Theorem atcvatlem 11957

Description: Lemma for atcvati 11958.

Hypothesis

Ref	Expression
atoml.1	|- A e. CH

Assertion

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

Proof of Theorem atcvatlem

Step	Hyp	Ref	Expression
1		atnem0 11949	. . . . . . . . . . . . . . . 16 |- ((x e. Atoms /\ B e. Atoms) -> (x =/= B <-> (x i^i B) = 0H))
2		sseq1 2637	. . . . . . . . . . . . . . . . . . . 20 |- (x = B -> (x C_ A <-> B C_ A))
3	2	biimpcd 172	. . . . . . . . . . . . . . . . . . 19 |- (x C_ A -> (x = B -> B C_ A))
4	3	necon3bd 2039	. . . . . . . . . . . . . . . . . 18 |- (x C_ A -> (-. B C_ A -> x =/= B))
5	4	imp 377	. . . . . . . . . . . . . . . . 17 |- ((x C_ A /\ -. B C_ A) -> x =/= B)
6	5	adantrl 430	. . . . . . . . . . . . . . . 16 |- ((x C_ A /\ (A C. (B vH C) /\ -. B C_ A)) -> x =/= B)
7	1, 6	syl5bi 225	. . . . . . . . . . . . . . 15 |- ((x e. Atoms /\ B e. Atoms) -> ((x C_ A /\ (A C. (B vH C) /\ -. B C_ A)) -> (x i^i B) = 0H))
8		cvp 11947	. . . . . . . . . . . . . . . . 17 |- ((x e. CH /\ B e. Atoms) -> ((x i^i B) = 0H <-> x <o (x vH B)))
9		chjcom 11062	. . . . . . . . . . . . . . . . . . 19 |- ((x e. CH /\ B e. CH) -> (x vH B) = (B vH x))
10		atelch 11916	. . . . . . . . . . . . . . . . . . 19 |- (B e. Atoms -> B e. CH)
11	9, 10	sylan2 500	. . . . . . . . . . . . . . . . . 18 |- ((x e. CH /\ B e. Atoms) -> (x vH B) = (B vH x))
12	11	breq2d 3350	. . . . . . . . . . . . . . . . 17 |- ((x e. CH /\ B e. Atoms) -> (x <o (x vH B) <-> x <o (B vH x)))
13	8, 12	bitrd 587	. . . . . . . . . . . . . . . 16 |- ((x e. CH /\ B e. Atoms) -> ((x i^i B) = 0H <-> x <o (B vH x)))
14		atelch 11916	. . . . . . . . . . . . . . . 16 |- (x e. Atoms -> x e. CH)
15	13, 14	sylan 497	. . . . . . . . . . . . . . 15 |- ((x e. Atoms /\ B e. Atoms) -> ((x i^i B) = 0H <-> x <o (B vH x)))
16	7, 15	sylibd 219	. . . . . . . . . . . . . 14 |- ((x e. Atoms /\ B e. Atoms) -> ((x C_ A /\ (A C. (B vH C) /\ -. B C_ A)) -> x <o (B vH x)))
17	16	ancoms 484	. . . . . . . . . . . . 13 |- ((B e. Atoms /\ x e. Atoms) -> ((x C_ A /\ (A C. (B vH C) /\ -. B C_ A)) -> x <o (B vH x)))
18	17	adantlr 429	. . . . . . . . . . . 12 |- (((B e. Atoms /\ C e. Atoms) /\ x e. Atoms) -> ((x C_ A /\ (A C. (B vH C) /\ -. B C_ A)) -> x <o (B vH x)))
19	18	imp 377	. . . . . . . . . . 11 |- ((((B e. Atoms /\ C e. Atoms) /\ x e. Atoms) /\ (x C_ A /\ (A C. (B vH C) /\ -. B C_ A))) -> x <o (B vH x))
20		simp1 876	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ((B e. CH /\ x e. CH /\ C e. CH) -> B e. CH)
21		simp3 878	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ((B e. CH /\ x e. CH /\ C e. CH) -> C e. CH)
22		chjcl 10962	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ((B e. CH /\ x e. CH) -> (B vH x) e. CH)
