binom4 - Intuitionistic Logic Explorer

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Theorem binom4 13497

Description: Work out a quartic binomial. (You would think that by this point it would be faster to use binom 11421, 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 8914	. . . 4
2	1	oveq2i 5852	. . 3
3		addcl 7874	. . . 4 $/\$
4		3nn0 9128	. . . 4
5		expp1 10458	. . . 4 $/\$
6	3, 4, 5	sylancl 410	. . 3 $/\$
7	2, 6	syl5eq 2210	. 2 $/\$
8		binom3 10568	. . 3 $/\$
9	8	oveq1d 5856	. 2 $/\$
10		simpl 108	. . . . . . 7 $/\$
11		expcl 10469	. . . . . . 7 $/\$
12	10, 4, 11	sylancl 410	. . . . . 6 $/\$
13		3cn 8928	. . . . . . 7
14	10	sqcld 10582	. . . . . . . 8 $/\$
15		simpr 109	. . . . . . . 8 $/\$
16	14, 15	mulcld 7915	. . . . . . 7 $/\$
17		mulcl 7876	. . . . . . 7 $/\$
18	13, 16, 17	sylancr 411	. . . . . 6 $/\$
19	12, 18	addcld 7914	. . . . 5 $/\$
20	15	sqcld 10582	. . . . . . . 8 $/\$
21	10, 20	mulcld 7915	. . . . . . 7 $/\$
22		mulcl 7876	. . . . . . 7 $/\$
23	13, 21, 22	sylancr 411	. . . . . 6 $/\$
24		expcl 10469	. . . . . . 7 $/\$
25	15, 4, 24	sylancl 410	. . . . . 6 $/\$
26	23, 25	addcld 7914	. . . . 5 $/\$
27	19, 26	addcld 7914	. . . 4 $/\$
28	27, 10, 15	adddid 7919	. . 3 $/\$
29	1	oveq2i 5852	. . . . . . . . 9
30		expp1 10458	. . . . . . . . . 10 $/\$
31	10, 4, 30	sylancl 410	. . . . . . . . 9 $/\$
32	29, 31	eqtr2id 2211	. . . . . . . 8 $/\$
33	13	a1i 9	. . . . . . . . . 10 $/\$
34	33, 16, 10	mulassd 7918	. . . . . . . . 9 $/\$
35	14, 15, 10	mul32d 8047	. . . . . . . . . . 11 $/\$
36		df-3 8913	. . . . . . . . . . . . . 14
37	36	oveq2i 5852	. . . . . . . . . . . . 13
38		2nn0 9127	. . . . . . . . . . . . . 14
39		expp1 10458	. . . . . . . . . . . . . 14 $/\$
40	10, 38, 39	sylancl 410	. . . . . . . . . . . . 13 $/\$
41	37, 40	eqtr2id 2211	. . . . . . . . . . . 12 $/\$
42	41	oveq1d 5856	. . . . . . . . . . 11 $/\$
43	35, 42	eqtrd 2198	. . . . . . . . . 10 $/\$
44	43	oveq2d 5857	. . . . . . . . 9 $/\$
45	34, 44	eqtrd 2198	. . . . . . . 8 $/\$
46	32, 45	oveq12d 5859	. . . . . . 7 $/\$
47	12, 10, 18, 46	joinlmuladdmuld 7922	. . . . . 6 $/\$
48	33, 21, 10	mulassd 7918	. . . . . . . . 9 $/\$
49	10, 20, 10	mul32d 8047	. . . . . . . . . . 11 $/\$
50	10	sqvald 10581	. . . . . . . . . . . 12 $/\$
51	50	oveq1d 5856	. . . . . . . . . . 11 $/\$
52	49, 51	eqtr4d 2201	. . . . . . . . . 10 $/\$
53	52	oveq2d 5857	. . . . . . . . 9 $/\$
54	48, 53	eqtrd 2198	. . . . . . . 8 $/\$
55	25, 10	mulcomd 7916	. . . . . . . 8 $/\$
56	54, 55	oveq12d 5859	. . . . . . 7 $/\$
57	23, 10, 25, 56	joinlmuladdmuld 7922	. . . . . 6 $/\$
58	47, 57	oveq12d 5859	. . . . 5 $/\$
59	19, 10, 26, 58	joinlmuladdmuld 7922	. . . 4 $/\$
60	19, 26, 15	adddird 7920	. . . . 5 $/\$
61	33, 16, 15	mulassd 7918	. . . . . . . . 9 $/\$
62	14, 15, 15	mulassd 7918	. . . . . . . . . . 11 $/\$
63	15	sqvald 10581	. . . . . . . . . . . 12 $/\$
64	63	oveq2d 5857	. . . . . . . . . . 11 $/\$
65	62, 64	eqtr4d 2201	. . . . . . . . . 10 $/\$
66	65	oveq2d 5857	. . . . . . . . 9 $/\$
67	61, 66	eqtrd 2198	. . . . . . . 8 $/\$
68	67	oveq2d 5857	. . . . . . 7 $/\$
69	12, 15, 18, 68	joinlmuladdmuld 7922	. . . . . 6 $/\$
