bcval5 - Intuitionistic Logic Explorer

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Theorem bcval5 10697

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, although it is no longer valid for nonpositive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 10684	. . . 4
2	1	adantl 275	. . 3 $/\$ $/\$
3		simprl 526	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
4		simprr 527	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
5	3, 4	mulcld 7940	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
6		simpr1 998	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
7		simpr2 999	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
8		simpr3 1000	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
9	6, 7, 8	mulassd 7943	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
10		simpll 524	. . . . . . . . . . . . 13 $/\$ $/\$
11	10	nn0zd 9332	. . . . . . . . . . . 12 $/\$ $/\$
12		simplr 525	. . . . . . . . . . . . 13 $/\$ $/\$
13	12	nnzd 9333	. . . . . . . . . . . 12 $/\$ $/\$
14	11, 13	zsubcld 9339	. . . . . . . . . . 11 $/\$ $/\$
15	14	peano2zd 9337	. . . . . . . . . 10 $/\$ $/\$
16		1red 7935	. . . . . . . . . . . 12 $/\$ $/\$
17	12	nnred 8891	. . . . . . . . . . . 12 $/\$ $/\$
18	10	nn0red 9189	. . . . . . . . . . . 12 $/\$ $/\$
19	12	nnge1d 8921	. . . . . . . . . . . 12 $/\$ $/\$
20	16, 17, 18, 19	lesub2dd 8481	. . . . . . . . . . 11 $/\$ $/\$
21	14	zred 9334	. . . . . . . . . . . 12 $/\$ $/\$
22		leaddsub 8357	. . . . . . . . . . . 12 $/\$ $/\$
23	21, 16, 18, 22	syl3anc 1233	. . . . . . . . . . 11 $/\$ $/\$
24	20, 23	mpbird 166	. . . . . . . . . 10 $/\$ $/\$
25		eluz2 9493	. . . . . . . . . 10 $/\$ $/\$
26	15, 11, 24, 25	syl3anbrc 1176	. . . . . . . . 9 $/\$ $/\$
27	26	adantrr 476	. . . . . . . 8 $/\$ $/\$ $/\$
28		simprr 527	. . . . . . . . 9 $/\$ $/\$ $/\$
29		nnuz 9522	. . . . . . . . 9
30	28, 29	eleqtrdi 2263	. . . . . . . 8 $/\$ $/\$ $/\$
31		fvi 5553	. . . . . . . . . 10
32	31	elv 2734	. . . . . . . . 9
33		eluzelcn 9498	. . . . . . . . . 10
34	33	adantl 275	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
35	32, 34	eqeltrid 2257	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
36	5, 9, 27, 30, 35	seq3split 10435	. . . . . . 7 $/\$ $/\$ $/\$
37		elfzuz3 9978	. . . . . . . . . . 11
38	37	adantl 275	. . . . . . . . . 10 $/\$ $/\$
39		eluznn 9559	. . . . . . . . . 10 $/\$
40	12, 38, 39	syl2anc 409	. . . . . . . . 9 $/\$ $/\$
41	40	adantrr 476	. . . . . . . 8 $/\$ $/\$ $/\$
42		facnn 10661	. . . . . . . 8
43	41, 42	syl 14	. . . . . . 7 $/\$ $/\$ $/\$
44		facnn 10661	. . . . . . . . 9
45	28, 44	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$
46	45	oveq1d 5868	. . . . . . 7 $/\$ $/\$ $/\$
47	36, 43, 46	3eqtr4d 2213	. . . . . 6 $/\$ $/\$ $/\$
48	47	expr 373	. . . . 5 $/\$ $/\$
49	10	faccld 10670	. . . . . . . . 9 $/\$ $/\$
50	49	nncnd 8892	. . . . . . . 8 $/\$ $/\$
51	50	mulid2d 7938	. . . . . . 7 $/\$ $/\$
52	40, 42	syl 14	. . . . . . . 8 $/\$ $/\$
53	52	oveq2d 5869	. . . . . . 7 $/\$ $/\$
54	51, 53	eqtr3d 2205	. . . . . 6 $/\$ $/\$
55		fveq2 5496	. . . . . . . . 9
56		fac0 10662	. . . . . . . . 9
57	55, 56	eqtrdi 2219	. . . . . . . 8
58		oveq1 5860	. . . . . . . . . . 11
59		0p1e1 8992	. . . . . . . . . . 11
60	58, 59	eqtrdi 2219	. . . . . . . . . 10
61	60	seqeq1d 10407	. . . . . . . . 9
62	61	fveq1d 5498	. . . . . . . 8
63	57, 62	oveq12d 5871	. . . . . . 7
64	63	eqeq2d 2182	. . . . . 6
65	54, 64	syl5ibrcom 156	. . . . 5 $/\$ $/\$
66		fznn0sub 10013	. . . . . . 7
67	66	adantl 275	. . . . . 6 $/\$ $/\$
68		elnn0 9137	. . . . . 6 $\/$
69	67, 68	sylib 121	. . . . 5 $/\$ $/\$ $\/$
70	48, 65, 69	mpjaod 713	. . . 4 $/\$ $/\$
