isltrn2N - Mathbox for Norm Megill

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Theorem isltrn2N 35406

Description: The predicate "is a lattice translation". Version of isltrn 35405 that considers only different

and

. TODO: Can this eliminate some separate proofs for the

case? (Contributed by NM, 22-Apr-2013.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
ltrnset.l
ltrnset.j	$.\/$
ltrnset.m	$./\$
ltrnset.a
ltrnset.h
ltrnset.d
ltrnset.t

**Assertion**
Ref	Expression
isltrn2N	$/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$

Distinct variable groups:

Allowed substitution hints:

(

)

(

)

(

)

(

) $.\/$ (

)

(

) $./\$ (

)

**Proof of Theorem isltrn2N**
Step	Hyp	Ref	Expression
1		ltrnset.l	. . 3
2		ltrnset.j	. . 3 $.\/$
3		ltrnset.m	. . 3 $./\$
4		ltrnset.a	. . 3
5		ltrnset.h	. . 3
6		ltrnset.d	. . 3
7		ltrnset.t	. . 3
8	1, 2, 3, 4, 5, 6, 7	isltrn 35405	. 2 $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$
9		3simpa 1058	. . . . . 6 $/\$ $/\$ $/\$
10	9	imim1i 63	. . . . 5 $/\$ $.\/$ $./\$ $.\/$ $./\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$
11		3anass 1042	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
12		3anrot 1043	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
13		df-ne 2795	. . . . . . . . . 10
14	13	anbi1i 731	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
15	11, 12, 14	3bitr3i 290	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
16	15	imbi1i 339	. . . . . . 7 $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$
17		impexp 462	. . . . . . 7 $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$ $/\$ $.\/$ $./\$ $.\/$ $./\$
18	16, 17	bitri 264	. . . . . 6 $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$ $/\$ $.\/$ $./\$ $.\/$ $./\$
19		id 22	. . . . . . . . . 10
20		fveq2 6191	. . . . . . . . . 10
21	19, 20	oveq12d 6668	. . . . . . . . 9 $.\/$ $.\/$
22	21	oveq1d 6665	. . . . . . . 8 $.\/$ $./\$ $.\/$ $./\$
23	22	a1d 25	. . . . . . 7 $/\$ $.\/$ $./\$ $.\/$ $./\$
24		pm2.61 183	. . . . . . 7 $/\$ $.\/$ $./\$ $.\/$ $./\$ $/\$ $.\/$ $./\$ $.\/$ $./\$ $/\$ $.\/$ $./\$ $.\/$ $./\$
25	23, 24	ax-mp 5	. . . . . 6 $/\$ $.\/$ $./\$ $.\/$ $./\$ $/\$ $.\/$ $./\$ $.\/$ $./\$
26	18, 25	sylbi 207	. . . . 5 $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$ $/\$ $.\/$ $./\$ $.\/$ $./\$
27	10, 26	impbii 199	. . . 4 $/\$ $.\/$ $./\$ $.\/$ $./\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$
28	27	2ralbii 2981	. . 3 $/\$ $.\/$ $./\$ $.\/$ $./\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$
29	28	anbi2i 730	. 2 $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$
30	8, 29	syl6bb 276	1 $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4

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

wceq 1483

wcel 1990

wne 2794

wral 2912 class class class wbr 4653

cfv 5888 (class class class)co 6650

cple 15948

cjn 16944

cmee 16945

catm 34550

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-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-pr 4906

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-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-ov 6653 df-ltrn 35391

This theorem is referenced by: (None)

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