Theorem projlem3 10821

Description: Part of Lemma 3.6 of [Beran] p. 100, bottom inequality. Used by projlem6 10824.

Hypotheses

Ref	Expression
projlem3.1	|- R e. RR
projlem3.2	|- D e. NN
projlem3.3	|- G e. NN

Assertion

Ref	Expression
projlem3	|- (((2 x. ((R + (1 / D))^2)) + (2 x. ((R + (1 / G))^2))) - (4 x. (R^2))) <_ (((4 x. R) + 2) x. ((1 / D) + (1 / G)))

Proof of Theorem projlem3

Step	Hyp	Ref	Expression
1		2cn 7164	. . . . . 6 |- 2 e. CC
2		projlem3.1	. . . . . . . . 9 |- R e. RR
3		projlem3.2	. . . . . . . . . . 11 |- D e. NN
4	3	nnrei 7114	. . . . . . . . . 10 |- D e. RR
5	3	nnne0i 7134	. . . . . . . . . 10 |- D =/= 0
6	4, 5	rereccli 6979	. . . . . . . . 9 |- (1 / D) e. RR
7	2, 6	readdcli 6487	. . . . . . . 8 |- (R + (1 / D)) e. RR
8	7	resqcli 7868	. . . . . . 7 |- ((R + (1 / D))^2) e. RR
9	8	recni 6467	. . . . . 6 |- ((R + (1 / D))^2) e. CC
10		projlem3.3	. . . . . . . . . . 11 |- G e. NN
11	10	nnrei 7114	. . . . . . . . . 10 |- G e. RR
12	10	nnne0i 7134	. . . . . . . . . 10 |- G =/= 0
13	11, 12	rereccli 6979	. . . . . . . . 9 |- (1 / G) e. RR
14	2, 13	readdcli 6487	. . . . . . . 8 |- (R + (1 / G)) e. RR
15	14	resqcli 7868	. . . . . . 7 |- ((R + (1 / G))^2) e. RR
16	15	recni 6467	. . . . . 6 |- ((R + (1 / G))^2) e. CC
17	1, 9, 16	adddii 6479	. . . . 5 |- (2 x. (((R + (1 / D))^2) + ((R + (1 / G))^2))) = ((2 x. ((R + (1 / D))^2)) + (2 x. ((R + (1 / G))^2)))
18	2	recni 6467	. . . . . . . . . 10 |- R e. CC
19	6	recni 6467	. . . . . . . . . 10 |- (1 / D) e. CC
20	18, 19	binom2i 7890	. . . . . . . . 9 |- ((R + (1 / D))^2) = (((R^2) + (2 x. (R x. (1 / D)))) + ((1 / D)^2))
21	18	sqcli 7860	. . . . . . . . . 10 |- (R^2) e. CC
22	18, 19	mulcli 6474	. . . . . . . . . . 11 |- (R x. (1 / D)) e. CC
23	1, 22	mulcli 6474	. . . . . . . . . 10 |- (2 x. (R x. (1 / D))) e. CC
24	19	sqcli 7860	. . . . . . . . . 10 |- ((1 / D)^2) e. CC
25	21, 23, 24	addassi 6477	. . . . . . . . 9 |- (((R^2) + (2 x. (R x. (1 / D)))) + ((1 / D)^2)) = ((R^2) + ((2 x. (R x. (1 / D))) + ((1 / D)^2)))
26	20, 25	eqtri 1908	. . . . . . . 8 |- ((R + (1 / D))^2) = ((R^2) + ((2 x. (R x. (1 / D))) + ((1 / D)^2)))
27	13	recni 6467	. . . . . . . . . 10 |- (1 / G) e. CC
28	18, 27	binom2i 7890	. . . . . . . . 9 |- ((R + (1 / G))^2) = (((R^2) + (2 x. (R x. (1 / G)))) + ((1 / G)^2))
