binom4 - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > binom4		Unicode version

Theorem binom4 15451

Description: Work out a quartic binomial. (You would think that by this point it would be faster to use binom 11795, 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 9097	. . . 4
2	1	oveq2i 5955	. . 3
3		addcl 8050	. . . 4 $/\$
4		3nn0 9313	. . . 4
5		expp1 10691	. . . 4 $/\$
6	3, 4, 5	sylancl 413	. . 3 $/\$
7	2, 6	eqtrid 2250	. 2 $/\$
8		binom3 10802	. . 3 $/\$
9	8	oveq1d 5959	. 2 $/\$
10		simpl 109	. . . . . . 7 $/\$
11		expcl 10702	. . . . . . 7 $/\$
12	10, 4, 11	sylancl 413	. . . . . 6 $/\$
13		3cn 9111	. . . . . . 7
14	10	sqcld 10816	. . . . . . . 8 $/\$
15		simpr 110	. . . . . . . 8 $/\$
16	14, 15	mulcld 8093	. . . . . . 7 $/\$
17		mulcl 8052	. . . . . . 7 $/\$
18	13, 16, 17	sylancr 414	. . . . . 6 $/\$
19	12, 18	addcld 8092	. . . . 5 $/\$
20	15	sqcld 10816	. . . . . . . 8 $/\$
21	10, 20	mulcld 8093	. . . . . . 7 $/\$
22		mulcl 8052	. . . . . . 7 $/\$
23	13, 21, 22	sylancr 414	. . . . . 6 $/\$
24		expcl 10702	. . . . . . 7 $/\$
25	15, 4, 24	sylancl 413	. . . . . 6 $/\$
26	23, 25	addcld 8092	. . . . 5 $/\$
27	19, 26	addcld 8092	. . . 4 $/\$
28	27, 10, 15	adddid 8097	. . 3 $/\$
29	1	oveq2i 5955	. . . . . . . . 9
30		expp1 10691	. . . . . . . . . 10 $/\$
31	10, 4, 30	sylancl 413	. . . . . . . . 9 $/\$
32	29, 31	eqtr2id 2251	. . . . . . . 8 $/\$
33	13	a1i 9	. . . . . . . . . 10 $/\$
34	33, 16, 10	mulassd 8096	. . . . . . . . 9 $/\$
35	14, 15, 10	mul32d 8225	. . . . . . . . . . 11 $/\$
36		df-3 9096	. . . . . . . . . . . . . 14
37	36	oveq2i 5955	. . . . . . . . . . . . 13
38		2nn0 9312	. . . . . . . . . . . . . 14
39		expp1 10691	. . . . . . . . . . . . . 14 $/\$
40	10, 38, 39	sylancl 413	. . . . . . . . . . . . 13 $/\$
41	37, 40	eqtr2id 2251	. . . . . . . . . . . 12 $/\$
42	41	oveq1d 5959	. . . . . . . . . . 11 $/\$
43	35, 42	eqtrd 2238	. . . . . . . . . 10 $/\$
44	43	oveq2d 5960	. . . . . . . . 9 $/\$
45	34, 44	eqtrd 2238	. . . . . . . 8 $/\$
46	32, 45	oveq12d 5962	. . . . . . 7 $/\$
47	12, 10, 18, 46	joinlmuladdmuld 8100	. . . . . 6 $/\$
48	33, 21, 10	mulassd 8096	. . . . . . . . 9 $/\$
49	10, 20, 10	mul32d 8225	. . . . . . . . . . 11 $/\$
50	10	sqvald 10815	. . . . . . . . . . . 12 $/\$
51	50	oveq1d 5959	. . . . . . . . . . 11 $/\$
52	49, 51	eqtr4d 2241	. . . . . . . . . 10 $/\$
53	52	oveq2d 5960	. . . . . . . . 9 $/\$
54	48, 53	eqtrd 2238	. . . . . . . 8 $/\$
55	25, 10	mulcomd 8094	. . . . . . . 8 $/\$
56	54, 55	oveq12d 5962	. . . . . . 7 $/\$
57	23, 10, 25, 56	joinlmuladdmuld 8100	. . . . . 6 $/\$
58	47, 57	oveq12d 5962	. . . . 5 $/\$
59	19, 10, 26, 58	joinlmuladdmuld 8100	. . . 4 $/\$
60	19, 26, 15	adddird 8098	. . . . 5 $/\$
61	33, 16, 15	mulassd 8096	. . . . . . . . 9 $/\$
62	14, 15, 15	mulassd 8096	. . . . . . . . . . 11 $/\$
63	15	sqvald 10815	. . . . . . . . . . . 12 $/\$
64	63	oveq2d 5960	. . . . . . . . . . 11 $/\$
65	62, 64	eqtr4d 2241	. . . . . . . . . 10 $/\$
66	65	oveq2d 5960	. . . . . . . . 9 $/\$
67	61, 66	eqtrd 2238	. . . . . . . 8 $/\$
68	67	oveq2d 5960	. . . . . . 7 $/\$
69	12, 15, 18, 68	joinlmuladdmuld 8100	. . . . . 6 $/\$
