Theorem ipasslem1 27686

Description: Lemma for ipassi 27696. Show the inner product associative law for nonnegative 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
ipasslem1	|- ( ( N e. NN0 /\ A e. X ) -> ( ( N S A ) P B ) = ( N x. ( A P B ) ) )

Proof of Theorem ipasslem1

Dummy variables j k are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nn0cn 11302	. . . . . . . . . . 11 |- ( k e. NN0 -> k e. CC )
2		ax-1cn 9994	. . . . . . . . . . . 12 |- 1 e. CC
3		ip1i.9	. . . . . . . . . . . . . 14 |- U e. CPreHil OLD
4	3	phnvi 27671	. . . . . . . . . . . . 13 |- U e. NrmCVec
5		ip1i.1	. . . . . . . . . . . . . 14 |- X = ( BaseSet ` U )
6		ip1i.2	. . . . . . . . . . . . . 14 |- G = ( +v ` U )
7		ip1i.4	. . . . . . . . . . . . . 14 |- S = ( .sOLD ` U )
8	5, 6, 7	nvdir 27486	. . . . . . . . . . . . 13 |- ( ( U e. NrmCVec /\ ( k e. CC /\ 1 e. CC /\ A e. X ) ) -> ( ( k + 1 ) S A ) = ( ( k S A ) G ( 1 S A ) ) )
9	4, 8	mpan 706	. . . . . . . . . . . 12 |- ( ( k e. CC /\ 1 e. CC /\ A e. X ) -> ( ( k + 1 ) S A ) = ( ( k S A ) G ( 1 S A ) ) )
10	2, 9	mp3an2 1412	. . . . . . . . . . 11 |- ( ( k e. CC /\ A e. X ) -> ( ( k + 1 ) S A ) = ( ( k S A ) G ( 1 S A ) ) )
11	1, 10	sylan 488	. . . . . . . . . 10 |- ( ( k e. NN0 /\ A e. X ) -> ( ( k + 1 ) S A ) = ( ( k S A ) G ( 1 S A ) ) )
12	5, 7	nvsid 27482	. . . . . . . . . . . . 13 |- ( ( U e. NrmCVec /\ A e. X ) -> ( 1 S A ) = A )
13	4, 12	mpan 706	. . . . . . . . . . . 12 |- ( A e. X -> ( 1 S A ) = A )
14	13	adantl 482	. . . . . . . . . . 11 |- ( ( k e. NN0 /\ A e. X ) -> ( 1 S A ) = A )
15	14	oveq2d 6666	. . . . . . . . . 10 |- ( ( k e. NN0 /\ A e. X ) -> ( ( k S A ) G ( 1 S A ) ) = ( ( k S A ) G A ) )
16	11, 15	eqtrd 2656	. . . . . . . . 9 |- ( ( k e. NN0 /\ A e. X ) -> ( ( k + 1 ) S A ) = ( ( k S A ) G A ) )
17	16	oveq1d 6665	. . . . . . . 8 |- ( ( k e. NN0 /\ A e. X ) -> ( ( ( k + 1 ) S A ) P B ) = ( ( ( k S A ) G A ) P B ) )
18		ipasslem1.b	. . . . . . . . . . . . 13 |- B e. X
19		ip1i.7	. . . . . . . . . . . . . 14 |- P = ( .iOLD ` U )
20	5, 19	dipcl 27567	. . . . . . . . . . . . 13 |- ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( A P B ) e. CC )
21	4, 18, 20	mp3an13 1415	. . . . . . . . . . . 12 |- ( A e. X -> ( A P B ) e. CC )
22	21	mulid2d 10058	. . . . . . . . . . 11 |- ( A e. X -> ( 1 x. ( A P B ) ) = ( A P B ) )
23	22	adantl 482	. . . . . . . . . 10 |- ( ( k e. NN0 /\ A e. X ) -> ( 1 x. ( A P B ) ) = ( A P B ) )
24	23	oveq2d 6666	. . . . . . . . 9 |- ( ( k e. NN0 /\ A e. X ) -> ( ( ( k S A ) P B ) + ( 1 x. ( A P B ) ) ) = ( ( ( k S A ) P B ) + ( A P B ) ) )
25	5, 7	nvscl 27481	. . . . . . . . . . . 12 |- ( ( U e. NrmCVec /\ k e. CC /\ A e. X ) -> ( k S A ) e. X )
26	4, 25	mp3an1 1411	. . . . . . . . . . 11 |- ( ( k e. CC /\ A e. X ) -> ( k S A ) e. X )
