bcval5 - Intuitionistic Logic Explorer

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Theorem bcval5 10834

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, although it is no longer valid for nonpositive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 10821	. . . 4
2	1	adantl 277	. . 3 $/\$ $/\$
3		simprl 529	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
4		simprr 531	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
5	3, 4	mulcld 8040	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
6		simpr1 1005	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
7		simpr2 1006	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
8		simpr3 1007	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
9	6, 7, 8	mulassd 8043	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
10		simpll 527	. . . . . . . . . . . . 13 $/\$ $/\$
11	10	nn0zd 9437	. . . . . . . . . . . 12 $/\$ $/\$
12		simplr 528	. . . . . . . . . . . . 13 $/\$ $/\$
13	12	nnzd 9438	. . . . . . . . . . . 12 $/\$ $/\$
14	11, 13	zsubcld 9444	. . . . . . . . . . 11 $/\$ $/\$
15	14	peano2zd 9442	. . . . . . . . . 10 $/\$ $/\$
16		1red 8034	. . . . . . . . . . . 12 $/\$ $/\$
17	12	nnred 8995	. . . . . . . . . . . 12 $/\$ $/\$
18	10	nn0red 9294	. . . . . . . . . . . 12 $/\$ $/\$
19	12	nnge1d 9025	. . . . . . . . . . . 12 $/\$ $/\$
20	16, 17, 18, 19	lesub2dd 8581	. . . . . . . . . . 11 $/\$ $/\$
21	14	zred 9439	. . . . . . . . . . . 12 $/\$ $/\$
22		leaddsub 8457	. . . . . . . . . . . 12 $/\$ $/\$
23	21, 16, 18, 22	syl3anc 1249	. . . . . . . . . . 11 $/\$ $/\$
24	20, 23	mpbird 167	. . . . . . . . . 10 $/\$ $/\$
25		eluz2 9598	. . . . . . . . . 10 $/\$ $/\$
26	15, 11, 24, 25	syl3anbrc 1183	. . . . . . . . 9 $/\$ $/\$
27	26	adantrr 479	. . . . . . . 8 $/\$ $/\$ $/\$
28		simprr 531	. . . . . . . . 9 $/\$ $/\$ $/\$
29		nnuz 9628	. . . . . . . . 9
30	28, 29	eleqtrdi 2286	. . . . . . . 8 $/\$ $/\$ $/\$
31		fvi 5614	. . . . . . . . . 10
32	31	elv 2764	. . . . . . . . 9
33		eluzelcn 9603	. . . . . . . . . 10
34	33	adantl 277	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
35	32, 34	eqeltrid 2280	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
36	5, 9, 27, 30, 35	seq3split 10559	. . . . . . 7 $/\$ $/\$ $/\$
37		elfzuz3 10088	. . . . . . . . . . 11
38	37	adantl 277	. . . . . . . . . 10 $/\$ $/\$
39		eluznn 9665	. . . . . . . . . 10 $/\$
40	12, 38, 39	syl2anc 411	. . . . . . . . 9 $/\$ $/\$
41	40	adantrr 479	. . . . . . . 8 $/\$ $/\$ $/\$
42		facnn 10798	. . . . . . . 8
43	41, 42	syl 14	. . . . . . 7 $/\$ $/\$ $/\$
44		facnn 10798	. . . . . . . . 9
45	28, 44	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$
46	45	oveq1d 5933	. . . . . . 7 $/\$ $/\$ $/\$
47	36, 43, 46	3eqtr4d 2236	. . . . . 6 $/\$ $/\$ $/\$
48	47	expr 375	. . . . 5 $/\$ $/\$
49	10	faccld 10807	. . . . . . . . 9 $/\$ $/\$
50	49	nncnd 8996	. . . . . . . 8 $/\$ $/\$
51	50	mulid2d 8038	. . . . . . 7 $/\$ $/\$
52	40, 42	syl 14	. . . . . . . 8 $/\$ $/\$
53	52	oveq2d 5934	. . . . . . 7 $/\$ $/\$
54	51, 53	eqtr3d 2228	. . . . . 6 $/\$ $/\$
55		fveq2 5554	. . . . . . . . 9
56		fac0 10799	. . . . . . . . 9
57	55, 56	eqtrdi 2242	. . . . . . . 8
58		oveq1 5925	. . . . . . . . . . 11
59		0p1e1 9096	. . . . . . . . . . 11
60	58, 59	eqtrdi 2242	. . . . . . . . . 10
61	60	seqeq1d 10524	. . . . . . . . 9
62	61	fveq1d 5556	. . . . . . . 8
63	57, 62	oveq12d 5936	. . . . . . 7
64	63	eqeq2d 2205	. . . . . 6
65	54, 64	syl5ibrcom 157	. . . . 5 $/\$ $/\$
66		fznn0sub 10123	. . . . . . 7
67	66	adantl 277	. . . . . 6 $/\$ $/\$
68		elnn0 9242	. . . . . 6 $\/$
69	67, 68	sylib 122	. . . . 5 $/\$ $/\$ $\/$
70	48, 65, 69	mpjaod 719	. . . 4 $/\$ $/\$
