Theorem sqoddm1div8 10629

Description: A squared odd number minus 1 divided by 8 is the odd number multiplied with its successor divided by 2. (Contributed by AV, 19-Jul-2021.)

Assertion

Ref	Expression
sqoddm1div8	|- ( ( N e. ZZ /\ M = ( ( 2 x. N ) + 1 ) ) -> ( ( ( M ^ 2 ) - 1 ) / 8 ) = ( ( N x. ( N + 1 ) ) / 2 ) )

Proof of Theorem sqoddm1div8

Step	Hyp	Ref	Expression
1		oveq1 5860	. . . . . 6 |- ( M = ( ( 2 x. N ) + 1 ) -> ( M ^ 2 ) = ( ( ( 2 x. N ) + 1 ) ^ 2 ) )
2		2z 9240	. . . . . . . . . 10 |- 2 e. ZZ
3	2	a1i 9	. . . . . . . . 9 |- ( N e. ZZ -> 2 e. ZZ )
4		id 19	. . . . . . . . 9 |- ( N e. ZZ -> N e. ZZ )
5	3, 4	zmulcld 9340	. . . . . . . 8 |- ( N e. ZZ -> ( 2 x. N ) e. ZZ )
6	5	zcnd 9335	. . . . . . 7 |- ( N e. ZZ -> ( 2 x. N ) e. CC )
7		binom21 10588	. . . . . . 7 |- ( ( 2 x. N ) e. CC -> ( ( ( 2 x. N ) + 1 ) ^ 2 ) = ( ( ( ( 2 x. N ) ^ 2 ) + ( 2 x. ( 2 x. N ) ) ) + 1 ) )
8	6, 7	syl 14	. . . . . 6 |- ( N e. ZZ -> ( ( ( 2 x. N ) + 1 ) ^ 2 ) = ( ( ( ( 2 x. N ) ^ 2 ) + ( 2 x. ( 2 x. N ) ) ) + 1 ) )
9	1, 8	sylan9eqr 2225	. . . . 5 |- ( ( N e. ZZ /\ M = ( ( 2 x. N ) + 1 ) ) -> ( M ^ 2 ) = ( ( ( ( 2 x. N ) ^ 2 ) + ( 2 x. ( 2 x. N ) ) ) + 1 ) )
10	9	oveq1d 5868	. . . 4 |- ( ( N e. ZZ /\ M = ( ( 2 x. N ) + 1 ) ) -> ( ( M ^ 2 ) - 1 ) = ( ( ( ( ( 2 x. N ) ^ 2 ) + ( 2 x. ( 2 x. N ) ) ) + 1 ) - 1 ) )
11		2cnd 8951	. . . . . . . . . . 11 |- ( N e. ZZ -> 2 e. CC )
12		zcn 9217	. . . . . . . . . . 11 |- ( N e. ZZ -> N e. CC )
13	11, 12	sqmuld 10621	. . . . . . . . . 10 |- ( N e. ZZ -> ( ( 2 x. N ) ^ 2 ) = ( ( 2 ^ 2 ) x. ( N ^ 2 ) ) )
14		sq2 10571	. . . . . . . . . . . 12 |- ( 2 ^ 2 ) = 4
15	14	a1i 9	. . . . . . . . . . 11 |- ( N e. ZZ -> ( 2 ^ 2 ) = 4 )
16	15	oveq1d 5868	. . . . . . . . . 10 |- ( N e. ZZ -> ( ( 2 ^ 2 ) x. ( N ^ 2 ) ) = ( 4 x. ( N ^ 2 ) ) )
17	13, 16	eqtrd 2203	. . . . . . . . 9 |- ( N e. ZZ -> ( ( 2 x. N ) ^ 2 ) = ( 4 x. ( N ^ 2 ) ) )