71	70	oveq1d 5868	. . 3 $/\$ $/\$
72		eqid 2170	. . . . . 6
73		fvi 5553	. . . . . . . 8
74	73	elv 2734	. . . . . . 7
75		eluzelcn 9498	. . . . . . . 8
76	75	adantl 275	. . . . . . 7 $/\$ $/\$ $/\$
77	74, 76	eqeltrid 2257	. . . . . 6 $/\$ $/\$ $/\$
78		mulcl 7901	. . . . . . 7 $/\$
79	78	adantl 275	. . . . . 6 $/\$ $/\$ $/\$ $/\$
80	72, 15, 77, 79	seqf 10417	. . . . 5 $/\$ $/\$
81	80, 26	ffvelrnd 5632	. . . 4 $/\$ $/\$
82	12	nnnn0d 9188	. . . . . 6 $/\$ $/\$
83	82	faccld 10670	. . . . 5 $/\$ $/\$
84	83	nncnd 8892	. . . 4 $/\$ $/\$
85	67	faccld 10670	. . . . 5 $/\$ $/\$
86	85	nncnd 8892	. . . 4 $/\$ $/\$
87	83	nnap0d 8924	. . . 4 $/\$ $/\$ #
88	85	nnap0d 8924	. . . 4 $/\$ $/\$ #
89	81, 84, 86, 87, 88	divcanap5d 8734	. . 3 $/\$ $/\$
90	2, 71, 89	3eqtrd 2207	. 2 $/\$ $/\$
91		simplr 525	. . . . . . 7 $/\$ $/\$
92	91	nnnn0d 9188	. . . . . 6 $/\$ $/\$
93	92	faccld 10670	. . . . 5 $/\$ $/\$
94	93	nncnd 8892	. . . 4 $/\$ $/\$
95	93	nnap0d 8924	. . . 4 $/\$ $/\$ #
96	94, 95	div0apd 8704	. . 3 $/\$ $/\$
97		mulcl 7901	. . . . . 6 $/\$
98	97	adantl 275	. . . . 5 $/\$ $/\$ $/\$ $/\$
99		eluzelcn 9498	. . . . . . 7
100	99	adantl 275	. . . . . 6 $/\$ $/\$ $/\$
101	32, 100	eqeltrid 2257	. . . . 5 $/\$ $/\$ $/\$
102		simpr 109	. . . . . 6 $/\$ $/\$ $/\$
103	102	mul02d 8311	. . . . 5 $/\$ $/\$ $/\$
104	102	mul01d 8312	. . . . 5 $/\$ $/\$ $/\$
105		simpr 109	. . . . . . . . 9 $/\$ $/\$
106		nn0uz 9521	. . . . . . . . . . . 12
107	92, 106	eleqtrdi 2263	. . . . . . . . . . 11 $/\$ $/\$
108		simpll 524	. . . . . . . . . . . 12 $/\$ $/\$
109	108	nn0zd 9332	. . . . . . . . . . 11 $/\$ $/\$
110		elfz5 9973	. . . . . . . . . . 11 $/\$
111	107, 109, 110	syl2anc 409	. . . . . . . . . 10 $/\$ $/\$
112		nn0re 9144	. . . . . . . . . . . 12
113	112	ad2antrr 485	. . . . . . . . . . 11 $/\$ $/\$
114		nnre 8885	. . . . . . . . . . . 12
115	114	ad2antlr 486	. . . . . . . . . . 11 $/\$ $/\$
116	113, 115	subge0d 8454	. . . . . . . . . 10 $/\$ $/\$
117	111, 116	bitr4d 190	. . . . . . . . 9 $/\$ $/\$
118	105, 117	mtbid 667	. . . . . . . 8 $/\$ $/\$
119		simpl 108	. . . . . . . . . . . 12 $/\$
120	119	nn0zd 9332	. . . . . . . . . . 11 $/\$
121		simpr 109	. . . . . . . . . . . 12 $/\$
122	121	nnzd 9333	. . . . . . . . . . 11 $/\$
123	120, 122	zsubcld 9339	. . . . . . . . . 10 $/\$
124	123	adantr 274	. . . . . . . . 9 $/\$ $/\$
125		0z 9223	. . . . . . . . 9
126		zltnle 9258	. . . . . . . . 9 $/\$
127	124, 125, 126	sylancl 411	. . . . . . . 8 $/\$ $/\$
128	118, 127	mpbird 166	. . . . . . 7 $/\$ $/\$
129		zltp1le 9266	. . . . . . . 8 $/\$
130	124, 125, 129	sylancl 411	. . . . . . 7 $/\$ $/\$
131	128, 130	mpbid 146	. . . . . 6 $/\$ $/\$
132		nn0ge0 9160	. . . . . . 7
133	132	ad2antrr 485	. . . . . 6 $/\$ $/\$
134		0zd 9224	. . . . . . 7 $/\$ $/\$
135	124	peano2zd 9337	. . . . . . 7 $/\$ $/\$
136		elfz 9971	. . . . . . 7 $/\$ $/\$ $/\$
137	134, 135, 109, 136	syl3anc 1233	. . . . . 6 $/\$ $/\$ $/\$
138	131, 133, 137	mpbir2and 939	. . . . 5 $/\$ $/\$
139		0cn 7912	. . . . . 6
140		fvi 5553	. . . . . 6
141	139, 140	mp1i 10	. . . . 5 $/\$ $/\$
142	98, 101, 103, 104, 138, 141	seq3z 10467	. . . 4 $/\$ $/\$
143	142	oveq1d 5868	. . 3 $/\$ $/\$
144		nnz 9231	. . . . 5
145		bcval3 10685	. . . . 5 $/\$ $/\$
146	144, 145	syl3an2 1267	. . . 4 $/\$ $/\$
147	146	3expa 1198	. . 3 $/\$ $/\$
148	96, 143, 147	3eqtr4rd 2214	. 2 $/\$ $/\$
149		0zd 9224	. . . 4 $/\$
150		fzdcel 9996	. . . 4 $/\$ $/\$ DECID
151	122, 149, 120, 150	syl3anc 1233	. . 3 $/\$ DECID
152		exmiddc 831	. . 3 DECID $\/$
153	151, 152	syl 14	. 2 $/\$ $\/$
154	90, 148, 153	mpjaodan 793	1 $/\$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 103

