binom4 - Intuitionistic Logic Explorer

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Theorem binom4 14400

Description: Work out a quartic binomial. (You would think that by this point it would be faster to use binom 11492, 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 8980	. . . 4
2	1	oveq2i 5886	. . 3
3		addcl 7936	. . . 4 $/\$
4		3nn0 9194	. . . 4
5		expp1 10527	. . . 4 $/\$
6	3, 4, 5	sylancl 413	. . 3 $/\$
7	2, 6	eqtrid 2222	. 2 $/\$
8		binom3 10638	. . 3 $/\$
9	8	oveq1d 5890	. 2 $/\$
10		simpl 109	. . . . . . 7 $/\$
11		expcl 10538	. . . . . . 7 $/\$
12	10, 4, 11	sylancl 413	. . . . . 6 $/\$
13		3cn 8994	. . . . . . 7
14	10	sqcld 10652	. . . . . . . 8 $/\$
15		simpr 110	. . . . . . . 8 $/\$
16	14, 15	mulcld 7978	. . . . . . 7 $/\$
17		mulcl 7938	. . . . . . 7 $/\$
18	13, 16, 17	sylancr 414	. . . . . 6 $/\$
19	12, 18	addcld 7977	. . . . 5 $/\$
20	15	sqcld 10652	. . . . . . . 8 $/\$
21	10, 20	mulcld 7978	. . . . . . 7 $/\$
22		mulcl 7938	. . . . . . 7 $/\$
23	13, 21, 22	sylancr 414	. . . . . 6 $/\$
24		expcl 10538	. . . . . . 7 $/\$
25	15, 4, 24	sylancl 413	. . . . . 6 $/\$
26	23, 25	addcld 7977	. . . . 5 $/\$
27	19, 26	addcld 7977	. . . 4 $/\$
28	27, 10, 15	adddid 7982	. . 3 $/\$
29	1	oveq2i 5886	. . . . . . . . 9
30		expp1 10527	. . . . . . . . . 10 $/\$
31	10, 4, 30	sylancl 413	. . . . . . . . 9 $/\$
32	29, 31	eqtr2id 2223	. . . . . . . 8 $/\$
33	13	a1i 9	. . . . . . . . . 10 $/\$
34	33, 16, 10	mulassd 7981	. . . . . . . . 9 $/\$
35	14, 15, 10	mul32d 8110	. . . . . . . . . . 11 $/\$
36		df-3 8979	. . . . . . . . . . . . . 14
37	36	oveq2i 5886	. . . . . . . . . . . . 13
38		2nn0 9193	. . . . . . . . . . . . . 14
39		expp1 10527	. . . . . . . . . . . . . 14 $/\$
40	10, 38, 39	sylancl 413	. . . . . . . . . . . . 13 $/\$
41	37, 40	eqtr2id 2223	. . . . . . . . . . . 12 $/\$
42	41	oveq1d 5890	. . . . . . . . . . 11 $/\$
43	35, 42	eqtrd 2210	. . . . . . . . . 10 $/\$
44	43	oveq2d 5891	. . . . . . . . 9 $/\$
45	34, 44	eqtrd 2210	. . . . . . . 8 $/\$
46	32, 45	oveq12d 5893	. . . . . . 7 $/\$
47	12, 10, 18, 46	joinlmuladdmuld 7985	. . . . . 6 $/\$
48	33, 21, 10	mulassd 7981	. . . . . . . . 9 $/\$
49	10, 20, 10	mul32d 8110	. . . . . . . . . . 11 $/\$
50	10	sqvald 10651	. . . . . . . . . . . 12 $/\$
51	50	oveq1d 5890	. . . . . . . . . . 11 $/\$
52	49, 51	eqtr4d 2213	. . . . . . . . . 10 $/\$
53	52	oveq2d 5891	. . . . . . . . 9 $/\$
54	48, 53	eqtrd 2210	. . . . . . . 8 $/\$
55	25, 10	mulcomd 7979	. . . . . . . 8 $/\$
56	54, 55	oveq12d 5893	. . . . . . 7 $/\$
57	23, 10, 25, 56	joinlmuladdmuld 7985	. . . . . 6 $/\$
58	47, 57	oveq12d 5893	. . . . 5 $/\$
59	19, 10, 26, 58	joinlmuladdmuld 7985	. . . 4 $/\$
60	19, 26, 15	adddird 7983	. . . . 5 $/\$
61	33, 16, 15	mulassd 7981	. . . . . . . . 9 $/\$
62	14, 15, 15	mulassd 7981	. . . . . . . . . . 11 $/\$
63	15	sqvald 10651	. . . . . . . . . . . 12 $/\$
64	63	oveq2d 5891	. . . . . . . . . . 11 $/\$
65	62, 64	eqtr4d 2213	. . . . . . . . . 10 $/\$
66	65	oveq2d 5891	. . . . . . . . 9 $/\$
67	61, 66	eqtrd 2210	. . . . . . . 8 $/\$
68	67	oveq2d 5891	. . . . . . 7 $/\$
