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Theorem normlem9 21697
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 21600 . . 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 21600 . . 3  |-  ( C  -h  D )  =  ( C  +h  ( -u 1  .h  D ) )
73, 6oveq12i 5870 . 2  |-  ( ( A  -h  B ) 
.ih  ( C  -h  D ) )  =  ( ( A  +h  ( -u 1  .h  B
) )  .ih  ( C  +h  ( -u 1  .h  D ) ) )
8 neg1cn 9813 . . . 4  |-  -u 1  e.  CC
98, 2hvmulcli 21594 . . 3  |-  ( -u
1  .h  B )  e.  ~H
108, 5hvmulcli 21594 . . 3  |-  ( -u
1  .h  D )  e.  ~H
111, 9, 4, 10normlem8 21696 . 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 21663 . . . . . . 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 21665 . . . . . . . 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 5869 . . . . . 6  |-  ( -u
1  x.  ( B 
.ih  ( -u 1  .h  D ) ) )  =  ( -u 1  x.  ( ( * `  -u 1 )  x.  ( B  .ih  D ) ) )
17 1re 8837 . . . . . . . . . . . 12  |-  1  e.  RR
1817renegcli 9108 . . . . . . . . . . 11  |-  -u 1  e.  RR
19 cjre 11624 . . . . . . . . . . 11  |-  ( -u
1  e.  RR  ->  ( * `  -u 1
)  =  -u 1
)
2018, 19ax-mp 8 . . . . . . . . . 10  |-  ( * `
 -u 1 )  = 
-u 1
2120oveq2i 5869 . . . . . . . . 9  |-  ( -u
1  x.  ( * `
 -u 1 ) )  =  ( -u 1  x.  -u 1 )
22 ax-1cn 8795 . . . . . . . . . 10  |-  1  e.  CC
2322, 22mul2negi 9227 . . . . . . . . 9  |-  ( -u
1  x.  -u 1
)  =  ( 1  x.  1 )
2422mulid2i 8840 . . . . . . . . 9  |-  ( 1  x.  1 )  =  1
2521, 23, 243eqtri 2307 . . . . . . . 8  |-  ( -u
1  x.  ( * `
 -u 1 ) )  =  1
2625oveq1i 5868 . . . . . . 7  |-  ( (
-u 1  x.  (
* `  -u 1 ) )  x.  ( B 
.ih  D ) )  =  ( 1  x.  ( B  .ih  D
) )
278cjcli 11654 . . . . . . . 8  |-  ( * `
 -u 1 )  e.  CC
282, 5hicli 21660 . . . . . . . 8  |-  ( B 
.ih  D )  e.  CC
298, 27, 28mulassi 8846 . . . . . . 7  |-  ( (
-u 1  x.  (
* `  -u 1 ) )  x.  ( B 
.ih  D ) )  =  ( -u 1  x.  ( ( * `  -u 1 )  x.  ( B  .ih  D ) ) )
3028mulid2i 8840 . . . . . . 7  |-  ( 1  x.  ( B  .ih  D ) )  =  ( B  .ih  D )
3126, 29, 303eqtr3i 2311 . . . . . 6  |-  ( -u
1  x.  ( ( * `  -u 1
)  x.  ( B 
.ih  D ) ) )  =  ( B 
.ih  D )
3213, 16, 313eqtri 2307 . . . . 5  |-  ( (
-u 1  .h  B
)  .ih  ( -u 1  .h  D ) )  =  ( B  .ih  D
)
3332oveq2i 5869 . . . 4  |-  ( ( A  .ih  C )  +  ( ( -u
1  .h  B ) 
.ih  ( -u 1  .h  D ) ) )  =  ( ( A 
.ih  C )  +  ( B  .ih  D
) )
34 his5 21665 . . . . . . . 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 5868 . . . . . . 7  |-  ( ( * `  -u 1
)  x.  ( A 
.ih  D ) )  =  ( -u 1  x.  ( A  .ih  D
) )
371, 5hicli 21660 . . . . . . . 8  |-  ( A 
.ih  D )  e.  CC
3837mulm1i 9224 . . . . . . 7  |-  ( -u
1  x.  ( A 
.ih  D ) )  =  -u ( A  .ih  D )
3935, 36, 383eqtri 2307 . . . . . 6  |-  ( A 
.ih  ( -u 1  .h  D ) )  = 
-u ( A  .ih  D )
40 ax-his3 21663 . . . . . . . 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 21660 . . . . . . . 8  |-  ( B 
.ih  C )  e.  CC
4342mulm1i 9224 . . . . . . 7  |-  ( -u
1  x.  ( B 
.ih  C ) )  =  -u ( B  .ih  C )
4441, 43eqtri 2303 . . . . . 6  |-  ( (
-u 1  .h  B
)  .ih  C )  =  -u ( B  .ih  C )
4539, 44oveq12i 5870 . . . . 5  |-  ( ( A  .ih  ( -u
1  .h  D ) )  +  ( (
-u 1  .h  B
)  .ih  C )
)  =  ( -u ( A  .ih  D )  +  -u ( B  .ih  C ) )
4637, 42negdii 9130 . . . . 5  |-  -u (
( A  .ih  D
)  +  ( B 
.ih  C ) )  =  ( -u ( A  .ih  D )  + 
-u ( B  .ih  C ) )
4745, 46eqtr4i 2306 . . . 4  |-  ( ( A  .ih  ( -u
1  .h  D ) )  +  ( (
-u 1  .h  B
)  .ih  C )
)  =  -u (
( A  .ih  D
)  +  ( B 
.ih  C ) )
4833, 47oveq12i 5870 . . 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 21660 . . . . 5  |-  ( A 
.ih  C )  e.  CC
5049, 28addcli 8841 . . . 4  |-  ( ( A  .ih  C )  +  ( B  .ih  D ) )  e.  CC
5137, 42addcli 8841 . . . 4  |-  ( ( A  .ih  D )  +  ( B  .ih  C ) )  e.  CC
5250, 51negsubi 9124 . . 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 2303 . 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 2307 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 1623    e. wcel 1684   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   1c1 8738    + caddc 8740    x. cmul 8742    - cmin 9037   -ucneg 9038   *ccj 11581   ~Hchil 21499    +h cva 21500    .h csm 21501    .ih csp 21502    -h cmv 21505
This theorem is referenced by:  bcseqi  21699  normlem9at  21700  normpari  21733  polid2i  21736
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-hfvadd 21580  ax-hfvmul 21585  ax-hfi 21658  ax-his1 21661  ax-his2 21662  ax-his3 21663
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-2 9804  df-cj 11584  df-re 11585  df-im 11586  df-hvsub 21551
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