binom3 - Intuitionistic Logic Explorer

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Theorem binom3 11115

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 9373	. . . 4
2	1	oveq2i 6099	. . 3
3		addcl 8310	. . . 4 $/\$
4		2nn0 9591	. . . 4
5		expp1 11004	. . . 4 $/\$
6	3, 4, 5	sylancl 417	. . 3 $/\$
7	2, 6	eqtrid 2283	. 2 $/\$
8		sqcl 11058	. . . . 5
9	3, 8	syl 14	. . . 4 $/\$
10		simpl 109	. . . 4 $/\$
11		simpr 110	. . . 4 $/\$
12	9, 10, 11	adddid 8356	. . 3 $/\$
13		binom2 11109	. . . . . 6 $/\$
14	13	oveq1d 6103	. . . . 5 $/\$
15		sqcl 11058	. . . . . . . 8
16	10, 15	syl 14	. . . . . . 7 $/\$
17		2cn 9384	. . . . . . . 8
18		mulcl 8312	. . . . . . . 8 $/\$
19		mulcl 8312	. . . . . . . 8 $/\$
20	17, 18, 19	sylancr 418	. . . . . . 7 $/\$
21	16, 20	addcld 8351	. . . . . 6 $/\$
22		sqcl 11058	. . . . . . 7
23	11, 22	syl 14	. . . . . 6 $/\$
24	21, 23, 10	adddird 8357	. . . . 5 $/\$
25	16, 20, 10	adddird 8357	. . . . . . 7 $/\$
26	1	oveq2i 6099	. . . . . . . . 9
27		expp1 11004	. . . . . . . . . 10 $/\$
28	10, 4, 27	sylancl 417	. . . . . . . . 9 $/\$
29	26, 28	eqtrid 2283	. . . . . . . 8 $/\$
30		sqval 11055	. . . . . . . . . . . . 13
31	10, 30	syl 14	. . . . . . . . . . . 12 $/\$
32	31	oveq1d 6103	. . . . . . . . . . 11 $/\$
33	10, 10, 11	mul32d 8486	. . . . . . . . . . 11 $/\$
34	32, 33	eqtrd 2271	. . . . . . . . . 10 $/\$
35	34	oveq2d 6104	. . . . . . . . 9 $/\$
36		2cnd 9386	. . . . . . . . . 10 $/\$
37	36, 18, 10	mulassd 8355	. . . . . . . . 9 $/\$
38	35, 37	eqtr4d 2274	. . . . . . . 8 $/\$
39	29, 38	oveq12d 6106	. . . . . . 7 $/\$
40	25, 39	eqtr4d 2274	. . . . . 6 $/\$
41	23, 10	mulcomd 8353	. . . . . 6 $/\$
42	40, 41	oveq12d 6106	. . . . 5 $/\$
43	14, 24, 42	3eqtrd 2275	. . . 4 $/\$
44	13	oveq1d 6103	. . . . 5 $/\$
45	21, 23, 11	adddird 8357	. . . . 5 $/\$
46		sqval 11055	. . . . . . . . . . . . . 14
47	11, 46	syl 14	. . . . . . . . . . . . 13 $/\$
48	47	oveq2d 6104	. . . . . . . . . . . 12 $/\$
49	10, 11, 11	mulassd 8355	. . . . . . . . . . . 12 $/\$
50	48, 49	eqtr4d 2274	. . . . . . . . . . 11 $/\$
51	50	oveq2d 6104	. . . . . . . . . 10 $/\$
52	36, 18, 11	mulassd 8355	. . . . . . . . . 10 $/\$
53	51, 52	eqtr4d 2274	. . . . . . . . 9 $/\$
54	53	oveq2d 6104	. . . . . . . 8 $/\$
55	16, 20, 11	adddird 8357	. . . . . . . 8 $/\$
56	54, 55	eqtr4d 2274	. . . . . . 7 $/\$
57	1	oveq2i 6099	. . . . . . . 8
58		expp1 11004	. . . . . . . . 9 $/\$
59	11, 4, 58	sylancl 417	. . . . . . . 8 $/\$
60	57, 59	eqtrid 2283	. . . . . . 7 $/\$
61	56, 60	oveq12d 6106	. . . . . 6 $/\$
62	16, 11	mulcld 8352	. . . . . . 7 $/\$
63	10, 23	mulcld 8352	. . . . . . . 8 $/\$
64		mulcl 8312	. . . . . . . 8 $/\$
65	17, 63, 64	sylancr 418	. . . . . . 7 $/\$
66		3nn0 9592	. . . . . . . 8
67		expcl 11015	. . . . . . . 8 $/\$
68	11, 66, 67	sylancl 417	. . . . . . 7 $/\$
69	62, 65, 68	addassd 8354	. . . . . 6 $/\$
70	61, 69	eqtr3d 2273	. . . . 5 $/\$
71	44, 45, 70	3eqtrd 2275	. . . 4 $/\$
72	43, 71	oveq12d 6106	. . 3 $/\$
73		expcl 11015	. . . . . 6 $/\$
74	10, 66, 73	sylancl 417	. . . . 5 $/\$
75		mulcl 8312	. . . . . 6 $/\$
76	17, 62, 75	sylancr 418	. . . . 5 $/\$
77	74, 76	addcld 8351	. . . 4 $/\$
78	65, 68	addcld 8351	. . . 4 $/\$
79	77, 63, 62, 78	add4d 8502	. . 3 $/\$
80	12, 72, 79	3eqtrd 2275	. 2 $/\$
81	74, 76, 62	addassd 8354	. . . 4 $/\$
82	1	oveq1i 6098	. . . . . . 7
83		1cnd 8348	. . . . . . . 8 $/\$
84	36, 83, 62	adddird 8357	. . . . . . 7 $/\$
85	82, 84	eqtrid 2283	. . . . . 6 $/\$
86	62	mullidd 8350	. . . . . . 7 $/\$
87	86	oveq2d 6104	. . . . . 6 $/\$
88	85, 87	eqtrd 2271	. . . . 5 $/\$
89	88	oveq2d 6104	. . . 4 $/\$
90	81, 89	eqtr4d 2274	. . 3 $/\$
91		1p2e3 9448	. . . . . . . 8
92	91	oveq1i 6098	. . . . . . 7
93	83, 36, 63	adddird 8357	. . . . . . 7 $/\$
94	92, 93	eqtr3id 2285	. . . . . 6 $/\$
95	63	mullidd 8350	. . . . . . 7 $/\$
96	95	oveq1d 6103	. . . . . 6 $/\$
97	94, 96	eqtrd 2271	. . . . 5 $/\$
98	97	oveq1d 6103	. . . 4 $/\$
99	63, 65, 68	addassd 8354	. . . 4 $/\$
100	98, 99	eqtr2d 2272	. . 3 $/\$
101	90, 100	oveq12d 6106	. 2 $/\$
102	7, 80, 101	3eqtrd 2275	1 $/\$

