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Theorem ip2di 18443
Description: Distributive law for inner product. (Contributed by NM, 17-Apr-2008.) (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.g  |-  .+  =  ( +g  `  W )
ipdir.p  |-  .+^  =  ( +g  `  F )
ip2di.1  |-  ( ph  ->  W  e.  PreHil )
ip2di.2  |-  ( ph  ->  A  e.  V )
ip2di.3  |-  ( ph  ->  B  e.  V )
ip2di.4  |-  ( ph  ->  C  e.  V )
ip2di.5  |-  ( ph  ->  D  e.  V )
Assertion
Ref Expression
ip2di  |-  ( ph  ->  ( ( A  .+  B )  .,  ( C  .+  D ) )  =  ( ( ( A  .,  C ) 
.+^  ( B  .,  D ) )  .+^  ( ( A  .,  D )  .+^  ( B 
.,  C ) ) ) )

Proof of Theorem ip2di
StepHypRef Expression
1 ip2di.1 . . 3  |-  ( ph  ->  W  e.  PreHil )
2 ip2di.2 . . 3  |-  ( ph  ->  A  e.  V )
3 ip2di.3 . . 3  |-  ( ph  ->  B  e.  V )
4 phllmod 18432 . . . . 5  |-  ( W  e.  PreHil  ->  W  e.  LMod )
51, 4syl 16 . . . 4  |-  ( ph  ->  W  e.  LMod )
6 ip2di.4 . . . 4  |-  ( ph  ->  C  e.  V )
7 ip2di.5 . . . 4  |-  ( ph  ->  D  e.  V )
8 phllmhm.v . . . . 5  |-  V  =  ( Base `  W
)
9 ipdir.g . . . . 5  |-  .+  =  ( +g  `  W )
108, 9lmodvacl 17309 . . . 4  |-  ( ( W  e.  LMod  /\  C  e.  V  /\  D  e.  V )  ->  ( C  .+  D )  e.  V )
115, 6, 7, 10syl3anc 1228 . . 3  |-  ( ph  ->  ( C  .+  D
)  e.  V )
12 phlsrng.f . . . 4  |-  F  =  (Scalar `  W )
13 phllmhm.h . . . 4  |-  .,  =  ( .i `  W )
14 ipdir.p . . . 4  |-  .+^  =  ( +g  `  F )
1512, 13, 8, 9, 14ipdir 18441 . . 3  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  B  e.  V  /\  ( C  .+  D )  e.  V ) )  ->  ( ( A 
.+  B )  .,  ( C  .+  D ) )  =  ( ( A  .,  ( C 
.+  D ) ) 
.+^  ( B  .,  ( C  .+  D ) ) ) )
161, 2, 3, 11, 15syl13anc 1230 . 2  |-  ( ph  ->  ( ( A  .+  B )  .,  ( C  .+  D ) )  =  ( ( A 
.,  ( C  .+  D ) )  .+^  ( B  .,  ( C 
.+  D ) ) ) )
1712, 13, 8, 9, 14ipdi 18442 . . . 4  |-  ( ( W  e.  PreHil  /\  ( A  e.  V  /\  C  e.  V  /\  D  e.  V )
)  ->  ( A  .,  ( C  .+  D
) )  =  ( ( A  .,  C
)  .+^  ( A  .,  D ) ) )
181, 2, 6, 7, 17syl13anc 1230 . . 3  |-  ( ph  ->  ( A  .,  ( C  .+  D ) )  =  ( ( A 
.,  C )  .+^  ( A  .,  D ) ) )
1912, 13, 8, 9, 14ipdi 18442 . . . . 5  |-  ( ( W  e.  PreHil  /\  ( B  e.  V  /\  C  e.  V  /\  D  e.  V )
)  ->  ( B  .,  ( C  .+  D
) )  =  ( ( B  .,  C
)  .+^  ( B  .,  D ) ) )
201, 3, 6, 7, 19syl13anc 1230 . . . 4  |-  ( ph  ->  ( B  .,  ( C  .+  D ) )  =  ( ( B 
.,  C )  .+^  ( B  .,  D ) ) )
2112phlsrng 18433 . . . . . 6  |-  ( W  e.  PreHil  ->  F  e.  *Ring )
22 srngrng 17284 . . . . . 6  |-  ( F  e.  *Ring  ->  F  e.  Ring )
23 rngcmn 17016 . . . . . 6  |-  ( F  e.  Ring  ->  F  e. CMnd
)
241, 21, 22, 234syl 21 . . . . 5  |-  ( ph  ->  F  e. CMnd )
25 eqid 2467 . . . . . . 7  |-  ( Base `  F )  =  (
Base `  F )
