dihglblem5apreN - Mathbox for Norm Megill

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Theorem dihglblem5apreN 36580

Description: A conjunction property of isomorphism H. TODO: reduce antecedent size; general review for shorter proof. (Contributed by NM, 21-Mar-2014.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
dihglblem5a.b
dihglblem5a.m	$./\$
dihglblem5a.h
dihglblem5a.i
dihglblem5a.l
dihglblem5a.j	$.\/$
dihglblem5a.a
dihglblem5a.p
dihglblem5a.t
dihglblem5a.r
dihglblem5a.e
dihglblem5a.g
dihglblem5a.o

**Assertion**
Ref	Expression
dihglblem5apreN	$/\$ $/\$ $/\$ $./\$

Distinct variable groups: $./\$ ,

Allowed substitution hints:

(

)

(

)

(

)

(

)

(

)

(

) $.\/$ (

) $./\$ (

)

(

)

(

)

Proof of Theorem dihglblem5apreN Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		hllat 34650	. . . . . 6
2	1	ad2antrr 762	. . . . 5 $/\$ $/\$ $/\$
3		simprl 794	. . . . 5 $/\$ $/\$ $/\$
4		dihglblem5a.b	. . . . . . 7
5		dihglblem5a.h	. . . . . . 7
6	4, 5	lhpbase 35284	. . . . . 6
7	6	ad2antlr 763	. . . . 5 $/\$ $/\$ $/\$
8		dihglblem5a.l	. . . . . 6
9		dihglblem5a.m	. . . . . 6 $./\$
10	4, 8, 9	latmle1 17076	. . . . 5 $/\$ $/\$ $./\$
11	2, 3, 7, 10	syl3anc 1326	. . . 4 $/\$ $/\$ $/\$ $./\$
12		simpl 473	. . . . 5 $/\$ $/\$ $/\$ $/\$
13	4, 9	latmcl 17052	. . . . . 6 $/\$ $/\$ $./\$
14	2, 3, 7, 13	syl3anc 1326	. . . . 5 $/\$ $/\$ $/\$ $./\$
15		dihglblem5a.i	. . . . . 6
16	4, 8, 5, 15	dihord 36553	. . . . 5 $/\$ $/\$ $./\$ $/\$ $./\$ $./\$
17	12, 14, 3, 16	syl3anc 1326	. . . 4 $/\$ $/\$ $/\$ $./\$ $./\$
18	11, 17	mpbird 247	. . 3 $/\$ $/\$ $/\$ $./\$
19	4, 8, 9	latmle2 17077	. . . . 5 $/\$ $/\$ $./\$
20	2, 3, 7, 19	syl3anc 1326	. . . 4 $/\$ $/\$ $/\$ $./\$
21	4, 8, 5, 15	dihord 36553	. . . . 5 $/\$ $/\$ $./\$ $/\$ $./\$ $./\$
22	12, 14, 7, 21	syl3anc 1326	. . . 4 $/\$ $/\$ $/\$ $./\$ $./\$
23	20, 22	mpbird 247	. . 3 $/\$ $/\$ $/\$ $./\$
24	18, 23	ssind 3837	. 2 $/\$ $/\$ $/\$ $./\$
25	5, 15	dihvalrel 36568	. . . . 5 $/\$
26		relin1 5236	. . . . 5
27	25, 26	syl 17	. . . 4 $/\$
28	27	adantr 481	. . 3 $/\$ $/\$ $/\$
29		elin 3796	. . . 4 $/\$
30		dihglblem5a.j	. . . . . 6 $.\/$
31		dihglblem5a.a	. . . . . 6
32	4, 8, 30, 9, 31, 5	lhpmcvr2 35310	. . . . 5 $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$
33		dihglblem5a.p	. . . . . . . . . . . 12
34		dihglblem5a.t	. . . . . . . . . . . 12
35		dihglblem5a.r	. . . . . . . . . . . 12
36		dihglblem5a.e	. . . . . . . . . . . 12
37		dihglblem5a.g	. . . . . . . . . . . 12
38		vex 3203	. . . . . . . . . . . 12
39		vex 3203	. . . . . . . . . . . 12
40	4, 8, 30, 9, 31, 5, 33, 34, 35, 36, 15, 37, 38, 39	dihopelvalc 36538	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $/\$
41		id 22	. . . . . . . . . . . . 13 $/\$ $/\$
42	6	adantl 482	. . . . . . . . . . . . 13 $/\$
43	4, 8	latref 17053	. . . . . . . . . . . . . 14 $/\$
44	1, 6, 43	syl2an 494	. . . . . . . . . . . . 13 $/\$
45		dihglblem5a.o	. . . . . . . . . . . . . 14
46	4, 8, 5, 34, 35, 45, 15	dihopelvalbN 36527	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$
47	41, 42, 44, 46	syl12anc 1324	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
48	47	3ad2ant1 1082	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $/\$
49	40, 48	anbi12d 747	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
50		simprll 802	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$
51	50	adantl 482	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
52		simprrr 805	. . . . . . . . . . . . . . . . . . . . . 22 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
53	52	fveq1d 6193	. . . . . . . . . . . . . . . . . . . . 21 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
54		simpl1 1064	. . . . . . . . . . . . . . . . . . . . . . 23 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
55	8, 31, 5, 33	lhpocnel2 35305	. . . . . . . . . . . . . . . . . . . . . . . 24 $/\$ $/\$
56	54, 55	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
57		simpl3l 1116	. . . . . . . . . . . . . . . . . . . . . . 23 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
