bcval5 - Intuitionistic Logic Explorer

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Theorem bcval5 10855

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, although it is no longer valid for nonpositive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 10842	. . . 4
2	1	adantl 277	. . 3 $/\$ $/\$
3		simprl 529	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
4		simprr 531	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
5	3, 4	mulcld 8047	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
6		simpr1 1005	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
7		simpr2 1006	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
8		simpr3 1007	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
9	6, 7, 8	mulassd 8050	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
10		simpll 527	. . . . . . . . . . . . 13 $/\$ $/\$
11	10	nn0zd 9446	. . . . . . . . . . . 12 $/\$ $/\$
12		simplr 528	. . . . . . . . . . . . 13 $/\$ $/\$
13	12	nnzd 9447	. . . . . . . . . . . 12 $/\$ $/\$
14	11, 13	zsubcld 9453	. . . . . . . . . . 11 $/\$ $/\$
15	14	peano2zd 9451	. . . . . . . . . 10 $/\$ $/\$
16		1red 8041	. . . . . . . . . . . 12 $/\$ $/\$
17	12	nnred 9003	. . . . . . . . . . . 12 $/\$ $/\$
18	10	nn0red 9303	. . . . . . . . . . . 12 $/\$ $/\$
19	12	nnge1d 9033	. . . . . . . . . . . 12 $/\$ $/\$
20	16, 17, 18, 19	lesub2dd 8589	. . . . . . . . . . 11 $/\$ $/\$
21	14	zred 9448	. . . . . . . . . . . 12 $/\$ $/\$
22		leaddsub 8465	. . . . . . . . . . . 12 $/\$ $/\$
23	21, 16, 18, 22	syl3anc 1249	. . . . . . . . . . 11 $/\$ $/\$
24	20, 23	mpbird 167	. . . . . . . . . 10 $/\$ $/\$
25		eluz2 9607	. . . . . . . . . 10 $/\$ $/\$
26	15, 11, 24, 25	syl3anbrc 1183	. . . . . . . . 9 $/\$ $/\$
27	26	adantrr 479	. . . . . . . 8 $/\$ $/\$ $/\$
28		simprr 531	. . . . . . . . 9 $/\$ $/\$ $/\$
29		nnuz 9637	. . . . . . . . 9
30	28, 29	eleqtrdi 2289	. . . . . . . 8 $/\$ $/\$ $/\$
31		fvi 5618	. . . . . . . . . 10
32	31	elv 2767	. . . . . . . . 9
33		eluzelcn 9612	. . . . . . . . . 10
34	33	adantl 277	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
35	32, 34	eqeltrid 2283	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
36	5, 9, 27, 30, 35	seq3split 10580	. . . . . . 7 $/\$ $/\$ $/\$
37		elfzuz3 10097	. . . . . . . . . . 11
38	37	adantl 277	. . . . . . . . . 10 $/\$ $/\$
39		eluznn 9674	. . . . . . . . . 10 $/\$
40	12, 38, 39	syl2anc 411	. . . . . . . . 9 $/\$ $/\$
41	40	adantrr 479	. . . . . . . 8 $/\$ $/\$ $/\$
42		facnn 10819	. . . . . . . 8
43	41, 42	syl 14	. . . . . . 7 $/\$ $/\$ $/\$
44		facnn 10819	. . . . . . . . 9
45	28, 44	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$
46	45	oveq1d 5937	. . . . . . 7 $/\$ $/\$ $/\$
47	36, 43, 46	3eqtr4d 2239	. . . . . 6 $/\$ $/\$ $/\$
48	47	expr 375	. . . . 5 $/\$ $/\$
49	10	faccld 10828	. . . . . . . . 9 $/\$ $/\$
50	49	nncnd 9004	. . . . . . . 8 $/\$ $/\$
51	50	mulid2d 8045	. . . . . . 7 $/\$ $/\$
52	40, 42	syl 14	. . . . . . . 8 $/\$ $/\$
53	52	oveq2d 5938	. . . . . . 7 $/\$ $/\$
54	51, 53	eqtr3d 2231	. . . . . 6 $/\$ $/\$
55		fveq2 5558	. . . . . . . . 9
56		fac0 10820	. . . . . . . . 9
57	55, 56	eqtrdi 2245	. . . . . . . 8
58		oveq1 5929	. . . . . . . . . . 11
59		0p1e1 9104	. . . . . . . . . . 11
60	58, 59	eqtrdi 2245	. . . . . . . . . 10
61	60	seqeq1d 10545	. . . . . . . . 9
62	61	fveq1d 5560	. . . . . . . 8
63	57, 62	oveq12d 5940	. . . . . . 7
64	63	eqeq2d 2208	. . . . . 6
65	54, 64	syl5ibrcom 157	. . . . 5 $/\$ $/\$
66		fznn0sub 10132	. . . . . . 7
67	66	adantl 277	. . . . . 6 $/\$ $/\$
68		elnn0 9251	. . . . . 6 $\/$
