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Theorem ipasslem4 24185
Description: Lemma for ipassi 24192. 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 10350 . . . . 5  |-  ( N  e.  NN  ->  (
1  /  N )  e.  RR )
21recnd 9404 . . . 4  |-  ( N  e.  NN  ->  (
1  /  N )  e.  CC )
3 ip1i.9 . . . . . 6  |-  U  e.  CPreHil
OLD
43phnvi 24167 . . . . 5  |-  U  e.  NrmCVec
5 ip1i.1 . . . . . 6  |-  X  =  ( BaseSet `  U )
6 ip1i.4 . . . . . 6  |-  S  =  ( .sOLD `  U )
75, 6nvscl 23957 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  (
1  /  N )  e.  CC  /\  A  e.  X )  ->  (
( 1  /  N
) S A )  e.  X )
84, 7mp3an1 1301 . . . 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 24061 . . . 4  |-  ( ( U  e.  NrmCVec  /\  (
( 1  /  N
) S A )  e.  X  /\  B  e.  X )  ->  (
( ( 1  /  N ) S A ) P B )  e.  CC )
134, 10, 12mp3an13 1305 . . 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 24061 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A P B )  e.  CC )
164, 10, 15mp3an13 1305 . . 3  |-  ( A  e.  X  ->  ( A P B )  e.  CC )
17 mulcl 9358 . . 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 10322 . . 3  |-  ( N  e.  NN  ->  N  e.  CC )
2019adantr 465 . 2  |-  ( ( N  e.  NN  /\  A  e.  X )  ->  N  e.  CC )
21 nnne0 10346 . . 3  |-  ( N  e.  NN  ->  N  =/=  0 )
2221adantr 465 . 2  |-  ( ( N  e.  NN  /\  A  e.  X )  ->  N  =/=  0 )
2319, 21recidd 10094 . . . . . 6  |-  ( N  e.  NN  ->  ( N  x.  ( 1  /  N ) )  =  1 )
2423oveq1d 6101 . . . . 5  |-  ( N  e.  NN  ->  (
( N  x.  (
1  /  N ) )  x.  ( A P B ) )  =  ( 1  x.  ( A P B ) ) )
2516mulid2d 9396 . . . . 5  |-  ( A  e.  X  ->  (
1  x.  ( A P B ) )  =  ( A P B ) )
2624, 25sylan9eq 2490 . . . 4  |-  ( ( N  e.  NN  /\  A  e.  X )  ->  ( ( N  x.  ( 1  /  N
) )  x.  ( A P B ) )  =  ( A P B ) )
2723oveq1d 6101 . . . . . . 7  |-  ( N  e.  NN  ->  (
( N  x.  (
1  /  N ) ) S A )  =  ( 1 S A ) )
285, 6nvsid 23958 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
1 S A )  =  A )
294, 28mpan 670 . . . . . . 7  |-  ( A  e.  X  ->  (
1 S A )  =  A )
3027, 29sylan9eq 2490 . . . . . 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 23959 . . . . . . . 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 1218 . . . . . 6  |-  ( ( N  e.  NN  /\  A  e.  X )  ->  ( ( N  x.  ( 1  /  N
) ) S A )  =  ( N S ( ( 1  /  N ) S A ) ) )
3630, 35eqtr3d 2472 . . . . 5  |-  ( ( N  e.  NN  /\  A  e.  X )  ->  A  =  ( N S ( ( 1  /  N ) S A ) ) )
3736oveq1d 6101 . . . 4  |-  ( ( N  e.  NN  /\  A  e.  X )  ->  ( A P B )  =  ( ( N S ( ( 1  /  N ) S A ) ) P B ) )
38 nnnn0 10578 . . . . . 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 24182 . . . . 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 2474 . . 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 9401 . . 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 2472 . 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 9963 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 1369    e. wcel 1756    =/= wne 2601   ` cfv 5413  (class class class)co 6086   CCcc 9272   0cc0 9274   1c1 9275    x. cmul 9279    / cdiv 9985   NNcn 10314   NN0cn0 10571   NrmCVeccnv 23913   +vcpv 23914   BaseSetcba 23915   .sOLDcns 23916   .iOLDcdip 24046   CPreHil OLDccphlo 24163
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-1o 6912  df-oadd 6916  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-sup 7683  df-oi 7716  df-card 8101  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-n0 10572  df-z 10639  df-uz 10854  df-rp 10984  df-fz 11430  df-fzo 11541  df-seq 11799  df-exp 11858  df-hash 12096  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-clim 12958  df-sum 13156  df-grpo 23629  df-gid 23630  df-ginv 23631  df-ablo 23720  df-vc 23875  df-nv 23921  df-va 23924  df-ba 23925  df-sm 23926  df-0v 23927  df-nmcv 23929  df-dip 24047  df-ph 24164
This theorem is referenced by:  ipasslem5  24186
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