70	33, 21, 15	mulassd 8096	. . . . . . . . 9 $/\$
71	10, 20, 15	mulassd 8096	. . . . . . . . . . 11 $/\$
72	36	oveq2i 5955	. . . . . . . . . . . . 13
73		expp1 10691	. . . . . . . . . . . . . 14 $/\$
74	15, 38, 73	sylancl 413	. . . . . . . . . . . . 13 $/\$
75	72, 74	eqtr2id 2251	. . . . . . . . . . . 12 $/\$
76	75	oveq2d 5960	. . . . . . . . . . 11 $/\$
77	71, 76	eqtrd 2238	. . . . . . . . . 10 $/\$
78	77	oveq2d 5960	. . . . . . . . 9 $/\$
79	70, 78	eqtrd 2238	. . . . . . . 8 $/\$
80	1	oveq2i 5955	. . . . . . . . 9
81		expp1 10691	. . . . . . . . . 10 $/\$
82	15, 4, 81	sylancl 413	. . . . . . . . 9 $/\$
83	80, 82	eqtr2id 2251	. . . . . . . 8 $/\$
84	79, 83	oveq12d 5962	. . . . . . 7 $/\$
85	23, 15, 25, 84	joinlmuladdmuld 8100	. . . . . 6 $/\$
86	69, 85	oveq12d 5962	. . . . 5 $/\$
87	12, 15	mulcld 8093	. . . . . 6 $/\$
88	14, 20	mulcld 8093	. . . . . . 7 $/\$
89		mulcl 8052	. . . . . . 7 $/\$
90	13, 88, 89	sylancr 414	. . . . . 6 $/\$
91	10, 25	mulcld 8093	. . . . . . . 8 $/\$
92		mulcl 8052	. . . . . . . 8 $/\$
93	13, 91, 92	sylancr 414	. . . . . . 7 $/\$
94		4nn0 9314	. . . . . . . 8
95		expcl 10702	. . . . . . . 8 $/\$
96	15, 94, 95	sylancl 413	. . . . . . 7 $/\$
97	93, 96	addcld 8092	. . . . . 6 $/\$
98	87, 90, 97	addassd 8095	. . . . 5 $/\$
99	60, 86, 98	3eqtrd 2242	. . . 4 $/\$
100	59, 99	oveq12d 5962	. . 3 $/\$
101		expcl 10702	. . . . . . 7 $/\$
102	10, 94, 101	sylancl 413	. . . . . 6 $/\$
103		mulcl 8052	. . . . . . 7 $/\$
104	13, 87, 103	sylancr 414	. . . . . 6 $/\$
105	102, 104	addcld 8092	. . . . 5 $/\$
106	90, 91	addcld 8092	. . . . 5 $/\$
107	90, 97	addcld 8092	. . . . 5 $/\$
108	105, 106, 87, 107	add4d 8241	. . . 4 $/\$
109	102, 104, 87	addassd 8095	. . . . . 6 $/\$
110	1	oveq1i 5954	. . . . . . . . 9
111		ax-1cn 8018	. . . . . . . . . . 11
112	111	a1i 9	. . . . . . . . . 10 $/\$
113	33, 112, 87	adddird 8098	. . . . . . . . 9 $/\$
114	110, 113	eqtrid 2250	. . . . . . . 8 $/\$
115	87	mulid2d 8091	. . . . . . . . 9 $/\$
116	115	oveq2d 5960	. . . . . . . 8 $/\$
117	114, 116	eqtrd 2238	. . . . . . 7 $/\$
118	117	oveq2d 5960	. . . . . 6 $/\$
119	109, 118	eqtr4d 2241	. . . . 5 $/\$
120	90, 91, 90, 97	add4d 8241	. . . . . 6 $/\$
121		3p3e6 9179	. . . . . . . . 9
122	121	oveq1i 5954	. . . . . . . 8
123	33, 33, 88	adddird 8098	. . . . . . . 8 $/\$
124	122, 123	eqtr3id 2252	. . . . . . 7 $/\$
125		3p1e4 9172	. . . . . . . . . . . . 13
126	13, 111, 125	addcomli 8217	. . . . . . . . . . . 12
127	126	oveq1i 5954	. . . . . . . . . . 11
128	112, 33, 91	adddird 8098	. . . . . . . . . . 11 $/\$
129	127, 128	eqtr3id 2252	. . . . . . . . . 10 $/\$
130	91	mulid2d 8091	. . . . . . . . . . 11 $/\$
131	130	oveq1d 5959	. . . . . . . . . 10 $/\$
132	129, 131	eqtrd 2238	. . . . . . . . 9 $/\$
133	132	oveq1d 5959	. . . . . . . 8 $/\$
134	91, 93, 96	addassd 8095	. . . . . . . 8 $/\$
135	133, 134	eqtrd 2238	. . . . . . 7 $/\$
136	124, 135	oveq12d 5962	. . . . . 6 $/\$
137	120, 136	eqtr4d 2241	. . . . 5 $/\$
138	119, 137	oveq12d 5962	. . . 4 $/\$
139	108, 138	eqtrd 2238	. . 3 $/\$
140	28, 100, 139	3eqtrd 2242	. 2 $/\$
141	7, 9, 140	3eqtrd 2242	1 $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wceq 1373

