bcval5 - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > bcval5		Unicode version

Theorem bcval5 10908

Description: Write out the top and bottom parts of the binomial coefficient

explicitly. In this form, it is valid even for

, although it is no longer valid for nonpositive

. (Contributed by Mario Carneiro, 22-May-2014.) (Revised by Jim Kingdon, 23-Apr-2023.)

**Assertion**
Ref	Expression
bcval5	$/\$

Proof of Theorem bcval5 Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		bcval2 10895	. . . 4
2	1	adantl 277	. . 3 $/\$ $/\$
3		simprl 529	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
4		simprr 531	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
5	3, 4	mulcld 8093	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
6		simpr1 1006	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
7		simpr2 1007	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
8		simpr3 1008	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
9	6, 7, 8	mulassd 8096	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
10		simpll 527	. . . . . . . . . . . . 13 $/\$ $/\$
11	10	nn0zd 9493	. . . . . . . . . . . 12 $/\$ $/\$
12		simplr 528	. . . . . . . . . . . . 13 $/\$ $/\$
13	12	nnzd 9494	. . . . . . . . . . . 12 $/\$ $/\$
14	11, 13	zsubcld 9500	. . . . . . . . . . 11 $/\$ $/\$
15	14	peano2zd 9498	. . . . . . . . . 10 $/\$ $/\$
16		1red 8087	. . . . . . . . . . . 12 $/\$ $/\$
17	12	nnred 9049	. . . . . . . . . . . 12 $/\$ $/\$
18	10	nn0red 9349	. . . . . . . . . . . 12 $/\$ $/\$
19	12	nnge1d 9079	. . . . . . . . . . . 12 $/\$ $/\$
20	16, 17, 18, 19	lesub2dd 8635	. . . . . . . . . . 11 $/\$ $/\$
21	14	zred 9495	. . . . . . . . . . . 12 $/\$ $/\$
22		leaddsub 8511	. . . . . . . . . . . 12 $/\$ $/\$
23	21, 16, 18, 22	syl3anc 1250	. . . . . . . . . . 11 $/\$ $/\$
24	20, 23	mpbird 167	. . . . . . . . . 10 $/\$ $/\$
25		eluz2 9654	. . . . . . . . . 10 $/\$ $/\$
26	15, 11, 24, 25	syl3anbrc 1184	. . . . . . . . 9 $/\$ $/\$
27	26	adantrr 479	. . . . . . . 8 $/\$ $/\$ $/\$
28		simprr 531	. . . . . . . . 9 $/\$ $/\$ $/\$
29		nnuz 9684	. . . . . . . . 9
30	28, 29	eleqtrdi 2298	. . . . . . . 8 $/\$ $/\$ $/\$
31		fvi 5636	. . . . . . . . . 10
32	31	elv 2776	. . . . . . . . 9
33		eluzelcn 9659	. . . . . . . . . 10
34	33	adantl 277	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
35	32, 34	eqeltrid 2292	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
36	5, 9, 27, 30, 35	seq3split 10633	. . . . . . 7 $/\$ $/\$ $/\$
37		elfzuz3 10144	. . . . . . . . . . 11
38	37	adantl 277	. . . . . . . . . 10 $/\$ $/\$
39		eluznn 9721	. . . . . . . . . 10 $/\$
40	12, 38, 39	syl2anc 411	. . . . . . . . 9 $/\$ $/\$
41	40	adantrr 479	. . . . . . . 8 $/\$ $/\$ $/\$
42		facnn 10872	. . . . . . . 8
43	41, 42	syl 14	. . . . . . 7 $/\$ $/\$ $/\$
44		facnn 10872	. . . . . . . . 9
45	28, 44	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$
46	45	oveq1d 5959	. . . . . . 7 $/\$ $/\$ $/\$
47	36, 43, 46	3eqtr4d 2248	. . . . . 6 $/\$ $/\$ $/\$
48	47	expr 375	. . . . 5 $/\$ $/\$
49	10	faccld 10881	. . . . . . . . 9 $/\$ $/\$
50	49	nncnd 9050	. . . . . . . 8 $/\$ $/\$
51	50	mulid2d 8091	. . . . . . 7 $/\$ $/\$
52	40, 42	syl 14	. . . . . . . 8 $/\$ $/\$
53	52	oveq2d 5960	. . . . . . 7 $/\$ $/\$
54	51, 53	eqtr3d 2240	. . . . . 6 $/\$ $/\$
55		fveq2 5576	. . . . . . . . 9
56		fac0 10873	. . . . . . . . 9
57	55, 56	eqtrdi 2254	. . . . . . . 8
58		oveq1 5951	. . . . . . . . . . 11
59		0p1e1 9150	. . . . . . . . . . 11
60	58, 59	eqtrdi 2254	. . . . . . . . . 10
61	60	seqeq1d 10598	. . . . . . . . 9
62	61	fveq1d 5578	. . . . . . . 8
63	57, 62	oveq12d 5962	. . . . . . 7
64	63	eqeq2d 2217	. . . . . 6
65	54, 64	syl5ibrcom 157	. . . . 5 $/\$ $/\$
66		fznn0sub 10179	. . . . . . 7
67	66	adantl 277	. . . . . 6 $/\$ $/\$
