bcval5 - Intuitionistic Logic Explorer

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Theorem bcval5 10734

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, although it is no longer valid for nonpositive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 10721	. . . 4
2	1	adantl 277	. . 3 $/\$ $/\$
3		simprl 529	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
4		simprr 531	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
5	3, 4	mulcld 7972	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
6		simpr1 1003	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
7		simpr2 1004	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
8		simpr3 1005	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
9	6, 7, 8	mulassd 7975	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
10		simpll 527	. . . . . . . . . . . . 13 $/\$ $/\$
11	10	nn0zd 9367	. . . . . . . . . . . 12 $/\$ $/\$
12		simplr 528	. . . . . . . . . . . . 13 $/\$ $/\$
13	12	nnzd 9368	. . . . . . . . . . . 12 $/\$ $/\$
14	11, 13	zsubcld 9374	. . . . . . . . . . 11 $/\$ $/\$
15	14	peano2zd 9372	. . . . . . . . . 10 $/\$ $/\$
16		1red 7967	. . . . . . . . . . . 12 $/\$ $/\$
17	12	nnred 8926	. . . . . . . . . . . 12 $/\$ $/\$
18	10	nn0red 9224	. . . . . . . . . . . 12 $/\$ $/\$
19	12	nnge1d 8956	. . . . . . . . . . . 12 $/\$ $/\$
20	16, 17, 18, 19	lesub2dd 8513	. . . . . . . . . . 11 $/\$ $/\$
21	14	zred 9369	. . . . . . . . . . . 12 $/\$ $/\$
22		leaddsub 8389	. . . . . . . . . . . 12 $/\$ $/\$
23	21, 16, 18, 22	syl3anc 1238	. . . . . . . . . . 11 $/\$ $/\$
24	20, 23	mpbird 167	. . . . . . . . . 10 $/\$ $/\$
25		eluz2 9528	. . . . . . . . . 10 $/\$ $/\$
26	15, 11, 24, 25	syl3anbrc 1181	. . . . . . . . 9 $/\$ $/\$
27	26	adantrr 479	. . . . . . . 8 $/\$ $/\$ $/\$
28		simprr 531	. . . . . . . . 9 $/\$ $/\$ $/\$
29		nnuz 9557	. . . . . . . . 9
30	28, 29	eleqtrdi 2270	. . . . . . . 8 $/\$ $/\$ $/\$
31		fvi 5570	. . . . . . . . . 10
32	31	elv 2741	. . . . . . . . 9
33		eluzelcn 9533	. . . . . . . . . 10
34	33	adantl 277	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
35	32, 34	eqeltrid 2264	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
36	5, 9, 27, 30, 35	seq3split 10472	. . . . . . 7 $/\$ $/\$ $/\$
37		elfzuz3 10015	. . . . . . . . . . 11
38	37	adantl 277	. . . . . . . . . 10 $/\$ $/\$
39		eluznn 9594	. . . . . . . . . 10 $/\$
40	12, 38, 39	syl2anc 411	. . . . . . . . 9 $/\$ $/\$
41	40	adantrr 479	. . . . . . . 8 $/\$ $/\$ $/\$
42		facnn 10698	. . . . . . . 8
43	41, 42	syl 14	. . . . . . 7 $/\$ $/\$ $/\$
44		facnn 10698	. . . . . . . . 9
45	28, 44	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$
46	45	oveq1d 5885	. . . . . . 7 $/\$ $/\$ $/\$
47	36, 43, 46	3eqtr4d 2220	. . . . . 6 $/\$ $/\$ $/\$
48	47	expr 375	. . . . 5 $/\$ $/\$
49	10	faccld 10707	. . . . . . . . 9 $/\$ $/\$
50	49	nncnd 8927	. . . . . . . 8 $/\$ $/\$
51	50	mulid2d 7970	. . . . . . 7 $/\$ $/\$
52	40, 42	syl 14	. . . . . . . 8 $/\$ $/\$
53	52	oveq2d 5886	. . . . . . 7 $/\$ $/\$
54	51, 53	eqtr3d 2212	. . . . . 6 $/\$ $/\$
55		fveq2 5512	. . . . . . . . 9
56		fac0 10699	. . . . . . . . 9
57	55, 56	eqtrdi 2226	. . . . . . . 8
58		oveq1 5877	. . . . . . . . . . 11
59		0p1e1 9027	. . . . . . . . . . 11
60	58, 59	eqtrdi 2226	. . . . . . . . . 10
61	60	seqeq1d 10444	. . . . . . . . 9
62	61	fveq1d 5514	. . . . . . . 8
63	57, 62	oveq12d 5888	. . . . . . 7
64	63	eqeq2d 2189	. . . . . 6
65	54, 64	syl5ibrcom 157	. . . . 5 $/\$ $/\$
66		fznn0sub 10050	. . . . . . 7
67	66	adantl 277	. . . . . 6 $/\$ $/\$
68		elnn0 9172	. . . . . 6 $\/$
