binom4 - Intuitionistic Logic Explorer

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Theorem binom4 14482

Description: Work out a quartic binomial. (You would think that by this point it would be faster to use binom 11494, 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 8982	. . . 4
2	1	oveq2i 5888	. . 3
3		addcl 7938	. . . 4 $/\$
4		3nn0 9196	. . . 4
5		expp1 10529	. . . 4 $/\$
6	3, 4, 5	sylancl 413	. . 3 $/\$
7	2, 6	eqtrid 2222	. 2 $/\$
8		binom3 10640	. . 3 $/\$
9	8	oveq1d 5892	. 2 $/\$
10		simpl 109	. . . . . . 7 $/\$
11		expcl 10540	. . . . . . 7 $/\$
12	10, 4, 11	sylancl 413	. . . . . 6 $/\$
13		3cn 8996	. . . . . . 7
14	10	sqcld 10654	. . . . . . . 8 $/\$
15		simpr 110	. . . . . . . 8 $/\$
16	14, 15	mulcld 7980	. . . . . . 7 $/\$
17		mulcl 7940	. . . . . . 7 $/\$
18	13, 16, 17	sylancr 414	. . . . . 6 $/\$
19	12, 18	addcld 7979	. . . . 5 $/\$
20	15	sqcld 10654	. . . . . . . 8 $/\$
21	10, 20	mulcld 7980	. . . . . . 7 $/\$
22		mulcl 7940	. . . . . . 7 $/\$
23	13, 21, 22	sylancr 414	. . . . . 6 $/\$
24		expcl 10540	. . . . . . 7 $/\$
25	15, 4, 24	sylancl 413	. . . . . 6 $/\$
26	23, 25	addcld 7979	. . . . 5 $/\$
27	19, 26	addcld 7979	. . . 4 $/\$
28	27, 10, 15	adddid 7984	. . 3 $/\$
29	1	oveq2i 5888	. . . . . . . . 9
30		expp1 10529	. . . . . . . . . 10 $/\$
31	10, 4, 30	sylancl 413	. . . . . . . . 9 $/\$
32	29, 31	eqtr2id 2223	. . . . . . . 8 $/\$
33	13	a1i 9	. . . . . . . . . 10 $/\$
34	33, 16, 10	mulassd 7983	. . . . . . . . 9 $/\$
35	14, 15, 10	mul32d 8112	. . . . . . . . . . 11 $/\$
36		df-3 8981	. . . . . . . . . . . . . 14
37	36	oveq2i 5888	. . . . . . . . . . . . 13
38		2nn0 9195	. . . . . . . . . . . . . 14
39		expp1 10529	. . . . . . . . . . . . . 14 $/\$
40	10, 38, 39	sylancl 413	. . . . . . . . . . . . 13 $/\$
41	37, 40	eqtr2id 2223	. . . . . . . . . . . 12 $/\$
42	41	oveq1d 5892	. . . . . . . . . . 11 $/\$
43	35, 42	eqtrd 2210	. . . . . . . . . 10 $/\$
44	43	oveq2d 5893	. . . . . . . . 9 $/\$
45	34, 44	eqtrd 2210	. . . . . . . 8 $/\$
46	32, 45	oveq12d 5895	. . . . . . 7 $/\$
47	12, 10, 18, 46	joinlmuladdmuld 7987	. . . . . 6 $/\$
48	33, 21, 10	mulassd 7983	. . . . . . . . 9 $/\$
49	10, 20, 10	mul32d 8112	. . . . . . . . . . 11 $/\$
50	10	sqvald 10653	. . . . . . . . . . . 12 $/\$
51	50	oveq1d 5892	. . . . . . . . . . 11 $/\$
52	49, 51	eqtr4d 2213	. . . . . . . . . 10 $/\$
53	52	oveq2d 5893	. . . . . . . . 9 $/\$
54	48, 53	eqtrd 2210	. . . . . . . 8 $/\$
55	25, 10	mulcomd 7981	. . . . . . . 8 $/\$
56	54, 55	oveq12d 5895	. . . . . . 7 $/\$
57	23, 10, 25, 56	joinlmuladdmuld 7987	. . . . . 6 $/\$
58	47, 57	oveq12d 5895	. . . . 5 $/\$
59	19, 10, 26, 58	joinlmuladdmuld 7987	. . . 4 $/\$
60	19, 26, 15	adddird 7985	. . . . 5 $/\$
61	33, 16, 15	mulassd 7983	. . . . . . . . 9 $/\$
62	14, 15, 15	mulassd 7983	. . . . . . . . . . 11 $/\$
63	15	sqvald 10653	. . . . . . . . . . . 12 $/\$
64	63	oveq2d 5893	. . . . . . . . . . 11 $/\$
65	62, 64	eqtr4d 2213	. . . . . . . . . 10 $/\$
66	65	oveq2d 5893	. . . . . . . . 9 $/\$
67	61, 66	eqtrd 2210	. . . . . . . 8 $/\$
68	67	oveq2d 5893	. . . . . . 7 $/\$