27	1, 26	sylan 488	. . . . . . . . . 10 |- ( ( k e. NN0 /\ A e. X ) -> ( k S A ) e. X )
28	5, 6, 7, 19, 3	ipdiri 27685	. . . . . . . . . . 11 |- ( ( ( k S A ) e. X /\ A e. X /\ B e. X ) -> ( ( ( k S A ) G A ) P B ) = ( ( ( k S A ) P B ) + ( A P B ) ) )
29	18, 28	mp3an3 1413	. . . . . . . . . 10 |- ( ( ( k S A ) e. X /\ A e. X ) -> ( ( ( k S A ) G A ) P B ) = ( ( ( k S A ) P B ) + ( A P B ) ) )
30	27, 29	sylancom 701	. . . . . . . . 9 |- ( ( k e. NN0 /\ A e. X ) -> ( ( ( k S A ) G A ) P B ) = ( ( ( k S A ) P B ) + ( A P B ) ) )
31	24, 30	eqtr4d 2659	. . . . . . . 8 |- ( ( k e. NN0 /\ A e. X ) -> ( ( ( k S A ) P B ) + ( 1 x. ( A P B ) ) ) = ( ( ( k S A ) G A ) P B ) )
32	17, 31	eqtr4d 2659	. . . . . . 7 |- ( ( k e. NN0 /\ A e. X ) -> ( ( ( k + 1 ) S A ) P B ) = ( ( ( k S A ) P B ) + ( 1 x. ( A P B ) ) ) )
33		oveq1 6657	. . . . . . 7 |- ( ( ( k S A ) P B ) = ( k x. ( A P B ) ) -> ( ( ( k S A ) P B ) + ( 1 x. ( A P B ) ) ) = ( ( k x. ( A P B ) ) + ( 1 x. ( A P B ) ) ) )
34	32, 33	sylan9eq 2676	. . . . . 6 |- ( ( ( k e. NN0 /\ A e. X ) /\ ( ( k S A ) P B ) = ( k x. ( A P B ) ) ) -> ( ( ( k + 1 ) S A ) P B ) = ( ( k x. ( A P B ) ) + ( 1 x. ( A P B ) ) ) )
35		adddir 10031	. . . . . . . . 9 |- ( ( k e. CC /\ 1 e. CC /\ ( A P B ) e. CC ) -> ( ( k + 1 ) x. ( A P B ) ) = ( ( k x. ( A P B ) ) + ( 1 x. ( A P B ) ) ) )
36	2, 35	mp3an2 1412	. . . . . . . 8 |- ( ( k e. CC /\ ( A P B ) e. CC ) -> ( ( k + 1 ) x. ( A P B ) ) = ( ( k x. ( A P B ) ) + ( 1 x. ( A P B ) ) ) )
37	1, 21, 36	syl2an 494	. . . . . . 7 |- ( ( k e. NN0 /\ A e. X ) -> ( ( k + 1 ) x. ( A P B ) ) = ( ( k x. ( A P B ) ) + ( 1 x. ( A P B ) ) ) )
38	37	adantr 481	. . . . . 6 |- ( ( ( k e. NN0 /\ A e. X ) /\ ( ( k S A ) P B ) = ( k x. ( A P B ) ) ) -> ( ( k + 1 ) x. ( A P B ) ) = ( ( k x. ( A P B ) ) + ( 1 x. ( A P B ) ) ) )
39	34, 38	eqtr4d 2659	. . . . 5 |- ( ( ( k e. NN0 /\ A e. X ) /\ ( ( k S A ) P B ) = ( k x. ( A P B ) ) ) -> ( ( ( k + 1 ) S A ) P B ) = ( ( k + 1 ) x. ( A P B ) ) )
40	39	exp31 630	. . . 4 |- ( k e. NN0 -> ( A e. X -> ( ( ( k S A ) P B ) = ( k x. ( A P B ) ) -> ( ( ( k + 1 ) S A ) P B ) = ( ( k + 1 ) x. ( A P B ) ) ) ) )
41	40	a2d 29	. . 3 |- ( k e. NN0 -> ( ( A e. X -> ( ( k S A ) P B ) = ( k x. ( A P B ) ) ) -> ( A e. X -> ( ( ( k + 1 ) S A ) P B ) = ( ( k + 1 ) x. ( A P B ) ) ) ) )
42		eqid 2622	. . . . . 6 |- ( 0vec ` U ) = ( 0vec ` U )
43	5, 42, 19	dip0l 27573	. . . . 5 |- ( ( U e. NrmCVec /\ B e. X ) -> ( ( 0vec ` U ) P B ) = 0 )
44	4, 18, 43	mp2an 708	. . . 4 |- ( ( 0vec ` U ) P B ) = 0
45	5, 7, 42	nv0 27492	. . . . . 6 |- ( ( U e. NrmCVec /\ A e. X ) -> ( 0 S A ) = ( 0vec ` U ) )
46	4, 45	mpan 706	. . . . 5 |- ( A e. X -> ( 0 S A ) = ( 0vec ` U ) )
47	46	oveq1d 6665	. . . 4 |- ( A e. X -> ( ( 0 S A ) P B ) = ( ( 0vec ` U ) P B ) )
