bcval5 - Intuitionistic Logic Explorer

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Theorem bcval5 10727

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, although it is no longer valid for nonpositive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 10714	. . . 4
2	1	adantl 277	. . 3 $/\$ $/\$
3		simprl 529	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
4		simprr 531	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
5	3, 4	mulcld 7968	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
6		simpr1 1003	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
7		simpr2 1004	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
8		simpr3 1005	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
9	6, 7, 8	mulassd 7971	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
10		simpll 527	. . . . . . . . . . . . 13 $/\$ $/\$
11	10	nn0zd 9362	. . . . . . . . . . . 12 $/\$ $/\$
12		simplr 528	. . . . . . . . . . . . 13 $/\$ $/\$
13	12	nnzd 9363	. . . . . . . . . . . 12 $/\$ $/\$
14	11, 13	zsubcld 9369	. . . . . . . . . . 11 $/\$ $/\$
15	14	peano2zd 9367	. . . . . . . . . 10 $/\$ $/\$
16		1red 7963	. . . . . . . . . . . 12 $/\$ $/\$
17	12	nnred 8921	. . . . . . . . . . . 12 $/\$ $/\$
18	10	nn0red 9219	. . . . . . . . . . . 12 $/\$ $/\$
19	12	nnge1d 8951	. . . . . . . . . . . 12 $/\$ $/\$
20	16, 17, 18, 19	lesub2dd 8509	. . . . . . . . . . 11 $/\$ $/\$
21	14	zred 9364	. . . . . . . . . . . 12 $/\$ $/\$
22		leaddsub 8385	. . . . . . . . . . . 12 $/\$ $/\$
23	21, 16, 18, 22	syl3anc 1238	. . . . . . . . . . 11 $/\$ $/\$
24	20, 23	mpbird 167	. . . . . . . . . 10 $/\$ $/\$
25		eluz2 9523	. . . . . . . . . 10 $/\$ $/\$
26	15, 11, 24, 25	syl3anbrc 1181	. . . . . . . . 9 $/\$ $/\$
27	26	adantrr 479	. . . . . . . 8 $/\$ $/\$ $/\$
28		simprr 531	. . . . . . . . 9 $/\$ $/\$ $/\$
29		nnuz 9552	. . . . . . . . 9
30	28, 29	eleqtrdi 2270	. . . . . . . 8 $/\$ $/\$ $/\$
31		fvi 5569	. . . . . . . . . 10
32	31	elv 2741	. . . . . . . . 9
33		eluzelcn 9528	. . . . . . . . . 10
34	33	adantl 277	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
35	32, 34	eqeltrid 2264	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
36	5, 9, 27, 30, 35	seq3split 10465	. . . . . . 7 $/\$ $/\$ $/\$
37		elfzuz3 10008	. . . . . . . . . . 11
38	37	adantl 277	. . . . . . . . . 10 $/\$ $/\$
39		eluznn 9589	. . . . . . . . . 10 $/\$
40	12, 38, 39	syl2anc 411	. . . . . . . . 9 $/\$ $/\$
41	40	adantrr 479	. . . . . . . 8 $/\$ $/\$ $/\$
42		facnn 10691	. . . . . . . 8
43	41, 42	syl 14	. . . . . . 7 $/\$ $/\$ $/\$
44		facnn 10691	. . . . . . . . 9
45	28, 44	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$
46	45	oveq1d 5884	. . . . . . 7 $/\$ $/\$ $/\$
47	36, 43, 46	3eqtr4d 2220	. . . . . 6 $/\$ $/\$ $/\$
48	47	expr 375	. . . . 5 $/\$ $/\$
49	10	faccld 10700	. . . . . . . . 9 $/\$ $/\$
50	49	nncnd 8922	. . . . . . . 8 $/\$ $/\$
51	50	mulid2d 7966	. . . . . . 7 $/\$ $/\$
52	40, 42	syl 14	. . . . . . . 8 $/\$ $/\$
53	52	oveq2d 5885	. . . . . . 7 $/\$ $/\$
54	51, 53	eqtr3d 2212	. . . . . 6 $/\$ $/\$
55		fveq2 5511	. . . . . . . . 9
56		fac0 10692	. . . . . . . . 9
57	55, 56	eqtrdi 2226	. . . . . . . 8
58		oveq1 5876	. . . . . . . . . . 11
59		0p1e1 9022	. . . . . . . . . . 11
60	58, 59	eqtrdi 2226	. . . . . . . . . 10
61	60	seqeq1d 10437	. . . . . . . . 9
62	61	fveq1d 5513	. . . . . . . 8
63	57, 62	oveq12d 5887	. . . . . . 7
64	63	eqeq2d 2189	. . . . . 6
65	54, 64	syl5ibrcom 157	. . . . 5 $/\$ $/\$
66		fznn0sub 10043	. . . . . . 7
67	66	adantl 277	. . . . . 6 $/\$ $/\$
68		elnn0 9167	. . . . . 6 $\/$
69	67, 68	sylib 122	. . . . 5 $/\$ $/\$ $\/$
70	48, 65, 69	mpjaod 718	. . . 4 $/\$ $/\$