18		mulass 7905	. . . . . . . . . . . 12 |- ( ( 2 e. CC /\ 2 e. CC /\ N e. CC ) -> ( ( 2 x. 2 ) x. N ) = ( 2 x. ( 2 x. N ) ) )
19	18	eqcomd 2176	. . . . . . . . . . 11 |- ( ( 2 e. CC /\ 2 e. CC /\ N e. CC ) -> ( 2 x. ( 2 x. N ) ) = ( ( 2 x. 2 ) x. N ) )
20	11, 11, 12, 19	syl3anc 1233	. . . . . . . . . 10 |- ( N e. ZZ -> ( 2 x. ( 2 x. N ) ) = ( ( 2 x. 2 ) x. N ) )
21		2t2e4 9032	. . . . . . . . . . . 12 |- ( 2 x. 2 ) = 4
22	21	a1i 9	. . . . . . . . . . 11 |- ( N e. ZZ -> ( 2 x. 2 ) = 4 )
23	22	oveq1d 5868	. . . . . . . . . 10 |- ( N e. ZZ -> ( ( 2 x. 2 ) x. N ) = ( 4 x. N ) )
24	20, 23	eqtrd 2203	. . . . . . . . 9 |- ( N e. ZZ -> ( 2 x. ( 2 x. N ) ) = ( 4 x. N ) )
25	17, 24	oveq12d 5871	. . . . . . . 8 |- ( N e. ZZ -> ( ( ( 2 x. N ) ^ 2 ) + ( 2 x. ( 2 x. N ) ) ) = ( ( 4 x. ( N ^ 2 ) ) + ( 4 x. N ) ) )
26	25	oveq1d 5868	. . . . . . 7 |- ( N e. ZZ -> ( ( ( ( 2 x. N ) ^ 2 ) + ( 2 x. ( 2 x. N ) ) ) + 1 ) = ( ( ( 4 x. ( N ^ 2 ) ) + ( 4 x. N ) ) + 1 ) )
27	26	oveq1d 5868	. . . . . 6 |- ( N e. ZZ -> ( ( ( ( ( 2 x. N ) ^ 2 ) + ( 2 x. ( 2 x. N ) ) ) + 1 ) - 1 ) = ( ( ( ( 4 x. ( N ^ 2 ) ) + ( 4 x. N ) ) + 1 ) - 1 ) )
28		4z 9242	. . . . . . . . . . 11 |- 4 e. ZZ
29	28	a1i 9	. . . . . . . . . 10 |- ( N e. ZZ -> 4 e. ZZ )
30		zsqcl 10546	. . . . . . . . . 10 |- ( N e. ZZ -> ( N ^ 2 ) e. ZZ )
31	29, 30	zmulcld 9340	. . . . . . . . 9 |- ( N e. ZZ -> ( 4 x. ( N ^ 2 ) ) e. ZZ )
32	31	zcnd 9335	. . . . . . . 8 |- ( N e. ZZ -> ( 4 x. ( N ^ 2 ) ) e. CC )
33	29, 4	zmulcld 9340	. . . . . . . . 9 |- ( N e. ZZ -> ( 4 x. N ) e. ZZ )
34	33	zcnd 9335	. . . . . . . 8 |- ( N e. ZZ -> ( 4 x. N ) e. CC )
35	32, 34	addcld 7939	. . . . . . 7 |- ( N e. ZZ -> ( ( 4 x. ( N ^ 2 ) ) + ( 4 x. N ) ) e. CC )
36		pncan1 8296	. . . . . . 7 |- ( ( ( 4 x. ( N ^ 2 ) ) + ( 4 x. N ) ) e. CC -> ( ( ( ( 4 x. ( N ^ 2 ) ) + ( 4 x. N ) ) + 1 ) - 1 ) = ( ( 4 x. ( N ^ 2 ) ) + ( 4 x. N ) ) )
37	35, 36	syl 14	. . . . . 6 |- ( N e. ZZ -> ( ( ( ( 4 x. ( N ^ 2 ) ) + ( 4 x. N ) ) + 1 ) - 1 ) = ( ( 4 x. ( N ^ 2 ) ) + ( 4 x. N ) ) )
