bcval5 - Intuitionistic Logic Explorer

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Theorem bcval5 10684

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, although it is no longer valid for nonpositive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 10671	. . . 4
2	1	adantl 275	. . 3 $/\$ $/\$
3		simprl 526	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
4		simprr 527	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
5	3, 4	mulcld 7927	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
6		simpr1 998	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
7		simpr2 999	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
8		simpr3 1000	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
9	6, 7, 8	mulassd 7930	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
10		simpll 524	. . . . . . . . . . . . 13 $/\$ $/\$
11	10	nn0zd 9319	. . . . . . . . . . . 12 $/\$ $/\$
12		simplr 525	. . . . . . . . . . . . 13 $/\$ $/\$
13	12	nnzd 9320	. . . . . . . . . . . 12 $/\$ $/\$
14	11, 13	zsubcld 9326	. . . . . . . . . . 11 $/\$ $/\$
15	14	peano2zd 9324	. . . . . . . . . 10 $/\$ $/\$
16		1red 7922	. . . . . . . . . . . 12 $/\$ $/\$
17	12	nnred 8878	. . . . . . . . . . . 12 $/\$ $/\$
18	10	nn0red 9176	. . . . . . . . . . . 12 $/\$ $/\$
19	12	nnge1d 8908	. . . . . . . . . . . 12 $/\$ $/\$
20	16, 17, 18, 19	lesub2dd 8468	. . . . . . . . . . 11 $/\$ $/\$
21	14	zred 9321	. . . . . . . . . . . 12 $/\$ $/\$
22		leaddsub 8344	. . . . . . . . . . . 12 $/\$ $/\$
23	21, 16, 18, 22	syl3anc 1233	. . . . . . . . . . 11 $/\$ $/\$
24	20, 23	mpbird 166	. . . . . . . . . 10 $/\$ $/\$
25		eluz2 9480	. . . . . . . . . 10 $/\$ $/\$
26	15, 11, 24, 25	syl3anbrc 1176	. . . . . . . . 9 $/\$ $/\$
27	26	adantrr 476	. . . . . . . 8 $/\$ $/\$ $/\$
28		simprr 527	. . . . . . . . 9 $/\$ $/\$ $/\$
29		nnuz 9509	. . . . . . . . 9
30	28, 29	eleqtrdi 2263	. . . . . . . 8 $/\$ $/\$ $/\$
31		fvi 5551	. . . . . . . . . 10
32	31	elv 2734	. . . . . . . . 9
33		eluzelcn 9485	. . . . . . . . . 10
34	33	adantl 275	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
35	32, 34	eqeltrid 2257	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
36	5, 9, 27, 30, 35	seq3split 10422	. . . . . . 7 $/\$ $/\$ $/\$
37		elfzuz3 9965	. . . . . . . . . . 11
38	37	adantl 275	. . . . . . . . . 10 $/\$ $/\$
39		eluznn 9546	. . . . . . . . . 10 $/\$
40	12, 38, 39	syl2anc 409	. . . . . . . . 9 $/\$ $/\$
41	40	adantrr 476	. . . . . . . 8 $/\$ $/\$ $/\$
42		facnn 10648	. . . . . . . 8
43	41, 42	syl 14	. . . . . . 7 $/\$ $/\$ $/\$
44		facnn 10648	. . . . . . . . 9
45	28, 44	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$
46	45	oveq1d 5865	. . . . . . 7 $/\$ $/\$ $/\$
47	36, 43, 46	3eqtr4d 2213	. . . . . 6 $/\$ $/\$ $/\$
48	47	expr 373	. . . . 5 $/\$ $/\$
49	10	faccld 10657	. . . . . . . . 9 $/\$ $/\$
50	49	nncnd 8879	. . . . . . . 8 $/\$ $/\$
51	50	mulid2d 7925	. . . . . . 7 $/\$ $/\$
52	40, 42	syl 14	. . . . . . . 8 $/\$ $/\$
53	52	oveq2d 5866	. . . . . . 7 $/\$ $/\$
54	51, 53	eqtr3d 2205	. . . . . 6 $/\$ $/\$
55		fveq2 5494	. . . . . . . . 9
56		fac0 10649	. . . . . . . . 9
57	55, 56	eqtrdi 2219	. . . . . . . 8
58		oveq1 5857	. . . . . . . . . . 11
59		0p1e1 8979	. . . . . . . . . . 11
60	58, 59	eqtrdi 2219	. . . . . . . . . 10
61	60	seqeq1d 10394	. . . . . . . . 9
62	61	fveq1d 5496	. . . . . . . 8
63	57, 62	oveq12d 5868	. . . . . . 7
64	63	eqeq2d 2182	. . . . . 6
65	54, 64	syl5ibrcom 156	. . . . 5 $/\$ $/\$
66		fznn0sub 10000	. . . . . . 7
67	66	adantl 275	. . . . . 6 $/\$ $/\$
68		elnn0 9124	. . . . . 6 $\/$