71	70	oveq1d 5933	. . 3 $/\$ $/\$
72		eqid 2193	. . . . . 6
73		fvi 5614	. . . . . . . 8
74	73	elv 2764	. . . . . . 7
75		eluzelcn 9603	. . . . . . . 8
76	75	adantl 277	. . . . . . 7 $/\$ $/\$ $/\$
77	74, 76	eqeltrid 2280	. . . . . 6 $/\$ $/\$ $/\$
78		mulcl 7999	. . . . . . 7 $/\$
79	78	adantl 277	. . . . . 6 $/\$ $/\$ $/\$ $/\$
80	72, 15, 77, 79	seqf 10535	. . . . 5 $/\$ $/\$
81	80, 26	ffvelcdmd 5694	. . . 4 $/\$ $/\$
82	12	nnnn0d 9293	. . . . . 6 $/\$ $/\$
83	82	faccld 10807	. . . . 5 $/\$ $/\$
84	83	nncnd 8996	. . . 4 $/\$ $/\$
85	67	faccld 10807	. . . . 5 $/\$ $/\$
86	85	nncnd 8996	. . . 4 $/\$ $/\$
87	83	nnap0d 9028	. . . 4 $/\$ $/\$ #
88	85	nnap0d 9028	. . . 4 $/\$ $/\$ #
89	81, 84, 86, 87, 88	divcanap5d 8836	. . 3 $/\$ $/\$
90	2, 71, 89	3eqtrd 2230	. 2 $/\$ $/\$
91		simplr 528	. . . . . . 7 $/\$ $/\$
92	91	nnnn0d 9293	. . . . . 6 $/\$ $/\$
93	92	faccld 10807	. . . . 5 $/\$ $/\$
94	93	nncnd 8996	. . . 4 $/\$ $/\$
95	93	nnap0d 9028	. . . 4 $/\$ $/\$ #
96	94, 95	div0apd 8806	. . 3 $/\$ $/\$
97		mulcl 7999	. . . . . 6 $/\$
98	97	adantl 277	. . . . 5 $/\$ $/\$ $/\$ $/\$
99		eluzelcn 9603	. . . . . . 7
100	99	adantl 277	. . . . . 6 $/\$ $/\$ $/\$
101	32, 100	eqeltrid 2280	. . . . 5 $/\$ $/\$ $/\$
102		simpr 110	. . . . . 6 $/\$ $/\$ $/\$
103	102	mul02d 8411	. . . . 5 $/\$ $/\$ $/\$
104	102	mul01d 8412	. . . . 5 $/\$ $/\$ $/\$
105		simpr 110	. . . . . . . . 9 $/\$ $/\$
106		nn0uz 9627	. . . . . . . . . . . 12
107	92, 106	eleqtrdi 2286	. . . . . . . . . . 11 $/\$ $/\$
108		simpll 527	. . . . . . . . . . . 12 $/\$ $/\$
109	108	nn0zd 9437	. . . . . . . . . . 11 $/\$ $/\$
110		elfz5 10083	. . . . . . . . . . 11 $/\$
111	107, 109, 110	syl2anc 411	. . . . . . . . . 10 $/\$ $/\$
112		nn0re 9249	. . . . . . . . . . . 12
113	112	ad2antrr 488	. . . . . . . . . . 11 $/\$ $/\$
114		nnre 8989	. . . . . . . . . . . 12
115	114	ad2antlr 489	. . . . . . . . . . 11 $/\$ $/\$
116	113, 115	subge0d 8554	. . . . . . . . . 10 $/\$ $/\$
117	111, 116	bitr4d 191	. . . . . . . . 9 $/\$ $/\$
118	105, 117	mtbid 673	. . . . . . . 8 $/\$ $/\$
119		simpl 109	. . . . . . . . . . . 12 $/\$
120	119	nn0zd 9437	. . . . . . . . . . 11 $/\$
121		simpr 110	. . . . . . . . . . . 12 $/\$
122	121	nnzd 9438	. . . . . . . . . . 11 $/\$
123	120, 122	zsubcld 9444	. . . . . . . . . 10 $/\$
124	123	adantr 276	. . . . . . . . 9 $/\$ $/\$
125		0z 9328	. . . . . . . . 9
126		zltnle 9363	. . . . . . . . 9 $/\$
127	124, 125, 126	sylancl 413	. . . . . . . 8 $/\$ $/\$
128	118, 127	mpbird 167	. . . . . . 7 $/\$ $/\$
129		zltp1le 9371	. . . . . . . 8 $/\$
130	124, 125, 129	sylancl 413	. . . . . . 7 $/\$ $/\$
131	128, 130	mpbid 147	. . . . . 6 $/\$ $/\$
132		nn0ge0 9265	. . . . . . 7
133	132	ad2antrr 488	. . . . . 6 $/\$ $/\$
134		0zd 9329	. . . . . . 7 $/\$ $/\$
135	124	peano2zd 9442	. . . . . . 7 $/\$ $/\$
136		elfz 10080	. . . . . . 7 $/\$ $/\$ $/\$
137	134, 135, 109, 136	syl3anc 1249	. . . . . 6 $/\$ $/\$ $/\$
138	131, 133, 137	mpbir2and 946	. . . . 5 $/\$ $/\$
139		0cn 8011	. . . . . 6
140		fvi 5614	. . . . . 6
141	139, 140	mp1i 10	. . . . 5 $/\$ $/\$
142	98, 101, 103, 104, 138, 141	seq3z 10599	. . . 4 $/\$ $/\$
143	142	oveq1d 5933	. . 3 $/\$ $/\$
144		nnz 9336	. . . . 5
145		bcval3 10822	. . . . 5 $/\$ $/\$
146	144, 145	syl3an2 1283	. . . 4 $/\$ $/\$
147	146	3expa 1205	. . 3 $/\$ $/\$
148	96, 143, 147	3eqtr4rd 2237	. 2 $/\$ $/\$
149		0zd 9329	. . . 4 $/\$
150		fzdcel 10106	. . . 4 $/\$ $/\$ DECID
151	122, 149, 120, 150	syl3anc 1249	. . 3 $/\$ DECID
152		exmiddc 837	. . 3 DECID $\/$
153	151, 152	syl 14	. 2 $/\$ $\/$
154	90, 148, 153	mpjaodan 799	1 $/\$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104