38	27, 37	eqtrd 2203	. . . . 5 |- ( N e. ZZ -> ( ( ( ( ( 2 x. N ) ^ 2 ) + ( 2 x. ( 2 x. N ) ) ) + 1 ) - 1 ) = ( ( 4 x. ( N ^ 2 ) ) + ( 4 x. N ) ) )
39	38	adantr 274	. . . 4 |- ( ( N e. ZZ /\ M = ( ( 2 x. N ) + 1 ) ) -> ( ( ( ( ( 2 x. N ) ^ 2 ) + ( 2 x. ( 2 x. N ) ) ) + 1 ) - 1 ) = ( ( 4 x. ( N ^ 2 ) ) + ( 4 x. N ) ) )
40	10, 39	eqtrd 2203	. . 3 |- ( ( N e. ZZ /\ M = ( ( 2 x. N ) + 1 ) ) -> ( ( M ^ 2 ) - 1 ) = ( ( 4 x. ( N ^ 2 ) ) + ( 4 x. N ) ) )
41	40	oveq1d 5868	. 2 |- ( ( N e. ZZ /\ M = ( ( 2 x. N ) + 1 ) ) -> ( ( ( M ^ 2 ) - 1 ) / 8 ) = ( ( ( 4 x. ( N ^ 2 ) ) + ( 4 x. N ) ) / 8 ) )
42		4cn 8956	. . . . . . 7 |- 4 e. CC
43	42	a1i 9	. . . . . 6 |- ( N e. ZZ -> 4 e. CC )
44	30	zcnd 9335	. . . . . 6 |- ( N e. ZZ -> ( N ^ 2 ) e. CC )
45	43, 44, 12	adddid 7944	. . . . 5 |- ( N e. ZZ -> ( 4 x. ( ( N ^ 2 ) + N ) ) = ( ( 4 x. ( N ^ 2 ) ) + ( 4 x. N ) ) )
46	45	eqcomd 2176	. . . 4 |- ( N e. ZZ -> ( ( 4 x. ( N ^ 2 ) ) + ( 4 x. N ) ) = ( 4 x. ( ( N ^ 2 ) + N ) ) )
47	46	oveq1d 5868	. . 3 |- ( N e. ZZ -> ( ( ( 4 x. ( N ^ 2 ) ) + ( 4 x. N ) ) / 8 ) = ( ( 4 x. ( ( N ^ 2 ) + N ) ) / 8 ) )
48	47	adantr 274	. 2 |- ( ( N e. ZZ /\ M = ( ( 2 x. N ) + 1 ) ) -> ( ( ( 4 x. ( N ^ 2 ) ) + ( 4 x. N ) ) / 8 ) = ( ( 4 x. ( ( N ^ 2 ) + N ) ) / 8 ) )
49		4t2e8 9036	. . . . . . 7 |- ( 4 x. 2 ) = 8
50	49	a1i 9	. . . . . 6 |- ( N e. ZZ -> ( 4 x. 2 ) = 8 )
51	50	eqcomd 2176	. . . . 5 |- ( N e. ZZ -> 8 = ( 4 x. 2 ) )
52	51	oveq2d 5869	. . . 4 |- ( N e. ZZ -> ( ( 4 x. ( ( N ^ 2 ) + N ) ) / 8 ) = ( ( 4 x. ( ( N ^ 2 ) + N ) ) / ( 4 x. 2 ) ) )
53	30, 4	zaddcld 9338	. . . . . 6 |- ( N e. ZZ -> ( ( N ^ 2 ) + N ) e. ZZ )
54	53	zcnd 9335	. . . . 5 |- ( N e. ZZ -> ( ( N ^ 2 ) + N ) e. CC )
55		2ap0 8971	. . . . . 6 |- 2 # 0
56	55	a1i 9	. . . . 5 |- ( N e. ZZ -> 2 # 0 )
57		4ap0 8977	. . . . . 6 |- 4 # 0
58	57	a1i 9	. . . . 5 |- ( N e. ZZ -> 4 # 0 )
