Theorem nvmul0or 9604

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

Hypotheses

Ref	Expression
nvmul0or.1	|- X = (BaseSet` 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 4890	. . . . . . . 8 |- ((ASB) = Z -> ((1 / A)S(ASB)) = ((1 / A)SZ))
2	1	ad2antlr 441	. . . . . . 7 |- ((((U e. NrmCVec /\ A e. CC /\ B e. X) /\ (ASB) = Z) /\ A =/= 0) -> ((1 / A)S(ASB)) = ((1 / A)SZ))
3		recid2 6919	. . . . . . . . . . 11 |- ((A e. CC /\ A =/= 0) -> ((1 / A) x. A) = 1)
4	3	opreq1d 4897	. . . . . . . . . 10 |- ((A e. CC /\ A =/= 0) -> (((1 / A) x. A)SB) = (1SB))
5	4	3ad2antl2 1039	. . . . . . . . 9 |- (((U e. NrmCVec /\ A e. CC /\ B e. X) /\ A =/= 0) -> (((1 / A) x. A)SB) = (1SB))
6		simpl1 879	. . . . . . . . . 10 |- (((U e. NrmCVec /\ A e. CC /\ B e. X) /\ A =/= 0) -> U e. NrmCVec)
7		reccl 6904	. . . . . . . . . . 11 |- ((A e. CC /\ A =/= 0) -> (1 / A) e. CC)
8	7	3ad2antl2 1039	. . . . . . . . . 10 |- (((U e. NrmCVec /\ A e. CC /\ B e. X) /\ A =/= 0) -> (1 / A) e. CC)
9		simpl2 880	. . . . . . . . . 10 |- (((U e. NrmCVec /\ A e. CC /\ B e. X) /\ A =/= 0) -> A e. CC)
10		simpl3 881	. . . . . . . . . 10 |- (((U e. NrmCVec /\ A e. CC /\ B e. X) /\ A =/= 0) -> B e. X)
11		nvmul0or.1	. . . . . . . . . . 11 |- X = (BaseSet` U)
12		nvmul0or.4	. . . . . . . . . . 11 |- S = (.s` U)
13	11, 12	nvsass 9581	. . . . . . . . . 10 |- ((U e. NrmCVec /\ ((1 / A) e. CC /\ A e. CC /\ B e. X)) -> (((1 / A) x. A)SB) = ((1 / A)S(ASB)))
14	6, 8, 9, 10, 13	syl13anc 1102	. . . . . . . . 9 |- (((U e. NrmCVec /\ A e. CC /\ B e. X) /\ A =/= 0) -> (((1 / A) x. A)SB) = ((1 / A)S(ASB)))
15	11, 12	nvsid 9580	. . . . . . . . . . 11 |- ((U e. NrmCVec /\ B e. X) -> (1SB) = B)
16	15	3adant2 895	. . . . . . . . . 10 |- ((U e. NrmCVec /\ A e. CC /\ B e. X) -> (1SB) = B)
17	16	adantr 425	. . . . . . . . 9 |- (((U e. NrmCVec /\ A e. CC /\ B e. X) /\ A =/= 0) -> (1SB) = B)
18	5, 14, 17	3eqtr3d 1934	. . . . . . . 8 |- (((U e. NrmCVec /\ A e. CC /\ B e. X) /\ A =/= 0) -> ((1 / A)S(ASB)) = B)
19	18	adantlr 429	. . . . . . 7 |- ((((U e. NrmCVec /\ A e. CC /\ B e. X) /\ (ASB) = Z) /\ A =/= 0) -> ((1 / A)S(ASB)) = B)
20		nvmul0or.6	. . . . . . . . . . . 12 |- Z = (0v` U)
21	12, 20	nvsz 9591	. . . . . . . . . . 11 |- ((U e. NrmCVec /\ (1 / A) e. CC) -> ((1 / A)SZ) = Z)
22	21, 7	sylan2 500	. . . . . . . . . 10 |- ((U e. NrmCVec /\ (A e. CC /\ A =/= 0)) -> ((1 / A)SZ) = Z)
