bcval5 - Intuitionistic Logic Explorer

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Theorem bcval5 10910

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, although it is no longer valid for nonpositive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 10897	. . . 4
2	1	adantl 277	. . 3 $/\$ $/\$
3		simprl 529	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
4		simprr 531	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
5	3, 4	mulcld 8095	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
6		simpr1 1006	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
7		simpr2 1007	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
8		simpr3 1008	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
9	6, 7, 8	mulassd 8098	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
10		simpll 527	. . . . . . . . . . . . 13 $/\$ $/\$
11	10	nn0zd 9495	. . . . . . . . . . . 12 $/\$ $/\$
12		simplr 528	. . . . . . . . . . . . 13 $/\$ $/\$
13	12	nnzd 9496	. . . . . . . . . . . 12 $/\$ $/\$
14	11, 13	zsubcld 9502	. . . . . . . . . . 11 $/\$ $/\$
15	14	peano2zd 9500	. . . . . . . . . 10 $/\$ $/\$
16		1red 8089	. . . . . . . . . . . 12 $/\$ $/\$
17	12	nnred 9051	. . . . . . . . . . . 12 $/\$ $/\$
18	10	nn0red 9351	. . . . . . . . . . . 12 $/\$ $/\$
19	12	nnge1d 9081	. . . . . . . . . . . 12 $/\$ $/\$
20	16, 17, 18, 19	lesub2dd 8637	. . . . . . . . . . 11 $/\$ $/\$
21	14	zred 9497	. . . . . . . . . . . 12 $/\$ $/\$
22		leaddsub 8513	. . . . . . . . . . . 12 $/\$ $/\$
23	21, 16, 18, 22	syl3anc 1250	. . . . . . . . . . 11 $/\$ $/\$
24	20, 23	mpbird 167	. . . . . . . . . 10 $/\$ $/\$
25		eluz2 9656	. . . . . . . . . 10 $/\$ $/\$
26	15, 11, 24, 25	syl3anbrc 1184	. . . . . . . . 9 $/\$ $/\$
27	26	adantrr 479	. . . . . . . 8 $/\$ $/\$ $/\$
28		simprr 531	. . . . . . . . 9 $/\$ $/\$ $/\$
29		nnuz 9686	. . . . . . . . 9
30	28, 29	eleqtrdi 2298	. . . . . . . 8 $/\$ $/\$ $/\$
31		fvi 5638	. . . . . . . . . 10
32	31	elv 2776	. . . . . . . . 9
33		eluzelcn 9661	. . . . . . . . . 10
34	33	adantl 277	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
35	32, 34	eqeltrid 2292	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
36	5, 9, 27, 30, 35	seq3split 10635	. . . . . . 7 $/\$ $/\$ $/\$
37		elfzuz3 10146	. . . . . . . . . . 11
38	37	adantl 277	. . . . . . . . . 10 $/\$ $/\$
39		eluznn 9723	. . . . . . . . . 10 $/\$
40	12, 38, 39	syl2anc 411	. . . . . . . . 9 $/\$ $/\$
41	40	adantrr 479	. . . . . . . 8 $/\$ $/\$ $/\$
42		facnn 10874	. . . . . . . 8
43	41, 42	syl 14	. . . . . . 7 $/\$ $/\$ $/\$
44		facnn 10874	. . . . . . . . 9
45	28, 44	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$
46	45	oveq1d 5961	. . . . . . 7 $/\$ $/\$ $/\$
47	36, 43, 46	3eqtr4d 2248	. . . . . 6 $/\$ $/\$ $/\$
48	47	expr 375	. . . . 5 $/\$ $/\$
49	10	faccld 10883	. . . . . . . . 9 $/\$ $/\$
50	49	nncnd 9052	. . . . . . . 8 $/\$ $/\$
51	50	mulid2d 8093	. . . . . . 7 $/\$ $/\$
52	40, 42	syl 14	. . . . . . . 8 $/\$ $/\$
53	52	oveq2d 5962	. . . . . . 7 $/\$ $/\$
54	51, 53	eqtr3d 2240	. . . . . 6 $/\$ $/\$
55		fveq2 5578	. . . . . . . . 9
56		fac0 10875	. . . . . . . . 9
57	55, 56	eqtrdi 2254	. . . . . . . 8
58		oveq1 5953	. . . . . . . . . . 11
59		0p1e1 9152	. . . . . . . . . . 11
60	58, 59	eqtrdi 2254	. . . . . . . . . 10
61	60	seqeq1d 10600	. . . . . . . . 9
62	61	fveq1d 5580	. . . . . . . 8
63	57, 62	oveq12d 5964	. . . . . . 7
64	63	eqeq2d 2217	. . . . . 6
65	54, 64	syl5ibrcom 157	. . . . 5 $/\$ $/\$
66		fznn0sub 10181	. . . . . . 7
67	66	adantl 277	. . . . . 6 $/\$ $/\$
68		elnn0 9299	. . . . . 6 $\/$
69	67, 68	sylib 122	. . . . 5 $/\$ $/\$ $\/$
70	48, 65, 69	mpjaod 720	. . . 4 $/\$ $/\$
