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Theorem normlem9 22621
Description: Lemma used to derive properties of norm. (Contributed by NM, 30-Jun-2005.) (New usage is discouraged.)
Hypotheses
Ref Expression
normlem8.1  |-  A  e. 
~H
normlem8.2  |-  B  e. 
~H
normlem8.3  |-  C  e. 
~H
normlem8.4  |-  D  e. 
~H
Assertion
Ref Expression
normlem9  |-  ( ( A  -h  B ) 
.ih  ( C  -h  D ) )  =  ( ( ( A 
.ih  C )  +  ( B  .ih  D
) )  -  (
( A  .ih  D
)  +  ( B 
.ih  C ) ) )

Proof of Theorem normlem9
StepHypRef Expression
1 normlem8.1 . . . 4  |-  A  e. 
~H
2 normlem8.2 . . . 4  |-  B  e. 
~H
31, 2hvsubvali 22524 . . 3  |-  ( A  -h  B )  =  ( A  +h  ( -u 1  .h  B ) )
4 normlem8.3 . . . 4  |-  C  e. 
~H
5 normlem8.4 . . . 4  |-  D  e. 
~H
64, 5hvsubvali 22524 . . 3  |-  ( C  -h  D )  =  ( C  +h  ( -u 1  .h  D ) )
73, 6oveq12i 6094 . 2  |-  ( ( A  -h  B ) 
.ih  ( C  -h  D ) )  =  ( ( A  +h  ( -u 1  .h  B
) )  .ih  ( C  +h  ( -u 1  .h  D ) ) )
8 neg1cn 10068 . . . 4  |-  -u 1  e.  CC
98, 2hvmulcli 22518 . . 3  |-  ( -u
1  .h  B )  e.  ~H
108, 5hvmulcli 22518 . . 3  |-  ( -u
1  .h  D )  e.  ~H
111, 9, 4, 10normlem8 22620 . 2  |-  ( ( A  +h  ( -u
1  .h  B ) )  .ih  ( C  +h  ( -u 1  .h  D ) ) )  =  ( ( ( A  .ih  C )  +  ( ( -u
1  .h  B ) 
.ih  ( -u 1  .h  D ) ) )  +  ( ( A 
.ih  ( -u 1  .h  D ) )  +  ( ( -u 1  .h  B )  .ih  C
) ) )
12 ax-his3 22587 . . . . . . 7  |-  ( (
-u 1  e.  CC  /\  B  e.  ~H  /\  ( -u 1  .h  D
)  e.  ~H )  ->  ( ( -u 1  .h  B )  .ih  ( -u 1  .h  D ) )  =  ( -u
1  x.  ( B 
.ih  ( -u 1  .h  D ) ) ) )
138, 2, 10, 12mp3an 1280 . . . . . 6  |-  ( (
-u 1  .h  B
)  .ih  ( -u 1  .h  D ) )  =  ( -u 1  x.  ( B  .ih  ( -u 1  .h  D ) ) )
14 his5 22589 . . . . . . . 8  |-  ( (
-u 1  e.  CC  /\  B  e.  ~H  /\  D  e.  ~H )  ->  ( B  .ih  ( -u 1  .h  D ) )  =  ( ( * `  -u 1
)  x.  ( B 
.ih  D ) ) )
158, 2, 5, 14mp3an 1280 . . . . . . 7  |-  ( B 
.ih  ( -u 1  .h  D ) )  =  ( ( * `  -u 1 )  x.  ( B  .ih  D ) )
1615oveq2i 6093 . . . . . 6  |-  ( -u
1  x.  ( B 
.ih  ( -u 1  .h  D ) ) )  =  ( -u 1  x.  ( ( * `  -u 1 )  x.  ( B  .ih  D ) ) )
17 1re 9091 . . . . . . . . . . . 12  |-  1  e.  RR
1817renegcli 9363 . . . . . . . . . . 11  |-  -u 1  e.  RR
19 cjre 11945 . . . . . . . . . . 11  |-  ( -u
1  e.  RR  ->  ( * `  -u 1
)  =  -u 1
)
2018, 19ax-mp 8 . . . . . . . . . 10  |-  ( * `
 -u 1 )  = 
-u 1
2120oveq2i 6093 . . . . . . . . 9  |-  ( -u
1  x.  ( * `
 -u 1 ) )  =  ( -u 1  x.  -u 1 )
22 ax-1cn 9049 . . . . . . . . . 10  |-  1  e.  CC