71	70	oveq1d 5884	. . 3 $/\$ $/\$
72		eqid 2177	. . . . . 6
73		fvi 5569	. . . . . . . 8
74	73	elv 2741	. . . . . . 7
75		eluzelcn 9528	. . . . . . . 8
76	75	adantl 277	. . . . . . 7 $/\$ $/\$ $/\$
77	74, 76	eqeltrid 2264	. . . . . 6 $/\$ $/\$ $/\$
78		mulcl 7929	. . . . . . 7 $/\$
79	78	adantl 277	. . . . . 6 $/\$ $/\$ $/\$ $/\$
80	72, 15, 77, 79	seqf 10447	. . . . 5 $/\$ $/\$
81	80, 26	ffvelcdmd 5648	. . . 4 $/\$ $/\$
82	12	nnnn0d 9218	. . . . . 6 $/\$ $/\$
83	82	faccld 10700	. . . . 5 $/\$ $/\$
84	83	nncnd 8922	. . . 4 $/\$ $/\$
85	67	faccld 10700	. . . . 5 $/\$ $/\$
86	85	nncnd 8922	. . . 4 $/\$ $/\$
87	83	nnap0d 8954	. . . 4 $/\$ $/\$ #
88	85	nnap0d 8954	. . . 4 $/\$ $/\$ #
89	81, 84, 86, 87, 88	divcanap5d 8763	. . 3 $/\$ $/\$
90	2, 71, 89	3eqtrd 2214	. 2 $/\$ $/\$
91		simplr 528	. . . . . . 7 $/\$ $/\$
92	91	nnnn0d 9218	. . . . . 6 $/\$ $/\$
93	92	faccld 10700	. . . . 5 $/\$ $/\$
94	93	nncnd 8922	. . . 4 $/\$ $/\$
95	93	nnap0d 8954	. . . 4 $/\$ $/\$ #
96	94, 95	div0apd 8733	. . 3 $/\$ $/\$
97		mulcl 7929	. . . . . 6 $/\$
98	97	adantl 277	. . . . 5 $/\$ $/\$ $/\$ $/\$
99		eluzelcn 9528	. . . . . . 7
100	99	adantl 277	. . . . . 6 $/\$ $/\$ $/\$
101	32, 100	eqeltrid 2264	. . . . 5 $/\$ $/\$ $/\$
102		simpr 110	. . . . . 6 $/\$ $/\$ $/\$
103	102	mul02d 8339	. . . . 5 $/\$ $/\$ $/\$
104	102	mul01d 8340	. . . . 5 $/\$ $/\$ $/\$
105		simpr 110	. . . . . . . . 9 $/\$ $/\$
106		nn0uz 9551	. . . . . . . . . . . 12
107	92, 106	eleqtrdi 2270	. . . . . . . . . . 11 $/\$ $/\$
108		simpll 527	. . . . . . . . . . . 12 $/\$ $/\$
109	108	nn0zd 9362	. . . . . . . . . . 11 $/\$ $/\$
110		elfz5 10003	. . . . . . . . . . 11 $/\$
111	107, 109, 110	syl2anc 411	. . . . . . . . . 10 $/\$ $/\$
112		nn0re 9174	. . . . . . . . . . . 12
113	112	ad2antrr 488	. . . . . . . . . . 11 $/\$ $/\$
114		nnre 8915	. . . . . . . . . . . 12
115	114	ad2antlr 489	. . . . . . . . . . 11 $/\$ $/\$
116	113, 115	subge0d 8482	. . . . . . . . . 10 $/\$ $/\$
117	111, 116	bitr4d 191	. . . . . . . . 9 $/\$ $/\$
118	105, 117	mtbid 672	. . . . . . . 8 $/\$ $/\$
119		simpl 109	. . . . . . . . . . . 12 $/\$
120	119	nn0zd 9362	. . . . . . . . . . 11 $/\$
121		simpr 110	. . . . . . . . . . . 12 $/\$
122	121	nnzd 9363	. . . . . . . . . . 11 $/\$
123	120, 122	zsubcld 9369	. . . . . . . . . 10 $/\$
124	123	adantr 276	. . . . . . . . 9 $/\$ $/\$
125		0z 9253	. . . . . . . . 9
126		zltnle 9288	. . . . . . . . 9 $/\$
127	124, 125, 126	sylancl 413	. . . . . . . 8 $/\$ $/\$
128	118, 127	mpbird 167	. . . . . . 7 $/\$ $/\$
129		zltp1le 9296	. . . . . . . 8 $/\$
130	124, 125, 129	sylancl 413	. . . . . . 7 $/\$ $/\$
131	128, 130	mpbid 147	. . . . . 6 $/\$ $/\$
132		nn0ge0 9190	. . . . . . 7
133	132	ad2antrr 488	. . . . . 6 $/\$ $/\$
134		0zd 9254	. . . . . . 7 $/\$ $/\$
135	124	peano2zd 9367	. . . . . . 7 $/\$ $/\$
136		elfz 10001	. . . . . . 7 $/\$ $/\$ $/\$
137	134, 135, 109, 136	syl3anc 1238	. . . . . 6 $/\$ $/\$ $/\$
138	131, 133, 137	mpbir2and 944	. . . . 5 $/\$ $/\$
139		0cn 7940	. . . . . 6
140		fvi 5569	. . . . . 6
141	139, 140	mp1i 10	. . . . 5 $/\$ $/\$
142	98, 101, 103, 104, 138, 141	seq3z 10497	. . . 4 $/\$ $/\$
143	142	oveq1d 5884	. . 3 $/\$ $/\$
144		nnz 9261	. . . . 5
145		bcval3 10715	. . . . 5 $/\$ $/\$
146	144, 145	syl3an2 1272	. . . 4 $/\$ $/\$
147	146	3expa 1203	. . 3 $/\$ $/\$
148	96, 143, 147	3eqtr4rd 2221	. 2 $/\$ $/\$
149		0zd 9254	. . . 4 $/\$
150		fzdcel 10026	. . . 4 $/\$ $/\$ DECID
151	122, 149, 120, 150	syl3anc 1238	. . 3 $/\$ DECID
152		exmiddc 836	. . 3 DECID $\/$
153	151, 152	syl 14	. 2 $/\$ $\/$
154	90, 148, 153	mpjaodan 798	1 $/\$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104