69	12, 15, 18, 68	joinlmuladdmuld 7987	. . . . . 6 $/\$
70	33, 21, 15	mulassd 7983	. . . . . . . . 9 $/\$
71	10, 20, 15	mulassd 7983	. . . . . . . . . . 11 $/\$
72	36	oveq2i 5888	. . . . . . . . . . . . 13
73		expp1 10529	. . . . . . . . . . . . . 14 $/\$
74	15, 38, 73	sylancl 413	. . . . . . . . . . . . 13 $/\$
75	72, 74	eqtr2id 2223	. . . . . . . . . . . 12 $/\$
76	75	oveq2d 5893	. . . . . . . . . . 11 $/\$
77	71, 76	eqtrd 2210	. . . . . . . . . 10 $/\$
78	77	oveq2d 5893	. . . . . . . . 9 $/\$
79	70, 78	eqtrd 2210	. . . . . . . 8 $/\$
80	1	oveq2i 5888	. . . . . . . . 9
81		expp1 10529	. . . . . . . . . 10 $/\$
82	15, 4, 81	sylancl 413	. . . . . . . . 9 $/\$
83	80, 82	eqtr2id 2223	. . . . . . . 8 $/\$
84	79, 83	oveq12d 5895	. . . . . . 7 $/\$
85	23, 15, 25, 84	joinlmuladdmuld 7987	. . . . . 6 $/\$
86	69, 85	oveq12d 5895	. . . . 5 $/\$
87	12, 15	mulcld 7980	. . . . . 6 $/\$
88	14, 20	mulcld 7980	. . . . . . 7 $/\$
89		mulcl 7940	. . . . . . 7 $/\$
90	13, 88, 89	sylancr 414	. . . . . 6 $/\$
91	10, 25	mulcld 7980	. . . . . . . 8 $/\$
92		mulcl 7940	. . . . . . . 8 $/\$
93	13, 91, 92	sylancr 414	. . . . . . 7 $/\$
94		4nn0 9197	. . . . . . . 8
95		expcl 10540	. . . . . . . 8 $/\$
96	15, 94, 95	sylancl 413	. . . . . . 7 $/\$
97	93, 96	addcld 7979	. . . . . 6 $/\$
98	87, 90, 97	addassd 7982	. . . . 5 $/\$
99	60, 86, 98	3eqtrd 2214	. . . 4 $/\$
100	59, 99	oveq12d 5895	. . 3 $/\$
101		expcl 10540	. . . . . . 7 $/\$
102	10, 94, 101	sylancl 413	. . . . . 6 $/\$
103		mulcl 7940	. . . . . . 7 $/\$
104	13, 87, 103	sylancr 414	. . . . . 6 $/\$
105	102, 104	addcld 7979	. . . . 5 $/\$
106	90, 91	addcld 7979	. . . . 5 $/\$
107	90, 97	addcld 7979	. . . . 5 $/\$
108	105, 106, 87, 107	add4d 8128	. . . 4 $/\$
109	102, 104, 87	addassd 7982	. . . . . 6 $/\$
110	1	oveq1i 5887	. . . . . . . . 9
111		ax-1cn 7906	. . . . . . . . . . 11
112	111	a1i 9	. . . . . . . . . 10 $/\$
113	33, 112, 87	adddird 7985	. . . . . . . . 9 $/\$
114	110, 113	eqtrid 2222	. . . . . . . 8 $/\$
115	87	mulid2d 7978	. . . . . . . . 9 $/\$
116	115	oveq2d 5893	. . . . . . . 8 $/\$
117	114, 116	eqtrd 2210	. . . . . . 7 $/\$
118	117	oveq2d 5893	. . . . . 6 $/\$
119	109, 118	eqtr4d 2213	. . . . 5 $/\$
120	90, 91, 90, 97	add4d 8128	. . . . . 6 $/\$
121		3p3e6 9063	. . . . . . . . 9
122	121	oveq1i 5887	. . . . . . . 8
123	33, 33, 88	adddird 7985	. . . . . . . 8 $/\$
124	122, 123	eqtr3id 2224	. . . . . . 7 $/\$
125		3p1e4 9056	. . . . . . . . . . . . 13
126	13, 111, 125	addcomli 8104	. . . . . . . . . . . 12
127	126	oveq1i 5887	. . . . . . . . . . 11
128	112, 33, 91	adddird 7985	. . . . . . . . . . 11 $/\$
129	127, 128	eqtr3id 2224	. . . . . . . . . 10 $/\$
130	91	mulid2d 7978	. . . . . . . . . . 11 $/\$
131	130	oveq1d 5892	. . . . . . . . . 10 $/\$
132	129, 131	eqtrd 2210	. . . . . . . . 9 $/\$
133	132	oveq1d 5892	. . . . . . . 8 $/\$
134	91, 93, 96	addassd 7982	. . . . . . . 8 $/\$
135	133, 134	eqtrd 2210	. . . . . . 7 $/\$
136	124, 135	oveq12d 5895	. . . . . 6 $/\$
137	120, 136	eqtr4d 2213	. . . . 5 $/\$
138	119, 137	oveq12d 5895	. . . 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 5877