48	21	mul02d 10234	. . . 4 |- ( A e. X -> ( 0 x. ( A P B ) ) = 0 )
49	44, 47, 48	3eqtr4a 2682	. . 3 |- ( A e. X -> ( ( 0 S A ) P B ) = ( 0 x. ( A P B ) ) )
50		oveq1 6657	. . . . . 6 |- ( j = 0 -> ( j S A ) = ( 0 S A ) )
51	50	oveq1d 6665	. . . . 5 |- ( j = 0 -> ( ( j S A ) P B ) = ( ( 0 S A ) P B ) )
52		oveq1 6657	. . . . 5 |- ( j = 0 -> ( j x. ( A P B ) ) = ( 0 x. ( A P B ) ) )
53	51, 52	eqeq12d 2637	. . . 4 |- ( j = 0 -> ( ( ( j S A ) P B ) = ( j x. ( A P B ) ) <-> ( ( 0 S A ) P B ) = ( 0 x. ( A P B ) ) ) )
54	53	imbi2d 330	. . 3 |- ( j = 0 -> ( ( A e. X -> ( ( j S A ) P B ) = ( j x. ( A P B ) ) ) <-> ( A e. X -> ( ( 0 S A ) P B ) = ( 0 x. ( A P B ) ) ) ) )
55		oveq1 6657	. . . . . 6 |- ( j = k -> ( j S A ) = ( k S A ) )
56	55	oveq1d 6665	. . . . 5 |- ( j = k -> ( ( j S A ) P B ) = ( ( k S A ) P B ) )
57		oveq1 6657	. . . . 5 |- ( j = k -> ( j x. ( A P B ) ) = ( k x. ( A P B ) ) )
58	56, 57	eqeq12d 2637	. . . 4 |- ( j = k -> ( ( ( j S A ) P B ) = ( j x. ( A P B ) ) <-> ( ( k S A ) P B ) = ( k x. ( A P B ) ) ) )
59	58	imbi2d 330	. . 3 |- ( j = k -> ( ( A e. X -> ( ( j S A ) P B ) = ( j x. ( A P B ) ) ) <-> ( A e. X -> ( ( k S A ) P B ) = ( k x. ( A P B ) ) ) ) )
60		oveq1 6657	. . . . . 6 |- ( j = ( k + 1 ) -> ( j S A ) = ( ( k + 1 ) S A ) )
61	60	oveq1d 6665	. . . . 5 |- ( j = ( k + 1 ) -> ( ( j S A ) P B ) = ( ( ( k + 1 ) S A ) P B ) )
62		oveq1 6657	. . . . 5 |- ( j = ( k + 1 ) -> ( j x. ( A P B ) ) = ( ( k + 1 ) x. ( A P B ) ) )
63	61, 62	eqeq12d 2637	. . . 4 |- ( j = ( k + 1 ) -> ( ( ( j S A ) P B ) = ( j x. ( A P B ) ) <-> ( ( ( k + 1 ) S A ) P B ) = ( ( k + 1 ) x. ( A P B ) ) ) )
64	63	imbi2d 330	. . 3 |- ( j = ( k + 1 ) -> ( ( A e. X -> ( ( j S A ) P B ) = ( j x. ( A P B ) ) ) <-> ( A e. X -> ( ( ( k + 1 ) S A ) P B ) = ( ( k + 1 ) x. ( A P B ) ) ) ) )
65		oveq1 6657	. . . . . 6 |- ( j = N -> ( j S A ) = ( N S A ) )
66	65	oveq1d 6665	. . . . 5 |- ( j = N -> ( ( j S A ) P B ) = ( ( N S A ) P B ) )
67		oveq1 6657	. . . . 5 |- ( j = N -> ( j x. ( A P B ) ) = ( N x. ( A P B ) ) )
68	66, 67	eqeq12d 2637	. . . 4 |- ( j = N -> ( ( ( j S A ) P B ) = ( j x. ( A P B ) ) <-> ( ( N S A ) P B ) = ( N x. ( A P B ) ) ) )
69	68	imbi2d 330	. . 3 |- ( j = N -> ( ( A e. X -> ( ( j S A ) P B ) = ( j x. ( A P B ) ) ) <-> ( A e. X -> ( ( N S A ) P B ) = ( N x. ( A P B ) ) ) ) )
70	41, 49, 54, 59, 64, 69	nn0indALT 11473	. 2 |- ( N e. NN0 -> ( A e. X -> ( ( N S A ) P B ) = ( N x. ( A P B ) ) ) )
71	70	imp 445	1 |- ( ( N e. NN0 /\ A e. X ) -> ( ( N S A ) P B ) = ( N x. ( A P B ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   NN0cn0 11292   NrmCVeccnv 27439   +vcpv 27440   BaseSetcba 27441   .sOLDcns 27442   0veccn0v 27443   .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:  ipasslem2  27687  ipasslem3  27688  ipasslem4  27689