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Theorem hvsubass 21625
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 9815 . . . 4  |-  -u 1  e.  CC
2 hvmulcl 21595 . . . 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 21622 . . 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 21598 . . . 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 5875 . 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 21594 . . . . 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 21598 . . . 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 21598 . . . . . . 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 21590 . . . . . . 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 2320 . . . 4  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( -u 1  .h  B
)  -h  C )  =  ( -u 1  .h  ( B  +h  C
) ) )
2120oveq2d 5876 . . 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 2320 . 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 2327 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 1625    e. wcel 1686  (class class class)co 5860   CCcc 8737   1c1 8740   -ucneg 9040   ~Hchil 21501    +h cva 21502    .h csm 21503    -h cmv 21507
This theorem is referenced by:  hvsub32  21626  hvsubassi  21636  pjhthlem1  21972
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-resscn 8796  ax-1cn 8797  ax-icn 8798  ax-addcl 8799  ax-addrcl 8800  ax-mulcl 8801  ax-mulrcl 8802  ax-mulcom 8803  ax-addass 8804  ax-mulass 8805  ax-distr 8806  ax-i2m1 8807  ax-1ne0 8808  ax-1rid 8809  ax-rnegex 8810  ax-rrecex 8811  ax-cnre 8812  ax-pre-lttri 8813  ax-pre-lttrn 8814  ax-pre-ltadd 8815  ax-hfvadd 21582  ax-hvass 21584  ax-hfvmul 21587  ax-hvdistr1 21590
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 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4311  df-po 4316  df-so 4317  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-riota 6306  df-er 6662  df-en 6866  df-dom 6867  df-sdom 6868  df-pnf 8871  df-mnf 8872  df-ltxr 8874  df-sub 9041  df-neg 9042  df-hvsub 21553
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