23	22	anassrs 489	. . . . . . . . 9 |- (((U e. NrmCVec /\ A e. CC) /\ A =/= 0) -> ((1 / A)SZ) = Z)
24	23	3adantl3 1034	. . . . . . . 8 |- (((U e. NrmCVec /\ A e. CC /\ B e. X) /\ A =/= 0) -> ((1 / A)SZ) = Z)
25	24	adantlr 429	. . . . . . 7 |- ((((U e. NrmCVec /\ A e. CC /\ B e. X) /\ (ASB) = Z) /\ A =/= 0) -> ((1 / A)SZ) = Z)
26	2, 19, 25	3eqtr3d 1934	. . . . . 6 |- ((((U e. NrmCVec /\ A e. CC /\ B e. X) /\ (ASB) = Z) /\ A =/= 0) -> B = Z)
27	26	ex 402	. . . . 5 |- (((U e. NrmCVec /\ A e. CC /\ B e. X) /\ (ASB) = Z) -> (A =/= 0 -> B = Z))
28		df-ne 2019	. . . . 5 |- (A =/= 0 <-> -. A = 0)
29	27, 28	syl5ibr 224	. . . 4 |- (((U e. NrmCVec /\ A e. CC /\ B e. X) /\ (ASB) = Z) -> (-. A = 0 -> B = Z))
30	29	orrd 250	. . 3 |- (((U e. NrmCVec /\ A e. CC /\ B e. X) /\ (ASB) = Z) -> (A = 0 \/ B = Z))
31	30	ex 402	. 2 |- ((U e. NrmCVec /\ A e. CC /\ B e. X) -> ((ASB) = Z -> (A = 0 \/ B = Z)))
32		opreq1 4889	. . . . . 6 |- (A = 0 -> (ASB) = (0SB))
33	32	eqeq1d 1892	. . . . 5 |- (A = 0 -> ((ASB) = Z <-> (0SB) = Z))
34	11, 12, 20	nv0 9590	. . . . 5 |- ((U e. NrmCVec /\ B e. X) -> (0SB) = Z)
35	33, 34	syl5cbir 228	. . . 4 |- ((U e. NrmCVec /\ B e. X) -> (A = 0 -> (ASB) = Z))
36	35	3adant2 895	. . 3 |- ((U e. NrmCVec /\ A e. CC /\ B e. X) -> (A = 0 -> (ASB) = Z))
37		opreq2 4890	. . . . . 6 |- (B = Z -> (ASB) = (ASZ))
38	37	eqeq1d 1892	. . . . 5 |- (B = Z -> ((ASB) = Z <-> (ASZ) = Z))
39	12, 20	nvsz 9591	. . . . 5 |- ((U e. NrmCVec /\ A e. CC) -> (ASZ) = Z)
40	38, 39	syl5cbir 228	. . . 4 |- ((U e. NrmCVec /\ A e. CC) -> (B = Z -> (ASB) = Z))
41	40	3adant3 896	. . 3 |- ((U e. NrmCVec /\ A e. CC /\ B e. X) -> (B = Z -> (ASB) = Z))
42	36, 41	jaod 469	. 2 |- ((U e. NrmCVec /\ A e. CC /\ B e. X) -> ((A = 0 \/ B = Z) -> (ASB) = Z))
43	31, 42	impbid 574	1 |- ((U e. NrmCVec /\ A e. CC /\ B e. X) -> ((ASB) = Z <-> (A = 0 \/ B = Z)))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   \/ wo 239   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300   =/= wne 2017  ` cfv 3998  (class class class)co 4884  CCcc 6384  0cc0 6386  1c1 6387   x. cmul 6391   / cdiv 6447  NrmCVeccnv 9535  BaseSetcba 9537  .scns 9538  0vcn0v 9539

This theorem is referenced by:  nmlno0lem 9793

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-div 6892  df-grp 9316  df-gid 9317  df-ginv 9318  df-abl 9408  df-vc 9497  df-nv 9543  df-va 9546  df-ba 9547  df-sm 9548  df-0v 9549  df-nm 9551

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