binom4 - Intuitionistic Logic Explorer

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Theorem binom4 15215

Description: Work out a quartic binomial. (You would think that by this point it would be faster to use binom 11649, 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 9051	. . . 4
2	1	oveq2i 5933	. . 3
3		addcl 8004	. . . 4 $/\$
4		3nn0 9267	. . . 4
5		expp1 10638	. . . 4 $/\$
6	3, 4, 5	sylancl 413	. . 3 $/\$
7	2, 6	eqtrid 2241	. 2 $/\$
8		binom3 10749	. . 3 $/\$
9	8	oveq1d 5937	. 2 $/\$
10		simpl 109	. . . . . . 7 $/\$
11		expcl 10649	. . . . . . 7 $/\$
12	10, 4, 11	sylancl 413	. . . . . 6 $/\$
13		3cn 9065	. . . . . . 7
14	10	sqcld 10763	. . . . . . . 8 $/\$
15		simpr 110	. . . . . . . 8 $/\$
16	14, 15	mulcld 8047	. . . . . . 7 $/\$
17		mulcl 8006	. . . . . . 7 $/\$
18	13, 16, 17	sylancr 414	. . . . . 6 $/\$
19	12, 18	addcld 8046	. . . . 5 $/\$
20	15	sqcld 10763	. . . . . . . 8 $/\$
21	10, 20	mulcld 8047	. . . . . . 7 $/\$
22		mulcl 8006	. . . . . . 7 $/\$
23	13, 21, 22	sylancr 414	. . . . . 6 $/\$
24		expcl 10649	. . . . . . 7 $/\$
25	15, 4, 24	sylancl 413	. . . . . 6 $/\$
26	23, 25	addcld 8046	. . . . 5 $/\$
27	19, 26	addcld 8046	. . . 4 $/\$
28	27, 10, 15	adddid 8051	. . 3 $/\$
29	1	oveq2i 5933	. . . . . . . . 9
30		expp1 10638	. . . . . . . . . 10 $/\$
31	10, 4, 30	sylancl 413	. . . . . . . . 9 $/\$
32	29, 31	eqtr2id 2242	. . . . . . . 8 $/\$
33	13	a1i 9	. . . . . . . . . 10 $/\$
34	33, 16, 10	mulassd 8050	. . . . . . . . 9 $/\$
35	14, 15, 10	mul32d 8179	. . . . . . . . . . 11 $/\$
36		df-3 9050	. . . . . . . . . . . . . 14
37	36	oveq2i 5933	. . . . . . . . . . . . 13
38		2nn0 9266	. . . . . . . . . . . . . 14
39		expp1 10638	. . . . . . . . . . . . . 14 $/\$
40	10, 38, 39	sylancl 413	. . . . . . . . . . . . 13 $/\$
41	37, 40	eqtr2id 2242	. . . . . . . . . . . 12 $/\$
42	41	oveq1d 5937	. . . . . . . . . . 11 $/\$
43	35, 42	eqtrd 2229	. . . . . . . . . 10 $/\$
44	43	oveq2d 5938	. . . . . . . . 9 $/\$
45	34, 44	eqtrd 2229	. . . . . . . 8 $/\$
46	32, 45	oveq12d 5940	. . . . . . 7 $/\$
47	12, 10, 18, 46	joinlmuladdmuld 8054	. . . . . 6 $/\$
48	33, 21, 10	mulassd 8050	. . . . . . . . 9 $/\$
49	10, 20, 10	mul32d 8179	. . . . . . . . . . 11 $/\$
50	10	sqvald 10762	. . . . . . . . . . . 12 $/\$
51	50	oveq1d 5937	. . . . . . . . . . 11 $/\$
52	49, 51	eqtr4d 2232	. . . . . . . . . 10 $/\$
53	52	oveq2d 5938	. . . . . . . . 9 $/\$
54	48, 53	eqtrd 2229	. . . . . . . 8 $/\$
55	25, 10	mulcomd 8048	. . . . . . . 8 $/\$
56	54, 55	oveq12d 5940	. . . . . . 7 $/\$
57	23, 10, 25, 56	joinlmuladdmuld 8054	. . . . . 6 $/\$
58	47, 57	oveq12d 5940	. . . . 5 $/\$
59	19, 10, 26, 58	joinlmuladdmuld 8054	. . . 4 $/\$
60	19, 26, 15	adddird 8052	. . . . 5 $/\$
61	33, 16, 15	mulassd 8050	. . . . . . . . 9 $/\$
62	14, 15, 15	mulassd 8050	. . . . . . . . . . 11 $/\$
63	15	sqvald 10762	. . . . . . . . . . . 12 $/\$
64	63	oveq2d 5938	. . . . . . . . . . 11 $/\$
65	62, 64	eqtr4d 2232	. . . . . . . . . 10 $/\$
66	65	oveq2d 5938	. . . . . . . . 9 $/\$
67	61, 66	eqtrd 2229	. . . . . . . 8 $/\$
68	67	oveq2d 5938	. . . . . . 7 $/\$
69	12, 15, 18, 68	joinlmuladdmuld 8054	. . . . . 6 $/\$