69	67, 68	sylib 121	. . . . 5 $/\$ $/\$ $\/$
70	48, 65, 69	mpjaod 713	. . . 4 $/\$ $/\$
71	70	oveq1d 5865	. . 3 $/\$ $/\$
72		eqid 2170	. . . . . 6
73		fvi 5551	. . . . . . . 8
74	73	elv 2734	. . . . . . 7
75		eluzelcn 9485	. . . . . . . 8
76	75	adantl 275	. . . . . . 7 $/\$ $/\$ $/\$
77	74, 76	eqeltrid 2257	. . . . . 6 $/\$ $/\$ $/\$
78		mulcl 7888	. . . . . . 7 $/\$
79	78	adantl 275	. . . . . 6 $/\$ $/\$ $/\$ $/\$
80	72, 15, 77, 79	seqf 10404	. . . . 5 $/\$ $/\$
81	80, 26	ffvelrnd 5629	. . . 4 $/\$ $/\$
82	12	nnnn0d 9175	. . . . . 6 $/\$ $/\$
83	82	faccld 10657	. . . . 5 $/\$ $/\$
84	83	nncnd 8879	. . . 4 $/\$ $/\$
85	67	faccld 10657	. . . . 5 $/\$ $/\$
86	85	nncnd 8879	. . . 4 $/\$ $/\$
87	83	nnap0d 8911	. . . 4 $/\$ $/\$ #
88	85	nnap0d 8911	. . . 4 $/\$ $/\$ #
89	81, 84, 86, 87, 88	divcanap5d 8721	. . 3 $/\$ $/\$
90	2, 71, 89	3eqtrd 2207	. 2 $/\$ $/\$
91		simplr 525	. . . . . . 7 $/\$ $/\$
92	91	nnnn0d 9175	. . . . . 6 $/\$ $/\$
93	92	faccld 10657	. . . . 5 $/\$ $/\$
94	93	nncnd 8879	. . . 4 $/\$ $/\$
95	93	nnap0d 8911	. . . 4 $/\$ $/\$ #
96	94, 95	div0apd 8691	. . 3 $/\$ $/\$
97		mulcl 7888	. . . . . 6 $/\$
98	97	adantl 275	. . . . 5 $/\$ $/\$ $/\$ $/\$
99		eluzelcn 9485	. . . . . . 7
100	99	adantl 275	. . . . . 6 $/\$ $/\$ $/\$
101	32, 100	eqeltrid 2257	. . . . 5 $/\$ $/\$ $/\$
102		simpr 109	. . . . . 6 $/\$ $/\$ $/\$
103	102	mul02d 8298	. . . . 5 $/\$ $/\$ $/\$
104	102	mul01d 8299	. . . . 5 $/\$ $/\$ $/\$
105		simpr 109	. . . . . . . . 9 $/\$ $/\$
106		nn0uz 9508	. . . . . . . . . . . 12
107	92, 106	eleqtrdi 2263	. . . . . . . . . . 11 $/\$ $/\$
108		simpll 524	. . . . . . . . . . . 12 $/\$ $/\$
109	108	nn0zd 9319	. . . . . . . . . . 11 $/\$ $/\$
110		elfz5 9960	. . . . . . . . . . 11 $/\$
111	107, 109, 110	syl2anc 409	. . . . . . . . . 10 $/\$ $/\$
112		nn0re 9131	. . . . . . . . . . . 12
113	112	ad2antrr 485	. . . . . . . . . . 11 $/\$ $/\$
114		nnre 8872	. . . . . . . . . . . 12
115	114	ad2antlr 486	. . . . . . . . . . 11 $/\$ $/\$
116	113, 115	subge0d 8441	. . . . . . . . . 10 $/\$ $/\$
117	111, 116	bitr4d 190	. . . . . . . . 9 $/\$ $/\$
118	105, 117	mtbid 667	. . . . . . . 8 $/\$ $/\$
119		simpl 108	. . . . . . . . . . . 12 $/\$
120	119	nn0zd 9319	. . . . . . . . . . 11 $/\$
121		simpr 109	. . . . . . . . . . . 12 $/\$
122	121	nnzd 9320	. . . . . . . . . . 11 $/\$
123	120, 122	zsubcld 9326	. . . . . . . . . 10 $/\$
124	123	adantr 274	. . . . . . . . 9 $/\$ $/\$
125		0z 9210	. . . . . . . . 9
126		zltnle 9245	. . . . . . . . 9 $/\$
127	124, 125, 126	sylancl 411	. . . . . . . 8 $/\$ $/\$
128	118, 127	mpbird 166	. . . . . . 7 $/\$ $/\$
129		zltp1le 9253	. . . . . . . 8 $/\$
130	124, 125, 129	sylancl 411	. . . . . . 7 $/\$ $/\$
131	128, 130	mpbid 146	. . . . . 6 $/\$ $/\$
132		nn0ge0 9147	. . . . . . 7
133	132	ad2antrr 485	. . . . . 6 $/\$ $/\$
134		0zd 9211	. . . . . . 7 $/\$ $/\$
135	124	peano2zd 9324	. . . . . . 7 $/\$ $/\$
136		elfz 9958	. . . . . . 7 $/\$ $/\$ $/\$
137	134, 135, 109, 136	syl3anc 1233	. . . . . 6 $/\$ $/\$ $/\$
138	131, 133, 137	mpbir2and 939	. . . . 5 $/\$ $/\$
139		0cn 7899	. . . . . 6
140		fvi 5551	. . . . . 6
141	139, 140	mp1i 10	. . . . 5 $/\$ $/\$
142	98, 101, 103, 104, 138, 141	seq3z 10454	. . . 4 $/\$ $/\$
143	142	oveq1d 5865	. . 3 $/\$ $/\$
144		nnz 9218	. . . . 5
145		bcval3 10672	. . . . 5 $/\$ $/\$
146	144, 145	syl3an2 1267	. . . 4 $/\$ $/\$
147	146	3expa 1198	. . 3 $/\$ $/\$
148	96, 143, 147	3eqtr4rd 2214	. 2 $/\$ $/\$
149		0zd 9211	. . . 4 $/\$
150		fzdcel 9983	. . . 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 3987

