binom4 - Intuitionistic Logic Explorer

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Theorem binom4 15732

Description: Work out a quartic binomial. (You would think that by this point it would be faster to use binom 12068, 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 9209	. . . 4
2	1	oveq2i 6034	. . 3
3		addcl 8162	. . . 4 $/\$
4		3nn0 9425	. . . 4
5		expp1 10814	. . . 4 $/\$
6	3, 4, 5	sylancl 413	. . 3 $/\$
7	2, 6	eqtrid 2275	. 2 $/\$
8		binom3 10925	. . 3 $/\$
9	8	oveq1d 6038	. 2 $/\$
10		simpl 109	. . . . . . 7 $/\$
11		expcl 10825	. . . . . . 7 $/\$
12	10, 4, 11	sylancl 413	. . . . . 6 $/\$
13		3cn 9223	. . . . . . 7
14	10	sqcld 10939	. . . . . . . 8 $/\$
15		simpr 110	. . . . . . . 8 $/\$
16	14, 15	mulcld 8205	. . . . . . 7 $/\$
17		mulcl 8164	. . . . . . 7 $/\$
18	13, 16, 17	sylancr 414	. . . . . 6 $/\$
19	12, 18	addcld 8204	. . . . 5 $/\$
20	15	sqcld 10939	. . . . . . . 8 $/\$
21	10, 20	mulcld 8205	. . . . . . 7 $/\$
22		mulcl 8164	. . . . . . 7 $/\$
23	13, 21, 22	sylancr 414	. . . . . 6 $/\$
24		expcl 10825	. . . . . . 7 $/\$
25	15, 4, 24	sylancl 413	. . . . . 6 $/\$
26	23, 25	addcld 8204	. . . . 5 $/\$
27	19, 26	addcld 8204	. . . 4 $/\$
28	27, 10, 15	adddid 8209	. . 3 $/\$
29	1	oveq2i 6034	. . . . . . . . 9
30		expp1 10814	. . . . . . . . . 10 $/\$
31	10, 4, 30	sylancl 413	. . . . . . . . 9 $/\$
32	29, 31	eqtr2id 2276	. . . . . . . 8 $/\$
33	13	a1i 9	. . . . . . . . . 10 $/\$
34	33, 16, 10	mulassd 8208	. . . . . . . . 9 $/\$
35	14, 15, 10	mul32d 8337	. . . . . . . . . . 11 $/\$
36		df-3 9208	. . . . . . . . . . . . . 14
37	36	oveq2i 6034	. . . . . . . . . . . . 13
38		2nn0 9424	. . . . . . . . . . . . . 14
39		expp1 10814	. . . . . . . . . . . . . 14 $/\$
40	10, 38, 39	sylancl 413	. . . . . . . . . . . . 13 $/\$
41	37, 40	eqtr2id 2276	. . . . . . . . . . . 12 $/\$
42	41	oveq1d 6038	. . . . . . . . . . 11 $/\$
43	35, 42	eqtrd 2263	. . . . . . . . . 10 $/\$
44	43	oveq2d 6039	. . . . . . . . 9 $/\$
45	34, 44	eqtrd 2263	. . . . . . . 8 $/\$
46	32, 45	oveq12d 6041	. . . . . . 7 $/\$
47	12, 10, 18, 46	joinlmuladdmuld 8212	. . . . . 6 $/\$
48	33, 21, 10	mulassd 8208	. . . . . . . . 9 $/\$
49	10, 20, 10	mul32d 8337	. . . . . . . . . . 11 $/\$
50	10	sqvald 10938	. . . . . . . . . . . 12 $/\$
51	50	oveq1d 6038	. . . . . . . . . . 11 $/\$
52	49, 51	eqtr4d 2266	. . . . . . . . . 10 $/\$
53	52	oveq2d 6039	. . . . . . . . 9 $/\$
54	48, 53	eqtrd 2263	. . . . . . . 8 $/\$
55	25, 10	mulcomd 8206	. . . . . . . 8 $/\$
56	54, 55	oveq12d 6041	. . . . . . 7 $/\$
57	23, 10, 25, 56	joinlmuladdmuld 8212	. . . . . 6 $/\$
58	47, 57	oveq12d 6041	. . . . 5 $/\$
59	19, 10, 26, 58	joinlmuladdmuld 8212	. . . 4 $/\$
60	19, 26, 15	adddird 8210	. . . . 5 $/\$
61	33, 16, 15	mulassd 8208	. . . . . . . . 9 $/\$
62	14, 15, 15	mulassd 8208	. . . . . . . . . . 11 $/\$
63	15	sqvald 10938	. . . . . . . . . . . 12 $/\$
64	63	oveq2d 6039	. . . . . . . . . . 11 $/\$
65	62, 64	eqtr4d 2266	. . . . . . . . . 10 $/\$
66	65	oveq2d 6039	. . . . . . . . 9 $/\$
67	61, 66	eqtrd 2263	. . . . . . . 8 $/\$
68	67	oveq2d 6039	. . . . . . 7 $/\$
69	12, 15, 18, 68	joinlmuladdmuld 8212	. . . . . 6 $/\$
