binom3 - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > binom3		Unicode version

Theorem binom3 11072

Description: The cube of a binomial. (Contributed by Mario Carneiro, 24-Apr-2015.)

**Assertion**
Ref	Expression
binom3	$/\$

**Proof of Theorem binom3**
Step	Hyp	Ref	Expression
1		df-3 9343	. . . 4
2	1	oveq2i 6086	. . 3
3		addcl 8294	. . . 4 $/\$
4		2nn0 9559	. . . 4
5		expp1 10961	. . . 4 $/\$
6	3, 4, 5	sylancl 417	. . 3 $/\$
7	2, 6	eqtrid 2283	. 2 $/\$
8		sqcl 11015	. . . . 5
9	3, 8	syl 14	. . . 4 $/\$
10		simpl 109	. . . 4 $/\$
11		simpr 110	. . . 4 $/\$
12	9, 10, 11	adddid 8340	. . 3 $/\$
13		binom2 11066	. . . . . 6 $/\$
14	13	oveq1d 6090	. . . . 5 $/\$
15		sqcl 11015	. . . . . . . 8
16	10, 15	syl 14	. . . . . . 7 $/\$
17		2cn 9354	. . . . . . . 8
18		mulcl 8296	. . . . . . . 8 $/\$
19		mulcl 8296	. . . . . . . 8 $/\$
20	17, 18, 19	sylancr 418	. . . . . . 7 $/\$
21	16, 20	addcld 8335	. . . . . 6 $/\$
22		sqcl 11015	. . . . . . 7
23	11, 22	syl 14	. . . . . 6 $/\$
24	21, 23, 10	adddird 8341	. . . . 5 $/\$
25	16, 20, 10	adddird 8341	. . . . . . 7 $/\$
26	1	oveq2i 6086	. . . . . . . . 9
27		expp1 10961	. . . . . . . . . 10 $/\$
28	10, 4, 27	sylancl 417	. . . . . . . . 9 $/\$
29	26, 28	eqtrid 2283	. . . . . . . 8 $/\$
30		sqval 11012	. . . . . . . . . . . . 13
31	10, 30	syl 14	. . . . . . . . . . . 12 $/\$
32	31	oveq1d 6090	. . . . . . . . . . 11 $/\$
33	10, 10, 11	mul32d 8469	. . . . . . . . . . 11 $/\$
34	32, 33	eqtrd 2271	. . . . . . . . . 10 $/\$
35	34	oveq2d 6091	. . . . . . . . 9 $/\$
36		2cnd 9356	. . . . . . . . . 10 $/\$
37	36, 18, 10	mulassd 8339	. . . . . . . . 9 $/\$
38	35, 37	eqtr4d 2274	. . . . . . . 8 $/\$
39	29, 38	oveq12d 6093	. . . . . . 7 $/\$
40	25, 39	eqtr4d 2274	. . . . . 6 $/\$
41	23, 10	mulcomd 8337	. . . . . 6 $/\$
42	40, 41	oveq12d 6093	. . . . 5 $/\$
43	14, 24, 42	3eqtrd 2275	. . . 4 $/\$
44	13	oveq1d 6090	. . . . 5 $/\$
45	21, 23, 11	adddird 8341	. . . . 5 $/\$
46		sqval 11012	. . . . . . . . . . . . . 14
47	11, 46	syl 14	. . . . . . . . . . . . 13 $/\$
48	47	oveq2d 6091	. . . . . . . . . . . 12 $/\$
49	10, 11, 11	mulassd 8339	. . . . . . . . . . . 12 $/\$
50	48, 49	eqtr4d 2274	. . . . . . . . . . 11 $/\$
51	50	oveq2d 6091	. . . . . . . . . 10 $/\$
52	36, 18, 11	mulassd 8339	. . . . . . . . . 10 $/\$
53	51, 52	eqtr4d 2274	. . . . . . . . 9 $/\$
54	53	oveq2d 6091	. . . . . . . 8 $/\$
55	16, 20, 11	adddird 8341	. . . . . . . 8 $/\$
56	54, 55	eqtr4d 2274	. . . . . . 7 $/\$
57	1	oveq2i 6086	. . . . . . . 8
58		expp1 10961	. . . . . . . . 9 $/\$
59	11, 4, 58	sylancl 417	. . . . . . . 8 $/\$
60	57, 59	eqtrid 2283	. . . . . . 7 $/\$
61	56, 60	oveq12d 6093	. . . . . 6 $/\$
62	16, 11	mulcld 8336	. . . . . . 7 $/\$
63	10, 23	mulcld 8336	. . . . . . . 8 $/\$
64		mulcl 8296	. . . . . . . 8 $/\$
65	17, 63, 64	sylancr 418	. . . . . . 7 $/\$
66		3nn0 9560	. . . . . . . 8
67		expcl 10972	. . . . . . . 8 $/\$
68	11, 66, 67	sylancl 417	. . . . . . 7 $/\$
69	62, 65, 68	addassd 8338	. . . . . 6 $/\$
70	61, 69	eqtr3d 2273	. . . . 5 $/\$
71	44, 45, 70	3eqtrd 2275	. . . 4 $/\$
72	43, 71	oveq12d 6093	. . 3 $/\$
73		expcl 10972	. . . . . 6 $/\$
74	10, 66, 73	sylancl 417	. . . . 5 $/\$
75		mulcl 8296	. . . . . 6 $/\$
76	17, 62, 75	sylancr 418	. . . . 5 $/\$
77	74, 76	addcld 8335	. . . 4 $/\$
78	65, 68	addcld 8335	. . . 4 $/\$
79	77, 63, 62, 78	add4d 8485	. . 3 $/\$
80	12, 72, 79	3eqtrd 2275	. 2 $/\$
81	74, 76, 62	addassd 8338	. . . 4 $/\$
82	1	oveq1i 6085	. . . . . . 7
83		1cnd 8332	. . . . . . . 8 $/\$
84	36, 83, 62	adddird 8341	. . . . . . 7 $/\$
85	82, 84	eqtrid 2283	. . . . . 6 $/\$
86	62	mullidd 8334	. . . . . . 7 $/\$
87	86	oveq2d 6091	. . . . . 6 $/\$
88	85, 87	eqtrd 2271	. . . . 5 $/\$
89	88	oveq2d 6091	. . . 4 $/\$
90	81, 89	eqtr4d 2274	. . 3 $/\$
91		1p2e3 9418	. . . . . . . 8
92	91	oveq1i 6085	. . . . . . 7
93	83, 36, 63	adddird 8341	. . . . . . 7 $/\$
94	92, 93	eqtr3id 2285	. . . . . 6 $/\$
95	63	mullidd 8334	. . . . . . 7 $/\$
96	95	oveq1d 6090	. . . . . 6 $/\$
97	94, 96	eqtrd 2271	. . . . 5 $/\$
98	97	oveq1d 6090	. . . 4 $/\$
99	63, 65, 68	addassd 8338	. . . 4 $/\$
100	98, 99	eqtr2d 2272	. . 3 $/\$
101	90, 100	oveq12d 6093	. 2 $/\$
102	7, 80, 101	3eqtrd 2275	1 $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wceq 1402