2322, 22mul2negi 9482 . . . . . . . . 9  |-  ( -u
1  x.  -u 1
)  =  ( 1  x.  1 )
2422mulid2i 9094 . . . . . . . . 9  |-  ( 1  x.  1 )  =  1
2521, 23, 243eqtri 2461 . . . . . . . 8  |-  ( -u
1  x.  ( * `
 -u 1 ) )  =  1
2625oveq1i 6092 . . . . . . 7  |-  ( (
-u 1  x.  (
* `  -u 1 ) )  x.  ( B 
.ih  D ) )  =  ( 1  x.  ( B  .ih  D
) )
278cjcli 11975 . . . . . . . 8  |-  ( * `
 -u 1 )  e.  CC
282, 5hicli 22584 . . . . . . . 8  |-  ( B 
.ih  D )  e.  CC
298, 27, 28mulassi 9100 . . . . . . 7  |-  ( (
-u 1  x.  (
* `  -u 1 ) )  x.  ( B 
.ih  D ) )  =  ( -u 1  x.  ( ( * `  -u 1 )  x.  ( B  .ih  D ) ) )
3028mulid2i 9094 . . . . . . 7  |-  ( 1  x.  ( B  .ih  D ) )  =  ( B  .ih  D )
3126, 29, 303eqtr3i 2465 . . . . . 6  |-  ( -u
1  x.  ( ( * `  -u 1
)  x.  ( B 
.ih  D ) ) )  =  ( B 
.ih  D )
3213, 16, 313eqtri 2461 . . . . 5  |-  ( (
-u 1  .h  B
)  .ih  ( -u 1  .h  D ) )  =  ( B  .ih  D
)
3332oveq2i 6093 . . . 4  |-  ( ( A  .ih  C )  +  ( ( -u
1  .h  B ) 
.ih  ( -u 1  .h  D ) ) )  =  ( ( A 
.ih  C )  +  ( B  .ih  D
) )
34 his5 22589 . . . . . . . 8  |-  ( (
-u 1  e.  CC  /\  A  e.  ~H  /\  D  e.  ~H )  ->  ( A  .ih  ( -u 1  .h  D ) )  =  ( ( * `  -u 1
)  x.  ( A 
.ih  D ) ) )
358, 1, 5, 34mp3an 1280 . . . . . . 7  |-  ( A 
.ih  ( -u 1  .h  D ) )  =  ( ( * `  -u 1 )  x.  ( A  .ih  D ) )
3620oveq1i 6092 . . . . . . 7  |-  ( ( * `  -u 1
)  x.  ( A 
.ih  D ) )  =  ( -u 1  x.  ( A  .ih  D
) )
371, 5hicli 22584 . . . . . . . 8  |-  ( A 
.ih  D )  e.  CC
3837mulm1i 9479 . . . . . . 7  |-  ( -u
1  x.  ( A 
.ih  D ) )  =  -u ( A  .ih  D )
3935, 36, 383eqtri 2461 . . . . . 6  |-  ( A 
.ih  ( -u 1  .h  D ) )  = 
-u ( A  .ih  D )
40 ax-his3 22587 . . . . . . . 8  |-  ( (
-u 1  e.  CC  /\  B  e.  ~H  /\  C  e.  ~H )  ->  ( ( -u 1  .h  B )  .ih  C
)  =  ( -u
1  x.  ( B 
.ih  C ) ) )
418, 2, 4, 40mp3an 1280 . . . . . . 7  |-  ( (
-u 1  .h  B
)  .ih  C )  =  ( -u 1  x.  ( B  .ih  C
) )
422, 4hicli 22584 . . . . . . . 8  |-  ( B 
.ih  C )  e.  CC
4342mulm1i 9479 . . . . . . 7  |-  ( -u
1  x.  ( B 
.ih  C ) )  =  -u ( B  .ih  C )
4441, 43eqtri 2457 . . . . . 6  |-  ( (
-u 1  .h  B
)  .ih  C )  =  -u ( B  .ih  C )
4539, 44oveq12i 6094 . . . . 5  |-  ( ( A  .ih  ( -u
1  .h  D ) )  +  ( (
-u 1  .h  B
)  .ih  C )
)  =  ( -u ( A  .ih  D )  +  -u ( B  .ih  C ) )
4637, 42negdii 9385 . . . . 5  |-  -u (
( A  .ih  D
)  +  ( B 
.ih  C ) )  =  ( -u ( A  .ih  D )  + 
-u ( B  .ih  C ) )
4745, 46eqtr4i 2460 . . . 4  |-  ( ( A  .ih  ( -u
1  .h  D ) )  +  ( (
-u 1  .h  B
