42, 48	syl3anc 1219	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$
50	39, 46, 49	3eqtr4rd 2503	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
51	50	ex 434	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
52	51	necon3ad 2658	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
53	1, 52	mpd 15	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

cmee 15217

clat 15317

catm 33214

chlt 33301

clh 33934

cltrn 34051

ctrl 34108

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-mpt2 6195 df-map 7316 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-oposet 33127 df-ol 33129 df-oml 33130 df-covers 33217 df-ats 33218 df-atl 33249 df-cvlat 33273 df-hlat 33302 df-lhyp 33938 df-laut 33939 df-ldil 34054 df-ltrn 34055 df-trl 34109

This theorem is referenced by: cdlemg43 34680

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