69	67, 68	sylib 122	. . . . 5 $/\$ $/\$ $\/$
70	48, 65, 69	mpjaod 719	. . . 4 $/\$ $/\$
71	70	oveq1d 5937	. . 3 $/\$ $/\$
72		eqid 2196	. . . . . 6
73		fvi 5618	. . . . . . . 8
74	73	elv 2767	. . . . . . 7
75		eluzelcn 9612	. . . . . . . 8
76	75	adantl 277	. . . . . . 7 $/\$ $/\$ $/\$
77	74, 76	eqeltrid 2283	. . . . . 6 $/\$ $/\$ $/\$
78		mulcl 8006	. . . . . . 7 $/\$
79	78	adantl 277	. . . . . 6 $/\$ $/\$ $/\$ $/\$
80	72, 15, 77, 79	seqf 10556	. . . . 5 $/\$ $/\$
81	80, 26	ffvelcdmd 5698	. . . 4 $/\$ $/\$
82	12	nnnn0d 9302	. . . . . 6 $/\$ $/\$
83	82	faccld 10828	. . . . 5 $/\$ $/\$
84	83	nncnd 9004	. . . 4 $/\$ $/\$
85	67	faccld 10828	. . . . 5 $/\$ $/\$
86	85	nncnd 9004	. . . 4 $/\$ $/\$
87	83	nnap0d 9036	. . . 4 $/\$ $/\$ #
88	85	nnap0d 9036	. . . 4 $/\$ $/\$ #
89	81, 84, 86, 87, 88	divcanap5d 8844	. . 3 $/\$ $/\$
90	2, 71, 89	3eqtrd 2233	. 2 $/\$ $/\$
91		simplr 528	. . . . . . 7 $/\$ $/\$
92	91	nnnn0d 9302	. . . . . 6 $/\$ $/\$
93	92	faccld 10828	. . . . 5 $/\$ $/\$
94	93	nncnd 9004	. . . 4 $/\$ $/\$
95	93	nnap0d 9036	. . . 4 $/\$ $/\$ #
96	94, 95	div0apd 8814	. . 3 $/\$ $/\$
97		mulcl 8006	. . . . . 6 $/\$
98	97	adantl 277	. . . . 5 $/\$ $/\$ $/\$ $/\$
99		eluzelcn 9612	. . . . . . 7
100	99	adantl 277	. . . . . 6 $/\$ $/\$ $/\$
101	32, 100	eqeltrid 2283	. . . . 5 $/\$ $/\$ $/\$
102		simpr 110	. . . . . 6 $/\$ $/\$ $/\$
103	102	mul02d 8418	. . . . 5 $/\$ $/\$ $/\$
104	102	mul01d 8419	. . . . 5 $/\$ $/\$ $/\$
105		simpr 110	. . . . . . . . 9 $/\$ $/\$
106		nn0uz 9636	. . . . . . . . . . . 12
107	92, 106	eleqtrdi 2289	. . . . . . . . . . 11 $/\$ $/\$
108		simpll 527	. . . . . . . . . . . 12 $/\$ $/\$
109	108	nn0zd 9446	. . . . . . . . . . 11 $/\$ $/\$
110		elfz5 10092	. . . . . . . . . . 11 $/\$
111	107, 109, 110	syl2anc 411	. . . . . . . . . 10 $/\$ $/\$
112		nn0re 9258	. . . . . . . . . . . 12
113	112	ad2antrr 488	. . . . . . . . . . 11 $/\$ $/\$
114		nnre 8997	. . . . . . . . . . . 12
115	114	ad2antlr 489	. . . . . . . . . . 11 $/\$ $/\$
116	113, 115	subge0d 8562	. . . . . . . . . 10 $/\$ $/\$
117	111, 116	bitr4d 191	. . . . . . . . 9 $/\$ $/\$
118	105, 117	mtbid 673	. . . . . . . 8 $/\$ $/\$
119		simpl 109	. . . . . . . . . . . 12 $/\$
120	119	nn0zd 9446	. . . . . . . . . . 11 $/\$
121		simpr 110	. . . . . . . . . . . 12 $/\$
122	121	nnzd 9447	. . . . . . . . . . 11 $/\$
123	120, 122	zsubcld 9453	. . . . . . . . . 10 $/\$
124	123	adantr 276	. . . . . . . . 9 $/\$ $/\$
125		0z 9337	. . . . . . . . 9
126		zltnle 9372	. . . . . . . . 9 $/\$
127	124, 125, 126	sylancl 413	. . . . . . . 8 $/\$ $/\$
128	118, 127	mpbird 167	. . . . . . 7 $/\$ $/\$
129		zltp1le 9380	. . . . . . . 8 $/\$
130	124, 125, 129	sylancl 413	. . . . . . 7 $/\$ $/\$
131	128, 130	mpbid 147	. . . . . 6 $/\$ $/\$
132		nn0ge0 9274	. . . . . . 7
133	132	ad2antrr 488	. . . . . 6 $/\$ $/\$
134		0zd 9338	. . . . . . 7 $/\$ $/\$
135	124	peano2zd 9451	. . . . . . 7 $/\$ $/\$
136		elfz 10089	. . . . . . 7 $/\$ $/\$ $/\$
137	134, 135, 109, 136	syl3anc 1249	. . . . . 6 $/\$ $/\$ $/\$
138	131, 133, 137	mpbir2and 946	. . . . 5 $/\$ $/\$
139		0cn 8018	. . . . . 6
140		fvi 5618	. . . . . 6
141	139, 140	mp1i 10	. . . . 5 $/\$ $/\$
142	98, 101, 103, 104, 138, 141	seq3z 10620	. . . 4 $/\$ $/\$
143	142	oveq1d 5937	. . 3 $/\$ $/\$
144		nnz 9345	. . . . 5
145		bcval3 10843	. . . . 5 $/\$ $/\$
146	144, 145	syl3an2 1283	. . . 4 $/\$ $/\$
147	146	3expa 1205	. . 3 $/\$ $/\$
148	96, 143, 147	3eqtr4rd 2240	. 2 $/\$ $/\$
149		0zd 9338	. . . 4 $/\$
150		fzdcel 10115	. . . 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 4033