58	8, 31, 5, 34, 37	ltrniotacl 35867	. . . . . . . . . . . . . . . . . . . . . . 23 $/\$ $/\$ $/\$ $/\$ $/\$
59	54, 56, 57, 58	syl3anc 1326	. . . . . . . . . . . . . . . . . . . . . 22 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
60	45, 4	tendo02 36075	. . . . . . . . . . . . . . . . . . . . . 22
61	59, 60	syl 17	. . . . . . . . . . . . . . . . . . . . 21 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
62	53, 61	eqtrd 2656	. . . . . . . . . . . . . . . . . . . 20 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
63	62	cnveqd 5298	. . . . . . . . . . . . . . . . . . 19 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
64		cnvresid 5968	. . . . . . . . . . . . . . . . . . 19
65	63, 64	syl6eq 2672	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
66	65	coeq2d 5284	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
67	4, 5, 34	ltrn1o 35410	. . . . . . . . . . . . . . . . . . 19 $/\$ $/\$
68	54, 51, 67	syl2anc 693	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
69		f1of 6137	. . . . . . . . . . . . . . . . . 18
70		fcoi1 6078	. . . . . . . . . . . . . . . . . 18
71	68, 69, 70	3syl 18	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
72	66, 71	eqtrd 2656	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
73	72	fveq2d 6195	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
74		simprlr 803	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
75	73, 74	eqbrtrrd 4677	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
76	8, 5, 34, 35	trlle 35471	. . . . . . . . . . . . . . 15 $/\$ $/\$
77	54, 51, 76	syl2anc 693	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
78		simpl1l 1112	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
79	78, 1	syl 17	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
80	4, 5, 34, 35	trlcl 35451	. . . . . . . . . . . . . . . 16 $/\$ $/\$
81	54, 51, 80	syl2anc 693	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
82		simpl2l 1114	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
83		simpl1r 1113	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
84	83, 6	syl 17	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
85	4, 8, 9	latlem12 17078	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $./\$
86	79, 81, 82, 84, 85	syl13anc 1328	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$
87	75, 77, 86	mpbi2and 956	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$
88	51, 87	jca 554	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$
89	79, 82, 84, 13	syl3anc 1326	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$
90	79, 82, 84, 19	syl3anc 1326	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$
91	4, 8, 5, 34, 35, 45, 15	dihopelvalbN 36527	. . . . . . . . . . . . 13 $/\$ $/\$ $./\$ $/\$ $./\$ $./\$ $/\$ $./\$ $/\$
92	54, 89, 90, 91	syl12anc 1324	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$ $/\$ $./\$ $/\$
93	88, 52, 92	mpbir2and 957	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$
94	93	ex 450	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $/\$ $/\$ $./\$
95	49, 94	sylbid 230	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $./\$
96	95	3expia 1267	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $./\$
97	96	exp4c 636	. . . . . . 7 $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $./\$
98	97	imp4a 614	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $./\$
99	98	rexlimdv 3030	. . . . 5 $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $/\$ $./\$
100	32, 99	mpd 15	. . . 4 $/\$ $/\$ $/\$ $/\$ $./\$
101	29, 100	syl5bi 232	. . 3 $/\$ $/\$ $/\$ $./\$
102	28, 101	relssdv 5212	. 2 $/\$ $/\$ $/\$ $./\$
103	24, 102	eqssd 3620	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

cin 3573

wss 3574

cop 4183 class class class wbr 4653

cmpt 4729

cid 5023

ccnv 5113

cres 5116

ccom 5118

wrel 5119

wf 5884

wf1o 5887

cfv 5888

crio 6610 (class class class)co 6650

cbs 15857

cple 15948

coc 15949

cjn 16944

cmee 16945

clat 17045

catm 34550

chlt 34637

clh 35270

cltrn 35387

ctrl 35445

ctendo 36040

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: dihglblem5aN 36581

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