bcval5 - Intuitionistic Logic Explorer

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Theorem bcval5 10665

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, although it is no longer valid for nonpositive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 10652	. . . 4
2	1	adantl 275	. . 3 $/\$ $/\$
3		simprl 521	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
4		simprr 522	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
5	3, 4	mulcld 7910	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
6		simpr1 992	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
7		simpr2 993	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
8		simpr3 994	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
9	6, 7, 8	mulassd 7913	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
10		simpll 519	. . . . . . . . . . . . 13 $/\$ $/\$
11	10	nn0zd 9302	. . . . . . . . . . . 12 $/\$ $/\$
12		simplr 520	. . . . . . . . . . . . 13 $/\$ $/\$
13	12	nnzd 9303	. . . . . . . . . . . 12 $/\$ $/\$
14	11, 13	zsubcld 9309	. . . . . . . . . . 11 $/\$ $/\$
15	14	peano2zd 9307	. . . . . . . . . 10 $/\$ $/\$
16		1red 7905	. . . . . . . . . . . 12 $/\$ $/\$
17	12	nnred 8861	. . . . . . . . . . . 12 $/\$ $/\$
18	10	nn0red 9159	. . . . . . . . . . . 12 $/\$ $/\$
19	12	nnge1d 8891	. . . . . . . . . . . 12 $/\$ $/\$
20	16, 17, 18, 19	lesub2dd 8451	. . . . . . . . . . 11 $/\$ $/\$
21	14	zred 9304	. . . . . . . . . . . 12 $/\$ $/\$
22		leaddsub 8327	. . . . . . . . . . . 12 $/\$ $/\$
23	21, 16, 18, 22	syl3anc 1227	. . . . . . . . . . 11 $/\$ $/\$
24	20, 23	mpbird 166	. . . . . . . . . 10 $/\$ $/\$
25		eluz2 9463	. . . . . . . . . 10 $/\$ $/\$
26	15, 11, 24, 25	syl3anbrc 1170	. . . . . . . . 9 $/\$ $/\$
27	26	adantrr 471	. . . . . . . 8 $/\$ $/\$ $/\$
28		simprr 522	. . . . . . . . 9 $/\$ $/\$ $/\$
29		nnuz 9492	. . . . . . . . 9
30	28, 29	eleqtrdi 2257	. . . . . . . 8 $/\$ $/\$ $/\$
31		fvi 5537	. . . . . . . . . 10
32	31	elv 2725	. . . . . . . . 9
33		eluzelcn 9468	. . . . . . . . . 10
34	33	adantl 275	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
35	32, 34	eqeltrid 2251	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
36	5, 9, 27, 30, 35	seq3split 10404	. . . . . . 7 $/\$ $/\$ $/\$
37		elfzuz3 9948	. . . . . . . . . . 11
38	37	adantl 275	. . . . . . . . . 10 $/\$ $/\$
39		eluznn 9529	. . . . . . . . . 10 $/\$
40	12, 38, 39	syl2anc 409	. . . . . . . . 9 $/\$ $/\$
41	40	adantrr 471	. . . . . . . 8 $/\$ $/\$ $/\$
42		facnn 10629	. . . . . . . 8
43	41, 42	syl 14	. . . . . . 7 $/\$ $/\$ $/\$
44		facnn 10629	. . . . . . . . 9
45	28, 44	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$
46	45	oveq1d 5851	. . . . . . 7 $/\$ $/\$ $/\$
47	36, 43, 46	3eqtr4d 2207	. . . . . 6 $/\$ $/\$ $/\$
48	47	expr 373	. . . . 5 $/\$ $/\$
49	10	faccld 10638	. . . . . . . . 9 $/\$ $/\$
50	49	nncnd 8862	. . . . . . . 8 $/\$ $/\$
51	50	mulid2d 7908	. . . . . . 7 $/\$ $/\$
52	40, 42	syl 14	. . . . . . . 8 $/\$ $/\$
53	52	oveq2d 5852	. . . . . . 7 $/\$ $/\$
54	51, 53	eqtr3d 2199	. . . . . 6 $/\$ $/\$
55		fveq2 5480	. . . . . . . . 9
56		fac0 10630	. . . . . . . . 9
57	55, 56	eqtrdi 2213	. . . . . . . 8
58		oveq1 5843	. . . . . . . . . . 11
59		0p1e1 8962	. . . . . . . . . . 11
60	58, 59	eqtrdi 2213	. . . . . . . . . 10
61	60	seqeq1d 10376	. . . . . . . . 9
62	61	fveq1d 5482	. . . . . . . 8
63	57, 62	oveq12d 5854	. . . . . . 7
64	63	eqeq2d 2176	. . . . . 6
65	54, 64	syl5ibrcom 156	. . . . 5 $/\$ $/\$
66		fznn0sub 9982	. . . . . . 7
67	66	adantl 275	. . . . . 6 $/\$ $/\$
68		elnn0 9107	. . . . . 6 $\/$
69	67, 68	sylib 121	. . . . 5 $/\$ $/\$ $\/$
70	48, 65, 69	mpjaod 708	. . . 4 $/\$ $/\$
