Theorem ipasslem11 27695

Description: Lemma for ipassi 27696. Show the inner product associative law for all complex numbers. (Contributed by NM, 25-Aug-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
ipasslem11.a	|- A e. X
ipasslem11.b	|- B e. X

Assertion

Ref	Expression
ipasslem11	|- ( C e. CC -> ( ( C S A ) P B ) = ( C x. ( A P B ) ) )

Proof of Theorem ipasslem11

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnre 10036	. 2 |- ( C e. CC -> E. x e. RR E. y e. RR C = ( x + ( _i x. y ) ) )
2		ax-icn 9995	. . . . . . . 8 |- _i e. CC
3		recn 10026	. . . . . . . 8 |- ( y e. RR -> y e. CC )
4		mulcom 10022	. . . . . . . 8 |- ( ( _i e. CC /\ y e. CC ) -> ( _i x. y ) = ( y x. _i ) )
5	2, 3, 4	sylancr 695	. . . . . . 7 |- ( y e. RR -> ( _i x. y ) = ( y x. _i ) )
6	5	adantl 482	. . . . . 6 |- ( ( x e. RR /\ y e. RR ) -> ( _i x. y ) = ( y x. _i ) )
7	6	oveq2d 6666	. . . . 5 |- ( ( x e. RR /\ y e. RR ) -> ( x + ( _i x. y ) ) = ( x + ( y x. _i ) ) )
8	7	eqeq2d 2632	. . . 4 |- ( ( x e. RR /\ y e. RR ) -> ( C = ( x + ( _i x. y ) ) <-> C = ( x + ( y x. _i ) ) ) )
9		recn 10026	. . . . . . . . 9 |- ( x e. RR -> x e. CC )
10		ip1i.9	. . . . . . . . . . 11 |- U e. CPreHil OLD
11	10	phnvi 27671	. . . . . . . . . 10 |- U e. NrmCVec
12		ipasslem11.a	. . . . . . . . . 10 |- A e. X
13		ip1i.1	. . . . . . . . . . 11 |- X = ( BaseSet ` U )
14		ip1i.4	. . . . . . . . . . 11 |- S = ( .sOLD ` U )
15	13, 14	nvscl 27481	. . . . . . . . . 10 |- ( ( U e. NrmCVec /\ x e. CC /\ A e. X ) -> ( x S A ) e. X )
16	11, 12, 15	mp3an13 1415	. . . . . . . . 9 |- ( x e. CC -> ( x S A ) e. X )
17	9, 16	syl 17	. . . . . . . 8 |- ( x e. RR -> ( x S A ) e. X )
18		mulcl 10020	. . . . . . . . . 10 |- ( ( y e. CC /\ _i e. CC ) -> ( y x. _i ) e. CC )
19	3, 2, 18	sylancl 694	. . . . . . . . 9 |- ( y e. RR -> ( y x. _i ) e. CC )
20	13, 14	nvscl 27481	. . . . . . . . . 10 |- ( ( U e. NrmCVec /\ ( y x. _i ) e. CC /\ A e. X ) -> ( ( y x. _i ) S A ) e. X )
21	11, 12, 20	mp3an13 1415	. . . . . . . . 9 |- ( ( y x. _i ) e. CC -> ( ( y x. _i ) S A ) e. X )
22	19, 21	syl 17	. . . . . . . 8 |- ( y e. RR -> ( ( y x. _i ) S A ) e. X )
23		ipasslem11.b	. . . . . . . . 9 |- B e. X
24		ip1i.2	. . . . . . . . . 10 |- G = ( +v ` U )
25		ip1i.7	. . . . . . . . . 10 |- P = ( .iOLD ` U )
26	13, 24, 14, 25, 10	ipdiri 27685	. . . . . . . . 9 |- ( ( ( x S A ) e. X /\ ( ( y x. _i ) S A ) e. X /\ B e. X ) -> ( ( ( x S A ) G ( ( y x. _i ) S A ) ) P B ) = ( ( ( x S A ) P B ) + ( ( ( y x. _i ) S A ) P B ) ) )
27	23, 26	mp3an3 1413	. . . . . . . 8 |- ( ( ( x S A ) e. X /\ ( ( y x. _i ) S A ) e. X ) -> ( ( ( x S A ) G ( ( y x. _i ) S A ) ) P B ) = ( ( ( x S A ) P B ) + ( ( ( y x. _i ) S A ) P B ) ) )
28	17, 22, 27	syl2an 494	. . . . . . 7 |- ( ( x e. RR /\ y e. RR ) -> ( ( ( x S A ) G ( ( y x. _i ) S A ) ) P B ) = ( ( ( x S A ) P B ) + ( ( ( y x. _i ) S A ) P B ) ) )
29	13, 24, 14, 25, 10, 12, 23	ipasslem9 27693	. . . . . . . 8 |- ( x e. RR -> ( ( x S A ) P B ) = ( x x. ( A P B ) ) )
