Theorem hatomistic 5755

Description: CH is atomistic, i.e. any element is the supremum of its atoms. Remark of [Kalmbach] p. 140.

Hypothesis

Ref	Expression
hatomistic.1	|- A e. CH

Assertion

Ref	Expression
hatomistic	|- A = ( \/H ` {x e. Atoms | x (_ A})

Distinct variable group(s):   x,A

Proof of Theorem hatomistic

Step	Hyp	Ref	Expression
1		ssrab 1556	. . . . 5 |- {x e. Atoms | x (_ A} (_ Atoms
2		atssch 5741	. . . . 5 |- Atoms (_ CH
3	1, 2	sstri 1512	. . . 4 |- {x e. Atoms | x (_ A} (_ CH
4		chsupclt 5309	. . . 4 |- ({x e. Atoms | x (_ A} (_ CH -> ( \/H ` {x e. Atoms | x (_ A}) e. CH)
5	3, 4	ax-mp 6	. . 3 |- ( \/H ` {x e. Atoms | x (_ A}) e. CH
6		hatomistic.1	. . . 4 |- A e. CH
7	6	chshi 5132	. . 3 |- A e. SH
8		atelch 5742	. . . . . . . 8 |- (y e. Atoms -> y e. CH)
9	8	anim1i 269	. . . . . . 7 |- ((y e. Atoms /\ y (_ A) -> (y e. CH /\ y (_ A))
10		sseq1 1521	. . . . . . . 8 |- (x = y -> (x (_ A <-> y (_ A))
11	10	elrab 1422	. . . . . . 7 |- (y e. {x e. Atoms | x (_ A} <-> (y e. Atoms /\ y (_ A))
12	10	elrab 1422	. . . . . . 7 |- (y e. {x e. CH | x (_ A} <-> (y e. CH /\ y (_ A))
13	9, 11, 12	3imtr4 192	. . . . . 6 |- (y e. {x e. Atoms | x (_ A} -> y e. {x e. CH | x (_ A})
14	13	ssriv 1508	. . . . 5 |- {x e. Atoms | x (_ A} (_ {x e. CH | x (_ A}
15		ssrab 1556	. . . . . 6 |- {x e. CH | x (_ A} (_ CH
16		chsupss 5311	. . . . . 6 |- (({x e. Atoms | x (_ A} (_ CH /\ {x e. CH | x (_ A} (_ CH) -> ({x e. Atoms | x (_ A} (_ {x e. CH | x (_ A} -> ( \/H ` {x e. Atoms | x (_ A}) (_ ( \/H ` {x e. CH | x (_ A})))
17	3, 15, 16	mp2an 520	. . . . 5 |- ({x e. Atoms | x (_ A} (_ {x e. CH | x (_ A} -> ( \/H ` {x e. Atoms | x (_ A}) (_ ( \/H ` {x e. CH | x (_ A}))
18	14, 17	ax-mp 6	. . . 4 |- ( \/H ` {x e. Atoms | x (_ A}) (_ ( \/H ` {x e. CH | x (_ A})
19		chsupid 5312	. . . . 5 |- (A e. CH -> ( \/H ` {x e. CH | x (_ A}) = A)
20	6, 19	ax-mp 6	. . . 4 |- ( \/H ` {x e. CH | x (_ A}) = A
21	18, 20	sseqtr 1532	. . 3 |- ( \/H ` {x e. Atoms | x (_ A}) (_ A
22		elssuni 1940	. . . . . . . . . . 11 |- (y e. {x e. Atoms | x (_ A} -> y (_ U.{x e. Atoms | x (_ A})
23	11, 22	sylbir 176	. . . . . . . . . 10 |- ((y e. Atoms /\ y (_ A) -> y (_ U.{x e. Atoms | x (_ A})
24		chsupunss 5317	. . . . . . . . . . . 12 |- ({x e. Atoms | x (_ A} (_ CH -> U.{x e. Atoms | x (_ A} (_ ( \/H ` {x e. Atoms | x (_ A}))