23	22	3adant3 896	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ((B e. CH /\ x e. CH /\ C e. CH) -> (B vH x) e. CH)
24	20, 21, 23	3jca 1050	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ((B e. CH /\ x e. CH /\ C e. CH) -> (B e. CH /\ C e. CH /\ (B vH x) e. CH))
25		atelch 11916	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (C e. Atoms -> C e. CH)
26	24, 10, 14, 25	syl3an 1139	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ((B e. Atoms /\ x e. Atoms /\ C e. Atoms) -> (B e. CH /\ C e. CH /\ (B vH x) e. CH))
27		chlub 11065	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ((B e. CH /\ C e. CH /\ (B vH x) e. CH) -> ((B C_ (B vH x) /\ C C_ (B vH x)) <-> (B vH C) C_ (B vH x)))
28	26, 27	syl 12	. . . . . . . . . . . . . . . . . . . . . . 23 |- ((B e. Atoms /\ x e. Atoms /\ C e. Atoms) -> ((B C_ (B vH x) /\ C C_ (B vH x)) <-> (B vH C) C_ (B vH x)))
29	28	adantr 425	. . . . . . . . . . . . . . . . . . . . . 22 |- (((B e. Atoms /\ x e. Atoms /\ C e. Atoms) /\ ((x C_ A /\ -. B C_ A) /\ A C. (B vH C))) -> ((B C_ (B vH x) /\ C C_ (B vH x)) <-> (B vH C) C_ (B vH x)))
30		chub1 11063	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ((B e. CH /\ x e. CH) -> B C_ (B vH x))
31	30, 10, 14	syl2an 503	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ((B e. Atoms /\ x e. Atoms) -> B C_ (B vH x))
32	31	3adant3 896	. . . . . . . . . . . . . . . . . . . . . . 23 |- ((B e. Atoms /\ x e. Atoms /\ C e. Atoms) -> B C_ (B vH x))
33	32	adantr 425	. . . . . . . . . . . . . . . . . . . . . 22 |- (((B e. Atoms /\ x e. Atoms /\ C e. Atoms) /\ ((x C_ A /\ -. B C_ A) /\ A C. (B vH C))) -> B C_ (B vH x))
34		sstr 2625	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ((x C_ A /\ A C_ (B vH C)) -> x C_ (B vH C))
35		pssss 2705	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (A C. (B vH C) -> A C_ (B vH C))
36	34, 35	sylan2 500	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ((x C_ A /\ A C. (B vH C)) -> x C_ (B vH C))
37	36	adantlr 429	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (((x C_ A /\ -. B C_ A) /\ A C. (B vH C)) -> x C_ (B vH C))
38	37	adantl 424	. . . . . . . . . . . . . . . . . . . . . . 23 |- (((B e. Atoms /\ x e. Atoms /\ C e. Atoms) /\ ((x C_ A /\ -. B C_ A) /\ A C. (B vH C))) -> x C_ (B vH C))
39	1, 5	syl5bi 225	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ((x e. Atoms /\ B e. Atoms) -> ((x C_ A /\ -. B C_ A) -> (x i^i B) = 0H))
40	39	ancoms 484	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ((B e. Atoms /\ x e. Atoms) -> ((x C_ A /\ -. B C_ A) -> (x i^i B) = 0H))
41	40	3adant3 896	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ((B e. Atoms /\ x e. Atoms /\ C e. Atoms) -> ((x C_ A /\ -. B C_ A) -> (x i^i B) = 0H))