30	13, 14	nvscl 27481	. . . . . . . . . . 11 |- ( ( U e. NrmCVec /\ _i e. CC /\ A e. X ) -> ( _i S A ) e. X )
31	11, 2, 12, 30	mp3an 1424	. . . . . . . . . 10 |- ( _i S A ) e. X
32	13, 24, 14, 25, 10, 31, 23	ipasslem9 27693	. . . . . . . . 9 |- ( y e. RR -> ( ( y S ( _i S A ) ) P B ) = ( y x. ( ( _i S A ) P B ) ) )
33	13, 14	nvsass 27483	. . . . . . . . . . . . 13 |- ( ( U e. NrmCVec /\ ( y e. CC /\ _i e. CC /\ A e. X ) ) -> ( ( y x. _i ) S A ) = ( y S ( _i S A ) ) )
34	11, 33	mpan 706	. . . . . . . . . . . 12 |- ( ( y e. CC /\ _i e. CC /\ A e. X ) -> ( ( y x. _i ) S A ) = ( y S ( _i S A ) ) )
35	2, 12, 34	mp3an23 1416	. . . . . . . . . . 11 |- ( y e. CC -> ( ( y x. _i ) S A ) = ( y S ( _i S A ) ) )
36	3, 35	syl 17	. . . . . . . . . 10 |- ( y e. RR -> ( ( y x. _i ) S A ) = ( y S ( _i S A ) ) )
37	36	oveq1d 6665	. . . . . . . . 9 |- ( y e. RR -> ( ( ( y x. _i ) S A ) P B ) = ( ( y S ( _i S A ) ) P B ) )
38	13, 25	dipcl 27567	. . . . . . . . . . . . 13 |- ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( A P B ) e. CC )
39	11, 12, 23, 38	mp3an 1424	. . . . . . . . . . . 12 |- ( A P B ) e. CC
40		mulass 10024	. . . . . . . . . . . 12 |- ( ( y e. CC /\ _i e. CC /\ ( A P B ) e. CC ) -> ( ( y x. _i ) x. ( A P B ) ) = ( y x. ( _i x. ( A P B ) ) ) )
41	2, 39, 40	mp3an23 1416	. . . . . . . . . . 11 |- ( y e. CC -> ( ( y x. _i ) x. ( A P B ) ) = ( y x. ( _i x. ( A P B ) ) ) )
42	3, 41	syl 17	. . . . . . . . . 10 |- ( y e. RR -> ( ( y x. _i ) x. ( A P B ) ) = ( y x. ( _i x. ( A P B ) ) ) )
43		eqid 2622	. . . . . . . . . . . 12 |- ( normCV ` U ) = ( normCV ` U )
44	13, 24, 14, 25, 10, 12, 23, 43	ipasslem10 27694	. . . . . . . . . . 11 |- ( ( _i S A ) P B ) = ( _i x. ( A P B ) )
45	44	oveq2i 6661	. . . . . . . . . 10 |- ( y x. ( ( _i S A ) P B ) ) = ( y x. ( _i x. ( A P B ) ) )
46	42, 45	syl6eqr 2674	. . . . . . . . 9 |- ( y e. RR -> ( ( y x. _i ) x. ( A P B ) ) = ( y x. ( ( _i S A ) P B ) ) )
47	32, 37, 46	3eqtr4d 2666	. . . . . . . 8 |- ( y e. RR -> ( ( ( y x. _i ) S A ) P B ) = ( ( y x. _i ) x. ( A P B ) ) )
48	29, 47	oveqan12d 6669	. . . . . . 7 |- ( ( x e. RR /\ y e. RR ) -> ( ( ( x S A ) P B ) + ( ( ( y x. _i ) S A ) P B ) ) = ( ( x x. ( A P B ) ) + ( ( y x. _i ) x. ( A P B ) ) ) )
49	28, 48	eqtrd 2656	. . . . . 6 |- ( ( x e. RR /\ y e. RR ) -> ( ( ( x S A ) G ( ( y x. _i ) S A ) ) P B ) = ( ( x x. ( A P B ) ) + ( ( y x. _i ) x. ( A P B ) ) ) )
50	13, 24, 14	nvdir 27486	. . . . . . . . . 10 |- ( ( U e. NrmCVec /\ ( x e. CC /\ ( y x. _i ) e. CC /\ A e. X ) ) -> ( ( x + ( y x. _i ) ) S A ) = ( ( x S A ) G ( ( y x. _i ) S A ) ) )
51	11, 50	mpan 706	. . . . . . . . 9 |- ( ( x e. CC /\ ( y x. _i ) e. CC /\ A e. X ) -> ( ( x + ( y x. _i ) ) S A ) = ( ( x S A ) G ( ( y x. _i ) S A ) ) )
52	12, 51	mp3an3 1413	. . . . . . . 8 |- ( ( x e. CC /\ ( y x. _i ) e. CC ) -> ( ( x + ( y x. _i ) ) S A ) = ( ( x S A ) G ( ( y x. _i ) S A ) ) )
53	9, 19, 52	syl2an 494	. . . . . . 7 |- ( ( x e. RR /\ y e. RR ) -> ( ( x + ( y x. _i ) ) S A ) = ( ( x S A ) G ( ( y x. _i ) S A ) ) )
