Theorem ipasslem4 27689

Description: Lemma for ipassi 27696. 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

Step	Hyp	Ref	Expression
1		nnrecre 11057	. . . . 5 |- ( N e. NN -> ( 1 / N ) e. RR )
2	1	recnd 10068	. . . 4 |- ( N e. NN -> ( 1 / N ) e. CC )
3		ip1i.9	. . . . . 6 |- U e. CPreHil OLD
4	3	phnvi 27671	. . . . 5 |- U e. NrmCVec
5		ip1i.1	. . . . . 6 |- X = ( BaseSet ` U )
6		ip1i.4	. . . . . 6 |- S = ( .sOLD ` U )
7	5, 6	nvscl 27481	. . . . 5 |- ( ( U e. NrmCVec /\ ( 1 / N ) e. CC /\ A e. X ) -> ( ( 1 / N ) S A ) e. X )
8	4, 7	mp3an1 1411	. . . 4 |- ( ( ( 1 / N ) e. CC /\ A e. X ) -> ( ( 1 / N ) S A ) e. X )
9	2, 8	sylan 488	. . 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 )
12	5, 11	dipcl 27567	. . . 4 |- ( ( U e. NrmCVec /\ ( ( 1 / N ) S A ) e. X /\ B e. X ) -> ( ( ( 1 / N ) S A ) P B ) e. CC )
13	4, 10, 12	mp3an13 1415	. . 3 |- ( ( ( 1 / N ) S A ) e. X -> ( ( ( 1 / N ) S A ) P B ) e. CC )
14	9, 13	syl 17	. 2 |- ( ( N e. NN /\ A e. X ) -> ( ( ( 1 / N ) S A ) P B ) e. CC )
15	5, 11	dipcl 27567	. . . 4 |- ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( A P B ) e. CC )
16	4, 10, 15	mp3an13 1415	. . 3 |- ( A e. X -> ( A P B ) e. CC )
17		mulcl 10020	. . 3 |- ( ( ( 1 / N ) e. CC /\ ( A P B ) e. CC ) -> ( ( 1 / N ) x. ( A P B ) ) e. CC )
18	2, 16, 17	syl2an 494	. 2 |- ( ( N e. NN /\ A e. X ) -> ( ( 1 / N ) x. ( A P B ) ) e. CC )
19		nncn 11028	. . 3 |- ( N e. NN -> N e. CC )
20	19	adantr 481	. 2 |- ( ( N e. NN /\ A e. X ) -> N e. CC )
21		nnne0 11053	. . 3 |- ( N e. NN -> N =/= 0 )
22	21	adantr 481	. 2 |- ( ( N e. NN /\ A e. X ) -> N =/= 0 )
23	19, 21	recidd 10796	. . . . . 6 |- ( N e. NN -> ( N x. ( 1 / N ) ) = 1 )
24	23	oveq1d 6665	. . . . 5 |- ( N e. NN -> ( ( N x. ( 1 / N ) ) x. ( A P B ) ) = ( 1 x. ( A P B ) ) )
25	16	mulid2d 10058	. . . . 5 |- ( A e. X -> ( 1 x. ( A P B ) ) = ( A P B ) )
26	24, 25	sylan9eq 2676	. . . 4 |- ( ( N e. NN /\ A e. X ) -> ( ( N x. ( 1 / N ) ) x. ( A P B ) ) = ( A P B ) )
27	23	oveq1d 6665	. . . . . . 7 |- ( N e. NN -> ( ( N x. ( 1 / N ) ) S A ) = ( 1 S A ) )
28	5, 6	nvsid 27482	. . . . . . . 8 |- ( ( U e. NrmCVec /\ A e. X ) -> ( 1 S A ) = A )
29	4, 28	mpan 706	. . . . . . 7 |- ( A e. X -> ( 1 S A ) = A )
30	27, 29	sylan9eq 2676	. . . . . 6 |- ( ( N e. NN /\ A e. X ) -> ( ( N x. ( 1 / N ) ) S A ) = A )
31	2	adantr 481	. . . . . . 7 |- ( ( N e. NN /\ A e. X ) -> ( 1 / N ) e. CC )
32		simpr 477	. . . . . . 7 |- ( ( N e. NN /\ A e. X ) -> A e. X )
33	5, 6	nvsass 27483	. . . . . . . 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 ) ) )
34	4, 33	mpan 706	. . . . . . 7 |- ( ( N e. CC /\ ( 1 / N ) e. CC /\ A e. X ) -> ( ( N x. ( 1 / N ) ) S A ) = ( N S ( ( 1 / N ) S A ) ) )
35	20, 31, 32, 34	syl3anc 1326	. . . . . 6 |- ( ( N e. NN /\ A e. X ) -> ( ( N x. ( 1 / N ) ) S A ) = ( N S ( ( 1 / N ) S A ) ) )
36	30, 35	eqtr3d 2658	. . . . 5 |- ( ( N e. NN /\ A e. X ) -> A = ( N S ( ( 1 / N ) S A ) ) )
37	36	oveq1d 6665	. . . 4 |- ( ( N e. NN /\ A e. X ) -> ( A P B ) = ( ( N S ( ( 1 / N ) S A ) ) P B ) )
38		nnnn0 11299	. . . . . 6 |- ( N e. NN -> N e. NN0 )
39	38	adantr 481	. . . . 5 |- ( ( N e. NN /\ A e. X ) -> N e. NN0 )
40		ip1i.2	. . . . . 6 |- G = ( +v ` U )
41	5, 40, 6, 11, 3, 10	ipasslem1 27686	. . . . 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 ) ) )
42	39, 9, 41	syl2anc 693	. . . 4 |- ( ( N e. NN /\ A e. X ) -> ( ( N S ( ( 1 / N ) S A ) ) P B ) = ( N x. ( ( ( 1 / N ) S A ) P B ) ) )
43	26, 37, 42	3eqtrd 2660	. . 3 |- ( ( N e. NN /\ A e. X ) -> ( ( N x. ( 1 / N ) ) x. ( A P B ) ) = ( N x. ( ( ( 1 / N ) S A ) P B ) ) )
44	16	adantl 482	. . . 4 |- ( ( N e. NN /\ A e. X ) -> ( A P B ) e. CC )
45	20, 31, 44	mulassd 10063	. . 3 |- ( ( N e. NN /\ A e. X ) -> ( ( N x. ( 1 / N ) ) x. ( A P B ) ) = ( N x. ( ( 1 / N ) x. ( A P B ) ) ) )
46	43, 45	eqtr3d 2658	. 2 |- ( ( N e. NN /\ A e. X ) -> ( N x. ( ( ( 1 / N ) S A ) P B ) ) = ( N x. ( ( 1 / N ) x. ( A P B ) ) ) )
47	14, 18, 20, 22, 46	mulcanad 10662	1 |- ( ( N e. NN /\ A e. X ) -> ( ( ( 1 / N ) S A ) P B ) = ( ( 1 / N ) x. ( A P B ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   x. cmul 9941   / cdiv 10684   NNcn 11020   NN0cn0 11292   NrmCVeccnv 27439   +vcpv 27440   BaseSetcba 27441   .sOLDcns 27442   .iOLDcdip 27555   CPreHil OLDccphlo 27667

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417  df-grpo 27347  df-gid 27348  df-ginv 27349  df-ablo 27399  df-vc 27414  df-nv 27447  df-va 27450  df-ba 27451  df-sm 27452  df-0v 27453  df-nmcv 27455  df-dip 27556  df-ph 27668

This theorem is referenced by:  ipasslem5  27690