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Theorem ipassr 18075
Description: "Associative" law for second argument of inner product (compare ipass 18074). (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 457 . . . . 5  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  K )
)  ->  W  e.  PreHil )
2 simpr3 996 . . . . 5  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  K )
)  ->  C  e.  K )
3 simpr2 995 . . . . 5  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  K )
)  ->  B  e.  V )
4 simpr1 994 . . . . 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 18074 . . . . 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 1220 . . . 4  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  K )
)  ->  ( ( C  .x.  B )  .,  A )  =  ( C  .X.  ( B  .,  A ) ) )
1312fveq2d 5695 . . 3  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  K )
)  ->  (  .*  `  ( ( C  .x.  B )  .,  A
) )  =  (  .*  `  ( C 
.X.  ( B  .,  A ) ) ) )
14 phllmod 18059 . . . . . 6  |-  ( W  e.  PreHil  ->  W  e.  LMod )
1514adantr 465 . . . . 5  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  K )
)  ->  W  e.  LMod )
167, 5, 9, 8lmodvscl 16965 . . . . 5  |-  ( ( W  e.  LMod  /\  C  e.  K  /\  B  e.  V )  ->  ( C  .x.  B )  e.  V )
1715, 2, 3, 16syl3anc 1218 . . . 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 18063 . . . 4  |-  ( ( W  e.  PreHil  /\  ( C  .x.  B )  e.  V  /\  A  e.  V )  ->  (  .*  `  ( ( C 
.x.  B )  .,  A ) )  =  ( A  .,  ( C  .x.  B ) ) )
201, 17, 4, 19syl3anc 1218 . . 3  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  K )
)  ->  (  .*  `  ( ( C  .x.  B )  .,  A
) )  =  ( A  .,  ( C 
.x.  B ) ) )
215phlsrng 18060 . . . . 5  |-  ( W  e.  PreHil  ->  F  e.  *Ring )
2221adantr 465 . . . 4  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  K )
)  ->  F  e.  *Ring
)
235, 6, 7, 8ipcl 18062 . . . . 5  |-  ( ( W  e.  PreHil  /\  B  e.  V  /\  A  e.  V )  ->  ( B  .,  A )  e.  K )
241, 3, 4, 23syl3anc 1218 . . . 4  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  K )
)  ->  ( B  .,  A )  e.  K
)
2518, 8, 10srngmul 16943 . . . 4  |-  ( ( F  e.  *Ring  /\  C  e.  K  /\  ( B  .,  A )  e.  K )  ->  (  .*  `  ( C  .X.  ( B  .,  A ) ) )  =  ( (  .*  `  ( B  .,  A ) ) 
.X.  (  .*  `  C ) ) )
2622, 2, 24, 25syl3anc 1218 . . 3  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  K )
)  ->  (  .*  `  ( C  .X.  ( B  .,  A ) ) )  =  ( (  .*  `  ( B 
.,  A ) ) 
.X.  (  .*  `  C ) ) )
2713, 20, 263eqtr3d 2483 . 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 18063 . . . 4  |-  ( ( W  e.  PreHil  /\  B  e.  V  /\  A  e.  V )  ->  (  .*  `  ( B  .,  A ) )  =  ( A  .,  B
) )
291, 3, 4, 28syl3anc 1218 . . 3  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  K )
)  ->  (  .*  `  ( B  .,  A
) )  =  ( A  .,  B ) )
3029oveq1d 6106 . 2  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  K )
)  ->  ( (  .*  `  ( B  .,  A ) )  .X.  (  .*  `  C ) )  =  ( ( A  .,  B ) 
.X.  (  .*  `  C ) ) )
3127, 30eqtrd 2475 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   ` cfv 5418  (class class class)co 6091   Basecbs 14174   .rcmulr 14239   *rcstv 14240  Scalarcsca 14241   .scvsca 14242   .icip 14243   *Ringcsr 16929   LModclmod 16948   PreHilcphl 18053
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-tpos 6745  df-recs 6832  df-rdg 6866  df-er 7101  df-map 7216  df-en 7311  df-dom 7312  df-sdom 7313  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-nn 10323  df-2 10380  df-3 10381  df-4 10382  df-5 10383  df-6 10384  df-7 10385  df-8 10386  df-ndx 14177  df-slot 14178  df-base 14179  df-sets 14180  df-plusg 14251  df-mulr 14252  df-sca 14254  df-vsca 14255  df-ip 14256  df-0g 14380  df-mnd 15415  df-mhm 15464  df-ghm 15745  df-mgp 16592  df-ur 16604  df-rng 16647  df-oppr 16715  df-rnghom 16806  df-staf 16930  df-srng 16931  df-lmod 16950  df-lmhm 17103  df-lvec 17184  df-sra 17253  df-rgmod 17254  df-phl 18055
This theorem is referenced by:  ipassr2  18076  cphassr  20730  tchcphlem2  20751
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