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Theorem ipasslem4 22327
Description: Lemma for ipassi 22334. Show the inner product associative law for positive integer reciprocals. (Contributed by NM, 27-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
ipasslem1.b  |-  B  e.  X
Assertion
Ref Expression
ipasslem4  |-  ( ( N  e.  NN  /\  A  e.  X )  ->  ( ( ( 1  /  N ) S A ) P B )  =  ( ( 1  /  N )  x.  ( A P B ) ) )

Proof of Theorem ipasslem4
StepHypRef Expression
1 nnrecre 10028 . . . . 5  |-  ( N  e.  NN  ->  (
1  /  N )  e.  RR )
21recnd 9106 . . . 4  |-  ( N  e.  NN  ->  (
1  /  N )  e.  CC )
3 ip1i.9 . . . . . 6  |-  U  e.  CPreHil
OLD
43phnvi 22309 . . . . 5  |-  U  e.  NrmCVec
5 ip1i.1 . . . . . 6  |-  X  =  ( BaseSet `  U )
6 ip1i.4 . . . . . 6  |-  S  =  ( .s OLD `  U
)
75, 6nvscl 22099 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  (
1  /  N )  e.  CC  /\  A  e.  X )  ->  (
( 1  /  N
) S A )  e.  X )
84, 7mp3an1 1266 . . . 4  |-  ( ( ( 1  /  N
)  e.  CC  /\  A  e.  X )  ->  ( ( 1  /  N ) S A )  e.  X )
92, 8sylan 458 . . 3  |-  ( ( N  e.  NN  /\  A  e.  X )  ->  ( ( 1  /  N ) S A )  e.  X )
10 ipasslem1.b . . . 4  |-  B  e.  X
11 ip1i.7 . . . . 5  |-  P  =  ( .i OLD `  U
)
125, 11dipcl 22203 . . . 4  |-  ( ( U  e.  NrmCVec  /\  (
( 1  /  N
) S A )  e.  X  /\  B  e.  X )  ->  (
( ( 1  /  N ) S A ) P B )  e.  CC )
134, 10, 12mp3an13 1270 . . 3  |-  ( ( ( 1  /  N
) S A )  e.  X  ->  (
( ( 1  /  N ) S A ) P B )  e.  CC )
149, 13syl 16 . 2  |-  ( ( N  e.  NN  /\  A  e.  X )  ->  ( ( ( 1  /  N ) S A ) P B )  e.  CC )
155, 11dipcl 22203 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A P B )  e.  CC )
164, 10, 15mp3an13 1270 . . 3  |-  ( A  e.  X  ->  ( A P B )  e.  CC )
17 mulcl 9066 . . 3  |-  ( ( ( 1  /  N
)  e.  CC  /\  ( A P B )  e.  CC )  -> 
( ( 1  /  N )  x.  ( A P B ) )  e.  CC )
182, 16, 17syl2an 464 . 2  |-  ( ( N  e.  NN  /\  A  e.  X )  ->  ( ( 1  /  N )  x.  ( A P B ) )  e.  CC )
19 nncn 10000 . . 3  |-  ( N  e.  NN  ->  N  e.  CC )
2019adantr 452 . 2  |-  ( ( N  e.  NN  /\  A  e.  X )  ->  N  e.  CC )
21 nnne0 10024 . . 3  |-  ( N  e.  NN  ->  N  =/=  0 )
2221adantr 452 . 2  |-  ( ( N  e.  NN  /\  A  e.  X )  ->  N  =/=  0 )
2319, 21recidd 9777 . . . . . 6  |-  ( N  e.  NN  ->  ( N  x.  ( 1  /  N ) )  =  1 )
2423oveq1d 6088 . . . . 5  |-  ( N  e.  NN  ->  (
( N  x.  (
1  /  N ) )  x.  ( A P B ) )  =  ( 1  x.  ( A P B ) ) )
2516mulid2d 9098 . . . . 5  |-  ( A  e.  X  ->  (
1  x.  ( A P B ) )  =  ( A P B ) )
2624, 25sylan9eq 2487 . . . 4  |-  ( ( N  e.  NN  /\  A  e.  X )  ->  ( ( N  x.  ( 1  /  N
) )  x.  ( A P B ) )  =  ( A P B ) )
2723oveq1d 6088 . . . . . . 7  |-  ( N  e.  NN  ->  (
( N  x.  (
1  /  N ) ) S A )  =  ( 1 S A ) )
285, 6nvsid 22100 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
1 S A )  =  A )
294, 28mpan 652 . . . . . . 7  |-  ( A  e.  X  ->  (
1 S A )  =  A )
3027, 29sylan9eq 2487 . . . . . 6  |-  ( ( N  e.  NN  /\  A  e.  X )  ->  ( ( N  x.  ( 1  /  N
) ) S A )  =  A )