70	33, 21, 15	mulassd 8208	. . . . . . . . 9 $/\$
71	10, 20, 15	mulassd 8208	. . . . . . . . . . 11 $/\$
72	36	oveq2i 6034	. . . . . . . . . . . . 13
73		expp1 10814	. . . . . . . . . . . . . 14 $/\$
74	15, 38, 73	sylancl 413	. . . . . . . . . . . . 13 $/\$
75	72, 74	eqtr2id 2276	. . . . . . . . . . . 12 $/\$
76	75	oveq2d 6039	. . . . . . . . . . 11 $/\$
77	71, 76	eqtrd 2263	. . . . . . . . . 10 $/\$
78	77	oveq2d 6039	. . . . . . . . 9 $/\$
79	70, 78	eqtrd 2263	. . . . . . . 8 $/\$
80	1	oveq2i 6034	. . . . . . . . 9
81		expp1 10814	. . . . . . . . . 10 $/\$
82	15, 4, 81	sylancl 413	. . . . . . . . 9 $/\$
83	80, 82	eqtr2id 2276	. . . . . . . 8 $/\$
84	79, 83	oveq12d 6041	. . . . . . 7 $/\$
85	23, 15, 25, 84	joinlmuladdmuld 8212	. . . . . 6 $/\$
86	69, 85	oveq12d 6041	. . . . 5 $/\$
87	12, 15	mulcld 8205	. . . . . 6 $/\$
88	14, 20	mulcld 8205	. . . . . . 7 $/\$
89		mulcl 8164	. . . . . . 7 $/\$
90	13, 88, 89	sylancr 414	. . . . . 6 $/\$
91	10, 25	mulcld 8205	. . . . . . . 8 $/\$
92		mulcl 8164	. . . . . . . 8 $/\$
93	13, 91, 92	sylancr 414	. . . . . . 7 $/\$
94		4nn0 9426	. . . . . . . 8
95		expcl 10825	. . . . . . . 8 $/\$
96	15, 94, 95	sylancl 413	. . . . . . 7 $/\$
97	93, 96	addcld 8204	. . . . . 6 $/\$
98	87, 90, 97	addassd 8207	. . . . 5 $/\$
99	60, 86, 98	3eqtrd 2267	. . . 4 $/\$
100	59, 99	oveq12d 6041	. . 3 $/\$
101		expcl 10825	. . . . . . 7 $/\$
102	10, 94, 101	sylancl 413	. . . . . 6 $/\$
103		mulcl 8164	. . . . . . 7 $/\$
104	13, 87, 103	sylancr 414	. . . . . 6 $/\$
105	102, 104	addcld 8204	. . . . 5 $/\$
106	90, 91	addcld 8204	. . . . 5 $/\$
107	90, 97	addcld 8204	. . . . 5 $/\$
108	105, 106, 87, 107	add4d 8353	. . . 4 $/\$
109	102, 104, 87	addassd 8207	. . . . . 6 $/\$
110	1	oveq1i 6033	. . . . . . . . 9
111		ax-1cn 8130	. . . . . . . . . . 11
112	111	a1i 9	. . . . . . . . . 10 $/\$
113	33, 112, 87	adddird 8210	. . . . . . . . 9 $/\$
114	110, 113	eqtrid 2275	. . . . . . . 8 $/\$
115	87	mulid2d 8203	. . . . . . . . 9 $/\$
116	115	oveq2d 6039	. . . . . . . 8 $/\$
117	114, 116	eqtrd 2263	. . . . . . 7 $/\$
118	117	oveq2d 6039	. . . . . 6 $/\$
119	109, 118	eqtr4d 2266	. . . . 5 $/\$
120	90, 91, 90, 97	add4d 8353	. . . . . 6 $/\$
121		3p3e6 9291	. . . . . . . . 9
122	121	oveq1i 6033	. . . . . . . 8
123	33, 33, 88	adddird 8210	. . . . . . . 8 $/\$
124	122, 123	eqtr3id 2277	. . . . . . 7 $/\$
125		3p1e4 9284	. . . . . . . . . . . . 13
126	13, 111, 125	addcomli 8329	. . . . . . . . . . . 12
127	126	oveq1i 6033	. . . . . . . . . . 11
128	112, 33, 91	adddird 8210	. . . . . . . . . . 11 $/\$
129	127, 128	eqtr3id 2277	. . . . . . . . . 10 $/\$
130	91	mulid2d 8203	. . . . . . . . . . 11 $/\$
131	130	oveq1d 6038	. . . . . . . . . 10 $/\$
132	129, 131	eqtrd 2263	. . . . . . . . 9 $/\$
133	132	oveq1d 6038	. . . . . . . 8 $/\$
134	91, 93, 96	addassd 8207	. . . . . . . 8 $/\$
135	133, 134	eqtrd 2263	. . . . . . 7 $/\$
136	124, 135	oveq12d 6041	. . . . . 6 $/\$
137	120, 136	eqtr4d 2266	. . . . 5 $/\$
138	119, 137	oveq12d 6041	. . . 4 $/\$
139	108, 138	eqtrd 2263	. . 3 $/\$
140	28, 100, 139	3eqtrd 2267	. 2 $/\$
141	7, 9, 140	3eqtrd 2267	1 $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wceq 1397