cid 4271

cfv 5196 (class class class)co 5850

cc 7759

cr 7760

cc0 7761

c1 7762

caddc 7764

cmul 7766

clt 7941

cle 7942

cmin 8077

cdiv 8576

cn 8865

cn0 9122

cz 9199

cuz 9474

cfz 9952

cseq 10388

cfa 10646

cbc 10668

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 4102 ax-sep 4105 ax-nul 4113 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-iinf 4570 ax-cnex 7852 ax-resscn 7853 ax-1cn 7854 ax-1re 7855 ax-icn 7856 ax-addcl 7857 ax-addrcl 7858 ax-mulcl 7859 ax-mulrcl 7860 ax-addcom 7861 ax-mulcom 7862 ax-addass 7863 ax-mulass 7864 ax-distr 7865 ax-i2m1 7866 ax-0lt1 7867 ax-1rid 7868 ax-0id 7869 ax-rnegex 7870 ax-precex 7871 ax-cnre 7872 ax-pre-ltirr 7873 ax-pre-ltwlin 7874 ax-pre-lttrn 7875 ax-pre-apti 7876 ax-pre-ltadd 7877 ax-pre-mulgt0 7878 ax-pre-mulext 7879

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 3526 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-iun 3873 df-br 3988 df-opab 4049 df-mpt 4050 df-tr 4086 df-id 4276 df-po 4279 df-iso 4280 df-iord 4349 df-on 4351 df-ilim 4352 df-suc 4354 df-iom 4573 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-f1 5201 df-fo 5202 df-f1o 5203 df-fv 5204 df-riota 5806 df-ov 5853 df-oprab 5854 df-mpo 5855 df-1st 6116 df-2nd 6117 df-recs 6281 df-frec 6367 df-pnf 7943 df-mnf 7944 df-xr 7945 df-ltxr 7946 df-le 7947 df-sub 8079 df-neg 8080 df-reap 8481 df-ap 8488 df-div 8577 df-inn 8866 df-n0 9123 df-z 9200 df-uz 9475 df-q 9566 df-fz 9953 df-seqfrec 10389 df-fac 10647 df-bc 10669

This theorem is referenced by: bcn2 10685

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