Theorem atexcht 10303

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. Also Definition 3.4-3(b) in [MegillPavicic] p. 2345 (PDF p. 8) (use atnem0 10299 to obtain atom inequality).

Assertion

Ref	Expression
atexcht	|- ((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 atexcht

Step	Hyp	Ref	Expression
1		chub2t 9426	. . . . . . 7 |- ((C e. CH /\ A e. CH) -> C (_ (A vH C))
2	1	ancoms 438	. . . . . 6 |- ((A e. CH /\ C e. CH) -> C (_ (A vH C))
3		atelch 10266	. . . . . 6 |- (C e. Atoms -> C e. CH)
4	2, 3	sylan2 453	. . . . 5 |- ((A e. CH /\ C e. Atoms) -> C (_ (A vH C))
5	4	3adant2 800	. . . 4 |- ((A e. CH /\ B e. Atoms /\ C e. Atoms) -> C (_ (A vH C))
6	5	adantr 391	. . 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 10297	. . . . . . . . 9 |- ((A e. CH /\ B e. Atoms) -> ((A i^i B) = 0H <-> A <o (A vH B)))
8		chjclt 9324	. . . . . . . . . . 11 |- ((A e. CH /\ B e. CH) -> (A vH B) e. CH)
9		atelch 10266	. . . . . . . . . . 11 |- (B e. Atoms -> B e. CH)
10	8, 9	sylan2 453	. . . . . . . . . 10 |- ((A e. CH /\ B e. Atoms) -> (A vH B) e. CH)
11		cvpsst 10207	. . . . . . . . . 10 |- ((A e. CH /\ (A vH B) e. CH) -> (A <o (A vH B) -> A (. (A vH B)))
12	10, 11	syldan 469	. . . . . . . . 9 |- ((A e. CH /\ B e. Atoms) -> (A <o (A vH B) -> A (. (A vH B)))
13	7, 12	sylbid 203	. . . . . . . 8 |- ((A e. CH /\ B e. Atoms) -> ((A i^i B) = 0H -> A (. (A vH B)))
14	13	3adant3 801	. . . . . . 7 |- ((A e. CH /\ B e. Atoms /\ C e. Atoms) -> ((A i^i B) = 0H -> A (. (A vH B)))
15	14	adantld 392	. . . . . 6 |- ((A e. CH /\ B e. Atoms /\ C e. Atoms) -> ((B (_ (A vH C) /\ (A i^i B) = 0H) -> A (. (A vH B)))
16		chub1t 9425	. . . . . . . . . . . 12 |- ((A e. CH /\ C e. CH) -> A (_ (A vH C))
17	16	3adant2 800	. . . . . . . . . . 11 |- ((A e. CH /\ B e. CH /\ C e. CH) -> A (_ (A vH C))
18	17	a1d 12	. . . . . . . . . 10 |- ((A e. CH /\ B e. CH /\ C e. CH) -> (B (_ (A vH C) -> A (_ (A vH C)))
19	18	ancrd 299	. . . . . . . . 9 |- ((A e. CH /\ B e. CH /\ C e. CH) -> (B (_ (A vH C) -> (A (_ (A vH C) /\ B (_ (A vH C))))
20		chlubt 9427	. . . . . . . . . 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)))
21		chjclt 9324	. . . . . . . . . . 11 |- ((A e. CH /\ C e. CH) -> (A vH C) e. CH)
22	21	3adant2 800	. . . . . . . . . 10 |- ((A e. CH /\ B e. CH /\ C e. CH) -> (A vH C) e. CH)
23	20, 22	syld3an3 872	. . . . . . . . 9 |- ((A e. CH /\ B e. CH /\ C e. CH) -> ((A (_ (A vH C) /\ B (_ (A vH C)) <-> (A vH B) (_ (A vH C)))
24	19, 23	sylibd 202	. . . . . . . 8 |- ((A e. CH /\ B e. CH /\ C e. CH) -> (B (_ (A vH C) -> (A vH B) (_ (A vH C)))
25		id 59	. . . . . . . 8 |- (A e. CH -> A e. CH)
26	24, 25, 9, 3	syl3an 870	. . . . . . 7 |- ((A e. CH /\ B e. Atoms /\ C e. Atoms) -> (B (_ (A vH C) -> (A vH B) (_ (A vH C)))
27	26	adantrd 393	. . . . . 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)))
28	15, 27	jcad 602	. . . . 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))))
29	28	imp 350	. . . 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)))
