Theorem hvmul0or 27882

Description: If a scalar product is zero, one of its factors must be zero. (Contributed by NM, 19-May-2005.) (New usage is discouraged.)

Assertion

Ref	Expression
hvmul0or	|- ( ( A e. CC /\ B e. ~H ) -> ( ( A .h B ) = 0h <-> ( A = 0 \/ B = 0h ) ) )

Proof of Theorem hvmul0or

Step	Hyp	Ref	Expression
1		df-ne 2795	. . . . 5 |- ( A =/= 0 <-> -. A = 0 )
2		oveq2 6658	. . . . . . . 8 |- ( ( A .h B ) = 0h -> ( ( 1 / A ) .h ( A .h B ) ) = ( ( 1 / A ) .h 0h ) )
3	2	ad2antlr 763	. . . . . . 7 |- ( ( ( ( A e. CC /\ B e. ~H ) /\ ( A .h B ) = 0h ) /\ A =/= 0 ) -> ( ( 1 / A ) .h ( A .h B ) ) = ( ( 1 / A ) .h 0h ) )
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 ) .h B ) = ( 1 .h B ) )
6	5	adantlr 751	. . . . . . . . 9 |- ( ( ( A e. CC /\ B e. ~H ) /\ A =/= 0 ) -> ( ( ( 1 / A ) x. A ) .h B ) = ( 1 .h B ) )
7		reccl 10692	. . . . . . . . . . 11 |- ( ( A e. CC /\ A =/= 0 ) -> ( 1 / A ) e. CC )
8	7	adantlr 751	. . . . . . . . . 10 |- ( ( ( A e. CC /\ B e. ~H ) /\ A =/= 0 ) -> ( 1 / A ) e. CC )
9		simpll 790	. . . . . . . . . 10 |- ( ( ( A e. CC /\ B e. ~H ) /\ A =/= 0 ) -> A e. CC )
10		simplr 792	. . . . . . . . . 10 |- ( ( ( A e. CC /\ B e. ~H ) /\ A =/= 0 ) -> B e. ~H )
11		ax-hvmulass 27864	. . . . . . . . . 10 |- ( ( ( 1 / A ) e. CC /\ A e. CC /\ B e. ~H ) -> ( ( ( 1 / A ) x. A ) .h B ) = ( ( 1 / A ) .h ( A .h B ) ) )
12	8, 9, 10, 11	syl3anc 1326	. . . . . . . . 9 |- ( ( ( A e. CC /\ B e. ~H ) /\ A =/= 0 ) -> ( ( ( 1 / A ) x. A ) .h B ) = ( ( 1 / A ) .h ( A .h B ) ) )
13		ax-hvmulid 27863	. . . . . . . . . 10 |- ( B e. ~H -> ( 1 .h B ) = B )
14	13	ad2antlr 763	. . . . . . . . 9 |- ( ( ( A e. CC /\ B e. ~H ) /\ A =/= 0 ) -> ( 1 .h B ) = B )
15	6, 12, 14	3eqtr3d 2664	. . . . . . . 8 |- ( ( ( A e. CC /\ B e. ~H ) /\ A =/= 0 ) -> ( ( 1 / A ) .h ( A .h B ) ) = B )
16	15	adantlr 751	. . . . . . 7 |- ( ( ( ( A e. CC /\ B e. ~H ) /\ ( A .h B ) = 0h ) /\ A =/= 0 ) -> ( ( 1 / A ) .h ( A .h B ) ) = B )
17		hvmul0 27881	. . . . . . . . . 10 |- ( ( 1 / A ) e. CC -> ( ( 1 / A ) .h 0h ) = 0h )
18	7, 17	syl 17	. . . . . . . . 9 |- ( ( A e. CC /\ A =/= 0 ) -> ( ( 1 / A ) .h 0h ) = 0h )
19	18	adantlr 751	. . . . . . . 8 |- ( ( ( A e. CC /\ B e. ~H ) /\ A =/= 0 ) -> ( ( 1 / A ) .h 0h ) = 0h )
20	19	adantlr 751	. . . . . . 7 |- ( ( ( ( A e. CC /\ B e. ~H ) /\ ( A .h B ) = 0h ) /\ A =/= 0 ) -> ( ( 1 / A ) .h 0h ) = 0h )
21	3, 16, 20	3eqtr3d 2664	. . . . . 6 |- ( ( ( ( A e. CC /\ B e. ~H ) /\ ( A .h B ) = 0h ) /\ A =/= 0 ) -> B = 0h )
22	21	ex 450	. . . . 5 |- ( ( ( A e. CC /\ B e. ~H ) /\ ( A .h B ) = 0h ) -> ( A =/= 0 -> B = 0h ) )
23	1, 22	syl5bir 233	. . . 4 |- ( ( ( A e. CC /\ B e. ~H ) /\ ( A .h B ) = 0h ) -> ( -. A = 0 -> B = 0h ) )
24	23	orrd 393	. . 3 |- ( ( ( A e. CC /\ B e. ~H ) /\ ( A .h B ) = 0h ) -> ( A = 0 \/ B = 0h ) )
25	24	ex 450	. 2 |- ( ( A e. CC /\ B e. ~H ) -> ( ( A .h B ) = 0h -> ( A = 0 \/ B = 0h ) ) )
26		ax-hvmul0 27867	. . . . 5 |- ( B e. ~H -> ( 0 .h B ) = 0h )
27		oveq1 6657	. . . . . 6 |- ( A = 0 -> ( A .h B ) = ( 0 .h B ) )
28	27	eqeq1d 2624	. . . . 5 |- ( A = 0 -> ( ( A .h B ) = 0h <-> ( 0 .h B ) = 0h ) )
29	26, 28	syl5ibrcom 237	. . . 4 |- ( B e. ~H -> ( A = 0 -> ( A .h B ) = 0h ) )
30	29	adantl 482	. . 3 |- ( ( A e. CC /\ B e. ~H ) -> ( A = 0 -> ( A .h B ) = 0h ) )
31		hvmul0 27881	. . . . 5 |- ( A e. CC -> ( A .h 0h ) = 0h )
32		oveq2 6658	. . . . . 6 |- ( B = 0h -> ( A .h B ) = ( A .h 0h ) )
33	32	eqeq1d 2624	. . . . 5 |- ( B = 0h -> ( ( A .h B ) = 0h <-> ( A .h 0h ) = 0h ) )
34	31, 33	syl5ibrcom 237	. . . 4 |- ( A e. CC -> ( B = 0h -> ( A .h B ) = 0h ) )
35	34	adantr 481	. . 3 |- ( ( A e. CC /\ B e. ~H ) -> ( B = 0h -> ( A .h B ) = 0h ) )
36	30, 35	jaod 395	. 2 |- ( ( A e. CC /\ B e. ~H ) -> ( ( A = 0 \/ B = 0h ) -> ( A .h B ) = 0h ) )
37	25, 36	impbid 202	1 |- ( ( A e. CC /\ B e. ~H ) -> ( ( A .h B ) = 0h <-> ( A = 0 \/ B = 0h ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   x. cmul 9941   / cdiv 10684   ~Hchil 27776   .h csm 27778   0hc0v 27781

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-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  ax-hv0cl 27860  ax-hvmulid 27863  ax-hvmulass 27864  ax-hvmul0 27867

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-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-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

This theorem is referenced by:  hvmulcan  27929  hvmulcan2  27930  nmlnop0iALT  28854