Theorem sqoddm1div8 13028

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 6657	. . . . . 6 |- ( M = ( ( 2 x. N ) + 1 ) -> ( M ^ 2 ) = ( ( ( 2 x. N ) + 1 ) ^ 2 ) )
2		2z 11409	. . . . . . . . . 10 |- 2 e. ZZ
3	2	a1i 11	. . . . . . . . 9 |- ( N e. ZZ -> 2 e. ZZ )
4		id 22	. . . . . . . . 9 |- ( N e. ZZ -> N e. ZZ )
5	3, 4	zmulcld 11488	. . . . . . . 8 |- ( N e. ZZ -> ( 2 x. N ) e. ZZ )
6	5	zcnd 11483	. . . . . . 7 |- ( N e. ZZ -> ( 2 x. N ) e. CC )
7		binom21 12980	. . . . . . 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 17	. . . . . 6 |- ( N e. ZZ -> ( ( ( 2 x. N ) + 1 ) ^ 2 ) = ( ( ( ( 2 x. N ) ^ 2 ) + ( 2 x. ( 2 x. N ) ) ) + 1 ) )
9	1, 8	sylan9eqr 2678	. . . . 5 |- ( ( N e. ZZ /\ M = ( ( 2 x. N ) + 1 ) ) -> ( M ^ 2 ) = ( ( ( ( 2 x. N ) ^ 2 ) + ( 2 x. ( 2 x. N ) ) ) + 1 ) )
10	9	oveq1d 6665	. . . 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 11093	. . . . . . . . . . 11 |- ( N e. ZZ -> 2 e. CC )
12		zcn 11382	. . . . . . . . . . 11 |- ( N e. ZZ -> N e. CC )
13	11, 12	sqmuld 13020	. . . . . . . . . 10 |- ( N e. ZZ -> ( ( 2 x. N ) ^ 2 ) = ( ( 2 ^ 2 ) x. ( N ^ 2 ) ) )
14		sq2 12960	. . . . . . . . . . . 12 |- ( 2 ^ 2 ) = 4
15	14	a1i 11	. . . . . . . . . . 11 |- ( N e. ZZ -> ( 2 ^ 2 ) = 4 )
16	15	oveq1d 6665	. . . . . . . . . 10 |- ( N e. ZZ -> ( ( 2 ^ 2 ) x. ( N ^ 2 ) ) = ( 4 x. ( N ^ 2 ) ) )
17	13, 16	eqtrd 2656	. . . . . . . . 9 |- ( N e. ZZ -> ( ( 2 x. N ) ^ 2 ) = ( 4 x. ( N ^ 2 ) ) )
18		mulass 10024	. . . . . . . . . . . 12 |- ( ( 2 e. CC /\ 2 e. CC /\ N e. CC ) -> ( ( 2 x. 2 ) x. N ) = ( 2 x. ( 2 x. N ) ) )
19	18	eqcomd 2628	. . . . . . . . . . 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 1326	. . . . . . . . . 10 |- ( N e. ZZ -> ( 2 x. ( 2 x. N ) ) = ( ( 2 x. 2 ) x. N ) )
21		2t2e4 11177	. . . . . . . . . . . 12 |- ( 2 x. 2 ) = 4
22	21	a1i 11	. . . . . . . . . . 11 |- ( N e. ZZ -> ( 2 x. 2 ) = 4 )
23	22	oveq1d 6665	. . . . . . . . . 10 |- ( N e. ZZ -> ( ( 2 x. 2 ) x. N ) = ( 4 x. N ) )
24	20, 23	eqtrd 2656	. . . . . . . . 9 |- ( N e. ZZ -> ( 2 x. ( 2 x. N ) ) = ( 4 x. N ) )
25	17, 24	oveq12d 6668	. . . . . . . 8 |- ( N e. ZZ -> ( ( ( 2 x. N ) ^ 2 ) + ( 2 x. ( 2 x. N ) ) ) = ( ( 4 x. ( N ^ 2 ) ) + ( 4 x. N ) ) )