70	33, 21, 15	mulassd 7918	. . . . . . . . 9 $/\$
71	10, 20, 15	mulassd 7918	. . . . . . . . . . 11 $/\$
72	36	oveq2i 5852	. . . . . . . . . . . . 13
73		expp1 10458	. . . . . . . . . . . . . 14 $/\$
74	15, 38, 73	sylancl 410	. . . . . . . . . . . . 13 $/\$
75	72, 74	eqtr2id 2211	. . . . . . . . . . . 12 $/\$
76	75	oveq2d 5857	. . . . . . . . . . 11 $/\$
77	71, 76	eqtrd 2198	. . . . . . . . . 10 $/\$
78	77	oveq2d 5857	. . . . . . . . 9 $/\$
79	70, 78	eqtrd 2198	. . . . . . . 8 $/\$
80	1	oveq2i 5852	. . . . . . . . 9
81		expp1 10458	. . . . . . . . . 10 $/\$
82	15, 4, 81	sylancl 410	. . . . . . . . 9 $/\$
83	80, 82	eqtr2id 2211	. . . . . . . 8 $/\$
84	79, 83	oveq12d 5859	. . . . . . 7 $/\$
85	23, 15, 25, 84	joinlmuladdmuld 7922	. . . . . 6 $/\$
86	69, 85	oveq12d 5859	. . . . 5 $/\$
87	12, 15	mulcld 7915	. . . . . 6 $/\$
88	14, 20	mulcld 7915	. . . . . . 7 $/\$
89		mulcl 7876	. . . . . . 7 $/\$
90	13, 88, 89	sylancr 411	. . . . . 6 $/\$
91	10, 25	mulcld 7915	. . . . . . . 8 $/\$
92		mulcl 7876	. . . . . . . 8 $/\$
93	13, 91, 92	sylancr 411	. . . . . . 7 $/\$
94		4nn0 9129	. . . . . . . 8
95		expcl 10469	. . . . . . . 8 $/\$
96	15, 94, 95	sylancl 410	. . . . . . 7 $/\$
97	93, 96	addcld 7914	. . . . . 6 $/\$
98	87, 90, 97	addassd 7917	. . . . 5 $/\$
99	60, 86, 98	3eqtrd 2202	. . . 4 $/\$
100	59, 99	oveq12d 5859	. . 3 $/\$
101		expcl 10469	. . . . . . 7 $/\$
102	10, 94, 101	sylancl 410	. . . . . 6 $/\$
103		mulcl 7876	. . . . . . 7 $/\$
104	13, 87, 103	sylancr 411	. . . . . 6 $/\$
105	102, 104	addcld 7914	. . . . 5 $/\$
106	90, 91	addcld 7914	. . . . 5 $/\$
107	90, 97	addcld 7914	. . . . 5 $/\$
108	105, 106, 87, 107	add4d 8063	. . . 4 $/\$
109	102, 104, 87	addassd 7917	. . . . . 6 $/\$
110	1	oveq1i 5851	. . . . . . . . 9
111		ax-1cn 7842	. . . . . . . . . . 11
112	111	a1i 9	. . . . . . . . . 10 $/\$
113	33, 112, 87	adddird 7920	. . . . . . . . 9 $/\$
114	110, 113	syl5eq 2210	. . . . . . . 8 $/\$
115	87	mulid2d 7913	. . . . . . . . 9 $/\$
116	115	oveq2d 5857	. . . . . . . 8 $/\$
117	114, 116	eqtrd 2198	. . . . . . 7 $/\$
118	117	oveq2d 5857	. . . . . 6 $/\$
119	109, 118	eqtr4d 2201	. . . . 5 $/\$
120	90, 91, 90, 97	add4d 8063	. . . . . 6 $/\$
121		3p3e6 8995	. . . . . . . . 9
122	121	oveq1i 5851	. . . . . . . 8
123	33, 33, 88	adddird 7920	. . . . . . . 8 $/\$
124	122, 123	eqtr3id 2212	. . . . . . 7 $/\$
125		3p1e4 8988	. . . . . . . . . . . . 13
126	13, 111, 125	addcomli 8039	. . . . . . . . . . . 12
127	126	oveq1i 5851	. . . . . . . . . . 11
128	112, 33, 91	adddird 7920	. . . . . . . . . . 11 $/\$
129	127, 128	eqtr3id 2212	. . . . . . . . . 10 $/\$
130	91	mulid2d 7913	. . . . . . . . . . 11 $/\$
131	130	oveq1d 5856	. . . . . . . . . 10 $/\$
132	129, 131	eqtrd 2198	. . . . . . . . 9 $/\$
133	132	oveq1d 5856	. . . . . . . 8 $/\$
134	91, 93, 96	addassd 7917	. . . . . . . 8 $/\$
135	133, 134	eqtrd 2198	. . . . . . 7 $/\$
136	124, 135	oveq12d 5859	. . . . . 6 $/\$
137	120, 136	eqtr4d 2201	. . . . 5 $/\$
138	119, 137	oveq12d 5859	. . . 4 $/\$
139	108, 138	eqtrd 2198	. . 3 $/\$
140	28, 100, 139	3eqtrd 2202	. 2 $/\$
141	7, 9, 140	3eqtrd 2202	1 $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 103

