Theorem bcpasci 8221

Description: Pascal's rule for the binomial coefficient, generalized to all integers K. Equation 2 of [Gleason] p. 295.

Hypotheses

Ref	Expression
bcpasc.1	|- N e. NN0
bcpasc.2	|- K e. ZZ

Assertion

Ref	Expression
bcpasci	|- ((N _C K) + (N _C (K - 1))) = ((N + 1) _C K)

Proof of Theorem bcpasci

Step	Hyp	Ref	Expression
1		bcpasc.1	. . . . 5 |- N e. NN0
2		elnn0 7310	. . . . 5 |- (N e. NN0 <-> (N e. NN \/ N = 0))
3	1, 2	mpbi 206	. . . 4 |- (N e. NN \/ N = 0)
4	3	ori 247	. . 3 |- (-. N e. NN -> N = 0)
5		ax1cn 6422	. . . . . . 7 |- 1 e. CC
6	5	addid1i 6483	. . . . . 6 |- (1 + 0) = 1
7		opreq2 4890	. . . . . . . 8 |- (K = 0 -> (0 _C K) = (0 _C 0))
8		0nn0 7322	. . . . . . . . 9 |- 0 e. NN0
9		bcn0 8215	. . . . . . . . 9 |- (0 e. NN0 -> (0 _C 0) = 1)
10	8, 9	ax-mp 7	. . . . . . . 8 |- (0 _C 0) = 1
11	7, 10	syl6eq 1944	. . . . . . 7 |- (K = 0 -> (0 _C K) = 1)
12		opreq1 4889	. . . . . . . . . 10 |- (K = 0 -> (K - 1) = (0 - 1))
13		df-neg 6513	. . . . . . . . . 10 |- -u1 = (0 - 1)
14	12, 13	syl6eqr 1946	. . . . . . . . 9 |- (K = 0 -> (K - 1) = -u1)
15	14	opreq2d 4898	. . . . . . . 8 |- (K = 0 -> (0 _C (K - 1)) = (0 _C -u1))
16		1z 7368	. . . . . . . . . 10 |- 1 e. ZZ
17		znegcl 7372	. . . . . . . . . 10 |- (1 e. ZZ -> -u1 e. ZZ)
18	16, 17	ax-mp 7	. . . . . . . . 9 |- -u1 e. ZZ
19		lt01 6871	. . . . . . . . . . 11 |- 0 < 1
20		1re 6598	. . . . . . . . . . . 12 |- 1 e. RR
21		lt0neg2 6858	. . . . . . . . . . . 12 |- (1 e. RR -> (0 < 1 <-> -u1 < 0))
22	20, 21	ax-mp 7	. . . . . . . . . . 11 |- (0 < 1 <-> -u1 < 0)
23	19, 22	mpbi 206	. . . . . . . . . 10 |- -u1 < 0
24	23	orci 292	. . . . . . . . 9 |- (-u1 < 0 \/ 0 < -u1)
25		bcval4 8213	. . . . . . . . 9 |- ((0 e. NN0 /\ -u1 e. ZZ /\ (-u1 < 0 \/ 0 < -u1)) -> (0 _C -u1) = 0)
26	8, 18, 24, 25	mp3an 1191	. . . . . . . 8 |- (0 _C -u1) = 0
27	15, 26	syl6eq 1944	. . . . . . 7 |- (K = 0 -> (0 _C (K - 1)) = 0)
28	11, 27	opreq12d 4900	. . . . . 6 |- (K = 0 -> ((0 _C K) + (0 _C (K - 1))) = (1 + 0))
29		opreq2 4890	. . . . . . 7 |- (K = 0 -> (1 _C K) = (1 _C 0))
30		1nn0 7323	. . . . . . . 8 |- 1 e. NN0
31		bcn0 8215	. . . . . . . 8 |- (1 e. NN0 -> (1 _C 0) = 1)
32	30, 31	ax-mp 7	. . . . . . 7 |- (1 _C 0) = 1
33	29, 32	syl6eq 1944	. . . . . 6 |- (K = 0 -> (1 _C K) = 1)
34	6, 28, 33	3eqtr4a 1954	. . . . 5 |- (K = 0 -> ((0 _C K) + (0 _C (K - 1))) = (1 _C K))
35		bcpasc.2	. . . . . . . 8 |- K e. ZZ
