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Theorem normlem9 21713
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 21616 . . 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 21616 . . 3  |-  ( C  -h  D )  =  ( C  +h  ( -u 1  .h  D ) )
73, 6oveq12i 5886 . 2  |-  ( ( A  -h  B ) 
.ih  ( C  -h  D ) )  =  ( ( A  +h  ( -u 1  .h  B
) )  .ih  ( C  +h  ( -u 1  .h  D ) ) )
8 neg1cn 9829 . . . 4  |-  -u 1  e.  CC
98, 2hvmulcli 21610 . . 3  |-  ( -u
1  .h  B )  e.  ~H
108, 5hvmulcli 21610 . . 3  |-  ( -u
1  .h  D )  e.  ~H
111, 9, 4, 10normlem8 21712 . 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 21679 . . . . . . 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 1277 . . . . . 6  |-  ( (
-u 1  .h  B
)  .ih  ( -u 1  .h  D ) )  =  ( -u 1  x.  ( B  .ih  ( -u 1  .h  D ) ) )
14 his5 21681 . . . . . . . 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 1277 . . . . . . 7  |-  ( B 
.ih  ( -u 1  .h  D ) )  =  ( ( * `  -u 1 )  x.  ( B  .ih  D ) )
1615oveq2i 5885 . . . . . 6  |-  ( -u
1  x.  ( B 
.ih  ( -u 1  .h  D ) ) )  =  ( -u 1  x.  ( ( * `  -u 1 )  x.  ( B  .ih  D ) ) )
17 1re 8853 . . . . . . . . . . . 12  |-  1  e.  RR
1817renegcli 9124 . . . . . . . . . . 11  |-  -u 1  e.  RR
19 cjre 11640 . . . . . . . . . . 11  |-  ( -u
1  e.  RR  ->  ( * `  -u 1
)  =  -u 1
)
2018, 19ax-mp 8 . . . . . . . . . 10  |-  ( * `
 -u 1 )  = 
-u 1
2120oveq2i 5885 . . . . . . . . 9  |-  ( -u
1  x.  ( * `
 -u 1 ) )  =  ( -u 1  x.  -u 1 )
22 ax-1cn 8811 . . . . . . . . . 10  |-  1  e.  CC
2322, 22mul2negi 9243 . . . . . . . . 9  |-  ( -u
1  x.  -u 1
)  =  ( 1  x.  1 )
2422mulid2i 8856 . . . . . . . . 9  |-  ( 1  x.  1 )  =  1
2521, 23, 243eqtri 2320 . . . . . . . 8  |-  ( -u
1  x.  ( * `
 -u 1 ) )  =  1
2625oveq1i 5884 . . . . . . 7  |-  ( (
-u 1  x.  (
* `  -u 1 ) )  x.  ( B 
.ih  D ) )  =  ( 1  x.  ( B  .ih  D
) )
278cjcli 11670 . . . . . . . 8  |-  ( * `
 -u 1 )  e.  CC
282, 5hicli 21676 . . . . . . . 8  |-  ( B 
.ih  D )  e.  CC
298, 27, 28mulassi 8862 . . . . . . 7  |-  ( (
-u 1  x.  (
* `  -u 1 ) )  x.  ( B 
.ih  D ) )  =  ( -u 1  x.  ( ( * `  -u 1 )  x.  ( B  .ih  D ) ) )
3028mulid2i 8856 . . . . . . 7  |-  ( 1  x.  ( B  .ih  D ) )  =  ( B  .ih  D )
3126, 29, 303eqtr3i 2324 . . . . . 6  |-  ( -u
1  x.  ( ( * `  -u 1
)  x.  ( B 
.ih  D ) ) )  =  ( B 
.ih  D )
3213, 16, 313eqtri 2320 . . . . 5  |-  ( (
-u 1  .h  B
)  .ih  ( -u 1  .h  D ) )  =  ( B  .ih  D
)
3332oveq2i 5885 . . . 4  |-  ( ( A  .ih  C )  +  ( ( -u
1  .h  B ) 
.ih  ( -u 1  .h  D ) ) )  =  ( ( A 
.ih  C )  +  ( B  .ih  D
) )
34 his5 21681 . . . . . . . 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 1277 . . . . . . 7  |-  ( A 
.ih  ( -u 1  .h  D ) )  =  ( ( * `  -u 1 )  x.  ( A  .ih  D ) )
3620oveq1i 5884 . . . . . . 7  |-  ( ( * `  -u 1
)  x.  ( A 
.ih  D ) )  =  ( -u 1  x.  ( A  .ih  D
) )
371, 5hicli 21676 . . . . . . . 8  |-  ( A 
.ih  D )  e.  CC
3837mulm1i 9240 . . . . . . 7  |-  ( -u
1  x.  ( A 
.ih  D ) )  =  -u ( A  .ih  D )
3935, 36, 383eqtri 2320 . . . . . 6  |-  ( A 
.ih  ( -u 1  .h  D ) )  = 
-u ( A  .ih  D )
40 ax-his3 21679 . . . . . . . 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 1277 . . . . . . 7  |-  ( (
-u 1  .h  B
)  .ih  C )  =  ( -u 1  x.  ( B  .ih  C
) )
422, 4hicli 21676 . . . . . . . 8  |-  ( B 
.ih  C )  e.  CC
4342mulm1i 9240 . . . . . . 7  |-  ( -u
1  x.  ( B 
.ih  C ) )  =  -u ( B  .ih  C )
4441, 43eqtri 2316 . . . . . 6  |-  ( (
-u 1  .h  B
)  .ih  C )  =  -u ( B  .ih  C )
4539, 44oveq12i 5886 . . . . 5  |-  ( ( A  .ih  ( -u
1  .h  D ) )  +  ( (
-u 1  .h  B
)  .ih  C )
)  =  ( -u ( A  .ih  D )  +  -u ( B  .ih  C ) )
4637, 42negdii 9146 . . . . 5  |-  -u (
( A  .ih  D
)  +  ( B 
.ih  C ) )  =  ( -u ( A  .ih  D )  + 
-u ( B  .ih  C ) )
4745, 46eqtr4i 2319 . . . 4  |-  ( ( A  .ih  ( -u
1  .h  D ) )  +  ( (
-u 1  .h  B
)  .ih  C )
)  =  -u (
( A  .ih  D
)  +  ( B 
.ih  C ) )
4833, 47oveq12i 5886 . . 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 21676 . . . . 5  |-  ( A 
.ih  C )  e.  CC
5049, 28addcli 8857 . . . 4  |-  ( ( A  .ih  C )  +  ( B  .ih  D ) )  e.  CC
5137, 42addcli 8857 . . . 4  |-  ( ( A  .ih  D )  +  ( B  .ih  C ) )  e.  CC
5250, 51negsubi 9140 . . 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 2316 . 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 2320 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 1632    e. wcel 1696   ` cfv 5271  (class class class)co 5874   CCcc 8751   RRcr 8752   1c1 8754    + caddc 8756    x. cmul 8758    - cmin 9053   -ucneg 9054   *ccj 11597   ~Hchil 21515    +h cva 21516    .h csm 21517    .ih csp 21518    -h cmv 21521
This theorem is referenced by:  bcseqi  21715  normlem9at  21716  normpari  21749  polid2i  21752
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-hfvadd 21596  ax-hfvmul 21601  ax-hfi 21674  ax-his1 21677  ax-his2 21678  ax-his3 21679
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-2 9820  df-cj 11600  df-re 11601  df-im 11602  df-hvsub 21567
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