42	41	imp 377	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (((B e. Atoms /\ x e. Atoms /\ C e. Atoms) /\ (x C_ A /\ -. B C_ A)) -> (x i^i B) = 0H)
43		incom 2787	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (B i^i x) = (x i^i B)
44	42, 43	syl5eq 1940	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (((B e. Atoms /\ x e. Atoms /\ C e. Atoms) /\ (x C_ A /\ -. B C_ A)) -> (B i^i x) = 0H)
45	44	adantrr 431	. . . . . . . . . . . . . . . . . . . . . . 23 |- (((B e. Atoms /\ x e. Atoms /\ C e. Atoms) /\ ((x C_ A /\ -. B C_ A) /\ A C. (B vH C))) -> (B i^i x) = 0H)
46		atexch 11953	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ((B e. CH /\ x e. Atoms /\ C e. Atoms) -> ((x C_ (B vH C) /\ (B i^i x) = 0H) -> C C_ (B vH x)))
47	46, 10	syl3an1 1130	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ((B e. Atoms /\ x e. Atoms /\ C e. Atoms) -> ((x C_ (B vH C) /\ (B i^i x) = 0H) -> C C_ (B vH x)))
48	47	adantr 425	. . . . . . . . . . . . . . . . . . . . . . 23 |- (((B e. Atoms /\ x e. Atoms /\ C e. Atoms) /\ ((x C_ A /\ -. B C_ A) /\ A C. (B vH C))) -> ((x C_ (B vH C) /\ (B i^i x) = 0H) -> C C_ (B vH x)))
49	38, 45, 48	mp2and 767	. . . . . . . . . . . . . . . . . . . . . 22 |- (((B e. Atoms /\ x e. Atoms /\ C e. Atoms) /\ ((x C_ A /\ -. B C_ A) /\ A C. (B vH C))) -> C C_ (B vH x))
50	29, 33, 49	mpbi2and 801	. . . . . . . . . . . . . . . . . . . . 21 |- (((B e. Atoms /\ x e. Atoms /\ C e. Atoms) /\ ((x C_ A /\ -. B C_ A) /\ A C. (B vH C))) -> (B vH C) C_ (B vH x))
51		chub1 11063	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ((B e. CH /\ C e. CH) -> B C_ (B vH C))
52	51	3adant2 895	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ((B e. CH /\ x e. CH /\ C e. CH) -> B C_ (B vH C))
53	52, 37	anim12i 360	. . . . . . . . . . . . . . . . . . . . . . 23 |- (((B e. CH /\ x e. CH /\ C e. CH) /\ ((x C_ A /\ -. B C_ A) /\ A C. (B vH C))) -> (B C_ (B vH C) /\ x C_ (B vH C)))
54		chlub 11065	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ((B e. CH /\ x e. CH /\ (B vH C) e. CH) -> ((B C_ (B vH C) /\ x C_ (B vH C)) <-> (B vH x) C_ (B vH C)))
55		chjcl 10962	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ((B e. CH /\ C e. CH) -> (B vH C) e. CH)
56	55	3adant2 895	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ((B e. CH /\ x e. CH /\ C e. CH) -> (B vH C) e. CH)
57	54, 56	syld3an3 1142	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ((B e. CH /\ x e. CH /\ C e. CH) -> ((B C_ (B vH C) /\ x C_ (B vH C)) <-> (B vH x) C_ (B vH C)))
58	57	adantr 425	. . . . . . . . . . . . . . . . . . . . . . 23 |- (((B e. CH /\ x e. CH /\ C e. CH) /\ ((x C_ A /\ -. B C_ A) /\ A C. (B vH C))) -> ((B C_ (B vH C) /\ x C_ (B vH C)) <-> (B vH x) C_ (B vH C)))
