Theorem sqoddm1div8 11152

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 6095	. . . . . 6 |- ( M = ( ( 2 x. N ) + 1 ) -> ( M ^ 2 ) = ( ( ( 2 x. N ) + 1 ) ^ 2 ) )
2		2z 9683	. . . . . . . . . 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 9785	. . . . . . . 8 |- ( N e. ZZ -> ( 2 x. N ) e. ZZ )
6	5	zcnd 9780	. . . . . . 7 |- ( N e. ZZ -> ( 2 x. N ) e. CC )
7		binom21 11110	. . . . . . 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 2293	. . . . 5 |- ( ( N e. ZZ /\ M = ( ( 2 x. N ) + 1 ) ) -> ( M ^ 2 ) = ( ( ( ( 2 x. N ) ^ 2 ) + ( 2 x. ( 2 x. N ) ) ) + 1 ) )
10	9	oveq1d 6103	. . . 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 9386	. . . . . . . . . . 11 |- ( N e. ZZ -> 2 e. CC )
12		zcn 9660	. . . . . . . . . . 11 |- ( N e. ZZ -> N e. CC )
13	11, 12	sqmuld 11144	. . . . . . . . . 10 |- ( N e. ZZ -> ( ( 2 x. N ) ^ 2 ) = ( ( 2 ^ 2 ) x. ( N ^ 2 ) ) )
14		sq2 11093	. . . . . . . . . . . 12 |- ( 2 ^ 2 ) = 4
15	14	a1i 9	. . . . . . . . . . 11 |- ( N e. ZZ -> ( 2 ^ 2 ) = 4 )
16	15	oveq1d 6103	. . . . . . . . . 10 |- ( N e. ZZ -> ( ( 2 ^ 2 ) x. ( N ^ 2 ) ) = ( 4 x. ( N ^ 2 ) ) )
17	13, 16	eqtrd 2271	. . . . . . . . 9 |- ( N e. ZZ -> ( ( 2 x. N ) ^ 2 ) = ( 4 x. ( N ^ 2 ) ) )
18		mulass 8316	. . . . . . . . . . . 12 |- ( ( 2 e. CC /\ 2 e. CC /\ N e. CC ) -> ( ( 2 x. 2 ) x. N ) = ( 2 x. ( 2 x. N ) ) )
19	18	eqcomd 2244	. . . . . . . . . . 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 1278	. . . . . . . . . 10 |- ( N e. ZZ -> ( 2 x. ( 2 x. N ) ) = ( ( 2 x. 2 ) x. N ) )
21		2t2e4 9468	. . . . . . . . . . . 12 |- ( 2 x. 2 ) = 4
22	21	a1i 9	. . . . . . . . . . 11 |- ( N e. ZZ -> ( 2 x. 2 ) = 4 )
23	22	oveq1d 6103	. . . . . . . . . 10 |- ( N e. ZZ -> ( ( 2 x. 2 ) x. N ) = ( 4 x. N ) )
24	20, 23	eqtrd 2271	. . . . . . . . 9 |- ( N e. ZZ -> ( 2 x. ( 2 x. N ) ) = ( 4 x. N ) )
25	17, 24	oveq12d 6106	. . . . . . . 8 |- ( N e. ZZ -> ( ( ( 2 x. N ) ^ 2 ) + ( 2 x. ( 2 x. N ) ) ) = ( ( 4 x. ( N ^ 2 ) ) + ( 4 x. N ) ) )
26	25	oveq1d 6103	. . . . . . 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 6103	. . . . . 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 9685	. . . . . . . . . . 11 |- 4 e. ZZ
29	28	a1i 9	. . . . . . . . . 10 |- ( N e. ZZ -> 4 e. ZZ )
30		zsqcl 11068	. . . . . . . . . 10 |- ( N e. ZZ -> ( N ^ 2 ) e. ZZ )
31	29, 30	zmulcld 9785	. . . . . . . . 9 |- ( N e. ZZ -> ( 4 x. ( N ^ 2 ) ) e. ZZ )
32	31	zcnd 9780	. . . . . . . 8 |- ( N e. ZZ -> ( 4 x. ( N ^ 2 ) ) e. CC )
33	29, 4	zmulcld 9785	. . . . . . . . 9 |- ( N e. ZZ -> ( 4 x. N ) e. ZZ )
34	33	zcnd 9780	. . . . . . . 8 |- ( N e. ZZ -> ( 4 x. N ) e. CC )
35	32, 34	addcld 8351	. . . . . . 7 |- ( N e. ZZ -> ( ( 4 x. ( N ^ 2 ) ) + ( 4 x. N ) ) e. CC )
36		pncan1 8711	. . . . . . 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 2271	. . . . 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 2271	. . 3 |- ( ( N e. ZZ /\ M = ( ( 2 x. N ) + 1 ) ) -> ( ( M ^ 2 ) - 1 ) = ( ( 4 x. ( N ^ 2 ) ) + ( 4 x. N ) ) )