26	25	oveq1d 6665	. . . . . . 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 6665	. . . . . 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 11411	. . . . . . . . . . 11 |- 4 e. ZZ
29	28	a1i 11	. . . . . . . . . 10 |- ( N e. ZZ -> 4 e. ZZ )
30		zsqcl 12934	. . . . . . . . . 10 |- ( N e. ZZ -> ( N ^ 2 ) e. ZZ )
31	29, 30	zmulcld 11488	. . . . . . . . 9 |- ( N e. ZZ -> ( 4 x. ( N ^ 2 ) ) e. ZZ )
32	31	zcnd 11483	. . . . . . . 8 |- ( N e. ZZ -> ( 4 x. ( N ^ 2 ) ) e. CC )
33	29, 4	zmulcld 11488	. . . . . . . . 9 |- ( N e. ZZ -> ( 4 x. N ) e. ZZ )
34	33	zcnd 11483	. . . . . . . 8 |- ( N e. ZZ -> ( 4 x. N ) e. CC )
35	32, 34	addcld 10059	. . . . . . 7 |- ( N e. ZZ -> ( ( 4 x. ( N ^ 2 ) ) + ( 4 x. N ) ) e. CC )
36		pncan1 10454	. . . . . . 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 17	. . . . . 6 |- ( N e. ZZ -> ( ( ( ( 4 x. ( N ^ 2 ) ) + ( 4 x. N ) ) + 1 ) - 1 ) = ( ( 4 x. ( N ^ 2 ) ) + ( 4 x. N ) ) )
38	27, 37	eqtrd 2656	. . . . 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 481	. . . 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 2656	. . 3 |- ( ( N e. ZZ /\ M = ( ( 2 x. N ) + 1 ) ) -> ( ( M ^ 2 ) - 1 ) = ( ( 4 x. ( N ^ 2 ) ) + ( 4 x. N ) ) )
41	40	oveq1d 6665	. 2 |- ( ( N e. ZZ /\ M = ( ( 2 x. N ) + 1 ) ) -> ( ( ( M ^ 2 ) - 1 ) / 8 ) = ( ( ( 4 x. ( N ^ 2 ) ) + ( 4 x. N ) ) / 8 ) )
42		4cn 11098	. . . . . . 7 |- 4 e. CC
43	42	a1i 11	. . . . . 6 |- ( N e. ZZ -> 4 e. CC )
44	30	zcnd 11483	. . . . . 6 |- ( N e. ZZ -> ( N ^ 2 ) e. CC )
45	43, 44, 12	adddid 10064	. . . . 5 |- ( N e. ZZ -> ( 4 x. ( ( N ^ 2 ) + N ) ) = ( ( 4 x. ( N ^ 2 ) ) + ( 4 x. N ) ) )
46	45	eqcomd 2628	. . . 4 |- ( N e. ZZ -> ( ( 4 x. ( N ^ 2 ) ) + ( 4 x. N ) ) = ( 4 x. ( ( N ^ 2 ) + N ) ) )
47	46	oveq1d 6665	. . 3 |- ( N e. ZZ -> ( ( ( 4 x. ( N ^ 2 ) ) + ( 4 x. N ) ) / 8 ) = ( ( 4 x. ( ( N ^ 2 ) + N ) ) / 8 ) )
48	47	adantr 481	. 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 11181	. . . . . . 7 |- ( 4 x. 2 ) = 8
50	49	a1i 11	. . . . . 6 |- ( N e. ZZ -> ( 4 x. 2 ) = 8 )
51	50	eqcomd 2628	. . . . 5 |- ( N e. ZZ -> 8 = ( 4 x. 2 ) )
52	51	oveq2d 6666	. . . 4 |- ( N e. ZZ -> ( ( 4 x. ( ( N ^ 2 ) + N ) ) / 8 ) = ( ( 4 x. ( ( N ^ 2 ) + N ) ) / ( 4 x. 2 ) ) )
53	30, 4	zaddcld 11486	. . . . . 6 |- ( N e. ZZ -> ( ( N ^ 2 ) + N ) e. ZZ )
54	53	zcnd 11483	. . . . 5 |- ( N e. ZZ -> ( ( N ^ 2 ) + N ) e. CC )