2612, 13, 8, 25ipcl 18435 . . . . . 6  |-  ( ( W  e.  PreHil  /\  B  e.  V  /\  C  e.  V )  ->  ( B  .,  C )  e.  ( Base `  F
) )
271, 3, 6, 26syl3anc 1228 . . . . 5  |-  ( ph  ->  ( B  .,  C
)  e.  ( Base `  F ) )
2812, 13, 8, 25ipcl 18435 . . . . . 6  |-  ( ( W  e.  PreHil  /\  B  e.  V  /\  D  e.  V )  ->  ( B  .,  D )  e.  ( Base `  F
) )
291, 3, 7, 28syl3anc 1228 . . . . 5  |-  ( ph  ->  ( B  .,  D
)  e.  ( Base `  F ) )
3025, 14cmncom 16610 . . . . 5  |-  ( ( F  e. CMnd  /\  ( B  .,  C )  e.  ( Base `  F
)  /\  ( B  .,  D )  e.  (
Base `  F )
)  ->  ( ( B  .,  C )  .+^  ( B  .,  D ) )  =  ( ( B  .,  D ) 
.+^  ( B  .,  C ) ) )
3124, 27, 29, 30syl3anc 1228 . . . 4  |-  ( ph  ->  ( ( B  .,  C )  .+^  ( B 
.,  D ) )  =  ( ( B 
.,  D )  .+^  ( B  .,  C ) ) )
3220, 31eqtrd 2508 . . 3  |-  ( ph  ->  ( B  .,  ( C  .+  D ) )  =  ( ( B 
.,  D )  .+^  ( B  .,  C ) ) )
3318, 32oveq12d 6300 . 2  |-  ( ph  ->  ( ( A  .,  ( C  .+  D ) )  .+^  ( B  .,  ( C  .+  D
) ) )  =  ( ( ( A 
.,  C )  .+^  ( A  .,  D ) )  .+^  ( ( B  .,  D )  .+^  ( B  .,  C ) ) ) )
3412, 13, 8, 25ipcl 18435 . . . 4  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  C  e.  V )  ->  ( A  .,  C )  e.  ( Base `  F
) )
351, 2, 6, 34syl3anc 1228 . . 3  |-  ( ph  ->  ( A  .,  C
)  e.  ( Base `  F ) )
3612, 13, 8, 25ipcl 18435 . . . 4  |-  ( ( W  e.  PreHil  /\  A  e.  V  /\  D  e.  V )  ->  ( A  .,  D )  e.  ( Base `  F
) )
371, 2, 7, 36syl3anc 1228 . . 3  |-  ( ph  ->  ( A  .,  D
)  e.  ( Base `  F ) )
3825, 14cmn4 16613 . . 3  |-  ( ( F  e. CMnd  /\  (
( A  .,  C
)  e.  ( Base `  F )  /\  ( A  .,  D )  e.  ( Base `  F
) )  /\  (
( B  .,  D
)  e.  ( Base `  F )  /\  ( B  .,  C )  e.  ( Base `  F
) ) )  -> 
( ( ( A 
.,  C )  .+^  ( A  .,  D ) )  .+^  ( ( B  .,  D )  .+^  ( B  .,  C ) ) )  =  ( ( ( A  .,  C )  .+^  ( B 
.,  D ) ) 
.+^  ( ( A 
.,  D )  .+^  ( B  .,  C ) ) ) )
3924, 35, 37, 29, 27, 38syl122anc 1237 . 2  |-  ( ph  ->  ( ( ( A 
.,  C )  .+^  ( A  .,  D ) )  .+^  ( ( B  .,  D )  .+^  ( B  .,  C ) ) )  =  ( ( ( A  .,  C )  .+^  ( B 
.,  D ) ) 
.+^  ( ( A 
.,  D )  .+^  ( B  .,  C ) ) ) )
4016, 33, 393eqtrd 2512 1  |-  ( ph  ->  ( ( A  .+  B )  .,  ( C  .+  D ) )  =  ( ( ( A  .,  C ) 
.+^  ( B  .,  D ) )  .+^  ( ( A  .,  D )  .+^  ( B 
.,  C ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   ` cfv 5586  (class class class)co 6282   Basecbs 14486   +g cplusg 14551  Scalarcsca 14554   .icip 14556  CMndccmn 16594   Ringcrg 16986   *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-rmo 2822  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-grp 15858  df-minusg 15859  df-ghm 16060  df-cmn 16596  df-abl 16597  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:  cph2di  21388
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