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Theorem ipasslem4 26475
Description: Lemma for ipassi 26482. 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 10646 . . . . 5  |-  ( N  e.  NN  ->  (
1  /  N )  e.  RR )
21recnd 9669 . . . 4  |-  ( N  e.  NN  ->  (
1  /  N )  e.  CC )
3 ip1i.9 . . . . . 6  |-  U  e.  CPreHil
OLD
43phnvi 26457 . . . . 5  |-  U  e.  NrmCVec
5 ip1i.1 . . . . . 6  |-  X  =  ( BaseSet `  U )
6 ip1i.4 . . . . . 6  |-  S  =  ( .sOLD `  U )
75, 6nvscl 26247 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  (
1  /  N )  e.  CC  /\  A  e.  X )  ->  (
( 1  /  N
) S A )  e.  X )
84, 7mp3an1 1351 . . . 4  |-  ( ( ( 1  /  N
)  e.  CC  /\  A  e.  X )  ->  ( ( 1  /  N ) S A )  e.  X )
92, 8sylan 474 . . 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 26351 . . . 4  |-  ( ( U  e.  NrmCVec  /\  (
( 1  /  N
) S A )  e.  X  /\  B  e.  X )  ->  (
( ( 1  /  N ) S A ) P B )  e.  CC )
134, 10, 12mp3an13 1355 . . 3  |-  ( ( ( 1  /  N
) S A )  e.  X  ->  (
( ( 1  /  N ) S A ) P B )  e.  CC )
149, 13syl 17 . 2  |-  ( ( N  e.  NN  /\  A  e.  X )  ->  ( ( ( 1  /  N ) S A ) P B )  e.  CC )
155, 11dipcl 26351 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A P B )  e.  CC )
164, 10, 15mp3an13 1355 . . 3  |-  ( A  e.  X  ->  ( A P B )  e.  CC )
17 mulcl 9623 . . 3  |-  ( ( ( 1  /  N
)  e.  CC  /\  ( A P B )  e.  CC )  -> 
( ( 1  /  N )  x.  ( A P B ) )  e.  CC )
182, 16, 17syl2an 480 . 2  |-  ( ( N  e.  NN  /\  A  e.  X )  ->  ( ( 1  /  N )  x.  ( A P B ) )  e.  CC )
19 nncn 10617 . . 3  |-  ( N  e.  NN  ->  N  e.  CC )
2019adantr 467 . 2  |-  ( ( N  e.  NN  /\  A  e.  X )  ->  N  e.  CC )
21 nnne0 10642 . . 3  |-  ( N  e.  NN  ->  N  =/=  0 )
2221adantr 467 . 2  |-  ( ( N  e.  NN  /\  A  e.  X )  ->  N  =/=  0 )
2319, 21recidd 10378 . . . . . 6  |-  ( N  e.  NN  ->  ( N  x.  ( 1  /  N ) )  =  1 )
2423oveq1d 6305 . . . . 5  |-  ( N  e.  NN  ->  (
( N  x.  (
1  /  N ) )  x.  ( A P B ) )  =  ( 1  x.  ( A P B ) ) )
2516mulid2d 9661 . . . . 5  |-  ( A  e.  X  ->  (
1  x.  ( A P B ) )  =  ( A P B ) )
2624, 25sylan9eq 2505 . . . 4  |-  ( ( N  e.  NN  /\  A  e.  X )  ->  ( ( N  x.  ( 1  /  N
) )  x.  ( A P B ) )  =  ( A P B ) )
2723oveq1d 6305 . . . . . . 7  |-  ( N  e.  NN  ->  (
( N  x.  (
1  /  N ) ) S A )  =  ( 1 S A ) )
285, 6nvsid 26248 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  A  e.  X )  ->  (
1 S A )  =  A )
294, 28mpan 676 . . . . . . 7  |-  ( A  e.  X  ->  (
1 S A )  =  A )
3027, 29sylan9eq 2505 . . . . . 6  |-  ( ( N  e.  NN  /\  A  e.  X )  ->  ( ( N  x.  ( 1  /  N
) ) S A )  =  A )
312adantr 467 . . . . . . 7  |-  ( ( N  e.  NN  /\  A  e.  X )  ->  ( 1  /  N
)  e.  CC )
32 simpr 463 . . . . . . 7  |-  ( ( N  e.  NN  /\  A  e.  X )  ->  A  e.  X )
335, 6nvsass 26249 . . . . . . . 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 676 . . . . . . 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 1268 . . . . . 6  |-  ( ( N  e.  NN  /\  A  e.  X )  ->  ( ( N  x.  ( 1  /  N
) ) S A )  =  ( N S ( ( 1  /  N ) S A ) ) )
3630, 35eqtr3d 2487 . . . . 5  |-  ( ( N  e.  NN  /\  A  e.  X )  ->  A  =  ( N S ( ( 1  /  N ) S A ) ) )
3736oveq1d 6305 . . . 4  |-  ( ( N  e.  NN  /\  A  e.  X )  ->  ( A P B )  =  ( ( N S ( ( 1  /  N ) S A ) ) P B ) )
38 nnnn0 10876 . . . . . 6  |-  ( N  e.  NN  ->  N  e.  NN0 )
3938adantr 467 . . . . 5  |-  ( ( N  e.  NN  /\  A  e.  X )  ->  N  e.  NN0 )
40 ip1i.2 . . . . . 6  |-  G  =  ( +v `  U
)
415, 40, 6, 11, 3, 10ipasslem1 26472 . . . . 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 667 . . . 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 2489 . . 3  |-  ( ( N  e.  NN  /\  A  e.  X )  ->  ( ( N  x.  ( 1  /  N
) )  x.  ( A P B ) )  =  ( N  x.  ( ( ( 1  /  N ) S A ) P B ) ) )
4416adantl 468 . . . 4  |-  ( ( N  e.  NN  /\  A  e.  X )  ->  ( A P B )  e.  CC )
4520, 31, 44mulassd 9666 . . 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 2487 . 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 10247 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 371    /\ w3a 985    = wceq 1444    e. wcel 1887    =/= wne 2622   ` cfv 5582  (class class class)co 6290   CCcc 9537   0cc0 9539   1c1 9540    x. cmul 9544    / cdiv 10269   NNcn 10609   NN0cn0 10869   NrmCVeccnv 26203   +vcpv 26204   BaseSetcba 26205   .sOLDcns 26206   .iOLDcdip 26336   CPreHil OLDccphlo 26453
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-sup 7956  df-oi 8025  df-card 8373  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-fz 11785  df-fzo 11916  df-seq 12214  df-exp 12273  df-hash 12516  df-cj 13162  df-re 13163  df-im 13164  df-sqrt 13298  df-abs 13299  df-clim 13552  df-sum 13753  df-grpo 25919  df-gid 25920  df-ginv 25921  df-ablo 26010  df-vc 26165  df-nv 26211  df-va 26214  df-ba 26215  df-sm 26216  df-0v 26217  df-nmcv 26219  df-dip 26337  df-ph 26454
This theorem is referenced by:  ipasslem5  26476
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