binom4 - Intuitionistic Logic Explorer

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

Theorem binom4 15152

Description: Work out a quartic binomial. (You would think that by this point it would be faster to use binom 11630, 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 9045	. . . 4
2	1	oveq2i 5930	. . 3
3		addcl 7999	. . . 4 $/\$
4		3nn0 9261	. . . 4
5		expp1 10620	. . . 4 $/\$
6	3, 4, 5	sylancl 413	. . 3 $/\$
7	2, 6	eqtrid 2238	. 2 $/\$
8		binom3 10731	. . 3 $/\$
9	8	oveq1d 5934	. 2 $/\$
10		simpl 109	. . . . . . 7 $/\$
11		expcl 10631	. . . . . . 7 $/\$
12	10, 4, 11	sylancl 413	. . . . . 6 $/\$
13		3cn 9059	. . . . . . 7
14	10	sqcld 10745	. . . . . . . 8 $/\$
15		simpr 110	. . . . . . . 8 $/\$
16	14, 15	mulcld 8042	. . . . . . 7 $/\$
17		mulcl 8001	. . . . . . 7 $/\$
18	13, 16, 17	sylancr 414	. . . . . 6 $/\$
19	12, 18	addcld 8041	. . . . 5 $/\$
20	15	sqcld 10745	. . . . . . . 8 $/\$
21	10, 20	mulcld 8042	. . . . . . 7 $/\$
22		mulcl 8001	. . . . . . 7 $/\$
23	13, 21, 22	sylancr 414	. . . . . 6 $/\$
24		expcl 10631	. . . . . . 7 $/\$
25	15, 4, 24	sylancl 413	. . . . . 6 $/\$
26	23, 25	addcld 8041	. . . . 5 $/\$
27	19, 26	addcld 8041	. . . 4 $/\$
28	27, 10, 15	adddid 8046	. . 3 $/\$
29	1	oveq2i 5930	. . . . . . . . 9
30		expp1 10620	. . . . . . . . . 10 $/\$
31	10, 4, 30	sylancl 413	. . . . . . . . 9 $/\$
32	29, 31	eqtr2id 2239	. . . . . . . 8 $/\$
33	13	a1i 9	. . . . . . . . . 10 $/\$
34	33, 16, 10	mulassd 8045	. . . . . . . . 9 $/\$
35	14, 15, 10	mul32d 8174	. . . . . . . . . . 11 $/\$
36		df-3 9044	. . . . . . . . . . . . . 14
37	36	oveq2i 5930	. . . . . . . . . . . . 13
38		2nn0 9260	. . . . . . . . . . . . . 14
39		expp1 10620	. . . . . . . . . . . . . 14 $/\$
40	10, 38, 39	sylancl 413	. . . . . . . . . . . . 13 $/\$
41	37, 40	eqtr2id 2239	. . . . . . . . . . . 12 $/\$
42	41	oveq1d 5934	. . . . . . . . . . 11 $/\$
43	35, 42	eqtrd 2226	. . . . . . . . . 10 $/\$
44	43	oveq2d 5935	. . . . . . . . 9 $/\$
45	34, 44	eqtrd 2226	. . . . . . . 8 $/\$
46	32, 45	oveq12d 5937	. . . . . . 7 $/\$
47	12, 10, 18, 46	joinlmuladdmuld 8049	. . . . . 6 $/\$
48	33, 21, 10	mulassd 8045	. . . . . . . . 9 $/\$
49	10, 20, 10	mul32d 8174	. . . . . . . . . . 11 $/\$
50	10	sqvald 10744	. . . . . . . . . . . 12 $/\$
51	50	oveq1d 5934	. . . . . . . . . . 11 $/\$
52	49, 51	eqtr4d 2229	. . . . . . . . . 10 $/\$
53	52	oveq2d 5935	. . . . . . . . 9 $/\$
54	48, 53	eqtrd 2226	. . . . . . . 8 $/\$
55	25, 10	mulcomd 8043	. . . . . . . 8 $/\$
56	54, 55	oveq12d 5937	. . . . . . 7 $/\$
57	23, 10, 25, 56	joinlmuladdmuld 8049	. . . . . 6 $/\$
58	47, 57	oveq12d 5937	. . . . 5 $/\$
59	19, 10, 26, 58	joinlmuladdmuld 8049	. . . 4 $/\$
60	19, 26, 15	adddird 8047	. . . . 5 $/\$
61	33, 16, 15	mulassd 8045	. . . . . . . . 9 $/\$
62	14, 15, 15	mulassd 8045	. . . . . . . . . . 11 $/\$
63	15	sqvald 10744	. . . . . . . . . . . 12 $/\$
64	63	oveq2d 5935	. . . . . . . . . . 11 $/\$
65	62, 64	eqtr4d 2229	. . . . . . . . . 10 $/\$
66	65	oveq2d 5935	. . . . . . . . 9 $/\$
67	61, 66	eqtrd 2226	. . . . . . . 8 $/\$
68	67	oveq2d 5935	. . . . . . 7 $/\$
69	12, 15, 18, 68	joinlmuladdmuld 8049	. . . . . 6 $/\$
