binom4 - Intuitionistic Logic Explorer

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Theorem binom4 15484

Description: Work out a quartic binomial. (You would think that by this point it would be faster to use binom 11828, 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 9099	. . . 4
2	1	oveq2i 5957	. . 3
3		addcl 8052	. . . 4 $/\$
4		3nn0 9315	. . . 4
5		expp1 10693	. . . 4 $/\$
6	3, 4, 5	sylancl 413	. . 3 $/\$
7	2, 6	eqtrid 2250	. 2 $/\$
8		binom3 10804	. . 3 $/\$
9	8	oveq1d 5961	. 2 $/\$
10		simpl 109	. . . . . . 7 $/\$
11		expcl 10704	. . . . . . 7 $/\$
12	10, 4, 11	sylancl 413	. . . . . 6 $/\$
13		3cn 9113	. . . . . . 7
14	10	sqcld 10818	. . . . . . . 8 $/\$
15		simpr 110	. . . . . . . 8 $/\$
16	14, 15	mulcld 8095	. . . . . . 7 $/\$
17		mulcl 8054	. . . . . . 7 $/\$
18	13, 16, 17	sylancr 414	. . . . . 6 $/\$
19	12, 18	addcld 8094	. . . . 5 $/\$
20	15	sqcld 10818	. . . . . . . 8 $/\$
21	10, 20	mulcld 8095	. . . . . . 7 $/\$
22		mulcl 8054	. . . . . . 7 $/\$
23	13, 21, 22	sylancr 414	. . . . . 6 $/\$
24		expcl 10704	. . . . . . 7 $/\$
25	15, 4, 24	sylancl 413	. . . . . 6 $/\$
26	23, 25	addcld 8094	. . . . 5 $/\$
27	19, 26	addcld 8094	. . . 4 $/\$
28	27, 10, 15	adddid 8099	. . 3 $/\$
29	1	oveq2i 5957	. . . . . . . . 9
30		expp1 10693	. . . . . . . . . 10 $/\$
31	10, 4, 30	sylancl 413	. . . . . . . . 9 $/\$
32	29, 31	eqtr2id 2251	. . . . . . . 8 $/\$
33	13	a1i 9	. . . . . . . . . 10 $/\$
34	33, 16, 10	mulassd 8098	. . . . . . . . 9 $/\$
35	14, 15, 10	mul32d 8227	. . . . . . . . . . 11 $/\$
36		df-3 9098	. . . . . . . . . . . . . 14
37	36	oveq2i 5957	. . . . . . . . . . . . 13
38		2nn0 9314	. . . . . . . . . . . . . 14
39		expp1 10693	. . . . . . . . . . . . . 14 $/\$
40	10, 38, 39	sylancl 413	. . . . . . . . . . . . 13 $/\$
41	37, 40	eqtr2id 2251	. . . . . . . . . . . 12 $/\$
42	41	oveq1d 5961	. . . . . . . . . . 11 $/\$
43	35, 42	eqtrd 2238	. . . . . . . . . 10 $/\$
44	43	oveq2d 5962	. . . . . . . . 9 $/\$
45	34, 44	eqtrd 2238	. . . . . . . 8 $/\$
46	32, 45	oveq12d 5964	. . . . . . 7 $/\$
47	12, 10, 18, 46	joinlmuladdmuld 8102	. . . . . 6 $/\$
48	33, 21, 10	mulassd 8098	. . . . . . . . 9 $/\$
49	10, 20, 10	mul32d 8227	. . . . . . . . . . 11 $/\$
50	10	sqvald 10817	. . . . . . . . . . . 12 $/\$
51	50	oveq1d 5961	. . . . . . . . . . 11 $/\$
52	49, 51	eqtr4d 2241	. . . . . . . . . 10 $/\$
53	52	oveq2d 5962	. . . . . . . . 9 $/\$
54	48, 53	eqtrd 2238	. . . . . . . 8 $/\$
55	25, 10	mulcomd 8096	. . . . . . . 8 $/\$
56	54, 55	oveq12d 5964	. . . . . . 7 $/\$
57	23, 10, 25, 56	joinlmuladdmuld 8102	. . . . . 6 $/\$
58	47, 57	oveq12d 5964	. . . . 5 $/\$
59	19, 10, 26, 58	joinlmuladdmuld 8102	. . . 4 $/\$
60	19, 26, 15	adddird 8100	. . . . 5 $/\$
61	33, 16, 15	mulassd 8098	. . . . . . . . 9 $/\$
62	14, 15, 15	mulassd 8098	. . . . . . . . . . 11 $/\$
63	15	sqvald 10817	. . . . . . . . . . . 12 $/\$
64	63	oveq2d 5962	. . . . . . . . . . 11 $/\$
65	62, 64	eqtr4d 2241	. . . . . . . . . 10 $/\$
66	65	oveq2d 5962	. . . . . . . . 9 $/\$
67	61, 66	eqtrd 2238	. . . . . . . 8 $/\$
68	67	oveq2d 5962	. . . . . . 7 $/\$
