Theorem atexch 5769

Description: The Hilbert lattice satisfies the atom exchange property. Proposition 1(i) of [Kalmbach] p. 140. A version of this theorem related to vector analysis was originally proved by Hermann Grassmann in 1862.

Assertion

Ref	Expression
atexch	|- ((A e. CH /\ B e. Atoms /\ C e. Atoms) -> ((B (_ (A vH C) /\ (A i^i B) = 0H) -> C (_ (A vH B)))

Proof of Theorem atexch

Step	Hyp	Ref	Expression
1		chub2t 5425	. . . . . . 7 |- ((C e. CH /\ A e. CH) -> C (_ (A vH C))
2	1	ancoms 334	. . . . . 6 |- ((A e. CH /\ C e. CH) -> C (_ (A vH C))
3		atelch 5742	. . . . . 6 |- (C e. Atoms -> C e. CH)
4	2, 3	sylan2 346	. . . . 5 |- ((A e. CH /\ C e. Atoms) -> C (_ (A vH C))
5	4	3adant2 598	. . . 4 |- ((A e. CH /\ B e. Atoms /\ C e. Atoms) -> C (_ (A vH C))
6	5	adantr 306	. . 3 |- (((A e. CH /\ B e. Atoms /\ C e. Atoms) /\ (B (_ (A vH C) /\ (A i^i B) = 0H)) -> C (_ (A vH C))
7		cvp 5764	. . . . . . . . 9 |- ((A e. CH /\ B e. Atoms) -> ((A i^i B) = 0H <-> A <o (A vH B)))
8		cvpsst 5717	. . . . . . . . . 10 |- ((A e. CH /\ (A vH B) e. CH) -> (A <o (A vH B) -> A (. (A vH B)))
9		pm3.26 256	. . . . . . . . . 10 |- ((A e. CH /\ B e. Atoms) -> A e. CH)
10		chjclt 5330	. . . . . . . . . . 11 |- ((A e. CH /\ B e. CH) -> (A vH B) e. CH)
11		atelch 5742	. . . . . . . . . . 11 |- (B e. Atoms -> B e. CH)
12	10, 11	sylan2 346	. . . . . . . . . 10 |- ((A e. CH /\ B e. Atoms) -> (A vH B) e. CH)
13	8, 9, 12	sylanc 361	. . . . . . . . 9 |- ((A e. CH /\ B e. Atoms) -> (A <o (A vH B) -> A (. (A vH B)))
14	7, 13	sylbid 178	. . . . . . . 8 |- ((A e. CH /\ B e. Atoms) -> ((A i^i B) = 0H -> A (. (A vH B)))
15	14	3adant3 599	. . . . . . 7 |- ((A e. CH /\ B e. Atoms /\ C e. Atoms) -> ((A i^i B) = 0H -> A (. (A vH B)))
16	15	adantld 307	. . . . . 6 |- ((A e. CH /\ B e. Atoms /\ C e. Atoms) -> ((B (_ (A vH C) /\ (A i^i B) = 0H) -> A (. (A vH B)))
17		chub1t 5424	. . . . . . . . . . . 12 |- ((A e. CH /\ C e. CH) -> A (_ (A vH C))
18	17	3adant2 598	. . . . . . . . . . 11 |- ((A e. CH /\ B e. CH /\ C e. CH) -> A (_ (A vH C))
19	18	a1d 14	. . . . . . . . . 10 |- ((A e. CH /\ B e. CH /\ C e. CH) -> (B (_ (A vH C) -> A (_ (A vH C)))
20	19	ancrd 247	. . . . . . . . 9 |- ((A e. CH /\ B e. CH /\ C e. CH) -> (B (_ (A vH C) -> (A (_ (A vH C) /\ B (_ (A vH C))))
21		chlubt 5426	. . . . . . . . . 10 |- ((A e. CH /\ B e. CH /\ (A vH C) e. CH) -> ((A (_ (A vH C) /\ B (_ (A vH C)) <-> (A vH B) (_ (A vH C)))
22		3simp1 594	. . . . . . . . . 10 |- ((A e. CH /\ B e. CH /\ C e. CH) -> A e. CH)
23		3simp2 595	. . . . . . . . . 10 |- ((A e. CH /\ B e. CH /\ C e. CH) -> B e. CH)
24		chjclt 5330	. . . . . . . . . . 11 |- ((A e. CH /\ C e. CH) -> (A vH C) e. CH)
25	24	3adant2 598	. . . . . . . . . 10 |- ((A e. CH /\ B e. CH /\ C e. CH) -> (A vH C) e. CH)
26	21, 22, 23, 25	syl3anc 629	. . . . . . . . 9 |- ((A e. CH /\ B e. CH /\ C e. CH) -> ((A (_ (A vH C) /\ B (_ (A vH C)) <-> (A vH B) (_ (A vH C)))
