Theorem atcvat4i 11969

Description: A condition implying existence of an atom with the properties shown. Lemma 3.2.20 of [PtakPulmannova] p. 68.

Hypothesis

Ref	Expression
atcvat3.1	|- A e. CH

Assertion

Ref	Expression
atcvat4i	|- ((B e. Atoms /\ C e. Atoms) -> ((A =/= 0H /\ B C_ (A vH C)) -> E.x e. Atoms (x C_ A /\ B C_ (C vH x))))

Distinct variable groups:   x,A   x,B   x,C

Proof of Theorem atcvat4i

Step	Hyp	Ref	Expression
1		sseq1 2637	. . . . . . . . . . . . . . 15 |- (B = C -> (B C_ (C vH x) <-> C C_ (C vH x)))
2		chub1 11063	. . . . . . . . . . . . . . . 16 |- ((C e. CH /\ x e. CH) -> C C_ (C vH x))
3		atelch 11916	. . . . . . . . . . . . . . . 16 |- (C e. Atoms -> C e. CH)
4		atelch 11916	. . . . . . . . . . . . . . . 16 |- (x e. Atoms -> x e. CH)
5	2, 3, 4	syl2an 503	. . . . . . . . . . . . . . 15 |- ((C e. Atoms /\ x e. Atoms) -> C C_ (C vH x))
6	1, 5	syl5bir 227	. . . . . . . . . . . . . 14 |- (B = C -> ((C e. Atoms /\ x e. Atoms) -> B C_ (C vH x)))
7	6	exp3a 405	. . . . . . . . . . . . 13 |- (B = C -> (C e. Atoms -> (x e. Atoms -> B C_ (C vH x))))
8	7	impcom 378	. . . . . . . . . . . 12 |- ((C e. Atoms /\ B = C) -> (x e. Atoms -> B C_ (C vH x)))
9	8	anim2d 620	. . . . . . . . . . 11 |- ((C e. Atoms /\ B = C) -> ((x C_ A /\ x e. Atoms) -> (x C_ A /\ B C_ (C vH x))))
10	9	exp3a 405	. . . . . . . . . 10 |- ((C e. Atoms /\ B = C) -> (x C_ A -> (x e. Atoms -> (x C_ A /\ B C_ (C vH x)))))
11	10	com23 36	. . . . . . . . 9 |- ((C e. Atoms /\ B = C) -> (x e. Atoms -> (x C_ A -> (x C_ A /\ B C_ (C vH x)))))
12	11	reximdvai 2201	. . . . . . . 8 |- ((C e. Atoms /\ B = C) -> (E.x e. Atoms x C_ A -> E.x e. Atoms (x C_ A /\ B C_ (C vH x))))
13		atcvat3.1	. . . . . . . . 9 |- A e. CH
14	13	hatomici 11931	. . . . . . . 8 |- (A =/= 0H -> E.x e. Atoms x C_ A)
15	12, 14	syl5 20	. . . . . . 7 |- ((C e. Atoms /\ B = C) -> (A =/= 0H -> E.x e. Atoms (x C_ A /\ B C_ (C vH x))))
16	15	ex 402	. . . . . 6 |- (C e. Atoms -> (B = C -> (A =/= 0H -> E.x e. Atoms (x C_ A /\ B C_ (C vH x)))))
17	16	a1i 8	. . . . 5 |- (B C_ (A vH C) -> (C e. Atoms -> (B = C -> (A =/= 0H -> E.x e. Atoms (x C_ A /\ B C_ (C vH x))))))
18	17	com4l 43	. . . 4 |- (C e. Atoms -> (B = C -> (A =/= 0H -> (B C_ (A vH C) -> E.x e. Atoms (x C_ A /\ B C_ (C vH x))))))
19	18	imp4a 391	. . 3 |- (C e. Atoms -> (B = C -> ((A =/= 0H /\ B C_ (A vH C)) -> E.x e. Atoms (x C_ A /\ B C_ (C vH x)))))
20	19	adantl 424	. 2 |- ((B e. Atoms /\ C e. Atoms) -> (B = C -> ((A =/= 0H /\ B C_ (A vH C)) -> E.x e. Atoms (x C_ A /\ B C_ (C vH x)))))
21		chlejb2 11069	. . . . . . . . . . . . . . . 16 |- ((C e. CH /\ A e. CH) -> (C C_ A <-> (A vH C) = A))
22	13, 21	mpan2 760	. . . . . . . . . . . . . . 15 |- (C e. CH -> (C C_ A <-> (A vH C) = A))
23	22	biimpa 460	. . . . . . . . . . . . . 14 |- ((C e. CH /\ C C_ A) -> (A vH C) = A)
24	23	sseq2d 2645	. . . . . . . . . . . . 13 |- ((C e. CH /\ C C_ A) -> (B C_ (A vH C) <-> B C_ A))
25	24	biimpa 460	. . . . . . . . . . . 12 |- (((C e. CH /\ C C_ A) /\ B C_ (A vH C)) -> B C_ A)