69	12, 15, 18, 68	joinlmuladdmuld 7985	. . . . . 6 $/\$
70	33, 21, 15	mulassd 7981	. . . . . . . . 9 $/\$
71	10, 20, 15	mulassd 7981	. . . . . . . . . . 11 $/\$
72	36	oveq2i 5886	. . . . . . . . . . . . 13
73		expp1 10527	. . . . . . . . . . . . . 14 $/\$
74	15, 38, 73	sylancl 413	. . . . . . . . . . . . 13 $/\$
75	72, 74	eqtr2id 2223	. . . . . . . . . . . 12 $/\$
76	75	oveq2d 5891	. . . . . . . . . . 11 $/\$
77	71, 76	eqtrd 2210	. . . . . . . . . 10 $/\$
78	77	oveq2d 5891	. . . . . . . . 9 $/\$
79	70, 78	eqtrd 2210	. . . . . . . 8 $/\$
80	1	oveq2i 5886	. . . . . . . . 9
81		expp1 10527	. . . . . . . . . 10 $/\$
82	15, 4, 81	sylancl 413	. . . . . . . . 9 $/\$
83	80, 82	eqtr2id 2223	. . . . . . . 8 $/\$
84	79, 83	oveq12d 5893	. . . . . . 7 $/\$
85	23, 15, 25, 84	joinlmuladdmuld 7985	. . . . . 6 $/\$
86	69, 85	oveq12d 5893	. . . . 5 $/\$
87	12, 15	mulcld 7978	. . . . . 6 $/\$
88	14, 20	mulcld 7978	. . . . . . 7 $/\$
89		mulcl 7938	. . . . . . 7 $/\$
90	13, 88, 89	sylancr 414	. . . . . 6 $/\$
91	10, 25	mulcld 7978	. . . . . . . 8 $/\$
92		mulcl 7938	. . . . . . . 8 $/\$
93	13, 91, 92	sylancr 414	. . . . . . 7 $/\$
94		4nn0 9195	. . . . . . . 8
95		expcl 10538	. . . . . . . 8 $/\$
96	15, 94, 95	sylancl 413	. . . . . . 7 $/\$
97	93, 96	addcld 7977	. . . . . 6 $/\$
98	87, 90, 97	addassd 7980	. . . . 5 $/\$
99	60, 86, 98	3eqtrd 2214	. . . 4 $/\$
100	59, 99	oveq12d 5893	. . 3 $/\$
101		expcl 10538	. . . . . . 7 $/\$
102	10, 94, 101	sylancl 413	. . . . . 6 $/\$
103		mulcl 7938	. . . . . . 7 $/\$
104	13, 87, 103	sylancr 414	. . . . . 6 $/\$
105	102, 104	addcld 7977	. . . . 5 $/\$
106	90, 91	addcld 7977	. . . . 5 $/\$
107	90, 97	addcld 7977	. . . . 5 $/\$
108	105, 106, 87, 107	add4d 8126	. . . 4 $/\$
109	102, 104, 87	addassd 7980	. . . . . 6 $/\$
110	1	oveq1i 5885	. . . . . . . . 9
111		ax-1cn 7904	. . . . . . . . . . 11
112	111	a1i 9	. . . . . . . . . 10 $/\$
113	33, 112, 87	adddird 7983	. . . . . . . . 9 $/\$
114	110, 113	eqtrid 2222	. . . . . . . 8 $/\$
115	87	mulid2d 7976	. . . . . . . . 9 $/\$
116	115	oveq2d 5891	. . . . . . . 8 $/\$
117	114, 116	eqtrd 2210	. . . . . . 7 $/\$
118	117	oveq2d 5891	. . . . . 6 $/\$
119	109, 118	eqtr4d 2213	. . . . 5 $/\$
120	90, 91, 90, 97	add4d 8126	. . . . . 6 $/\$
121		3p3e6 9061	. . . . . . . . 9
122	121	oveq1i 5885	. . . . . . . 8
123	33, 33, 88	adddird 7983	. . . . . . . 8 $/\$
124	122, 123	eqtr3id 2224	. . . . . . 7 $/\$
125		3p1e4 9054	. . . . . . . . . . . . 13
126	13, 111, 125	addcomli 8102	. . . . . . . . . . . 12
127	126	oveq1i 5885	. . . . . . . . . . 11
128	112, 33, 91	adddird 7983	. . . . . . . . . . 11 $/\$
129	127, 128	eqtr3id 2224	. . . . . . . . . 10 $/\$
130	91	mulid2d 7976	. . . . . . . . . . 11 $/\$
131	130	oveq1d 5890	. . . . . . . . . 10 $/\$
132	129, 131	eqtrd 2210	. . . . . . . . 9 $/\$
133	132	oveq1d 5890	. . . . . . . 8 $/\$
134	91, 93, 96	addassd 7980	. . . . . . . 8 $/\$
135	133, 134	eqtrd 2210	. . . . . . 7 $/\$
136	124, 135	oveq12d 5893	. . . . . 6 $/\$
137	120, 136	eqtr4d 2213	. . . . 5 $/\$
138	119, 137	oveq12d 5893	. . . 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 5875

