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Theorem hvsubass 21623
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 9813 . . . 4  |-  -u 1  e.  CC
2 hvmulcl 21593 . . . 4  |-  ( (
-u 1  e.  CC  /\  B  e.  ~H )  ->  ( -u 1  .h  B )  e.  ~H )
31, 2mpan 651 . . 3  |-  ( B  e.  ~H  ->  ( -u 1  .h  B )  e.  ~H )
4 hvaddsubass 21620 . . 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 1216 . 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 21596 . . . 4  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A  -h  B
)  =  ( A  +h  ( -u 1  .h  B ) ) )
763adant3 975 . . 3  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( A  -h  B )  =  ( A  +h  ( -u 1  .h  B ) ) )
87oveq1d 5873 . 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 955 . . . 4  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  A  e.  ~H )
10 hvaddcl 21592 . . . . 5  |-  ( ( B  e.  ~H  /\  C  e.  ~H )  ->  ( B  +h  C
)  e.  ~H )
11103adant1 973 . . . 4  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( B  +h  C )  e. 
~H )
12 hvsubval 21596 . . . 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 642 . . 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 21596 . . . . . . 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 457 . . . . . 6  |-  ( ( B  e.  ~H  /\  C  e.  ~H )  ->  ( ( -u 1  .h  B )  -h  C
)  =  ( (
-u 1  .h  B
)  +h  ( -u
1  .h  C ) ) )
16153adant1 973 . . . . 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 21588 . . . . . . 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 1264 . . . . . 6  |-  ( ( B  e.  ~H  /\  C  e.  ~H )  ->  ( -u 1  .h  ( B  +h  C
) )  =  ( ( -u 1  .h  B )  +h  ( -u 1  .h  C ) ) )
19183adant1 973 . . . . 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 2318 . . . 4  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( -u 1  .h  B
)  -h  C )  =  ( -u 1  .h  ( B  +h  C
) ) )
2120oveq2d 5874 . . 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 2318 . 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 2325 1  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( A  -h  B
)  -h  C )  =  ( A  -h  ( B  +h  C
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1623    e. wcel 1684  (class class class)co 5858   CCcc 8735   1c1 8738   -ucneg 9038   ~Hchil 21499    +h cva 21500    .h csm 21501    -h cmv 21505
This theorem is referenced by:  hvsub32  21624  hvsubassi  21634  pjhthlem1  21970
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-hfvadd 21580  ax-hvass 21582  ax-hfvmul 21585  ax-hvdistr1 21588
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-ltxr 8872  df-sub 9039  df-neg 9040  df-hvsub 21551
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