bcval5 - Intuitionistic Logic Explorer

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Theorem bcval5 11179

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, although it is no longer valid for nonpositive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 11166	. . . 4
2	1	adantl 277	. . 3 $/\$ $/\$
3		simprl 535	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
4		simprr 537	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
5	3, 4	mulcld 8336	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
6		simpr1 1034	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
7		simpr2 1035	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
8		simpr3 1036	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
9	6, 7, 8	mulassd 8339	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
10		simpll 531	. . . . . . . . . . . . 13 $/\$ $/\$
11	10	nn0zd 9745	. . . . . . . . . . . 12 $/\$ $/\$
12		simplr 533	. . . . . . . . . . . . 13 $/\$ $/\$
13	12	nnzd 9746	. . . . . . . . . . . 12 $/\$ $/\$
14	11, 13	zsubcld 9752	. . . . . . . . . . 11 $/\$ $/\$
15	14	peano2zd 9750	. . . . . . . . . 10 $/\$ $/\$
16		1red 8331	. . . . . . . . . . . 12 $/\$ $/\$
17	12	nnred 9296	. . . . . . . . . . . 12 $/\$ $/\$
18	10	nn0red 9600	. . . . . . . . . . . 12 $/\$ $/\$
19	12	nnge1d 9326	. . . . . . . . . . . 12 $/\$ $/\$
20	16, 17, 18, 19	lesub2dd 8880	. . . . . . . . . . 11 $/\$ $/\$
21	14	zred 9747	. . . . . . . . . . . 12 $/\$ $/\$
22		leaddsub 8756	. . . . . . . . . . . 12 $/\$ $/\$
23	21, 16, 18, 22	syl3anc 1278	. . . . . . . . . . 11 $/\$ $/\$
24	20, 23	mpbird 167	. . . . . . . . . 10 $/\$ $/\$
25		eluz2 9906	. . . . . . . . . 10 $/\$ $/\$
26	15, 11, 24, 25	syl3anbrc 1212	. . . . . . . . 9 $/\$ $/\$
27	26	adantrr 483	. . . . . . . 8 $/\$ $/\$ $/\$
28		simprr 537	. . . . . . . . 9 $/\$ $/\$ $/\$
29		nnuz 9937	. . . . . . . . 9
30	28, 29	eleqtrdi 2331	. . . . . . . 8 $/\$ $/\$ $/\$
31		fvi 5754	. . . . . . . . . 10
32	31	elv 2825	. . . . . . . . 9
33		eluzelcn 9912	. . . . . . . . . 10
34	33	adantl 277	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
35	32, 34	eqeltrid 2325	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
36	5, 9, 27, 30, 35	seq3split 10903	. . . . . . 7 $/\$ $/\$ $/\$
37		elfzuz3 10404	. . . . . . . . . . 11
38	37	adantl 277	. . . . . . . . . 10 $/\$ $/\$
39		eluznn 9979	. . . . . . . . . 10 $/\$
40	12, 38, 39	syl2anc 415	. . . . . . . . 9 $/\$ $/\$
41	40	adantrr 483	. . . . . . . 8 $/\$ $/\$ $/\$
42		facnn 11143	. . . . . . . 8
43	41, 42	syl 14	. . . . . . 7 $/\$ $/\$ $/\$
44		facnn 11143	. . . . . . . . 9
45	28, 44	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$
46	45	oveq1d 6090	. . . . . . 7 $/\$ $/\$ $/\$
47	36, 43, 46	3eqtr4d 2281	. . . . . 6 $/\$ $/\$ $/\$
48	47	expr 375	. . . . 5 $/\$ $/\$
49	10	faccld 11152	. . . . . . . . 9 $/\$ $/\$
50	49	nncnd 9297	. . . . . . . 8 $/\$ $/\$
51	50	mullidd 8334	. . . . . . 7 $/\$ $/\$
52	40, 42	syl 14	. . . . . . . 8 $/\$ $/\$
53	52	oveq2d 6091	. . . . . . 7 $/\$ $/\$
54	51, 53	eqtr3d 2273	. . . . . 6 $/\$ $/\$
55		fveq2 5690	. . . . . . . . 9
56		fac0 11144	. . . . . . . . 9
57	55, 56	eqtrdi 2287	. . . . . . . 8
58		oveq1 6082	. . . . . . . . . . 11
59		0p1e1 9397	. . . . . . . . . . 11
60	58, 59	eqtrdi 2287	. . . . . . . . . 10
61	60	seqeq1d 10868	. . . . . . . . 9
62	61	fveq1d 5692	. . . . . . . 8
63	57, 62	oveq12d 6093	. . . . . . 7
64	63	eqeq2d 2250	. . . . . 6
65	54, 64	syl5ibrcom 157	. . . . 5 $/\$ $/\$
66		fznn0sub 10441	. . . . . . 7
67	66	adantl 277	. . . . . 6 $/\$ $/\$
68		elnn0 9544	. . . . . 6 $\/$
69	67, 68	sylib 122	. . . . 5 $/\$ $/\$ $\/$
70	48, 65, 69	mpjaod 730	. . . 4 $/\$ $/\$