wb 105 $\/$ wo 709 DECID wdc 835 $/\$ w3a 980

wceq 1364

wcel 2164

cvv 2760 class class class wbr 4029

cid 4319

cfv 5254 (class class class)co 5918

cc 7870

cr 7871

cc0 7872

c1 7873

caddc 7875

cmul 7877

clt 8054

cle 8055

cmin 8190

cdiv 8691

cn 8982

cn0 9240

cz 9317

cuz 9592

cfz 10074

cseq 10518

cfa 10796

cbc 10818

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 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-coll 4144 ax-sep 4147 ax-nul 4155 ax-pow 4203 ax-pr 4238 ax-un 4464 ax-setind 4569 ax-iinf 4620 ax-cnex 7963 ax-resscn 7964 ax-1cn 7965 ax-1re 7966 ax-icn 7967 ax-addcl 7968 ax-addrcl 7969 ax-mulcl 7970 ax-mulrcl 7971 ax-addcom 7972 ax-mulcom 7973 ax-addass 7974 ax-mulass 7975 ax-distr 7976 ax-i2m1 7977 ax-0lt1 7978 ax-1rid 7979 ax-0id 7980 ax-rnegex 7981 ax-precex 7982 ax-cnre 7983 ax-pre-ltirr 7984 ax-pre-ltwlin 7985 ax-pre-lttrn 7986 ax-pre-apti 7987 ax-pre-ltadd 7988 ax-pre-mulgt0 7989 ax-pre-mulext 7990

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rmo 2480 df-rab 2481 df-v 2762 df-sbc 2986 df-csb 3081 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-nul 3447 df-if 3558 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-int 3871 df-iun 3914 df-br 4030 df-opab 4091 df-mpt 4092 df-tr 4128 df-id 4324 df-po 4327 df-iso 4328 df-iord 4397 df-on 4399 df-ilim 4400 df-suc 4402 df-iom 4623 df-xp 4665 df-rel 4666 df-cnv 4667 df-co 4668 df-dm 4669 df-rn 4670 df-res 4671 df-ima 4672 df-iota 5215 df-fun 5256 df-fn 5257 df-f 5258 df-f1 5259 df-fo 5260 df-f1o 5261 df-fv 5262 df-riota 5873 df-ov 5921 df-oprab 5922 df-mpo 5923 df-1st 6193 df-2nd 6194 df-recs 6358 df-frec 6444 df-pnf 8056 df-mnf 8057 df-xr 8058 df-ltxr 8059 df-le 8060 df-sub 8192 df-neg 8193 df-reap 8594 df-ap 8601 df-div 8692 df-inn 8983 df-n0 9241 df-z 9318 df-uz 9593 df-q 9685 df-fz 10075 df-seqfrec 10519 df-fac 10797 df-bc 10819

This theorem is referenced by: bcn2 10835

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