binom4 - Intuitionistic Logic Explorer

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Theorem binom4 13691

Description: Work out a quartic binomial. (You would think that by this point it would be faster to use binom 11447, 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 8939	. . . 4
2	1	oveq2i 5864	. . 3
3		addcl 7899	. . . 4 $/\$
4		3nn0 9153	. . . 4
5		expp1 10483	. . . 4 $/\$
6	3, 4, 5	sylancl 411	. . 3 $/\$
7	2, 6	eqtrid 2215	. 2 $/\$
8		binom3 10593	. . 3 $/\$
9	8	oveq1d 5868	. 2 $/\$
10		simpl 108	. . . . . . 7 $/\$
11		expcl 10494	. . . . . . 7 $/\$
12	10, 4, 11	sylancl 411	. . . . . 6 $/\$
13		3cn 8953	. . . . . . 7
14	10	sqcld 10607	. . . . . . . 8 $/\$
15		simpr 109	. . . . . . . 8 $/\$
16	14, 15	mulcld 7940	. . . . . . 7 $/\$
17		mulcl 7901	. . . . . . 7 $/\$
18	13, 16, 17	sylancr 412	. . . . . 6 $/\$
19	12, 18	addcld 7939	. . . . 5 $/\$
20	15	sqcld 10607	. . . . . . . 8 $/\$
21	10, 20	mulcld 7940	. . . . . . 7 $/\$
22		mulcl 7901	. . . . . . 7 $/\$
23	13, 21, 22	sylancr 412	. . . . . 6 $/\$
24		expcl 10494	. . . . . . 7 $/\$
25	15, 4, 24	sylancl 411	. . . . . 6 $/\$
26	23, 25	addcld 7939	. . . . 5 $/\$
27	19, 26	addcld 7939	. . . 4 $/\$
28	27, 10, 15	adddid 7944	. . 3 $/\$
29	1	oveq2i 5864	. . . . . . . . 9
30		expp1 10483	. . . . . . . . . 10 $/\$
31	10, 4, 30	sylancl 411	. . . . . . . . 9 $/\$
32	29, 31	eqtr2id 2216	. . . . . . . 8 $/\$
33	13	a1i 9	. . . . . . . . . 10 $/\$
34	33, 16, 10	mulassd 7943	. . . . . . . . 9 $/\$
35	14, 15, 10	mul32d 8072	. . . . . . . . . . 11 $/\$
36		df-3 8938	. . . . . . . . . . . . . 14
37	36	oveq2i 5864	. . . . . . . . . . . . 13
38		2nn0 9152	. . . . . . . . . . . . . 14
39		expp1 10483	. . . . . . . . . . . . . 14 $/\$
40	10, 38, 39	sylancl 411	. . . . . . . . . . . . 13 $/\$
41	37, 40	eqtr2id 2216	. . . . . . . . . . . 12 $/\$
42	41	oveq1d 5868	. . . . . . . . . . 11 $/\$
43	35, 42	eqtrd 2203	. . . . . . . . . 10 $/\$
44	43	oveq2d 5869	. . . . . . . . 9 $/\$
45	34, 44	eqtrd 2203	. . . . . . . 8 $/\$
46	32, 45	oveq12d 5871	. . . . . . 7 $/\$
47	12, 10, 18, 46	joinlmuladdmuld 7947	. . . . . 6 $/\$
48	33, 21, 10	mulassd 7943	. . . . . . . . 9 $/\$
49	10, 20, 10	mul32d 8072	. . . . . . . . . . 11 $/\$
50	10	sqvald 10606	. . . . . . . . . . . 12 $/\$
51	50	oveq1d 5868	. . . . . . . . . . 11 $/\$
52	49, 51	eqtr4d 2206	. . . . . . . . . 10 $/\$
53	52	oveq2d 5869	. . . . . . . . 9 $/\$
54	48, 53	eqtrd 2203	. . . . . . . 8 $/\$
55	25, 10	mulcomd 7941	. . . . . . . 8 $/\$
56	54, 55	oveq12d 5871	. . . . . . 7 $/\$
57	23, 10, 25, 56	joinlmuladdmuld 7947	. . . . . 6 $/\$
58	47, 57	oveq12d 5871	. . . . 5 $/\$
59	19, 10, 26, 58	joinlmuladdmuld 7947	. . . 4 $/\$
60	19, 26, 15	adddird 7945	. . . . 5 $/\$
61	33, 16, 15	mulassd 7943	. . . . . . . . 9 $/\$
62	14, 15, 15	mulassd 7943	. . . . . . . . . . 11 $/\$
63	15	sqvald 10606	. . . . . . . . . . . 12 $/\$
64	63	oveq2d 5869	. . . . . . . . . . 11 $/\$
65	62, 64	eqtr4d 2206	. . . . . . . . . 10 $/\$
66	65	oveq2d 5869	. . . . . . . . 9 $/\$
67	61, 66	eqtrd 2203	. . . . . . . 8 $/\$
68	67	oveq2d 5869	. . . . . . 7 $/\$
69	12, 15, 18, 68	joinlmuladdmuld 7947	. . . . . 6 $/\$
70	33, 21, 15	mulassd 7943	. . . . . . . . 9 $/\$
71	10, 20, 15	mulassd 7943	. . . . . . . . . . 11 $/\$
72	36	oveq2i 5864	. . . . . . . . . . . . 13
73		expp1 10483	. . . . . . . . . . . . . 14 $/\$
74	15, 38, 73	sylancl 411	. . . . . . . . . . . . 13 $/\$
75	72, 74	eqtr2id 2216	. . . . . . . . . . . 12 $/\$
76	75	oveq2d 5869	. . . . . . . . . . 11 $/\$
77	71, 76	eqtrd 2203	. . . . . . . . . 10 $/\$
78	77	oveq2d 5869	. . . . . . . . 9 $/\$
79	70, 78	eqtrd 2203	. . . . . . . 8 $/\$
80	1	oveq2i 5864	. . . . . . . . 9
81		expp1 10483	. . . . . . . . . 10 $/\$
