ltrncnv - Mathbox for Norm Megill

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Theorem ltrncnv 35432

Description: The converse of a lattice translation is a lattice translation. (Contributed by NM, 10-May-2013.)

**Hypotheses**
Ref	Expression
ltrncnv.h
ltrncnv.t

**Assertion**
Ref	Expression
ltrncnv	$/\$ $/\$

Proof of Theorem ltrncnv Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ltrncnv.h	. . . 4
2		eqid 2622	. . . 4
3		ltrncnv.t	. . . 4
4	1, 2, 3	ltrnldil 35408	. . 3 $/\$ $/\$
5	1, 2	ldilcnv 35401	. . 3 $/\$ $/\$
6	4, 5	syldan 487	. 2 $/\$ $/\$
7		simp1 1061	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
8		simp1l 1085	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
9		simp1r 1086	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
10		simp2l 1087	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
11		simp3l 1089	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
12		eqid 2622	. . . . . . . 8
13		eqid 2622	. . . . . . . 8
14	12, 13, 1, 3	ltrncnvel 35428	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
15	8, 9, 10, 11, 14	syl112anc 1330	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
16		simp2r 1088	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
17		simp3r 1090	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
18	12, 13, 1, 3	ltrncnvel 35428	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
19	8, 9, 16, 17, 18	syl112anc 1330	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
20		eqid 2622	. . . . . . 7
21		eqid 2622	. . . . . . 7
22	12, 20, 21, 13, 1, 3	ltrnu 35407	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
23	7, 15, 19, 22	syl3anc 1326	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
24		eqid 2622	. . . . . . . . . . 11
25	24, 1, 3	ltrn1o 35410	. . . . . . . . . 10 $/\$ $/\$
26	25	3ad2ant1 1082	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
27	24, 13	atbase 34576	. . . . . . . . . 10
28	10, 27	syl 17	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
29		f1ocnvfv2 6533	. . . . . . . . 9 $/\$
30	26, 28, 29	syl2anc 693	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
31	30	oveq2d 6666	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
32		simp1ll 1124	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
33	12, 13, 1, 3	ltrncnvat 35427	. . . . . . . . 9 $/\$ $/\$ $/\$
34	8, 9, 10, 33	syl3anc 1326	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
35	20, 13	hlatjcom 34654	. . . . . . . 8 $/\$ $/\$
36	32, 34, 10, 35	syl3anc 1326	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
37	31, 36	eqtrd 2656	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
38	37	oveq1d 6665	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
39	24, 13	atbase 34576	. . . . . . . . . 10
40	16, 39	syl 17	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
41		f1ocnvfv2 6533	. . . . . . . . 9 $/\$
42	26, 40, 41	syl2anc 693	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
43	42	oveq2d 6666	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
44	12, 13, 1, 3	ltrncnvat 35427	. . . . . . . . 9 $/\$ $/\$ $/\$
45	8, 9, 16, 44	syl3anc 1326	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
46	20, 13	hlatjcom 34654	. . . . . . . 8 $/\$ $/\$
47	32, 45, 16, 46	syl3anc 1326	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
48	43, 47	eqtrd 2656	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
49	48	oveq1d 6665	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
50	23, 38, 49	3eqtr3d 2664	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
51	50	3exp 1264	. . 3 $/\$ $/\$ $/\$ $/\$
52	51	ralrimivv 2970	. 2 $/\$ $/\$ $/\$
53	12, 20, 21, 13, 1, 2, 3	isltrn 35405	. . 3 $/\$ $/\$ $/\$
54	53	adantr 481	. 2 $/\$ $/\$ $/\$ $/\$
55	6, 52, 54	mpbir2and 957	1 $/\$ $/\$

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4

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

wceq 1483

wcel 1990

wral 2912 class class class wbr 4653

ccnv 5113

wf1o 5887

cfv 5888 (class class class)co 6650

cbs 15857

cple 15948

cjn 16944

cmee 16945

catm 34550

chlt 34637

clh 35270

cldil 35386

cltrn 35387

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

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 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-ral 2917 df-rex 2918 df-reu 2919 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-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 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-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-map 7859 df-preset 16928 df-poset 16946 df-plt 16958 df-lub 16974 df-glb 16975 df-join 16976 df-p0 17039 df-lat 17046 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-lhyp 35274 df-laut 35275 df-ldil 35390 df-ltrn 35391

This theorem is referenced by: trlcnv 35452 trlcocnv 36008 trlcoabs2N 36010 trlcoat 36011 trlcocnvat 36012 trlcone 36016 cdlemg46 36023 tgrpgrplem 36037 tendoicl 36084 cdlemh1 36103 cdlemh2 36104 cdlemh 36105 cdlemi2 36107 cdlemi 36108 cdlemk2 36120 cdlemk3 36121 cdlemk4 36122 cdlemk8 36126 cdlemk9 36127 cdlemk9bN 36128 cdlemkvcl 36130 cdlemk10 36131 cdlemk11 36137 cdlemk12 36138 cdlemk14 36142 cdlemk11u 36159 cdlemk12u 36160 cdlemk37 36202 cdlemkfid1N 36209 cdlemkid1 36210 cdlemkid2 36212 tendocnv 36310 tendospcanN 36312 dvhgrp 36396 cdlemn8 36493 dihopelvalcpre 36537 dih1 36575 dihglbcpreN 36589 dihjatcclem3 36709 dihjatcclem4 36710

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