binom4 - Metamath Proof Explorer

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Theorem binom4 24577

Description: Work out a quartic binomial. (You would think that by this point it would be faster to use binom 14562, but it turns out to be just as much work to put it into this form after clearing all the sums and calculating binomial coefficients.) (Contributed by Mario Carneiro, 6-May-2015.)

**Assertion**
Ref	Expression
binom4	$/\$

**Proof of Theorem binom4**
Step	Hyp	Ref	Expression
1		df-4 11081	. . . 4
2	1	oveq2i 6661	. . 3
3		addcl 10018	. . . 4 $/\$
4		3nn0 11310	. . . 4
5		expp1 12867	. . . 4 $/\$
6	3, 4, 5	sylancl 694	. . 3 $/\$
7	2, 6	syl5eq 2668	. 2 $/\$
8		binom3 12985	. . 3 $/\$
9	8	oveq1d 6665	. 2 $/\$
10		simpl 473	. . . . . . 7 $/\$
11		expcl 12878	. . . . . . 7 $/\$
12	10, 4, 11	sylancl 694	. . . . . 6 $/\$
13		3cn 11095	. . . . . . 7
14	10	sqcld 13006	. . . . . . . 8 $/\$
15		simpr 477	. . . . . . . 8 $/\$
16	14, 15	mulcld 10060	. . . . . . 7 $/\$
17		mulcl 10020	. . . . . . 7 $/\$
18	13, 16, 17	sylancr 695	. . . . . 6 $/\$
19	12, 18	addcld 10059	. . . . 5 $/\$
20	15	sqcld 13006	. . . . . . . 8 $/\$
21	10, 20	mulcld 10060	. . . . . . 7 $/\$
22		mulcl 10020	. . . . . . 7 $/\$
23	13, 21, 22	sylancr 695	. . . . . 6 $/\$
24		expcl 12878	. . . . . . 7 $/\$
25	15, 4, 24	sylancl 694	. . . . . 6 $/\$
26	23, 25	addcld 10059	. . . . 5 $/\$
27	19, 26	addcld 10059	. . . 4 $/\$
28	27, 10, 15	adddid 10064	. . 3 $/\$
29	19, 26, 10	adddird 10065	. . . . 5 $/\$
30	12, 18, 10	adddird 10065	. . . . . . 7 $/\$
31	1	oveq2i 6661	. . . . . . . . 9
32		expp1 12867	. . . . . . . . . 10 $/\$
33	10, 4, 32	sylancl 694	. . . . . . . . 9 $/\$
34	31, 33	syl5req 2669	. . . . . . . 8 $/\$
35	13	a1i 11	. . . . . . . . . 10 $/\$
36	35, 16, 10	mulassd 10063	. . . . . . . . 9 $/\$
37	14, 15, 10	mul32d 10246	. . . . . . . . . . 11 $/\$
38		df-3 11080	. . . . . . . . . . . . . 14
39	38	oveq2i 6661	. . . . . . . . . . . . 13
40		2nn0 11309	. . . . . . . . . . . . . 14
41		expp1 12867	. . . . . . . . . . . . . 14 $/\$
42	10, 40, 41	sylancl 694	. . . . . . . . . . . . 13 $/\$
43	39, 42	syl5req 2669	. . . . . . . . . . . 12 $/\$
44	43	oveq1d 6665	. . . . . . . . . . 11 $/\$
45	37, 44	eqtrd 2656	. . . . . . . . . 10 $/\$
46	45	oveq2d 6666	. . . . . . . . 9 $/\$
47	36, 46	eqtrd 2656	. . . . . . . 8 $/\$
48	34, 47	oveq12d 6668	. . . . . . 7 $/\$
49	30, 48	eqtrd 2656	. . . . . 6 $/\$
50	23, 25, 10	adddird 10065	. . . . . . 7 $/\$
51	35, 21, 10	mulassd 10063	. . . . . . . . 9 $/\$
52	10, 20, 10	mul32d 10246	. . . . . . . . . . 11 $/\$
53	10	sqvald 13005	. . . . . . . . . . . 12 $/\$
54	53	oveq1d 6665	. . . . . . . . . . 11 $/\$
55	52, 54	eqtr4d 2659	. . . . . . . . . 10 $/\$
56	55	oveq2d 6666	. . . . . . . . 9 $/\$
57	51, 56	eqtrd 2656	. . . . . . . 8 $/\$
58	25, 10	mulcomd 10061	. . . . . . . 8 $/\$
59	57, 58	oveq12d 6668	. . . . . . 7 $/\$
60	50, 59	eqtrd 2656	. . . . . 6 $/\$
61	49, 60	oveq12d 6668	. . . . 5 $/\$
62	29, 61	eqtrd 2656	. . . 4 $/\$
63	19, 26, 15	adddird 10065	. . . . 5 $/\$
64	12, 18, 15	adddird 10065	. . . . . . 7 $/\$
65	35, 16, 15	mulassd 10063	. . . . . . . . 9 $/\$
66	14, 15, 15	mulassd 10063	. . . . . . . . . . 11 $/\$
67	15	sqvald 13005	. . . . . . . . . . . 12 $/\$
68	67	oveq2d 6666	. . . . . . . . . . 11 $/\$
69	66, 68	eqtr4d 2659	. . . . . . . . . 10 $/\$
70	69	oveq2d 6666	. . . . . . . . 9 $/\$
71	65, 70	eqtrd 2656	. . . . . . . 8 $/\$
72	71	oveq2d 6666	. . . . . . 7 $/\$
73	64, 72	eqtrd 2656	. . . . . 6 $/\$
74	23, 25, 15	adddird 10065	. . . . . . 7 $/\$
75	35, 21, 15	mulassd 10063	. . . . . . . . 9 $/\$
76	10, 20, 15	mulassd 10063	. . . . . . . . . . 11 $/\$