wcel 2201 (class class class)co 6023

cc 8035

c1 8038

caddc 8040

cmul 8042

c2 9199

c3 9200

c4 9201

c6 9203

cn0 9407

cexp 10806

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 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-13 2203 ax-14 2204 ax-ext 2212 ax-coll 4205 ax-sep 4208 ax-nul 4216 ax-pow 4266 ax-pr 4301 ax-un 4532 ax-setind 4637 ax-iinf 4688 ax-cnex 8128 ax-resscn 8129 ax-1cn 8130 ax-1re 8131 ax-icn 8132 ax-addcl 8133 ax-addrcl 8134 ax-mulcl 8135 ax-mulrcl 8136 ax-addcom 8137 ax-mulcom 8138 ax-addass 8139 ax-mulass 8140 ax-distr 8141 ax-i2m1 8142 ax-0lt1 8143 ax-1rid 8144 ax-0id 8145 ax-rnegex 8146 ax-precex 8147 ax-cnre 8148 ax-pre-ltirr 8149 ax-pre-ltwlin 8150 ax-pre-lttrn 8151 ax-pre-apti 8152 ax-pre-ltadd 8153 ax-pre-mulgt0 8154 ax-pre-mulext 8155

This theorem depends on definitions: df-bi 117 df-dc 842 df-3or 1005 df-3an 1006 df-tru 1400 df-fal 1403 df-nf 1509 df-sb 1810 df-eu 2081 df-mo 2082 df-clab 2217 df-cleq 2223 df-clel 2226 df-nfc 2362 df-ne 2402 df-nel 2497 df-ral 2514 df-rex 2515 df-reu 2516 df-rmo 2517 df-rab 2518 df-v 2803 df-sbc 3031 df-csb 3127 df-dif 3201 df-un 3203 df-in 3205 df-ss 3212 df-nul 3494 df-if 3605 df-pw 3655 df-sn 3676 df-pr 3677 df-op 3679 df-uni 3895 df-int 3930 df-iun 3973 df-br 4090 df-opab 4152 df-mpt 4153 df-tr 4189 df-id 4392 df-po 4395 df-iso 4396 df-iord 4465 df-on 4467 df-ilim 4468 df-suc 4470 df-iom 4691 df-xp 4733 df-rel 4734 df-cnv 4735 df-co 4736 df-dm 4737 df-rn 4738 df-res 4739 df-ima 4740 df-iota 5288 df-fun 5330 df-fn 5331 df-f 5332 df-f1 5333 df-fo 5334 df-f1o 5335 df-fv 5336 df-riota 5976 df-ov 6026 df-oprab 6027 df-mpo 6028 df-1st 6308 df-2nd 6309 df-recs 6476 df-frec 6562 df-pnf 8221 df-mnf 8222 df-xr 8223 df-ltxr 8224 df-le 8225 df-sub 8357 df-neg 8358 df-reap 8760 df-ap 8767 df-div 8858 df-inn 9149 df-2 9207 df-3 9208 df-4 9209 df-5 9210 df-6 9211 df-n0 9408 df-z 9485 df-uz 9761 df-seqfrec 10716 df-exp 10807

This theorem is referenced by: (None)

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