Theorem nvmul0or 8272

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 3969	. . . . . . . 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 5736	. . . . . . . . . . 11 |- ((A e. CC /\ A =/= 0) -> ((1 / A) x. A) = 1)
4	3	opreq1d 3975	. . . . . . . . . 10 |- ((A e. CC /\ A =/= 0) -> (((1 / A) x. A)SB) = (1SB))
5	4	3ad2antl2 810	. . . . . . . . 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 8249	. . . . . . . . . 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 788	. . . . . . . . . . 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 5715	. . . . . . . . . . . 12 |- ((A e. CC /\ A =/= 0) -> (1 / A) e. CC)
12	11	3ad2antl2 810	. . . . . . . . . . 11 |- (((U e. NrmCVec /\ A e. CC /\ B e. X) /\ A =/= 0) -> (1 / A) e. CC)
13		3simp2 789	. . . . . . . . . . . 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 790	. . . . . . . . . . . 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 819	. . . . . . . . . 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 8248	. . . . . . . . . . 11 |- ((U e. NrmCVec /\ B e. X) -> (1SB) = B)
20	19	3adant2 798	. . . . . . . . . 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 1515	. . . . . . . 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 8259	. . . . . . . . . . 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 805	. . . . . . . 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 1515	. . . . . 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 1587	. . . . 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 3968	. . . . . 6 |- (A = 0 -> (ASB) = (0SB))
37	36	eqeq1d 1483	. . . . 5 |- (A = 0 -> ((ASB) = Z <-> (0SB) = Z))
38	6, 7, 24	nv0 8258	. . . . 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 798	. . 3 |- ((U e. NrmCVec /\ A e. CC /\ B e. X) -> (A = 0 -> (ASB) = Z))
41		opreq2 3969	. . . . . 6 |- (B = Z -> (ASB) = (ASZ))
42	41	eqeq1d 1483	. . . . 5 |- (B = Z -> ((ASB) = Z <-> (ASZ) = Z))
43	7, 24	nvsz 8259	. . . . 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 799	. . 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 516	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 775   = wceq 956   e. wcel 958   =/= wne 1585  ` cfv 3182  (class class class)co 3963  CCcc 5232  0cc0 5234  1c1 5235   x. cmul 5239   / cdiv 5294  NrmCVeccnv 8203  Basecba 8205  .scns 8206  0vcn0v 8207

This theorem is referenced by:  nmlno0lem 8453

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-inf2 4625

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-nel 1588  df-ral 1649  df-rex 1650  df-reu 1651  df-rab 1652  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-pss 2055  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-int 2534  df-iun 2568  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-om 3132  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197  df-fv 3198  df-rdg 3932  df-opr 3965  df-oprab 3966  df-1st 4079  df-2nd 4080  df-1o 4133  df-oadd 4135  df-omul 4136  df-er 4261  df-ec 4263  df-qs 4266  df-en 4368  df-dom 4369  df-sdom 4370  df-ni 5000  df-pli 5001  df-mi 5002  df-lti 5003  df-plpq 5035  df-mpq 5036  df-enq 5037  df-nq 5038  df-plq 5039  df-mq 5040  df-rq 5041  df-ltq 5042  df-1q 5043  df-np 5086  df-1p 5087  df-plp 5088  df-mp 5089  df-ltp 5090  df-plpr 5164  df-mpr 5165  df-enr 5166  df-nr 5167  df-plr 5168  df-mr 5169  df-ltr 5170  df-0r 5171  df-1r 5172  df-m1r 5173  df-c 5240  df-0 5241  df-1 5242  df-i 5243  df-r 5244  df-plus 5245  df-mul 5246  df-lt 5247  df-sub 5356  df-neg 5358  df-pnf 5487  df-mnf 5488  df-xr 5489  df-ltxr 5490  df-le 5491  df-div 5703  df-grp 8037  df-gid 8038  df-ginv 8039  df-abl 8100  df-vc 8165  df-nv 8211  df-va 8214  df-ba 8215  df-sm 8216  df-0v 8217  df-nm 8219

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