27	20, 26	sylibd 177	. . . . . . . 8 |- ((A e. CH /\ B e. CH /\ C e. CH) -> (B (_ (A vH C) -> (A vH B) (_ (A vH C)))
28		id 9	. . . . . . . 8 |- (A e. CH -> A e. CH)
29	27, 28, 11, 3	syl3an 628	. . . . . . 7 |- ((A e. CH /\ B e. Atoms /\ C e. Atoms) -> (B (_ (A vH C) -> (A vH B) (_ (A vH C)))
30	29	adantrd 308	. . . . . 6 |- ((A e. CH /\ B e. Atoms /\ C e. Atoms) -> ((B (_ (A vH C) /\ (A i^i B) = 0H) -> (A vH B) (_ (A vH C)))
31	16, 30	jcad 455	. . . . 5 |- ((A e. CH /\ B e. Atoms /\ C e. Atoms) -> ((B (_ (A vH C) /\ (A i^i B) = 0H) -> (A (. (A vH B) /\ (A vH B) (_ (A vH C))))
32	31	imp 277	. . . 4 |- (((A e. CH /\ B e. Atoms /\ C e. Atoms) /\ (B (_ (A vH C) /\ (A i^i B) = 0H)) -> (A (. (A vH B) /\ (A vH B) (_ (A vH C)))
33	15, 29	anim12d 431	. . . . . . . . 9 |- ((A e. CH /\ B e. Atoms /\ C e. Atoms) -> (((A i^i B) = 0H /\ B (_ (A vH C)) -> (A (. (A vH B) /\ (A vH B) (_ (A vH C))))
34	33	ancomsd 335	. . . . . . . 8 |- ((A e. CH /\ B e. Atoms /\ C e. Atoms) -> ((B (_ (A vH C) /\ (A i^i B) = 0H) -> (A (. (A vH B) /\ (A vH B) (_ (A vH C))))
35		psssstr 1576	. . . . . . . 8 |- ((A (. (A vH B) /\ (A vH B) (_ (A vH C)) -> A (. (A vH C))
36	34, 35	syl6 23	. . . . . . 7 |- ((A e. CH /\ B e. Atoms /\ C e. Atoms) -> ((B (_ (A vH C) /\ (A i^i B) = 0H) -> A (. (A vH C)))
37		chcv2t 5752	. . . . . . . 8 |- ((A e. CH /\ C e. Atoms) -> (A (. (A vH C) <-> A <o (A vH C)))
38	37	3adant2 598	. . . . . . 7 |- ((A e. CH /\ B e. Atoms /\ C e. Atoms) -> (A (. (A vH C) <-> A <o (A vH C)))
39	36, 38	sylibd 177	. . . . . 6 |- ((A e. CH /\ B e. Atoms /\ C e. Atoms) -> ((B (_ (A vH C) /\ (A i^i B) = 0H) -> A <o (A vH C)))
40	10	3adant3 599	. . . . . . . . 9 |- ((A e. CH /\ B e. CH /\ C e. CH) -> (A vH B) e. CH)
41	22, 25, 40	3jca 604	. . . . . . . 8 |- ((A e. CH /\ B e. CH /\ C e. CH) -> (A e. CH /\ (A vH C) e. CH /\ (A vH B) e. CH))
42	41, 28, 11, 3	syl3an 628	. . . . . . 7 |- ((A e. CH /\ B e. Atoms /\ C e. Atoms) -> (A e. CH /\ (A vH C) e. CH /\ (A vH B) e. CH))
43		cvnbtwn2t 5719	. . . . . . 7 |- ((A e. CH /\ (A vH C) e. CH /\ (A vH B) e. CH) -> (A <o (A vH C) -> ((A (. (A vH B) /\ (A vH B) (_ (A vH C)) -> (A vH B) = (A vH C))))
44	42, 43	syl 12	. . . . . 6 |- ((A e. CH /\ B e. Atoms /\ C e. Atoms) -> (A <o (A vH C) -> ((A (. (A vH B) /\ (A vH B) (_ (A vH C)) -> (A vH B) = (A vH C))))
45	39, 44	syld 27	. . . . 5 |- ((A e. CH /\ B e. Atoms /\ C e. Atoms) -> ((B (_ (A vH C) /\ (A i^i B) = 0H) -> ((A (. (A vH B) /\ (A vH B) (_ (A vH C)) -> (A vH B) = (A vH C))))
46	45	imp 277	. . . 4 |- (((A e. CH /\ B e. Atoms /\ C e. Atoms) /\ (B (_ (A vH C) /\ (A i^i B) = 0H)) -> ((A (. (A vH B) /\ (A vH B) (_ (A vH C)) -> (A vH B) = (A vH C)))
47	32, 46	mpd 46	. . 3 |- (((A e. CH /\ B e. Atoms /\ C e. Atoms) /\ (B (_ (A vH C) /\ (A i^i B) = 0H)) -> (A vH B) = (A vH C))
48	6, 47	sseqtr4d 1537	. 2 |- (((A e. CH /\ B e. Atoms /\ C e. Atoms) /\ (B (_ (A vH C) /\ (A i^i B) = 0H)) -> C (_ (A vH B))
49	48	exp 291	1 |- ((A e. CH /\ B e. Atoms /\ C e. Atoms) -> ((B (_ (A vH C) /\ (A i^i B) = 0H) -> C (_ (A vH B)))

