Theorem dalemswapyzps 34976

Description: Lemma for dath 35022. Swap the Y and Z planes, along with dummy concurrency (center of perspectivity) atoms c and d, to allow reuse of analogous proofs. (Contributed by NM, 17-Aug-2012.)

Hypotheses

Ref	Expression
dalem.ph	|- ( ph <-> ( ( ( K e. HL /\ C e. ( Base ` K ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( Y e. O /\ Z e. O ) /\ ( ( -. C .<_ ( P .\/ Q ) /\ -. C .<_ ( Q .\/ R ) /\ -. C .<_ ( R .\/ P ) ) /\ ( -. C .<_ ( S .\/ T ) /\ -. C .<_ ( T .\/ U ) /\ -. C .<_ ( U .\/ S ) ) /\ ( C .<_ ( P .\/ S ) /\ C .<_ ( Q .\/ T ) /\ C .<_ ( R .\/ U ) ) ) ) )
dalem.l	|- .<_ = ( le ` K )
dalem.j	|- .\/ = ( join ` K )
dalem.a	|- A = ( Atoms ` K )
dalem.ps	|- ( ps <-> ( ( c e. A /\ d e. A ) /\ -. c .<_ Y /\ ( d =/= c /\ -. d .<_ Y /\ C .<_ ( c .\/ d ) ) ) )

Assertion

Ref	Expression
dalemswapyzps	|- ( ( ph /\ Y = Z /\ ps ) -> ( ( d e. A /\ c e. A ) /\ -. d .<_ Z /\ ( c =/= d /\ -. c .<_ Z /\ C .<_ ( d .\/ c ) ) ) )

Proof of Theorem dalemswapyzps

Step	Hyp	Ref	Expression
1		dalem.ps	. . . . 5 |- ( ps <-> ( ( c e. A /\ d e. A ) /\ -. c .<_ Y /\ ( d =/= c /\ -. d .<_ Y /\ C .<_ ( c .\/ d ) ) ) )
2	1	dalemddea 34970	. . . 4 |- ( ps -> d e. A )
3	1	dalemccea 34969	. . . 4 |- ( ps -> c e. A )
4	2, 3	jca 554	. . 3 |- ( ps -> ( d e. A /\ c e. A ) )
5	4	3ad2ant3 1084	. 2 |- ( ( ph /\ Y = Z /\ ps ) -> ( d e. A /\ c e. A ) )
6	1	dalem-ddly 34972	. . . 4 |- ( ps -> -. d .<_ Y )
7	6	3ad2ant3 1084	. . 3 |- ( ( ph /\ Y = Z /\ ps ) -> -. d .<_ Y )
8		simp2 1062	. . . 4 |- ( ( ph /\ Y = Z /\ ps ) -> Y = Z )
9	8	breq2d 4665	. . 3 |- ( ( ph /\ Y = Z /\ ps ) -> ( d .<_ Y <-> d .<_ Z ) )
10	7, 9	mtbid 314	. 2 |- ( ( ph /\ Y = Z /\ ps ) -> -. d .<_ Z )
11	1	dalemccnedd 34973	. . . 4 |- ( ps -> c =/= d )
12	11	3ad2ant3 1084	. . 3 |- ( ( ph /\ Y = Z /\ ps ) -> c =/= d )
13	1	dalem-ccly 34971	. . . . 5 |- ( ps -> -. c .<_ Y )
14	13	3ad2ant3 1084	. . . 4 |- ( ( ph /\ Y = Z /\ ps ) -> -. c .<_ Y )
15	8	breq2d 4665	. . . 4 |- ( ( ph /\ Y = Z /\ ps ) -> ( c .<_ Y <-> c .<_ Z ) )
16	14, 15	mtbid 314	. . 3 |- ( ( ph /\ Y = Z /\ ps ) -> -. c .<_ Z )
17	1	dalemclccjdd 34974	. . . . 5 |- ( ps -> C .<_ ( c .\/ d ) )
18	17	3ad2ant3 1084	. . . 4 |- ( ( ph /\ Y = Z /\ ps ) -> C .<_ ( c .\/ d ) )
19		dalem.ph	. . . . . . 7 |- ( ph <-> ( ( ( K e. HL /\ C e. ( Base ` K ) ) /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( S e. A /\ T e. A /\ U e. A ) ) /\ ( Y e. O /\ Z e. O ) /\ ( ( -. C .<_ ( P .\/ Q ) /\ -. C .<_ ( Q .\/ R ) /\ -. C .<_ ( R .\/ P ) ) /\ ( -. C .<_ ( S .\/ T ) /\ -. C .<_ ( T .\/ U ) /\ -. C .<_ ( U .\/ S ) ) /\ ( C .<_ ( P .\/ S ) /\ C .<_ ( Q .\/ T ) /\ C .<_ ( R .\/ U ) ) ) ) )
20	19	dalemkehl 34909	. . . . . 6 |- ( ph -> K e. HL )
21	20	3ad2ant1 1082	. . . . 5 |- ( ( ph /\ Y = Z /\ ps ) -> K e. HL )
22	3	3ad2ant3 1084	. . . . 5 |- ( ( ph /\ Y = Z /\ ps ) -> c e. A )
23	2	3ad2ant3 1084	. . . . 5 |- ( ( ph /\ Y = Z /\ ps ) -> d e. A )
24		dalem.j	. . . . . 6 |- .\/ = ( join ` K )
25		dalem.a	. . . . . 6 |- A = ( Atoms ` K )
26	24, 25	hlatjcom 34654	. . . . 5 |- ( ( K e. HL /\ c e. A /\ d e. A ) -> ( c .\/ d ) = ( d .\/ c ) )
27	21, 22, 23, 26	syl3anc 1326	. . . 4 |- ( ( ph /\ Y = Z /\ ps ) -> ( c .\/ d ) = ( d .\/ c ) )
28	18, 27	breqtrd 4679	. . 3 |- ( ( ph /\ Y = Z /\ ps ) -> C .<_ ( d .\/ c ) )
29	12, 16, 28	3jca 1242	. 2 |- ( ( ph /\ Y = Z /\ ps ) -> ( c =/= d /\ -. c .<_ Z /\ C .<_ ( d .\/ c ) ) )
30	5, 10, 29	3jca 1242	1 |- ( ( ph /\ Y = Z /\ ps ) -> ( ( d e. A /\ c e. A ) /\ -. d .<_ Z /\ ( c =/= d /\ -. c .<_ Z /\ C .<_ ( d .\/ c ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Basecbs 15857   lecple 15948   joincjn 16944   Atomscatm 34550   HLchlt 34637

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-lub 16974  df-join 16976  df-lat 17046  df-ats 34554  df-atl 34585  df-cvlat 34609  df-hlat 34638

This theorem is referenced by:  dalem56  35014