Theorem diameetN 36345

Description: Partial isomorphism A of a lattice meet. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
diam.m	|- ./\ = ( meet ` K )
diam.h	|- H = ( LHyp ` K )
diam.i	|- I = ( ( DIsoA ` K ) ` W )

Assertion

Ref	Expression
diameetN	|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> ( I ` ( X ./\ Y ) ) = ( ( I ` X ) i^i ( I ` Y ) ) )

Proof of Theorem diameetN

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . . 4 |- ( glb ` K ) = ( glb ` K )
2		diam.m	. . . 4 |- ./\ = ( meet ` K )
3		simpll 790	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> K e. HL )
4		eqid 2622	. . . . . 6 |- ( Base ` K ) = ( Base ` K )
5		diam.h	. . . . . 6 |- H = ( LHyp ` K )
6		diam.i	. . . . . 6 |- I = ( ( DIsoA ` K ) ` W )
7	4, 5, 6	diadmclN 36326	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ X e. dom I ) -> X e. ( Base ` K ) )
8	7	adantrr 753	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> X e. ( Base ` K ) )
9	4, 5, 6	diadmclN 36326	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ Y e. dom I ) -> Y e. ( Base ` K ) )
10	9	adantrl 752	. . . 4 |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> Y e. ( Base ` K ) )
11	1, 2, 3, 8, 10	meetval 17019	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> ( X ./\ Y ) = ( ( glb ` K ) ` { X , Y } ) )
12	11	fveq2d 6195	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> ( I ` ( X ./\ Y ) ) = ( I ` ( ( glb ` K ) ` { X , Y } ) ) )
13		simpl 473	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> ( K e. HL /\ W e. H ) )
14		prssi 4353	. . . 4 |- ( ( X e. dom I /\ Y e. dom I ) -> { X , Y } C_ dom I )
15	14	adantl 482	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> { X , Y } C_ dom I )
16		prnzg 4311	. . . 4 |- ( X e. dom I -> { X , Y } =/= (/) )
17	16	ad2antrl 764	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> { X , Y } =/= (/) )
18	1, 5, 6	diaglbN 36344	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( { X , Y } C_ dom I /\ { X , Y } =/= (/) ) ) -> ( I ` ( ( glb ` K ) ` { X , Y } ) ) = |^|_ x e. { X , Y } ( I ` x ) )
19	13, 15, 17, 18	syl12anc 1324	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> ( I ` ( ( glb ` K ) ` { X , Y } ) ) = |^|_ x e. { X , Y } ( I ` x ) )
20		fveq2 6191	. . . 4 |- ( x = X -> ( I ` x ) = ( I ` X ) )
21		fveq2 6191	. . . 4 |- ( x = Y -> ( I ` x ) = ( I ` Y ) )
22	20, 21	iinxprg 4601	. . 3 |- ( ( X e. dom I /\ Y e. dom I ) -> |^|_ x e. { X , Y } ( I ` x ) = ( ( I ` X ) i^i ( I ` Y ) ) )
23	22	adantl 482	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> |^|_ x e. { X , Y } ( I ` x ) = ( ( I ` X ) i^i ( I ` Y ) ) )
24	12, 19, 23	3eqtrd 2660	1 |- ( ( ( K e. HL /\ W e. H ) /\ ( X e. dom I /\ Y e. dom I ) ) -> ( I ` ( X ./\ Y ) ) = ( ( I ` X ) i^i ( I ` Y ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   i^i cin 3573   C_ wss 3574   (/)c0 3915   {cpr 4179   |^|_ciin 4521   dom cdm 5114   ` cfv 5888  (class class class)co 6650   Basecbs 15857   glbcglb 16943   meetcmee 16945   HLchlt 34637   LHypclh 35270   DIsoAcdia 36317

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-iin 4523  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-mpt2 6655  df-map 7859  df-preset 16928  df-poset 16946  df-plt 16958  df-lub 16974  df-glb 16975  df-join 16976  df-meet 16977  df-p0 17039  df-p1 17040  df-lat 17046  df-clat 17108  df-oposet 34463  df-ol 34465  df-oml 34466  df-covers 34553  df-ats 34554  df-atl 34585  df-cvlat 34609  df-hlat 34638  df-lhyp 35274  df-laut 35275  df-ldil 35390  df-ltrn 35391  df-trl 35446  df-disoa 36318

This theorem is referenced by:  diainN  36346  djajN  36426