Theorem nvmul0or 27505

Description: If a scalar product is zero, one of its factors must be zero. (Contributed by NM, 6-Dec-2007.) (New usage is discouraged.)

Hypotheses

Ref	Expression
nvmul0or.1	|- X = ( BaseSet ` U )
nvmul0or.4	|- S = ( .sOLD ` U )
nvmul0or.6	|- Z = ( 0vec ` U )

Assertion

Ref	Expression
nvmul0or	|- ( ( U e. NrmCVec /\ A e. CC /\ B e. X ) -> ( ( A S B ) = Z <-> ( A = 0 \/ B = Z ) ) )

Proof of Theorem nvmul0or

Step	Hyp	Ref	Expression
1		df-ne 2795	. . . . 5 |- ( A =/= 0 <-> -. A = 0 )
2		oveq2 6658	. . . . . . . 8 |- ( ( A S B ) = Z -> ( ( 1 / A ) S ( A S B ) ) = ( ( 1 / A ) S Z ) )
3	2	ad2antlr 763	. . . . . . 7 |- ( ( ( ( U e. NrmCVec /\ A e. CC /\ B e. X ) /\ ( A S B ) = Z ) /\ A =/= 0 ) -> ( ( 1 / A ) S ( A S B ) ) = ( ( 1 / A ) S Z ) )
4		recid2 10700	. . . . . . . . . . 11 |- ( ( A e. CC /\ A =/= 0 ) -> ( ( 1 / A ) x. A ) = 1 )
5	4	oveq1d 6665	. . . . . . . . . 10 |- ( ( A e. CC /\ A =/= 0 ) -> ( ( ( 1 / A ) x. A ) S B ) = ( 1 S B ) )
6	5	3ad2antl2 1224	. . . . . . . . 9 |- ( ( ( U e. NrmCVec /\ A e. CC /\ B e. X ) /\ A =/= 0 ) -> ( ( ( 1 / A ) x. A ) S B ) = ( 1 S B ) )
7		simpl1 1064	. . . . . . . . . 10 |- ( ( ( U e. NrmCVec /\ A e. CC /\ B e. X ) /\ A =/= 0 ) -> U e. NrmCVec )
8		reccl 10692	. . . . . . . . . . 11 |- ( ( A e. CC /\ A =/= 0 ) -> ( 1 / A ) e. CC )
9	8	3ad2antl2 1224	. . . . . . . . . 10 |- ( ( ( U e. NrmCVec /\ A e. CC /\ B e. X ) /\ A =/= 0 ) -> ( 1 / A ) e. CC )
10		simpl2 1065	. . . . . . . . . 10 |- ( ( ( U e. NrmCVec /\ A e. CC /\ B e. X ) /\ A =/= 0 ) -> A e. CC )
11		simpl3 1066	. . . . . . . . . 10 |- ( ( ( U e. NrmCVec /\ A e. CC /\ B e. X ) /\ A =/= 0 ) -> B e. X )
12		nvmul0or.1	. . . . . . . . . . 11 |- X = ( BaseSet ` U )
13		nvmul0or.4	. . . . . . . . . . 11 |- S = ( .sOLD ` U )
14	12, 13	nvsass 27483	. . . . . . . . . 10 |- ( ( U e. NrmCVec /\ ( ( 1 / A ) e. CC /\ A e. CC /\ B e. X ) ) -> ( ( ( 1 / A ) x. A ) S B ) = ( ( 1 / A ) S ( A S B ) ) )
15	7, 9, 10, 11, 14	syl13anc 1328	. . . . . . . . 9 |- ( ( ( U e. NrmCVec /\ A e. CC /\ B e. X ) /\ A =/= 0 ) -> ( ( ( 1 / A ) x. A ) S B ) = ( ( 1 / A ) S ( A S B ) ) )
16	12, 13	nvsid 27482	. . . . . . . . . . 11 |- ( ( U e. NrmCVec /\ B e. X ) -> ( 1 S B ) = B )
17	16	3adant2 1080	. . . . . . . . . 10 |- ( ( U e. NrmCVec /\ A e. CC /\ B e. X ) -> ( 1 S B ) = B )
18	17	adantr 481	. . . . . . . . 9 |- ( ( ( U e. NrmCVec /\ A e. CC /\ B e. X ) /\ A =/= 0 ) -> ( 1 S B ) = B )
19	6, 15, 18	3eqtr3d 2664	. . . . . . . 8 |- ( ( ( U e. NrmCVec /\ A e. CC /\ B e. X ) /\ A =/= 0 ) -> ( ( 1 / A ) S ( A S B ) ) = B )
20	19	adantlr 751	. . . . . . 7 |- ( ( ( ( U e. NrmCVec /\ A e. CC /\ B e. X ) /\ ( A S B ) = Z ) /\ A =/= 0 ) -> ( ( 1 / A ) S ( A S B ) ) = B )
21		nvmul0or.6	. . . . . . . . . . . 12 |- Z = ( 0vec ` U )
22	13, 21	nvsz 27493	. . . . . . . . . . 11 |- ( ( U e. NrmCVec /\ ( 1 / A ) e. CC ) -> ( ( 1 / A ) S Z ) = Z )
