Theorem cdleml7 36270

Description: Part of proof of Lemma L of [Crawley] p. 120. TODO: fix comment. (Contributed by NM, 11-Aug-2013.)

Hypotheses

Ref	Expression
cdleml6.b	|- B = ( Base ` K )
cdleml6.j	|- .\/ = ( join ` K )
cdleml6.m	|- ./\ = ( meet ` K )
cdleml6.h	|- H = ( LHyp ` K )
cdleml6.t	|- T = ( ( LTrn ` K ) ` W )
cdleml6.r	|- R = ( ( trL ` K ) ` W )
cdleml6.p	|- Q = ( ( oc ` K ) ` W )
cdleml6.z	|- Z = ( ( Q .\/ ( R ` b ) ) ./\ ( ( h ` Q ) .\/ ( R ` ( b o. `' ( s ` h ) ) ) ) )
cdleml6.y	|- Y = ( ( Q .\/ ( R ` g ) ) ./\ ( Z .\/ ( R ` ( g o. `' b ) ) ) )
cdleml6.x	|- X = ( iota_ z e. T A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` ( s ` h ) ) /\ ( R ` b ) =/= ( R ` g ) ) -> ( z ` Q ) = Y ) )
cdleml6.u	|- U = ( g e. T |-> if ( ( s ` h ) = h , g , X ) )
cdleml6.e	|- E = ( ( TEndo ` K ) ` W )
cdleml6.o	|- .0. = ( f e. T |-> ( _I |` B ) )

Assertion

Ref	Expression
cdleml7	|- ( ( ( K e. HL /\ W e. H ) /\ h e. T /\ ( s e. E /\ s =/= .0. ) ) -> ( ( U o. s ) ` h ) = ( ( _I |` T ) ` h ) )

Distinct variable groups:   g, b, z, ./\   .\/ , b, g, z   B, b, f, g, z   h, b, g, z   s, b, g, z   H, b, g, z   K, b, g, z   Q, b, g, z   R, b, g, z   T, b, f, g, z   W, b, g, z   z, Y   g, Z

Allowed substitution hints:   B( h, s)   Q( f, h, s)   R( f, h, s)   T( h, s)   U( z, f, g, h, s, b)   E( z, f, g, h, s, b)   H( f, h, s)   .\/ ( f, h, s)   K( f, h, s)   ./\ ( f, h, s)   W( f, h, s)   X( z, f, g, h, s, b)   Y( f, g, h, s, b)   .0. ( z, f, g, h, s, b)   Z( z, f, h, s, b)

Proof of Theorem cdleml7

Step	Hyp	Ref	Expression
1		cdleml6.b	. . . 4 |- B = ( Base ` K )
2		cdleml6.j	. . . 4 |- .\/ = ( join ` K )
3		cdleml6.m	. . . 4 |- ./\ = ( meet ` K )
4		cdleml6.h	. . . 4 |- H = ( LHyp ` K )
5		cdleml6.t	. . . 4 |- T = ( ( LTrn ` K ) ` W )
6		cdleml6.r	. . . 4 |- R = ( ( trL ` K ) ` W )
7		cdleml6.p	. . . 4 |- Q = ( ( oc ` K ) ` W )
8		cdleml6.z	. . . 4 |- Z = ( ( Q .\/ ( R ` b ) ) ./\ ( ( h ` Q ) .\/ ( R ` ( b o. `' ( s ` h ) ) ) ) )
9		cdleml6.y	. . . 4 |- Y = ( ( Q .\/ ( R ` g ) ) ./\ ( Z .\/ ( R ` ( g o. `' b ) ) ) )
10		cdleml6.x	. . . 4 |- X = ( iota_ z e. T A. b e. T ( ( b =/= ( _I |` B ) /\ ( R ` b ) =/= ( R ` ( s ` h ) ) /\ ( R ` b ) =/= ( R ` g ) ) -> ( z ` Q ) = Y ) )
11		cdleml6.u	. . . 4 |- U = ( g e. T |-> if ( ( s ` h ) = h , g , X ) )
12		cdleml6.e	. . . 4 |- E = ( ( TEndo ` K ) ` W )
13		cdleml6.o	. . . 4 |- .0. = ( f e. T |-> ( _I |` B ) )
14	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13	cdleml6 36269	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ h e. T /\ ( s e. E /\ s =/= .0. ) ) -> ( U e. E /\ ( U ` ( s ` h ) ) = h ) )
15	14	simprd 479	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ h e. T /\ ( s e. E /\ s =/= .0. ) ) -> ( U ` ( s ` h ) ) = h )
16		simp1 1061	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ h e. T /\ ( s e. E /\ s =/= .0. ) ) -> ( K e. HL /\ W e. H ) )
17	14	simpld 475	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ h e. T /\ ( s e. E /\ s =/= .0. ) ) -> U e. E )
18		simp3l 1089	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ h e. T /\ ( s e. E /\ s =/= .0. ) ) -> s e. E )
19		simp2 1062	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ h e. T /\ ( s e. E /\ s =/= .0. ) ) -> h e. T )
20	4, 5, 12	tendocoval 36054	. . 3 |- ( ( ( K e. HL /\ W e. H ) /\ ( U e. E /\ s e. E ) /\ h e. T ) -> ( ( U o. s ) ` h ) = ( U ` ( s ` h ) ) )
21	16, 17, 18, 19, 20	syl121anc 1331	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ h e. T /\ ( s e. E /\ s =/= .0. ) ) -> ( ( U o. s ) ` h ) = ( U ` ( s ` h ) ) )
22		fvresi 6439	. . 3 |- ( h e. T -> ( ( _I |` T ) ` h ) = h )
23	22	3ad2ant2 1083	. 2 |- ( ( ( K e. HL /\ W e. H ) /\ h e. T /\ ( s e. E /\ s =/= .0. ) ) -> ( ( _I |` T ) ` h ) = h )
24	15, 21, 23	3eqtr4d 2666	1 |- ( ( ( K e. HL /\ W e. H ) /\ h e. T /\ ( s e. E /\ s =/= .0. ) ) -> ( ( U o. s ) ` h ) = ( ( _I |` T ) ` h ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   ifcif 4086   |-> cmpt 4729   _I cid 5023   `'ccnv 5113   |` cres 5116   o. ccom 5118   ` cfv 5888   iota_crio 6610  (class class class)co 6650   Basecbs 15857   occoc 15949   joincjn 16944   meetcmee 16945   HLchlt 34637   LHypclh 35270   LTrncltrn 35387   trLctrl 35445   TEndoctendo 36040

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  ax-riotaBAD 34239

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  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-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  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-1st 7168  df-2nd 7169  df-undef 7399  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-llines 34784  df-lplanes 34785  df-lvols 34786  df-lines 34787  df-psubsp 34789  df-pmap 34790  df-padd 35082  df-lhyp 35274  df-laut 35275  df-ldil 35390  df-ltrn 35391  df-trl 35446  df-tendo 36043

This theorem is referenced by:  cdleml8  36271