312adantr 452 . . . . . . 7  |-  ( ( N  e.  NN  /\  A  e.  X )  ->  ( 1  /  N
)  e.  CC )
32 simpr 448 . . . . . . 7  |-  ( ( N  e.  NN  /\  A  e.  X )  ->  A  e.  X )
335, 6nvsass 22101 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  ( N  e.  CC  /\  (
1  /  N )  e.  CC  /\  A  e.  X ) )  -> 
( ( N  x.  ( 1  /  N
) ) S A )  =  ( N S ( ( 1  /  N ) S A ) ) )
344, 33mpan 652 . . . . . . 7  |-  ( ( N  e.  CC  /\  ( 1  /  N
)  e.  CC  /\  A  e.  X )  ->  ( ( N  x.  ( 1  /  N
) ) S A )  =  ( N S ( ( 1  /  N ) S A ) ) )
3520, 31, 32, 34syl3anc 1184 . . . . . 6  |-  ( ( N  e.  NN  /\  A  e.  X )  ->  ( ( N  x.  ( 1  /  N
) ) S A )  =  ( N S ( ( 1  /  N ) S A ) ) )
3630, 35eqtr3d 2469 . . . . 5  |-  ( ( N  e.  NN  /\  A  e.  X )  ->  A  =  ( N S ( ( 1  /  N ) S A ) ) )
3736oveq1d 6088 . . . 4  |-  ( ( N  e.  NN  /\  A  e.  X )  ->  ( A P B )  =  ( ( N S ( ( 1  /  N ) S A ) ) P B ) )
38 nnnn0 10220 . . . . . 6  |-  ( N  e.  NN  ->  N  e.  NN0 )
3938adantr 452 . . . . 5  |-  ( ( N  e.  NN  /\  A  e.  X )  ->  N  e.  NN0 )
40 ip1i.2 . . . . . 6  |-  G  =  ( +v `  U
)
415, 40, 6, 11, 3, 10ipasslem1 22324 . . . . 5  |-  ( ( N  e.  NN0  /\  ( ( 1  /  N ) S A )  e.  X )  ->  ( ( N S ( ( 1  /  N ) S A ) ) P B )  =  ( N  x.  ( ( ( 1  /  N
) S A ) P B ) ) )
4239, 9, 41syl2anc 643 . . . 4  |-  ( ( N  e.  NN  /\  A  e.  X )  ->  ( ( N S ( ( 1  /  N ) S A ) ) P B )  =  ( N  x.  ( ( ( 1  /  N ) S A ) P B ) ) )
4326, 37, 423eqtrd 2471 . . 3  |-  ( ( N  e.  NN  /\  A  e.  X )  ->  ( ( N  x.  ( 1  /  N
) )  x.  ( A P B ) )  =  ( N  x.  ( ( ( 1  /  N ) S A ) P B ) ) )
4416adantl 453 . . . 4  |-  ( ( N  e.  NN  /\  A  e.  X )  ->  ( A P B )  e.  CC )
4520, 31, 44mulassd 9103 . . 3  |-  ( ( N  e.  NN  /\  A  e.  X )  ->  ( ( N  x.  ( 1  /  N
) )  x.  ( A P B ) )  =  ( N  x.  ( ( 1  /  N )  x.  ( A P B ) ) ) )
4643, 45eqtr3d 2469 . 2  |-  ( ( N  e.  NN  /\  A  e.  X )  ->  ( N  x.  (
( ( 1  /  N ) S A ) P B ) )  =  ( N  x.  ( ( 1  /  N )  x.  ( A P B ) ) ) )
4714, 18, 20, 22, 46mulcanad 9649 1  |-  ( ( N  e.  NN  /\  A  e.  X )  ->  ( ( ( 1  /  N ) S A ) P B )  =  ( ( 1  /  N )  x.  ( A P B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   ` cfv 5446  (class class class)co 6073   CCcc 8980   0cc0 8982   1c1 8983    x. cmul 8987    / cdiv 9669   NNcn 9992   NN0cn0 10213   NrmCVeccnv 22055   +vcpv 22056   BaseSetcba 22057   .s
OLDcns 22058   .i OLDcdip 22188   CPreHil OLDccphlo 22305
This theorem is referenced by:  ipasslem5  22328
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-oi 7471  df-card 7818  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-n0 10214  df-z 10275  df-uz 10481  df-rp 10605  df-fz 11036  df-fzo 11128  df-seq 11316  df-exp 11375  df-hash 11611  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-clim 12274  df-sum 12472  df-grpo 21771  df-gid 21772  df-ginv 21773  df-ablo 21862  df-vc 22017  df-nv 22063  df-va 22066  df-ba 22067  df-sm 22068  df-0v 22069  df-nmcv 22071  df-dip 22189  df-ph 22306
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