dihord5apre - Mathbox for Norm Megill

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Theorem dihord5apre 36551

Description: Part of proof that isomorphism H is order-preserving . (Contributed by NM, 7-Mar-2014.)

**Hypotheses**
Ref	Expression
dihord5apre.b
dihord5apre.l
dihord5apre.h
dihord5apre.j	$.\/$
dihord5apre.m	$./\$
dihord5apre.a
dihord5apre.u
dihord5apre.s
dihord5apre.i

**Assertion**
Ref	Expression
dihord5apre	$/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Proof of Theorem dihord5apre Dummy variable

is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl1 1064	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
2		simpl3 1066	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
3		dihord5apre.b	. . . 4
4		dihord5apre.l	. . . 4
5		dihord5apre.j	. . . 4 $.\/$
6		dihord5apre.m	. . . 4 $./\$
7		dihord5apre.a	. . . 4
8		dihord5apre.h	. . . 4
9	3, 4, 5, 6, 7, 8	lhpmcvr2 35310	. . 3 $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$
10	1, 2, 9	syl2anc 693	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$
11		simp11l 1172	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$
12		hllat 34650	. . . . . . . 8
13	11, 12	syl 17	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$
14		simp12l 1174	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$
15		simp3ll 1132	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$
16	3, 7	atbase 34576	. . . . . . . . 9
17	15, 16	syl 17	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$
18	3, 5	latjcl 17051	. . . . . . . 8 $/\$ $/\$ $.\/$
19	13, 17, 14, 18	syl3anc 1326	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$
20		simp13l 1176	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$
21	3, 4, 5	latlej2 17061	. . . . . . . 8 $/\$ $/\$ $.\/$
22	13, 17, 14, 21	syl3anc 1326	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$
23		simp11 1091	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$
24		simp3lr 1133	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$
25	3, 4, 5	latlej1 17060	. . . . . . . . . . . . 13 $/\$ $/\$ $.\/$
26	13, 17, 14, 25	syl3anc 1326	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$
27		simp11r 1173	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$
28	3, 8	lhpbase 35284	. . . . . . . . . . . . . 14
29	27, 28	syl 17	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$
30	3, 4	lattr 17056	. . . . . . . . . . . . 13 $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $.\/$
31	13, 17, 19, 29, 30	syl13anc 1328	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $/\$ $.\/$
32	26, 31	mpand 711	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$
33	24, 32	mtod 189	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$
34		simp3l 1089	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$
35		simp12 1092	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$
36	3, 4, 5, 6, 7, 8	lhple 35328	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$
37	23, 34, 35, 36	syl3anc 1326	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$
38	37	oveq2d 6666	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $.\/$
39		dihord5apre.i	. . . . . . . . . . 11
40		eqid 2622	. . . . . . . . . . 11
41		eqid 2622	. . . . . . . . . . 11
42		dihord5apre.u	. . . . . . . . . . 11
43		dihord5apre.s	. . . . . . . . . . 11
44	3, 4, 5, 6, 7, 8, 39, 40, 41, 42, 43	dihvalcq 36525	. . . . . . . . . 10 $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $.\/$ $./\$
45	23, 19, 33, 34, 38, 44	syl122anc 1335	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $.\/$ $./\$
46	8, 42, 23	dvhlmod 36399	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$
47		eqid 2622	. . . . . . . . . . . . . . 15
48	47	lsssssubg 18958	. . . . . . . . . . . . . 14 SubGrp
49	46, 48	syl 17	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ SubGrp
50	4, 7, 8, 42, 41, 47	diclss 36482	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
51	23, 34, 50	syl2anc 693	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$
52	49, 51	sseldd 3604	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ SubGrp
53	3, 6	latmcl 17052	. . . . . . . . . . . . . . 15 $/\$ $/\$ $./\$
54	13, 20, 29, 53	syl3anc 1326	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $./\$
55	3, 4, 6	latmle2 17077	. . . . . . . . . . . . . . 15 $/\$ $/\$ $./\$
56	13, 20, 29, 55	syl3anc 1326	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $./\$
57	3, 4, 8, 42, 40, 47	diblss 36459	. . . . . . . . . . . . . 14 $/\$ $/\$ $./\$ $/\$ $./\$ $./\$
58	23, 54, 56, 57	syl12anc 1324	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $./\$
59	49, 58	sseldd 3604	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $./\$ SubGrp
60	43	lsmub1 18071	. . . . . . . . . . . 12 SubGrp $/\$ $./\$ SubGrp $./\$
61	52, 59, 60	syl2anc 693	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $./\$
62		simp13 1093	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$
63		simp3r 1090	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$
64	3, 4, 5, 6, 7, 8, 39, 40, 41, 42, 43	dihvalcq 36525	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $./\$
65	23, 62, 34, 63, 64	syl112anc 1330	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $./\$
66	61, 65	sseqtr4d 3642	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$
67	37	fveq2d 6195	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$
68	3, 4, 8, 39, 40	dihvalb 36526	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$
69	23, 35, 68	syl2anc 693	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$
70	67, 69	eqtr4d 2659	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$
71		simp2 1062	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$
72	70, 71	eqsstrd 3639	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$
73	3, 6	latmcl 17052	. . . . . . . . . . . . . 14 $/\$ $.\/$ $/\$ $.\/$ $./\$
74	13, 19, 29, 73	syl3anc 1326	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$
75	3, 4, 6	latmle2 17077	. . . . . . . . . . . . . 14 $/\$ $.\/$ $/\$ $.\/$ $./\$
76	13, 19, 29, 75	syl3anc 1326	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$
77	3, 4, 8, 42, 40, 47	diblss 36459	. . . . . . . . . . . . 13 $/\$ $/\$ $.\/$ $./\$ $/\$ $.\/$ $./\$ $.\/$ $./\$
78	23, 74, 76, 77	syl12anc 1324	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$
79	49, 78	sseldd 3604	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$ SubGrp
80	3, 8, 39, 42, 47	dihlss 36539	. . . . . . . . . . . . 13 $/\$ $/\$
81	23, 20, 80	syl2anc 693	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$
82	49, 81	sseldd 3604	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ SubGrp
83	43	lsmlub 18078	. . . . . . . . . . 11 SubGrp $/\$ $.\/$ $./\$ SubGrp $/\$ SubGrp $/\$ $.\/$ $./\$ $.\/$ $./\$
84	52, 79, 82, 83	syl3anc 1326	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $.\/$ $./\$ $.\/$ $./\$
85	66, 72, 84	mpbi2and 956	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$
86	45, 85	eqsstrd 3639	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$
87	3, 4, 8, 39	dihord4 36547	. . . . . . . . 9 $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$
88	23, 19, 33, 62, 87	syl121anc 1331	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $.\/$
89	86, 88	mpbid 222	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$
90	3, 4, 13, 14, 19, 20, 22, 89	lattrd 17058	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$
91	90	3expia 1267	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$
92	91	exp4c 636	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$
93	92	imp4a 614	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$
94	93	rexlimdv 3030	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$
95	10, 94	mpd 15	1 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4