70	33, 21, 15	mulassd 8045	. . . . . . . . 9 $/\$
71	10, 20, 15	mulassd 8045	. . . . . . . . . . 11 $/\$
72	36	oveq2i 5930	. . . . . . . . . . . . 13
73		expp1 10620	. . . . . . . . . . . . . 14 $/\$
74	15, 38, 73	sylancl 413	. . . . . . . . . . . . 13 $/\$
75	72, 74	eqtr2id 2239	. . . . . . . . . . . 12 $/\$
76	75	oveq2d 5935	. . . . . . . . . . 11 $/\$
77	71, 76	eqtrd 2226	. . . . . . . . . 10 $/\$
78	77	oveq2d 5935	. . . . . . . . 9 $/\$
79	70, 78	eqtrd 2226	. . . . . . . 8 $/\$
80	1	oveq2i 5930	. . . . . . . . 9
81		expp1 10620	. . . . . . . . . 10 $/\$
82	15, 4, 81	sylancl 413	. . . . . . . . 9 $/\$
83	80, 82	eqtr2id 2239	. . . . . . . 8 $/\$
84	79, 83	oveq12d 5937	. . . . . . 7 $/\$
85	23, 15, 25, 84	joinlmuladdmuld 8049	. . . . . 6 $/\$
86	69, 85	oveq12d 5937	. . . . 5 $/\$
87	12, 15	mulcld 8042	. . . . . 6 $/\$
88	14, 20	mulcld 8042	. . . . . . 7 $/\$
89		mulcl 8001	. . . . . . 7 $/\$
90	13, 88, 89	sylancr 414	. . . . . 6 $/\$
91	10, 25	mulcld 8042	. . . . . . . 8 $/\$
92		mulcl 8001	. . . . . . . 8 $/\$
93	13, 91, 92	sylancr 414	. . . . . . 7 $/\$
94		4nn0 9262	. . . . . . . 8
95		expcl 10631	. . . . . . . 8 $/\$
96	15, 94, 95	sylancl 413	. . . . . . 7 $/\$
97	93, 96	addcld 8041	. . . . . 6 $/\$
98	87, 90, 97	addassd 8044	. . . . 5 $/\$
99	60, 86, 98	3eqtrd 2230	. . . 4 $/\$
100	59, 99	oveq12d 5937	. . 3 $/\$
101		expcl 10631	. . . . . . 7 $/\$
102	10, 94, 101	sylancl 413	. . . . . 6 $/\$
103		mulcl 8001	. . . . . . 7 $/\$
104	13, 87, 103	sylancr 414	. . . . . 6 $/\$
105	102, 104	addcld 8041	. . . . 5 $/\$
106	90, 91	addcld 8041	. . . . 5 $/\$
107	90, 97	addcld 8041	. . . . 5 $/\$
108	105, 106, 87, 107	add4d 8190	. . . 4 $/\$
109	102, 104, 87	addassd 8044	. . . . . 6 $/\$
110	1	oveq1i 5929	. . . . . . . . 9
111		ax-1cn 7967	. . . . . . . . . . 11
112	111	a1i 9	. . . . . . . . . 10 $/\$
113	33, 112, 87	adddird 8047	. . . . . . . . 9 $/\$
114	110, 113	eqtrid 2238	. . . . . . . 8 $/\$
115	87	mulid2d 8040	. . . . . . . . 9 $/\$
116	115	oveq2d 5935	. . . . . . . 8 $/\$
117	114, 116	eqtrd 2226	. . . . . . 7 $/\$
118	117	oveq2d 5935	. . . . . 6 $/\$
119	109, 118	eqtr4d 2229	. . . . 5 $/\$
120	90, 91, 90, 97	add4d 8190	. . . . . 6 $/\$
121		3p3e6 9127	. . . . . . . . 9
122	121	oveq1i 5929	. . . . . . . 8
123	33, 33, 88	adddird 8047	. . . . . . . 8 $/\$
124	122, 123	eqtr3id 2240	. . . . . . 7 $/\$
125		3p1e4 9120	. . . . . . . . . . . . 13
126	13, 111, 125	addcomli 8166	. . . . . . . . . . . 12
127	126	oveq1i 5929	. . . . . . . . . . 11
128	112, 33, 91	adddird 8047	. . . . . . . . . . 11 $/\$
129	127, 128	eqtr3id 2240	. . . . . . . . . 10 $/\$
130	91	mulid2d 8040	. . . . . . . . . . 11 $/\$
131	130	oveq1d 5934	. . . . . . . . . 10 $/\$
132	129, 131	eqtrd 2226	. . . . . . . . 9 $/\$
133	132	oveq1d 5934	. . . . . . . 8 $/\$
134	91, 93, 96	addassd 8044	. . . . . . . 8 $/\$
135	133, 134	eqtrd 2226	. . . . . . 7 $/\$
136	124, 135	oveq12d 5937	. . . . . 6 $/\$
137	120, 136	eqtr4d 2229	. . . . 5 $/\$
138	119, 137	oveq12d 5937	. . . 4 $/\$
139	108, 138	eqtrd 2226	. . 3 $/\$
140	28, 100, 139	3eqtrd 2230	. 2 $/\$
141	7, 9, 140	3eqtrd 2230	1 $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wceq 1364

