Theorem nvmul0or 8212

Description: If a scalar product is zero, one of its factors must be zero.

Hypotheses

Ref	Expression
nvmul0or.1	|- X = (Base` U)
nvmul0or.4	|- S = (.s` U)
nvmul0or.6	|- Z = (0v` U)

Assertion

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

Proof of Theorem nvmul0or

Step	Hyp	Ref	Expression
1		opreq2 3954	. . . . . . . 8 |- ((ASB) = Z -> ((1 / A)S(ASB)) = ((1 / A)SZ))
2	1	ad2antlr 405	. . . . . . 7 |- ((((U e. NrmCVec /\ A e. CC /\ B e. X) /\ (ASB) = Z) /\ A =/= 0) -> ((1 / A)S(ASB)) = ((1 / A)SZ))
3		recid2t 5699	. . . . . . . . . . 11 |- ((A e. CC /\ A =/= 0) -> ((1 / A) x. A) = 1)
4	3	opreq1d 3960	. . . . . . . . . 10 |- ((A e. CC /\ A =/= 0) -> (((1 / A) x. A)SB) = (1SB))
5	4	3ad2antl2 808	. . . . . . . . 9 |- (((U e. NrmCVec /\ A e. CC /\ B e. X) /\ A =/= 0) -> (((1 / A) x. A)SB) = (1SB))
6		nvmul0or.1	. . . . . . . . . . 11 |- X = (Base` U)
7		nvmul0or.4	. . . . . . . . . . 11 |- S = (.s` U)
8	6, 7	nvsass 8189	. . . . . . . . . 10 |- ((U e. NrmCVec /\ ((1 / A) e. CC /\ A e. CC /\ B e. X)) -> (((1 / A) x. A)SB) = ((1 / A)S(ASB)))
9		3simp1 786	. . . . . . . . . . 11 |- ((U e. NrmCVec /\ A e. CC /\ B e. X) -> U e. NrmCVec)
10	9	adantr 389	. . . . . . . . . 10 |- (((U e. NrmCVec /\ A e. CC /\ B e. X) /\ A =/= 0) -> U e. NrmCVec)
11		recclt 5684	. . . . . . . . . . . 12 |- ((A e. CC /\ A =/= 0) -> (1 / A) e. CC)
12	11	3ad2antl2 808	. . . . . . . . . . 11 |- (((U e. NrmCVec /\ A e. CC /\ B e. X) /\ A =/= 0) -> (1 / A) e. CC)
13		3simp2 787	. . . . . . . . . . . 12 |- ((U e. NrmCVec /\ A e. CC /\ B e. X) -> A e. CC)
14	13	adantr 389	. . . . . . . . . . 11 |- (((U e. NrmCVec /\ A e. CC /\ B e. X) /\ A =/= 0) -> A e. CC)
15		3simp3 788	. . . . . . . . . . . 12 |- ((U e. NrmCVec /\ A e. CC /\ B e. X) -> B e. X)
16	15	adantr 389	. . . . . . . . . . 11 |- (((U e. NrmCVec /\ A e. CC /\ B e. X) /\ A =/= 0) -> B e. X)
17	12, 14, 16	3jca 817	. . . . . . . . . 10 |- (((U e. NrmCVec /\ A e. CC /\ B e. X) /\ A =/= 0) -> ((1 / A) e. CC /\ A e. CC /\ B e. X))
18	8, 10, 17	sylanc 471	. . . . . . . . 9 |- (((U e. NrmCVec /\ A e. CC /\ B e. X) /\ A =/= 0) -> (((1 / A) x. A)SB) = ((1 / A)S(ASB)))
19	6, 7	nvsid 8188	. . . . . . . . . . 11 |- ((U e. NrmCVec /\ B e. X) -> (1SB) = B)
20	19	3adant2 796	. . . . . . . . . 10 |- ((U e. NrmCVec /\ A e. CC /\ B e. X) -> (1SB) = B)
21	20	adantr 389	. . . . . . . . 9 |- (((U e. NrmCVec /\ A e. CC /\ B e. X) /\ A =/= 0) -> (1SB) = B)
22	5, 18, 21	3eqtr3d 1507	. . . . . . . 8 |- (((U e. NrmCVec /\ A e. CC /\ B e. X) /\ A =/= 0) -> ((1 / A)S(ASB)) = B)
23	22	adantlr 393	. . . . . . 7 |- ((((U e. NrmCVec /\ A e. CC /\ B e. X) /\ (ASB) = Z) /\ A =/= 0) -> ((1 / A)S(ASB)) = B)
24		nvmul0or.6	. . . . . . . . . . . 12 |- Z = (0v` U)
25	7, 24	nvsz 8199	. . . . . . . . . . 11 |- ((U e. NrmCVec /\ (1 / A) e. CC) -> ((1 / A)SZ) = Z)
26	25, 11	sylan2 451	. . . . . . . . . 10 |- ((U e. NrmCVec /\ (A e. CC /\ A =/= 0)) -> ((1 / A)SZ) = Z)
27	26	anassrs 441	. . . . . . . . 9 |- (((U e. NrmCVec /\ A e. CC) /\ A =/= 0) -> ((1 / A)SZ) = Z)
28	27	3adantl3 803	. . . . . . . 8 |- (((U e. NrmCVec /\ A e. CC /\ B e. X) /\ A =/= 0) -> ((1 / A)SZ) = Z)
29	28	adantlr 393	. . . . . . 7 |- ((((U e. NrmCVec /\ A e. CC /\ B e. X) /\ (ASB) = Z) /\ A =/= 0) -> ((1 / A)SZ) = Z)
30	2, 23, 29	3eqtr3d 1507	. . . . . 6 |- ((((U e. NrmCVec /\ A e. CC /\ B e. X) /\ (ASB) = Z) /\ A =/= 0) -> B = Z)
31	30	ex 373	. . . . 5 |- (((U e. NrmCVec /\ A e. CC /\ B e. X) /\ (ASB) = Z) -> (A =/= 0 -> B = Z))
32		df-ne 1579	. . . . 5 |- (A =/= 0 <-> -. A = 0)
33	31, 32	syl5ibr 207	. . . 4 |- (((U e. NrmCVec /\ A e. CC /\ B e. X) /\ (ASB) = Z) -> (-. A = 0 -> B = Z))
34	33	orrd 233	. . 3 |- (((U e. NrmCVec /\ A e. CC /\ B e. X) /\ (ASB) = Z) -> (A = 0 \/ B = Z))
35	34	ex 373	. 2 |- ((U e. NrmCVec /\ A e. CC /\ B e. X) -> ((ASB) = Z -> (A = 0 \/ B = Z)))
36		opreq1 3953	. . . . . 6 |- (A = 0 -> (ASB) = (0SB))
37	36	eqeq1d 1475	. . . . 5 |- (A = 0 -> ((ASB) = Z <-> (0SB) = Z))
38	6, 7, 24	nv0 8198	. . . . 5 |- ((U e. NrmCVec /\ B e. X) -> (0SB) = Z)
39	37, 38	syl5cbir 211	. . . 4 |- ((U e. NrmCVec /\ B e. X) -> (A = 0 -> (ASB) = Z))
40	39	3adant2 796	. . 3 |- ((U e. NrmCVec /\ A e. CC /\ B e. X) -> (A = 0 -> (ASB) = Z))
41		opreq2 3954	. . . . . 6 |- (B = Z -> (ASB) = (ASZ))
42	41	eqeq1d 1475	. . . . 5 |- (B = Z -> ((ASB) = Z <-> (ASZ) = Z))
43	7, 24	nvsz 8199	. . . . 5 |- ((U e. NrmCVec /\ A e. CC) -> (ASZ) = Z)
44	42, 43	syl5cbir 211	. . . 4 |- ((U e. NrmCVec /\ A e. CC) -> (B = Z -> (ASB) = Z))
45	44	3adant3 797	. . 3 |- ((U e. NrmCVec /\ A e. CC /\ B e. X) -> (B = Z -> (ASB) = Z))
46	40, 45	jaod 424	. 2 |- ((U e. NrmCVec /\ A e. CC /\ B e. X) -> ((A = 0 \/ B = Z) -> (ASB) = Z))
47	35, 46	impbid 514	1 |- ((U e. NrmCVec /\ A e. CC /\ B e. X) -> ((ASB) = Z <-> (A = 0 \/ B = Z)))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   \/ wo 222   /\ wa 223   /\ w3a 773   = wceq 953   e. wcel 955   =/= wne 1577  ` cfv 3172  (class class class)co 3948  CCcc 5204  0cc0 5206  1c1 5207   x. cmul 5211   / cdiv 5266  NrmCVeccnv 8141  Basecba 8143  .scns 8144  0vcn0v 8145