36	35	zrei 7350	. . . . . . 7 |- K e. RR
37		0re 6603	. . . . . . 7 |- 0 e. RR
38	36, 37	lttri2i 6747	. . . . . 6 |- (K =/= 0 <-> (K < 0 \/ 0 < K))
39		0cn 6481	. . . . . . . . 9 |- 0 e. CC
40	39	addid1i 6483	. . . . . . . 8 |- (0 + 0) = 0
41		bcval4 8213	. . . . . . . . . . 11 |- ((0 e. NN0 /\ K e. ZZ /\ (K < 0 \/ 0 < K)) -> (0 _C K) = 0)
42	8, 35, 41	mp3an12 1181	. . . . . . . . . 10 |- ((K < 0 \/ 0 < K) -> (0 _C K) = 0)
43	42	orcs 296	. . . . . . . . 9 |- (K < 0 -> (0 _C K) = 0)
44	36	leidi 6790	. . . . . . . . . . . 12 |- K <_ K
45		zlem1lt 7392	. . . . . . . . . . . . 13 |- ((K e. ZZ /\ K e. ZZ) -> (K <_ K <-> (K - 1) < K))
46	35, 35, 45	mp2an 761	. . . . . . . . . . . 12 |- (K <_ K <-> (K - 1) < K)
47	44, 46	mpbi 206	. . . . . . . . . . 11 |- (K - 1) < K
48	36, 20	resubcli 6602	. . . . . . . . . . . 12 |- (K - 1) e. RR
49	48, 36, 37	lttri 6760	. . . . . . . . . . 11 |- (((K - 1) < K /\ K < 0) -> (K - 1) < 0)
50	47, 49	mpan 759	. . . . . . . . . 10 |- (K < 0 -> (K - 1) < 0)
51		orc 291	. . . . . . . . . 10 |- ((K - 1) < 0 -> ((K - 1) < 0 \/ 0 < (K - 1)))
52		zsubcl 7377	. . . . . . . . . . . 12 |- ((K e. ZZ /\ 1 e. ZZ) -> (K - 1) e. ZZ)
53	35, 16, 52	mp2an 761	. . . . . . . . . . 11 |- (K - 1) e. ZZ
54		bcval4 8213	. . . . . . . . . . 11 |- ((0 e. NN0 /\ (K - 1) e. ZZ /\ ((K - 1) < 0 \/ 0 < (K - 1))) -> (0 _C (K - 1)) = 0)
55	8, 53, 54	mp3an12 1181	. . . . . . . . . 10 |- (((K - 1) < 0 \/ 0 < (K - 1)) -> (0 _C (K - 1)) = 0)
56	50, 51, 55	3syl 24	. . . . . . . . 9 |- (K < 0 -> (0 _C (K - 1)) = 0)
57	43, 56	opreq12d 4900	. . . . . . . 8 |- (K < 0 -> ((0 _C K) + (0 _C (K - 1))) = (0 + 0))
58		bcval4 8213	. . . . . . . . . 10 |- ((1 e. NN0 /\ K e. ZZ /\ (K < 0 \/ 1 < K)) -> (1 _C K) = 0)
59	30, 35, 58	mp3an12 1181	. . . . . . . . 9 |- ((K < 0 \/ 1 < K) -> (1 _C K) = 0)
60	59	orcs 296	. . . . . . . 8 |- (K < 0 -> (1 _C K) = 0)
61	40, 57, 60	3eqtr4a 1954	. . . . . . 7 |- (K < 0 -> ((0 _C K) + (0 _C (K - 1))) = (1 _C K))
62	42	olcs 297	. . . . . . . . 9 |- (0 < K -> (0 _C K) = 0)
63	62	opreq1d 4897	. . . . . . . 8 |- (0 < K -> ((0 _C K) + (0 _C (K - 1))) = (0 + (0 _C (K - 1))))
64		0z 7355	. . . . . . . . . . 11 |- 0 e. ZZ
65		zltp1le 7390	. . . . . . . . . . 11 |- ((0 e. ZZ /\ K e. ZZ) -> (0 < K <-> (0 + 1) <_ K))
66	64, 35, 65	mp2an 761	. . . . . . . . . 10 |- (0 < K <-> (0 + 1) <_ K)
67	5	addid2i 6484	. . . . . . . . . . 11 |- (0 + 1) = 1
