Theorem nonbool 9513

Description: A Hilbert lattice with two or more dimensions fails the distributive law and therefore cannot be a Boolean algebra. This counterexample demonstrates a condition where ((H i^i F) vH (H i^i G)) = 0H but (H i^i (F vH G)) =/= 0H. The antecedent specifies that the vectors A and B are nonzero and non-colinear. The last three hypotheses assign one-dimensional subspaces to F, G, and H.

Hypotheses

Ref	Expression
nonbool.1	|- A e. H~
nonbool.2	|- B e. H~
nonbool.3	|- F = (span` {A})
nonbool.4	|- G = (span` {B})
nonbool.5	|- H = (span` {(A +h B)})

Assertion

Ref	Expression
nonbool	|- (-. (A e. G \/ B e. F) -> (H i^i (F vH G)) =/= ((H i^i F) vH (H i^i G)))

Proof of Theorem nonbool

Step	Hyp	Ref	Expression
1		eqeq2 1476	. . . . 5 |- (((H i^i F) vH (H i^i G)) = 0H -> ((H i^i (F vH G)) = ((H i^i F) vH (H i^i G)) <-> (H i^i (F vH G)) = 0H))
2	1	negbid 609	. . . 4 |- (((H i^i F) vH (H i^i G)) = 0H -> (-. (H i^i (F vH G)) = ((H i^i F) vH (H i^i G)) <-> -. (H i^i (F vH G)) = 0H))
3	2	biimparc 419	. . 3 |- ((-. (H i^i (F vH G)) = 0H /\ ((H i^i F) vH (H i^i G)) = 0H) -> -. (H i^i (F vH G)) = ((H i^i F) vH (H i^i G)))
4		elin 2197	. . . . . . . . . 10 |- ((A +h B) e. (H i^i (F vH G)) <-> ((A +h B) e. H /\ (A +h B) e. (F vH G)))
5		nonbool.1	. . . . . . . . . . . . 13 |- A e. H~
6		nonbool.2	. . . . . . . . . . . . 13 |- B e. H~
7	5, 6	hvaddcl 8809	. . . . . . . . . . . 12 |- (A +h B) e. H~
8		spansnid 9402	. . . . . . . . . . . 12 |- ((A +h B) e. H~ -> (A +h B) e. (span` {(A +h B)}))
9	7, 8	ax-mp 7	. . . . . . . . . . 11 |- (A +h B) e. (span` {(A +h B)})
10		nonbool.5	. . . . . . . . . . 11 |- H = (span` {(A +h B)})
11	9, 10	eleqtrr 1539	. . . . . . . . . 10 |- (A +h B) e. H
12		nonbool.3	. . . . . . . . . . . . 13 |- F = (span` {A})
13	5	spansnch 9401	. . . . . . . . . . . . . 14 |- (span` {A}) e. CH
14	13	chshi 9018	. . . . . . . . . . . . 13 |- (span` {A}) e. SH
15	12, 14	eqeltr 1536	. . . . . . . . . . . 12 |- F e. SH
16		nonbool.4	. . . . . . . . . . . . 13 |- G = (span` {B})
17	6	spansnch 9401	. . . . . . . . . . . . . 14 |- (span` {B}) e. CH
18	17	chshi 9018	. . . . . . . . . . . . 13 |- (span` {B}) e. SH
19	16, 18	eqeltr 1536	. . . . . . . . . . . 12 |- G e. SH
20	15, 19	shslej 9253	. . . . . . . . . . 11 |- (F +H G) (_ (F vH G)
21		spansnid 9402	. . . . . . . . . . . . . 14 |- (A e. H~ -> A e. (span` {A}))
22	5, 21	ax-mp 7	. . . . . . . . . . . . 13 |- A e. (span` {A})
23	22, 12	eleqtrr 1539	. . . . . . . . . . . 12 |- A e. F
24		spansnid 9402	. . . . . . . . . . . . . 14 |- (B e. H~ -> B e. (span` {B}))
25	6, 24	ax-mp 7	. . . . . . . . . . . . 13 |- B e. (span` {B})
26	25, 16	eleqtrr 1539	. . . . . . . . . . . 12 |- B e. G
27	15, 19	shsva 9248	. . . . . . . . . . . 12 |- ((A e. F /\ B e. G) -> (A +h B) e. (F +H G))
28	23, 26, 27	mp2an 695	. . . . . . . . . . 11 |- (A +h B) e. (F +H G)
29	20, 28	sselii 2056	. . . . . . . . . 10 |- (A +h B) e. (F vH G)
30	4, 11, 29	mpbir2an 728	. . . . . . . . 9 |- (A +h B) e. (H i^i (F vH G))
31		eleq2 1527	. . . . . . . . 9 |- ((H i^i (F vH G)) = 0H -> ((A +h B) e. (H i^i (F vH G)) <-> (A +h B) e. 0H))
32	30, 31	mpbii 193	. . . . . . . 8 |- ((H i^i (F vH G)) = 0H -> (A +h B) e. 0H)
33		elch0 9047	. . . . . . . 8 |- ((A +h B) e. 0H <-> (A +h B) = 0h)
34	32, 33	sylib 198	. . . . . . 7 |- ((H i^i (F vH G)) = 0H -> (A +h B) = 0h)
35		ch0 9019	. . . . . . . 8 |- ((span` {A}) e. CH -> 0h e. (span` {A}))
36	13, 35	ax-mp 7	. . . . . . 7 |- 0h e. (span` {A})
37	34, 36	syl6eqel 1548	. . . . . 6 |- ((H i^i (F vH G)) = 0H -> (A +h B) e. (span` {A}))
38	12	eleq2i 1530	. . . . . . 7 |- (B e. F <-> B e. (span` {A}))
