bcval5 - Intuitionistic Logic Explorer

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Theorem bcval5 10676

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, although it is no longer valid for nonpositive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 10663	. . . 4
2	1	adantl 275	. . 3 $/\$ $/\$
3		simprl 521	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
4		simprr 522	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
5	3, 4	mulcld 7919	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
6		simpr1 993	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
7		simpr2 994	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
8		simpr3 995	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
9	6, 7, 8	mulassd 7922	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
10		simpll 519	. . . . . . . . . . . . 13 $/\$ $/\$
11	10	nn0zd 9311	. . . . . . . . . . . 12 $/\$ $/\$
12		simplr 520	. . . . . . . . . . . . 13 $/\$ $/\$
13	12	nnzd 9312	. . . . . . . . . . . 12 $/\$ $/\$
14	11, 13	zsubcld 9318	. . . . . . . . . . 11 $/\$ $/\$
15	14	peano2zd 9316	. . . . . . . . . 10 $/\$ $/\$
16		1red 7914	. . . . . . . . . . . 12 $/\$ $/\$
17	12	nnred 8870	. . . . . . . . . . . 12 $/\$ $/\$
18	10	nn0red 9168	. . . . . . . . . . . 12 $/\$ $/\$
19	12	nnge1d 8900	. . . . . . . . . . . 12 $/\$ $/\$
20	16, 17, 18, 19	lesub2dd 8460	. . . . . . . . . . 11 $/\$ $/\$
21	14	zred 9313	. . . . . . . . . . . 12 $/\$ $/\$
22		leaddsub 8336	. . . . . . . . . . . 12 $/\$ $/\$
23	21, 16, 18, 22	syl3anc 1228	. . . . . . . . . . 11 $/\$ $/\$
24	20, 23	mpbird 166	. . . . . . . . . 10 $/\$ $/\$
25		eluz2 9472	. . . . . . . . . 10 $/\$ $/\$
26	15, 11, 24, 25	syl3anbrc 1171	. . . . . . . . 9 $/\$ $/\$
27	26	adantrr 471	. . . . . . . 8 $/\$ $/\$ $/\$
28		simprr 522	. . . . . . . . 9 $/\$ $/\$ $/\$
29		nnuz 9501	. . . . . . . . 9
30	28, 29	eleqtrdi 2259	. . . . . . . 8 $/\$ $/\$ $/\$
31		fvi 5543	. . . . . . . . . 10
32	31	elv 2730	. . . . . . . . 9
33		eluzelcn 9477	. . . . . . . . . 10
34	33	adantl 275	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
35	32, 34	eqeltrid 2253	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
36	5, 9, 27, 30, 35	seq3split 10414	. . . . . . 7 $/\$ $/\$ $/\$
37		elfzuz3 9957	. . . . . . . . . . 11
38	37	adantl 275	. . . . . . . . . 10 $/\$ $/\$
39		eluznn 9538	. . . . . . . . . 10 $/\$
40	12, 38, 39	syl2anc 409	. . . . . . . . 9 $/\$ $/\$
41	40	adantrr 471	. . . . . . . 8 $/\$ $/\$ $/\$
42		facnn 10640	. . . . . . . 8
43	41, 42	syl 14	. . . . . . 7 $/\$ $/\$ $/\$
44		facnn 10640	. . . . . . . . 9
45	28, 44	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$
46	45	oveq1d 5857	. . . . . . 7 $/\$ $/\$ $/\$
47	36, 43, 46	3eqtr4d 2208	. . . . . 6 $/\$ $/\$ $/\$
48	47	expr 373	. . . . 5 $/\$ $/\$
49	10	faccld 10649	. . . . . . . . 9 $/\$ $/\$
50	49	nncnd 8871	. . . . . . . 8 $/\$ $/\$
51	50	mulid2d 7917	. . . . . . 7 $/\$ $/\$
52	40, 42	syl 14	. . . . . . . 8 $/\$ $/\$
53	52	oveq2d 5858	. . . . . . 7 $/\$ $/\$
54	51, 53	eqtr3d 2200	. . . . . 6 $/\$ $/\$
55		fveq2 5486	. . . . . . . . 9
56		fac0 10641	. . . . . . . . 9
57	55, 56	eqtrdi 2215	. . . . . . . 8
58		oveq1 5849	. . . . . . . . . . 11
59		0p1e1 8971	. . . . . . . . . . 11
60	58, 59	eqtrdi 2215	. . . . . . . . . 10
61	60	seqeq1d 10386	. . . . . . . . 9
62	61	fveq1d 5488	. . . . . . . 8
63	57, 62	oveq12d 5860	. . . . . . 7
64	63	eqeq2d 2177	. . . . . 6
65	54, 64	syl5ibrcom 156	. . . . 5 $/\$ $/\$
66		fznn0sub 9992	. . . . . . 7
67	66	adantl 275	. . . . . 6 $/\$ $/\$
68		elnn0 9116	. . . . . 6 $\/$
69	67, 68	sylib 121	. . . . 5 $/\$ $/\$ $\/$
