binom4 - Intuitionistic Logic Explorer

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Theorem binom4 15653

Description: Work out a quartic binomial. (You would think that by this point it would be faster to use binom 11995, 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 9171	. . . 4
2	1	oveq2i 6012	. . 3
3		addcl 8124	. . . 4 $/\$
4		3nn0 9387	. . . 4
5		expp1 10768	. . . 4 $/\$
6	3, 4, 5	sylancl 413	. . 3 $/\$
7	2, 6	eqtrid 2274	. 2 $/\$
8		binom3 10879	. . 3 $/\$
9	8	oveq1d 6016	. 2 $/\$
10		simpl 109	. . . . . . 7 $/\$
11		expcl 10779	. . . . . . 7 $/\$
12	10, 4, 11	sylancl 413	. . . . . 6 $/\$
13		3cn 9185	. . . . . . 7
14	10	sqcld 10893	. . . . . . . 8 $/\$
15		simpr 110	. . . . . . . 8 $/\$
16	14, 15	mulcld 8167	. . . . . . 7 $/\$
17		mulcl 8126	. . . . . . 7 $/\$
18	13, 16, 17	sylancr 414	. . . . . 6 $/\$
19	12, 18	addcld 8166	. . . . 5 $/\$
20	15	sqcld 10893	. . . . . . . 8 $/\$
21	10, 20	mulcld 8167	. . . . . . 7 $/\$
22		mulcl 8126	. . . . . . 7 $/\$
23	13, 21, 22	sylancr 414	. . . . . 6 $/\$
24		expcl 10779	. . . . . . 7 $/\$
25	15, 4, 24	sylancl 413	. . . . . 6 $/\$
26	23, 25	addcld 8166	. . . . 5 $/\$
27	19, 26	addcld 8166	. . . 4 $/\$
28	27, 10, 15	adddid 8171	. . 3 $/\$
29	1	oveq2i 6012	. . . . . . . . 9
30		expp1 10768	. . . . . . . . . 10 $/\$
31	10, 4, 30	sylancl 413	. . . . . . . . 9 $/\$
32	29, 31	eqtr2id 2275	. . . . . . . 8 $/\$
33	13	a1i 9	. . . . . . . . . 10 $/\$
34	33, 16, 10	mulassd 8170	. . . . . . . . 9 $/\$
35	14, 15, 10	mul32d 8299	. . . . . . . . . . 11 $/\$
36		df-3 9170	. . . . . . . . . . . . . 14
37	36	oveq2i 6012	. . . . . . . . . . . . 13
38		2nn0 9386	. . . . . . . . . . . . . 14
39		expp1 10768	. . . . . . . . . . . . . 14 $/\$
40	10, 38, 39	sylancl 413	. . . . . . . . . . . . 13 $/\$
41	37, 40	eqtr2id 2275	. . . . . . . . . . . 12 $/\$
42	41	oveq1d 6016	. . . . . . . . . . 11 $/\$
43	35, 42	eqtrd 2262	. . . . . . . . . 10 $/\$
44	43	oveq2d 6017	. . . . . . . . 9 $/\$
45	34, 44	eqtrd 2262	. . . . . . . 8 $/\$
46	32, 45	oveq12d 6019	. . . . . . 7 $/\$
47	12, 10, 18, 46	joinlmuladdmuld 8174	. . . . . 6 $/\$
48	33, 21, 10	mulassd 8170	. . . . . . . . 9 $/\$
49	10, 20, 10	mul32d 8299	. . . . . . . . . . 11 $/\$
50	10	sqvald 10892	. . . . . . . . . . . 12 $/\$
51	50	oveq1d 6016	. . . . . . . . . . 11 $/\$
52	49, 51	eqtr4d 2265	. . . . . . . . . 10 $/\$
53	52	oveq2d 6017	. . . . . . . . 9 $/\$
54	48, 53	eqtrd 2262	. . . . . . . 8 $/\$
55	25, 10	mulcomd 8168	. . . . . . . 8 $/\$
56	54, 55	oveq12d 6019	. . . . . . 7 $/\$
57	23, 10, 25, 56	joinlmuladdmuld 8174	. . . . . 6 $/\$
58	47, 57	oveq12d 6019	. . . . 5 $/\$
59	19, 10, 26, 58	joinlmuladdmuld 8174	. . . 4 $/\$
60	19, 26, 15	adddird 8172	. . . . 5 $/\$
61	33, 16, 15	mulassd 8170	. . . . . . . . 9 $/\$
62	14, 15, 15	mulassd 8170	. . . . . . . . . . 11 $/\$
63	15	sqvald 10892	. . . . . . . . . . . 12 $/\$
64	63	oveq2d 6017	. . . . . . . . . . 11 $/\$
65	62, 64	eqtr4d 2265	. . . . . . . . . 10 $/\$
66	65	oveq2d 6017	. . . . . . . . 9 $/\$
67	61, 66	eqtrd 2262	. . . . . . . 8 $/\$
68	67	oveq2d 6017	. . . . . . 7 $/\$
69	12, 15, 18, 68	joinlmuladdmuld 8174	. . . . . 6 $/\$