wcel 2164 (class class class)co 5919

cc 7872

c1 7875

caddc 7877

cmul 7879

c2 9035

c3 9036

c4 9037

c6 9039

cn0 9243

cexp 10612

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 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-coll 4145 ax-sep 4148 ax-nul 4156 ax-pow 4204 ax-pr 4239 ax-un 4465 ax-setind 4570 ax-iinf 4621 ax-cnex 7965 ax-resscn 7966 ax-1cn 7967 ax-1re 7968 ax-icn 7969 ax-addcl 7970 ax-addrcl 7971 ax-mulcl 7972 ax-mulrcl 7973 ax-addcom 7974 ax-mulcom 7975 ax-addass 7976 ax-mulass 7977 ax-distr 7978 ax-i2m1 7979 ax-0lt1 7980 ax-1rid 7981 ax-0id 7982 ax-rnegex 7983 ax-precex 7984 ax-cnre 7985 ax-pre-ltirr 7986 ax-pre-ltwlin 7987 ax-pre-lttrn 7988 ax-pre-apti 7989 ax-pre-ltadd 7990 ax-pre-mulgt0 7991 ax-pre-mulext 7992

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rmo 2480 df-rab 2481 df-v 2762 df-sbc 2987 df-csb 3082 df-dif 3156 df-un 3158 df-in 3160 df-ss 3167 df-nul 3448 df-if 3559 df-pw 3604 df-sn 3625 df-pr 3626 df-op 3628 df-uni 3837 df-int 3872 df-iun 3915 df-br 4031 df-opab 4092 df-mpt 4093 df-tr 4129 df-id 4325 df-po 4328 df-iso 4329 df-iord 4398 df-on 4400 df-ilim 4401 df-suc 4403 df-iom 4624 df-xp 4666 df-rel 4667 df-cnv 4668 df-co 4669 df-dm 4670 df-rn 4671 df-res 4672 df-ima 4673 df-iota 5216 df-fun 5257 df-fn 5258 df-f 5259 df-f1 5260 df-fo 5261 df-f1o 5262 df-fv 5263 df-riota 5874 df-ov 5922 df-oprab 5923 df-mpo 5924 df-1st 6195 df-2nd 6196 df-recs 6360 df-frec 6446 df-pnf 8058 df-mnf 8059 df-xr 8060 df-ltxr 8061 df-le 8062 df-sub 8194 df-neg 8195 df-reap 8596 df-ap 8603 df-div 8694 df-inn 8985 df-2 9043 df-3 9044 df-4 9045 df-5 9046 df-6 9047 df-n0 9244 df-z 9321 df-uz 9596 df-seqfrec 10522 df-exp 10613

This theorem is referenced by: (None)

W3C validator