55		2cnne0 11242	. . . . . 6 |- ( 2 e. CC /\ 2 =/= 0 )
56	55	a1i 11	. . . . 5 |- ( N e. ZZ -> ( 2 e. CC /\ 2 =/= 0 ) )
57		4ne0 11117	. . . . . . 7 |- 4 =/= 0
58	42, 57	pm3.2i 471	. . . . . 6 |- ( 4 e. CC /\ 4 =/= 0 )
59	58	a1i 11	. . . . 5 |- ( N e. ZZ -> ( 4 e. CC /\ 4 =/= 0 ) )
60		divcan5 10727	. . . . 5 |- ( ( ( ( N ^ 2 ) + N ) e. CC /\ ( 2 e. CC /\ 2 =/= 0 ) /\ ( 4 e. CC /\ 4 =/= 0 ) ) -> ( ( 4 x. ( ( N ^ 2 ) + N ) ) / ( 4 x. 2 ) ) = ( ( ( N ^ 2 ) + N ) / 2 ) )
61	54, 56, 59, 60	syl3anc 1326	. . . 4 |- ( N e. ZZ -> ( ( 4 x. ( ( N ^ 2 ) + N ) ) / ( 4 x. 2 ) ) = ( ( ( N ^ 2 ) + N ) / 2 ) )
62	12	sqvald 13005	. . . . . . 7 |- ( N e. ZZ -> ( N ^ 2 ) = ( N x. N ) )
63	62	oveq1d 6665	. . . . . 6 |- ( N e. ZZ -> ( ( N ^ 2 ) + N ) = ( ( N x. N ) + N ) )
64	12	mulid1d 10057	. . . . . . . 8 |- ( N e. ZZ -> ( N x. 1 ) = N )
65	64	eqcomd 2628	. . . . . . 7 |- ( N e. ZZ -> N = ( N x. 1 ) )
66	65	oveq2d 6666	. . . . . 6 |- ( N e. ZZ -> ( ( N x. N ) + N ) = ( ( N x. N ) + ( N x. 1 ) ) )
67		1cnd 10056	. . . . . . 7 |- ( N e. ZZ -> 1 e. CC )
68		adddi 10025	. . . . . . . 8 |- ( ( N e. CC /\ N e. CC /\ 1 e. CC ) -> ( N x. ( N + 1 ) ) = ( ( N x. N ) + ( N x. 1 ) ) )
69	68	eqcomd 2628	. . . . . . 7 |- ( ( N e. CC /\ N e. CC /\ 1 e. CC ) -> ( ( N x. N ) + ( N x. 1 ) ) = ( N x. ( N + 1 ) ) )
70	12, 12, 67, 69	syl3anc 1326	. . . . . 6 |- ( N e. ZZ -> ( ( N x. N ) + ( N x. 1 ) ) = ( N x. ( N + 1 ) ) )
71	63, 66, 70	3eqtrd 2660	. . . . 5 |- ( N e. ZZ -> ( ( N ^ 2 ) + N ) = ( N x. ( N + 1 ) ) )
72	71	oveq1d 6665	. . . 4 |- ( N e. ZZ -> ( ( ( N ^ 2 ) + N ) / 2 ) = ( ( N x. ( N + 1 ) ) / 2 ) )
73	52, 61, 72	3eqtrd 2660	. . 3 |- ( N e. ZZ -> ( ( 4 x. ( ( N ^ 2 ) + N ) ) / 8 ) = ( ( N x. ( N + 1 ) ) / 2 ) )
74	73	adantr 481	. 2 |- ( ( N e. ZZ /\ M = ( ( 2 x. N ) + 1 ) ) -> ( ( 4 x. ( ( N ^ 2 ) + N ) ) / 8 ) = ( ( N x. ( N + 1 ) ) / 2 ) )
75	41, 48, 74	3eqtrd 2660	1 |- ( ( N e. ZZ /\ M = ( ( 2 x. N ) + 1 ) ) -> ( ( ( M ^ 2 ) - 1 ) / 8 ) = ( ( N x. ( N + 1 ) ) / 2 ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   - cmin 10266   / cdiv 10684   2c2 11070   4c4 11072   8c8 11076   ZZcz 11377   ^cexp 12860

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-n0 11293  df-z 11378  df-uz 11688  df-seq 12802  df-exp 12861

This theorem is referenced by:  sqoddm1div8z  15078