54	53	oveq1d 6665	. . . . . 6 |- ( ( x e. RR /\ y e. RR ) -> ( ( ( x + ( y x. _i ) ) S A ) P B ) = ( ( ( x S A ) G ( ( y x. _i ) S A ) ) P B ) )
55		adddir 10031	. . . . . . . 8 |- ( ( x e. CC /\ ( y x. _i ) e. CC /\ ( A P B ) e. CC ) -> ( ( x + ( y x. _i ) ) x. ( A P B ) ) = ( ( x x. ( A P B ) ) + ( ( y x. _i ) x. ( A P B ) ) ) )
56	39, 55	mp3an3 1413	. . . . . . 7 |- ( ( x e. CC /\ ( y x. _i ) e. CC ) -> ( ( x + ( y x. _i ) ) x. ( A P B ) ) = ( ( x x. ( A P B ) ) + ( ( y x. _i ) x. ( A P B ) ) ) )
57	9, 19, 56	syl2an 494	. . . . . 6 |- ( ( x e. RR /\ y e. RR ) -> ( ( x + ( y x. _i ) ) x. ( A P B ) ) = ( ( x x. ( A P B ) ) + ( ( y x. _i ) x. ( A P B ) ) ) )
58	49, 54, 57	3eqtr4d 2666	. . . . 5 |- ( ( x e. RR /\ y e. RR ) -> ( ( ( x + ( y x. _i ) ) S A ) P B ) = ( ( x + ( y x. _i ) ) x. ( A P B ) ) )
59		oveq1 6657	. . . . . . 7 |- ( C = ( x + ( y x. _i ) ) -> ( C S A ) = ( ( x + ( y x. _i ) ) S A ) )
60	59	oveq1d 6665	. . . . . 6 |- ( C = ( x + ( y x. _i ) ) -> ( ( C S A ) P B ) = ( ( ( x + ( y x. _i ) ) S A ) P B ) )
61		oveq1 6657	. . . . . 6 |- ( C = ( x + ( y x. _i ) ) -> ( C x. ( A P B ) ) = ( ( x + ( y x. _i ) ) x. ( A P B ) ) )
62	60, 61	eqeq12d 2637	. . . . 5 |- ( C = ( x + ( y x. _i ) ) -> ( ( ( C S A ) P B ) = ( C x. ( A P B ) ) <-> ( ( ( x + ( y x. _i ) ) S A ) P B ) = ( ( x + ( y x. _i ) ) x. ( A P B ) ) ) )
63	58, 62	syl5ibrcom 237	. . . 4 |- ( ( x e. RR /\ y e. RR ) -> ( C = ( x + ( y x. _i ) ) -> ( ( C S A ) P B ) = ( C x. ( A P B ) ) ) )
64	8, 63	sylbid 230	. . 3 |- ( ( x e. RR /\ y e. RR ) -> ( C = ( x + ( _i x. y ) ) -> ( ( C S A ) P B ) = ( C x. ( A P B ) ) ) )
65	64	rexlimivv 3036	. 2 |- ( E. x e. RR E. y e. RR C = ( x + ( _i x. y ) ) -> ( ( C S A ) P B ) = ( C x. ( A P B ) ) )
66	1, 65	syl 17	1 |- ( C e. CC -> ( ( C S A ) P B ) = ( C x. ( A P B ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   E.wrex 2913   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   _ici 9938   + caddc 9939   x. cmul 9941   NrmCVeccnv 27439   +vcpv 27440   BaseSetcba 27441   .sOLDcns 27442   norm_CVcnmcv 27445   .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  ax-addf 10015  ax-mulf 10016

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-iin 4523  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-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  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-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-icc 12182  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-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-cn 21031  df-cnp 21032  df-t1 21118  df-haus 21119  df-tx 21365  df-hmeo 21558  df-xms 22125  df-ms 22126  df-tms 22127  df-grpo 27347  df-gid 27348  df-ginv 27349  df-gdiv 27350  df-ablo 27399  df-vc 27414  df-nv 27447  df-va 27450  df-ba 27451  df-sm 27452  df-0v 27453  df-vs 27454  df-nmcv 27455  df-ims 27456  df-dip 27556  df-ph 27668

This theorem is referenced by:  ipassi  27696