69	67, 68	sylib 122	. . . . 5 $/\$ $/\$ $\/$
70	48, 65, 69	mpjaod 718	. . . 4 $/\$ $/\$
71	70	oveq1d 5885	. . 3 $/\$ $/\$
72		eqid 2177	. . . . . 6
73		fvi 5570	. . . . . . . 8
74	73	elv 2741	. . . . . . 7
75		eluzelcn 9533	. . . . . . . 8
76	75	adantl 277	. . . . . . 7 $/\$ $/\$ $/\$
77	74, 76	eqeltrid 2264	. . . . . 6 $/\$ $/\$ $/\$
78		mulcl 7933	. . . . . . 7 $/\$
79	78	adantl 277	. . . . . 6 $/\$ $/\$ $/\$ $/\$
80	72, 15, 77, 79	seqf 10454	. . . . 5 $/\$ $/\$
81	80, 26	ffvelcdmd 5649	. . . 4 $/\$ $/\$
82	12	nnnn0d 9223	. . . . . 6 $/\$ $/\$
83	82	faccld 10707	. . . . 5 $/\$ $/\$
84	83	nncnd 8927	. . . 4 $/\$ $/\$
85	67	faccld 10707	. . . . 5 $/\$ $/\$
86	85	nncnd 8927	. . . 4 $/\$ $/\$
87	83	nnap0d 8959	. . . 4 $/\$ $/\$ #
88	85	nnap0d 8959	. . . 4 $/\$ $/\$ #
89	81, 84, 86, 87, 88	divcanap5d 8768	. . 3 $/\$ $/\$
90	2, 71, 89	3eqtrd 2214	. 2 $/\$ $/\$
91		simplr 528	. . . . . . 7 $/\$ $/\$
92	91	nnnn0d 9223	. . . . . 6 $/\$ $/\$
93	92	faccld 10707	. . . . 5 $/\$ $/\$
94	93	nncnd 8927	. . . 4 $/\$ $/\$
95	93	nnap0d 8959	. . . 4 $/\$ $/\$ #
96	94, 95	div0apd 8738	. . 3 $/\$ $/\$
97		mulcl 7933	. . . . . 6 $/\$
98	97	adantl 277	. . . . 5 $/\$ $/\$ $/\$ $/\$
99		eluzelcn 9533	. . . . . . 7
100	99	adantl 277	. . . . . 6 $/\$ $/\$ $/\$
101	32, 100	eqeltrid 2264	. . . . 5 $/\$ $/\$ $/\$
102		simpr 110	. . . . . 6 $/\$ $/\$ $/\$
103	102	mul02d 8343	. . . . 5 $/\$ $/\$ $/\$
104	102	mul01d 8344	. . . . 5 $/\$ $/\$ $/\$
105		simpr 110	. . . . . . . . 9 $/\$ $/\$
106		nn0uz 9556	. . . . . . . . . . . 12
107	92, 106	eleqtrdi 2270	. . . . . . . . . . 11 $/\$ $/\$
108		simpll 527	. . . . . . . . . . . 12 $/\$ $/\$
109	108	nn0zd 9367	. . . . . . . . . . 11 $/\$ $/\$
110		elfz5 10010	. . . . . . . . . . 11 $/\$
111	107, 109, 110	syl2anc 411	. . . . . . . . . 10 $/\$ $/\$
112		nn0re 9179	. . . . . . . . . . . 12
113	112	ad2antrr 488	. . . . . . . . . . 11 $/\$ $/\$
114		nnre 8920	. . . . . . . . . . . 12
115	114	ad2antlr 489	. . . . . . . . . . 11 $/\$ $/\$
116	113, 115	subge0d 8486	. . . . . . . . . 10 $/\$ $/\$
117	111, 116	bitr4d 191	. . . . . . . . 9 $/\$ $/\$
118	105, 117	mtbid 672	. . . . . . . 8 $/\$ $/\$
119		simpl 109	. . . . . . . . . . . 12 $/\$
120	119	nn0zd 9367	. . . . . . . . . . 11 $/\$
121		simpr 110	. . . . . . . . . . . 12 $/\$
122	121	nnzd 9368	. . . . . . . . . . 11 $/\$
123	120, 122	zsubcld 9374	. . . . . . . . . 10 $/\$
124	123	adantr 276	. . . . . . . . 9 $/\$ $/\$
125		0z 9258	. . . . . . . . 9
126		zltnle 9293	. . . . . . . . 9 $/\$
127	124, 125, 126	sylancl 413	. . . . . . . 8 $/\$ $/\$
128	118, 127	mpbird 167	. . . . . . 7 $/\$ $/\$
129		zltp1le 9301	. . . . . . . 8 $/\$
130	124, 125, 129	sylancl 413	. . . . . . 7 $/\$ $/\$
131	128, 130	mpbid 147	. . . . . 6 $/\$ $/\$
132		nn0ge0 9195	. . . . . . 7
133	132	ad2antrr 488	. . . . . 6 $/\$ $/\$
134		0zd 9259	. . . . . . 7 $/\$ $/\$
135	124	peano2zd 9372	. . . . . . 7 $/\$ $/\$
136		elfz 10008	. . . . . . 7 $/\$ $/\$ $/\$
137	134, 135, 109, 136	syl3anc 1238	. . . . . 6 $/\$ $/\$ $/\$
138	131, 133, 137	mpbir2and 944	. . . . 5 $/\$ $/\$
139		0cn 7944	. . . . . 6
140		fvi 5570	. . . . . 6
141	139, 140	mp1i 10	. . . . 5 $/\$ $/\$
142	98, 101, 103, 104, 138, 141	seq3z 10504	. . . 4 $/\$ $/\$
143	142	oveq1d 5885	. . 3 $/\$ $/\$
144		nnz 9266	. . . . 5
145		bcval3 10722	. . . . 5 $/\$ $/\$
146	144, 145	syl3an2 1272	. . . 4 $/\$ $/\$
147	146	3expa 1203	. . 3 $/\$ $/\$
148	96, 143, 147	3eqtr4rd 2221	. 2 $/\$ $/\$
149		0zd 9259	. . . 4 $/\$
150		fzdcel 10033	. . . 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 4001

