binom4 - Intuitionistic Logic Explorer

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Theorem binom4 14064

Description: Work out a quartic binomial. (You would think that by this point it would be faster to use binom 11476, 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 8969	. . . 4
2	1	oveq2i 5880	. . 3
3		addcl 7927	. . . 4 $/\$
4		3nn0 9183	. . . 4
5		expp1 10513	. . . 4 $/\$
6	3, 4, 5	sylancl 413	. . 3 $/\$
7	2, 6	eqtrid 2222	. 2 $/\$
8		binom3 10623	. . 3 $/\$
9	8	oveq1d 5884	. 2 $/\$
10		simpl 109	. . . . . . 7 $/\$
11		expcl 10524	. . . . . . 7 $/\$
12	10, 4, 11	sylancl 413	. . . . . 6 $/\$
13		3cn 8983	. . . . . . 7
14	10	sqcld 10637	. . . . . . . 8 $/\$
15		simpr 110	. . . . . . . 8 $/\$
16	14, 15	mulcld 7968	. . . . . . 7 $/\$
17		mulcl 7929	. . . . . . 7 $/\$
18	13, 16, 17	sylancr 414	. . . . . 6 $/\$
19	12, 18	addcld 7967	. . . . 5 $/\$
20	15	sqcld 10637	. . . . . . . 8 $/\$
21	10, 20	mulcld 7968	. . . . . . 7 $/\$
22		mulcl 7929	. . . . . . 7 $/\$
23	13, 21, 22	sylancr 414	. . . . . 6 $/\$
24		expcl 10524	. . . . . . 7 $/\$
25	15, 4, 24	sylancl 413	. . . . . 6 $/\$
26	23, 25	addcld 7967	. . . . 5 $/\$
27	19, 26	addcld 7967	. . . 4 $/\$
28	27, 10, 15	adddid 7972	. . 3 $/\$
29	1	oveq2i 5880	. . . . . . . . 9
30		expp1 10513	. . . . . . . . . 10 $/\$
31	10, 4, 30	sylancl 413	. . . . . . . . 9 $/\$
32	29, 31	eqtr2id 2223	. . . . . . . 8 $/\$
33	13	a1i 9	. . . . . . . . . 10 $/\$
34	33, 16, 10	mulassd 7971	. . . . . . . . 9 $/\$
35	14, 15, 10	mul32d 8100	. . . . . . . . . . 11 $/\$
36		df-3 8968	. . . . . . . . . . . . . 14
37	36	oveq2i 5880	. . . . . . . . . . . . 13
38		2nn0 9182	. . . . . . . . . . . . . 14
39		expp1 10513	. . . . . . . . . . . . . 14 $/\$
40	10, 38, 39	sylancl 413	. . . . . . . . . . . . 13 $/\$
41	37, 40	eqtr2id 2223	. . . . . . . . . . . 12 $/\$
42	41	oveq1d 5884	. . . . . . . . . . 11 $/\$
43	35, 42	eqtrd 2210	. . . . . . . . . 10 $/\$
44	43	oveq2d 5885	. . . . . . . . 9 $/\$
45	34, 44	eqtrd 2210	. . . . . . . 8 $/\$
46	32, 45	oveq12d 5887	. . . . . . 7 $/\$
47	12, 10, 18, 46	joinlmuladdmuld 7975	. . . . . 6 $/\$
48	33, 21, 10	mulassd 7971	. . . . . . . . 9 $/\$
49	10, 20, 10	mul32d 8100	. . . . . . . . . . 11 $/\$
50	10	sqvald 10636	. . . . . . . . . . . 12 $/\$
51	50	oveq1d 5884	. . . . . . . . . . 11 $/\$
52	49, 51	eqtr4d 2213	. . . . . . . . . 10 $/\$
53	52	oveq2d 5885	. . . . . . . . 9 $/\$
54	48, 53	eqtrd 2210	. . . . . . . 8 $/\$
55	25, 10	mulcomd 7969	. . . . . . . 8 $/\$
56	54, 55	oveq12d 5887	. . . . . . 7 $/\$
57	23, 10, 25, 56	joinlmuladdmuld 7975	. . . . . 6 $/\$
58	47, 57	oveq12d 5887	. . . . 5 $/\$
59	19, 10, 26, 58	joinlmuladdmuld 7975	. . . 4 $/\$
60	19, 26, 15	adddird 7973	. . . . 5 $/\$
61	33, 16, 15	mulassd 7971	. . . . . . . . 9 $/\$
62	14, 15, 15	mulassd 7971	. . . . . . . . . . 11 $/\$
63	15	sqvald 10636	. . . . . . . . . . . 12 $/\$
64	63	oveq2d 5885	. . . . . . . . . . 11 $/\$
65	62, 64	eqtr4d 2213	. . . . . . . . . 10 $/\$
66	65	oveq2d 5885	. . . . . . . . 9 $/\$
67	61, 66	eqtrd 2210	. . . . . . . 8 $/\$
68	67	oveq2d 5885	. . . . . . 7 $/\$
69	12, 15, 18, 68	joinlmuladdmuld 7975	. . . . . 6 $/\$