Colors of variables: wff set class

This proof depends on syntax axioms:

wi 4 $/\$ wa 104

wceq 1402

wcel 2209 (class class class)co 6088

cc 8183

c1 8186

caddc 8188

cmul 8190

c2 9364

c3 9365

cn0 9574

cexp 10996

This proof depends on 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 4248 ax-sep 4251 ax-nul 4261 ax-pow 4313 ax-pr 4348 ax-un 4580 ax-setind 4686 ax-iinf 4737 ax-cnex 8276 ax-resscn 8277 ax-1cn 8278 ax-1re 8279 ax-icn 8280 ax-addcl 8281 ax-addrcl 8282 ax-mulcl 8283 ax-mulrcl 8284 ax-addcom 8285 ax-mulcom 8286 ax-addass 8287 ax-mulass 8288 ax-distr 8289 ax-i2m1 8290 ax-0lt1 8291 ax-1rid 8292 ax-0id 8293 ax-rnegex 8294 ax-precex 8295 ax-cnre 8296 ax-pre-ltirr 8297 ax-pre-ltwlin 8298 ax-pre-lttrn 8299 ax-pre-apti 8300 ax-pre-ltadd 8301 ax-pre-mulgt0 8302 ax-pre-mulext 8303

This proof 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 3639 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-iun 4014 df-br 4132 df-opab 4194 df-mpt 4195 df-tr 4232 df-id 4440 df-po 4443 df-iso 4444 df-iord 4513 df-on 4515 df-ilim 4516 df-suc 4518 df-iom 4740 df-xp 4782 df-rel 4783 df-cnv 4784 df-co 4785 df-dm 4786 df-rn 4787 df-res 4788 df-ima 4789 df-iota 5340 df-fun 5382 df-fn 5383 df-f 5384 df-f1 5385 df-fo 5386 df-f1o 5387 df-fv 5388 df-riota 6041 df-ov 6091 df-oprab 6092 df-mpo 6093 df-1st 6378 df-2nd 6379 df-recs 6580 df-frec 6666 df-pnf 8368 df-mnf 8369 df-xr 8370 df-ltxr 8371 df-le 8372 df-sub 8506 df-neg 8507 df-reap 8911 df-ap 8918 df-div 9011 df-inn 9314 df-2 9372 df-3 9373 df-n0 9575 df-z 9656 df-uz 9938 df-seqfrec 10906 df-exp 10997

This theorem is used by: binom4 16219

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