29	18, 27	mulcli 6474	. . . . . . . . . . 11 |- (R x. (1 / G)) e. CC
30	1, 29	mulcli 6474	. . . . . . . . . 10 |- (2 x. (R x. (1 / G))) e. CC
31	27	sqcli 7860	. . . . . . . . . 10 |- ((1 / G)^2) e. CC
32	21, 30, 31	addassi 6477	. . . . . . . . 9 |- (((R^2) + (2 x. (R x. (1 / G)))) + ((1 / G)^2)) = ((R^2) + ((2 x. (R x. (1 / G))) + ((1 / G)^2)))
33	28, 32	eqtri 1908	. . . . . . . 8 |- ((R + (1 / G))^2) = ((R^2) + ((2 x. (R x. (1 / G))) + ((1 / G)^2)))
34	26, 33	opreq12i 4894	. . . . . . 7 |- (((R + (1 / D))^2) + ((R + (1 / G))^2)) = (((R^2) + ((2 x. (R x. (1 / D))) + ((1 / D)^2))) + ((R^2) + ((2 x. (R x. (1 / G))) + ((1 / G)^2))))
35	23, 24	addcli 6473	. . . . . . . 8 |- ((2 x. (R x. (1 / D))) + ((1 / D)^2)) e. CC
36	30, 31	addcli 6473	. . . . . . . 8 |- ((2 x. (R x. (1 / G))) + ((1 / G)^2)) e. CC
37	21, 35, 21, 36	add4i 6496	. . . . . . 7 |- (((R^2) + ((2 x. (R x. (1 / D))) + ((1 / D)^2))) + ((R^2) + ((2 x. (R x. (1 / G))) + ((1 / G)^2)))) = (((R^2) + (R^2)) + (((2 x. (R x. (1 / D))) + ((1 / D)^2)) + ((2 x. (R x. (1 / G))) + ((1 / G)^2))))
38	34, 37	eqtri 1908	. . . . . 6 |- (((R + (1 / D))^2) + ((R + (1 / G))^2)) = (((R^2) + (R^2)) + (((2 x. (R x. (1 / D))) + ((1 / D)^2)) + ((2 x. (R x. (1 / G))) + ((1 / G)^2))))
39	38	opreq2i 4893	. . . . 5 |- (2 x. (((R + (1 / D))^2) + ((R + (1 / G))^2))) = (2 x. (((R^2) + (R^2)) + (((2 x. (R x. (1 / D))) + ((1 / D)^2)) + ((2 x. (R x. (1 / G))) + ((1 / G)^2)))))
40	17, 39	eqtr3i 1910	. . . 4 |- ((2 x. ((R + (1 / D))^2)) + (2 x. ((R + (1 / G))^2))) = (2 x. (((R^2) + (R^2)) + (((2 x. (R x. (1 / D))) + ((1 / D)^2)) + ((2 x. (R x. (1 / G))) + ((1 / G)^2)))))
41	1, 1, 21	mulassi 6478	. . . . 5 |- ((2 x. 2) x. (R^2)) = (2 x. (2 x. (R^2)))
42		2t2e4 7206	. . . . . 6 |- (2 x. 2) = 4
43	42	opreq1i 4892	. . . . 5 |- ((2 x. 2) x. (R^2)) = (4 x. (R^2))
44	21	2timesi 7187	. . . . . 6 |- (2 x. (R^2)) = ((R^2) + (R^2))
45	44	opreq2i 4893	. . . . 5 |- (2 x. (2 x. (R^2))) = (2 x. ((R^2) + (R^2)))
46	41, 43, 45	3eqtr3i 1918	. . . 4 |- (4 x. (R^2)) = (2 x. ((R^2) + (R^2)))
47	40, 46	opreq12i 4894	. . 3 |- (((2 x. ((R + (1 / D))^2)) + (2 x. ((R + (1 / G))^2))) - (4 x. (R^2))) = ((2 x. (((R^2) + (R^2)) + (((2 x. (R x. (1 / D))) + ((1 / D)^2)) + ((2 x. (R x. (1 / G))) + ((1 / G)^2))))) - (2 x. ((R^2) + (R^2))))
