Theorem ltrn11at 35433

Description: Frequently used one-to-one property of lattice translation atoms. (Contributed by NM, 5-May-2013.)

Hypotheses

Ref	Expression
ltrneq2.a	|- A = ( Atoms ` K )
ltrneq2.h	|- H = ( LHyp ` K )
ltrneq2.t	|- T = ( ( LTrn ` K ) ` W )

Assertion

Ref	Expression
ltrn11at	|- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ Q e. A /\ P =/= Q ) ) -> ( F ` P ) =/= ( F ` Q ) )

Proof of Theorem ltrn11at

Step	Hyp	Ref	Expression
1		simp33 1099	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ Q e. A /\ P =/= Q ) ) -> P =/= Q )
2		simp1 1061	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ Q e. A /\ P =/= Q ) ) -> ( K e. HL /\ W e. H ) )
3		simp2 1062	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ Q e. A /\ P =/= Q ) ) -> F e. T )
4		simp31 1097	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ Q e. A /\ P =/= Q ) ) -> P e. A )
5		eqid 2622	. . . . . 6 |- ( Base ` K ) = ( Base ` K )
6		ltrneq2.a	. . . . . 6 |- A = ( Atoms ` K )
7	5, 6	atbase 34576	. . . . 5 |- ( P e. A -> P e. ( Base ` K ) )
8	4, 7	syl 17	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ Q e. A /\ P =/= Q ) ) -> P e. ( Base ` K ) )
9		simp32 1098	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ Q e. A /\ P =/= Q ) ) -> Q e. A )
10	5, 6	atbase 34576	. . . . 5 |- ( Q e. A -> Q e. ( Base ` K ) )
11	9, 10	syl 17	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ Q e. A /\ P =/= Q ) ) -> Q e. ( Base ` K ) )
12		ltrneq2.h	. . . . 5 |- H = ( LHyp ` K )
13		ltrneq2.t	. . . . 5 |- T = ( ( LTrn ` K ) ` W )
14	5, 12, 13	ltrn11 35412	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. ( Base ` K ) /\ Q e. ( Base ` K ) ) ) -> ( ( F ` P ) = ( F ` Q ) <-> P = Q ) )
15	2, 3, 8, 11, 14	syl112anc 1330	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ Q e. A /\ P =/= Q ) ) -> ( ( F ` P ) = ( F ` Q ) <-> P = Q ) )
16	15	necon3bid 2838	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ Q e. A /\ P =/= Q ) ) -> ( ( F ` P ) =/= ( F ` Q ) <-> P =/= Q ) )
17	1, 16	mpbird 247	1 |- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ Q e. A /\ P =/= Q ) ) -> ( F ` P ) =/= ( F ` Q ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   ` cfv 5888   Basecbs 15857   Atomscatm 34550   HLchlt 34637   LHypclh 35270   LTrncltrn 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-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-ats 34554  df-laut 35275  df-ldil 35390  df-ltrn 35391

This theorem is referenced by:  cdlemg10a  35928  cdlemg12d  35934  cdlemg18a  35966