Theorem sqoddm1div8 10659

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 5876	. . . . . 6 |- ( M = ( ( 2 x. N ) + 1 ) -> ( M ^ 2 ) = ( ( ( 2 x. N ) + 1 ) ^ 2 ) )
2		2z 9270	. . . . . . . . . 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 9370	. . . . . . . 8 |- ( N e. ZZ -> ( 2 x. N ) e. ZZ )
6	5	zcnd 9365	. . . . . . 7 |- ( N e. ZZ -> ( 2 x. N ) e. CC )
7		binom21 10618	. . . . . . 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 2232	. . . . 5 |- ( ( N e. ZZ /\ M = ( ( 2 x. N ) + 1 ) ) -> ( M ^ 2 ) = ( ( ( ( 2 x. N ) ^ 2 ) + ( 2 x. ( 2 x. N ) ) ) + 1 ) )
10	9	oveq1d 5884	. . . 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 8981	. . . . . . . . . . 11 |- ( N e. ZZ -> 2 e. CC )
12		zcn 9247	. . . . . . . . . . 11 |- ( N e. ZZ -> N e. CC )
13	11, 12	sqmuld 10651	. . . . . . . . . 10 |- ( N e. ZZ -> ( ( 2 x. N ) ^ 2 ) = ( ( 2 ^ 2 ) x. ( N ^ 2 ) ) )
14		sq2 10601	. . . . . . . . . . . 12 |- ( 2 ^ 2 ) = 4
15	14	a1i 9	. . . . . . . . . . 11 |- ( N e. ZZ -> ( 2 ^ 2 ) = 4 )
16	15	oveq1d 5884	. . . . . . . . . 10 |- ( N e. ZZ -> ( ( 2 ^ 2 ) x. ( N ^ 2 ) ) = ( 4 x. ( N ^ 2 ) ) )
17	13, 16	eqtrd 2210	. . . . . . . . 9 |- ( N e. ZZ -> ( ( 2 x. N ) ^ 2 ) = ( 4 x. ( N ^ 2 ) ) )
18		mulass 7933	. . . . . . . . . . . 12 |- ( ( 2 e. CC /\ 2 e. CC /\ N e. CC ) -> ( ( 2 x. 2 ) x. N ) = ( 2 x. ( 2 x. N ) ) )
19	18	eqcomd 2183	. . . . . . . . . . 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 1238	. . . . . . . . . 10 |- ( N e. ZZ -> ( 2 x. ( 2 x. N ) ) = ( ( 2 x. 2 ) x. N ) )
21		2t2e4 9062	. . . . . . . . . . . 12 |- ( 2 x. 2 ) = 4
22	21	a1i 9	. . . . . . . . . . 11 |- ( N e. ZZ -> ( 2 x. 2 ) = 4 )
23	22	oveq1d 5884	. . . . . . . . . 10 |- ( N e. ZZ -> ( ( 2 x. 2 ) x. N ) = ( 4 x. N ) )
24	20, 23	eqtrd 2210	. . . . . . . . 9 |- ( N e. ZZ -> ( 2 x. ( 2 x. N ) ) = ( 4 x. N ) )
25	17, 24	oveq12d 5887	. . . . . . . 8 |- ( N e. ZZ -> ( ( ( 2 x. N ) ^ 2 ) + ( 2 x. ( 2 x. N ) ) ) = ( ( 4 x. ( N ^ 2 ) ) + ( 4 x. N ) ) )
26	25	oveq1d 5884	. . . . . . 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 5884	. . . . . 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 9272	. . . . . . . . . . 11 |- 4 e. ZZ
29	28	a1i 9	. . . . . . . . . 10 |- ( N e. ZZ -> 4 e. ZZ )
30		zsqcl 10576	. . . . . . . . . 10 |- ( N e. ZZ -> ( N ^ 2 ) e. ZZ )
31	29, 30	zmulcld 9370	. . . . . . . . 9 |- ( N e. ZZ -> ( 4 x. ( N ^ 2 ) ) e. ZZ )
32	31	zcnd 9365	. . . . . . . 8 |- ( N e. ZZ -> ( 4 x. ( N ^ 2 ) ) e. CC )
33	29, 4	zmulcld 9370	. . . . . . . . 9 |- ( N e. ZZ -> ( 4 x. N ) e. ZZ )
34	33	zcnd 9365	. . . . . . . 8 |- ( N e. ZZ -> ( 4 x. N ) e. CC )
35	32, 34	addcld 7967	. . . . . . 7 |- ( N e. ZZ -> ( ( 4 x. ( N ^ 2 ) ) + ( 4 x. N ) ) e. CC )
36		pncan1 8324	. . . . . . 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 2210	. . . . 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 276	. . . 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 2210	. . 3 |- ( ( N e. ZZ /\ M = ( ( 2 x. N ) + 1 ) ) -> ( ( M ^ 2 ) - 1 ) = ( ( 4 x. ( N ^ 2 ) ) + ( 4 x. N ) ) )
41	40	oveq1d 5884	. 2 |- ( ( N e. ZZ /\ M = ( ( 2 x. N ) + 1 ) ) -> ( ( ( M ^ 2 ) - 1 ) / 8 ) = ( ( ( 4 x. ( N ^ 2 ) ) + ( 4 x. N ) ) / 8 ) )
42		4cn 8986	. . . . . . 7 |- 4 e. CC
43	42	a1i 9	. . . . . 6 |- ( N e. ZZ -> 4 e. CC )
44	30	zcnd 9365	. . . . . 6 |- ( N e. ZZ -> ( N ^ 2 ) e. CC )
