bcval5 - Intuitionistic Logic Explorer

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Theorem bcval5 10872

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, although it is no longer valid for nonpositive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 10859	. . . 4
2	1	adantl 277	. . 3 $/\$ $/\$
3		simprl 529	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
4		simprr 531	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
5	3, 4	mulcld 8064	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
6		simpr1 1005	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
7		simpr2 1006	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
8		simpr3 1007	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
9	6, 7, 8	mulassd 8067	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
10		simpll 527	. . . . . . . . . . . . 13 $/\$ $/\$
11	10	nn0zd 9463	. . . . . . . . . . . 12 $/\$ $/\$
12		simplr 528	. . . . . . . . . . . . 13 $/\$ $/\$
13	12	nnzd 9464	. . . . . . . . . . . 12 $/\$ $/\$
14	11, 13	zsubcld 9470	. . . . . . . . . . 11 $/\$ $/\$
15	14	peano2zd 9468	. . . . . . . . . 10 $/\$ $/\$
16		1red 8058	. . . . . . . . . . . 12 $/\$ $/\$
17	12	nnred 9020	. . . . . . . . . . . 12 $/\$ $/\$
18	10	nn0red 9320	. . . . . . . . . . . 12 $/\$ $/\$
19	12	nnge1d 9050	. . . . . . . . . . . 12 $/\$ $/\$
20	16, 17, 18, 19	lesub2dd 8606	. . . . . . . . . . 11 $/\$ $/\$
21	14	zred 9465	. . . . . . . . . . . 12 $/\$ $/\$
22		leaddsub 8482	. . . . . . . . . . . 12 $/\$ $/\$
23	21, 16, 18, 22	syl3anc 1249	. . . . . . . . . . 11 $/\$ $/\$
24	20, 23	mpbird 167	. . . . . . . . . 10 $/\$ $/\$
25		eluz2 9624	. . . . . . . . . 10 $/\$ $/\$
26	15, 11, 24, 25	syl3anbrc 1183	. . . . . . . . 9 $/\$ $/\$
27	26	adantrr 479	. . . . . . . 8 $/\$ $/\$ $/\$
28		simprr 531	. . . . . . . . 9 $/\$ $/\$ $/\$
29		nnuz 9654	. . . . . . . . 9
30	28, 29	eleqtrdi 2289	. . . . . . . 8 $/\$ $/\$ $/\$
31		fvi 5621	. . . . . . . . . 10
32	31	elv 2767	. . . . . . . . 9
33		eluzelcn 9629	. . . . . . . . . 10
34	33	adantl 277	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
35	32, 34	eqeltrid 2283	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
36	5, 9, 27, 30, 35	seq3split 10597	. . . . . . 7 $/\$ $/\$ $/\$
37		elfzuz3 10114	. . . . . . . . . . 11
38	37	adantl 277	. . . . . . . . . 10 $/\$ $/\$
39		eluznn 9691	. . . . . . . . . 10 $/\$
40	12, 38, 39	syl2anc 411	. . . . . . . . 9 $/\$ $/\$
41	40	adantrr 479	. . . . . . . 8 $/\$ $/\$ $/\$
42		facnn 10836	. . . . . . . 8
43	41, 42	syl 14	. . . . . . 7 $/\$ $/\$ $/\$
44		facnn 10836	. . . . . . . . 9
45	28, 44	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$
46	45	oveq1d 5940	. . . . . . 7 $/\$ $/\$ $/\$
47	36, 43, 46	3eqtr4d 2239	. . . . . 6 $/\$ $/\$ $/\$
48	47	expr 375	. . . . 5 $/\$ $/\$
49	10	faccld 10845	. . . . . . . . 9 $/\$ $/\$
50	49	nncnd 9021	. . . . . . . 8 $/\$ $/\$
51	50	mulid2d 8062	. . . . . . 7 $/\$ $/\$
52	40, 42	syl 14	. . . . . . . 8 $/\$ $/\$
53	52	oveq2d 5941	. . . . . . 7 $/\$ $/\$
54	51, 53	eqtr3d 2231	. . . . . 6 $/\$ $/\$
55		fveq2 5561	. . . . . . . . 9
56		fac0 10837	. . . . . . . . 9
57	55, 56	eqtrdi 2245	. . . . . . . 8
58		oveq1 5932	. . . . . . . . . . 11
59		0p1e1 9121	. . . . . . . . . . 11
60	58, 59	eqtrdi 2245	. . . . . . . . . 10
61	60	seqeq1d 10562	. . . . . . . . 9
62	61	fveq1d 5563	. . . . . . . 8
63	57, 62	oveq12d 5943	. . . . . . 7
64	63	eqeq2d 2208	. . . . . 6
65	54, 64	syl5ibrcom 157	. . . . 5 $/\$ $/\$
66		fznn0sub 10149	. . . . . . 7
67	66	adantl 277	. . . . . 6 $/\$ $/\$
68		elnn0 9268	. . . . . 6 $\/$
