MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ip2i Unicode version

Theorem ip2i 22286
Description: Equation 6.48 of [Ponnusamy] p. 362. (Contributed by NM, 26-Apr-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
ip1i.1  |-  X  =  ( BaseSet `  U )
ip1i.2  |-  G  =  ( +v `  U
)
ip1i.4  |-  S  =  ( .s OLD `  U
)
ip1i.7  |-  P  =  ( .i OLD `  U
)
ip1i.9  |-  U  e.  CPreHil
OLD
ip2i.8  |-  A  e.  X
ip2i.9  |-  B  e.  X
Assertion
Ref Expression
ip2i  |-  ( ( 2 S A ) P B )  =  ( 2  x.  ( A P B ) )

Proof of Theorem ip2i
StepHypRef Expression
1 ip1i.9 . . . . . 6  |-  U  e.  CPreHil
OLD
21phnvi 22274 . . . . 5  |-  U  e.  NrmCVec
3 ip2i.8 . . . . . 6  |-  A  e.  X
4 ip1i.1 . . . . . . 7  |-  X  =  ( BaseSet `  U )
5 ip1i.2 . . . . . . 7  |-  G  =  ( +v `  U
)
64, 5nvgcl 22056 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  A  e.  X )  ->  ( A G A )  e.  X )
72, 3, 3, 6mp3an 1279 . . . . 5  |-  ( A G A )  e.  X
8 ip2i.9 . . . . 5  |-  B  e.  X
9 ip1i.7 . . . . . 6  |-  P  =  ( .i OLD `  U
)
104, 9dipcl 22168 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  ( A G A )  e.  X  /\  B  e.  X )  ->  (
( A G A ) P B )  e.  CC )
112, 7, 8, 10mp3an 1279 . . . 4  |-  ( ( A G A ) P B )  e.  CC
1211addid1i 9213 . . 3  |-  ( ( ( A G A ) P B )  +  0 )  =  ( ( A G A ) P B )
13 ip1i.4 . . . . . . . 8  |-  S  =  ( .s OLD `  U
)
14 eqid 2408 . . . . . . . 8  |-  ( 0vec `  U )  =  (
0vec `  U )
154, 5, 13, 14nvrinv 22091 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  ( A G ( -u 1 S A ) )  =  ( 0vec `  U
) )
162, 3, 15mp2an 654 . . . . . 6  |-  ( A G ( -u 1 S A ) )  =  ( 0vec `  U
)
1716oveq1i 6054 . . . . 5  |-  ( ( A G ( -u
1 S A ) ) P B )  =  ( ( 0vec `  U ) P B )
184, 14, 9dip0l 22174 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  (
( 0vec `  U ) P B )  =  0 )
192, 8, 18mp2an 654 . . . . 5  |-  ( (
0vec `  U ) P B )  =  0
2017, 19eqtri 2428 . . . 4  |-  ( ( A G ( -u
1 S A ) ) P B )  =  0
2120oveq2i 6055 . . 3  |-  ( ( ( A G A ) P B )  +  ( ( A G ( -u 1 S A ) ) P B ) )  =  ( ( ( A G A ) P B )  +  0 )
22 df-2 10018 . . . . . 6  |-  2  =  ( 1  +  1 )
2322oveq1i 6054 . . . . 5  |-  ( 2 S A )  =  ( ( 1  +  1 ) S A )
24 ax-1cn 9008 . . . . . . . 8  |-  1  e.  CC
2524, 24, 33pm3.2i 1132 . . . . . . 7  |-  ( 1  e.  CC  /\  1  e.  CC  /\  A  e.  X )
264, 5, 13nvdir 22069 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  (
1  e.  CC  /\  1  e.  CC  /\  A  e.  X ) )  -> 
( ( 1  +  1 ) S A )  =  ( ( 1 S A ) G ( 1 S A ) ) )
272, 25, 26mp2an 654 . . . . . 6  |-  ( ( 1  +  1 ) S A )  =  ( ( 1 S A ) G ( 1 S A ) )
284, 13nvsid 22065 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
1 S A )  =  A )
292, 3, 28mp2an 654 . . . . . . 7  |-  ( 1 S A )  =  A
3029, 29oveq12i 6056 . . . . . 6  |-  ( ( 1 S A ) G ( 1 S A ) )  =  ( A G A )
3127, 30eqtri 2428 . . . . 5  |-  ( ( 1  +  1 ) S A )  =  ( A G A )
3223, 31eqtri 2428 . . . 4  |-  ( 2 S A )  =  ( A G A )
3332oveq1i 6054 . . 3  |-  ( ( 2 S A ) P B )  =  ( ( A G A ) P B )
3412, 21, 333eqtr4ri 2439 . 2  |-  ( ( 2 S A ) P B )  =  ( ( ( A G A ) P B )  +  ( ( A G (
-u 1 S A ) ) P B ) )
354, 5, 13, 9, 1, 3, 3, 8ip1i 22285 . 2  |-  ( ( ( A G A ) P B )  +  ( ( A G ( -u 1 S A ) ) P B ) )  =  ( 2  x.  ( A P B ) )
3634, 35eqtri 2428 1  |-  ( ( 2 S A ) P B )  =  ( 2  x.  ( A P B ) )
Colors of variables: wff set class
Syntax hints:    /\ w3a 936    = wceq 1649    e. wcel 1721   ` cfv 5417  (class class class)co 6044   CCcc 8948   0cc0 8950   1c1 8951    + caddc 8953    x. cmul 8955   -ucneg 9252   2c2 10009   NrmCVeccnv 22020   +vcpv 22021   BaseSetcba 22022   .s
OLDcns 22023   0veccn0v 22024   .i
OLDcdip 22153   CPreHil OLDccphlo 22270
This theorem is referenced by:  ipdirilem  22287
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-inf2 7556  ax-cnex 9006  ax-resscn 9007  ax-1cn 9008  ax-icn 9009  ax-addcl 9010  ax-addrcl 9011  ax-mulcl 9012  ax-mulrcl 9013  ax-mulcom 9014  ax-addass 9015  ax-mulass 9016  ax-distr 9017  ax-i2m1 9018  ax-1ne0 9019  ax-1rid 9020  ax-rnegex 9021  ax-rrecex 9022  ax-cnre 9023  ax-pre-lttri 9024  ax-pre-lttrn 9025  ax-pre-ltadd 9026  ax-pre-mulgt0 9027  ax-pre-sup 9028
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-int 4015  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-se 4506  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-isom 5426  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-riota 6512  df-recs 6596  df-rdg 6631  df-1o 6687  df-oadd 6691  df-er 6868  df-en 7073  df-dom 7074  df-sdom 7075  df-fin 7076  df-sup 7408  df-oi 7439  df-card 7786  df-pnf 9082  df-mnf 9083  df-xr 9084  df-ltxr 9085  df-le 9086  df-sub 9253  df-neg 9254  df-div 9638  df-nn 9961  df-2 10018  df-3 10019  df-4 10020  df-n0 10182  df-z 10243  df-uz 10449  df-rp 10573  df-fz 11004  df-fzo 11095  df-seq 11283  df-exp 11342  df-hash 11578  df-cj 11863  df-re 11864  df-im 11865  df-sqr 11999  df-abs 12000  df-clim 12241  df-sum 12439  df-grpo 21736  df-gid 21737  df-ginv 21738  df-ablo 21827  df-vc 21982  df-nv 22028  df-va 22031  df-ba 22032  df-sm 22033  df-0v 22034  df-nmcv 22036  df-dip 22154  df-ph 22271
  Copyright terms: Public domain W3C validator