45	43, 44, 12	adddid 7972	. . . . 5 |- ( N e. ZZ -> ( 4 x. ( ( N ^ 2 ) + N ) ) = ( ( 4 x. ( N ^ 2 ) ) + ( 4 x. N ) ) )
46	45	eqcomd 2183	. . . 4 |- ( N e. ZZ -> ( ( 4 x. ( N ^ 2 ) ) + ( 4 x. N ) ) = ( 4 x. ( ( N ^ 2 ) + N ) ) )
47	46	oveq1d 5884	. . 3 |- ( N e. ZZ -> ( ( ( 4 x. ( N ^ 2 ) ) + ( 4 x. N ) ) / 8 ) = ( ( 4 x. ( ( N ^ 2 ) + N ) ) / 8 ) )
48	47	adantr 276	. 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 9066	. . . . . . 7 |- ( 4 x. 2 ) = 8
50	49	a1i 9	. . . . . 6 |- ( N e. ZZ -> ( 4 x. 2 ) = 8 )
51	50	eqcomd 2183	. . . . 5 |- ( N e. ZZ -> 8 = ( 4 x. 2 ) )
52	51	oveq2d 5885	. . . 4 |- ( N e. ZZ -> ( ( 4 x. ( ( N ^ 2 ) + N ) ) / 8 ) = ( ( 4 x. ( ( N ^ 2 ) + N ) ) / ( 4 x. 2 ) ) )
53	30, 4	zaddcld 9368	. . . . . 6 |- ( N e. ZZ -> ( ( N ^ 2 ) + N ) e. ZZ )
54	53	zcnd 9365	. . . . 5 |- ( N e. ZZ -> ( ( N ^ 2 ) + N ) e. CC )
55		2ap0 9001	. . . . . 6 |- 2 # 0
56	55	a1i 9	. . . . 5 |- ( N e. ZZ -> 2 # 0 )
57		4ap0 9007	. . . . . 6 |- 4 # 0
58	57	a1i 9	. . . . 5 |- ( N e. ZZ -> 4 # 0 )
59	54, 11, 43, 56, 58	divcanap5d 8763	. . . 4 |- ( N e. ZZ -> ( ( 4 x. ( ( N ^ 2 ) + N ) ) / ( 4 x. 2 ) ) = ( ( ( N ^ 2 ) + N ) / 2 ) )
60	12	sqvald 10636	. . . . . . 7 |- ( N e. ZZ -> ( N ^ 2 ) = ( N x. N ) )
61	60	oveq1d 5884	. . . . . 6 |- ( N e. ZZ -> ( ( N ^ 2 ) + N ) = ( ( N x. N ) + N ) )
62	12	mulid1d 7965	. . . . . . . 8 |- ( N e. ZZ -> ( N x. 1 ) = N )
63	62	eqcomd 2183	. . . . . . 7 |- ( N e. ZZ -> N = ( N x. 1 ) )
64	63	oveq2d 5885	. . . . . 6 |- ( N e. ZZ -> ( ( N x. N ) + N ) = ( ( N x. N ) + ( N x. 1 ) ) )
65		1cnd 7964	. . . . . . 7 |- ( N e. ZZ -> 1 e. CC )
66		adddi 7934	. . . . . . . 8 |- ( ( N e. CC /\ N e. CC /\ 1 e. CC ) -> ( N x. ( N + 1 ) ) = ( ( N x. N ) + ( N x. 1 ) ) )
67	66	eqcomd 2183	. . . . . . 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 1238	. . . . . 6 |- ( N e. ZZ -> ( ( N x. N ) + ( N x. 1 ) ) = ( N x. ( N + 1 ) ) )
69	61, 64, 68	3eqtrd 2214	. . . . 5 |- ( N e. ZZ -> ( ( N ^ 2 ) + N ) = ( N x. ( N + 1 ) ) )
70	69	oveq1d 5884	. . . 4 |- ( N e. ZZ -> ( ( ( N ^ 2 ) + N ) / 2 ) = ( ( N x. ( N + 1 ) ) / 2 ) )
71	52, 59, 70	3eqtrd 2214	. . 3 |- ( N e. ZZ -> ( ( 4 x. ( ( N ^ 2 ) + N ) ) / 8 ) = ( ( N x. ( N + 1 ) ) / 2 ) )
72	71	adantr 276	. 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 2214	1 |- ( ( N e. ZZ /\ M = ( ( 2 x. N ) + 1 ) ) -> ( ( ( M ^ 2 ) - 1 ) / 8 ) = ( ( N x. ( N + 1 ) ) / 2 ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 978   = wceq 1353   e. wcel 2148   class class class wbr 4000  (class class class)co 5869   CCcc 7800   0cc0 7802   1c1 7803   + caddc 7805   x. cmul 7807   - cmin 8118   # cap 8528   / cdiv 8618   2c2 8959   4c4 8961   8c8 8965   ZZcz 9242   ^cexp 10505

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-mulrcl 7901  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-precex 7912  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918  ax-pre-mulgt0 7919  ax-pre-mulext 7920

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-ilim 4366  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-frec 6386  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-reap 8522  df-ap 8529  df-div 8619  df-inn 8909  df-2 8967  df-3 8968  df-4 8969  df-5 8970  df-6 8971  df-7 8972  df-8 8973  df-n0 9166  df-z 9243  df-uz 9518  df-seqfrec 10432  df-exp 10506

This theorem is referenced by:  sqoddm1div8z  11874