41	40	oveq1d 6103	. 2 |- ( ( N e. ZZ /\ M = ( ( 2 x. N ) + 1 ) ) -> ( ( ( M ^ 2 ) - 1 ) / 8 ) = ( ( ( 4 x. ( N ^ 2 ) ) + ( 4 x. N ) ) / 8 ) )
42		4cn 9391	. . . . . . 7 |- 4 e. CC
43	42	a1i 9	. . . . . 6 |- ( N e. ZZ -> 4 e. CC )
44	30	zcnd 9780	. . . . . 6 |- ( N e. ZZ -> ( N ^ 2 ) e. CC )
45	43, 44, 12	adddid 8356	. . . . 5 |- ( N e. ZZ -> ( 4 x. ( ( N ^ 2 ) + N ) ) = ( ( 4 x. ( N ^ 2 ) ) + ( 4 x. N ) ) )
46	45	eqcomd 2244	. . . 4 |- ( N e. ZZ -> ( ( 4 x. ( N ^ 2 ) ) + ( 4 x. N ) ) = ( 4 x. ( ( N ^ 2 ) + N ) ) )
47	46	oveq1d 6103	. . 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 9473	. . . . . . 7 |- ( 4 x. 2 ) = 8
50	49	a1i 9	. . . . . 6 |- ( N e. ZZ -> ( 4 x. 2 ) = 8 )
51	50	eqcomd 2244	. . . . 5 |- ( N e. ZZ -> 8 = ( 4 x. 2 ) )
52	51	oveq2d 6104	. . . 4 |- ( N e. ZZ -> ( ( 4 x. ( ( N ^ 2 ) + N ) ) / 8 ) = ( ( 4 x. ( ( N ^ 2 ) + N ) ) / ( 4 x. 2 ) ) )
53	30, 4	zaddcld 9783	. . . . . 6 |- ( N e. ZZ -> ( ( N ^ 2 ) + N ) e. ZZ )
54	53	zcnd 9780	. . . . 5 |- ( N e. ZZ -> ( ( N ^ 2 ) + N ) e. CC )
55		2ap0 9406	. . . . . 6 |- 2 # 0
56	55	a1i 9	. . . . 5 |- ( N e. ZZ -> 2 # 0 )
57		4ap0 9412	. . . . . 6 |- 4 # 0
58	57	a1i 9	. . . . 5 |- ( N e. ZZ -> 4 # 0 )
59	54, 11, 43, 56, 58	divcanap5d 9155	. . . 4 |- ( N e. ZZ -> ( ( 4 x. ( ( N ^ 2 ) + N ) ) / ( 4 x. 2 ) ) = ( ( ( N ^ 2 ) + N ) / 2 ) )
60	12	sqvald 11129	. . . . . . 7 |- ( N e. ZZ -> ( N ^ 2 ) = ( N x. N ) )
61	60	oveq1d 6103	. . . . . 6 |- ( N e. ZZ -> ( ( N ^ 2 ) + N ) = ( ( N x. N ) + N ) )
62	12	mulridd 8349	. . . . . . . 8 |- ( N e. ZZ -> ( N x. 1 ) = N )
63	62	eqcomd 2244	. . . . . . 7 |- ( N e. ZZ -> N = ( N x. 1 ) )
64	63	oveq2d 6104	. . . . . 6 |- ( N e. ZZ -> ( ( N x. N ) + N ) = ( ( N x. N ) + ( N x. 1 ) ) )
65		1cnd 8348	. . . . . . 7 |- ( N e. ZZ -> 1 e. CC )
66		adddi 8317	. . . . . . . 8 |- ( ( N e. CC /\ N e. CC /\ 1 e. CC ) -> ( N x. ( N + 1 ) ) = ( ( N x. N ) + ( N x. 1 ) ) )
67	66	eqcomd 2244	. . . . . . 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 1278	. . . . . 6 |- ( N e. ZZ -> ( ( N x. N ) + ( N x. 1 ) ) = ( N x. ( N + 1 ) ) )
69	61, 64, 68	3eqtrd 2275	. . . . 5 |- ( N e. ZZ -> ( ( N ^ 2 ) + N ) = ( N x. ( N + 1 ) ) )
70	69	oveq1d 6103	. . . 4 |- ( N e. ZZ -> ( ( ( N ^ 2 ) + N ) / 2 ) = ( ( N x. ( N + 1 ) ) / 2 ) )
71	52, 59, 70	3eqtrd 2275	. . 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 2275	1 |- ( ( N e. ZZ /\ M = ( ( 2 x. N ) + 1 ) ) -> ( ( ( M ^ 2 ) - 1 ) / 8 ) = ( ( N x. ( N + 1 ) ) / 2 ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209   class class class wbr 4131  (class class class)co 6088   CCcc 8183   0cc0 8185   1c1 8186   + caddc 8188   x. cmul 8190   - cmin 8504   # cap 8917   / cdiv 9010   2c2 9364   4c4 9366   8c8 9370   ZZcz 9655   ^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-4 9374  df-5 9375  df-6 9376  df-7 9377  df-8 9378  df-n0 9575  df-z 9656  df-uz 9938  df-seqfrec 10906  df-exp 10997

This theorem is used by:  sqoddm1div8z  12678