26	25	expl 420	. . . . . . . . . . 11 |- (C e. CH -> ((C C_ A /\ B C_ (A vH C)) -> B C_ A))
27	26	adantl 424	. . . . . . . . . 10 |- ((B e. CH /\ C e. CH) -> ((C C_ A /\ B C_ (A vH C)) -> B C_ A))
28		chub2 11064	. . . . . . . . . 10 |- ((B e. CH /\ C e. CH) -> B C_ (C vH B))
29	27, 28	jctird 663	. . . . . . . . 9 |- ((B e. CH /\ C e. CH) -> ((C C_ A /\ B C_ (A vH C)) -> (B C_ A /\ B C_ (C vH B))))
30		atelch 11916	. . . . . . . . 9 |- (B e. Atoms -> B e. CH)
31	29, 30, 3	syl2an 503	. . . . . . . 8 |- ((B e. Atoms /\ C e. Atoms) -> ((C C_ A /\ B C_ (A vH C)) -> (B C_ A /\ B C_ (C vH B))))
32		simpl 346	. . . . . . . 8 |- ((B e. Atoms /\ C e. Atoms) -> B e. Atoms)
33	31, 32	jctild 662	. . . . . . 7 |- ((B e. Atoms /\ C e. Atoms) -> ((C C_ A /\ B C_ (A vH C)) -> (B e. Atoms /\ (B C_ A /\ B C_ (C vH B)))))
34	33	imp 377	. . . . . 6 |- (((B e. Atoms /\ C e. Atoms) /\ (C C_ A /\ B C_ (A vH C))) -> (B e. Atoms /\ (B C_ A /\ B C_ (C vH B))))
35	34	anassrs 489	. . . . 5 |- ((((B e. Atoms /\ C e. Atoms) /\ C C_ A) /\ B C_ (A vH C)) -> (B e. Atoms /\ (B C_ A /\ B C_ (C vH B))))
36		sseq1 2637	. . . . . . 7 |- (x = B -> (x C_ A <-> B C_ A))
37		opreq2 4890	. . . . . . . 8 |- (x = B -> (C vH x) = (C vH B))
38	37	sseq2d 2645	. . . . . . 7 |- (x = B -> (B C_ (C vH x) <-> B C_ (C vH B)))
39	36, 38	anbi12d 690	. . . . . 6 |- (x = B -> ((x C_ A /\ B C_ (C vH x)) <-> (B C_ A /\ B C_ (C vH B))))
40	39	rcla4ev 2381	. . . . 5 |- ((B e. Atoms /\ (B C_ A /\ B C_ (C vH B))) -> E.x e. Atoms (x C_ A /\ B C_ (C vH x)))
41	35, 40	syl 12	. . . 4 |- ((((B e. Atoms /\ C e. Atoms) /\ C C_ A) /\ B C_ (A vH C)) -> E.x e. Atoms (x C_ A /\ B C_ (C vH x)))
42	41	adantrl 430	. . 3 |- ((((B e. Atoms /\ C e. Atoms) /\ C C_ A) /\ (A =/= 0H /\ B C_ (A vH C))) -> E.x e. Atoms (x C_ A /\ B C_ (C vH x)))
43	42	exp31 407	. 2 |- ((B e. Atoms /\ C e. Atoms) -> (C C_ A -> ((A =/= 0H /\ B C_ (A vH C)) -> E.x e. Atoms (x C_ A /\ B C_ (C vH x)))))
44	13	atcvat3i 11968	. . . . . . 7 |- ((B e. Atoms /\ C e. Atoms) -> (((-. B = C /\ -. C C_ A) /\ B C_ (A vH C)) -> (A i^i (B vH C)) e. Atoms))
45	3	ad2antlr 441	. . . . . . . . . . 11 |- (((B e. Atoms /\ C e. Atoms) /\ ((-. B = C /\ -. C C_ A) /\ B C_ (A vH C))) -> C e. CH)
46	44	imp 377	. . . . . . . . . . 11 |- (((B e. Atoms /\ C e. Atoms) /\ ((-. B = C /\ -. C C_ A) /\ B C_ (A vH C))) -> (A i^i (B vH C)) e. Atoms)
47		simpll 448	. . . . . . . . . . 11 |- (((B e. Atoms /\ C e. Atoms) /\ ((-. B = C /\ -. C C_ A) /\ B C_ (A vH C))) -> B e. Atoms)
48	45, 46, 47	3jca 1050	. . . . . . . . . 10 |- (((B e. Atoms /\ C e. Atoms) /\ ((-. B = C /\ -. C C_ A) /\ B C_ (A vH C))) -> (C e. CH /\ (A i^i (B vH C)) e. Atoms /\ B e. Atoms))
49		inss2 2813	. . . . . . . . . . . . . 14 |- (A i^i (B vH C)) C_ (B vH C)
50	49	a1i 8	. . . . . . . . . . . . 13 |- ((B e. Atoms /\ C e. Atoms) -> (A i^i (B vH C)) C_ (B vH C))
51		chjcom 11062	. . . . . . . . . . . . . 14 |- ((B e. CH /\ C e. CH) -> (B vH C) = (C vH B))
52	51, 30, 3	syl2an 503	. . . . . . . . . . . . 13 |- ((B e. Atoms /\ C e. Atoms) -> (B vH C) = (C vH B))