77	38	oveq2i 6661	. . . . . . . . . . . . 13
78		expp1 12867	. . . . . . . . . . . . . 14 $/\$
79	15, 40, 78	sylancl 694	. . . . . . . . . . . . 13 $/\$
80	77, 79	syl5req 2669	. . . . . . . . . . . 12 $/\$
81	80	oveq2d 6666	. . . . . . . . . . 11 $/\$
82	76, 81	eqtrd 2656	. . . . . . . . . 10 $/\$
83	82	oveq2d 6666	. . . . . . . . 9 $/\$
84	75, 83	eqtrd 2656	. . . . . . . 8 $/\$
85	1	oveq2i 6661	. . . . . . . . 9
86		expp1 12867	. . . . . . . . . 10 $/\$
87	15, 4, 86	sylancl 694	. . . . . . . . 9 $/\$
88	85, 87	syl5req 2669	. . . . . . . 8 $/\$
89	84, 88	oveq12d 6668	. . . . . . 7 $/\$
90	74, 89	eqtrd 2656	. . . . . 6 $/\$
91	73, 90	oveq12d 6668	. . . . 5 $/\$
92	12, 15	mulcld 10060	. . . . . 6 $/\$
93	14, 20	mulcld 10060	. . . . . . 7 $/\$
94		mulcl 10020	. . . . . . 7 $/\$
95	13, 93, 94	sylancr 695	. . . . . 6 $/\$
96	10, 25	mulcld 10060	. . . . . . . 8 $/\$
97		mulcl 10020	. . . . . . . 8 $/\$
98	13, 96, 97	sylancr 695	. . . . . . 7 $/\$
99		4nn0 11311	. . . . . . . 8
100		expcl 12878	. . . . . . . 8 $/\$
101	15, 99, 100	sylancl 694	. . . . . . 7 $/\$
102	98, 101	addcld 10059	. . . . . 6 $/\$
103	92, 95, 102	addassd 10062	. . . . 5 $/\$
104	63, 91, 103	3eqtrd 2660	. . . 4 $/\$
105	62, 104	oveq12d 6668	. . 3 $/\$
106		expcl 12878	. . . . . . 7 $/\$
107	10, 99, 106	sylancl 694	. . . . . 6 $/\$
108		mulcl 10020	. . . . . . 7 $/\$
109	13, 92, 108	sylancr 695	. . . . . 6 $/\$
110	107, 109	addcld 10059	. . . . 5 $/\$
111	95, 96	addcld 10059	. . . . 5 $/\$
112	95, 102	addcld 10059	. . . . 5 $/\$
113	110, 111, 92, 112	add4d 10264	. . . 4 $/\$
114	107, 109, 92	addassd 10062	. . . . . 6 $/\$
115	1	oveq1i 6660	. . . . . . . . 9
116		ax-1cn 9994	. . . . . . . . . . 11
117	116	a1i 11	. . . . . . . . . 10 $/\$
118	35, 117, 92	adddird 10065	. . . . . . . . 9 $/\$
119	115, 118	syl5eq 2668	. . . . . . . 8 $/\$
120	92	mulid2d 10058	. . . . . . . . 9 $/\$
121	120	oveq2d 6666	. . . . . . . 8 $/\$
122	119, 121	eqtrd 2656	. . . . . . 7 $/\$
123	122	oveq2d 6666	. . . . . 6 $/\$
124	114, 123	eqtr4d 2659	. . . . 5 $/\$
125	95, 96, 95, 102	add4d 10264	. . . . . 6 $/\$
126		3p3e6 11161	. . . . . . . . 9
127	126	oveq1i 6660	. . . . . . . 8
128	35, 35, 93	adddird 10065	. . . . . . . 8 $/\$
129	127, 128	syl5eqr 2670	. . . . . . 7 $/\$
130		3p1e4 11153	. . . . . . . . . . . . 13
131	13, 116, 130	addcomli 10228	. . . . . . . . . . . 12
132	131	oveq1i 6660	. . . . . . . . . . 11
133	117, 35, 96	adddird 10065	. . . . . . . . . . 11 $/\$
134	132, 133	syl5eqr 2670	. . . . . . . . . 10 $/\$
135	96	mulid2d 10058	. . . . . . . . . . 11 $/\$
136	135	oveq1d 6665	. . . . . . . . . 10 $/\$
137	134, 136	eqtrd 2656	. . . . . . . . 9 $/\$
138	137	oveq1d 6665	. . . . . . . 8 $/\$
139	96, 98, 101	addassd 10062	. . . . . . . 8 $/\$
140	138, 139	eqtrd 2656	. . . . . . 7 $/\$
141	129, 140	oveq12d 6668	. . . . . 6 $/\$
142	125, 141	eqtr4d 2659	. . . . 5 $/\$
143	124, 142	oveq12d 6668	. . . 4 $/\$
144	113, 143	eqtrd 2656	. . 3 $/\$
145	28, 105, 144	3eqtrd 2660	. 2 $/\$
146	7, 9, 145	3eqtrd 2660	1 $/\$

Colors of variables: wff setvar class

Syntax hints:

wi 4 $/\$ wa 384

wceq 1483

wcel 1990 (class class class)co 6650

cc 9934

c1 9937

caddc 9939

cmul 9941

c2 11070

c3 11071

c4 11072

c6 11074

cn0 11292

cexp 12860

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-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-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-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-n0 11293 df-z 11378 df-uz 11688 df-seq 12802 df-exp 12861

This theorem is referenced by: quart1 24583

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