48	21, 21	addcli 6473	. . . . 5 |- ((R^2) + (R^2)) e. CC
49	35, 36	addcli 6473	. . . . 5 |- (((2 x. (R x. (1 / D))) + ((1 / D)^2)) + ((2 x. (R x. (1 / G))) + ((1 / G)^2))) e. CC
50	48, 49	addcli 6473	. . . 4 |- (((R^2) + (R^2)) + (((2 x. (R x. (1 / D))) + ((1 / D)^2)) + ((2 x. (R x. (1 / G))) + ((1 / G)^2)))) e. CC
51	1, 50, 48	subdii 6592	. . 3 |- (2 x. ((((R^2) + (R^2)) + (((2 x. (R x. (1 / D))) + ((1 / D)^2)) + ((2 x. (R x. (1 / G))) + ((1 / G)^2)))) - ((R^2) + (R^2)))) = ((2 x. (((R^2) + (R^2)) + (((2 x. (R x. (1 / D))) + ((1 / D)^2)) + ((2 x. (R x. (1 / G))) + ((1 / G)^2))))) - (2 x. ((R^2) + (R^2))))
52	48, 49, 48	addsubassi 6546	. . . . 5 |- ((((R^2) + (R^2)) + (((2 x. (R x. (1 / D))) + ((1 / D)^2)) + ((2 x. (R x. (1 / G))) + ((1 / G)^2)))) - ((R^2) + (R^2))) = (((R^2) + (R^2)) + ((((2 x. (R x. (1 / D))) + ((1 / D)^2)) + ((2 x. (R x. (1 / G))) + ((1 / G)^2))) - ((R^2) + (R^2))))
53	48, 49	pncan3i 6535	. . . . 5 |- (((R^2) + (R^2)) + ((((2 x. (R x. (1 / D))) + ((1 / D)^2)) + ((2 x. (R x. (1 / G))) + ((1 / G)^2))) - ((R^2) + (R^2)))) = (((2 x. (R x. (1 / D))) + ((1 / D)^2)) + ((2 x. (R x. (1 / G))) + ((1 / G)^2)))
54	23, 24, 30, 31	add4i 6496	. . . . . 6 |- (((2 x. (R x. (1 / D))) + ((1 / D)^2)) + ((2 x. (R x. (1 / G))) + ((1 / G)^2))) = (((2 x. (R x. (1 / D))) + (2 x. (R x. (1 / G)))) + (((1 / D)^2) + ((1 / G)^2)))
55	18, 19, 27	adddii 6479	. . . . . . . . 9 |- (R x. ((1 / D) + (1 / G))) = ((R x. (1 / D)) + (R x. (1 / G)))
56	55	opreq2i 4893	. . . . . . . 8 |- (2 x. (R x. ((1 / D) + (1 / G)))) = (2 x. ((R x. (1 / D)) + (R x. (1 / G))))
57	1, 22, 29	adddii 6479	. . . . . . . 8 |- (2 x. ((R x. (1 / D)) + (R x. (1 / G)))) = ((2 x. (R x. (1 / D))) + (2 x. (R x. (1 / G))))
58	56, 57	eqtri 1908	. . . . . . 7 |- (2 x. (R x. ((1 / D) + (1 / G)))) = ((2 x. (R x. (1 / D))) + (2 x. (R x. (1 / G))))
59	58	opreq1i 4892	. . . . . 6 |- ((2 x. (R x. ((1 / D) + (1 / G)))) + (((1 / D)^2) + ((1 / G)^2))) = (((2 x. (R x. (1 / D))) + (2 x. (R x. (1 / G)))) + (((1 / D)^2) + ((1 / G)^2)))
60	54, 59	eqtr4i 1911	. . . . 5 |- (((2 x. (R x. (1 / D))) + ((1 / D)^2)) + ((2 x. (R x. (1 / G))) + ((1 / G)^2))) = ((2 x. (R x. ((1 / D) + (1 / G)))) + (((1 / D)^2) + ((1 / G)^2)))