wcel 2176 (class class class)co 5944

cc 7923

c1 7926

caddc 7928

cmul 7930

c2 9087

c3 9088

c4 9089

c6 9091

cn0 9295

cexp 10683

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 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-13 2178 ax-14 2179 ax-ext 2187 ax-coll 4159 ax-sep 4162 ax-nul 4170 ax-pow 4218 ax-pr 4253 ax-un 4480 ax-setind 4585 ax-iinf 4636 ax-cnex 8016 ax-resscn 8017 ax-1cn 8018 ax-1re 8019 ax-icn 8020 ax-addcl 8021 ax-addrcl 8022 ax-mulcl 8023 ax-mulrcl 8024 ax-addcom 8025 ax-mulcom 8026 ax-addass 8027 ax-mulass 8028 ax-distr 8029 ax-i2m1 8030 ax-0lt1 8031 ax-1rid 8032 ax-0id 8033 ax-rnegex 8034 ax-precex 8035 ax-cnre 8036 ax-pre-ltirr 8037 ax-pre-ltwlin 8038 ax-pre-lttrn 8039 ax-pre-apti 8040 ax-pre-ltadd 8041 ax-pre-mulgt0 8042 ax-pre-mulext 8043

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 1484 df-sb 1786 df-eu 2057 df-mo 2058 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-ne 2377 df-nel 2472 df-ral 2489 df-rex 2490 df-reu 2491 df-rmo 2492 df-rab 2493 df-v 2774 df-sbc 2999 df-csb 3094 df-dif 3168 df-un 3170 df-in 3172 df-ss 3179 df-nul 3461 df-if 3572 df-pw 3618 df-sn 3639 df-pr 3640 df-op 3642 df-uni 3851 df-int 3886 df-iun 3929 df-br 4045 df-opab 4106 df-mpt 4107 df-tr 4143 df-id 4340 df-po 4343 df-iso 4344 df-iord 4413 df-on 4415 df-ilim 4416 df-suc 4418 df-iom 4639 df-xp 4681 df-rel 4682 df-cnv 4683 df-co 4684 df-dm 4685 df-rn 4686 df-res 4687 df-ima 4688 df-iota 5232 df-fun 5273 df-fn 5274 df-f 5275 df-f1 5276 df-fo 5277 df-f1o 5278 df-fv 5279 df-riota 5899 df-ov 5947 df-oprab 5948 df-mpo 5949 df-1st 6226 df-2nd 6227 df-recs 6391 df-frec 6477 df-pnf 8109 df-mnf 8110 df-xr 8111 df-ltxr 8112 df-le 8113 df-sub 8245 df-neg 8246 df-reap 8648 df-ap 8655 df-div 8746 df-inn 9037 df-2 9095 df-3 9096 df-4 9097 df-5 9098 df-6 9099 df-n0 9296 df-z 9373 df-uz 9649 df-seqfrec 10593 df-exp 10684

This theorem is referenced by: (None)

W3C validator