25	3, 24	ax-mp 6	. . . . . . . . . . 11 |- U.{x e. Atoms | x (_ A} (_ ( \/H ` {x e. Atoms | x (_ A})
26		sstr2 1510	. . . . . . . . . . 11 |- (y (_ U.{x e. Atoms | x (_ A} -> (U.{x e. Atoms | x (_ A} (_ ( \/H ` {x e. Atoms | x (_ A}) -> y (_ ( \/H ` {x e. Atoms | x (_ A})))
27	25, 26	mpi 44	. . . . . . . . . 10 |- (y (_ U.{x e. Atoms | x (_ A} -> y (_ ( \/H ` {x e. Atoms | x (_ A}))
28	23, 27	syl 12	. . . . . . . . 9 |- ((y e. Atoms /\ y (_ A) -> y (_ ( \/H ` {x e. Atoms | x (_ A}))
29	28	exp 291	. . . . . . . 8 |- (y e. Atoms -> (y (_ A -> y (_ ( \/H ` {x e. Atoms | x (_ A})))
30		atn0 5743	. . . . . . . . . . 11 |- (y e. Atoms -> -. y = 0H)
31	30	adantr 306	. . . . . . . . . 10 |- ((y e. Atoms /\ y (_ ( \/H ` {x e. Atoms | x (_ A})) -> -. y = 0H)
32		chle0t 5368	. . . . . . . . . . . . . . 15 |- (y e. CH -> (y (_ 0H <-> y = 0H))
33	8, 32	syl 12	. . . . . . . . . . . . . 14 |- (y e. Atoms -> (y (_ 0H <-> y = 0H))
34		ssin 1659	. . . . . . . . . . . . . . 15 |- ((y (_ ( \/H ` {x e. Atoms | x (_ A}) /\ y (_ (_|_` ( \/H ` {x e. Atoms | x (_ A}))) <-> y (_ (( \/H ` {x e. Atoms | x (_ A}) i^i (_|_` ( \/H ` {x e. Atoms | x (_ A}))))
35	5	chocin 5376	. . . . . . . . . . . . . . . 16 |- (( \/H ` {x e. Atoms | x (_ A}) i^i (_|_` ( \/H ` {x e. Atoms | x (_ A}))) = 0H
36	35	sseq2i 1525	. . . . . . . . . . . . . . 15 |- (y (_ (( \/H ` {x e. Atoms | x (_ A}) i^i (_|_` ( \/H ` {x e. Atoms | x (_ A}))) <-> y (_ 0H)
37	34, 36	bitr2 152	. . . . . . . . . . . . . 14 |- (y (_ 0H <-> (y (_ ( \/H ` {x e. Atoms | x (_ A}) /\ y (_ (_|_` ( \/H ` {x e. Atoms | x (_ A}))))
38	33, 37	syl5bbr 412	. . . . . . . . . . . . 13 |- (y e. Atoms -> ((y (_ ( \/H ` {x e. Atoms | x (_ A}) /\ y (_ (_|_` ( \/H ` {x e. Atoms | x (_ A}))) <-> y = 0H))
39	38	biimpa 324	. . . . . . . . . . . 12 |- ((y e. Atoms /\ (y (_ ( \/H ` {x e. Atoms | x (_ A}) /\ y (_ (_|_` ( \/H ` {x e. Atoms | x (_ A})))) -> y = 0H)
40	39	exp32 294	. . . . . . . . . . 11 |- (y e. Atoms -> (y (_ ( \/H ` {x e. Atoms | x (_ A}) -> (y (_ (_|_` ( \/H ` {x e. Atoms | x (_ A})) -> y = 0H)))
41	40	imp 277	. . . . . . . . . 10 |- ((y e. Atoms /\ y (_ ( \/H ` {x e. Atoms | x (_ A})) -> (y (_ (_|_` ( \/H ` {x e. Atoms | x (_ A})) -> y = 0H))