61	52, 53, 60	3eqtri 1912	. . . 4 |- ((((R^2) + (R^2)) + (((2 x. (R x. (1 / D))) + ((1 / D)^2)) + ((2 x. (R x. (1 / G))) + ((1 / G)^2)))) - ((R^2) + (R^2))) = ((2 x. (R x. ((1 / D) + (1 / G)))) + (((1 / D)^2) + ((1 / G)^2)))
62	61	opreq2i 4893	. . 3 |- (2 x. ((((R^2) + (R^2)) + (((2 x. (R x. (1 / D))) + ((1 / D)^2)) + ((2 x. (R x. (1 / G))) + ((1 / G)^2)))) - ((R^2) + (R^2)))) = (2 x. ((2 x. (R x. ((1 / D) + (1 / G)))) + (((1 / D)^2) + ((1 / G)^2))))
63	47, 51, 62	3eqtr2i 1915	. 2 |- (((2 x. ((R + (1 / D))^2)) + (2 x. ((R + (1 / G))^2))) - (4 x. (R^2))) = (2 x. ((2 x. (R x. ((1 / D) + (1 / G)))) + (((1 / D)^2) + ((1 / G)^2))))
64	3	nncni 7115	. . . . . . . 8 |- D e. CC
65	64, 5	sqrecii 7864	. . . . . . 7 |- ((1 / D)^2) = (1 / (D^2))
66	3	nnlesqi 7911	. . . . . . . 8 |- D <_ (D^2)
67	3	nngt0i 7133	. . . . . . . . 9 |- 0 < D
68	4	sqgt0i 7872	. . . . . . . . . 10 |- (D =/= 0 -> 0 < (D^2))
69	5, 68	ax-mp 7	. . . . . . . . 9 |- 0 < (D^2)
70	4	resqcli 7868	. . . . . . . . . 10 |- (D^2) e. RR
71	4, 70	lereci 7063	. . . . . . . . 9 |- ((0 < D /\ 0 < (D^2)) -> (D <_ (D^2) <-> (1 / (D^2)) <_ (1 / D)))
72	67, 69, 71	mp2an 761	. . . . . . . 8 |- (D <_ (D^2) <-> (1 / (D^2)) <_ (1 / D))
73	66, 72	mpbi 206	. . . . . . 7 |- (1 / (D^2)) <_ (1 / D)
74	65, 73	eqbrtri 3356	. . . . . 6 |- ((1 / D)^2) <_ (1 / D)
75	10	nncni 7115	. . . . . . . 8 |- G e. CC
76	75, 12	sqrecii 7864	. . . . . . 7 |- ((1 / G)^2) = (1 / (G^2))
77	10	nnlesqi 7911	. . . . . . . 8 |- G <_ (G^2)
78	10	nngt0i 7133	. . . . . . . . 9 |- 0 < G
79	11	sqgt0i 7872	. . . . . . . . . 10 |- (G =/= 0 -> 0 < (G^2))
80	12, 79	ax-mp 7	. . . . . . . . 9 |- 0 < (G^2)
81	11	resqcli 7868	. . . . . . . . . 10 |- (G^2) e. RR
82	11, 81	lereci 7063	. . . . . . . . 9 |- ((0 < G /\ 0 < (G^2)) -> (G <_ (G^2) <-> (1 / (G^2)) <_ (1 / G)))
83	78, 80, 82	mp2an 761	. . . . . . . 8 |- (G <_ (G^2) <-> (1 / (G^2)) <_ (1 / G))
84	77, 83	mpbi 206	. . . . . . 7 |- (1 / (G^2)) <_ (1 / G)
85	76, 84	eqbrtri 3356	. . . . . 6 |- ((1 / G)^2) <_ (1 / G)
86	6	resqcli 7868	. . . . . . 7 |- ((1 / D)^2) e. RR
87	13	resqcli 7868	. . . . . . 7 |- ((1 / G)^2) e. RR
88	86, 87, 6, 13	le2addi 6774	. . . . . 6 |- ((((1 / D)^2) <_ (1 / D) /\ ((1 / G)^2) <_ (1 / G)) -> (((1 / D)^2) + ((1 / G)^2)) <_ ((1 / D) + (1 / G)))