59	53, 58	mpbid 212	. . . . . . . . . . . . . . . . . . . . . 22 |- (((B e. CH /\ x e. CH /\ C e. CH) /\ ((x C_ A /\ -. B C_ A) /\ A C. (B vH C))) -> (B vH x) C_ (B vH C))
60	59, 10, 14, 25	syl3anl 1148	. . . . . . . . . . . . . . . . . . . . 21 |- (((B e. Atoms /\ x e. Atoms /\ C e. Atoms) /\ ((x C_ A /\ -. B C_ A) /\ A C. (B vH C))) -> (B vH x) C_ (B vH C))
61	50, 60	eqssd 2633	. . . . . . . . . . . . . . . . . . . 20 |- (((B e. Atoms /\ x e. Atoms /\ C e. Atoms) /\ ((x C_ A /\ -. B C_ A) /\ A C. (B vH C))) -> (B vH C) = (B vH x))
62	61	anassrs 489	. . . . . . . . . . . . . . . . . . 19 |- ((((B e. Atoms /\ x e. Atoms /\ C e. Atoms) /\ (x C_ A /\ -. B C_ A)) /\ A C. (B vH C)) -> (B vH C) = (B vH x))
63	62	psseq2d 2703	. . . . . . . . . . . . . . . . . 18 |- ((((B e. Atoms /\ x e. Atoms /\ C e. Atoms) /\ (x C_ A /\ -. B C_ A)) /\ A C. (B vH C)) -> (A C. (B vH C) <-> A C. (B vH x)))
64	63	ex 402	. . . . . . . . . . . . . . . . 17 |- (((B e. Atoms /\ x e. Atoms /\ C e. Atoms) /\ (x C_ A /\ -. B C_ A)) -> (A C. (B vH C) -> (A C. (B vH C) <-> A C. (B vH x))))
65	64	ibd 654	. . . . . . . . . . . . . . . 16 |- (((B e. Atoms /\ x e. Atoms /\ C e. Atoms) /\ (x C_ A /\ -. B C_ A)) -> (A C. (B vH C) -> A C. (B vH x)))
66	65	exp32 408	. . . . . . . . . . . . . . 15 |- ((B e. Atoms /\ x e. Atoms /\ C e. Atoms) -> (x C_ A -> (-. B C_ A -> (A C. (B vH C) -> A C. (B vH x)))))
67	66	3expa 1067	. . . . . . . . . . . . . 14 |- (((B e. Atoms /\ x e. Atoms) /\ C e. Atoms) -> (x C_ A -> (-. B C_ A -> (A C. (B vH C) -> A C. (B vH x)))))
68	67	an1rs 547	. . . . . . . . . . . . 13 |- (((B e. Atoms /\ C e. Atoms) /\ x e. Atoms) -> (x C_ A -> (-. B C_ A -> (A C. (B vH C) -> A C. (B vH x)))))
69	68	com34 40	. . . . . . . . . . . 12 |- (((B e. Atoms /\ C e. Atoms) /\ x e. Atoms) -> (x C_ A -> (A C. (B vH C) -> (-. B C_ A -> A C. (B vH x)))))
70	69	imp45 399	. . . . . . . . . . 11 |- ((((B e. Atoms /\ C e. Atoms) /\ x e. Atoms) /\ (x C_ A /\ (A C. (B vH C) /\ -. B C_ A))) -> A C. (B vH x))
71		simpr 350	. . . . . . . . . . . . . . . . 17 |- ((B e. CH /\ x e. CH) -> x e. CH)
72	71, 22	jca 310	. . . . . . . . . . . . . . . 16 |- ((B e. CH /\ x e. CH) -> (x e. CH /\ (B vH x) e. CH))
73	72, 10, 14	syl2an 503	. . . . . . . . . . . . . . 15 |- ((B e. Atoms /\ x e. Atoms) -> (x e. CH /\ (B vH x) e. CH))
74		atoml.1	. . . . . . . . . . . . . . . . . . 19 |- A e. CH
75		cvnbtwn3 11860	. . . . . . . . . . . . . . . . . . 19 |- ((x e. CH /\ (B vH x) e. CH /\ A e. CH) -> (x <o (B vH x) -> ((x C_ A /\ A C. (B vH x)) -> A = x)))
76	74, 75	mp3an3 1180	. . . . . . . . . . . . . . . . . 18 |- ((x e. CH /\ (B vH x) e. CH) -> (x <o (B vH x) -> ((x C_ A /\ A C. (B vH x)) -> A = x)))
77	76	exp4a 409	. . . . . . . . . . . . . . . . 17 |- ((x e. CH /\ (B vH x) e. CH) -> (x <o (B vH x) -> (x C_ A -> (A C. (B vH x) -> A = x))))