53	50, 52	sseqtrd 2653	. . . . . . . . . . . 12 |- ((B e. Atoms /\ C e. Atoms) -> (A i^i (B vH C)) C_ (C vH B))
54	53	adantr 425	. . . . . . . . . . 11 |- (((B e. Atoms /\ C e. Atoms) /\ ((-. B = C /\ -. C C_ A) /\ B C_ (A vH C))) -> (A i^i (B vH C)) C_ (C vH B))
55		atnssm0 11948	. . . . . . . . . . . . . . . . 17 |- ((A e. CH /\ C e. Atoms) -> (-. C C_ A <-> (A i^i C) = 0H))
56	13, 55	mpan 759	. . . . . . . . . . . . . . . 16 |- (C e. Atoms -> (-. C C_ A <-> (A i^i C) = 0H))
57	56	adantl 424	. . . . . . . . . . . . . . 15 |- ((B e. Atoms /\ C e. Atoms) -> (-. C C_ A <-> (A i^i C) = 0H))
58		simpr 350	. . . . . . . . . . . . . . . . . . 19 |- ((B e. CH /\ C e. CH) -> C e. CH)
59		chjcl 10962	. . . . . . . . . . . . . . . . . . . 20 |- ((B e. CH /\ C e. CH) -> (B vH C) e. CH)
60		chincl 11055	. . . . . . . . . . . . . . . . . . . . 21 |- ((A e. CH /\ (B vH C) e. CH) -> (A i^i (B vH C)) e. CH)
61	13, 60	mpan 759	. . . . . . . . . . . . . . . . . . . 20 |- ((B vH C) e. CH -> (A i^i (B vH C)) e. CH)
62	59, 61	syl 12	. . . . . . . . . . . . . . . . . . 19 |- ((B e. CH /\ C e. CH) -> (A i^i (B vH C)) e. CH)
63		chincl 11055	. . . . . . . . . . . . . . . . . . 19 |- ((C e. CH /\ (A i^i (B vH C)) e. CH) -> (C i^i (A i^i (B vH C))) e. CH)
64	58, 62, 63	syl11anc 524	. . . . . . . . . . . . . . . . . 18 |- ((B e. CH /\ C e. CH) -> (C i^i (A i^i (B vH C))) e. CH)
65	64, 30, 3	syl2an 503	. . . . . . . . . . . . . . . . 17 |- ((B e. Atoms /\ C e. Atoms) -> (C i^i (A i^i (B vH C))) e. CH)
66		chle0 11000	. . . . . . . . . . . . . . . . 17 |- ((C i^i (A i^i (B vH C))) e. CH -> ((C i^i (A i^i (B vH C))) C_ 0H <-> (C i^i (A i^i (B vH C))) = 0H))
67	65, 66	syl 12	. . . . . . . . . . . . . . . 16 |- ((B e. Atoms /\ C e. Atoms) -> ((C i^i (A i^i (B vH C))) C_ 0H <-> (C i^i (A i^i (B vH C))) = 0H))
68		inss1 2812	. . . . . . . . . . . . . . . . . . 19 |- (A i^i (B vH C)) C_ A
69		sslin 2819	. . . . . . . . . . . . . . . . . . 19 |- ((A i^i (B vH C)) C_ A -> (C i^i (A i^i (B vH C))) C_ (C i^i A))
70	68, 69	ax-mp 7	. . . . . . . . . . . . . . . . . 18 |- (C i^i (A i^i (B vH C))) C_ (C i^i A)
71		incom 2787	. . . . . . . . . . . . . . . . . 18 |- (C i^i A) = (A i^i C)
72	70, 71	sseqtri 2649	. . . . . . . . . . . . . . . . 17 |- (C i^i (A i^i (B vH C))) C_ (A i^i C)
73		sseq2 2639	. . . . . . . . . . . . . . . . 17 |- ((A i^i C) = 0H -> ((C i^i (A i^i (B vH C))) C_ (A i^i C) <-> (C i^i (A i^i (B vH C))) C_ 0H))
74	72, 73	mpbii 210	. . . . . . . . . . . . . . . 16 |- ((A i^i C) = 0H -> (C i^i (A i^i (B vH C))) C_ 0H)
75	67, 74	syl5bi 225	. . . . . . . . . . . . . . 15 |- ((B e. Atoms /\ C e. Atoms) -> ((A i^i C) = 0H -> (C i^i (A i^i (B vH C))) = 0H))
76	57, 75	sylbid 220	. . . . . . . . . . . . . 14 |- ((B e. Atoms /\ C e. Atoms) -> (-. C C_ A -> (C i^i (A i^i (B vH C))) = 0H))
77	76	imp 377	. . . . . . . . . . . . 13 |- (((B e. Atoms /\ C e. Atoms) /\ -. C C_ A) -> (C i^i (A i^i (B vH C))) = 0H)
78	77	adantrl 430	. . . . . . . . . . . 12 |- (((B e. Atoms /\ C e. Atoms) /\ (-. B = C /\ -. C C_ A)) -> (C i^i (A i^i (B vH C))) = 0H)
