ltrnu - Mathbox for Norm Megill

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Theorem ltrnu 35407

Description: Uniqueness property of a lattice translation value for atoms not under the fiducial co-atom

. Similar to definition of translation in [Crawley] p. 111. (Contributed by NM, 20-May-2012.)

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

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

Proof of Theorem ltrnu Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		an4 865	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
2		simpr 477	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
3		simplr 792	. . . . . 6 $/\$ $/\$ $/\$ $/\$
4		ltrnu.l	. . . . . . . . 9
5		ltrnu.j	. . . . . . . . 9 $.\/$
6		ltrnu.m	. . . . . . . . 9 $./\$
7		ltrnu.a	. . . . . . . . 9
8		ltrnu.h	. . . . . . . . 9
9		eqid 2622	. . . . . . . . 9
10		ltrnu.t	. . . . . . . . 9
11	4, 5, 6, 7, 8, 9, 10	isltrn 35405	. . . . . . . 8 $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$
12	11	ad2antrr 762	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$
13		simpr 477	. . . . . . 7 $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$ $/\$ $.\/$ $./\$ $.\/$ $./\$
14	12, 13	syl6bi 243	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$
15	3, 14	mpd 15	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$
16		breq1 4656	. . . . . . . . 9
17	16	notbid 308	. . . . . . . 8
18	17	anbi1d 741	. . . . . . 7 $/\$ $/\$
19		id 22	. . . . . . . . . 10
20		fveq2 6191	. . . . . . . . . 10
21	19, 20	oveq12d 6668	. . . . . . . . 9 $.\/$ $.\/$
22	21	oveq1d 6665	. . . . . . . 8 $.\/$ $./\$ $.\/$ $./\$
23	22	eqeq1d 2624	. . . . . . 7 $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$
24	18, 23	imbi12d 334	. . . . . 6 $/\$ $.\/$ $./\$ $.\/$ $./\$ $/\$ $.\/$ $./\$ $.\/$ $./\$
25		breq1 4656	. . . . . . . . 9
26	25	notbid 308	. . . . . . . 8
27	26	anbi2d 740	. . . . . . 7 $/\$ $/\$
28		id 22	. . . . . . . . . 10
29		fveq2 6191	. . . . . . . . . 10
30	28, 29	oveq12d 6668	. . . . . . . . 9 $.\/$ $.\/$
31	30	oveq1d 6665	. . . . . . . 8 $.\/$ $./\$ $.\/$ $./\$
32	31	eqeq2d 2632	. . . . . . 7 $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$ $.\/$ $./\$
33	27, 32	imbi12d 334	. . . . . 6 $/\$ $.\/$ $./\$ $.\/$ $./\$ $/\$ $.\/$ $./\$ $.\/$ $./\$
34	24, 33	rspc2v 3322	. . . . 5 $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$ $/\$ $.\/$ $./\$ $.\/$ $./\$
35	2, 15, 34	sylc 65	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$
36	35	impr 649	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$
37	1, 36	sylan2b 492	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $./\$ $.\/$ $./\$
38	37	3impb 1260	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

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: ltrncnv 35432 trlval2 35450 cdlemg14f 35941 cdlemg14g 35942

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