binom4 - Intuitionistic Logic Explorer

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Theorem binom4 15970

Description: Work out a quartic binomial. (You would think that by this point it would be faster to use binom 12195, 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 9315	. . . 4
2	1	oveq2i 6069	. . 3
3		addcl 8268	. . . 4 $/\$
4		3nn0 9531	. . . 4
5		expp1 10932	. . . 4 $/\$
6	3, 4, 5	sylancl 413	. . 3 $/\$
7	2, 6	eqtrid 2279	. 2 $/\$
8		binom3 11043	. . 3 $/\$
9	8	oveq1d 6073	. 2 $/\$
10		simpl 109	. . . . . . 7 $/\$
11		expcl 10943	. . . . . . 7 $/\$
12	10, 4, 11	sylancl 413	. . . . . 6 $/\$
13		3cn 9329	. . . . . . 7
14	10	sqcld 11058	. . . . . . . 8 $/\$
15		simpr 110	. . . . . . . 8 $/\$
16	14, 15	mulcld 8310	. . . . . . 7 $/\$
17		mulcl 8270	. . . . . . 7 $/\$
18	13, 16, 17	sylancr 414	. . . . . 6 $/\$
19	12, 18	addcld 8309	. . . . 5 $/\$
20	15	sqcld 11058	. . . . . . . 8 $/\$
21	10, 20	mulcld 8310	. . . . . . 7 $/\$
22		mulcl 8270	. . . . . . 7 $/\$
23	13, 21, 22	sylancr 414	. . . . . 6 $/\$
24		expcl 10943	. . . . . . 7 $/\$
25	15, 4, 24	sylancl 413	. . . . . 6 $/\$
26	23, 25	addcld 8309	. . . . 5 $/\$
27	19, 26	addcld 8309	. . . 4 $/\$
28	27, 10, 15	adddid 8314	. . 3 $/\$
29	1	oveq2i 6069	. . . . . . . . 9
30		expp1 10932	. . . . . . . . . 10 $/\$
31	10, 4, 30	sylancl 413	. . . . . . . . 9 $/\$
32	29, 31	eqtr2id 2280	. . . . . . . 8 $/\$
33	13	a1i 9	. . . . . . . . . 10 $/\$
34	33, 16, 10	mulassd 8313	. . . . . . . . 9 $/\$
35	14, 15, 10	mul32d 8442	. . . . . . . . . . 11 $/\$
36		df-3 9314	. . . . . . . . . . . . . 14
37	36	oveq2i 6069	. . . . . . . . . . . . 13
38		2nn0 9530	. . . . . . . . . . . . . 14
39		expp1 10932	. . . . . . . . . . . . . 14 $/\$
40	10, 38, 39	sylancl 413	. . . . . . . . . . . . 13 $/\$
41	37, 40	eqtr2id 2280	. . . . . . . . . . . 12 $/\$
42	41	oveq1d 6073	. . . . . . . . . . 11 $/\$
43	35, 42	eqtrd 2267	. . . . . . . . . 10 $/\$
44	43	oveq2d 6074	. . . . . . . . 9 $/\$
45	34, 44	eqtrd 2267	. . . . . . . 8 $/\$
46	32, 45	oveq12d 6076	. . . . . . 7 $/\$
47	12, 10, 18, 46	joinlmuladdmuld 8317	. . . . . 6 $/\$
48	33, 21, 10	mulassd 8313	. . . . . . . . 9 $/\$
49	10, 20, 10	mul32d 8442	. . . . . . . . . . 11 $/\$
50	10	sqvald 11057	. . . . . . . . . . . 12 $/\$
51	50	oveq1d 6073	. . . . . . . . . . 11 $/\$
52	49, 51	eqtr4d 2270	. . . . . . . . . 10 $/\$
53	52	oveq2d 6074	. . . . . . . . 9 $/\$
54	48, 53	eqtrd 2267	. . . . . . . 8 $/\$
55	25, 10	mulcomd 8311	. . . . . . . 8 $/\$
56	54, 55	oveq12d 6076	. . . . . . 7 $/\$
57	23, 10, 25, 56	joinlmuladdmuld 8317	. . . . . 6 $/\$
58	47, 57	oveq12d 6076	. . . . 5 $/\$
59	19, 10, 26, 58	joinlmuladdmuld 8317	. . . 4 $/\$
60	19, 26, 15	adddird 8315	. . . . 5 $/\$
61	33, 16, 15	mulassd 8313	. . . . . . . . 9 $/\$
62	14, 15, 15	mulassd 8313	. . . . . . . . . . 11 $/\$
63	15	sqvald 11057	. . . . . . . . . . . 12 $/\$
64	63	oveq2d 6074	. . . . . . . . . . 11 $/\$
65	62, 64	eqtr4d 2270	. . . . . . . . . 10 $/\$
66	65	oveq2d 6074	. . . . . . . . 9 $/\$
67	61, 66	eqtrd 2267	. . . . . . . 8 $/\$
68	67	oveq2d 6074	. . . . . . 7 $/\$
69	12, 15, 18, 68	joinlmuladdmuld 8317	. . . . . 6 $/\$