71	70	oveq1d 5961	. . 3 $/\$ $/\$
72		eqid 2205	. . . . . 6
73		fvi 5638	. . . . . . . 8
74	73	elv 2776	. . . . . . 7
75		eluzelcn 9661	. . . . . . . 8
76	75	adantl 277	. . . . . . 7 $/\$ $/\$ $/\$
77	74, 76	eqeltrid 2292	. . . . . 6 $/\$ $/\$ $/\$
78		mulcl 8054	. . . . . . 7 $/\$
79	78	adantl 277	. . . . . 6 $/\$ $/\$ $/\$ $/\$
80	72, 15, 77, 79	seqf 10611	. . . . 5 $/\$ $/\$
81	80, 26	ffvelcdmd 5718	. . . 4 $/\$ $/\$
82	12	nnnn0d 9350	. . . . . 6 $/\$ $/\$
83	82	faccld 10883	. . . . 5 $/\$ $/\$
84	83	nncnd 9052	. . . 4 $/\$ $/\$
85	67	faccld 10883	. . . . 5 $/\$ $/\$
86	85	nncnd 9052	. . . 4 $/\$ $/\$
87	83	nnap0d 9084	. . . 4 $/\$ $/\$ #
88	85	nnap0d 9084	. . . 4 $/\$ $/\$ #
89	81, 84, 86, 87, 88	divcanap5d 8892	. . 3 $/\$ $/\$
90	2, 71, 89	3eqtrd 2242	. 2 $/\$ $/\$
91		simplr 528	. . . . . . 7 $/\$ $/\$
92	91	nnnn0d 9350	. . . . . 6 $/\$ $/\$
93	92	faccld 10883	. . . . 5 $/\$ $/\$
94	93	nncnd 9052	. . . 4 $/\$ $/\$
95	93	nnap0d 9084	. . . 4 $/\$ $/\$ #
96	94, 95	div0apd 8862	. . 3 $/\$ $/\$
97		mulcl 8054	. . . . . 6 $/\$
98	97	adantl 277	. . . . 5 $/\$ $/\$ $/\$ $/\$
99		eluzelcn 9661	. . . . . . 7
100	99	adantl 277	. . . . . 6 $/\$ $/\$ $/\$
101	32, 100	eqeltrid 2292	. . . . 5 $/\$ $/\$ $/\$
102		simpr 110	. . . . . 6 $/\$ $/\$ $/\$
103	102	mul02d 8466	. . . . 5 $/\$ $/\$ $/\$
104	102	mul01d 8467	. . . . 5 $/\$ $/\$ $/\$
105		simpr 110	. . . . . . . . 9 $/\$ $/\$
106		nn0uz 9685	. . . . . . . . . . . 12
107	92, 106	eleqtrdi 2298	. . . . . . . . . . 11 $/\$ $/\$
108		simpll 527	. . . . . . . . . . . 12 $/\$ $/\$
109	108	nn0zd 9495	. . . . . . . . . . 11 $/\$ $/\$
110		elfz5 10141	. . . . . . . . . . 11 $/\$
111	107, 109, 110	syl2anc 411	. . . . . . . . . 10 $/\$ $/\$
112		nn0re 9306	. . . . . . . . . . . 12
113	112	ad2antrr 488	. . . . . . . . . . 11 $/\$ $/\$
114		nnre 9045	. . . . . . . . . . . 12
115	114	ad2antlr 489	. . . . . . . . . . 11 $/\$ $/\$
116	113, 115	subge0d 8610	. . . . . . . . . 10 $/\$ $/\$
117	111, 116	bitr4d 191	. . . . . . . . 9 $/\$ $/\$
118	105, 117	mtbid 674	. . . . . . . 8 $/\$ $/\$
119		simpl 109	. . . . . . . . . . . 12 $/\$
120	119	nn0zd 9495	. . . . . . . . . . 11 $/\$
121		simpr 110	. . . . . . . . . . . 12 $/\$
122	121	nnzd 9496	. . . . . . . . . . 11 $/\$
123	120, 122	zsubcld 9502	. . . . . . . . . 10 $/\$
124	123	adantr 276	. . . . . . . . 9 $/\$ $/\$
125		0z 9385	. . . . . . . . 9
126		zltnle 9420	. . . . . . . . 9 $/\$
127	124, 125, 126	sylancl 413	. . . . . . . 8 $/\$ $/\$
128	118, 127	mpbird 167	. . . . . . 7 $/\$ $/\$
129		zltp1le 9429	. . . . . . . 8 $/\$
130	124, 125, 129	sylancl 413	. . . . . . 7 $/\$ $/\$
131	128, 130	mpbid 147	. . . . . 6 $/\$ $/\$
132		nn0ge0 9322	. . . . . . 7
133	132	ad2antrr 488	. . . . . 6 $/\$ $/\$
134		0zd 9386	. . . . . . 7 $/\$ $/\$
135	124	peano2zd 9500	. . . . . . 7 $/\$ $/\$
136		elfz 10138	. . . . . . 7 $/\$ $/\$ $/\$
137	134, 135, 109, 136	syl3anc 1250	. . . . . 6 $/\$ $/\$ $/\$
138	131, 133, 137	mpbir2and 947	. . . . 5 $/\$ $/\$
139		0cn 8066	. . . . . 6
140		fvi 5638	. . . . . 6
141	139, 140	mp1i 10	. . . . 5 $/\$ $/\$
142	98, 101, 103, 104, 138, 141	seq3z 10675	. . . 4 $/\$ $/\$
143	142	oveq1d 5961	. . 3 $/\$ $/\$
144		nnz 9393	. . . . 5
145		bcval3 10898	. . . . 5 $/\$ $/\$
146	144, 145	syl3an2 1284	. . . 4 $/\$ $/\$
147	146	3expa 1206	. . 3 $/\$ $/\$
148	96, 143, 147	3eqtr4rd 2249	. 2 $/\$ $/\$
149		0zd 9386	. . . 4 $/\$
150		fzdcel 10164	. . . 4 $/\$ $/\$ DECID
151	122, 149, 120, 150	syl3anc 1250	. . 3 $/\$ DECID
152		exmiddc 838	. . 3 DECID $\/$
153	151, 152	syl 14	. 2 $/\$ $\/$
154	90, 148, 153	mpjaodan 800	1 $/\$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104