79	78	adantrr 431	. . . . . . . . . . 11 |- (((B e. Atoms /\ C e. Atoms) /\ ((-. B = C /\ -. C C_ A) /\ B C_ (A vH C))) -> (C i^i (A i^i (B vH C))) = 0H)
80	54, 79	jca 310	. . . . . . . . . 10 |- (((B e. Atoms /\ C e. Atoms) /\ ((-. B = C /\ -. C C_ A) /\ B C_ (A vH C))) -> ((A i^i (B vH C)) C_ (C vH B) /\ (C i^i (A i^i (B vH C))) = 0H))
81		atexch 11953	. . . . . . . . . 10 |- ((C e. CH /\ (A i^i (B vH C)) e. Atoms /\ B e. Atoms) -> (((A i^i (B vH C)) C_ (C vH B) /\ (C i^i (A i^i (B vH C))) = 0H) -> B C_ (C vH (A i^i (B vH C)))))
82	48, 80, 81	sylc 83	. . . . . . . . 9 |- (((B e. Atoms /\ C e. Atoms) /\ ((-. B = C /\ -. C C_ A) /\ B C_ (A vH C))) -> B C_ (C vH (A i^i (B vH C))))
83	82, 68	jctil 316	. . . . . . . 8 |- (((B e. Atoms /\ C e. Atoms) /\ ((-. B = C /\ -. C C_ A) /\ B C_ (A vH C))) -> ((A i^i (B vH C)) C_ A /\ B C_ (C vH (A i^i (B vH C)))))
84	83	ex 402	. . . . . . 7 |- ((B e. Atoms /\ C e. Atoms) -> (((-. B = C /\ -. C C_ A) /\ B C_ (A vH C)) -> ((A i^i (B vH C)) C_ A /\ B C_ (C vH (A i^i (B vH C))))))
85	44, 84	jcad 661	. . . . . 6 |- ((B e. Atoms /\ C e. Atoms) -> (((-. B = C /\ -. C C_ A) /\ B C_ (A vH C)) -> ((A i^i (B vH C)) e. Atoms /\ ((A i^i (B vH C)) C_ A /\ B C_ (C vH (A i^i (B vH C)))))))
86		sseq1 2637	. . . . . . . 8 |- (x = (A i^i (B vH C)) -> (x C_ A <-> (A i^i (B vH C)) C_ A))
87		opreq2 4890	. . . . . . . . 9 |- (x = (A i^i (B vH C)) -> (C vH x) = (C vH (A i^i (B vH C))))
88	87	sseq2d 2645	. . . . . . . 8 |- (x = (A i^i (B vH C)) -> (B C_ (C vH x) <-> B C_ (C vH (A i^i (B vH C)))))
89	86, 88	anbi12d 690	. . . . . . 7 |- (x = (A i^i (B vH C)) -> ((x C_ A /\ B C_ (C vH x)) <-> ((A i^i (B vH C)) C_ A /\ B C_ (C vH (A i^i (B vH C))))))
90	89	rcla4ev 2381	. . . . . 6 |- (((A i^i (B vH C)) e. Atoms /\ ((A i^i (B vH C)) C_ A /\ B C_ (C vH (A i^i (B vH C))))) -> E.x e. Atoms (x C_ A /\ B C_ (C vH x)))
91	85, 90	syl6 25	. . . . 5 |- ((B e. Atoms /\ C e. Atoms) -> (((-. B = C /\ -. C C_ A) /\ B C_ (A vH C)) -> E.x e. Atoms (x C_ A /\ B C_ (C vH x))))
92	91	exp3a 405	. . . 4 |- ((B e. Atoms /\ C e. Atoms) -> ((-. B = C /\ -. C C_ A) -> (B C_ (A vH C) -> E.x e. Atoms (x C_ A /\ B C_ (C vH x)))))
93		ioran 331	. . . 4 |- (-. (B = C \/ C C_ A) <-> (-. B = C /\ -. C C_ A))
94	92, 93	syl5ib 223	. . 3 |- ((B e. Atoms /\ C e. Atoms) -> (-. (B = C \/ C C_ A) -> (B C_ (A vH C) -> E.x e. Atoms (x C_ A /\ B C_ (C vH x)))))
95		simpr 350	. . 3 |- ((A =/= 0H /\ B C_ (A vH C)) -> B C_ (A vH C))
96	94, 95	syl7 26	. 2 |- ((B e. Atoms /\ C e. Atoms) -> (-. (B = C \/ C C_ A) -> ((A =/= 0H /\ B C_ (A vH C)) -> E.x e. Atoms (x C_ A /\ B C_ (C vH x)))))
97	20, 43, 96	ecase3d 827	1 |- ((B e. Atoms /\ C e. Atoms) -> ((A =/= 0H /\ B C_ (A vH C)) -> E.x e. Atoms (x C_ A /\ B C_ (C vH x))))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   \/ wo 239   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300   =/= wne 2017  E.wrex 2106   i^i cin 2592   C_ wss 2593  (class class class)co 4884  CHcch 10430   vH chj 10434  0Hc0h 10436  Atomscat 10465