69	67, 68	sylib 122	. . . . 5 $/\$ $/\$ $\/$
70	48, 65, 69	mpjaod 719	. . . 4 $/\$ $/\$
71	70	oveq1d 5940	. . 3 $/\$ $/\$
72		eqid 2196	. . . . . 6
73		fvi 5621	. . . . . . . 8
74	73	elv 2767	. . . . . . 7
75		eluzelcn 9629	. . . . . . . 8
76	75	adantl 277	. . . . . . 7 $/\$ $/\$ $/\$
77	74, 76	eqeltrid 2283	. . . . . 6 $/\$ $/\$ $/\$
78		mulcl 8023	. . . . . . 7 $/\$
79	78	adantl 277	. . . . . 6 $/\$ $/\$ $/\$ $/\$
80	72, 15, 77, 79	seqf 10573	. . . . 5 $/\$ $/\$
81	80, 26	ffvelcdmd 5701	. . . 4 $/\$ $/\$
82	12	nnnn0d 9319	. . . . . 6 $/\$ $/\$
83	82	faccld 10845	. . . . 5 $/\$ $/\$
84	83	nncnd 9021	. . . 4 $/\$ $/\$
85	67	faccld 10845	. . . . 5 $/\$ $/\$
86	85	nncnd 9021	. . . 4 $/\$ $/\$
87	83	nnap0d 9053	. . . 4 $/\$ $/\$ #
88	85	nnap0d 9053	. . . 4 $/\$ $/\$ #
89	81, 84, 86, 87, 88	divcanap5d 8861	. . 3 $/\$ $/\$
90	2, 71, 89	3eqtrd 2233	. 2 $/\$ $/\$
91		simplr 528	. . . . . . 7 $/\$ $/\$
92	91	nnnn0d 9319	. . . . . 6 $/\$ $/\$
93	92	faccld 10845	. . . . 5 $/\$ $/\$
94	93	nncnd 9021	. . . 4 $/\$ $/\$
95	93	nnap0d 9053	. . . 4 $/\$ $/\$ #
96	94, 95	div0apd 8831	. . 3 $/\$ $/\$
97		mulcl 8023	. . . . . 6 $/\$
98	97	adantl 277	. . . . 5 $/\$ $/\$ $/\$ $/\$
99		eluzelcn 9629	. . . . . . 7
100	99	adantl 277	. . . . . 6 $/\$ $/\$ $/\$
101	32, 100	eqeltrid 2283	. . . . 5 $/\$ $/\$ $/\$
102		simpr 110	. . . . . 6 $/\$ $/\$ $/\$
103	102	mul02d 8435	. . . . 5 $/\$ $/\$ $/\$
104	102	mul01d 8436	. . . . 5 $/\$ $/\$ $/\$
105		simpr 110	. . . . . . . . 9 $/\$ $/\$
106		nn0uz 9653	. . . . . . . . . . . 12
107	92, 106	eleqtrdi 2289	. . . . . . . . . . 11 $/\$ $/\$
108		simpll 527	. . . . . . . . . . . 12 $/\$ $/\$
109	108	nn0zd 9463	. . . . . . . . . . 11 $/\$ $/\$
110		elfz5 10109	. . . . . . . . . . 11 $/\$
111	107, 109, 110	syl2anc 411	. . . . . . . . . 10 $/\$ $/\$
112		nn0re 9275	. . . . . . . . . . . 12
113	112	ad2antrr 488	. . . . . . . . . . 11 $/\$ $/\$
114		nnre 9014	. . . . . . . . . . . 12
115	114	ad2antlr 489	. . . . . . . . . . 11 $/\$ $/\$
116	113, 115	subge0d 8579	. . . . . . . . . 10 $/\$ $/\$
117	111, 116	bitr4d 191	. . . . . . . . 9 $/\$ $/\$
118	105, 117	mtbid 673	. . . . . . . 8 $/\$ $/\$
119		simpl 109	. . . . . . . . . . . 12 $/\$
120	119	nn0zd 9463	. . . . . . . . . . 11 $/\$
121		simpr 110	. . . . . . . . . . . 12 $/\$
122	121	nnzd 9464	. . . . . . . . . . 11 $/\$
123	120, 122	zsubcld 9470	. . . . . . . . . 10 $/\$
124	123	adantr 276	. . . . . . . . 9 $/\$ $/\$
125		0z 9354	. . . . . . . . 9
126		zltnle 9389	. . . . . . . . 9 $/\$
127	124, 125, 126	sylancl 413	. . . . . . . 8 $/\$ $/\$
128	118, 127	mpbird 167	. . . . . . 7 $/\$ $/\$
129		zltp1le 9397	. . . . . . . 8 $/\$
130	124, 125, 129	sylancl 413	. . . . . . 7 $/\$ $/\$
131	128, 130	mpbid 147	. . . . . 6 $/\$ $/\$
132		nn0ge0 9291	. . . . . . 7
133	132	ad2antrr 488	. . . . . 6 $/\$ $/\$
134		0zd 9355	. . . . . . 7 $/\$ $/\$
135	124	peano2zd 9468	. . . . . . 7 $/\$ $/\$
136		elfz 10106	. . . . . . 7 $/\$ $/\$ $/\$
137	134, 135, 109, 136	syl3anc 1249	. . . . . 6 $/\$ $/\$ $/\$
138	131, 133, 137	mpbir2and 946	. . . . 5 $/\$ $/\$
139		0cn 8035	. . . . . 6
140		fvi 5621	. . . . . 6
141	139, 140	mp1i 10	. . . . 5 $/\$ $/\$
142	98, 101, 103, 104, 138, 141	seq3z 10637	. . . 4 $/\$ $/\$
143	142	oveq1d 5940	. . 3 $/\$ $/\$
144		nnz 9362	. . . . 5
145		bcval3 10860	. . . . 5 $/\$ $/\$
146	144, 145	syl3an2 1283	. . . 4 $/\$ $/\$
147	146	3expa 1205	. . 3 $/\$ $/\$
148	96, 143, 147	3eqtr4rd 2240	. 2 $/\$ $/\$
149		0zd 9355	. . . 4 $/\$
150		fzdcel 10132	. . . 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 2167