wceq 1343

wcel 2136 (class class class)co 5841

cc 7747

c1 7750

caddc 7752

cmul 7754

c2 8904

c3 8905

c4 8906

c6 8908

cn0 9110

cexp 10450

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 604 ax-in2 605 ax-io 699 ax-5 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-coll 4096 ax-sep 4099 ax-nul 4107 ax-pow 4152 ax-pr 4186 ax-un 4410 ax-setind 4513 ax-iinf 4564 ax-cnex 7840 ax-resscn 7841 ax-1cn 7842 ax-1re 7843 ax-icn 7844 ax-addcl 7845 ax-addrcl 7846 ax-mulcl 7847 ax-mulrcl 7848 ax-addcom 7849 ax-mulcom 7850 ax-addass 7851 ax-mulass 7852 ax-distr 7853 ax-i2m1 7854 ax-0lt1 7855 ax-1rid 7856 ax-0id 7857 ax-rnegex 7858 ax-precex 7859 ax-cnre 7860 ax-pre-ltirr 7861 ax-pre-ltwlin 7862 ax-pre-lttrn 7863 ax-pre-apti 7864 ax-pre-ltadd 7865 ax-pre-mulgt0 7866 ax-pre-mulext 7867

This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 969 df-3an 970 df-tru 1346 df-fal 1349 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2296 df-ne 2336 df-nel 2431 df-ral 2448 df-rex 2449 df-reu 2450 df-rmo 2451 df-rab 2452 df-v 2727 df-sbc 2951 df-csb 3045 df-dif 3117 df-un 3119 df-in 3121 df-ss 3128 df-nul 3409 df-if 3520 df-pw 3560 df-sn 3581 df-pr 3582 df-op 3584 df-uni 3789 df-int 3824 df-iun 3867 df-br 3982 df-opab 4043 df-mpt 4044 df-tr 4080 df-id 4270 df-po 4273 df-iso 4274 df-iord 4343 df-on 4345 df-ilim 4346 df-suc 4348 df-iom 4567 df-xp 4609 df-rel 4610 df-cnv 4611 df-co 4612 df-dm 4613 df-rn 4614 df-res 4615 df-ima 4616 df-iota 5152 df-fun 5189 df-fn 5190 df-f 5191 df-f1 5192 df-fo 5193 df-f1o 5194 df-fv 5195 df-riota 5797 df-ov 5844 df-oprab 5845 df-mpo 5846 df-1st 6105 df-2nd 6106 df-recs 6269 df-frec 6355 df-pnf 7931 df-mnf 7932 df-xr 7933 df-ltxr 7934 df-le 7935 df-sub 8067 df-neg 8068 df-reap 8469 df-ap 8476 df-div 8565 df-inn 8854 df-2 8912 df-3 8913 df-4 8914 df-5 8915 df-6 8916 df-n0 9111 df-z 9188 df-uz 9463 df-seqfrec 10377 df-exp 10451

This theorem is referenced by: (None)

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