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Theorem tendoplcom 36070
Description: The endomorphism sum operation is commutative. (Contributed by NM, 11-Jun-2013.)
Hypotheses
Ref Expression
tendopl.h |- H = ( LHyp ` K )
tendopl.t |- T = ( ( LTrn ` K ) ` W )
tendopl.e |- E = ( ( TEndo ` K ) ` W )
tendopl.p |- P = ( s e. E , t e. E |-> ( f e. T |-> ( ( s ` f ) o. ( t ` f ) ) ) )
Assertion
Ref Expression
tendoplcom |- ( ( ( K e. HL /\ W e. H ) /\ U e. E /\ V e. E ) -> ( U P V ) = ( V P U ) )
Distinct variable groups:   t, s, E   f, s, t, T   f, W, s, t
Allowed substitution hints:   P( t, f, s)   U( t, f, s)   E( f)   H( t, f, s)   K( t, f, s)   V( t, f, s)
Proof of Theorem tendoplcom
Dummy variable g is distinct from all other variables.
StepHypRef Expression
1 simp1 1061 . 2 |- ( ( ( K e. HL /\ W e. H ) /\ U e. E /\ V e. E ) -> ( K e. HL /\ W e. H ) )
2 tendopl.h . . 3 |- H = ( LHyp ` K )
3 tendopl.t . . 3 |- T = ( ( LTrn ` K ) ` W )
4 tendopl.e . . 3 |- E = ( ( TEndo ` K ) ` W )
5 tendopl.p . . 3 |- P = ( s e. E , t e. E |-> ( f e. T |-> ( ( s ` f ) o. ( t ` f ) ) ) )
62, 3, 4, 5tendoplcl 36069 . 2 |- ( ( ( K e. HL /\ W e. H ) /\ U e. E /\ V e. E ) -> ( U P V ) e. E )
72, 3, 4, 5tendoplcl 36069 . . 3 |- ( ( ( K e. HL /\ W e. H ) /\ V e. E /\ U e. E ) -> ( V P U ) e. E )
873com23 1271 . 2 |- ( ( ( K e. HL /\ W e. H ) /\ U e. E /\ V e. E ) -> ( V P U ) e. E )
9 simpl1 1064 . . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ U e. E /\ V e. E ) /\ g e. T ) -> ( K e. HL /\ W e. H ) )
10 simpl2 1065 . . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ U e. E /\ V e. E ) /\ g e. T ) -> U e. E )
11 simpr 477 . . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ U e. E /\ V e. E ) /\ g e. T ) -> g e. T )
122, 3, 4tendocl 36055 . . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ U e. E /\ g e. T ) -> ( U ` g ) e. T )
139, 10, 11, 12syl3anc 1326 . . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ U e. E /\ V e. E ) /\ g e. T ) -> ( U ` g ) e. T )
14 simpl3 1066 . . . . . 6 |- ( ( ( ( K e. HL /\ W e. H ) /\ U e. E /\ V e. E ) /\ g e. T ) -> V e. E )
152, 3, 4tendocl 36055 . . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ V e. E /\ g e. T ) -> ( V ` g ) e. T )
169, 14, 11, 15syl3anc 1326 . . . . 5 |- ( ( ( ( K e. HL /\ W e. H ) /\ U e. E /\ V e. E ) /\ g e. T ) -> ( V ` g ) e. T )
172, 3ltrncom 36026 . . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( U ` g ) e. T /\ ( V ` g ) e. T ) -> ( ( U ` g ) o. ( V ` g ) ) = ( ( V ` g ) o. ( U ` g ) ) )
189, 13, 16, 17syl3anc 1326 . . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ U e. E /\ V e. E ) /\ g e. T ) -> ( ( U ` g ) o. ( V ` g ) ) = ( ( V ` g ) o. ( U ` g ) ) )
195, 3tendopl2 36065 . . . . 5 |- ( ( U e. E /\ V e. E /\ g e. T ) -> ( ( U P V ) ` g ) = ( ( U ` g ) o. ( V ` g ) ) )
2010, 14, 11, 19syl3anc 1326 . . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ U e. E /\ V e. E ) /\ g e. T ) -> ( ( U P V ) ` g ) = ( ( U ` g ) o. ( V ` g ) ) )
215, 3tendopl2 36065 . . . . 5 |- ( ( V e. E /\ U e. E /\ g e. T ) -> ( ( V P U ) ` g ) = ( ( V ` g ) o. ( U ` g ) ) )
2214, 10, 11, 21syl3anc 1326 . . . 4 |- ( ( ( ( K e. HL /\ W e. H ) /\ U e. E /\ V e. E ) /\ g e. T ) -> ( ( V P U ) ` g ) = ( ( V ` g ) o. ( U ` g ) ) )
2318, 20, 223eqtr4d 2666 . . 3 |- ( ( ( ( K e. HL /\ W e. H ) /\ U e. E /\ V e. E ) /\ g e. T ) -> ( ( U P V ) ` g ) = ( ( V P U ) ` g ) )
2423ralrimiva 2966 . 2 |- ( ( ( K e. HL /\ W e. H ) /\ U e. E /\ V e. E ) -> A. g e. T ( ( U P V ) ` g ) = ( ( V P U ) ` g ) )
252, 3, 4tendoeq1 36052 . 2 |- ( ( ( K e. HL /\ W e. H ) /\ ( ( U P V ) e. E /\ ( V P U ) e. E ) /\ A. g e. T ( ( U P V ) ` g ) = ( ( V P U ) ` g ) ) -> ( U P V ) = ( V P U ) )
261, 6, 8, 24, 25syl121anc 1331 1 |- ( ( ( K e. HL /\ W e. H ) /\ U e. E /\ V e. E ) -> ( U P V ) = ( V P U ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   |-> cmpt 4729   o. ccom 5118   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   HLchlt 34637   LHypclh 35270   LTrncltrn 35387   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-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:  tendo0plr  36080  tendoipl2  36086  erngdvlem2N  36277  erngdvlem2-rN  36285  dvhvaddcomN  36385
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