70	33, 21, 15	mulassd 8313	. . . . . . . . 9 $/\$
71	10, 20, 15	mulassd 8313	. . . . . . . . . . 11 $/\$
72	36	oveq2i 6069	. . . . . . . . . . . . 13
73		expp1 10932	. . . . . . . . . . . . . 14 $/\$
74	15, 38, 73	sylancl 413	. . . . . . . . . . . . 13 $/\$
75	72, 74	eqtr2id 2280	. . . . . . . . . . . 12 $/\$
76	75	oveq2d 6074	. . . . . . . . . . 11 $/\$
77	71, 76	eqtrd 2267	. . . . . . . . . 10 $/\$
78	77	oveq2d 6074	. . . . . . . . 9 $/\$
79	70, 78	eqtrd 2267	. . . . . . . 8 $/\$
80	1	oveq2i 6069	. . . . . . . . 9
81		expp1 10932	. . . . . . . . . 10 $/\$
82	15, 4, 81	sylancl 413	. . . . . . . . 9 $/\$
83	80, 82	eqtr2id 2280	. . . . . . . 8 $/\$
84	79, 83	oveq12d 6076	. . . . . . 7 $/\$
85	23, 15, 25, 84	joinlmuladdmuld 8317	. . . . . 6 $/\$
86	69, 85	oveq12d 6076	. . . . 5 $/\$
87	12, 15	mulcld 8310	. . . . . 6 $/\$
88	14, 20	mulcld 8310	. . . . . . 7 $/\$
89		mulcl 8270	. . . . . . 7 $/\$
90	13, 88, 89	sylancr 414	. . . . . 6 $/\$
91	10, 25	mulcld 8310	. . . . . . . 8 $/\$
92		mulcl 8270	. . . . . . . 8 $/\$
93	13, 91, 92	sylancr 414	. . . . . . 7 $/\$
94		4nn0 9532	. . . . . . . 8
95		expcl 10943	. . . . . . . 8 $/\$
96	15, 94, 95	sylancl 413	. . . . . . 7 $/\$
97	93, 96	addcld 8309	. . . . . 6 $/\$
98	87, 90, 97	addassd 8312	. . . . 5 $/\$
99	60, 86, 98	3eqtrd 2271	. . . 4 $/\$
100	59, 99	oveq12d 6076	. . 3 $/\$
101		expcl 10943	. . . . . . 7 $/\$
102	10, 94, 101	sylancl 413	. . . . . 6 $/\$
103		mulcl 8270	. . . . . . 7 $/\$
104	13, 87, 103	sylancr 414	. . . . . 6 $/\$
105	102, 104	addcld 8309	. . . . 5 $/\$
106	90, 91	addcld 8309	. . . . 5 $/\$
107	90, 97	addcld 8309	. . . . 5 $/\$
108	105, 106, 87, 107	add4d 8458	. . . 4 $/\$
109	102, 104, 87	addassd 8312	. . . . . 6 $/\$
110	1	oveq1i 6068	. . . . . . . . 9
111		ax-1cn 8236	. . . . . . . . . . 11
112	111	a1i 9	. . . . . . . . . 10 $/\$
113	33, 112, 87	adddird 8315	. . . . . . . . 9 $/\$
114	110, 113	eqtrid 2279	. . . . . . . 8 $/\$
115	87	mullidd 8308	. . . . . . . . 9 $/\$
116	115	oveq2d 6074	. . . . . . . 8 $/\$
117	114, 116	eqtrd 2267	. . . . . . 7 $/\$
118	117	oveq2d 6074	. . . . . 6 $/\$
119	109, 118	eqtr4d 2270	. . . . 5 $/\$
120	90, 91, 90, 97	add4d 8458	. . . . . 6 $/\$
121		3p3e6 9397	. . . . . . . . 9
122	121	oveq1i 6068	. . . . . . . 8
123	33, 33, 88	adddird 8315	. . . . . . . 8 $/\$
124	122, 123	eqtr3id 2281	. . . . . . 7 $/\$
125		3p1e4 9390	. . . . . . . . . . . . 13
126	13, 111, 125	addcomli 8434	. . . . . . . . . . . 12
127	126	oveq1i 6068	. . . . . . . . . . 11
128	112, 33, 91	adddird 8315	. . . . . . . . . . 11 $/\$
129	127, 128	eqtr3id 2281	. . . . . . . . . 10 $/\$
130	91	mullidd 8308	. . . . . . . . . . 11 $/\$
131	130	oveq1d 6073	. . . . . . . . . 10 $/\$
132	129, 131	eqtrd 2267	. . . . . . . . 9 $/\$
133	132	oveq1d 6073	. . . . . . . 8 $/\$
134	91, 93, 96	addassd 8312	. . . . . . . 8 $/\$
135	133, 134	eqtrd 2267	. . . . . . 7 $/\$
136	124, 135	oveq12d 6076	. . . . . 6 $/\$
137	120, 136	eqtr4d 2270	. . . . 5 $/\$
138	119, 137	oveq12d 6076	. . . 4 $/\$
139	108, 138	eqtrd 2267	. . 3 $/\$
140	28, 100, 139	3eqtrd 2271	. 2 $/\$
141	7, 9, 140	3eqtrd 2271	1 $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wceq 1398