cid 4286

cfv 5213 (class class class)co 5870

cc 7804

cr 7805

cc0 7806

c1 7807

caddc 7809

cmul 7811

clt 7986

cle 7987

cmin 8122

cdiv 8623

cn 8913

cn0 9170

cz 9247

cuz 9522

cfz 10002

cseq 10438

cfa 10696

cbc 10718

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 4116 ax-sep 4119 ax-nul 4127 ax-pow 4172 ax-pr 4207 ax-un 4431 ax-setind 4534 ax-iinf 4585 ax-cnex 7897 ax-resscn 7898 ax-1cn 7899 ax-1re 7900 ax-icn 7901 ax-addcl 7902 ax-addrcl 7903 ax-mulcl 7904 ax-mulrcl 7905 ax-addcom 7906 ax-mulcom 7907 ax-addass 7908 ax-mulass 7909 ax-distr 7910 ax-i2m1 7911 ax-0lt1 7912 ax-1rid 7913 ax-0id 7914 ax-rnegex 7915 ax-precex 7916 ax-cnre 7917 ax-pre-ltirr 7918 ax-pre-ltwlin 7919 ax-pre-lttrn 7920 ax-pre-apti 7921 ax-pre-ltadd 7922 ax-pre-mulgt0 7923 ax-pre-mulext 7924

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 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3809 df-int 3844 df-iun 3887 df-br 4002 df-opab 4063 df-mpt 4064 df-tr 4100 df-id 4291 df-po 4294 df-iso 4295 df-iord 4364 df-on 4366 df-ilim 4367 df-suc 4369 df-iom 4588 df-xp 4630 df-rel 4631 df-cnv 4632 df-co 4633 df-dm 4634 df-rn 4635 df-res 4636 df-ima 4637 df-iota 5175 df-fun 5215 df-fn 5216 df-f 5217 df-f1 5218 df-fo 5219 df-f1o 5220 df-fv 5221 df-riota 5826 df-ov 5873 df-oprab 5874 df-mpo 5875 df-1st 6136 df-2nd 6137 df-recs 6301 df-frec 6387 df-pnf 7988 df-mnf 7989 df-xr 7990 df-ltxr 7991 df-le 7992 df-sub 8124 df-neg 8125 df-reap 8526 df-ap 8533 df-div 8624 df-inn 8914 df-n0 9171 df-z 9248 df-uz 9523 df-q 9614 df-fz 10003 df-seqfrec 10439 df-fac 10697 df-bc 10719

This theorem is referenced by: bcn2 10735

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