42	31, 41	mtod 95	. . . . . . . . 9 |- ((y e. Atoms /\ y (_ ( \/H ` {x e. Atoms | x (_ A})) -> -. y (_ (_|_` ( \/H ` {x e. Atoms | x (_ A})))
43	42	exp 291	. . . . . . . 8 |- (y e. Atoms -> (y (_ ( \/H ` {x e. Atoms | x (_ A}) -> -. y (_ (_|_` ( \/H ` {x e. Atoms | x (_ A}))))
44	29, 43	syld 27	. . . . . . 7 |- (y e. Atoms -> (y (_ A -> -. y (_ (_|_` ( \/H ` {x e. Atoms | x (_ A}))))
45		imnan 207	. . . . . . 7 |- ((y (_ A -> -. y (_ (_|_` ( \/H ` {x e. Atoms | x (_ A}))) <-> -. (y (_ A /\ y (_ (_|_` ( \/H ` {x e. Atoms | x (_ A}))))
46	44, 45	sylib 173	. . . . . 6 |- (y e. Atoms -> -. (y (_ A /\ y (_ (_|_` ( \/H ` {x e. Atoms | x (_ A}))))
47		ssin 1659	. . . . . . 7 |- ((y (_ A /\ y (_ (_|_` ( \/H ` {x e. Atoms | x (_ A}))) <-> y (_ (A i^i (_|_` ( \/H ` {x e. Atoms | x (_ A}))))
48	47	negbii 162	. . . . . 6 |- (-. (y (_ A /\ y (_ (_|_` ( \/H ` {x e. Atoms | x (_ A}))) <-> -. y (_ (A i^i (_|_` ( \/H ` {x e. Atoms | x (_ A}))))
49	46, 48	sylib 173	. . . . 5 |- (y e. Atoms -> -. y (_ (A i^i (_|_` ( \/H ` {x e. Atoms | x (_ A}))))
50	49	nrex 1270	. . . 4 |- -. E.y e. Atoms y (_ (A i^i (_|_` ( \/H ` {x e. Atoms | x (_ A})))
51	5	chocl 5192	. . . . . 6 |- (_|_` ( \/H ` {x e. Atoms | x (_ A})) e. CH
52	6, 51	chincl 5382	. . . . 5 |- (A i^i (_|_` ( \/H ` {x e. Atoms | x (_ A}))) e. CH
53	52	hatomic 5754	. . . 4 |- (-. (A i^i (_|_` ( \/H ` {x e. Atoms | x (_ A}))) = 0H -> E.y e. Atoms y (_ (A i^i (_|_` ( \/H ` {x e. Atoms | x (_ A}))))
54	50, 53	mt3 99	. . 3 |- (A i^i (_|_` ( \/H ` {x e. Atoms | x (_ A}))) = 0H
55	5, 7, 21, 54	omlsi 5250	. 2 |- ( \/H ` {x e. Atoms | x (_ A}) = A
56	55	cleqcomi 1105	1 |- A = ( \/H ` {x e. Atoms | x (_ A})

Syntax hints:  -. wn 1   -> wi 2   <-> wb 127   /\ wa 196   = wceq 1091   e. wcel 1092  E.wrex 1202  {crab 1204   i^i cin 1486   (_ wss 1487  U.cuni 1919  ` cfv 2422  CHcch 4968  _|_cort 4969   \/H chsup 4973  0Hc0h 4974  Atomscat 4980

This theorem is referenced by:  chpssat 5756

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 3880  df-1p 3881  df-plp 3882  df-mp 3883  df-ltp 3884  df-plpr 3958  df-mpr 3959  df-enr 3960  df-nr 3961  df-plr 3962  df-mr 3963  df-ltr 3964  df-0r 3965  df-1r 3966  df-m1r 3967  df-c 4034  df-0 4035  df-1 4036  df-i 4037  df-r 4038  df-plus 4039  df-mul 4040  df-lt 4041  d