binom4 - Intuitionistic Logic Explorer

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Theorem binom4 15566

Description: Work out a quartic binomial. (You would think that by this point it would be faster to use binom 11910, 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 9132	. . . 4
2	1	oveq2i 5978	. . 3
3		addcl 8085	. . . 4 $/\$
4		3nn0 9348	. . . 4
5		expp1 10728	. . . 4 $/\$
6	3, 4, 5	sylancl 413	. . 3 $/\$
7	2, 6	eqtrid 2252	. 2 $/\$
8		binom3 10839	. . 3 $/\$
9	8	oveq1d 5982	. 2 $/\$
10		simpl 109	. . . . . . 7 $/\$
11		expcl 10739	. . . . . . 7 $/\$
12	10, 4, 11	sylancl 413	. . . . . 6 $/\$
13		3cn 9146	. . . . . . 7
14	10	sqcld 10853	. . . . . . . 8 $/\$
15		simpr 110	. . . . . . . 8 $/\$
16	14, 15	mulcld 8128	. . . . . . 7 $/\$
17		mulcl 8087	. . . . . . 7 $/\$
18	13, 16, 17	sylancr 414	. . . . . 6 $/\$
19	12, 18	addcld 8127	. . . . 5 $/\$
20	15	sqcld 10853	. . . . . . . 8 $/\$
21	10, 20	mulcld 8128	. . . . . . 7 $/\$
22		mulcl 8087	. . . . . . 7 $/\$
23	13, 21, 22	sylancr 414	. . . . . 6 $/\$
24		expcl 10739	. . . . . . 7 $/\$
25	15, 4, 24	sylancl 413	. . . . . 6 $/\$
26	23, 25	addcld 8127	. . . . 5 $/\$
27	19, 26	addcld 8127	. . . 4 $/\$
28	27, 10, 15	adddid 8132	. . 3 $/\$
29	1	oveq2i 5978	. . . . . . . . 9
30		expp1 10728	. . . . . . . . . 10 $/\$
31	10, 4, 30	sylancl 413	. . . . . . . . 9 $/\$
32	29, 31	eqtr2id 2253	. . . . . . . 8 $/\$
33	13	a1i 9	. . . . . . . . . 10 $/\$
34	33, 16, 10	mulassd 8131	. . . . . . . . 9 $/\$
35	14, 15, 10	mul32d 8260	. . . . . . . . . . 11 $/\$
36		df-3 9131	. . . . . . . . . . . . . 14
37	36	oveq2i 5978	. . . . . . . . . . . . 13
38		2nn0 9347	. . . . . . . . . . . . . 14
39		expp1 10728	. . . . . . . . . . . . . 14 $/\$
40	10, 38, 39	sylancl 413	. . . . . . . . . . . . 13 $/\$
41	37, 40	eqtr2id 2253	. . . . . . . . . . . 12 $/\$
42	41	oveq1d 5982	. . . . . . . . . . 11 $/\$
43	35, 42	eqtrd 2240	. . . . . . . . . 10 $/\$
44	43	oveq2d 5983	. . . . . . . . 9 $/\$
45	34, 44	eqtrd 2240	. . . . . . . 8 $/\$
46	32, 45	oveq12d 5985	. . . . . . 7 $/\$
47	12, 10, 18, 46	joinlmuladdmuld 8135	. . . . . 6 $/\$
48	33, 21, 10	mulassd 8131	. . . . . . . . 9 $/\$
49	10, 20, 10	mul32d 8260	. . . . . . . . . . 11 $/\$
50	10	sqvald 10852	. . . . . . . . . . . 12 $/\$
51	50	oveq1d 5982	. . . . . . . . . . 11 $/\$
52	49, 51	eqtr4d 2243	. . . . . . . . . 10 $/\$
53	52	oveq2d 5983	. . . . . . . . 9 $/\$
54	48, 53	eqtrd 2240	. . . . . . . 8 $/\$
55	25, 10	mulcomd 8129	. . . . . . . 8 $/\$
56	54, 55	oveq12d 5985	. . . . . . 7 $/\$
57	23, 10, 25, 56	joinlmuladdmuld 8135	. . . . . 6 $/\$
58	47, 57	oveq12d 5985	. . . . 5 $/\$
59	19, 10, 26, 58	joinlmuladdmuld 8135	. . . 4 $/\$
60	19, 26, 15	adddird 8133	. . . . 5 $/\$
61	33, 16, 15	mulassd 8131	. . . . . . . . 9 $/\$
62	14, 15, 15	mulassd 8131	. . . . . . . . . . 11 $/\$
63	15	sqvald 10852	. . . . . . . . . . . 12 $/\$
64	63	oveq2d 5983	. . . . . . . . . . 11 $/\$
65	62, 64	eqtr4d 2243	. . . . . . . . . 10 $/\$
66	65	oveq2d 5983	. . . . . . . . 9 $/\$
67	61, 66	eqtrd 2240	. . . . . . . 8 $/\$
68	67	oveq2d 5983	. . . . . . 7 $/\$
69	12, 15, 18, 68	joinlmuladdmuld 8135	. . . . . 6 $/\$