71	70	oveq1d 5851	. . 3 $/\$ $/\$
72		eqid 2164	. . . . . 6
73		fvi 5537	. . . . . . . 8
74	73	elv 2725	. . . . . . 7
75		eluzelcn 9468	. . . . . . . 8
76	75	adantl 275	. . . . . . 7 $/\$ $/\$ $/\$
77	74, 76	eqeltrid 2251	. . . . . 6 $/\$ $/\$ $/\$
78		mulcl 7871	. . . . . . 7 $/\$
79	78	adantl 275	. . . . . 6 $/\$ $/\$ $/\$ $/\$
80	72, 15, 77, 79	seqf 10386	. . . . 5 $/\$ $/\$
81	80, 26	ffvelrnd 5615	. . . 4 $/\$ $/\$
82	12	nnnn0d 9158	. . . . . 6 $/\$ $/\$
83	82	faccld 10638	. . . . 5 $/\$ $/\$
84	83	nncnd 8862	. . . 4 $/\$ $/\$
85	67	faccld 10638	. . . . 5 $/\$ $/\$
86	85	nncnd 8862	. . . 4 $/\$ $/\$
87	83	nnap0d 8894	. . . 4 $/\$ $/\$ #
88	85	nnap0d 8894	. . . 4 $/\$ $/\$ #
89	81, 84, 86, 87, 88	divcanap5d 8704	. . 3 $/\$ $/\$
90	2, 71, 89	3eqtrd 2201	. 2 $/\$ $/\$
91		simplr 520	. . . . . . 7 $/\$ $/\$
92	91	nnnn0d 9158	. . . . . 6 $/\$ $/\$
93	92	faccld 10638	. . . . 5 $/\$ $/\$
94	93	nncnd 8862	. . . 4 $/\$ $/\$
95	93	nnap0d 8894	. . . 4 $/\$ $/\$ #
96	94, 95	div0apd 8674	. . 3 $/\$ $/\$
97		mulcl 7871	. . . . . 6 $/\$
98	97	adantl 275	. . . . 5 $/\$ $/\$ $/\$ $/\$
99		eluzelcn 9468	. . . . . . 7
100	99	adantl 275	. . . . . 6 $/\$ $/\$ $/\$
101	32, 100	eqeltrid 2251	. . . . 5 $/\$ $/\$ $/\$
102		simpr 109	. . . . . 6 $/\$ $/\$ $/\$
103	102	mul02d 8281	. . . . 5 $/\$ $/\$ $/\$
104	102	mul01d 8282	. . . . 5 $/\$ $/\$ $/\$
105		simpr 109	. . . . . . . . 9 $/\$ $/\$
106		nn0uz 9491	. . . . . . . . . . . 12
107	92, 106	eleqtrdi 2257	. . . . . . . . . . 11 $/\$ $/\$
108		simpll 519	. . . . . . . . . . . 12 $/\$ $/\$
109	108	nn0zd 9302	. . . . . . . . . . 11 $/\$ $/\$
110		elfz5 9943	. . . . . . . . . . 11 $/\$
111	107, 109, 110	syl2anc 409	. . . . . . . . . 10 $/\$ $/\$
112		nn0re 9114	. . . . . . . . . . . 12
113	112	ad2antrr 480	. . . . . . . . . . 11 $/\$ $/\$
114		nnre 8855	. . . . . . . . . . . 12
115	114	ad2antlr 481	. . . . . . . . . . 11 $/\$ $/\$
116	113, 115	subge0d 8424	. . . . . . . . . 10 $/\$ $/\$
117	111, 116	bitr4d 190	. . . . . . . . 9 $/\$ $/\$
118	105, 117	mtbid 662	. . . . . . . 8 $/\$ $/\$
119		simpl 108	. . . . . . . . . . . 12 $/\$
120	119	nn0zd 9302	. . . . . . . . . . 11 $/\$
121		simpr 109	. . . . . . . . . . . 12 $/\$
122	121	nnzd 9303	. . . . . . . . . . 11 $/\$
123	120, 122	zsubcld 9309	. . . . . . . . . 10 $/\$
124	123	adantr 274	. . . . . . . . 9 $/\$ $/\$
125		0z 9193	. . . . . . . . 9
126		zltnle 9228	. . . . . . . . 9 $/\$
127	124, 125, 126	sylancl 410	. . . . . . . 8 $/\$ $/\$
128	118, 127	mpbird 166	. . . . . . 7 $/\$ $/\$
129		zltp1le 9236	. . . . . . . 8 $/\$
130	124, 125, 129	sylancl 410	. . . . . . 7 $/\$ $/\$
131	128, 130	mpbid 146	. . . . . 6 $/\$ $/\$
132		nn0ge0 9130	. . . . . . 7
133	132	ad2antrr 480	. . . . . 6 $/\$ $/\$
134		0zd 9194	. . . . . . 7 $/\$ $/\$
135	124	peano2zd 9307	. . . . . . 7 $/\$ $/\$
136		elfz 9941	. . . . . . 7 $/\$ $/\$ $/\$
137	134, 135, 109, 136	syl3anc 1227	. . . . . 6 $/\$ $/\$ $/\$
138	131, 133, 137	mpbir2and 933	. . . . 5 $/\$ $/\$
139		0cn 7882	. . . . . 6
140		fvi 5537	. . . . . 6
141	139, 140	mp1i 10	. . . . 5 $/\$ $/\$
142	98, 101, 103, 104, 138, 141	seq3z 10436	. . . 4 $/\$ $/\$
143	142	oveq1d 5851	. . 3 $/\$ $/\$
144		nnz 9201	. . . . 5
145		bcval3 10653	. . . . 5 $/\$ $/\$
146	144, 145	syl3an2 1261	. . . 4 $/\$ $/\$
147	146	3expa 1192	. . 3 $/\$ $/\$
148	96, 143, 147	3eqtr4rd 2208	. 2 $/\$ $/\$
149		0zd 9194	. . . 4 $/\$
150		fzdcel 9965	. . . 4 $/\$ $/\$ DECID
151	122, 149, 120, 150	syl3anc 1227	. . 3 $/\$ DECID
152		exmiddc 826	. . 3 DECID $\/$
153	151, 152	syl 14	. 2 $/\$ $\/$
154	90, 148, 153	mpjaodan 788	1 $/\$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 103