wb 104 $\/$ wo 703 DECID wdc 829 $/\$ w3a 973

wceq 1348

wcel 2141

cvv 2730 class class class wbr 3989

cid 4273

cfv 5198 (class class class)co 5853

cc 7772

cr 7773

cc0 7774

c1 7775

caddc 7777

cmul 7779

clt 7954

cle 7955

cmin 8090

cdiv 8589

cn 8878

cn0 9135

cz 9212

cuz 9487

cfz 9965

cseq 10401

cfa 10659

cbc 10681

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-mulrcl 7873 ax-addcom 7874 ax-mulcom 7875 ax-addass 7876 ax-mulass 7877 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-1rid 7881 ax-0id 7882 ax-rnegex 7883 ax-precex 7884 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-apti 7889 ax-pre-ltadd 7890 ax-pre-mulgt0 7891 ax-pre-mulext 7892

This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3527 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-id 4278 df-po 4281 df-iso 4282 df-iord 4351 df-on 4353 df-ilim 4354 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-recs 6284 df-frec 6370 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-reap 8494 df-ap 8501 df-div 8590 df-inn 8879 df-n0 9136 df-z 9213 df-uz 9488 df-q 9579 df-fz 9966 df-seqfrec 10402 df-fac 10660 df-bc 10682

This theorem is referenced by: bcn2 10698

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