78	77	com23 36	. . . . . . . . . . . . . . . 16 |- ((x e. CH /\ (B vH x) e. CH) -> (x C_ A -> (x <o (B vH x) -> (A C. (B vH x) -> A = x))))
79	78	imp4a 391	. . . . . . . . . . . . . . 15 |- ((x e. CH /\ (B vH x) e. CH) -> (x C_ A -> ((x <o (B vH x) /\ A C. (B vH x)) -> A = x)))
80	73, 79	syl 12	. . . . . . . . . . . . . 14 |- ((B e. Atoms /\ x e. Atoms) -> (x C_ A -> ((x <o (B vH x) /\ A C. (B vH x)) -> A = x)))
81	80	adantlr 429	. . . . . . . . . . . . 13 |- (((B e. Atoms /\ C e. Atoms) /\ x e. Atoms) -> (x C_ A -> ((x <o (B vH x) /\ A C. (B vH x)) -> A = x)))
82	81	imp 377	. . . . . . . . . . . 12 |- ((((B e. Atoms /\ C e. Atoms) /\ x e. Atoms) /\ x C_ A) -> ((x <o (B vH x) /\ A C. (B vH x)) -> A = x))
83	82	adantrr 431	. . . . . . . . . . 11 |- ((((B e. Atoms /\ C e. Atoms) /\ x e. Atoms) /\ (x C_ A /\ (A C. (B vH C) /\ -. B C_ A))) -> ((x <o (B vH x) /\ A C. (B vH x)) -> A = x))
84	19, 70, 83	mp2and 767	. . . . . . . . . 10 |- ((((B e. Atoms /\ C e. Atoms) /\ x e. Atoms) /\ (x C_ A /\ (A C. (B vH C) /\ -. B C_ A))) -> A = x)
85	84	eleq1d 1963	. . . . . . . . 9 |- ((((B e. Atoms /\ C e. Atoms) /\ x e. Atoms) /\ (x C_ A /\ (A C. (B vH C) /\ -. B C_ A))) -> (A e. Atoms <-> x e. Atoms))
86	85	biimprcd 173	. . . . . . . 8 |- (x e. Atoms -> ((((B e. Atoms /\ C e. Atoms) /\ x e. Atoms) /\ (x C_ A /\ (A C. (B vH C) /\ -. B C_ A))) -> A e. Atoms))
87	86	exp4c 411	. . . . . . 7 |- (x e. Atoms -> ((B e. Atoms /\ C e. Atoms) -> (x e. Atoms -> ((x C_ A /\ (A C. (B vH C) /\ -. B C_ A)) -> A e. Atoms))))
88	87	pm2.43b 81	. . . . . 6 |- ((B e. Atoms /\ C e. Atoms) -> (x e. Atoms -> ((x C_ A /\ (A C. (B vH C) /\ -. B C_ A)) -> A e. Atoms)))
89	88	imp 377	. . . . 5 |- (((B e. Atoms /\ C e. Atoms) /\ x e. Atoms) -> ((x C_ A /\ (A C. (B vH C) /\ -. B C_ A)) -> A e. Atoms))
90	89	exp4d 412	. . . 4 |- (((B e. Atoms /\ C e. Atoms) /\ x e. Atoms) -> (x C_ A -> (A C. (B vH C) -> (-. B C_ A -> A e. Atoms))))
91	90	r19.23adva 2216	. . 3 |- ((B e. Atoms /\ C e. Atoms) -> (E.x e. Atoms x C_ A -> (A C. (B vH C) -> (-. B C_ A -> A e. Atoms))))
92	74	hatomici 11931	. . 3 |- (A =/= 0H -> E.x e. Atoms x C_ A)
93	91, 92	syl5 20	. 2 |- ((B e. Atoms /\ C e. Atoms) -> (A =/= 0H -> (A C. (B vH C) -> (-. B C_ A -> A e. Atoms))))
94	93	imp32 390	1 |- (((B e. Atoms /\ C e. Atoms) /\ (A =/= 0H /\ A C. (B vH C))) -> (-. B C_ A -> A e. Atoms))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300   =/= wne 2017  E.wrex 2106   i^i cin 2592   C_ wss 2593   C. wpss 2594   class class class wbr 3338  (class class class)co 4884  CHcch 10430   vH chj 10434  0Hc0h 10436  Atomscat 10465   <o ccv 10466

This theorem is referenced by:  atcvati 11958

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

Copyright terms: Public domain