69	12, 15, 18, 68	joinlmuladdmuld 8102	. . . . . 6 $/\$
70	33, 21, 15	mulassd 8098	. . . . . . . . 9 $/\$
71	10, 20, 15	mulassd 8098	. . . . . . . . . . 11 $/\$
72	36	oveq2i 5957	. . . . . . . . . . . . 13
73		expp1 10693	. . . . . . . . . . . . . 14 $/\$
74	15, 38, 73	sylancl 413	. . . . . . . . . . . . 13 $/\$
75	72, 74	eqtr2id 2251	. . . . . . . . . . . 12 $/\$
76	75	oveq2d 5962	. . . . . . . . . . 11 $/\$
77	71, 76	eqtrd 2238	. . . . . . . . . 10 $/\$
78	77	oveq2d 5962	. . . . . . . . 9 $/\$
79	70, 78	eqtrd 2238	. . . . . . . 8 $/\$
80	1	oveq2i 5957	. . . . . . . . 9
81		expp1 10693	. . . . . . . . . 10 $/\$
82	15, 4, 81	sylancl 413	. . . . . . . . 9 $/\$
83	80, 82	eqtr2id 2251	. . . . . . . 8 $/\$
84	79, 83	oveq12d 5964	. . . . . . 7 $/\$
85	23, 15, 25, 84	joinlmuladdmuld 8102	. . . . . 6 $/\$
86	69, 85	oveq12d 5964	. . . . 5 $/\$
87	12, 15	mulcld 8095	. . . . . 6 $/\$
88	14, 20	mulcld 8095	. . . . . . 7 $/\$
89		mulcl 8054	. . . . . . 7 $/\$
90	13, 88, 89	sylancr 414	. . . . . 6 $/\$
91	10, 25	mulcld 8095	. . . . . . . 8 $/\$
92		mulcl 8054	. . . . . . . 8 $/\$
93	13, 91, 92	sylancr 414	. . . . . . 7 $/\$
94		4nn0 9316	. . . . . . . 8
95		expcl 10704	. . . . . . . 8 $/\$
96	15, 94, 95	sylancl 413	. . . . . . 7 $/\$
97	93, 96	addcld 8094	. . . . . 6 $/\$
98	87, 90, 97	addassd 8097	. . . . 5 $/\$
99	60, 86, 98	3eqtrd 2242	. . . 4 $/\$
100	59, 99	oveq12d 5964	. . 3 $/\$
101		expcl 10704	. . . . . . 7 $/\$
102	10, 94, 101	sylancl 413	. . . . . 6 $/\$
103		mulcl 8054	. . . . . . 7 $/\$
104	13, 87, 103	sylancr 414	. . . . . 6 $/\$
105	102, 104	addcld 8094	. . . . 5 $/\$
106	90, 91	addcld 8094	. . . . 5 $/\$
107	90, 97	addcld 8094	. . . . 5 $/\$
108	105, 106, 87, 107	add4d 8243	. . . 4 $/\$
109	102, 104, 87	addassd 8097	. . . . . 6 $/\$
110	1	oveq1i 5956	. . . . . . . . 9
111		ax-1cn 8020	. . . . . . . . . . 11
112	111	a1i 9	. . . . . . . . . 10 $/\$
113	33, 112, 87	adddird 8100	. . . . . . . . 9 $/\$
114	110, 113	eqtrid 2250	. . . . . . . 8 $/\$
115	87	mulid2d 8093	. . . . . . . . 9 $/\$
116	115	oveq2d 5962	. . . . . . . 8 $/\$
117	114, 116	eqtrd 2238	. . . . . . 7 $/\$
118	117	oveq2d 5962	. . . . . 6 $/\$
119	109, 118	eqtr4d 2241	. . . . 5 $/\$
120	90, 91, 90, 97	add4d 8243	. . . . . 6 $/\$
121		3p3e6 9181	. . . . . . . . 9
122	121	oveq1i 5956	. . . . . . . 8
123	33, 33, 88	adddird 8100	. . . . . . . 8 $/\$
124	122, 123	eqtr3id 2252	. . . . . . 7 $/\$
125		3p1e4 9174	. . . . . . . . . . . . 13
126	13, 111, 125	addcomli 8219	. . . . . . . . . . . 12
127	126	oveq1i 5956	. . . . . . . . . . 11
128	112, 33, 91	adddird 8100	. . . . . . . . . . 11 $/\$
129	127, 128	eqtr3id 2252	. . . . . . . . . 10 $/\$
130	91	mulid2d 8093	. . . . . . . . . . 11 $/\$
131	130	oveq1d 5961	. . . . . . . . . 10 $/\$
132	129, 131	eqtrd 2238	. . . . . . . . 9 $/\$
133	132	oveq1d 5961	. . . . . . . 8 $/\$
134	91, 93, 96	addassd 8097	. . . . . . . 8 $/\$
135	133, 134	eqtrd 2238	. . . . . . 7 $/\$
136	124, 135	oveq12d 5964	. . . . . 6 $/\$
137	120, 136	eqtr4d 2241	. . . . 5 $/\$
138	119, 137	oveq12d 5964	. . . 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 5946

