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Theorem ipassr 16566
Description: "Associative" law for second argument of inner product (compare ipass 16565). (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 443 . . . . 5  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  K )
)  ->  W  e.  PreHil )
2 simpr3 963 . . . . 5  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  K )
)  ->  C  e.  K )
3 simpr2 962 . . . . 5  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  K )
)  ->  B  e.  V )
4 simpr1 961 . . . . 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 16565 . . . . 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 1184 . . . 4  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  K )
)  ->  ( ( C  .x.  B )  .,  A )  =  ( C  .X.  ( B  .,  A ) ) )
1312fveq2d 5545 . . 3  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  K )
)  ->  (  .*  `  ( ( C  .x.  B )  .,  A
) )  =  (  .*  `  ( C 
.X.  ( B  .,  A ) ) ) )
14 phllmod 16550 . . . . . 6  |-  ( W  e.  PreHil  ->  W  e.  LMod )
1514adantr 451 . . . . 5  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  K )
)  ->  W  e.  LMod )
167, 5, 9, 8lmodvscl 15660 . . . . 5  |-  ( ( W  e.  LMod  /\  C  e.  K  /\  B  e.  V )  ->  ( C  .x.  B )  e.  V )
1715, 2, 3, 16syl3anc 1182 . . . 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 16554 . . . 4  |-  ( ( W  e.  PreHil  /\  ( C  .x.  B )  e.  V  /\  A  e.  V )  ->  (  .*  `  ( ( C 
.x.  B )  .,  A ) )  =  ( A  .,  ( C  .x.  B ) ) )
201, 17, 4, 19syl3anc 1182 . . 3  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  K )
)  ->  (  .*  `  ( ( C  .x.  B )  .,  A
) )  =  ( A  .,  ( C 
.x.  B ) ) )
215phlsrng 16551 . . . . 5  |-  ( W  e.  PreHil  ->  F  e.  *Ring )
2221adantr 451 . . . 4  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  K )
)  ->  F  e.  *Ring
)
235, 6, 7, 8ipcl 16553 . . . . 5  |-  ( ( W  e.  PreHil  /\  B  e.  V  /\  A  e.  V )  ->  ( B  .,  A )  e.  K )
241, 3, 4, 23syl3anc 1182 . . . 4  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  K )
)  ->  ( B  .,  A )  e.  K
)
2518, 8, 10srngmul 15639 . . . 4  |-  ( ( F  e.  *Ring  /\  C  e.  K  /\  ( B  .,  A )  e.  K )  ->  (  .*  `  ( C  .X.  ( B  .,  A ) ) )  =  ( (  .*  `  ( B  .,  A ) ) 
.X.  (  .*  `  C ) ) )
2622, 2, 24, 25syl3anc 1182 . . 3  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  K )
)  ->  (  .*  `  ( C  .X.  ( B  .,  A ) ) )  =  ( (  .*  `  ( B 
.,  A ) ) 
.X.  (  .*  `  C ) ) )
2713, 20, 263eqtr3d 2336 . 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 16554 . . . 4  |-  ( ( W  e.  PreHil  /\  B  e.  V  /\  A  e.  V )  ->  (  .*  `  ( B  .,  A ) )  =  ( A  .,  B
) )
291, 3, 4, 28syl3anc 1182 . . 3  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  K )
)  ->  (  .*  `  ( B  .,  A
) )  =  ( A  .,  B ) )
3029oveq1d 5889 . 2  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  K )
)  ->  ( (  .*  `  ( B  .,  A ) )  .X.  (  .*  `  C ) )  =  ( ( A  .,  B ) 
.X.  (  .*  `  C ) ) )
3127, 30eqtrd 2328 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 358    /\ w3a 934    = wceq 1632    e. wcel 1696   ` cfv 5271  (class class class)co 5874   Basecbs 13164   .rcmulr 13225   * rcstv 13226  Scalarcsca 13227   .scvsca 13228   .icip 13229   *Ringcsr 15625   LModclmod 15643   PreHilcphl 16544
This theorem is referenced by:  ipassr2  16567  cphassr  18663  tchcphlem2  18682
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-tpos 6250  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-plusg 13237  df-mulr 13238  df-sca 13240  df-vsca 13241  df-0g 13420  df-mnd 14383  df-mhm 14431  df-ghm 14697  df-mgp 15342  df-rng 15356  df-ur 15358  df-oppr 15421  df-rnghom 15512  df-staf 15626  df-srng 15627  df-lmod 15645  df-lmhm 15795  df-lvec 15872  df-sra 15941  df-rgmod 15942  df-phl 16546
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