68	67	breq1i 3345	. . . . . . . . . 10 |- ((0 + 1) <_ K <-> 1 <_ K)
69	20, 36	leloei 6750	. . . . . . . . . 10 |- (1 <_ K <-> (1 < K \/ 1 = K))
70	66, 68, 69	3bitri 194	. . . . . . . . 9 |- (0 < K <-> (1 < K \/ 1 = K))
71	59	olcs 297	. . . . . . . . . . 11 |- (1 < K -> (1 _C K) = 0)
72	20, 36	posdifi 6854	. . . . . . . . . . . . . . . 16 |- (1 < K <-> 0 < (K - 1))
73		olc 290	. . . . . . . . . . . . . . . 16 |- (0 < (K - 1) -> ((K - 1) < 0 \/ 0 < (K - 1)))
74	72, 73	sylbi 216	. . . . . . . . . . . . . . 15 |- (1 < K -> ((K - 1) < 0 \/ 0 < (K - 1)))
75	74, 55	syl 12	. . . . . . . . . . . . . 14 |- (1 < K -> (0 _C (K - 1)) = 0)
76	75	eqcomd 1889	. . . . . . . . . . . . 13 |- (1 < K -> 0 = (0 _C (K - 1)))
77	76	opreq2d 4898	. . . . . . . . . . . 12 |- (1 < K -> (0 + 0) = (0 + (0 _C (K - 1))))
78	77, 40	syl5eqr 1942	. . . . . . . . . . 11 |- (1 < K -> 0 = (0 + (0 _C (K - 1))))
79	71, 78	eqtr2d 1926	. . . . . . . . . 10 |- (1 < K -> (0 + (0 _C (K - 1))) = (1 _C K))
80		opreq1 4889	. . . . . . . . . . . . . . . 16 |- (1 = K -> (1 - 1) = (K - 1))
81	5	subidi 6551	. . . . . . . . . . . . . . . 16 |- (1 - 1) = 0
82	80, 81	syl5eqr 1942	. . . . . . . . . . . . . . 15 |- (1 = K -> 0 = (K - 1))
83	82	opreq2d 4898	. . . . . . . . . . . . . 14 |- (1 = K -> (0 _C 0) = (0 _C (K - 1)))
84	83, 10	syl5eqr 1942	. . . . . . . . . . . . 13 |- (1 = K -> 1 = (0 _C (K - 1)))
85	84	opreq2d 4898	. . . . . . . . . . . 12 |- (1 = K -> (0 + 1) = (0 + (0 _C (K - 1))))
86	85, 67	syl5eqr 1942	. . . . . . . . . . 11 |- (1 = K -> 1 = (0 + (0 _C (K - 1))))
87		opreq2 4890	. . . . . . . . . . . 12 |- (1 = K -> (1 _C 1) = (1 _C K))
88		bcnn 8216	. . . . . . . . . . . . 13 |- (1 e. NN0 -> (1 _C 1) = 1)
89	30, 88	ax-mp 7	. . . . . . . . . . . 12 |- (1 _C 1) = 1
90	87, 89	syl5eqr 1942	. . . . . . . . . . 11 |- (1 = K -> 1 = (1 _C K))
91	86, 90	eqtr3d 1927	. . . . . . . . . 10 |- (1 = K -> (0 + (0 _C (K - 1))) = (1 _C K))
92	79, 91	jaoi 368	. . . . . . . . 9 |- ((1 < K \/ 1 = K) -> (0 + (0 _C (K - 1))) = (1 _C K))
93	70, 92	sylbi 216	. . . . . . . 8 |- (0 < K -> (0 + (0 _C (K - 1))) = (1 _C K))
94	63, 93	eqtrd 1925	. . . . . . 7 |- (0 < K -> ((0 _C K) + (0 _C (K - 1))) = (1 _C K))
95	61, 94	jaoi 368	. . . . . 6 |- ((K < 0 \/ 0 < K) -> ((0 _C K) + (0 _C (K - 1))) = (1 _C K))
96	38, 95	sylbi 216	. . . . 5 |- (K =/= 0 -> ((0 _C K) + (0 _C (K - 1))) = (1 _C K))