70	48, 65, 69	mpjaod 708	. . . 4 $/\$ $/\$
71	70	oveq1d 5857	. . 3 $/\$ $/\$
72		eqid 2165	. . . . . 6
73		fvi 5543	. . . . . . . 8
74	73	elv 2730	. . . . . . 7
75		eluzelcn 9477	. . . . . . . 8
76	75	adantl 275	. . . . . . 7 $/\$ $/\$ $/\$
77	74, 76	eqeltrid 2253	. . . . . 6 $/\$ $/\$ $/\$
78		mulcl 7880	. . . . . . 7 $/\$
79	78	adantl 275	. . . . . 6 $/\$ $/\$ $/\$ $/\$
80	72, 15, 77, 79	seqf 10396	. . . . 5 $/\$ $/\$
81	80, 26	ffvelrnd 5621	. . . 4 $/\$ $/\$
82	12	nnnn0d 9167	. . . . . 6 $/\$ $/\$
83	82	faccld 10649	. . . . 5 $/\$ $/\$
84	83	nncnd 8871	. . . 4 $/\$ $/\$
85	67	faccld 10649	. . . . 5 $/\$ $/\$
86	85	nncnd 8871	. . . 4 $/\$ $/\$
87	83	nnap0d 8903	. . . 4 $/\$ $/\$ #
88	85	nnap0d 8903	. . . 4 $/\$ $/\$ #
89	81, 84, 86, 87, 88	divcanap5d 8713	. . 3 $/\$ $/\$
90	2, 71, 89	3eqtrd 2202	. 2 $/\$ $/\$
91		simplr 520	. . . . . . 7 $/\$ $/\$
92	91	nnnn0d 9167	. . . . . 6 $/\$ $/\$
93	92	faccld 10649	. . . . 5 $/\$ $/\$
94	93	nncnd 8871	. . . 4 $/\$ $/\$
95	93	nnap0d 8903	. . . 4 $/\$ $/\$ #
96	94, 95	div0apd 8683	. . 3 $/\$ $/\$
97		mulcl 7880	. . . . . 6 $/\$
98	97	adantl 275	. . . . 5 $/\$ $/\$ $/\$ $/\$
99		eluzelcn 9477	. . . . . . 7
100	99	adantl 275	. . . . . 6 $/\$ $/\$ $/\$
101	32, 100	eqeltrid 2253	. . . . 5 $/\$ $/\$ $/\$
102		simpr 109	. . . . . 6 $/\$ $/\$ $/\$
103	102	mul02d 8290	. . . . 5 $/\$ $/\$ $/\$
104	102	mul01d 8291	. . . . 5 $/\$ $/\$ $/\$
105		simpr 109	. . . . . . . . 9 $/\$ $/\$
106		nn0uz 9500	. . . . . . . . . . . 12
107	92, 106	eleqtrdi 2259	. . . . . . . . . . 11 $/\$ $/\$
108		simpll 519	. . . . . . . . . . . 12 $/\$ $/\$
109	108	nn0zd 9311	. . . . . . . . . . 11 $/\$ $/\$
110		elfz5 9952	. . . . . . . . . . 11 $/\$
111	107, 109, 110	syl2anc 409	. . . . . . . . . 10 $/\$ $/\$
112		nn0re 9123	. . . . . . . . . . . 12
113	112	ad2antrr 480	. . . . . . . . . . 11 $/\$ $/\$
114		nnre 8864	. . . . . . . . . . . 12
115	114	ad2antlr 481	. . . . . . . . . . 11 $/\$ $/\$
116	113, 115	subge0d 8433	. . . . . . . . . 10 $/\$ $/\$
117	111, 116	bitr4d 190	. . . . . . . . 9 $/\$ $/\$
118	105, 117	mtbid 662	. . . . . . . 8 $/\$ $/\$
119		simpl 108	. . . . . . . . . . . 12 $/\$
120	119	nn0zd 9311	. . . . . . . . . . 11 $/\$
121		simpr 109	. . . . . . . . . . . 12 $/\$
122	121	nnzd 9312	. . . . . . . . . . 11 $/\$
123	120, 122	zsubcld 9318	. . . . . . . . . 10 $/\$
124	123	adantr 274	. . . . . . . . 9 $/\$ $/\$
125		0z 9202	. . . . . . . . 9
126		zltnle 9237	. . . . . . . . 9 $/\$
127	124, 125, 126	sylancl 410	. . . . . . . 8 $/\$ $/\$
128	118, 127	mpbird 166	. . . . . . 7 $/\$ $/\$
129		zltp1le 9245	. . . . . . . 8 $/\$
130	124, 125, 129	sylancl 410	. . . . . . 7 $/\$ $/\$
131	128, 130	mpbid 146	. . . . . 6 $/\$ $/\$
132		nn0ge0 9139	. . . . . . 7
133	132	ad2antrr 480	. . . . . 6 $/\$ $/\$
134		0zd 9203	. . . . . . 7 $/\$ $/\$
135	124	peano2zd 9316	. . . . . . 7 $/\$ $/\$
136		elfz 9950	. . . . . . 7 $/\$ $/\$ $/\$
137	134, 135, 109, 136	syl3anc 1228	. . . . . 6 $/\$ $/\$ $/\$
138	131, 133, 137	mpbir2and 934	. . . . 5 $/\$ $/\$
139		0cn 7891	. . . . . 6
140		fvi 5543	. . . . . 6
141	139, 140	mp1i 10	. . . . 5 $/\$ $/\$
142	98, 101, 103, 104, 138, 141	seq3z 10446	. . . 4 $/\$ $/\$
143	142	oveq1d 5857	. . 3 $/\$ $/\$
144		nnz 9210	. . . . 5
145		bcval3 10664	. . . . 5 $/\$ $/\$
146	144, 145	syl3an2 1262	. . . 4 $/\$ $/\$
147	146	3expa 1193	. . 3 $/\$ $/\$
148	96, 143, 147	3eqtr4rd 2209	. 2 $/\$ $/\$
149		0zd 9203	. . . 4 $/\$
150		fzdcel 9975	. . . 4 $/\$ $/\$ DECID
151	122, 149, 120, 150	syl3anc 1228	. . 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 968