68		elnn0 9297	. . . . . 6 $\/$
69	67, 68	sylib 122	. . . . 5 $/\$ $/\$ $\/$
70	48, 65, 69	mpjaod 720	. . . 4 $/\$ $/\$
71	70	oveq1d 5959	. . 3 $/\$ $/\$
72		eqid 2205	. . . . . 6
73		fvi 5636	. . . . . . . 8
74	73	elv 2776	. . . . . . 7
75		eluzelcn 9659	. . . . . . . 8
76	75	adantl 277	. . . . . . 7 $/\$ $/\$ $/\$
77	74, 76	eqeltrid 2292	. . . . . 6 $/\$ $/\$ $/\$
78		mulcl 8052	. . . . . . 7 $/\$
79	78	adantl 277	. . . . . 6 $/\$ $/\$ $/\$ $/\$
80	72, 15, 77, 79	seqf 10609	. . . . 5 $/\$ $/\$
81	80, 26	ffvelcdmd 5716	. . . 4 $/\$ $/\$
82	12	nnnn0d 9348	. . . . . 6 $/\$ $/\$
83	82	faccld 10881	. . . . 5 $/\$ $/\$
84	83	nncnd 9050	. . . 4 $/\$ $/\$
85	67	faccld 10881	. . . . 5 $/\$ $/\$
86	85	nncnd 9050	. . . 4 $/\$ $/\$
87	83	nnap0d 9082	. . . 4 $/\$ $/\$ #
88	85	nnap0d 9082	. . . 4 $/\$ $/\$ #
89	81, 84, 86, 87, 88	divcanap5d 8890	. . 3 $/\$ $/\$
90	2, 71, 89	3eqtrd 2242	. 2 $/\$ $/\$
91		simplr 528	. . . . . . 7 $/\$ $/\$
92	91	nnnn0d 9348	. . . . . 6 $/\$ $/\$
93	92	faccld 10881	. . . . 5 $/\$ $/\$
94	93	nncnd 9050	. . . 4 $/\$ $/\$
95	93	nnap0d 9082	. . . 4 $/\$ $/\$ #
96	94, 95	div0apd 8860	. . 3 $/\$ $/\$
97		mulcl 8052	. . . . . 6 $/\$
98	97	adantl 277	. . . . 5 $/\$ $/\$ $/\$ $/\$
99		eluzelcn 9659	. . . . . . 7
100	99	adantl 277	. . . . . 6 $/\$ $/\$ $/\$
101	32, 100	eqeltrid 2292	. . . . 5 $/\$ $/\$ $/\$
102		simpr 110	. . . . . 6 $/\$ $/\$ $/\$
103	102	mul02d 8464	. . . . 5 $/\$ $/\$ $/\$
104	102	mul01d 8465	. . . . 5 $/\$ $/\$ $/\$
105		simpr 110	. . . . . . . . 9 $/\$ $/\$
106		nn0uz 9683	. . . . . . . . . . . 12
107	92, 106	eleqtrdi 2298	. . . . . . . . . . 11 $/\$ $/\$
108		simpll 527	. . . . . . . . . . . 12 $/\$ $/\$
109	108	nn0zd 9493	. . . . . . . . . . 11 $/\$ $/\$
110		elfz5 10139	. . . . . . . . . . 11 $/\$
111	107, 109, 110	syl2anc 411	. . . . . . . . . 10 $/\$ $/\$
112		nn0re 9304	. . . . . . . . . . . 12
113	112	ad2antrr 488	. . . . . . . . . . 11 $/\$ $/\$
114		nnre 9043	. . . . . . . . . . . 12
115	114	ad2antlr 489	. . . . . . . . . . 11 $/\$ $/\$
116	113, 115	subge0d 8608	. . . . . . . . . 10 $/\$ $/\$
117	111, 116	bitr4d 191	. . . . . . . . 9 $/\$ $/\$
118	105, 117	mtbid 674	. . . . . . . 8 $/\$ $/\$
119		simpl 109	. . . . . . . . . . . 12 $/\$
120	119	nn0zd 9493	. . . . . . . . . . 11 $/\$
121		simpr 110	. . . . . . . . . . . 12 $/\$
122	121	nnzd 9494	. . . . . . . . . . 11 $/\$
123	120, 122	zsubcld 9500	. . . . . . . . . 10 $/\$
124	123	adantr 276	. . . . . . . . 9 $/\$ $/\$
125		0z 9383	. . . . . . . . 9
126		zltnle 9418	. . . . . . . . 9 $/\$
127	124, 125, 126	sylancl 413	. . . . . . . 8 $/\$ $/\$
128	118, 127	mpbird 167	. . . . . . 7 $/\$ $/\$
129		zltp1le 9427	. . . . . . . 8 $/\$
130	124, 125, 129	sylancl 413	. . . . . . 7 $/\$ $/\$
131	128, 130	mpbid 147	. . . . . 6 $/\$ $/\$
132		nn0ge0 9320	. . . . . . 7
133	132	ad2antrr 488	. . . . . 6 $/\$ $/\$
134		0zd 9384	. . . . . . 7 $/\$ $/\$
135	124	peano2zd 9498	. . . . . . 7 $/\$ $/\$
136		elfz 10136	. . . . . . 7 $/\$ $/\$ $/\$
137	134, 135, 109, 136	syl3anc 1250	. . . . . 6 $/\$ $/\$ $/\$
138	131, 133, 137	mpbir2and 947	. . . . 5 $/\$ $/\$
139		0cn 8064	. . . . . 6
140		fvi 5636	. . . . . 6
141	139, 140	mp1i 10	. . . . 5 $/\$ $/\$
142	98, 101, 103, 104, 138, 141	seq3z 10673	. . . 4 $/\$ $/\$
143	142	oveq1d 5959	. . 3 $/\$ $/\$
144		nnz 9391	. . . . 5
145		bcval3 10896	. . . . 5 $/\$ $/\$
146	144, 145	syl3an2 1284	. . . 4 $/\$ $/\$
147	146	3expa 1206	. . 3 $/\$ $/\$
148	96, 143, 147	3eqtr4rd 2249	. 2 $/\$ $/\$
149		0zd 9384	. . . 4 $/\$
150		fzdcel 10162	. . . 4 $/\$ $/\$ DECID
151	122, 149, 120, 150	syl3anc 1250	. . 3 $/\$ DECID
152		exmiddc 838	. . 3 DECID $\/$
153	151, 152	syl 14	. 2 $/\$ $\/$
154	90, 148, 153	mpjaodan 800	1 $/\$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104