wb 105 $\/$ wo 710 DECID wdc 836 $/\$ w3a 981

wceq 1373

wcel 2176

cvv 2772 class class class wbr 4045

cid 4336

cfv 5272 (class class class)co 5946

cc 7925

cr 7926

cc0 7927

c1 7928

caddc 7930

cmul 7932

clt 8109

cle 8110

cmin 8245

cdiv 8747

cn 9038

cn0 9297

cz 9374

cuz 9650

cfz 10132

cseq 10594

cfa 10872

cbc 10894

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 711 ax-5 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-13 2178 ax-14 2179 ax-ext 2187 ax-coll 4160 ax-sep 4163 ax-nul 4171 ax-pow 4219 ax-pr 4254 ax-un 4481 ax-setind 4586 ax-iinf 4637 ax-cnex 8018 ax-resscn 8019 ax-1cn 8020 ax-1re 8021 ax-icn 8022 ax-addcl 8023 ax-addrcl 8024 ax-mulcl 8025 ax-mulrcl 8026 ax-addcom 8027 ax-mulcom 8028 ax-addass 8029 ax-mulass 8030 ax-distr 8031 ax-i2m1 8032 ax-0lt1 8033 ax-1rid 8034 ax-0id 8035 ax-rnegex 8036 ax-precex 8037 ax-cnre 8038 ax-pre-ltirr 8039 ax-pre-ltwlin 8040 ax-pre-lttrn 8041 ax-pre-apti 8042 ax-pre-ltadd 8043 ax-pre-mulgt0 8044 ax-pre-mulext 8045

This theorem depends on definitions: df-bi 117 df-dc 837 df-3or 982 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1484 df-sb 1786 df-eu 2057 df-mo 2058 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-ne 2377 df-nel 2472 df-ral 2489 df-rex 2490 df-reu 2491 df-rmo 2492 df-rab 2493 df-v 2774 df-sbc 2999 df-csb 3094 df-dif 3168 df-un 3170 df-in 3172 df-ss 3179 df-nul 3461 df-if 3572 df-pw 3618 df-sn 3639 df-pr 3640 df-op 3642 df-uni 3851 df-int 3886 df-iun 3929 df-br 4046 df-opab 4107 df-mpt 4108 df-tr 4144 df-id 4341 df-po 4344 df-iso 4345 df-iord 4414 df-on 4416 df-ilim 4417 df-suc 4419 df-iom 4640 df-xp 4682 df-rel 4683 df-cnv 4684 df-co 4685 df-dm 4686 df-rn 4687 df-res 4688 df-ima 4689 df-iota 5233 df-fun 5274 df-fn 5275 df-f 5276 df-f1 5277 df-fo 5278 df-f1o 5279 df-fv 5280 df-riota 5901 df-ov 5949 df-oprab 5950 df-mpo 5951 df-1st 6228 df-2nd 6229 df-recs 6393 df-frec 6479 df-pnf 8111 df-mnf 8112 df-xr 8113 df-ltxr 8114 df-le 8115 df-sub 8247 df-neg 8248 df-reap 8650 df-ap 8657 df-div 8748 df-inn 9039 df-n0 9298 df-z 9375 df-uz 9651 df-q 9743 df-fz 10133 df-seqfrec 10595 df-fac 10873 df-bc 10895

This theorem is referenced by: bcn2 10911

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