97	34, 96	pm2.61ine 2089	. . . 4 |- ((0 _C K) + (0 _C (K - 1))) = (1 _C K)
98		opreq1 4889	. . . . 5 |- (N = 0 -> (N _C K) = (0 _C K))
99		opreq1 4889	. . . . 5 |- (N = 0 -> (N _C (K - 1)) = (0 _C (K - 1)))
100	98, 99	opreq12d 4900	. . . 4 |- (N = 0 -> ((N _C K) + (N _C (K - 1))) = ((0 _C K) + (0 _C (K - 1))))
101		opreq1 4889	. . . . . 6 |- (N = 0 -> (N + 1) = (0 + 1))
102	101, 67	syl6eq 1944	. . . . 5 |- (N = 0 -> (N + 1) = 1)
103	102	opreq1d 4897	. . . 4 |- (N = 0 -> ((N + 1) _C K) = (1 _C K))
104	97, 100, 103	3eqtr4a 1954	. . 3 |- (N = 0 -> ((N _C K) + (N _C (K - 1))) = ((N + 1) _C K))
105	4, 104	syl 12	. 2 |- (-. N e. NN -> ((N _C K) + (N _C (K - 1))) = ((N + 1) _C K))
106		znnnlt1 7365	. . . . 5 |- (K e. ZZ -> (-. K e. NN <-> K < 1))
107	35, 106	ax-mp 7	. . . 4 |- (-. K e. NN <-> K < 1)
108		zleltp1 7391	. . . . . 6 |- ((K e. ZZ /\ 0 e. ZZ) -> (K <_ 0 <-> K < (0 + 1)))
109	35, 64, 108	mp2an 761	. . . . 5 |- (K <_ 0 <-> K < (0 + 1))
110	36, 37	leloei 6750	. . . . 5 |- (K <_ 0 <-> (K < 0 \/ K = 0))
111	67	breq2i 3346	. . . . 5 |- (K < (0 + 1) <-> K < 1)
112	109, 110, 111	3bitr3ri 199	. . . 4 |- (K < 1 <-> (K < 0 \/ K = 0))
113	107, 112	bitri 190	. . 3 |- (-. K e. NN <-> (K < 0 \/ K = 0))
114		bcval4 8213	. . . . . . . 8 |- ((N e. NN0 /\ K e. ZZ /\ (K < 0 \/ N < K)) -> (N _C K) = 0)
115	1, 35, 114	mp3an12 1181	. . . . . . 7 |- ((K < 0 \/ N < K) -> (N _C K) = 0)
116	115	orcs 296	. . . . . 6 |- (K < 0 -> (N _C K) = 0)
117		orc 291	. . . . . . 7 |- ((K - 1) < 0 -> ((K - 1) < 0 \/ N < (K - 1)))
118		bcval4 8213	. . . . . . . 8 |- ((N e. NN0 /\ (K - 1) e. ZZ /\ ((K - 1) < 0 \/ N < (K - 1))) -> (N _C (K - 1)) = 0)
119	1, 53, 118	mp3an12 1181	. . . . . . 7 |- (((K - 1) < 0 \/ N < (K - 1)) -> (N _C (K - 1)) = 0)
120	50, 117, 119	3syl 24	. . . . . 6 |- (K < 0 -> (N _C (K - 1)) = 0)
121	116, 120	opreq12d 4900	. . . . 5 |- (K < 0 -> ((N _C K) + (N _C (K - 1))) = (0 + 0))
122	1, 30	nn0addcli 7330	. . . . . . 7 |- (N + 1) e. NN0
123		bcval4 8213	. . . . . . 7 |- (((N + 1) e. NN0 /\ K e. ZZ /\ (K < 0 \/ (N + 1) < K)) -> ((N + 1) _C K) = 0)
124	122, 35, 123	mp3an12 1181	. . . . . 6 |- ((K < 0 \/ (N + 1) < K) -> ((N + 1) _C K) = 0)
125	124	orcs 296	. . . . 5 |- (K < 0 -> ((N + 1) _C K) = 0)
126	40, 121, 125	3eqtr4a 1954	. . . 4 |- (K < 0 -> ((N _C K) + (N _C (K - 1))) = ((N + 1) _C K))
127		opreq2 4890	. . . . . . . 8 |- (K = 0 -> (N _C K) = (N _C 0))