wceq 1343

wcel 2136

cvv 2726 class class class wbr 3982

cid 4266

cfv 5188 (class class class)co 5842

cc 7751

cr 7752

cc0 7753

c1 7754

caddc 7756

cmul 7758

clt 7933

cle 7934

cmin 8069

cdiv 8568

cn 8857

cn0 9114

cz 9191

cuz 9466

cfz 9944

cseq 10380

cfa 10638

cbc 10660

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 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-coll 4097 ax-sep 4100 ax-nul 4108 ax-pow 4153 ax-pr 4187 ax-un 4411 ax-setind 4514 ax-iinf 4565 ax-cnex 7844 ax-resscn 7845 ax-1cn 7846 ax-1re 7847 ax-icn 7848 ax-addcl 7849 ax-addrcl 7850 ax-mulcl 7851 ax-mulrcl 7852 ax-addcom 7853 ax-mulcom 7854 ax-addass 7855 ax-mulass 7856 ax-distr 7857 ax-i2m1 7858 ax-0lt1 7859 ax-1rid 7860 ax-0id 7861 ax-rnegex 7862 ax-precex 7863 ax-cnre 7864 ax-pre-ltirr 7865 ax-pre-ltwlin 7866 ax-pre-lttrn 7867 ax-pre-apti 7868 ax-pre-ltadd 7869 ax-pre-mulgt0 7870 ax-pre-mulext 7871

This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 969 df-3an 970 df-tru 1346 df-fal 1349 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ne 2337 df-nel 2432 df-ral 2449 df-rex 2450 df-reu 2451 df-rmo 2452 df-rab 2453 df-v 2728 df-sbc 2952 df-csb 3046 df-dif 3118 df-un 3120 df-in 3122 df-ss 3129 df-nul 3410 df-if 3521 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-int 3825 df-iun 3868 df-br 3983 df-opab 4044 df-mpt 4045 df-tr 4081 df-id 4271 df-po 4274 df-iso 4275 df-iord 4344 df-on 4346 df-ilim 4347 df-suc 4349 df-iom 4568 df-xp 4610 df-rel 4611 df-cnv 4612 df-co 4613 df-dm 4614 df-rn 4615 df-res 4616 df-ima 4617 df-iota 5153 df-fun 5190 df-fn 5191 df-f 5192 df-f1 5193 df-fo 5194 df-f1o 5195 df-fv 5196 df-riota 5798 df-ov 5845 df-oprab 5846 df-mpo 5847 df-1st 6108 df-2nd 6109 df-recs 6273 df-frec 6359 df-pnf 7935 df-mnf 7936 df-xr 7937 df-ltxr 7938 df-le 7939 df-sub 8071 df-neg 8072 df-reap 8473 df-ap 8480 df-div 8569 df-inn 8858 df-n0 9115 df-z 9192 df-uz 9467 df-q 9558 df-fz 9945 df-seqfrec 10381 df-fac 10639 df-bc 10661

This theorem is referenced by: bcn2 10677

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