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Theorem hvaddsub12 24445
Description: Commutative/associative law. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.)
Assertion
Ref Expression
hvaddsub12  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( A  +h  ( B  -h  C ) )  =  ( B  +h  ( A  -h  C ) ) )

Proof of Theorem hvaddsub12
StepHypRef Expression
1 neg1cn 10430 . . . 4  |-  -u 1  e.  CC
2 hvmulcl 24420 . . . 4  |-  ( (
-u 1  e.  CC  /\  C  e.  ~H )  ->  ( -u 1  .h  C )  e.  ~H )
31, 2mpan 670 . . 3  |-  ( C  e.  ~H  ->  ( -u 1  .h  C )  e.  ~H )
4 hvadd12 24442 . . 3  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  ( -u 1  .h  C )  e.  ~H )  -> 
( A  +h  ( B  +h  ( -u 1  .h  C ) ) )  =  ( B  +h  ( A  +h  ( -u 1  .h  C ) ) ) )
53, 4syl3an3 1253 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( A  +h  ( B  +h  ( -u 1  .h  C
) ) )  =  ( B  +h  ( A  +h  ( -u 1  .h  C ) ) ) )
6 hvsubval 24423 . . . 4  |-  ( ( B  e.  ~H  /\  C  e.  ~H )  ->  ( B  -h  C
)  =  ( B  +h  ( -u 1  .h  C ) ) )
76oveq2d 6112 . . 3  |-  ( ( B  e.  ~H  /\  C  e.  ~H )  ->  ( A  +h  ( B  -h  C ) )  =  ( A  +h  ( B  +h  ( -u 1  .h  C ) ) ) )
873adant1 1006 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( A  +h  ( B  -h  C ) )  =  ( A  +h  ( B  +h  ( -u 1  .h  C ) ) ) )
9 hvsubval 24423 . . . 4  |-  ( ( A  e.  ~H  /\  C  e.  ~H )  ->  ( A  -h  C
)  =  ( A  +h  ( -u 1  .h  C ) ) )
109oveq2d 6112 . . 3  |-  ( ( A  e.  ~H  /\  C  e.  ~H )  ->  ( B  +h  ( A  -h  C ) )  =  ( B  +h  ( A  +h  ( -u 1  .h  C ) ) ) )
11103adant2 1007 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( B  +h  ( A  -h  C ) )  =  ( B  +h  ( A  +h  ( -u 1  .h  C ) ) ) )
125, 8, 113eqtr4d 2485 1  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( A  +h  ( B  -h  C ) )  =  ( B  +h  ( A  -h  C ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756  (class class class)co 6096   CCcc 9285   1c1 9288   -ucneg 9601   ~Hchil 24326    +h cva 24327    .h csm 24328    -h cmv 24332
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 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-hvcom 24408  ax-hvass 24409  ax-hfvmul 24412
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 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-po 4646  df-so 4647  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-pnf 9425  df-mnf 9426  df-ltxr 9428  df-sub 9602  df-neg 9603  df-hvsub 24378
This theorem is referenced by:  5oalem1  25062  3oalem2  25071  pjcji  25092  pjclem4  25608  pj3si  25616
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