cvv 2763 class class class wbr 4034

cid 4324

cfv 5259 (class class class)co 5925

cc 7894

cr 7895

cc0 7896

c1 7897

caddc 7899

cmul 7901

clt 8078

cle 8079

cmin 8214

cdiv 8716

cn 9007

cn0 9266

cz 9343

cuz 9618

cfz 10100

cseq 10556

cfa 10834

cbc 10856

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 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-coll 4149 ax-sep 4152 ax-nul 4160 ax-pow 4208 ax-pr 4243 ax-un 4469 ax-setind 4574 ax-iinf 4625 ax-cnex 7987 ax-resscn 7988 ax-1cn 7989 ax-1re 7990 ax-icn 7991 ax-addcl 7992 ax-addrcl 7993 ax-mulcl 7994 ax-mulrcl 7995 ax-addcom 7996 ax-mulcom 7997 ax-addass 7998 ax-mulass 7999 ax-distr 8000 ax-i2m1 8001 ax-0lt1 8002 ax-1rid 8003 ax-0id 8004 ax-rnegex 8005 ax-precex 8006 ax-cnre 8007 ax-pre-ltirr 8008 ax-pre-ltwlin 8009 ax-pre-lttrn 8010 ax-pre-apti 8011 ax-pre-ltadd 8012 ax-pre-mulgt0 8013 ax-pre-mulext 8014

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 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-ral 2480 df-rex 2481 df-reu 2482 df-rmo 2483 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-nul 3452 df-if 3563 df-pw 3608 df-sn 3629 df-pr 3630 df-op 3632 df-uni 3841 df-int 3876 df-iun 3919 df-br 4035 df-opab 4096 df-mpt 4097 df-tr 4133 df-id 4329 df-po 4332 df-iso 4333 df-iord 4402 df-on 4404 df-ilim 4405 df-suc 4407 df-iom 4628 df-xp 4670 df-rel 4671 df-cnv 4672 df-co 4673 df-dm 4674 df-rn 4675 df-res 4676 df-ima 4677 df-iota 5220 df-fun 5261 df-fn 5262 df-f 5263 df-f1 5264 df-fo 5265 df-f1o 5266 df-fv 5267 df-riota 5880 df-ov 5928 df-oprab 5929 df-mpo 5930 df-1st 6207 df-2nd 6208 df-recs 6372 df-frec 6458 df-pnf 8080 df-mnf 8081 df-xr 8082 df-ltxr 8083 df-le 8084 df-sub 8216 df-neg 8217 df-reap 8619 df-ap 8626 df-div 8717 df-inn 9008 df-n0 9267 df-z 9344 df-uz 9619 df-q 9711 df-fz 10101 df-seqfrec 10557 df-fac 10835 df-bc 10857

This theorem is referenced by: bcn2 10873

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