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196   /\ w3a 581   = wceq 1091   e. wcel 1092   i^i cin 1486   (_ wss 1487   (. wpss 1488   class class class wbr 2054  (class class class)co 3001  CHcch 4968   vH chj 4972  0Hc0h 4974  Atomscat 4980   <o ccv 4981

This theorem is referenced by:  atcvatlem 5770  atcvat4 5775  mdsymlem3 5778  mdsymlem5 5780

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079  ax-ac 1080  ax-hilex 4983  ax-hvaddcl 4984  ax-hvcom 4985  ax-hvass 4986  ax-hvzercl 4987  ax-hvaddid 4988  ax-hvmulcl 4989  ax-hvmulid 4991  ax-hvmulass 4992  ax-hvdistr1 4993  ax-hvdistr2 4994  ax-hvmulzer 4995  ax-hicl 5043  ax-his1 5045  ax-his2 5046  ax-his3 5047  ax-his4 5048  ax-hcompl 5113

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-sup 2154  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-fv 2438  df-rdg 2970  df-opr 3003  df-oprab 3004  df-1st 3087  df-2nd 3088  df-1o 3104  df-oadd 3106  df-omul 3107  df-er 3200  df-ec 3202  df-qs 3205  df-ni 3794  df-pli 3795  df-mi 3796  df-lti 3797  df-plpq 3829  df-mpq 3830  df-enq 3831  df-nq 3832  df-plq 3833  df-mq 3834  df-rq 3835  df-ltq 3836  df-1q 3837  df-np&nb