cc 7809

c1 7812

caddc 7814

cmul 7816

c2 8970

c3 8971

c4 8972

c6 8974

cn0 9176

cexp 10519

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 4119 ax-sep 4122 ax-nul 4130 ax-pow 4175 ax-pr 4210 ax-un 4434 ax-setind 4537 ax-iinf 4588 ax-cnex 7902 ax-resscn 7903 ax-1cn 7904 ax-1re 7905 ax-icn 7906 ax-addcl 7907 ax-addrcl 7908 ax-mulcl 7909 ax-mulrcl 7910 ax-addcom 7911 ax-mulcom 7912 ax-addass 7913 ax-mulass 7914 ax-distr 7915 ax-i2m1 7916 ax-0lt1 7917 ax-1rid 7918 ax-0id 7919 ax-rnegex 7920 ax-precex 7921 ax-cnre 7922 ax-pre-ltirr 7923 ax-pre-ltwlin 7924 ax-pre-lttrn 7925 ax-pre-apti 7926 ax-pre-ltadd 7927 ax-pre-mulgt0 7928 ax-pre-mulext 7929

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 2740 df-sbc 2964 df-csb 3059 df-dif 3132 df-un 3134 df-in 3136 df-ss 3143 df-nul 3424 df-if 3536 df-pw 3578 df-sn 3599 df-pr 3600 df-op 3602 df-uni 3811 df-int 3846 df-iun 3889 df-br 4005 df-opab 4066 df-mpt 4067 df-tr 4103 df-id 4294 df-po 4297 df-iso 4298 df-iord 4367 df-on 4369 df-ilim 4370 df-suc 4372 df-iom 4591 df-xp 4633 df-rel 4634 df-cnv 4635 df-co 4636 df-dm 4637 df-rn 4638 df-res 4639 df-ima 4640 df-iota 5179 df-fun 5219 df-fn 5220 df-f 5221 df-f1 5222 df-fo 5223 df-f1o 5224 df-fv 5225 df-riota 5831 df-ov 5878 df-oprab 5879 df-mpo 5880 df-1st 6141 df-2nd 6142 df-recs 6306 df-frec 6392 df-pnf 7994 df-mnf 7995 df-xr 7996 df-ltxr 7997 df-le 7998 df-sub 8130 df-neg 8131 df-reap 8532 df-ap 8539 df-div 8630 df-inn 8920 df-2 8978 df-3 8979 df-4 8980 df-5 8981 df-6 8982 df-n0 9177 df-z 9254 df-uz 9529 df-seqfrec 10446 df-exp 10520

This theorem is referenced by: (None)

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