cdlemg42 - Mathbox for Norm Megill

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Theorem cdlemg42 34341

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 1052	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
2		simpl1l 1065	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
3		simp31l 1137	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
4	3	adantr 471	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
5		simp1 1014	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
6		simp2l 1040	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
7		cdlemg42.l	. . . . . . . . . . . 12
8		cdlemg42.a	. . . . . . . . . . . 12
9		cdlemg42.h	. . . . . . . . . . . 12
10		cdlemg42.t	. . . . . . . . . . . 12
11	7, 8, 9, 10	ltrnat 33750	. . . . . . . . . . 11 $/\$ $/\$ $/\$
12	5, 6, 3, 11	syl3anc 1276	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
13	12	adantr 471	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
14		cdlemg42.j	. . . . . . . . . 10 $.\/$
15	7, 14, 8	hlatlej1 32985	. . . . . . . . 9 $/\$ $/\$ $.\/$
16	2, 4, 13, 15	syl3anc 1276	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$
17		simpr 467	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$
18		hllat 32974	. . . . . . . . . 10
19	2, 18	syl 17	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
20		eqid 2462	. . . . . . . . . . 11
21	20, 8	atbase 32900	. . . . . . . . . 10
22	4, 21	syl 17	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
23		simp2r 1041	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
24	7, 8, 9, 10	ltrnat 33750	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
25	5, 23, 3, 24	syl3anc 1276	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
26	25	adantr 471	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
27	20, 8	atbase 32900	. . . . . . . . . 10
28	26, 27	syl 17	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
29	20, 14, 8	hlatjcl 32977	. . . . . . . . . 10 $/\$ $/\$ $.\/$
30	2, 4, 13, 29	syl3anc 1276	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$
31	20, 7, 14	latjle12 16357	. . . . . . . . 9 $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$
32	19, 22, 28, 30, 31	syl13anc 1278	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $/\$ $.\/$ $.\/$ $.\/$
33	16, 17, 32	mpbi2and 937	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$
34		simpl32 1096	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
35	34	necomd 2691	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
36	7, 14, 8	ps-1 33087	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$
37	2, 4, 26, 35, 4, 13, 36	syl132anc 1294	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $.\/$ $.\/$
38	33, 37	mpbid 215	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$
39	38	oveq1d 6330	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$
40		simpl1 1017	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$
41		simpl2r 1068	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
42		simpl31 1095	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$
43		eqid 2462	. . . . . . 7
44		cdlemg42.r	. . . . . . 7
45	7, 14, 43, 8, 9, 10, 44	trlval2 33774	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $.\/$
46	40, 41, 42, 45	syl3anc 1276	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$
47		simpl2l 1067	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
48	7, 14, 43, 8, 9, 10, 44	trlval2 33774	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $.\/$
49	40, 47, 42, 48	syl3anc 1276	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $.\/$
50	39, 46, 49	3eqtr4rd 2507	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
51	50	ex 440	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
52	51	necon3ad 2649	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$
53	1, 52	mpd 15	1 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4

wb 189 $/\$ wa 375 $/\$ w3a 991

wceq 1455

wcel 1898

wne 2633 class class class wbr 4416

cfv 5601 (class class class)co 6315

cbs 15170

cple 15246

cjn 16238

cmee 16239

clat 16340

catm 32874

chlt 32961

clh 33594

cltrn 33711

ctrl 33769

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1680 ax-4 1693 ax-5 1769 ax-6 1816 ax-7 1862 ax-8 1900 ax-9 1907 ax-10 1926 ax-11 1931 ax-12 1944 ax-13 2102 ax-ext 2442 ax-rep 4529 ax-sep 4539 ax-nul 4548 ax-pow 4595 ax-pr 4653 ax-un 6610

This theorem depends on definitions: df-bi 190 df-or 376 df-an 377 df-3an 993 df-tru 1458 df-ex 1675 df-nf 1679 df-sb 1809 df-eu 2314 df-mo 2315 df-clab 2449 df-cleq 2455 df-clel 2458 df-nfc 2592 df-ne 2635 df-ral 2754 df-rex 2755 df-reu 2756 df-rab 2758 df-v 3059 df-sbc 3280 df-csb 3376 df-dif 3419 df-un 3421 df-in 3423 df-ss 3430 df-nul 3744 df-if 3894 df-pw 3965 df-sn 3981 df-pr 3983 df-op 3987 df-uni 4213 df-iun 4294 df-br 4417 df-opab 4476 df-mpt 4477 df-id 4768 df-xp 4859 df-rel 4860 df-cnv 4861 df-co 4862 df-dm 4863 df-rn 4864 df-res 4865 df-ima 4866 df-iota 5565 df-fun 5603 df-fn 5604 df-f 5605 df-f1 5606 df-fo 5607 df-f1o 5608 df-fv 5609 df-riota 6277 df-ov 6318 df-oprab 6319 df-mpt2 6320 df-map 7500 df-preset 16222 df-poset 16240 df-plt 16253 df-lub 16269 df-glb 16270 df-join 16271 df-meet 16272 df-p0 16334 df-lat 16341 df-oposet 32787 df-ol 32789 df-oml 32790 df-covers 32877 df-ats 32878 df-atl 32909 df-cvlat 32933 df-hlat 32962 df-lhyp 33598 df-laut 33599 df-ldil 33714 df-ltrn 33715 df-trl 33770

This theorem is referenced by: cdlemg43 34342

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