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Theorem ipasslem3 24238
Description: Lemma for ipassi 24246. Show the inner product associative law for all integers. (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
ipasslem3  |-  ( ( N  e.  ZZ  /\  A  e.  X )  ->  ( ( N S A ) P B )  =  ( N  x.  ( A P B ) ) )

Proof of Theorem ipasslem3
StepHypRef Expression
1 elznn0nn 10665 . 2  |-  ( N  e.  ZZ  <->  ( N  e.  NN0  \/  ( N  e.  RR  /\  -u N  e.  NN ) ) )
2 ip1i.1 . . . 4  |-  X  =  ( BaseSet `  U )
3 ip1i.2 . . . 4  |-  G  =  ( +v `  U
)
4 ip1i.4 . . . 4  |-  S  =  ( .sOLD `  U )
5 ip1i.7 . . . 4  |-  P  =  ( .iOLD `  U )
6 ip1i.9 . . . 4  |-  U  e.  CPreHil
OLD
7 ipasslem1.b . . . 4  |-  B  e.  X
82, 3, 4, 5, 6, 7ipasslem1 24236 . . 3  |-  ( ( N  e.  NN0  /\  A  e.  X )  ->  ( ( N S A ) P B )  =  ( N  x.  ( A P B ) ) )
9 nnnn0 10591 . . . . . 6  |-  ( -u N  e.  NN  ->  -u N  e.  NN0 )
102, 3, 4, 5, 6, 7ipasslem2 24237 . . . . . 6  |-  ( (
-u N  e.  NN0  /\  A  e.  X )  ->  ( ( -u -u N S A ) P B )  =  ( -u -u N  x.  ( A P B ) ) )
119, 10sylan 471 . . . . 5  |-  ( (
-u N  e.  NN  /\  A  e.  X )  ->  ( ( -u -u N S A ) P B )  =  ( -u -u N  x.  ( A P B ) ) )
1211adantll 713 . . . 4  |-  ( ( ( N  e.  RR  /\  -u N  e.  NN )  /\  A  e.  X
)  ->  ( ( -u -u N S A ) P B )  =  ( -u -u N  x.  ( A P B ) ) )
13 recn 9377 . . . . . . . 8  |-  ( N  e.  RR  ->  N  e.  CC )
1413negnegd 9715 . . . . . . 7  |-  ( N  e.  RR  ->  -u -u N  =  N )
1514oveq1d 6111 . . . . . 6  |-  ( N  e.  RR  ->  ( -u -u N S A )  =  ( N S A ) )
1615oveq1d 6111 . . . . 5  |-  ( N  e.  RR  ->  (
( -u -u N S A ) P B )  =  ( ( N S A ) P B ) )
1716ad2antrr 725 . . . 4  |-  ( ( ( N  e.  RR  /\  -u N  e.  NN )  /\  A  e.  X
)  ->  ( ( -u -u N S A ) P B )  =  ( ( N S A ) P B ) )
1814oveq1d 6111 . . . . 5  |-  ( N  e.  RR  ->  ( -u -u N  x.  ( A P B ) )  =  ( N  x.  ( A P B ) ) )
1918ad2antrr 725 . . . 4  |-  ( ( ( N  e.  RR  /\  -u N  e.  NN )  /\  A  e.  X
)  ->  ( -u -u N  x.  ( A P B ) )  =  ( N  x.  ( A P B ) ) )
2012, 17, 193eqtr3d 2483 . . 3  |-  ( ( ( N  e.  RR  /\  -u N  e.  NN )  /\  A  e.  X
)  ->  ( ( N S A ) P B )  =  ( N  x.  ( A P B ) ) )
218, 20jaoian 782 . 2  |-  ( ( ( N  e.  NN0  \/  ( N  e.  RR  /\  -u N  e.  NN ) )  /\  A  e.  X )  ->  (
( N S A ) P B )  =  ( N  x.  ( A P B ) ) )
221, 21sylanb 472 1  |-  ( ( N  e.  ZZ  /\  A  e.  X )  ->  ( ( N S A ) P B )  =  ( N  x.  ( A P B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1369    e. wcel 1756   ` cfv 5423  (class class class)co 6096   RRcr 9286    x. cmul 9292   -ucneg 9601   NNcn 10327   NN0cn0 10584   ZZcz 10651   +vcpv 23968   BaseSetcba 23969   .sOLDcns 23970   .iOLDcdip 24100   CPreHil OLDccphlo 24217
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 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-inf2 7852  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-pre-sup 9365
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 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-se 4685  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-isom 5432  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-1o 6925  df-oadd 6929  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-sup 7696  df-oi 7729  df-card 8114  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999  df-nn 10328  df-2 10385  df-3 10386  df-4 10387  df-n0 10585  df-z 10652  df-uz 10867  df-rp 10997  df-fz 11443  df-fzo 11554  df-seq 11812  df-exp 11871  df-hash 12109  df-cj 12593  df-re 12594  df-im 12595  df-sqr 12729  df-abs 12730  df-clim 12971  df-sum 13169  df-grpo 23683  df-gid 23684  df-ginv 23685  df-ablo 23774  df-vc 23929  df-nv 23975  df-va 23978  df-ba 23979  df-sm 23980  df-0v 23981  df-nmcv 23983  df-dip 24101  df-ph 24218
This theorem is referenced by:  ipasslem5  24240
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