wb 104 $\/$ wo 698 DECID wdc 824 $/\$ w3a 967

wceq 1342

wcel 2135

cvv 2721 class class class wbr 3976

cid 4260

cfv 5182 (class class class)co 5836

cc 7742

cr 7743

cc0 7744

c1 7745

caddc 7747

cmul 7749

clt 7924

cle 7925

cmin 8060

cdiv 8559

cn 8848

cn0 9105

cz 9182

cuz 9457

cfz 9935

cseq 10370

cfa 10627

cbc 10649

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-mulrcl 7843 ax-addcom 7844 ax-mulcom 7845 ax-addass 7846 ax-mulass 7847 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-1rid 7851 ax-0id 7852 ax-rnegex 7853 ax-precex 7854 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-apti 7859 ax-pre-ltadd 7860 ax-pre-mulgt0 7861 ax-pre-mulext 7862

This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rmo 2450 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-if 3516 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-id 4265 df-po 4268 df-iso 4269 df-iord 4338 df-on 4340 df-ilim 4341 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-recs 6264 df-frec 6350 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-reap 8464 df-ap 8471 df-div 8560 df-inn 8849 df-n0 9106 df-z 9183 df-uz 9458 df-q 9549 df-fz 9936 df-seqfrec 10371 df-fac 10628 df-bc 10650

This theorem is referenced by: bcn2 10666

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