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Theorem ipasslem4 24379
Description: Lemma for ipassi 24386. 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  =  ( .sOLD `  U )
ip1i.7  |-  P  =  ( .iOLD `  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 10462 . . . . 5  |-  ( N  e.  NN  ->  (
1  /  N )  e.  RR )
21recnd 9516 . . . 4  |-  ( N  e.  NN  ->  (
1  /  N )  e.  CC )
3 ip1i.9 . . . . . 6  |-  U  e.  CPreHil
OLD
43phnvi 24361 . . . . 5  |-  U  e.  NrmCVec
5 ip1i.1 . . . . . 6  |-  X  =  ( BaseSet `  U )
6 ip1i.4 . . . . . 6  |-  S  =  ( .sOLD `  U )
75, 6nvscl 24151 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  (
1  /  N )  e.  CC  /\  A  e.  X )  ->  (
( 1  /  N
) S A )  e.  X )
84, 7mp3an1 1302 . . . 4  |-  ( ( ( 1  /  N
)  e.  CC  /\  A  e.  X )  ->  ( ( 1  /  N ) S A )  e.  X )
92, 8sylan 471 . . 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  =  ( .iOLD `  U )
125, 11dipcl 24255 . . . 4  |-  ( ( U  e.  NrmCVec  /\  (
( 1  /  N
) S A )  e.  X  /\  B  e.  X )  ->  (
( ( 1  /  N ) S A ) P B )  e.  CC )
134, 10, 12mp3an13 1306 . . 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 24255 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A P B )  e.  CC )
164, 10, 15mp3an13 1306 . . 3  |-  ( A  e.  X  ->  ( A P B )  e.  CC )
17 mulcl 9470 . . 3  |-  ( ( ( 1  /  N
)  e.  CC  /\  ( A P B )  e.  CC )  -> 
( ( 1  /  N )  x.  ( A P B ) )  e.  CC )
182, 16, 17syl2an 477 . 2  |-  ( ( N  e.  NN  /\  A  e.  X )  ->  ( ( 1  /  N )  x.  ( A P B ) )  e.  CC )
19 nncn 10434 . . 3  |-  ( N  e.  NN  ->  N  e.  CC )
2019adantr 465 . 2  |-  ( ( N  e.  NN  /\  A  e.  X )  ->  N  e.  CC )
21 nnne0 10458 . . 3  |-  ( N  e.  NN  ->  N  =/=  0 )
2221adantr 465 . 2  |-  ( ( N  e.  NN  /\  A  e.  X )  ->  N  =/=  0 )
2319, 21recidd 10206 . . . . . 6  |-  ( N  e.  NN  ->  ( N  x.  ( 1  /  N ) )  =  1 )
2423oveq1d 6208 . . . . 5  |-  ( N  e.  NN  ->  (
( N  x.  (
1  /  N ) )  x.  ( A P B ) )  =  ( 1  x.  ( A P B ) ) )
2516mulid2d 9508 . . . . 5  |-  ( A  e.  X  ->  (
1  x.  ( A P B ) )  =  ( A P B ) )
2624, 25sylan9eq 2512 . . . 4  |-  ( ( N  e.  NN  /\  A  e.  X )  ->  ( ( N  x.  ( 1  /  N
) )  x.  ( A P B ) )  =  ( A P B ) )
2723oveq1d 6208 . . . . . . 7  |-  ( N  e.  NN  ->  (
( N  x.  (
1  /  N ) ) S A )  =  ( 1 S A ) )
285, 6nvsid 24152 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
1 S A )  =  A )
294, 28mpan 670 . . . . . . 7  |-  ( A  e.  X  ->  (
1 S A )  =  A )
3027, 29sylan9eq 2512 . . . . . 6  |-  ( ( N  e.  NN  /\  A  e.  X )  ->  ( ( N  x.  ( 1  /  N
) ) S A )  =  A )
312adantr 465 . . . . . . 7  |-  ( ( N  e.  NN  /\  A  e.  X )  ->  ( 1  /  N
)  e.  CC )
32 simpr 461 . . . . . . 7  |-  ( ( N  e.  NN  /\  A  e.  X )  ->  A  e.  X )
335, 6nvsass 24153 . . . . . . . 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 670 . . . . . . 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 1219 . . . . . 6  |-  ( ( N  e.  NN  /\  A  e.  X )  ->  ( ( N  x.  ( 1  /  N
) ) S A )  =  ( N S ( ( 1  /  N ) S A ) ) )
3630, 35eqtr3d 2494 . . . . 5  |-  ( ( N  e.  NN  /\  A  e.  X )  ->  A  =  ( N S ( ( 1  /  N ) S A ) ) )
3736oveq1d 6208 . . . 4  |-  ( ( N  e.  NN  /\  A  e.  X )  ->  ( A P B )  =  ( ( N S ( ( 1  /  N ) S A ) ) P B ) )
38 nnnn0 10690 . . . . . 6  |-  ( N  e.  NN  ->  N  e.  NN0 )
3938adantr 465 . . . . 5  |-  ( ( N  e.  NN  /\  A  e.  X )  ->  N  e.  NN0 )
40 ip1i.2 . . . . . 6  |-  G  =  ( +v `  U
)
415, 40, 6, 11, 3, 10ipasslem1 24376 . . . . 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 661 . . . 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 2496 . . 3  |-  ( ( N  e.  NN  /\  A  e.  X )  ->  ( ( N  x.  ( 1  /  N
) )  x.  ( A P B ) )  =  ( N  x.  ( ( ( 1  /  N ) S A ) P B ) ) )
4416adantl 466 . . . 4  |-  ( ( N  e.  NN  /\  A  e.  X )  ->  ( A P B )  e.  CC )
4520, 31, 44mulassd 9513 . . 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 2494 . 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 10075 1  |-  ( ( N  e.  NN  /\  A  e.  X )  ->  ( ( ( 1  /  N ) S A ) P B )  =  ( ( 1  /  N )  x.  ( A P B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2644   ` cfv 5519  (class class class)co 6193   CCcc 9384   0cc0 9386   1c1 9387    x. cmul 9391    / cdiv 10097   NNcn 10426   NN0cn0 10683   NrmCVeccnv 24107   +vcpv 24108   BaseSetcba 24109   .sOLDcns 24110   .iOLDcdip 24240   CPreHil OLDccphlo 24357
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-inf2 7951  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463  ax-pre-sup 9464
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-int 4230  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-se 4781  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-isom 5528  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-om 6580  df-1st 6680  df-2nd 6681  df-recs 6935  df-rdg 6969  df-1o 7023  df-oadd 7027  df-er 7204  df-en 7414  df-dom 7415  df-sdom 7416  df-fin 7417  df-sup 7795  df-oi 7828  df-card 8213  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-div 10098  df-nn 10427  df-2 10484  df-3 10485  df-4 10486  df-n0 10684  df-z 10751  df-uz 10966  df-rp 11096  df-fz 11548  df-fzo 11659  df-seq 11917  df-exp 11976  df-hash 12214  df-cj 12699  df-re 12700  df-im 12701  df-sqr 12835  df-abs 12836  df-clim 13077  df-sum 13275  df-grpo 23823  df-gid 23824  df-ginv 23825  df-ablo 23914  df-vc 24069  df-nv 24115  df-va 24118  df-ba 24119  df-sm 24120  df-0v 24121  df-nmcv 24123  df-dip 24241  df-ph 24358
This theorem is referenced by:  ipasslem5  24380
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