89	74, 85, 88	mp2an 761	. . . . 5 |- (((1 / D)^2) + ((1 / G)^2)) <_ ((1 / D) + (1 / G))
90	86, 87	readdcli 6487	. . . . . 6 |- (((1 / D)^2) + ((1 / G)^2)) e. RR
91	6, 13	readdcli 6487	. . . . . 6 |- ((1 / D) + (1 / G)) e. RR
92		2re 7163	. . . . . . 7 |- 2 e. RR
93	2, 91	remulcli 6488	. . . . . . 7 |- (R x. ((1 / D) + (1 / G))) e. RR
94	92, 93	remulcli 6488	. . . . . 6 |- (2 x. (R x. ((1 / D) + (1 / G)))) e. RR
95	90, 91, 94	leadd2i 6768	. . . . 5 |- ((((1 / D)^2) + ((1 / G)^2)) <_ ((1 / D) + (1 / G)) <-> ((2 x. (R x. ((1 / D) + (1 / G)))) + (((1 / D)^2) + ((1 / G)^2))) <_ ((2 x. (R x. ((1 / D) + (1 / G)))) + ((1 / D) + (1 / G))))
96	89, 95	mpbi 206	. . . 4 |- ((2 x. (R x. ((1 / D) + (1 / G)))) + (((1 / D)^2) + ((1 / G)^2))) <_ ((2 x. (R x. ((1 / D) + (1 / G)))) + ((1 / D) + (1 / G)))
97		2pos 7173	. . . . 5 |- 0 < 2
98	94, 90	readdcli 6487	. . . . . 6 |- ((2 x. (R x. ((1 / D) + (1 / G)))) + (((1 / D)^2) + ((1 / G)^2))) e. RR
99	94, 91	readdcli 6487	. . . . . 6 |- ((2 x. (R x. ((1 / D) + (1 / G)))) + ((1 / D) + (1 / G))) e. RR
100	98, 99, 92	lemul2i 7018	. . . . 5 |- (0 < 2 -> (((2 x. (R x. ((1 / D) + (1 / G)))) + (((1 / D)^2) + ((1 / G)^2))) <_ ((2 x. (R x. ((1 / D) + (1 / G)))) + ((1 / D) + (1 / G))) <-> (2 x. ((2 x. (R x. ((1 / D) + (1 / G)))) + (((1 / D)^2) + ((1 / G)^2)))) <_ (2 x. ((2 x. (R x. ((1 / D) + (1 / G)))) + ((1 / D) + (1 / G))))))
101	97, 100	ax-mp 7	. . . 4 |- (((2 x. (R x. ((1 / D) + (1 / G)))) + (((1 / D)^2) + ((1 / G)^2))) <_ ((2 x. (R x. ((1 / D) + (1 / G)))) + ((1 / D) + (1 / G))) <-> (2 x. ((2 x. (R x. ((1 / D) + (1 / G)))) + (((1 / D)^2) + ((1 / G)^2)))) <_ (2 x. ((2 x. (R x. ((1 / D) + (1 / G)))) + ((1 / D) + (1 / G)))))
102	96, 101	mpbi 206	. . 3 |- (2 x. ((2 x. (R x. ((1 / D) + (1 / G)))) + (((1 / D)^2) + ((1 / G)^2)))) <_ (2 x. ((2 x. (R x. ((1 / D) + (1 / G)))) + ((1 / D) + (1 / G))))
103		4re 7166	. . . . . . . 8 |- 4 e. RR
104	103	recni 6467	. . . . . . 7 |- 4 e. CC
105	91	recni 6467	. . . . . . 7 |- ((1 / D) + (1 / G)) e. CC
106	104, 18, 105	mulassi 6478	. . . . . 6 |- ((4 x. R) x. ((1 / D) + (1 / G))) = (4 x. (R x. ((1 / D) + (1 / G))))