70	33, 21, 15	mulassd 8131	. . . . . . . . 9 $/\$
71	10, 20, 15	mulassd 8131	. . . . . . . . . . 11 $/\$
72	36	oveq2i 5978	. . . . . . . . . . . . 13
73		expp1 10728	. . . . . . . . . . . . . 14 $/\$
74	15, 38, 73	sylancl 413	. . . . . . . . . . . . 13 $/\$
75	72, 74	eqtr2id 2253	. . . . . . . . . . . 12 $/\$
76	75	oveq2d 5983	. . . . . . . . . . 11 $/\$
77	71, 76	eqtrd 2240	. . . . . . . . . 10 $/\$
78	77	oveq2d 5983	. . . . . . . . 9 $/\$
79	70, 78	eqtrd 2240	. . . . . . . 8 $/\$
80	1	oveq2i 5978	. . . . . . . . 9
81		expp1 10728	. . . . . . . . . 10 $/\$
82	15, 4, 81	sylancl 413	. . . . . . . . 9 $/\$
83	80, 82	eqtr2id 2253	. . . . . . . 8 $/\$
84	79, 83	oveq12d 5985	. . . . . . 7 $/\$
85	23, 15, 25, 84	joinlmuladdmuld 8135	. . . . . 6 $/\$
86	69, 85	oveq12d 5985	. . . . 5 $/\$
87	12, 15	mulcld 8128	. . . . . 6 $/\$
88	14, 20	mulcld 8128	. . . . . . 7 $/\$
89		mulcl 8087	. . . . . . 7 $/\$
90	13, 88, 89	sylancr 414	. . . . . 6 $/\$
91	10, 25	mulcld 8128	. . . . . . . 8 $/\$
92		mulcl 8087	. . . . . . . 8 $/\$
93	13, 91, 92	sylancr 414	. . . . . . 7 $/\$
94		4nn0 9349	. . . . . . . 8
95		expcl 10739	. . . . . . . 8 $/\$
96	15, 94, 95	sylancl 413	. . . . . . 7 $/\$
97	93, 96	addcld 8127	. . . . . 6 $/\$
98	87, 90, 97	addassd 8130	. . . . 5 $/\$
99	60, 86, 98	3eqtrd 2244	. . . 4 $/\$
100	59, 99	oveq12d 5985	. . 3 $/\$
101		expcl 10739	. . . . . . 7 $/\$
102	10, 94, 101	sylancl 413	. . . . . 6 $/\$
103		mulcl 8087	. . . . . . 7 $/\$
104	13, 87, 103	sylancr 414	. . . . . 6 $/\$
105	102, 104	addcld 8127	. . . . 5 $/\$
106	90, 91	addcld 8127	. . . . 5 $/\$
107	90, 97	addcld 8127	. . . . 5 $/\$
108	105, 106, 87, 107	add4d 8276	. . . 4 $/\$
109	102, 104, 87	addassd 8130	. . . . . 6 $/\$
110	1	oveq1i 5977	. . . . . . . . 9
111		ax-1cn 8053	. . . . . . . . . . 11
112	111	a1i 9	. . . . . . . . . 10 $/\$
113	33, 112, 87	adddird 8133	. . . . . . . . 9 $/\$
114	110, 113	eqtrid 2252	. . . . . . . 8 $/\$
115	87	mulid2d 8126	. . . . . . . . 9 $/\$
116	115	oveq2d 5983	. . . . . . . 8 $/\$
117	114, 116	eqtrd 2240	. . . . . . 7 $/\$
118	117	oveq2d 5983	. . . . . 6 $/\$
119	109, 118	eqtr4d 2243	. . . . 5 $/\$
120	90, 91, 90, 97	add4d 8276	. . . . . 6 $/\$
121		3p3e6 9214	. . . . . . . . 9
122	121	oveq1i 5977	. . . . . . . 8
123	33, 33, 88	adddird 8133	. . . . . . . 8 $/\$
124	122, 123	eqtr3id 2254	. . . . . . 7 $/\$
125		3p1e4 9207	. . . . . . . . . . . . 13
126	13, 111, 125	addcomli 8252	. . . . . . . . . . . 12
127	126	oveq1i 5977	. . . . . . . . . . 11
128	112, 33, 91	adddird 8133	. . . . . . . . . . 11 $/\$
129	127, 128	eqtr3id 2254	. . . . . . . . . 10 $/\$
130	91	mulid2d 8126	. . . . . . . . . . 11 $/\$
131	130	oveq1d 5982	. . . . . . . . . 10 $/\$
132	129, 131	eqtrd 2240	. . . . . . . . 9 $/\$
133	132	oveq1d 5982	. . . . . . . 8 $/\$
134	91, 93, 96	addassd 8130	. . . . . . . 8 $/\$
135	133, 134	eqtrd 2240	. . . . . . 7 $/\$
136	124, 135	oveq12d 5985	. . . . . 6 $/\$
137	120, 136	eqtr4d 2243	. . . . 5 $/\$
138	119, 137	oveq12d 5985	. . . 4 $/\$
139	108, 138	eqtrd 2240	. . 3 $/\$
140	28, 100, 139	3eqtrd 2244	. 2 $/\$
141	7, 9, 140	3eqtrd 2244	1 $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wceq 1373