30	14, 26	anim12d 560	. . . . . . . . 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))))
31	30	ancomsd 439	. . . . . . . 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))))
32		psssstr 2155	. . . . . . . 8 |- ((A (. (A vH B) /\ (A vH B) (_ (A vH C)) -> A (. (A vH C))
33	31, 32	syl6 22	. . . . . . 7 |- ((A e. CH /\ B e. Atoms /\ C e. Atoms) -> ((B (_ (A vH C) /\ (A i^i B) = 0H) -> A (. (A vH C)))
34		chcv2t 10278	. . . . . . . 8 |- ((A e. CH /\ C e. Atoms) -> (A (. (A vH C) <-> A <o (A vH C)))
35	34	3adant2 800	. . . . . . 7 |- ((A e. CH /\ B e. Atoms /\ C e. Atoms) -> (A (. (A vH C) <-> A <o (A vH C)))
36	33, 35	sylibd 202	. . . . . 6 |- ((A e. CH /\ B e. Atoms /\ C e. Atoms) -> ((B (_ (A vH C) /\ (A i^i B) = 0H) -> A <o (A vH C)))
37		3simp1 790	. . . . . . . . 9 |- ((A e. CH /\ B e. CH /\ C e. CH) -> A e. CH)
38	8	3adant3 801	. . . . . . . . 9 |- ((A e. CH /\ B e. CH /\ C e. CH) -> (A vH B) e. CH)
39	37, 22, 38	3jca 821	. . . . . . . 8 |- ((A e. CH /\ B e. CH /\ C e. CH) -> (A e. CH /\ (A vH C) e. CH /\ (A vH B) e. CH))
40	39, 25, 9, 3	syl3an 870	. . . . . . 7 |- ((A e. CH /\ B e. Atoms /\ C e. Atoms) -> (A e. CH /\ (A vH C) e. CH /\ (A vH B) e. CH))
41		cvnbtwn2t 10209	. . . . . . 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))))
42	40, 41	syl 10	. . . . . 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))))
43	36, 42	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))))
44	43	imp 350	. . . 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)))
45	29, 44	mpd 26	. . 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))
46	6, 45	sseqtr4d 2101	. 2 |- (((A e. CH /\ B e. Atoms /\ C e. Atoms) /\ (B (_ (A vH C) /\ (A i^i B) = 0H)) -> C (_ (A vH B))
47	46	ex 373	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 3   <-> wb 146   /\ wa 223   /\ w3a 777   = wceq 958   e. wcel 960   i^i cin 2049   (_ wss 2050   (. wpss 2051   class class class wbr 2624  (class class class)co 3969  CHcch 8793   vH chj 8797  0Hc0h 8799  Atomscat 8828   <o ccv 8829

This theorem is referenced by:  atoml 10304  atcvatlem 10307  atcvat4 10319  mdsymlem3 10327  mdsymlem5 10329

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-reg 4602  ax-inf2 4634  ax-ac 4754  ax-hilex 8864  ax-hfvadd 8865  ax-hvcom 8866  ax-hvass 8867  ax-hv0cl 8868  ax-hvaddid 8869  ax-hfvmul 8870  ax-hvmulid 8871  ax-hvmulass 8872  ax-hvdistr1 8873  ax-hvdistr2 8874  ax-hvmul0 8875  ax-hfi 8941  ax-his1 8944  ax-his2 8945  ax-his3 8946  ax-his4 8947  ax-hcompl 9066

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-nel 1591  df-ral 1652  df-rex 1653  df-reu 1654  df-rab 1655  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-pss 2058  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-iun 2572  df-iin 2573  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-om 3138  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-f1 3201  df-fo 3202  df-f1o 3203  df-fv 3204  df-rdg 3938  df-opr 3971  df-oprab 3972  df-1st 4085  df-2nd 4086  df-1o 4139  df-oadd 4141  df-omul 4142  df-er