70	33, 21, 15	mulassd 8050	. . . . . . . . 9 $/\$
71	10, 20, 15	mulassd 8050	. . . . . . . . . . 11 $/\$
72	36	oveq2i 5933	. . . . . . . . . . . . 13
73		expp1 10638	. . . . . . . . . . . . . 14 $/\$
74	15, 38, 73	sylancl 413	. . . . . . . . . . . . 13 $/\$
75	72, 74	eqtr2id 2242	. . . . . . . . . . . 12 $/\$
76	75	oveq2d 5938	. . . . . . . . . . 11 $/\$
77	71, 76	eqtrd 2229	. . . . . . . . . 10 $/\$
78	77	oveq2d 5938	. . . . . . . . 9 $/\$
79	70, 78	eqtrd 2229	. . . . . . . 8 $/\$
80	1	oveq2i 5933	. . . . . . . . 9
81		expp1 10638	. . . . . . . . . 10 $/\$
82	15, 4, 81	sylancl 413	. . . . . . . . 9 $/\$
83	80, 82	eqtr2id 2242	. . . . . . . 8 $/\$
84	79, 83	oveq12d 5940	. . . . . . 7 $/\$
85	23, 15, 25, 84	joinlmuladdmuld 8054	. . . . . 6 $/\$
86	69, 85	oveq12d 5940	. . . . 5 $/\$
87	12, 15	mulcld 8047	. . . . . 6 $/\$
88	14, 20	mulcld 8047	. . . . . . 7 $/\$
89		mulcl 8006	. . . . . . 7 $/\$
90	13, 88, 89	sylancr 414	. . . . . 6 $/\$
91	10, 25	mulcld 8047	. . . . . . . 8 $/\$
92		mulcl 8006	. . . . . . . 8 $/\$
93	13, 91, 92	sylancr 414	. . . . . . 7 $/\$
94		4nn0 9268	. . . . . . . 8
95		expcl 10649	. . . . . . . 8 $/\$
96	15, 94, 95	sylancl 413	. . . . . . 7 $/\$
97	93, 96	addcld 8046	. . . . . 6 $/\$
98	87, 90, 97	addassd 8049	. . . . 5 $/\$
99	60, 86, 98	3eqtrd 2233	. . . 4 $/\$
100	59, 99	oveq12d 5940	. . 3 $/\$
101		expcl 10649	. . . . . . 7 $/\$
102	10, 94, 101	sylancl 413	. . . . . 6 $/\$
103		mulcl 8006	. . . . . . 7 $/\$
104	13, 87, 103	sylancr 414	. . . . . 6 $/\$
105	102, 104	addcld 8046	. . . . 5 $/\$
106	90, 91	addcld 8046	. . . . 5 $/\$
107	90, 97	addcld 8046	. . . . 5 $/\$
108	105, 106, 87, 107	add4d 8195	. . . 4 $/\$
109	102, 104, 87	addassd 8049	. . . . . 6 $/\$
110	1	oveq1i 5932	. . . . . . . . 9
111		ax-1cn 7972	. . . . . . . . . . 11
112	111	a1i 9	. . . . . . . . . 10 $/\$
113	33, 112, 87	adddird 8052	. . . . . . . . 9 $/\$
114	110, 113	eqtrid 2241	. . . . . . . 8 $/\$
115	87	mulid2d 8045	. . . . . . . . 9 $/\$
116	115	oveq2d 5938	. . . . . . . 8 $/\$
117	114, 116	eqtrd 2229	. . . . . . 7 $/\$
118	117	oveq2d 5938	. . . . . 6 $/\$
119	109, 118	eqtr4d 2232	. . . . 5 $/\$
120	90, 91, 90, 97	add4d 8195	. . . . . 6 $/\$
121		3p3e6 9133	. . . . . . . . 9
122	121	oveq1i 5932	. . . . . . . 8
123	33, 33, 88	adddird 8052	. . . . . . . 8 $/\$
124	122, 123	eqtr3id 2243	. . . . . . 7 $/\$
125		3p1e4 9126	. . . . . . . . . . . . 13
126	13, 111, 125	addcomli 8171	. . . . . . . . . . . 12
127	126	oveq1i 5932	. . . . . . . . . . 11
128	112, 33, 91	adddird 8052	. . . . . . . . . . 11 $/\$
129	127, 128	eqtr3id 2243	. . . . . . . . . 10 $/\$
130	91	mulid2d 8045	. . . . . . . . . . 11 $/\$
131	130	oveq1d 5937	. . . . . . . . . 10 $/\$
132	129, 131	eqtrd 2229	. . . . . . . . 9 $/\$
133	132	oveq1d 5937	. . . . . . . 8 $/\$
134	91, 93, 96	addassd 8049	. . . . . . . 8 $/\$
135	133, 134	eqtrd 2229	. . . . . . 7 $/\$
136	124, 135	oveq12d 5940	. . . . . 6 $/\$
137	120, 136	eqtr4d 2232	. . . . 5 $/\$
138	119, 137	oveq12d 5940	. . . 4 $/\$
139	108, 138	eqtrd 2229	. . 3 $/\$
140	28, 100, 139	3eqtrd 2233	. 2 $/\$
141	7, 9, 140	3eqtrd 2233	1 $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wceq 1364