71	70	oveq1d 6090	. . 3 $/\$ $/\$
72		eqid 2238	. . . . . 6
73		fvi 5754	. . . . . . . 8
74	73	elv 2825	. . . . . . 7
75		eluzelcn 9912	. . . . . . . 8
76	75	adantl 277	. . . . . . 7 $/\$ $/\$ $/\$
77	74, 76	eqeltrid 2325	. . . . . 6 $/\$ $/\$ $/\$
78		mulcl 8296	. . . . . . 7 $/\$
79	78	adantl 277	. . . . . 6 $/\$ $/\$ $/\$ $/\$
80	72, 15, 77, 79	seqf 10879	. . . . 5 $/\$ $/\$
81	80, 26	ffvelcdmd 5835	. . . 4 $/\$ $/\$
82	12	nnnn0d 9599	. . . . . 6 $/\$ $/\$
83	82	faccld 11152	. . . . 5 $/\$ $/\$
84	83	nncnd 9297	. . . 4 $/\$ $/\$
85	67	faccld 11152	. . . . 5 $/\$ $/\$
86	85	nncnd 9297	. . . 4 $/\$ $/\$
87	83	nnap0d 9329	. . . 4 $/\$ $/\$ #
88	85	nnap0d 9329	. . . 4 $/\$ $/\$ #
89	81, 84, 86, 87, 88	divcanap5d 9137	. . 3 $/\$ $/\$
90	2, 71, 89	3eqtrd 2275	. 2 $/\$ $/\$
91		simplr 533	. . . . . . 7 $/\$ $/\$
92	91	nnnn0d 9599	. . . . . 6 $/\$ $/\$
93	92	faccld 11152	. . . . 5 $/\$ $/\$
94	93	nncnd 9297	. . . 4 $/\$ $/\$
95	93	nnap0d 9329	. . . 4 $/\$ $/\$ #
96	94, 95	div0apd 9107	. . 3 $/\$ $/\$
97		mulcl 8296	. . . . . 6 $/\$
98	97	adantl 277	. . . . 5 $/\$ $/\$ $/\$ $/\$
99		eluzelcn 9912	. . . . . . 7
100	99	adantl 277	. . . . . 6 $/\$ $/\$ $/\$
101	32, 100	eqeltrid 2325	. . . . 5 $/\$ $/\$ $/\$
102		simpr 110	. . . . . 6 $/\$ $/\$ $/\$
103	102	mul02d 8709	. . . . 5 $/\$ $/\$ $/\$
104	102	mul01d 8710	. . . . 5 $/\$ $/\$ $/\$
105		simpr 110	. . . . . . . . 9 $/\$ $/\$
106		nn0uz 9936	. . . . . . . . . . . 12
107	92, 106	eleqtrdi 2331	. . . . . . . . . . 11 $/\$ $/\$
108		simpll 531	. . . . . . . . . . . 12 $/\$ $/\$
109	108	nn0zd 9745	. . . . . . . . . . 11 $/\$ $/\$
110		elfz5 10399	. . . . . . . . . . 11 $/\$
111	107, 109, 110	syl2anc 415	. . . . . . . . . 10 $/\$ $/\$
112		nn0re 9551	. . . . . . . . . . . 12
113	112	ad2antrr 492	. . . . . . . . . . 11 $/\$ $/\$
114		nnre 9290	. . . . . . . . . . . 12
115	114	ad2antlr 493	. . . . . . . . . . 11 $/\$ $/\$
116	113, 115	subge0d 8853	. . . . . . . . . 10 $/\$ $/\$
117	111, 116	bitr4d 191	. . . . . . . . 9 $/\$ $/\$
118	105, 117	mtbid 683	. . . . . . . 8 $/\$ $/\$
119		simpl 109	. . . . . . . . . . . 12 $/\$
120	119	nn0zd 9745	. . . . . . . . . . 11 $/\$
121		simpr 110	. . . . . . . . . . . 12 $/\$
122	121	nnzd 9746	. . . . . . . . . . 11 $/\$
123	120, 122	zsubcld 9752	. . . . . . . . . 10 $/\$
124	123	adantr 276	. . . . . . . . 9 $/\$ $/\$
125		0z 9634	. . . . . . . . 9
126		zltnle 9669	. . . . . . . . 9 $/\$
127	124, 125, 126	sylancl 417	. . . . . . . 8 $/\$ $/\$
128	118, 127	mpbird 167	. . . . . . 7 $/\$ $/\$
129		zltp1le 9678	. . . . . . . 8 $/\$
130	124, 125, 129	sylancl 417	. . . . . . 7 $/\$ $/\$
131	128, 130	mpbid 147	. . . . . 6 $/\$ $/\$
132		nn0ge0 9567	. . . . . . 7
133	132	ad2antrr 492	. . . . . 6 $/\$ $/\$
134		0zd 9635	. . . . . . 7 $/\$ $/\$
135	124	peano2zd 9750	. . . . . . 7 $/\$ $/\$
136		elfz 10396	. . . . . . 7 $/\$ $/\$ $/\$
137	134, 135, 109, 136	syl3anc 1278	. . . . . 6 $/\$ $/\$ $/\$
138	131, 133, 137	mpbir2and 957	. . . . 5 $/\$ $/\$
139		0cn 8308	. . . . . 6
140		fvi 5754	. . . . . 6
141	139, 140	mp1i 10	. . . . 5 $/\$ $/\$
142	98, 101, 103, 104, 138, 141	seq3z 10943	. . . 4 $/\$ $/\$
143	142	oveq1d 6090	. . 3 $/\$ $/\$
144		nnz 9642	. . . . 5
145		bcval3 11167	. . . . 5 $/\$ $/\$
146	144, 145	syl3an2 1312	. . . 4 $/\$ $/\$
147	146	3expa 1234	. . 3 $/\$ $/\$
148	96, 143, 147	3eqtr4rd 2282	. 2 $/\$ $/\$
149		0zd 9635	. . . 4 $/\$
150		fzdcel 10423	. . . 4 $/\$ $/\$ DECID
151	122, 149, 120, 150	syl3anc 1278	. . 3 $/\$ DECID
152		exmiddc 848	. . 3 DECID $\/$
153	151, 152	syl 14	. 2 $/\$ $\/$
154	90, 148, 153	mpjaodan 810	1 $/\$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104