39		sumspansnt 9511	. . . . . . . 8 |- ((A e. H~ /\ B e. H~) -> ((A +h B) e. (span` {A}) <-> B e. (span` {A})))
40	5, 6, 39	mp2an 695	. . . . . . 7 |- ((A +h B) e. (span` {A}) <-> B e. (span` {A}))
41	38, 40	bitr4 176	. . . . . 6 |- (B e. F <-> (A +h B) e. (span` {A}))
42	37, 41	sylibr 200	. . . . 5 |- ((H i^i (F vH G)) = 0H -> B e. F)
43	42	con3i 98	. . . 4 |- (-. B e. F -> -. (H i^i (F vH G)) = 0H)
44	43	adantl 388	. . 3 |- ((-. A e. G /\ -. B e. F) -> -. (H i^i (F vH G)) = 0H)
45	7, 5	spansnm0 9512	. . . . . 6 |- (-. (A +h B) e. (span` {A}) -> ((span` {(A +h B)}) i^i (span` {A})) = 0H)
46	41	negbii 187	. . . . . 6 |- (-. B e. F <-> -. (A +h B) e. (span` {A}))
47	10, 12	ineq12i 2205	. . . . . . 7 |- (H i^i F) = ((span` {(A +h B)}) i^i (span` {A}))
48	47	eqeq1i 1474	. . . . . 6 |- ((H i^i F) = 0H <-> ((span` {(A +h B)}) i^i (span` {A})) = 0H)
49	45, 46, 48	3imtr4 219	. . . . 5 |- (-. B e. F -> (H i^i F) = 0H)
50	7, 6	spansnm0 9512	. . . . . 6 |- (-. (A +h B) e. (span` {B}) -> ((span` {(A +h B)}) i^i (span` {B})) = 0H)
51		sumspansnt 9511	. . . . . . . . 9 |- ((B e. H~ /\ A e. H~) -> ((B +h A) e. (span` {B}) <-> A e. (span` {B})))
52	6, 5, 51	mp2an 695	. . . . . . . 8 |- ((B +h A) e. (span` {B}) <-> A e. (span` {B}))
53	5, 6	hvcom 8810	. . . . . . . . 9 |- (A +h B) = (B +h A)
54	53	eleq1i 1529	. . . . . . . 8 |- ((A +h B) e. (span` {B}) <-> (B +h A) e. (span` {B}))
55	16	eleq2i 1530	. . . . . . . 8 |- (A e. G <-> A e. (span` {B}))
56	52, 54, 55	3bitr4r 184	. . . . . . 7 |- (A e. G <-> (A +h B) e. (span` {B}))
57	56	negbii 187	. . . . . 6 |- (-. A e. G <-> -. (A +h B) e. (span` {B}))
58	10, 16	ineq12i 2205	. . . . . . 7 |- (H i^i G) = ((span` {(A +h B)}) i^i (span` {B}))
59	58	eqeq1i 1474	. . . . . 6 |- ((H i^i G) = 0H <-> ((span` {(A +h B)}) i^i (span` {B})) = 0H)
60	50, 57, 59	3imtr4 219	. . . . 5 |- (-. A e. G -> (H i^i G) = 0H)
61	49, 60	opreqan12rd 3965	. . . 4 |- ((-. A e. G /\ -. B e. F) -> ((H i^i F) vH (H i^i G)) = (0H vH 0H))
62		h0elch 9048	. . . . 5 |- 0H e. CH
63	62	chj0 9293	. . . 4 |- (0H vH 0H) = 0H
64	61, 63	syl6eq 1515	. . 3 |- ((-. A e. G /\ -. B e. F) -> ((H i^i F) vH (H i^i G)) = 0H)
65	3, 44, 64	sylanc 471	. 2 |- ((-. A e. G /\ -. B e. F) -> -. (H i^i (F vH G)) = ((H i^i F) vH (H i^i G)))
66		ioran 306	. 2 |- (-. (A e. G \/ B e. F) <-> (-. A e. G /\ -. B e. F))
67		df-ne 1579	. 2 |- ((H i^i (F vH G)) =/= ((H i^i F) vH (H i^i G)) <-> -. (H i^i (F vH G)) = ((H i^i F) vH (H i^i G)))
68	65, 66, 67	3imtr4 219	1 |- (-. (A e. G \/ B e. F) -> (H i^i (F vH G)) =/= ((H i^i F) vH (H i^i G)))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   \/ wo 222   /\ wa 223   = wceq 953   e. wcel 955   =/= wne 1577   i^i cin 2036  {csn 2399  ` cfv 3172  (class class class)co 3948  H~chil 8727   +h cva 8728  0hc0v 8730  SHcsh 8736  CHcch 8737   +H cph 8739  spancspn 8740   vH chj 8741  0Hc0h 8743

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-reg 4565  ax-inf2 4597  ax-ac 4716  ax-hilex 8790  ax-hfvadd 8791  ax-hvcom 8792  ax-hvass 8793  ax-hv0cl 8794  ax-hvaddid 8795  ax-hfvmul 8796  ax-hvmulid 8797  ax-hvmulass 8798  ax-hvdistr1 8799  ax-hvdistr2 8800  ax-hvmul0 8801  ax-hfi 8867  ax-his1 8870  ax-his2 8871  ax-his3 8872  ax-his4 8873  ax-hcompl 8992

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-iin 2559  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-map 4308  df-en 4351  df-dom 4352  df-sdom 4353  df-sup 4548  df-r1 4615  df-rank 4616  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-n 5873  df-2 5917  df-3 5918  df-4 5919  df-n0 6047  df-z 6083  df-fl 6172  df-q 6194  df-seq1 6245  df-shft 6278  df-ioo 6298  df-uz 6350  df-fz 6400  df-seq