59	54, 11, 43, 56, 58	divcanap5d 8734	. . . 4 |- ( N e. ZZ -> ( ( 4 x. ( ( N ^ 2 ) + N ) ) / ( 4 x. 2 ) ) = ( ( ( N ^ 2 ) + N ) / 2 ) )
60	12	sqvald 10606	. . . . . . 7 |- ( N e. ZZ -> ( N ^ 2 ) = ( N x. N ) )
61	60	oveq1d 5868	. . . . . 6 |- ( N e. ZZ -> ( ( N ^ 2 ) + N ) = ( ( N x. N ) + N ) )
62	12	mulid1d 7937	. . . . . . . 8 |- ( N e. ZZ -> ( N x. 1 ) = N )
63	62	eqcomd 2176	. . . . . . 7 |- ( N e. ZZ -> N = ( N x. 1 ) )
64	63	oveq2d 5869	. . . . . 6 |- ( N e. ZZ -> ( ( N x. N ) + N ) = ( ( N x. N ) + ( N x. 1 ) ) )
65		1cnd 7936	. . . . . . 7 |- ( N e. ZZ -> 1 e. CC )
66		adddi 7906	. . . . . . . 8 |- ( ( N e. CC /\ N e. CC /\ 1 e. CC ) -> ( N x. ( N + 1 ) ) = ( ( N x. N ) + ( N x. 1 ) ) )
67	66	eqcomd 2176	. . . . . . 7 |- ( ( N e. CC /\ N e. CC /\ 1 e. CC ) -> ( ( N x. N ) + ( N x. 1 ) ) = ( N x. ( N + 1 ) ) )
68	12, 12, 65, 67	syl3anc 1233	. . . . . 6 |- ( N e. ZZ -> ( ( N x. N ) + ( N x. 1 ) ) = ( N x. ( N + 1 ) ) )
69	61, 64, 68	3eqtrd 2207	. . . . 5 |- ( N e. ZZ -> ( ( N ^ 2 ) + N ) = ( N x. ( N + 1 ) ) )
70	69	oveq1d 5868	. . . 4 |- ( N e. ZZ -> ( ( ( N ^ 2 ) + N ) / 2 ) = ( ( N x. ( N + 1 ) ) / 2 ) )
71	52, 59, 70	3eqtrd 2207	. . 3 |- ( N e. ZZ -> ( ( 4 x. ( ( N ^ 2 ) + N ) ) / 8 ) = ( ( N x. ( N + 1 ) ) / 2 ) )
72	71	adantr 274	. 2 |- ( ( N e. ZZ /\ M = ( ( 2 x. N ) + 1 ) ) -> ( ( 4 x. ( ( N ^ 2 ) + N ) ) / 8 ) = ( ( N x. ( N + 1 ) ) / 2 ) )
73	41, 48, 72	3eqtrd 2207	1 |- ( ( N e. ZZ /\ M = ( ( 2 x. N ) + 1 ) ) -> ( ( ( M ^ 2 ) - 1 ) / 8 ) = ( ( N x. ( N + 1 ) ) / 2 ) )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 973   = wceq 1348   e. wcel 2141   class class class wbr 3989  (class class class)co 5853   CCcc 7772   0cc0 7774   1c1 7775   + caddc 7777   x. cmul 7779   - cmin 8090   # cap 8500   / cdiv 8589   2c2 8929   4c4 8931   8c8 8935   ZZcz 9212   ^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-7 8942  df-8 8943  df-n0 9136  df-z 9213  df-uz 9488  df-seqfrec 10402  df-exp 10476

This theorem is referenced by:  sqoddm1div8z  11845