107	42	opreq1i 4892	. . . . . 6 |- ((2 x. 2) x. (R x. ((1 / D) + (1 / G)))) = (4 x. (R x. ((1 / D) + (1 / G))))
108	18, 105	mulcli 6474	. . . . . . 7 |- (R x. ((1 / D) + (1 / G))) e. CC
109	1, 1, 108	mulassi 6478	. . . . . 6 |- ((2 x. 2) x. (R x. ((1 / D) + (1 / G)))) = (2 x. (2 x. (R x. ((1 / D) + (1 / G)))))
110	106, 107, 109	3eqtr2ri 1916	. . . . 5 |- (2 x. (2 x. (R x. ((1 / D) + (1 / G))))) = ((4 x. R) x. ((1 / D) + (1 / G)))
111	110	opreq1i 4892	. . . 4 |- ((2 x. (2 x. (R x. ((1 / D) + (1 / G))))) + (2 x. ((1 / D) + (1 / G)))) = (((4 x. R) x. ((1 / D) + (1 / G))) + (2 x. ((1 / D) + (1 / G))))
112	1, 108	mulcli 6474	. . . . 5 |- (2 x. (R x. ((1 / D) + (1 / G)))) e. CC
113	1, 112, 105	adddii 6479	. . . 4 |- (2 x. ((2 x. (R x. ((1 / D) + (1 / G)))) + ((1 / D) + (1 / G)))) = ((2 x. (2 x. (R x. ((1 / D) + (1 / G))))) + (2 x. ((1 / D) + (1 / G))))
114	103, 2	remulcli 6488	. . . . . 6 |- (4 x. R) e. RR
115	114	recni 6467	. . . . 5 |- (4 x. R) e. CC
116	115, 1, 105	adddiri 6480	. . . 4 |- (((4 x. R) + 2) x. ((1 / D) + (1 / G))) = (((4 x. R) x. ((1 / D) + (1 / G))) + (2 x. ((1 / D) + (1 / G))))
117	111, 113, 116	3eqtr4i 1921	. . 3 |- (2 x. ((2 x. (R x. ((1 / D) + (1 / G)))) + ((1 / D) + (1 / G)))) = (((4 x. R) + 2) x. ((1 / D) + (1 / G)))
118	102, 117	breqtri 3360	. 2 |- (2 x. ((2 x. (R x. ((1 / D) + (1 / G)))) + (((1 / D)^2) + ((1 / G)^2)))) <_ (((4 x. R) + 2) x. ((1 / D) + (1 / G)))
119	63, 118	eqbrtri 3356	1 |- (((2 x. ((R + (1 / D))^2)) + (2 x. ((R + (1 / G))^2))) - (4 x. (R^2))) <_ (((4 x. R) + 2) x. ((1 / D) + (1 / G)))

Syntax hints:   <-> wb 163   e. wcel 1300   =/= wne 2017   class class class wbr 3338  (class class class)co 4884  RRcr 6385  0cc0 6386  1c1 6387   + caddc 6389   x. cmul 6391   - cmin 6445   / cdiv 6447   <_ cle 6448  NNcn 6449   < clt 6653  2c2 7145  4c4 7147  ^cexp 7811

This theorem is referenced by:  projlem6 10824

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  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-inf2 5731

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-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-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  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-seq1 7721  df-exp 7812

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