bcm1k - Metamath Proof Explorer

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Theorem bcm1k 13102

Description: The proportion of one binomial coefficient to another with

decreased by 1. (Contributed by Mario Carneiro, 10-Mar-2014.)

**Assertion**
Ref	Expression
bcm1k

**Proof of Theorem bcm1k**
Step	Hyp	Ref	Expression
1		elfzuz2 12346	. . . . . . . . 9
2		nnuz 11723	. . . . . . . . 9
3	1, 2	syl6eleqr 2712	. . . . . . . 8
4	3	nnnn0d 11351	. . . . . . 7
5		faccl 13070	. . . . . . 7
6	4, 5	syl 17	. . . . . 6
7	6	nncnd 11036	. . . . 5
8		fznn0sub 12373	. . . . . . 7
9		nn0p1nn 11332	. . . . . . 7
10	8, 9	syl 17	. . . . . 6
11	10	nncnd 11036	. . . . 5
12	10	nnnn0d 11351	. . . . . . . 8
13		faccl 13070	. . . . . . . 8
14	12, 13	syl 17	. . . . . . 7
15		elfznn 12370	. . . . . . . 8
16		nnm1nn0 11334	. . . . . . . 8
17		faccl 13070	. . . . . . . 8
18	15, 16, 17	3syl 18	. . . . . . 7
19	14, 18	nnmulcld 11068	. . . . . 6
20		nncn 11028	. . . . . . 7
21		nnne0 11053	. . . . . . 7
22	20, 21	jca 554	. . . . . 6 $/\$
23	19, 22	syl 17	. . . . 5 $/\$
24	15	nncnd 11036	. . . . . 6
25	15	nnne0d 11065	. . . . . 6
26	24, 25	jca 554	. . . . 5 $/\$
27		divmuldiv 10725	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
28	7, 11, 23, 26, 27	syl22anc 1327	. . . 4
29		elfzel2 12340	. . . . . . . . . 10
30	29	zcnd 11483	. . . . . . . . 9
31		1cnd 10056	. . . . . . . . 9
32	30, 24, 31	subsubd 10420	. . . . . . . 8
33	32	fveq2d 6195	. . . . . . 7
34	33	oveq1d 6665	. . . . . 6
35	34	oveq2d 6666	. . . . 5
36	32	oveq1d 6665	. . . . 5
37	35, 36	oveq12d 6668	. . . 4
38		facp1 13065	. . . . . . . . 9
39	8, 38	syl 17	. . . . . . . 8
40	39	eqcomd 2628	. . . . . . 7
41		facnn2 13069	. . . . . . . 8
42	15, 41	syl 17	. . . . . . 7
43	40, 42	oveq12d 6668	. . . . . 6
44		faccl 13070	. . . . . . . . 9
45	8, 44	syl 17	. . . . . . . 8
46	45	nncnd 11036	. . . . . . 7
47	15	nnnn0d 11351	. . . . . . . . 9
48		faccl 13070	. . . . . . . . 9
49	47, 48	syl 17	. . . . . . . 8
50	49	nncnd 11036	. . . . . . 7
51	46, 50, 11	mul32d 10246	. . . . . 6
52	14	nncnd 11036	. . . . . . 7
53	18	nncnd 11036	. . . . . . 7
54	52, 53, 24	mulassd 10063	. . . . . 6
55	43, 51, 54	3eqtr4d 2666	. . . . 5
56	55	oveq2d 6666	. . . 4
57	28, 37, 56	3eqtr4d 2666	. . 3
58	7, 11	mulcomd 10061	. . . 4
59	45, 49	nnmulcld 11068	. . . . . 6
60	59	nncnd 11036	. . . . 5
61	60, 11	mulcomd 10061	. . . 4
62	58, 61	oveq12d 6668	. . 3
63	59	nnne0d 11065	. . . 4
64	10	nnne0d 11065	. . . 4
65	7, 60, 11, 63, 64	divcan5d 10827	. . 3
66	57, 62, 65	3eqtrrd 2661	. 2
67		0p1e1 11132	. . . . . 6
68	67	oveq1i 6660	. . . . 5
69		0z 11388	. . . . . 6
70		fzp1ss 12392	. . . . . 6
71	69, 70	ax-mp 5	. . . . 5
72	68, 71	eqsstr3i 3636	. . . 4
73	72	sseli 3599	. . 3
74		bcval2 13092	. . 3
75	73, 74	syl 17	. 2
76		ax-1cn 9994	. . . . . . . 8
77		npcan 10290	. . . . . . . 8 $/\$
78	30, 76, 77	sylancl 694	. . . . . . 7
79		peano2zm 11420	. . . . . . . 8
80		uzid 11702	. . . . . . . 8
81		peano2uz 11741	. . . . . . . 8
82	29, 79, 80, 81	4syl 19	. . . . . . 7
83	78, 82	eqeltrrd 2702	. . . . . 6
84		fzss2 12381	. . . . . 6
85	83, 84	syl 17	. . . . 5
86		elfzmlbm 12449	. . . . 5
87	85, 86	sseldd 3604	. . . 4
88		bcval2 13092	. . . 4
89	87, 88	syl 17	. . 3
90	89	oveq1d 6665	. 2
91	66, 75, 90	3eqtr4d 2666	1

Colors of variables: wff setvar class

Syntax hints:

wi 4 $/\$ wa 384

wceq 1483

wcel 1990

wne 2794

wss 3574

cfv 5888 (class class class)co 6650

cc 9934

cc0 9936

c1 9937

caddc 9939

cmul 9941

cmin 10266

cdiv 10684

cn 11020

cn0 11292

cz 11377

cuz 11687

cfz 12326

cfa 13060

cbc 13089

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-seq 12802 df-fac 13061 df-bc 13090

This theorem is referenced by: bcp1nk 13104 bcpasc 13108 bpolydiflem 14785 basellem5 24811

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