cc 7811

c1 7814

caddc 7816

cmul 7818

c2 8972

c3 8973

c4 8974

c6 8976

cn0 9178

cexp 10521

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 4120 ax-sep 4123 ax-nul 4131 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-iinf 4589 ax-cnex 7904 ax-resscn 7905 ax-1cn 7906 ax-1re 7907 ax-icn 7908 ax-addcl 7909 ax-addrcl 7910 ax-mulcl 7911 ax-mulrcl 7912 ax-addcom 7913 ax-mulcom 7914 ax-addass 7915 ax-mulass 7916 ax-distr 7917 ax-i2m1 7918 ax-0lt1 7919 ax-1rid 7920 ax-0id 7921 ax-rnegex 7922 ax-precex 7923 ax-cnre 7924 ax-pre-ltirr 7925 ax-pre-ltwlin 7926 ax-pre-lttrn 7927 ax-pre-apti 7928 ax-pre-ltadd 7929 ax-pre-mulgt0 7930 ax-pre-mulext 7931

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 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-nul 3425 df-if 3537 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-iun 3890 df-br 4006 df-opab 4067 df-mpt 4068 df-tr 4104 df-id 4295 df-po 4298 df-iso 4299 df-iord 4368 df-on 4370 df-ilim 4371 df-suc 4373 df-iom 4592 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-f1 5223 df-fo 5224 df-f1o 5225 df-fv 5226 df-riota 5833 df-ov 5880 df-oprab 5881 df-mpo 5882 df-1st 6143 df-2nd 6144 df-recs 6308 df-frec 6394 df-pnf 7996 df-mnf 7997 df-xr 7998 df-ltxr 7999 df-le 8000 df-sub 8132 df-neg 8133 df-reap 8534 df-ap 8541 df-div 8632 df-inn 8922 df-2 8980 df-3 8981 df-4 8982 df-5 8983 df-6 8984 df-n0 9179 df-z 9256 df-uz 9531 df-seqfrec 10448 df-exp 10522

This theorem is referenced by: (None)

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