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Theorem hvsubass 24445
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 10424 . . . 4  |-  -u 1  e.  CC
2 hvmulcl 24414 . . . 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 24442 . . 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 1252 . 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 24417 . . . 4  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A  -h  B
)  =  ( A  +h  ( -u 1  .h  B ) ) )
763adant3 1008 . . 3  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( A  -h  B )  =  ( A  +h  ( -u 1  .h  B ) ) )
87oveq1d 6105 . 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 988 . . . 4  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  A  e.  ~H )
10 hvaddcl 24413 . . . . 5  |-  ( ( B  e.  ~H  /\  C  e.  ~H )  ->  ( B  +h  C
)  e.  ~H )
11103adant1 1006 . . . 4  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( B  +h  C )  e. 
~H )
12 hvsubval 24417 . . . 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 24417 . . . . . . 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 1006 . . . . 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 24409 . . . . . . 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 1301 . . . . . 6  |-  ( ( B  e.  ~H  /\  C  e.  ~H )  ->  ( -u 1  .h  ( B  +h  C
) )  =  ( ( -u 1  .h  B )  +h  ( -u 1  .h  C ) ) )
19183adant1 1006 . . . . 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 2477 . . . 4  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( -u 1  .h  B
)  -h  C )  =  ( -u 1  .h  ( B  +h  C
) ) )
2120oveq2d 6106 . . 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 2477 . 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 2484 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 965    = wceq 1369    e. wcel 1756  (class class class)co 6090   CCcc 9279   1c1 9282   -ucneg 9595   ~Hchil 24320    +h cva 24321    .h csm 24322    -h cmv 24326
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-mulcom 9345  ax-addass 9346  ax-mulass 9347  ax-distr 9348  ax-i2m1 9349  ax-1ne0 9350  ax-1rid 9351  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356  ax-pre-ltadd 9357  ax-hfvadd 24401  ax-hvass 24403  ax-hfvmul 24406  ax-hvdistr1 24409
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-id 4635  df-po 4640  df-so 4641  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-riota 6051  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-er 7100  df-en 7310  df-dom 7311  df-sdom 7312  df-pnf 9419  df-mnf 9420  df-ltxr 9422  df-sub 9596  df-neg 9597  df-hvsub 24372
This theorem is referenced by:  hvsub32  24446  hvsubassi  24456  pjhthlem1  24793
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