This theorem is referenced by:  mdsymlem3 11977

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-5 1302  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-reg 5695  ax-inf2 5731  ax-ac 5906  ax-hilex 10501  ax-hfvadd 10502  ax-hvcom 10503  ax-hvass 10504  ax-hv0cl 10505  ax-hvaddid 10506  ax-hfvmul 10507  ax-hvmulid 10508  ax-hvmulass 10509  ax-hvdistr1 10510  ax-hvdistr2 10511  ax-hvmul0 10512  ax-hfi 10579  ax-his1 10582  ax-his2 10583  ax-his3 10584  ax-his4 10585  ax-hcompl 10704

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-iin 3258  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-map 5383  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-sup 5664  df-r1 5750  df-rank 5751  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-n 7108  df-2 7154  df-3 7155  df-4 7156  df-n0 7309  df-z 7345  df-q 7436  df-fl 7463  df-ioo 7528  df-uz 7587  df-fz 7638  df-seq1 7721  df-shft 7754  df-seqz 7776  df-exp 7812  df-sqr 7920  df-re 8001  df-im 8002  df-cj 8003  df-abs 8004  df-clim 8235  df-sum 8240  df-top 8861  df-bases 8863  df-topgen 8864  df-cld 8939  df-ntr 8940  df-cls 8941  df-cn 9030  df-cnp 9031  df-haus 9059  df-met 9070  df-bl 9072  df-opn 9073  df-lm 9200  df-grp 9316  df-gid 9317  df-ginv 9318  df-gdiv 9319  df-abl 9408  df-vc 9497  df-nv 9543  df-va 9546  df-ba 9547  df-sm 9548  df-0v 9549  df-vs 9550  df-nm 9551  df-ims 9552  df-ip 9689  df-ph 9813  df-hnorm 10469  df-hvsub 10472  df-hlim 10473  df-hcau 10474  df-sh 10709  df-ch 10725  df-oc 10757  df-ch0 10758  df-pj 10870  df-shsum 10906  df-span 10907  df-chj 10908  df-chsup 10909  df-cv 11851  df-at 11910

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