82	15, 4, 81	sylancl 411	. . . . . . . . 9 $/\$
83	80, 82	eqtr2id 2216	. . . . . . . 8 $/\$
84	79, 83	oveq12d 5871	. . . . . . 7 $/\$
85	23, 15, 25, 84	joinlmuladdmuld 7947	. . . . . 6 $/\$
86	69, 85	oveq12d 5871	. . . . 5 $/\$
87	12, 15	mulcld 7940	. . . . . 6 $/\$
88	14, 20	mulcld 7940	. . . . . . 7 $/\$
89		mulcl 7901	. . . . . . 7 $/\$
90	13, 88, 89	sylancr 412	. . . . . 6 $/\$
91	10, 25	mulcld 7940	. . . . . . . 8 $/\$
92		mulcl 7901	. . . . . . . 8 $/\$
93	13, 91, 92	sylancr 412	. . . . . . 7 $/\$
94		4nn0 9154	. . . . . . . 8
95		expcl 10494	. . . . . . . 8 $/\$
96	15, 94, 95	sylancl 411	. . . . . . 7 $/\$
97	93, 96	addcld 7939	. . . . . 6 $/\$
98	87, 90, 97	addassd 7942	. . . . 5 $/\$
99	60, 86, 98	3eqtrd 2207	. . . 4 $/\$
100	59, 99	oveq12d 5871	. . 3 $/\$
101		expcl 10494	. . . . . . 7 $/\$
102	10, 94, 101	sylancl 411	. . . . . 6 $/\$
103		mulcl 7901	. . . . . . 7 $/\$
104	13, 87, 103	sylancr 412	. . . . . 6 $/\$
105	102, 104	addcld 7939	. . . . 5 $/\$
106	90, 91	addcld 7939	. . . . 5 $/\$
107	90, 97	addcld 7939	. . . . 5 $/\$
108	105, 106, 87, 107	add4d 8088	. . . 4 $/\$
109	102, 104, 87	addassd 7942	. . . . . 6 $/\$
110	1	oveq1i 5863	. . . . . . . . 9
111		ax-1cn 7867	. . . . . . . . . . 11
112	111	a1i 9	. . . . . . . . . 10 $/\$
113	33, 112, 87	adddird 7945	. . . . . . . . 9 $/\$
114	110, 113	eqtrid 2215	. . . . . . . 8 $/\$
115	87	mulid2d 7938	. . . . . . . . 9 $/\$
116	115	oveq2d 5869	. . . . . . . 8 $/\$
117	114, 116	eqtrd 2203	. . . . . . 7 $/\$
118	117	oveq2d 5869	. . . . . 6 $/\$
119	109, 118	eqtr4d 2206	. . . . 5 $/\$
120	90, 91, 90, 97	add4d 8088	. . . . . 6 $/\$
121		3p3e6 9020	. . . . . . . . 9
122	121	oveq1i 5863	. . . . . . . 8
123	33, 33, 88	adddird 7945	. . . . . . . 8 $/\$
124	122, 123	eqtr3id 2217	. . . . . . 7 $/\$
125		3p1e4 9013	. . . . . . . . . . . . 13
126	13, 111, 125	addcomli 8064	. . . . . . . . . . . 12
127	126	oveq1i 5863	. . . . . . . . . . 11
128	112, 33, 91	adddird 7945	. . . . . . . . . . 11 $/\$
129	127, 128	eqtr3id 2217	. . . . . . . . . 10 $/\$
130	91	mulid2d 7938	. . . . . . . . . . 11 $/\$
131	130	oveq1d 5868	. . . . . . . . . 10 $/\$
132	129, 131	eqtrd 2203	. . . . . . . . 9 $/\$
133	132	oveq1d 5868	. . . . . . . 8 $/\$
134	91, 93, 96	addassd 7942	. . . . . . . 8 $/\$
135	133, 134	eqtrd 2203	. . . . . . 7 $/\$
136	124, 135	oveq12d 5871	. . . . . 6 $/\$
137	120, 136	eqtr4d 2206	. . . . 5 $/\$
138	119, 137	oveq12d 5871	. . . 4 $/\$
139	108, 138	eqtrd 2203	. . 3 $/\$
140	28, 100, 139	3eqtrd 2207	. 2 $/\$
141	7, 9, 140	3eqtrd 2207	1 $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 103

wceq 1348

wcel 2141 (class class class)co 5853

cc 7772

c1 7775

caddc 7777

cmul 7779

c2 8929

c3 8930

c4 8931

c6 8933

cn0 9135

cexp 10475

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-mulrcl 7873 ax-addcom 7874 ax-mulcom 7875 ax-addass 7876 ax-mulass 7877 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-1rid 7881 ax-0id 7882 ax-rnegex 7883 ax-precex 7884 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-apti 7889 ax-pre-ltadd 7890 ax-pre-mulgt0 7891 ax-pre-mulext 7892

This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3527 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-id 4278 df-po 4281 df-iso 4282 df-iord 4351 df-on 4353 df-ilim 4354 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-recs 6284 df-frec 6370 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-reap 8494 df-ap 8501 df-div 8590 df-inn 8879 df-2 8937 df-3 8938 df-4 8939 df-5 8940 df-6 8941 df-n0 9136 df-z 9213 df-uz 9488 df-seqfrec 10402 df-exp 10476

This theorem is referenced by: (None)

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