23	8, 22	sylan2 491	. . . . . . . . . 10 |- ( ( U e. NrmCVec /\ ( A e. CC /\ A =/= 0 ) ) -> ( ( 1 / A ) S Z ) = Z )
24	23	anassrs 680	. . . . . . . . 9 |- ( ( ( U e. NrmCVec /\ A e. CC ) /\ A =/= 0 ) -> ( ( 1 / A ) S Z ) = Z )
25	24	3adantl3 1219	. . . . . . . 8 |- ( ( ( U e. NrmCVec /\ A e. CC /\ B e. X ) /\ A =/= 0 ) -> ( ( 1 / A ) S Z ) = Z )
26	25	adantlr 751	. . . . . . 7 |- ( ( ( ( U e. NrmCVec /\ A e. CC /\ B e. X ) /\ ( A S B ) = Z ) /\ A =/= 0 ) -> ( ( 1 / A ) S Z ) = Z )
27	3, 20, 26	3eqtr3d 2664	. . . . . 6 |- ( ( ( ( U e. NrmCVec /\ A e. CC /\ B e. X ) /\ ( A S B ) = Z ) /\ A =/= 0 ) -> B = Z )
28	27	ex 450	. . . . 5 |- ( ( ( U e. NrmCVec /\ A e. CC /\ B e. X ) /\ ( A S B ) = Z ) -> ( A =/= 0 -> B = Z ) )
29	1, 28	syl5bir 233	. . . 4 |- ( ( ( U e. NrmCVec /\ A e. CC /\ B e. X ) /\ ( A S B ) = Z ) -> ( -. A = 0 -> B = Z ) )
30	29	orrd 393	. . 3 |- ( ( ( U e. NrmCVec /\ A e. CC /\ B e. X ) /\ ( A S B ) = Z ) -> ( A = 0 \/ B = Z ) )
31	30	ex 450	. 2 |- ( ( U e. NrmCVec /\ A e. CC /\ B e. X ) -> ( ( A S B ) = Z -> ( A = 0 \/ B = Z ) ) )
32	12, 13, 21	nv0 27492	. . . . 5 |- ( ( U e. NrmCVec /\ B e. X ) -> ( 0 S B ) = Z )
33		oveq1 6657	. . . . . 6 |- ( A = 0 -> ( A S B ) = ( 0 S B ) )
34	33	eqeq1d 2624	. . . . 5 |- ( A = 0 -> ( ( A S B ) = Z <-> ( 0 S B ) = Z ) )
35	32, 34	syl5ibrcom 237	. . . 4 |- ( ( U e. NrmCVec /\ B e. X ) -> ( A = 0 -> ( A S B ) = Z ) )
36	35	3adant2 1080	. . 3 |- ( ( U e. NrmCVec /\ A e. CC /\ B e. X ) -> ( A = 0 -> ( A S B ) = Z ) )
37	13, 21	nvsz 27493	. . . . 5 |- ( ( U e. NrmCVec /\ A e. CC ) -> ( A S Z ) = Z )
38		oveq2 6658	. . . . . 6 |- ( B = Z -> ( A S B ) = ( A S Z ) )
39	38	eqeq1d 2624	. . . . 5 |- ( B = Z -> ( ( A S B ) = Z <-> ( A S Z ) = Z ) )
40	37, 39	syl5ibrcom 237	. . . 4 |- ( ( U e. NrmCVec /\ A e. CC ) -> ( B = Z -> ( A S B ) = Z ) )
41	40	3adant3 1081	. . 3 |- ( ( U e. NrmCVec /\ A e. CC /\ B e. X ) -> ( B = Z -> ( A S B ) = Z ) )
42	36, 41	jaod 395	. 2 |- ( ( U e. NrmCVec /\ A e. CC /\ B e. X ) -> ( ( A = 0 \/ B = Z ) -> ( A S B ) = Z ) )
43	31, 42	impbid 202	1 |- ( ( U e. NrmCVec /\ A e. CC /\ B e. X ) -> ( ( A S B ) = Z <-> ( A = 0 \/ B = Z ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ 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   NrmCVeccnv 27439   BaseSetcba 27441   .sOLDcns 27442   0veccn0v 27443

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-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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  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-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  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-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  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-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

This theorem is referenced by:  nmlno0lem  27648