wb 105 $\/$ wo 720 DECID wdc 846 $/\$ w3a 1009

wceq 1402

wcel 2209

cvv 2821 class class class wbr 4125

cid 4428

cfv 5372 (class class class)co 6075

cc 8167

cr 8168

cc0 8169

c1 8170

caddc 8172

cmul 8174

clt 8350

cle 8351

cmin 8487

cdiv 8992

cn 9283

cn0 9542

cz 9623

cuz 9900

cfz 10390

cseq 10862

cfa 11141

cbc 11163

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 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4241 ax-sep 4244 ax-nul 4254 ax-pow 4306 ax-pr 4341 ax-un 4573 ax-setind 4679 ax-iinf 4730 ax-cnex 8260 ax-resscn 8261 ax-1cn 8262 ax-1re 8263 ax-icn 8264 ax-addcl 8265 ax-addrcl 8266 ax-mulcl 8267 ax-mulrcl 8268 ax-addcom 8269 ax-mulcom 8270 ax-addass 8271 ax-mulass 8272 ax-distr 8273 ax-i2m1 8274 ax-0lt1 8275 ax-1rid 8276 ax-0id 8277 ax-rnegex 8278 ax-precex 8279 ax-cnre 8280 ax-pre-ltirr 8281 ax-pre-ltwlin 8282 ax-pre-lttrn 8283 ax-pre-apti 8284 ax-pre-ltadd 8285 ax-pre-mulgt0 8286 ax-pre-mulext 8287

This theorem depends on definitions: df-bi 117 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rmo 2536 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3636 df-pw 3687 df-sn 3711 df-pr 3712 df-op 3714 df-uni 3931 df-int 3966 df-iun 4009 df-br 4126 df-opab 4188 df-mpt 4189 df-tr 4225 df-id 4433 df-po 4436 df-iso 4437 df-iord 4506 df-on 4508 df-ilim 4509 df-suc 4511 df-iom 4733 df-xp 4775 df-rel 4776 df-cnv 4777 df-co 4778 df-dm 4779 df-rn 4780 df-res 4781 df-ima 4782 df-iota 5332 df-fun 5374 df-fn 5375 df-f 5376 df-f1 5377 df-fo 5378 df-f1o 5379 df-fv 5380 df-riota 6028 df-ov 6078 df-oprab 6079 df-mpo 6080 df-1st 6364 df-2nd 6365 df-recs 6566 df-frec 6652 df-pnf 8352 df-mnf 8353 df-xr 8354 df-ltxr 8355 df-le 8356 df-sub 8489 df-neg 8490 df-reap 8893 df-ap 8900 df-div 8993 df-inn 9284 df-n0 9543 df-z 9624 df-uz 9901 df-q 9999 df-fz 10391 df-seqfrec 10863 df-fac 11142 df-bc 11164

This theorem is referenced by: bcn2 11180

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