70	33, 21, 15	mulassd 8170	. . . . . . . . 9 $/\$
71	10, 20, 15	mulassd 8170	. . . . . . . . . . 11 $/\$
72	36	oveq2i 6012	. . . . . . . . . . . . 13
73		expp1 10768	. . . . . . . . . . . . . 14 $/\$
74	15, 38, 73	sylancl 413	. . . . . . . . . . . . 13 $/\$
75	72, 74	eqtr2id 2275	. . . . . . . . . . . 12 $/\$
76	75	oveq2d 6017	. . . . . . . . . . 11 $/\$
77	71, 76	eqtrd 2262	. . . . . . . . . 10 $/\$
78	77	oveq2d 6017	. . . . . . . . 9 $/\$
79	70, 78	eqtrd 2262	. . . . . . . 8 $/\$
80	1	oveq2i 6012	. . . . . . . . 9
81		expp1 10768	. . . . . . . . . 10 $/\$
82	15, 4, 81	sylancl 413	. . . . . . . . 9 $/\$
83	80, 82	eqtr2id 2275	. . . . . . . 8 $/\$
84	79, 83	oveq12d 6019	. . . . . . 7 $/\$
85	23, 15, 25, 84	joinlmuladdmuld 8174	. . . . . 6 $/\$
86	69, 85	oveq12d 6019	. . . . 5 $/\$
87	12, 15	mulcld 8167	. . . . . 6 $/\$
88	14, 20	mulcld 8167	. . . . . . 7 $/\$
89		mulcl 8126	. . . . . . 7 $/\$
90	13, 88, 89	sylancr 414	. . . . . 6 $/\$
91	10, 25	mulcld 8167	. . . . . . . 8 $/\$
92		mulcl 8126	. . . . . . . 8 $/\$
93	13, 91, 92	sylancr 414	. . . . . . 7 $/\$
94		4nn0 9388	. . . . . . . 8
95		expcl 10779	. . . . . . . 8 $/\$
96	15, 94, 95	sylancl 413	. . . . . . 7 $/\$
97	93, 96	addcld 8166	. . . . . 6 $/\$
98	87, 90, 97	addassd 8169	. . . . 5 $/\$
99	60, 86, 98	3eqtrd 2266	. . . 4 $/\$
100	59, 99	oveq12d 6019	. . 3 $/\$
101		expcl 10779	. . . . . . 7 $/\$
102	10, 94, 101	sylancl 413	. . . . . 6 $/\$
103		mulcl 8126	. . . . . . 7 $/\$
104	13, 87, 103	sylancr 414	. . . . . 6 $/\$
105	102, 104	addcld 8166	. . . . 5 $/\$
106	90, 91	addcld 8166	. . . . 5 $/\$
107	90, 97	addcld 8166	. . . . 5 $/\$
108	105, 106, 87, 107	add4d 8315	. . . 4 $/\$
109	102, 104, 87	addassd 8169	. . . . . 6 $/\$
110	1	oveq1i 6011	. . . . . . . . 9
111		ax-1cn 8092	. . . . . . . . . . 11
112	111	a1i 9	. . . . . . . . . 10 $/\$
113	33, 112, 87	adddird 8172	. . . . . . . . 9 $/\$
114	110, 113	eqtrid 2274	. . . . . . . 8 $/\$
115	87	mulid2d 8165	. . . . . . . . 9 $/\$
116	115	oveq2d 6017	. . . . . . . 8 $/\$
117	114, 116	eqtrd 2262	. . . . . . 7 $/\$
118	117	oveq2d 6017	. . . . . 6 $/\$
119	109, 118	eqtr4d 2265	. . . . 5 $/\$
120	90, 91, 90, 97	add4d 8315	. . . . . 6 $/\$
121		3p3e6 9253	. . . . . . . . 9
122	121	oveq1i 6011	. . . . . . . 8
123	33, 33, 88	adddird 8172	. . . . . . . 8 $/\$
124	122, 123	eqtr3id 2276	. . . . . . 7 $/\$
125		3p1e4 9246	. . . . . . . . . . . . 13
126	13, 111, 125	addcomli 8291	. . . . . . . . . . . 12
127	126	oveq1i 6011	. . . . . . . . . . 11
128	112, 33, 91	adddird 8172	. . . . . . . . . . 11 $/\$
129	127, 128	eqtr3id 2276	. . . . . . . . . 10 $/\$
130	91	mulid2d 8165	. . . . . . . . . . 11 $/\$
131	130	oveq1d 6016	. . . . . . . . . 10 $/\$
132	129, 131	eqtrd 2262	. . . . . . . . 9 $/\$
133	132	oveq1d 6016	. . . . . . . 8 $/\$
134	91, 93, 96	addassd 8169	. . . . . . . 8 $/\$
135	133, 134	eqtrd 2262	. . . . . . 7 $/\$
136	124, 135	oveq12d 6019	. . . . . 6 $/\$
137	120, 136	eqtr4d 2265	. . . . 5 $/\$
138	119, 137	oveq12d 6019	. . . 4 $/\$
139	108, 138	eqtrd 2262	. . 3 $/\$
140	28, 100, 139	3eqtrd 2266	. 2 $/\$
141	7, 9, 140	3eqtrd 2266	1 $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wceq 1395

