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Theorem hvsubass 25665
Description: Hilbert vector space associative law for subtraction. (Contributed by Mario Carneiro, 15-May-2014.) (New usage is discouraged.)
Assertion
Ref Expression
hvsubass  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( A  -h  B
)  -h  C )  =  ( A  -h  ( B  +h  C
) ) )

Proof of Theorem hvsubass
StepHypRef Expression
1 neg1cn 10639 . . . 4  |-  -u 1  e.  CC
2 hvmulcl 25634 . . . 4  |-  ( (
-u 1  e.  CC  /\  B  e.  ~H )  ->  ( -u 1  .h  B )  e.  ~H )
31, 2mpan 670 . . 3  |-  ( B  e.  ~H  ->  ( -u 1  .h  B )  e.  ~H )
4 hvaddsubass 25662 . . 3  |-  ( ( A  e.  ~H  /\  ( -u 1  .h  B
)  e.  ~H  /\  C  e.  ~H )  ->  ( ( A  +h  ( -u 1  .h  B
) )  -h  C
)  =  ( A  +h  ( ( -u
1  .h  B )  -h  C ) ) )
53, 4syl3an2 1262 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( A  +h  ( -u 1  .h  B ) )  -h  C )  =  ( A  +h  ( ( -u 1  .h  B )  -h  C
) ) )
6 hvsubval 25637 . . . 4  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A  -h  B
)  =  ( A  +h  ( -u 1  .h  B ) ) )
763adant3 1016 . . 3  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( A  -h  B )  =  ( A  +h  ( -u 1  .h  B ) ) )
87oveq1d 6299 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( A  -h  B
)  -h  C )  =  ( ( A  +h  ( -u 1  .h  B ) )  -h  C ) )
9 simp1 996 . . . 4  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  A  e.  ~H )
10 hvaddcl 25633 . . . . 5  |-  ( ( B  e.  ~H  /\  C  e.  ~H )  ->  ( B  +h  C
)  e.  ~H )
11103adant1 1014 . . . 4  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( B  +h  C )  e. 
~H )
12 hvsubval 25637 . . . 4  |-  ( ( A  e.  ~H  /\  ( B  +h  C
)  e.  ~H )  ->  ( A  -h  ( B  +h  C ) )  =  ( A  +h  ( -u 1  .h  ( B  +h  C ) ) ) )
139, 11, 12syl2anc 661 . . 3  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( A  -h  ( B  +h  C ) )  =  ( A  +h  ( -u 1  .h  ( B  +h  C ) ) ) )
14 hvsubval 25637 . . . . . . 7  |-  ( ( ( -u 1  .h  B )  e.  ~H  /\  C  e.  ~H )  ->  ( ( -u 1  .h  B )  -h  C
)  =  ( (
-u 1  .h  B
)  +h  ( -u
1  .h  C ) ) )
153, 14sylan 471 . . . . . 6  |-  ( ( B  e.  ~H  /\  C  e.  ~H )  ->  ( ( -u 1  .h  B )  -h  C
)  =  ( (
-u 1  .h  B
)  +h  ( -u
1  .h  C ) ) )
16153adant1 1014 . . . . 5  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( -u 1  .h  B
)  -h  C )  =  ( ( -u
1  .h  B )  +h  ( -u 1  .h  C ) ) )
17 ax-hvdistr1 25629 . . . . . . 7  |-  ( (
-u 1  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( -u 1  .h  ( B  +h  C
) )  =  ( ( -u 1  .h  B )  +h  ( -u 1  .h  C ) ) )
181, 17mp3an1 1311 . . . . . 6  |-  ( ( B  e.  ~H  /\  C  e.  ~H )  ->  ( -u 1  .h  ( B  +h  C
) )  =  ( ( -u 1  .h  B )  +h  ( -u 1  .h  C ) ) )
19183adant1 1014 . . . . 5  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( -u 1  .h  ( B  +h  C ) )  =  ( ( -u
1  .h  B )  +h  ( -u 1  .h  C ) ) )
2016, 19eqtr4d 2511 . . . 4  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( -u 1  .h  B
)  -h  C )  =  ( -u 1  .h  ( B  +h  C
) ) )
2120oveq2d 6300 . . 3  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( A  +h  ( ( -u
1  .h  B )  -h  C ) )  =  ( A  +h  ( -u 1  .h  ( B  +h  C ) ) ) )
2213, 21eqtr4d 2511 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( A  -h  ( B  +h  C ) )  =  ( A  +h  (
( -u 1  .h  B
)  -h  C ) ) )
235, 8, 223eqtr4d 2518 1  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( A  -h  B
)  -h  C )  =  ( A  -h  ( B  +h  C
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 973    = wceq 1379    e. wcel 1767  (class class class)co 6284   CCcc 9490   1c1 9493   -ucneg 9806   ~Hchil 25540    +h cva 25541    .h csm 25542    -h cmv 25546
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-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-hfvadd 25621  ax-hvass 25623  ax-hfvmul 25626  ax-hvdistr1 25629
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-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  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 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-pnf 9630  df-mnf 9631  df-ltxr 9633  df-sub 9807  df-neg 9808  df-hvsub 25592
This theorem is referenced by:  hvsub32  25666  hvsubassi  25676  pjhthlem1  26013
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