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Theorem ipassr 18448
Description: "Associative" law for second argument of inner product (compare ipass 18447). (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 1004 . . . . 5  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  K )
)  ->  C  e.  K )
3 simpr2 1003 . . . . 5  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  K )
)  ->  B  e.  V )
4 simpr1 1002 . . . . 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 18447 . . . . 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 1230 . . . 4  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  K )
)  ->  ( ( C  .x.  B )  .,  A )  =  ( C  .X.  ( B  .,  A ) ) )
1312fveq2d 5868 . . 3  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  K )
)  ->  (  .*  `  ( ( C  .x.  B )  .,  A
) )  =  (  .*  `  ( C 
.X.  ( B  .,  A ) ) ) )
14 phllmod 18432 . . . . . 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 17312 . . . . 5  |-  ( ( W  e.  LMod  /\  C  e.  K  /\  B  e.  V )  ->  ( C  .x.  B )  e.  V )
1715, 2, 3, 16syl3anc 1228 . . . 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 18436 . . . 4  |-  ( ( W  e.  PreHil  /\  ( C  .x.  B )  e.  V  /\  A  e.  V )  ->  (  .*  `  ( ( C 
.x.  B )  .,  A ) )  =  ( A  .,  ( C  .x.  B ) ) )
201, 17, 4, 19syl3anc 1228 . . 3  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  K )
)  ->  (  .*  `  ( ( C  .x.  B )  .,  A
) )  =  ( A  .,  ( C 
.x.  B ) ) )
215phlsrng 18433 . . . . 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 18435 . . . . 5  |-  ( ( W  e.  PreHil  /\  B  e.  V  /\  A  e.  V )  ->  ( B  .,  A )  e.  K )
241, 3, 4, 23syl3anc 1228 . . . 4  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  K )
)  ->  ( B  .,  A )  e.  K
)
2518, 8, 10srngmul 17290 . . . 4  |-  ( ( F  e.  *Ring  /\  C  e.  K  /\  ( B  .,  A )  e.  K )  ->  (  .*  `  ( C  .X.  ( B  .,  A ) ) )  =  ( (  .*  `  ( B  .,  A ) ) 
.X.  (  .*  `  C ) ) )
2622, 2, 24, 25syl3anc 1228 . . 3  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  K )
)  ->  (  .*  `  ( C  .X.  ( B  .,  A ) ) )  =  ( (  .*  `  ( B 
.,  A ) ) 
.X.  (  .*  `  C ) ) )
2713, 20, 263eqtr3d 2516 . 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 18436 . . . 4  |-  ( ( W  e.  PreHil  /\  B  e.  V  /\  A  e.  V )  ->  (  .*  `  ( B  .,  A ) )  =  ( A  .,  B
) )
291, 3, 4, 28syl3anc 1228 . . 3  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  K )
)  ->  (  .*  `  ( B  .,  A
) )  =  ( A  .,  B ) )
3029oveq1d 6297 . 2  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  K )
)  ->  ( (  .*  `  ( B  .,  A ) )  .X.  (  .*  `  C ) )  =  ( ( A  .,  B ) 
.X.  (  .*  `  C ) ) )
3127, 30eqtrd 2508 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 973    = wceq 1379    e. wcel 1767   ` cfv 5586  (class class class)co 6282   Basecbs 14486   .rcmulr 14552   *rcstv 14553  Scalarcsca 14554   .scvsca 14555   .icip 14556   *Ringcsr 17276   LModclmod 17295   PreHilcphl 18426
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-tpos 6952  df-recs 7039  df-rdg 7073  df-er 7308  df-map 7419  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-ndx 14489  df-slot 14490  df-base 14491  df-sets 14492  df-plusg 14564  df-mulr 14565  df-sca 14567  df-vsca 14568  df-ip 14569  df-0g 14693  df-mnd 15728  df-mhm 15777  df-ghm 16060  df-mgp 16932  df-ur 16944  df-rng 16988  df-oppr 17056  df-rnghom 17148  df-staf 17277  df-srng 17278  df-lmod 17297  df-lmhm 17451  df-lvec 17532  df-sra 17601  df-rgmod 17602  df-phl 18428
This theorem is referenced by:  ipassr2  18449  cphassr  21393  tchcphlem2  21414
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