cc 7925

c1 7928

caddc 7930

cmul 7932

c2 9089

c3 9090

c4 9091

c6 9093

cn0 9297

cexp 10685

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 4160 ax-sep 4163 ax-nul 4171 ax-pow 4219 ax-pr 4254 ax-un 4481 ax-setind 4586 ax-iinf 4637 ax-cnex 8018 ax-resscn 8019 ax-1cn 8020 ax-1re 8021 ax-icn 8022 ax-addcl 8023 ax-addrcl 8024 ax-mulcl 8025 ax-mulrcl 8026 ax-addcom 8027 ax-mulcom 8028 ax-addass 8029 ax-mulass 8030 ax-distr 8031 ax-i2m1 8032 ax-0lt1 8033 ax-1rid 8034 ax-0id 8035 ax-rnegex 8036 ax-precex 8037 ax-cnre 8038 ax-pre-ltirr 8039 ax-pre-ltwlin 8040 ax-pre-lttrn 8041 ax-pre-apti 8042 ax-pre-ltadd 8043 ax-pre-mulgt0 8044 ax-pre-mulext 8045

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 4046 df-opab 4107 df-mpt 4108 df-tr 4144 df-id 4341 df-po 4344 df-iso 4345 df-iord 4414 df-on 4416 df-ilim 4417 df-suc 4419 df-iom 4640 df-xp 4682 df-rel 4683 df-cnv 4684 df-co 4685 df-dm 4686 df-rn 4687 df-res 4688 df-ima 4689 df-iota 5233 df-fun 5274 df-fn 5275 df-f 5276 df-f1 5277 df-fo 5278 df-f1o 5279 df-fv 5280 df-riota 5901 df-ov 5949 df-oprab 5950 df-mpo 5951 df-1st 6228 df-2nd 6229 df-recs 6393 df-frec 6479 df-pnf 8111 df-mnf 8112 df-xr 8113 df-ltxr 8114 df-le 8115 df-sub 8247 df-neg 8248 df-reap 8650 df-ap 8657 df-div 8748 df-inn 9039 df-2 9097 df-3 9098 df-4 9099 df-5 9100 df-6 9101 df-n0 9298 df-z 9375 df-uz 9651 df-seqfrec 10595 df-exp 10686

This theorem is referenced by: (None)

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