70	33, 21, 15	mulassd 7971	. . . . . . . . 9 $/\$
71	10, 20, 15	mulassd 7971	. . . . . . . . . . 11 $/\$
72	36	oveq2i 5880	. . . . . . . . . . . . 13
73		expp1 10513	. . . . . . . . . . . . . 14 $/\$
74	15, 38, 73	sylancl 413	. . . . . . . . . . . . 13 $/\$
75	72, 74	eqtr2id 2223	. . . . . . . . . . . 12 $/\$
76	75	oveq2d 5885	. . . . . . . . . . 11 $/\$
77	71, 76	eqtrd 2210	. . . . . . . . . 10 $/\$
78	77	oveq2d 5885	. . . . . . . . 9 $/\$
79	70, 78	eqtrd 2210	. . . . . . . 8 $/\$
80	1	oveq2i 5880	. . . . . . . . 9
81		expp1 10513	. . . . . . . . . 10 $/\$
82	15, 4, 81	sylancl 413	. . . . . . . . 9 $/\$
83	80, 82	eqtr2id 2223	. . . . . . . 8 $/\$
84	79, 83	oveq12d 5887	. . . . . . 7 $/\$
85	23, 15, 25, 84	joinlmuladdmuld 7975	. . . . . 6 $/\$
86	69, 85	oveq12d 5887	. . . . 5 $/\$
87	12, 15	mulcld 7968	. . . . . 6 $/\$
88	14, 20	mulcld 7968	. . . . . . 7 $/\$
89		mulcl 7929	. . . . . . 7 $/\$
90	13, 88, 89	sylancr 414	. . . . . 6 $/\$
91	10, 25	mulcld 7968	. . . . . . . 8 $/\$
92		mulcl 7929	. . . . . . . 8 $/\$
93	13, 91, 92	sylancr 414	. . . . . . 7 $/\$
94		4nn0 9184	. . . . . . . 8
95		expcl 10524	. . . . . . . 8 $/\$
96	15, 94, 95	sylancl 413	. . . . . . 7 $/\$
97	93, 96	addcld 7967	. . . . . 6 $/\$
98	87, 90, 97	addassd 7970	. . . . 5 $/\$
99	60, 86, 98	3eqtrd 2214	. . . 4 $/\$
100	59, 99	oveq12d 5887	. . 3 $/\$
101		expcl 10524	. . . . . . 7 $/\$
102	10, 94, 101	sylancl 413	. . . . . 6 $/\$
103		mulcl 7929	. . . . . . 7 $/\$
104	13, 87, 103	sylancr 414	. . . . . 6 $/\$
105	102, 104	addcld 7967	. . . . 5 $/\$
106	90, 91	addcld 7967	. . . . 5 $/\$
107	90, 97	addcld 7967	. . . . 5 $/\$
108	105, 106, 87, 107	add4d 8116	. . . 4 $/\$
109	102, 104, 87	addassd 7970	. . . . . 6 $/\$
110	1	oveq1i 5879	. . . . . . . . 9
111		ax-1cn 7895	. . . . . . . . . . 11
112	111	a1i 9	. . . . . . . . . 10 $/\$
113	33, 112, 87	adddird 7973	. . . . . . . . 9 $/\$
114	110, 113	eqtrid 2222	. . . . . . . 8 $/\$
115	87	mulid2d 7966	. . . . . . . . 9 $/\$
116	115	oveq2d 5885	. . . . . . . 8 $/\$
117	114, 116	eqtrd 2210	. . . . . . 7 $/\$
118	117	oveq2d 5885	. . . . . 6 $/\$
119	109, 118	eqtr4d 2213	. . . . 5 $/\$
120	90, 91, 90, 97	add4d 8116	. . . . . 6 $/\$
121		3p3e6 9050	. . . . . . . . 9
122	121	oveq1i 5879	. . . . . . . 8
123	33, 33, 88	adddird 7973	. . . . . . . 8 $/\$
124	122, 123	eqtr3id 2224	. . . . . . 7 $/\$
125		3p1e4 9043	. . . . . . . . . . . . 13
126	13, 111, 125	addcomli 8092	. . . . . . . . . . . 12
127	126	oveq1i 5879	. . . . . . . . . . 11
128	112, 33, 91	adddird 7973	. . . . . . . . . . 11 $/\$
129	127, 128	eqtr3id 2224	. . . . . . . . . 10 $/\$
130	91	mulid2d 7966	. . . . . . . . . . 11 $/\$
131	130	oveq1d 5884	. . . . . . . . . 10 $/\$
132	129, 131	eqtrd 2210	. . . . . . . . 9 $/\$
133	132	oveq1d 5884	. . . . . . . 8 $/\$
134	91, 93, 96	addassd 7970	. . . . . . . 8 $/\$
135	133, 134	eqtrd 2210	. . . . . . 7 $/\$
136	124, 135	oveq12d 5887	. . . . . 6 $/\$
137	120, 136	eqtr4d 2213	. . . . 5 $/\$
138	119, 137	oveq12d 5887	. . . 4 $/\$
139	108, 138	eqtrd 2210	. . 3 $/\$
140	28, 100, 139	3eqtrd 2214	. 2 $/\$
141	7, 9, 140	3eqtrd 2214	1 $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wceq 1353