wcel 2209 (class class class)co 6075

cc 8167

c1 8170

caddc 8172

cmul 8174

c2 9334

c3 9335

cn0 9542

cexp 10953

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 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4241 ax-sep 4244 ax-nul 4254 ax-pow 4306 ax-pr 4341 ax-un 4573 ax-setind 4679 ax-iinf 4730 ax-cnex 8260 ax-resscn 8261 ax-1cn 8262 ax-1re 8263 ax-icn 8264 ax-addcl 8265 ax-addrcl 8266 ax-mulcl 8267 ax-mulrcl 8268 ax-addcom 8269 ax-mulcom 8270 ax-addass 8271 ax-mulass 8272 ax-distr 8273 ax-i2m1 8274 ax-0lt1 8275 ax-1rid 8276 ax-0id 8277 ax-rnegex 8278 ax-precex 8279 ax-cnre 8280 ax-pre-ltirr 8281 ax-pre-ltwlin 8282 ax-pre-lttrn 8283 ax-pre-apti 8284 ax-pre-ltadd 8285 ax-pre-mulgt0 8286 ax-pre-mulext 8287

This theorem depends on definitions: df-bi 117 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rmo 2536 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3636 df-pw 3687 df-sn 3711 df-pr 3712 df-op 3714 df-uni 3931 df-int 3966 df-iun 4009 df-br 4126 df-opab 4188 df-mpt 4189 df-tr 4225 df-id 4433 df-po 4436 df-iso 4437 df-iord 4506 df-on 4508 df-ilim 4509 df-suc 4511 df-iom 4733 df-xp 4775 df-rel 4776 df-cnv 4777 df-co 4778 df-dm 4779 df-rn 4780 df-res 4781 df-ima 4782 df-iota 5332 df-fun 5374 df-fn 5375 df-f 5376 df-f1 5377 df-fo 5378 df-f1o 5379 df-fv 5380 df-riota 6028 df-ov 6078 df-oprab 6079 df-mpo 6080 df-1st 6364 df-2nd 6365 df-recs 6566 df-frec 6652 df-pnf 8352 df-mnf 8353 df-xr 8354 df-ltxr 8355 df-le 8356 df-sub 8489 df-neg 8490 df-reap 8893 df-ap 8900 df-div 8993 df-inn 9284 df-2 9342 df-3 9343 df-n0 9543 df-z 9624 df-uz 9901 df-seqfrec 10863 df-exp 10954

This theorem is referenced by: binom4 16004

W3C validator