ltrnfset - Mathbox for Norm Megill

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Theorem ltrnfset 35403

Description: The set of all lattice translations for a lattice

. (Contributed by NM, 11-May-2012.)

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

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

Distinct variable groups:

Allowed substitution hints:

(

)

(

)

(

) $.\/$ (

)

(

) $./\$ (

)

Proof of Theorem ltrnfset Dummy variable

is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3212	. 2
2		fveq2 6191	. . . . 5
3		ltrnset.h	. . . . 5
4	2, 3	syl6eqr 2674	. . . 4
5		fveq2 6191	. . . . . 6
6	5	fveq1d 6193	. . . . 5
7		fveq2 6191	. . . . . . 7
8		ltrnset.a	. . . . . . 7
9	7, 8	syl6eqr 2674	. . . . . 6
10		fveq2 6191	. . . . . . . . . . . 12
11		ltrnset.l	. . . . . . . . . . . 12
12	10, 11	syl6eqr 2674	. . . . . . . . . . 11
13	12	breqd 4664	. . . . . . . . . 10
14	13	notbid 308	. . . . . . . . 9
15	12	breqd 4664	. . . . . . . . . 10
16	15	notbid 308	. . . . . . . . 9
17	14, 16	anbi12d 747	. . . . . . . 8 $/\$ $/\$
18		fveq2 6191	. . . . . . . . . . 11
19		ltrnset.m	. . . . . . . . . . 11 $./\$
20	18, 19	syl6eqr 2674	. . . . . . . . . 10 $./\$
21		fveq2 6191	. . . . . . . . . . . 12
22		ltrnset.j	. . . . . . . . . . . 12 $.\/$
23	21, 22	syl6eqr 2674	. . . . . . . . . . 11 $.\/$
24	23	oveqd 6667	. . . . . . . . . 10 $.\/$
25		eqidd 2623	. . . . . . . . . 10
26	20, 24, 25	oveq123d 6671	. . . . . . . . 9 $.\/$ $./\$
27	23	oveqd 6667	. . . . . . . . . 10 $.\/$
28	20, 27, 25	oveq123d 6671	. . . . . . . . 9 $.\/$ $./\$
29	26, 28	eqeq12d 2637	. . . . . . . 8 $.\/$ $./\$ $.\/$ $./\$
30	17, 29	imbi12d 334	. . . . . . 7 $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$
31	9, 30	raleqbidv 3152	. . . . . 6 $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$
32	9, 31	raleqbidv 3152	. . . . 5 $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$
33	6, 32	rabeqbidv 3195	. . . 4 $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$
34	4, 33	mpteq12dv 4733	. . 3 $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$
35		df-ltrn 35391	. . 3 $/\$
36		fvex 6201	. . . . 5
37	3, 36	eqeltri 2697	. . . 4
38	37	mptex 6486	. . 3 $/\$ $.\/$ $./\$ $.\/$ $./\$
39	34, 35, 38	fvmpt 6282	. 2 $/\$ $.\/$ $./\$ $.\/$ $./\$
40	1, 39	syl 17	1 $/\$ $.\/$ $./\$ $.\/$ $./\$

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4 $/\$ wa 384

wceq 1483

wcel 1990

wral 2912

crab 2916

cvv 3200 class class class wbr 4653

cmpt 4729

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: ltrnset 35404

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