128		bcn0 8215	. . . . . . . . 9 |- (N e. NN0 -> (N _C 0) = 1)
129	1, 128	ax-mp 7	. . . . . . . 8 |- (N _C 0) = 1
130	127, 129	syl6eq 1944	. . . . . . 7 |- (K = 0 -> (N _C K) = 1)
131	13, 23	eqbrtrri 3358	. . . . . . . . 9 |- (0 - 1) < 0
132	12, 131	syl6eqbr 3374	. . . . . . . 8 |- (K = 0 -> (K - 1) < 0)
133	132, 117, 119	3syl 24	. . . . . . 7 |- (K = 0 -> (N _C (K - 1)) = 0)
134	130, 133	opreq12d 4900	. . . . . 6 |- (K = 0 -> ((N _C K) + (N _C (K - 1))) = (1 + 0))
135	134, 6	syl6eq 1944	. . . . 5 |- (K = 0 -> ((N _C K) + (N _C (K - 1))) = 1)
136		opreq2 4890	. . . . . 6 |- (K = 0 -> ((N + 1) _C K) = ((N + 1) _C 0))
137		bcn0 8215	. . . . . . 7 |- ((N + 1) e. NN0 -> ((N + 1) _C 0) = 1)
138	122, 137	ax-mp 7	. . . . . 6 |- ((N + 1) _C 0) = 1
139	136, 138	syl6req 1945	. . . . 5 |- (K = 0 -> 1 = ((N + 1) _C K))
140	135, 139	eqtrd 1925	. . . 4 |- (K = 0 -> ((N _C K) + (N _C (K - 1))) = ((N + 1) _C K))
141	126, 140	jaoi 368	. . 3 |- ((K < 0 \/ K = 0) -> ((N _C K) + (N _C (K - 1))) = ((N + 1) _C K))
142	113, 141	sylbi 216	. 2 |- (-. K e. NN -> ((N _C K) + (N _C (K - 1))) = ((N + 1) _C K))
143	1	nn0rei 7319	. . . 4 |- N e. RR
144	143, 36	ltnlei 6754	. . 3 |- (N < K <-> -. K <_ N)
145	115	olcs 297	. . . . 5 |- (N < K -> (N _C K) = 0)
146	145	opreq1d 4897	. . . 4 |- (N < K -> ((N _C K) + (N _C (K - 1))) = (0 + (N _C (K - 1))))
147		nn0z 7363	. . . . . . . 8 |- (N e. NN0 -> N e. ZZ)
148	1, 147	ax-mp 7	. . . . . . 7 |- N e. ZZ
149		zltp1le 7390	. . . . . . 7 |- ((N e. ZZ /\ K e. ZZ) -> (N < K <-> (N + 1) <_ K))
150	148, 35, 149	mp2an 761	. . . . . 6 |- (N < K <-> (N + 1) <_ K)
151	143, 20	readdcli 6487	. . . . . . 7 |- (N + 1) e. RR
152	151, 36	leloei 6750	. . . . . 6 |- ((N + 1) <_ K <-> ((N + 1) < K \/ (N + 1) = K))
153	150, 152	bitri 190	. . . . 5 |- (N < K <-> ((N + 1) < K \/ (N + 1) = K))
154	143, 20, 36	ltaddsubi 6823	. . . . . . . . . 10 |- ((N + 1) < K <-> N < (K - 1))
155		olc 290	. . . . . . . . . 10 |- (N < (K - 1) -> ((K - 1) < 0 \/ N < (K - 1)))
156	154, 155	sylbi 216	. . . . . . . . 9 |- ((N + 1) < K -> ((K - 1) < 0 \/ N < (K - 1)))
157	156, 119	syl 12	. . . . . . . 8 |- ((N + 1) < K -> (N _C (K - 1)) = 0)
158	157	opreq2d 4898	. . . . . . 7 |- ((N + 1) < K -> (0 + (N _C (K - 1))) = (0 + 0))
159	124	olcs 297	. . . . . . 7 |- ((N + 1) < K -> ((N + 1) _C K) = 0)
160	40, 158, 159	3eqtr4a 1954	. . . . . 6 |- ((N + 1) < K -> (0 + (N _C (K - 1))) = ((N + 1) _C K))