wcel 2200 (class class class)co 6001

cc 7997

c1 8000

caddc 8002

cmul 8004

c2 9161

c3 9162

c4 9163

c6 9165

cn0 9369

cexp 10760

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4199 ax-sep 4202 ax-nul 4210 ax-pow 4258 ax-pr 4293 ax-un 4524 ax-setind 4629 ax-iinf 4680 ax-cnex 8090 ax-resscn 8091 ax-1cn 8092 ax-1re 8093 ax-icn 8094 ax-addcl 8095 ax-addrcl 8096 ax-mulcl 8097 ax-mulrcl 8098 ax-addcom 8099 ax-mulcom 8100 ax-addass 8101 ax-mulass 8102 ax-distr 8103 ax-i2m1 8104 ax-0lt1 8105 ax-1rid 8106 ax-0id 8107 ax-rnegex 8108 ax-precex 8109 ax-cnre 8110 ax-pre-ltirr 8111 ax-pre-ltwlin 8112 ax-pre-lttrn 8113 ax-pre-apti 8114 ax-pre-ltadd 8115 ax-pre-mulgt0 8116 ax-pre-mulext 8117

This theorem depends on definitions: df-bi 117 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rmo 2516 df-rab 2517 df-v 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-nul 3492 df-if 3603 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3889 df-int 3924 df-iun 3967 df-br 4084 df-opab 4146 df-mpt 4147 df-tr 4183 df-id 4384 df-po 4387 df-iso 4388 df-iord 4457 df-on 4459 df-ilim 4460 df-suc 4462 df-iom 4683 df-xp 4725 df-rel 4726 df-cnv 4727 df-co 4728 df-dm 4729 df-rn 4730 df-res 4731 df-ima 4732 df-iota 5278 df-fun 5320 df-fn 5321 df-f 5322 df-f1 5323 df-fo 5324 df-f1o 5325 df-fv 5326 df-riota 5954 df-ov 6004 df-oprab 6005 df-mpo 6006 df-1st 6286 df-2nd 6287 df-recs 6451 df-frec 6537 df-pnf 8183 df-mnf 8184 df-xr 8185 df-ltxr 8186 df-le 8187 df-sub 8319 df-neg 8320 df-reap 8722 df-ap 8729 df-div 8820 df-inn 9111 df-2 9169 df-3 9170 df-4 9171 df-5 9172 df-6 9173 df-n0 9370 df-z 9447 df-uz 9723 df-seqfrec 10670 df-exp 10761

This theorem is referenced by: (None)

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