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Theorem hvsubass 26540
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 10713 . . . 4  |-  -u 1  e.  CC
2 hvmulcl 26509 . . . 4  |-  ( (
-u 1  e.  CC  /\  B  e.  ~H )  ->  ( -u 1  .h  B )  e.  ~H )
31, 2mpan 674 . . 3  |-  ( B  e.  ~H  ->  ( -u 1  .h  B )  e.  ~H )
4 hvaddsubass 26537 . . 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 1298 . 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 26512 . . . 4  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A  -h  B
)  =  ( A  +h  ( -u 1  .h  B ) ) )
763adant3 1025 . . 3  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( A  -h  B )  =  ( A  +h  ( -u 1  .h  B ) ) )
87oveq1d 6320 . 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 1005 . . . 4  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  A  e.  ~H )
10 hvaddcl 26508 . . . . 5  |-  ( ( B  e.  ~H  /\  C  e.  ~H )  ->  ( B  +h  C
)  e.  ~H )
11103adant1 1023 . . . 4  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( B  +h  C )  e. 
~H )
12 hvsubval 26512 . . . 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 665 . . 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 26512 . . . . . . 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 473 . . . . . 6  |-  ( ( B  e.  ~H  /\  C  e.  ~H )  ->  ( ( -u 1  .h  B )  -h  C
)  =  ( (
-u 1  .h  B
)  +h  ( -u
1  .h  C ) ) )
16153adant1 1023 . . . . 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 26504 . . . . . . 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 1347 . . . . . 6  |-  ( ( B  e.  ~H  /\  C  e.  ~H )  ->  ( -u 1  .h  ( B  +h  C
) )  =  ( ( -u 1  .h  B )  +h  ( -u 1  .h  C ) ) )
19183adant1 1023 . . . . 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 2473 . . . 4  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( -u 1  .h  B
)  -h  C )  =  ( -u 1  .h  ( B  +h  C
) ) )
2120oveq2d 6321 . . 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 2473 . 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 2480 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 982    = wceq 1437    e. wcel 1870  (class class class)co 6305   CCcc 9536   1c1 9539   -ucneg 9860   ~Hchil 26415    +h cva 26416    .h csm 26417    -h cmv 26421
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-hfvadd 26496  ax-hvass 26498  ax-hfvmul 26501  ax-hvdistr1 26504
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-po 4775  df-so 4776  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-pnf 9676  df-mnf 9677  df-ltxr 9679  df-sub 9861  df-neg 9862  df-hvsub 26467
This theorem is referenced by:  hvsub32  26541  hvsubassi  26551  pjhthlem1  26887
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