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Theorem ipassr 16793
Description: "Associative" law for second argument of inner product (compare ipass 16792). (Contributed by NM, 25-Aug-2007.) (Revised by Mario Carneiro, 7-Oct-2015.)
Hypotheses
Ref Expression
phlsrng.f  |-  F  =  (Scalar `  W )
phllmhm.h  |-  .,  =  ( .i `  W )
phllmhm.v  |-  V  =  ( Base `  W
)
ipdir.f  |-  K  =  ( Base `  F
)
ipass.s  |-  .x.  =  ( .s `  W )
ipass.p  |-  .X.  =  ( .r `  F )
ipassr.i  |-  .*  =  ( * r `  F )
Assertion
Ref Expression
ipassr  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  K )
)  ->  ( A  .,  ( C  .x.  B
) )  =  ( ( A  .,  B
)  .X.  (  .*  `  C ) ) )

Proof of Theorem ipassr
StepHypRef Expression
1 simpl 444 . . . . 5  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  K )
)  ->  W  e.  PreHil )
2 simpr3 965 . . . . 5  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  K )
)  ->  C  e.  K )
3 simpr2 964 . . . . 5  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  K )
)  ->  B  e.  V )
4 simpr1 963 . . . . 5  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  K )
)  ->  A  e.  V )
5 phlsrng.f . . . . . 6  |-  F  =  (Scalar `  W )
6 phllmhm.h . . . . . 6  |-  .,  =  ( .i `  W )
7 phllmhm.v . . . . . 6  |-  V  =  ( Base `  W
)
8 ipdir.f . . . . . 6  |-  K  =  ( Base `  F
)
9 ipass.s . . . . . 6  |-  .x.  =  ( .s `  W )
10 ipass.p . . . . . 6  |-  .X.  =  ( .r `  F )
115, 6, 7, 8, 9, 10ipass 16792 . . . . 5  |-  ( ( W  e.  PreHil  /\  ( C  e.  K  /\  B  e.  V  /\  A  e.  V )
)  ->  ( ( C  .x.  B )  .,  A )  =  ( C  .X.  ( B  .,  A ) ) )
121, 2, 3, 4, 11syl13anc 1186 . . . 4  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  K )
)  ->  ( ( C  .x.  B )  .,  A )  =  ( C  .X.  ( B  .,  A ) ) )
1312fveq2d 5665 . . 3  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  K )
)  ->  (  .*  `  ( ( C  .x.  B )  .,  A
) )  =  (  .*  `  ( C 
.X.  ( B  .,  A ) ) ) )
14 phllmod 16777 . . . . . 6  |-  ( W  e.  PreHil  ->  W  e.  LMod )
1514adantr 452 . . . . 5  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  K )
)  ->  W  e.  LMod )
167, 5, 9, 8lmodvscl 15887 . . . . 5  |-  ( ( W  e.  LMod  /\  C  e.  K  /\  B  e.  V )  ->  ( C  .x.  B )  e.  V )
1715, 2, 3, 16syl3anc 1184 . . . 4  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  K )
)  ->  ( C  .x.  B )  e.  V
)
18 ipassr.i . . . . 5  |-  .*  =  ( * r `  F )
195, 6, 7, 18ipcj 16781 . . . 4  |-  ( ( W  e.  PreHil  /\  ( C  .x.  B )  e.  V  /\  A  e.  V )  ->  (  .*  `  ( ( C 
.x.  B )  .,  A ) )  =  ( A  .,  ( C  .x.  B ) ) )
201, 17, 4, 19syl3anc 1184 . . 3  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  K )
)  ->  (  .*  `  ( ( C  .x.  B )  .,  A
) )  =  ( A  .,  ( C 
.x.  B ) ) )
215phlsrng 16778 . . . . 5  |-  ( W  e.  PreHil  ->  F  e.  *Ring )
2221adantr 452 . . . 4  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  K )
)  ->  F  e.  *Ring
)
235, 6, 7, 8ipcl 16780 . . . . 5  |-  ( ( W  e.  PreHil  /\  B  e.  V  /\  A  e.  V )  ->  ( B  .,  A )  e.  K )
241, 3, 4, 23syl3anc 1184 . . . 4  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  K )
)  ->  ( B  .,  A )  e.  K
)
2518, 8, 10srngmul 15866 . . . 4  |-  ( ( F  e.  *Ring  /\  C  e.  K  /\  ( B  .,  A )  e.  K )  ->  (  .*  `  ( C  .X.  ( B  .,  A ) ) )  =  ( (  .*  `  ( B  .,  A ) ) 
.X.  (  .*  `  C ) ) )
2622, 2, 24, 25syl3anc 1184 . . 3  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  K )
)  ->  (  .*  `  ( C  .X.  ( B  .,  A ) ) )  =  ( (  .*  `  ( B 
.,  A ) ) 
.X.  (  .*  `  C ) ) )
2713, 20, 263eqtr3d 2420 . 2  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  K )
)  ->  ( A  .,  ( C  .x.  B
) )  =  ( (  .*  `  ( B  .,  A ) ) 
.X.  (  .*  `  C ) ) )
285, 6, 7, 18ipcj 16781 . . . 4  |-  ( ( W  e.  PreHil  /\  B  e.  V  /\  A  e.  V )  ->  (  .*  `  ( B  .,  A ) )  =  ( A  .,  B
) )
291, 3, 4, 28syl3anc 1184 . . 3  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  K )
)  ->  (  .*  `  ( B  .,  A
) )  =  ( A  .,  B ) )
3029oveq1d 6028 . 2  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  K )
)  ->  ( (  .*  `  ( B  .,  A ) )  .X.  (  .*  `  C ) )  =  ( ( A  .,  B ) 
.X.  (  .*  `  C ) ) )
3127, 30eqtrd 2412 1  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  K )
)  ->  ( A  .,  ( C  .x.  B
) )  =  ( ( A  .,  B
)  .X.  (  .*  `  C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   ` cfv 5387  (class class class)co 6013   Basecbs 13389   .rcmulr 13450   * rcstv 13451  Scalarcsca 13452   .scvsca 13453   .icip 13454   *Ringcsr 15852   LModclmod 15870   PreHilcphl 16771
This theorem is referenced by:  ipassr2  16794  cphassr  19038  tchcphlem2  19057
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-tpos 6408  df-riota 6478  df-recs 6562  df-rdg 6597  df-er 6834  df-map 6949  df-en 7039  df-dom 7040  df-sdom 7041  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-nn 9926  df-2 9983  df-3 9984  df-4 9985  df-5 9986  df-6 9987  df-ndx 13392  df-slot 13393  df-base 13394  df-sets 13395  df-plusg 13462  df-mulr 13463  df-sca 13465  df-vsca 13466  df-0g 13647  df-mnd 14610  df-mhm 14658  df-ghm 14924  df-mgp 15569  df-rng 15583  df-ur 15585  df-oppr 15648  df-rnghom 15739  df-staf 15853  df-srng 15854  df-lmod 15872  df-lmhm 16018  df-lvec 16095  df-sra 16164  df-rgmod 16165  df-phl 16773
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