)  .ih  C )
)  =  -u (
( A  .ih  D
)  +  ( B 
.ih  C ) )
4833, 47oveq12i 6094 . . 3  |-  ( ( ( A  .ih  C
)  +  ( (
-u 1  .h  B
)  .ih  ( -u 1  .h  D ) ) )  +  ( ( A 
.ih  ( -u 1  .h  D ) )  +  ( ( -u 1  .h  B )  .ih  C
) ) )  =  ( ( ( A 
.ih  C )  +  ( B  .ih  D
) )  +  -u ( ( A  .ih  D )  +  ( B 
.ih  C ) ) )
491, 4hicli 22584 . . . . 5  |-  ( A 
.ih  C )  e.  CC
5049, 28addcli 9095 . . . 4  |-  ( ( A  .ih  C )  +  ( B  .ih  D ) )  e.  CC
5137, 42addcli 9095 . . . 4  |-  ( ( A  .ih  D )  +  ( B  .ih  C ) )  e.  CC
5250, 51negsubi 9379 . . 3  |-  ( ( ( A  .ih  C
)  +  ( B 
.ih  D ) )  +  -u ( ( A 
.ih  D )  +  ( B  .ih  C
) ) )  =  ( ( ( A 
.ih  C )  +  ( B  .ih  D
) )  -  (
( A  .ih  D
)  +  ( B 
.ih  C ) ) )
5348, 52eqtri 2457 . 2  |-  ( ( ( A  .ih  C
)  +  ( (
-u 1  .h  B
)  .ih  ( -u 1  .h  D ) ) )  +  ( ( A 
.ih  ( -u 1  .h  D ) )  +  ( ( -u 1  .h  B )  .ih  C
) ) )  =  ( ( ( A 
.ih  C )  +  ( B  .ih  D
) )  -  (
( A  .ih  D
)  +  ( B 
.ih  C ) ) )
547, 11, 533eqtri 2461 1  |-  ( ( A  -h  B ) 
.ih  ( C  -h  D ) )  =  ( ( ( A 
.ih  C )  +  ( B  .ih  D
) )  -  (
( A  .ih  D
)  +  ( B 
.ih  C ) ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1653    e. wcel 1726   ` cfv 5455  (class class class)co 6082   CCcc 8989   RRcr 8990   1c1 8992    + caddc 8994    x. cmul 8996    - cmin 9292   -ucneg 9293   *ccj 11902   ~Hchil 22423    +h cva 22424    .h csm 22425    .ih csp 22426    -h cmv 22429
This theorem is referenced by:  bcseqi  22623  normlem9at  22624  normpari  22657  polid2i  22660
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-resscn 9048  ax-1cn 9049  ax-icn 9050  ax-addcl 9051  ax-addrcl 9052  ax-mulcl 9053  ax-mulrcl 9054  ax-mulcom 9055  ax-addass 9056  ax-mulass 9057  ax-distr 9058  ax-i2m1 9059  ax-1ne0 9060  ax-1rid 9061  ax-rnegex 9062  ax-rrecex 9063  ax-cnre 9064  ax-pre-lttri 9065  ax-pre-lttrn 9066  ax-pre-ltadd 9067  ax-pre-mulgt0 9068  ax-hfvadd 22504  ax-hfvmul 22509  ax-hfi 22582  ax-his1 22585  ax-his2 22586  ax-his3 22587
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rmo 2714  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-po 4504  df-so 4505  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-riota 6550  df-er 6906  df-en 7111  df-dom 7112  df-sdom 7113  df-pnf 9123  df-mnf 9124  df-xr 9125  df-ltxr 9126  df-le 9127  df-sub 9294  df-neg 9295  df-div 9679  df-2 10059  df-cj 11905  df-re 11906  df-im 11907  df-hvsub 22475
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