wcel 2178 (class class class)co 5967

cc 7958

c1 7961

caddc 7963

cmul 7965

c2 9122

c3 9123

c4 9124

c6 9126

cn0 9330

cexp 10720

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 615 ax-in2 616 ax-io 711 ax-5 1471 ax-7 1472 ax-gen 1473 ax-ie1 1517 ax-ie2 1518 ax-8 1528 ax-10 1529 ax-11 1530 ax-i12 1531 ax-bndl 1533 ax-4 1534 ax-17 1550 ax-i9 1554 ax-ial 1558 ax-i5r 1559 ax-13 2180 ax-14 2181 ax-ext 2189 ax-coll 4175 ax-sep 4178 ax-nul 4186 ax-pow 4234 ax-pr 4269 ax-un 4498 ax-setind 4603 ax-iinf 4654 ax-cnex 8051 ax-resscn 8052 ax-1cn 8053 ax-1re 8054 ax-icn 8055 ax-addcl 8056 ax-addrcl 8057 ax-mulcl 8058 ax-mulrcl 8059 ax-addcom 8060 ax-mulcom 8061 ax-addass 8062 ax-mulass 8063 ax-distr 8064 ax-i2m1 8065 ax-0lt1 8066 ax-1rid 8067 ax-0id 8068 ax-rnegex 8069 ax-precex 8070 ax-cnre 8071 ax-pre-ltirr 8072 ax-pre-ltwlin 8073 ax-pre-lttrn 8074 ax-pre-apti 8075 ax-pre-ltadd 8076 ax-pre-mulgt0 8077 ax-pre-mulext 8078

This theorem depends on definitions: df-bi 117 df-dc 837 df-3or 982 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1485 df-sb 1787 df-eu 2058 df-mo 2059 df-clab 2194 df-cleq 2200 df-clel 2203 df-nfc 2339 df-ne 2379 df-nel 2474 df-ral 2491 df-rex 2492 df-reu 2493 df-rmo 2494 df-rab 2495 df-v 2778 df-sbc 3006 df-csb 3102 df-dif 3176 df-un 3178 df-in 3180 df-ss 3187 df-nul 3469 df-if 3580 df-pw 3628 df-sn 3649 df-pr 3650 df-op 3652 df-uni 3865 df-int 3900 df-iun 3943 df-br 4060 df-opab 4122 df-mpt 4123 df-tr 4159 df-id 4358 df-po 4361 df-iso 4362 df-iord 4431 df-on 4433 df-ilim 4434 df-suc 4436 df-iom 4657 df-xp 4699 df-rel 4700 df-cnv 4701 df-co 4702 df-dm 4703 df-rn 4704 df-res 4705 df-ima 4706 df-iota 5251 df-fun 5292 df-fn 5293 df-f 5294 df-f1 5295 df-fo 5296 df-f1o 5297 df-fv 5298 df-riota 5922 df-ov 5970 df-oprab 5971 df-mpo 5972 df-1st 6249 df-2nd 6250 df-recs 6414 df-frec 6500 df-pnf 8144 df-mnf 8145 df-xr 8146 df-ltxr 8147 df-le 8148 df-sub 8280 df-neg 8281 df-reap 8683 df-ap 8690 df-div 8781 df-inn 9072 df-2 9130 df-3 9131 df-4 9132 df-5 9133 df-6 9134 df-n0 9331 df-z 9408 df-uz 9684 df-seqfrec 10630 df-exp 10721

This theorem is referenced by: (None)

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