cid 4323

cfv 5258 (class class class)co 5922

cc 7877

cr 7878

cc0 7879

c1 7880

caddc 7882

cmul 7884

clt 8061

cle 8062

cmin 8197

cdiv 8699

cn 8990

cn0 9249

cz 9326

cuz 9601

cfz 10083

cseq 10539

cfa 10817

cbc 10839

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 4148 ax-sep 4151 ax-nul 4159 ax-pow 4207 ax-pr 4242 ax-un 4468 ax-setind 4573 ax-iinf 4624 ax-cnex 7970 ax-resscn 7971 ax-1cn 7972 ax-1re 7973 ax-icn 7974 ax-addcl 7975 ax-addrcl 7976 ax-mulcl 7977 ax-mulrcl 7978 ax-addcom 7979 ax-mulcom 7980 ax-addass 7981 ax-mulass 7982 ax-distr 7983 ax-i2m1 7984 ax-0lt1 7985 ax-1rid 7986 ax-0id 7987 ax-rnegex 7988 ax-precex 7989 ax-cnre 7990 ax-pre-ltirr 7991 ax-pre-ltwlin 7992 ax-pre-lttrn 7993 ax-pre-apti 7994 ax-pre-ltadd 7995 ax-pre-mulgt0 7996 ax-pre-mulext 7997

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 3451 df-if 3562 df-pw 3607 df-sn 3628 df-pr 3629 df-op 3631 df-uni 3840 df-int 3875 df-iun 3918 df-br 4034 df-opab 4095 df-mpt 4096 df-tr 4132 df-id 4328 df-po 4331 df-iso 4332 df-iord 4401 df-on 4403 df-ilim 4404 df-suc 4406 df-iom 4627 df-xp 4669 df-rel 4670 df-cnv 4671 df-co 4672 df-dm 4673 df-rn 4674 df-res 4675 df-ima 4676 df-iota 5219 df-fun 5260 df-fn 5261 df-f 5262 df-f1 5263 df-fo 5264 df-f1o 5265 df-fv 5266 df-riota 5877 df-ov 5925 df-oprab 5926 df-mpo 5927 df-1st 6198 df-2nd 6199 df-recs 6363 df-frec 6449 df-pnf 8063 df-mnf 8064 df-xr 8065 df-ltxr 8066 df-le 8067 df-sub 8199 df-neg 8200 df-reap 8602 df-ap 8609 df-div 8700 df-inn 8991 df-n0 9250 df-z 9327 df-uz 9602 df-q 9694 df-fz 10084 df-seqfrec 10540 df-fac 10818 df-bc 10840

This theorem is referenced by: bcn2 10856

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