wcel 2148 (class class class)co 5869

cc 7800

c1 7803

caddc 7805

cmul 7807

c2 8959

c3 8960

c4 8961

c6 8963

cn0 9165

cexp 10505

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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4115 ax-sep 4118 ax-nul 4126 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 ax-iinf 4584 ax-cnex 7893 ax-resscn 7894 ax-1cn 7895 ax-1re 7896 ax-icn 7897 ax-addcl 7898 ax-addrcl 7899 ax-mulcl 7900 ax-mulrcl 7901 ax-addcom 7902 ax-mulcom 7903 ax-addass 7904 ax-mulass 7905 ax-distr 7906 ax-i2m1 7907 ax-0lt1 7908 ax-1rid 7909 ax-0id 7910 ax-rnegex 7911 ax-precex 7912 ax-cnre 7913 ax-pre-ltirr 7914 ax-pre-ltwlin 7915 ax-pre-lttrn 7916 ax-pre-apti 7917 ax-pre-ltadd 7918 ax-pre-mulgt0 7919 ax-pre-mulext 7920

This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-if 3535 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-tr 4099 df-id 4290 df-po 4293 df-iso 4294 df-iord 4363 df-on 4365 df-ilim 4366 df-suc 4368 df-iom 4587 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-res 4635 df-ima 4636 df-iota 5174 df-fun 5214 df-fn 5215 df-f 5216 df-f1 5217 df-fo 5218 df-f1o 5219 df-fv 5220 df-riota 5825 df-ov 5872 df-oprab 5873 df-mpo 5874 df-1st 6135 df-2nd 6136 df-recs 6300 df-frec 6386 df-pnf 7984 df-mnf 7985 df-xr 7986 df-ltxr 7987 df-le 7988 df-sub 8120 df-neg 8121 df-reap 8522 df-ap 8529 df-div 8619 df-inn 8909 df-2 8967 df-3 8968 df-4 8969 df-5 8970 df-6 8971 df-n0 9166 df-z 9243 df-uz 9518 df-seqfrec 10432 df-exp 10506

This theorem is referenced by: (None)

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