wcel 2167 (class class class)co 5922

cc 7877

c1 7880

caddc 7882

cmul 7884

c2 9041

c3 9042

c4 9043

c6 9045

cn0 9249

cexp 10630

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 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-coll 4148 ax-sep 4151 ax-nul 4159 ax-pow 4207 ax-pr 4242 ax-un 4468 ax-setind 4573 ax-iinf 4624 ax-cnex 7970 ax-resscn 7971 ax-1cn 7972 ax-1re 7973 ax-icn 7974 ax-addcl 7975 ax-addrcl 7976 ax-mulcl 7977 ax-mulrcl 7978 ax-addcom 7979 ax-mulcom 7980 ax-addass 7981 ax-mulass 7982 ax-distr 7983 ax-i2m1 7984 ax-0lt1 7985 ax-1rid 7986 ax-0id 7987 ax-rnegex 7988 ax-precex 7989 ax-cnre 7990 ax-pre-ltirr 7991 ax-pre-ltwlin 7992 ax-pre-lttrn 7993 ax-pre-apti 7994 ax-pre-ltadd 7995 ax-pre-mulgt0 7996 ax-pre-mulext 7997

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 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-ral 2480 df-rex 2481 df-reu 2482 df-rmo 2483 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-nul 3451 df-if 3562 df-pw 3607 df-sn 3628 df-pr 3629 df-op 3631 df-uni 3840 df-int 3875 df-iun 3918 df-br 4034 df-opab 4095 df-mpt 4096 df-tr 4132 df-id 4328 df-po 4331 df-iso 4332 df-iord 4401 df-on 4403 df-ilim 4404 df-suc 4406 df-iom 4627 df-xp 4669 df-rel 4670 df-cnv 4671 df-co 4672 df-dm 4673 df-rn 4674 df-res 4675 df-ima 4676 df-iota 5219 df-fun 5260 df-fn 5261 df-f 5262 df-f1 5263 df-fo 5264 df-f1o 5265 df-fv 5266 df-riota 5877 df-ov 5925 df-oprab 5926 df-mpo 5927 df-1st 6198 df-2nd 6199 df-recs 6363 df-frec 6449 df-pnf 8063 df-mnf 8064 df-xr 8065 df-ltxr 8066 df-le 8067 df-sub 8199 df-neg 8200 df-reap 8602 df-ap 8609 df-div 8700 df-inn 8991 df-2 9049 df-3 9050 df-4 9051 df-5 9052 df-6 9053 df-n0 9250 df-z 9327 df-uz 9602 df-seqfrec 10540 df-exp 10631

This theorem is referenced by: (None)

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