wb 105 $\/$ wo 708 DECID wdc 834 $/\$ w3a 978

wceq 1353

wcel 2148

cvv 2737 class class class wbr 4000

cid 4285

cfv 5212 (class class class)co 5869

cc 7800

cr 7801

cc0 7802

c1 7803

caddc 7805

cmul 7807

clt 7982

cle 7983

cmin 8118

cdiv 8618

cn 8908

cn0 9165

cz 9242

cuz 9517

cfz 9995

cseq 10431

cfa 10689

cbc 10711

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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4115 ax-sep 4118 ax-nul 4126 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 ax-iinf 4584 ax-cnex 7893 ax-resscn 7894 ax-1cn 7895 ax-1re 7896 ax-icn 7897 ax-addcl 7898 ax-addrcl 7899 ax-mulcl 7900 ax-mulrcl 7901 ax-addcom 7902 ax-mulcom 7903 ax-addass 7904 ax-mulass 7905 ax-distr 7906 ax-i2m1 7907 ax-0lt1 7908 ax-1rid 7909 ax-0id 7910 ax-rnegex 7911 ax-precex 7912 ax-cnre 7913 ax-pre-ltirr 7914 ax-pre-ltwlin 7915 ax-pre-lttrn 7916 ax-pre-apti 7917 ax-pre-ltadd 7918 ax-pre-mulgt0 7919 ax-pre-mulext 7920

This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-if 3535 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-tr 4099 df-id 4290 df-po 4293 df-iso 4294 df-iord 4363 df-on 4365 df-ilim 4366 df-suc 4368 df-iom 4587 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-res 4635 df-ima 4636 df-iota 5174 df-fun 5214 df-fn 5215 df-f 5216 df-f1 5217 df-fo 5218 df-f1o 5219 df-fv 5220 df-riota 5825 df-ov 5872 df-oprab 5873 df-mpo 5874 df-1st 6135 df-2nd 6136 df-recs 6300 df-frec 6386 df-pnf 7984 df-mnf 7985 df-xr 7986 df-ltxr 7987 df-le 7988 df-sub 8120 df-neg 8121 df-reap 8522 df-ap 8529 df-div 8619 df-inn 8909 df-n0 9166 df-z 9243 df-uz 9518 df-q 9609 df-fz 9996 df-seqfrec 10432 df-fac 10690 df-bc 10712

This theorem is referenced by: bcn2 10728

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