161		opreq1 4889	. . . . . . . . . . . 12 |- ((N + 1) = K -> ((N + 1) - 1) = (K - 1))
162	1	nn0cni 7320	. . . . . . . . . . . . 13 |- N e. CC
163		pncan 6557	. . . . . . . . . . . . 13 |- ((N e. CC /\ 1 e. CC) -> ((N + 1) - 1) = N)
164	162, 5, 163	mp2an 761	. . . . . . . . . . . 12 |- ((N + 1) - 1) = N
165	161, 164	syl5eqr 1942	. . . . . . . . . . 11 |- ((N + 1) = K -> N = (K - 1))
166	165	opreq2d 4898	. . . . . . . . . 10 |- ((N + 1) = K -> (N _C N) = (N _C (K - 1)))
167		bcnn 8216	. . . . . . . . . . 11 |- (N e. NN0 -> (N _C N) = 1)
168	1, 167	ax-mp 7	. . . . . . . . . 10 |- (N _C N) = 1
169	166, 168	syl5eqr 1942	. . . . . . . . 9 |- ((N + 1) = K -> 1 = (N _C (K - 1)))
170	169	opreq2d 4898	. . . . . . . 8 |- ((N + 1) = K -> (0 + 1) = (0 + (N _C (K - 1))))
171	170, 67	syl5eqr 1942	. . . . . . 7 |- ((N + 1) = K -> 1 = (0 + (N _C (K - 1))))
172		opreq2 4890	. . . . . . . 8 |- ((N + 1) = K -> ((N + 1) _C (N + 1)) = ((N + 1) _C K))
173		bcnn 8216	. . . . . . . . 9 |- ((N + 1) e. NN0 -> ((N + 1) _C (N + 1)) = 1)
174	122, 173	ax-mp 7	. . . . . . . 8 |- ((N + 1) _C (N + 1)) = 1
175	172, 174	syl5eqr 1942	. . . . . . 7 |- ((N + 1) = K -> 1 = ((N + 1) _C K))
176	171, 175	eqtr3d 1927	. . . . . 6 |- ((N + 1) = K -> (0 + (N _C (K - 1))) = ((N + 1) _C K))
177	160, 176	jaoi 368	. . . . 5 |- (((N + 1) < K \/ (N + 1) = K) -> (0 + (N _C (K - 1))) = ((N + 1) _C K))
178	153, 177	sylbi 216	. . . 4 |- (N < K -> (0 + (N _C (K - 1))) = ((N + 1) _C K))
179	146, 178	eqtrd 1925	. . 3 |- (N < K -> ((N _C K) + (N _C (K - 1))) = ((N + 1) _C K))
180	144, 179	sylbir 218	. 2 |- (-. K <_ N -> ((N _C K) + (N _C (K - 1))) = ((N + 1) _C K))
181		bcpasc2 8220	. 2 |- ((N e. NN /\ K e. NN /\ K <_ N) -> ((N _C K) + (N _C (K - 1))) = ((N + 1) _C K))
182	105, 142, 180, 181	3ecase 1199	1 |- ((N _C K) + (N _C (K - 1))) = ((N + 1) _C K)

Syntax hints:  -. wn 2   <-> wb 163   \/ wo 239   = wceq 1298   e. wcel 1300   =/= wne 2017   class class class wbr 3338  (class class class)co 4884  CCcc 6384  RRcr 6385  0cc0 6386  1c1 6387   + caddc 6389   - cmin 6445  -ucneg 6446   <_ cle 6448  NNcn 6449  NN0cn0 6450  ZZcz 6451   < clt 6653   _C cbc 8208

This theorem is referenced by:  bcpasc 8222

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-n0 7309  df-z 7345  df-seq1 7721  df-fac 8184  df-bc 8209

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