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Theorem ipassr 18977
Description: "Associative" law for second argument of inner product (compare ipass 18976). (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 455 . . . . 5  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  K )
)  ->  W  e.  PreHil )
2 simpr3 1005 . . . . 5  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  K )
)  ->  C  e.  K )
3 simpr2 1004 . . . . 5  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  K )
)  ->  B  e.  V )
4 simpr1 1003 . . . . 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 18976 . . . . 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 1232 . . . 4  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  K )
)  ->  ( ( C  .x.  B )  .,  A )  =  ( C  .X.  ( B  .,  A ) ) )
1312fveq2d 5852 . . 3  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  K )
)  ->  (  .*  `  ( ( C  .x.  B )  .,  A
) )  =  (  .*  `  ( C 
.X.  ( B  .,  A ) ) ) )
14 phllmod 18961 . . . . . 6  |-  ( W  e.  PreHil  ->  W  e.  LMod )
1514adantr 463 . . . . 5  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  K )
)  ->  W  e.  LMod )
167, 5, 9, 8lmodvscl 17847 . . . . 5  |-  ( ( W  e.  LMod  /\  C  e.  K  /\  B  e.  V )  ->  ( C  .x.  B )  e.  V )
1715, 2, 3, 16syl3anc 1230 . . . 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 18965 . . . 4  |-  ( ( W  e.  PreHil  /\  ( C  .x.  B )  e.  V  /\  A  e.  V )  ->  (  .*  `  ( ( C 
.x.  B )  .,  A ) )  =  ( A  .,  ( C  .x.  B ) ) )
201, 17, 4, 19syl3anc 1230 . . 3  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  K )
)  ->  (  .*  `  ( ( C  .x.  B )  .,  A
) )  =  ( A  .,  ( C 
.x.  B ) ) )
215phlsrng 18962 . . . . 5  |-  ( W  e.  PreHil  ->  F  e.  *Ring )
2221adantr 463 . . . 4  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  K )
)  ->  F  e.  *Ring
)
235, 6, 7, 8ipcl 18964 . . . . 5  |-  ( ( W  e.  PreHil  /\  B  e.  V  /\  A  e.  V )  ->  ( B  .,  A )  e.  K )
241, 3, 4, 23syl3anc 1230 . . . 4  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  K )
)  ->  ( B  .,  A )  e.  K
)
2518, 8, 10srngmul 17825 . . . 4  |-  ( ( F  e.  *Ring  /\  C  e.  K  /\  ( B  .,  A )  e.  K )  ->  (  .*  `  ( C  .X.  ( B  .,  A ) ) )  =  ( (  .*  `  ( B  .,  A ) ) 
.X.  (  .*  `  C ) ) )
2622, 2, 24, 25syl3anc 1230 . . 3  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  K )
)  ->  (  .*  `  ( C  .X.  ( B  .,  A ) ) )  =  ( (  .*  `  ( B 
.,  A ) ) 
.X.  (  .*  `  C ) ) )
2713, 20, 263eqtr3d 2451 . 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 18965 . . . 4  |-  ( ( W  e.  PreHil  /\  B  e.  V  /\  A  e.  V )  ->  (  .*  `  ( B  .,  A ) )  =  ( A  .,  B
) )
291, 3, 4, 28syl3anc 1230 . . 3  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  K )
)  ->  (  .*  `  ( B  .,  A
) )  =  ( A  .,  B ) )
3029oveq1d 6292 . 2  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  K )
)  ->  ( (  .*  `  ( B  .,  A ) )  .X.  (  .*  `  C ) )  =  ( ( A  .,  B ) 
.X.  (  .*  `  C ) ) )
3127, 30eqtrd 2443 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 367    /\ w3a 974    = wceq 1405    e. wcel 1842   ` cfv 5568  (class class class)co 6277   Basecbs 14839   .rcmulr 14908   *rcstv 14909  Scalarcsca 14910   .scvsca 14911   .icip 14912   *Ringcsr 17811   LModclmod 17830   PreHilcphl 18955
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6683  df-tpos 6957  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-er 7347  df-map 7458  df-en 7554  df-dom 7555  df-sdom 7556  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-nn 10576  df-2 10634  df-3 10635  df-4 10636  df-5 10637  df-6 10638  df-7 10639  df-8 10640  df-ndx 14842  df-slot 14843  df-base 14844  df-sets 14845  df-plusg 14920  df-mulr 14921  df-sca 14923  df-vsca 14924  df-ip 14925  df-0g 15054  df-mgm 16194  df-sgrp 16233  df-mnd 16243  df-mhm 16288  df-ghm 16587  df-mgp 17460  df-ur 17472  df-ring 17518  df-oppr 17590  df-rnghom 17682  df-staf 17812  df-srng 17813  df-lmod 17832  df-lmhm 17986  df-lvec 18067  df-sra 18136  df-rgmod 18137  df-phl 18957
This theorem is referenced by:  ipassr2  18978  cphassr  21948  tchcphlem2  21969
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