This theorem is referenced by:  nmlno0lem 8385

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-inf2 4597

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-nel 1580  df-ral 1641  df-rex 1642  df-reu 1643  df-rab 1644  df-v 1803  df-sbc 1932  df-csb 1992  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-pss 2045  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-int 2524  df-iun 2558  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-om 3122  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187  df-fv 3188  df-rdg 3917  df-opr 3950  df-oprab 3951  df-1st 4063  df-2nd 4064  df-1o 4117  df-oadd 4119  df-omul 4120  df-er 4245  df-ec 4247  df-qs 4250  df-en 4351  df-dom 4352  df-sdom 4353  df-ni 4972  df-pli 4973  df-mi 4974  df-lti 4975  df-plpq 5007  df-mpq 5008  df-enq 5009  df-nq 5010  df-plq 5011  df-mq 5012  df-rq 5013  df-ltq 5014  df-1q 5015  df-np 5058  df-1p 5059  df-plp 5060  df-mp 5061  df-ltp 5062  df-plpr 5136  df-mpr 5137  df-enr 5138  df-nr 5139  df-plr 5140  df-mr 5141  df-ltr 5142  df-0r 5143  df-1r 5144  df-m1r 5145  df-c 5212  df-0 5213  df-1 5214  df-i 5215  df-r 5216  df-plus 5217  df-mul 5218  df-lt 5219  df-sub 5328  df-neg 5330  df-pnf 5459  df-mnf 5460  df-xr 5461  df-ltxr 5462  df-le 5463  df-div 5672  df-grp 7971  df-gid 7972  df-ginv 7973  df-abl 8036  df-vc 8102  df-nv 8149  df-va 8152  df-ba 8153  df-sm 8154  df-0v 8155  df-nm 8157

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