wb 196 $/\$ wa 384 $/\$ w3a 1037

wceq 1483

wcel 1990

wrex 2913

wss 3574 class class class wbr 4653

cfv 5888 (class class class)co 6650

cbs 15857

cple 15948

cjn 16944

cmee 16945

clat 17045 SubGrpcsubg 17588

clsm 18049

clmod 18863

clss 18932

catm 34550

chlt 34637

clh 35270

cdvh 36367

cdib 36427

cdic 36461

cdih 36517

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-riotaBAD 34239

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-tpos 7352 df-undef 7399 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-sca 15957 df-vsca 15958 df-0g 16102 df-preset 16928 df-poset 16946 df-plt 16958 df-lub 16974 df-glb 16975 df-join 16976 df-meet 16977 df-p0 17039 df-p1 17040 df-lat 17046 df-clat 17108 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-grp 17425 df-minusg 17426 df-sbg 17427 df-subg 17591 df-cntz 17750 df-lsm 18051 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-ring 18549 df-oppr 18623 df-dvdsr 18641 df-unit 18642 df-invr 18672 df-dvr 18683 df-drng 18749 df-lmod 18865 df-lss 18933 df-lsp 18972 df-lvec 19103 df-oposet 34463 df-ol 34465 df-oml 34466 df-covers 34553 df-ats 34554 df-atl 34585 df-cvlat 34609 df-hlat 34638 df-llines 34784 df-lplanes 34785 df-lvols 34786 df-lines 34787 df-psubsp 34789 df-pmap 34790 df-padd 35082 df-lhyp 35274 df-laut 35275 df-ldil 35390 df-ltrn 35391 df-trl 35446 df-tendo 36043 df-edring 36045 df-disoa 36318 df-dvech 36368 df-dib 36428 df-dic 36462 df-dih 36518

This theorem is referenced by: dihord5a 36552

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