wb 105 $\/$ wo 710 DECID wdc 836 $/\$ w3a 981

wceq 1373

wcel 2176

cvv 2772 class class class wbr 4044

cid 4335

cfv 5271 (class class class)co 5944

cc 7923

cr 7924

cc0 7925

c1 7926

caddc 7928

cmul 7930

clt 8107

cle 8108

cmin 8243

cdiv 8745

cn 9036

cn0 9295

cz 9372

cuz 9648

cfz 10130

cseq 10592

cfa 10870

cbc 10892

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 711 ax-5 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-13 2178 ax-14 2179 ax-ext 2187 ax-coll 4159 ax-sep 4162 ax-nul 4170 ax-pow 4218 ax-pr 4253 ax-un 4480 ax-setind 4585 ax-iinf 4636 ax-cnex 8016 ax-resscn 8017 ax-1cn 8018 ax-1re 8019 ax-icn 8020 ax-addcl 8021 ax-addrcl 8022 ax-mulcl 8023 ax-mulrcl 8024 ax-addcom 8025 ax-mulcom 8026 ax-addass 8027 ax-mulass 8028 ax-distr 8029 ax-i2m1 8030 ax-0lt1 8031 ax-1rid 8032 ax-0id 8033 ax-rnegex 8034 ax-precex 8035 ax-cnre 8036 ax-pre-ltirr 8037 ax-pre-ltwlin 8038 ax-pre-lttrn 8039 ax-pre-apti 8040 ax-pre-ltadd 8041 ax-pre-mulgt0 8042 ax-pre-mulext 8043

This theorem depends on definitions: df-bi 117 df-dc 837 df-3or 982 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1484 df-sb 1786 df-eu 2057 df-mo 2058 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-ne 2377 df-nel 2472 df-ral 2489 df-rex 2490 df-reu 2491 df-rmo 2492 df-rab 2493 df-v 2774 df-sbc 2999 df-csb 3094 df-dif 3168 df-un 3170 df-in 3172 df-ss 3179 df-nul 3461 df-if 3572 df-pw 3618 df-sn 3639 df-pr 3640 df-op 3642 df-uni 3851 df-int 3886 df-iun 3929 df-br 4045 df-opab 4106 df-mpt 4107 df-tr 4143 df-id 4340 df-po 4343 df-iso 4344 df-iord 4413 df-on 4415 df-ilim 4416 df-suc 4418 df-iom 4639 df-xp 4681 df-rel 4682 df-cnv 4683 df-co 4684 df-dm 4685 df-rn 4686 df-res 4687 df-ima 4688 df-iota 5232 df-fun 5273 df-fn 5274 df-f 5275 df-f1 5276 df-fo 5277 df-f1o 5278 df-fv 5279 df-riota 5899 df-ov 5947 df-oprab 5948 df-mpo 5949 df-1st 6226 df-2nd 6227 df-recs 6391 df-frec 6477 df-pnf 8109 df-mnf 8110 df-xr 8111 df-ltxr 8112 df-le 8113 df-sub 8245 df-neg 8246 df-reap 8648 df-ap 8655 df-div 8746 df-inn 9037 df-n0 9296 df-z 9373 df-uz 9649 df-q 9741 df-fz 10131 df-seqfrec 10593 df-fac 10871 df-bc 10893

This theorem is referenced by: bcn2 10909

W3C validator