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Theorem ip2i 22170
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 22158 . . . . 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 21940 . . . . . 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 22052 . . . . 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 9178 . . 3  |-  ( ( ( A G A ) P B )  +  0 )  =  ( ( A G A ) P B )
13 ip1i.4 . . . . . . . 8  |-  S  =  ( .s OLD `  U
)
14 eqid 2380 . . . . . . . 8  |-  ( 0vec `  U )  =  (
0vec `  U )
154, 5, 13, 14nvrinv 21975 . . . . . . 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 6023 . . . . 5  |-  ( ( A G ( -u
1 S A ) ) P B )  =  ( ( 0vec `  U ) P B )
184, 14, 9dip0l 22058 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  B  e.  X )  ->  (
( 0vec `  U ) P B )  =  0 )
192, 8, 18mp2an 654 . . . . 5  |-  ( (
0vec `  U ) P B )  =  0
2017, 19eqtri 2400 . . . 4  |-  ( ( A G ( -u
1 S A ) ) P B )  =  0
2120oveq2i 6024 . . 3  |-  ( ( ( A G A ) P B )  +  ( ( A G ( -u 1 S A ) ) P B ) )  =  ( ( ( A G A ) P B )  +  0 )
22 df-2 9983 . . . . . 6  |-  2  =  ( 1  +  1 )
2322oveq1i 6023 . . . . 5  |-  ( 2 S A )  =  ( ( 1  +  1 ) S A )
24 ax-1cn 8974 . . . . . . . 8  |-  1  e.  CC
2524, 24, 33pm3.2i 1132 . . . . . . 7  |-  ( 1  e.  CC  /\  1  e.  CC  /\  A  e.  X )
264, 5, 13nvdir 21953 . . . . . . 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 21949 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
1 S A )  =  A )
292, 3, 28mp2an 654 . . . . . . 7  |-  ( 1 S A )  =  A
3029, 29oveq12i 6025 . . . . . 6  |-  ( ( 1 S A ) G ( 1 S A ) )  =  ( A G A )
3127, 30eqtri 2400 . . . . 5  |-  ( ( 1  +  1 ) S A )  =  ( A G A )
3223, 31eqtri 2400 . . . 4  |-  ( 2 S A )  =  ( A G A )
3332oveq1i 6023 . . 3  |-  ( ( 2 S A ) P B )  =  ( ( A G A ) P B )
3412, 21, 333eqtr4ri 2411 . 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 22169 . 2  |-  ( ( ( A G A ) P B )  +  ( ( A G ( -u 1 S A ) ) P B ) )  =  ( 2  x.  ( A P B ) )
3634, 35eqtri 2400 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 1717   ` cfv 5387  (class class class)co 6013   CCcc 8914   0cc0 8916   1c1 8917    + caddc 8919    x. cmul 8921   -ucneg 9217   2c2 9974   NrmCVeccnv 21904   +vcpv 21905   BaseSetcba 21906   .s
OLDcns 21907   0veccn0v 21908   .i
OLDcdip 22037   CPreHil OLDccphlo 22154
This theorem is referenced by:  ipdirilem  22171
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-inf2 7522  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993  ax-pre-sup 8994
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-int 3986  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-se 4476  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-isom 5396  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-riota 6478  df-recs 6562  df-rdg 6597  df-1o 6653  df-oadd 6657  df-er 6834  df-en 7039  df-dom 7040  df-sdom 7041  df-fin 7042  df-sup 7374  df-oi 7405  df-card 7752  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-div 9603  df-nn 9926  df-2 9983  df-3 9984  df-4 9985  df-n0 10147  df-z 10208  df-uz 10414  df-rp 10538  df-fz 10969  df-fzo 11059  df-seq 11244  df-exp 11303  df-hash 11539  df-cj 11824  df-re 11825  df-im 11826  df-sqr 11960  df-abs 11961  df-clim 12202  df-sum 12400  df-grpo 21620  df-gid 21621  df-ginv 21622  df-ablo 21711  df-vc 21866  df-nv 21912  df-va 21915  df-ba 21916  df-sm 21917  df-0v 21918  df-nmcv 21920  df-dip 22038  df-ph 22155
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