wcel 2205 (class class class)co 6058

cc 8141

c1 8144

caddc 8146

cmul 8148

c2 9305

c3 9306

c4 9307

c6 9309

cn0 9513

cexp 10924

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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2207 ax-14 2208 ax-ext 2216 ax-coll 4230 ax-sep 4233 ax-nul 4241 ax-pow 4292 ax-pr 4327 ax-un 4559 ax-setind 4664 ax-iinf 4715 ax-cnex 8234 ax-resscn 8235 ax-1cn 8236 ax-1re 8237 ax-icn 8238 ax-addcl 8239 ax-addrcl 8240 ax-mulcl 8241 ax-mulrcl 8242 ax-addcom 8243 ax-mulcom 8244 ax-addass 8245 ax-mulass 8246 ax-distr 8247 ax-i2m1 8248 ax-0lt1 8249 ax-1rid 8250 ax-0id 8251 ax-rnegex 8252 ax-precex 8253 ax-cnre 8254 ax-pre-ltirr 8255 ax-pre-ltwlin 8256 ax-pre-lttrn 8257 ax-pre-apti 8258 ax-pre-ltadd 8259 ax-pre-mulgt0 8260 ax-pre-mulext 8261

This theorem depends on definitions: df-bi 117 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ne 2415 df-nel 2510 df-ral 2527 df-rex 2528 df-reu 2529 df-rmo 2530 df-rab 2531 df-v 2817 df-sbc 3046 df-csb 3142 df-dif 3216 df-un 3218 df-in 3220 df-ss 3227 df-nul 3513 df-if 3625 df-pw 3676 df-sn 3700 df-pr 3701 df-op 3703 df-uni 3920 df-int 3955 df-iun 3998 df-br 4115 df-opab 4177 df-mpt 4178 df-tr 4214 df-id 4419 df-po 4422 df-iso 4423 df-iord 4492 df-on 4494 df-ilim 4495 df-suc 4497 df-iom 4718 df-xp 4760 df-rel 4761 df-cnv 4762 df-co 4763 df-dm 4764 df-rn 4765 df-res 4766 df-ima 4767 df-iota 5317 df-fun 5359 df-fn 5360 df-f 5361 df-f1 5362 df-fo 5363 df-f1o 5364 df-fv 5365 df-riota 6011 df-ov 6061 df-oprab 6062 df-mpo 6063 df-1st 6347 df-2nd 6348 df-recs 6549 df-frec 6635 df-pnf 8326 df-mnf 8327 df-xr 8328 df-ltxr 8329 df-le 8330 df-sub 8462 df-neg 8463 df-reap 8866 df-ap 8873 df-div 8964 df-inn 9255 df-2 9313 df-3 9314 df-4 9315 df-5 9316 df-6 9317 df-n0 9514 df-z 9595 df-uz 9872 df-seqfrec 10834 df-exp 10925

This theorem is referenced by: (None)

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