Theorem 2lgsoddprmlem1 15784

Description: Lemma 1 for 2lgsoddprm . (Contributed by AV, 19-Jul-2021.)

Assertion

Ref	Expression
2lgsoddprmlem1	|- ( ( A e. ZZ /\ B e. ZZ /\ N = ( ( 8 x. A ) + B ) ) -> ( ( ( N ^ 2 ) - 1 ) / 8 ) = ( ( ( 8 x. ( A ^ 2 ) ) + ( 2 x. ( A x. B ) ) ) + ( ( ( B ^ 2 ) - 1 ) / 8 ) ) )

Proof of Theorem 2lgsoddprmlem1

Step	Hyp	Ref	Expression
1		oveq1 6008	. . . . 5 |- ( N = ( ( 8 x. A ) + B ) -> ( N ^ 2 ) = ( ( ( 8 x. A ) + B ) ^ 2 ) )
2	1	3ad2ant3 1044	. . . 4 |- ( ( A e. ZZ /\ B e. ZZ /\ N = ( ( 8 x. A ) + B ) ) -> ( N ^ 2 ) = ( ( ( 8 x. A ) + B ) ^ 2 ) )
3	2	oveq1d 6016	. . 3 |- ( ( A e. ZZ /\ B e. ZZ /\ N = ( ( 8 x. A ) + B ) ) -> ( ( N ^ 2 ) - 1 ) = ( ( ( ( 8 x. A ) + B ) ^ 2 ) - 1 ) )
4	3	oveq1d 6016	. 2 |- ( ( A e. ZZ /\ B e. ZZ /\ N = ( ( 8 x. A ) + B ) ) -> ( ( ( N ^ 2 ) - 1 ) / 8 ) = ( ( ( ( ( 8 x. A ) + B ) ^ 2 ) - 1 ) / 8 ) )
5		zcn 9451	. . . . 5 |- ( A e. ZZ -> A e. CC )
6	5	adantr 276	. . . 4 |- ( ( A e. ZZ /\ B e. ZZ ) -> A e. CC )
7		zcn 9451	. . . . 5 |- ( B e. ZZ -> B e. CC )
8	7	adantl 277	. . . 4 |- ( ( A e. ZZ /\ B e. ZZ ) -> B e. CC )
9		1cnd 8162	. . . 4 |- ( ( A e. ZZ /\ B e. ZZ ) -> 1 e. CC )
10		8cn 9196	. . . . . 6 |- 8 e. CC
11		8re 9195	. . . . . . 7 |- 8 e. RR
12		8pos 9213	. . . . . . 7 |- 0 < 8
13	11, 12	gt0ap0ii 8775	. . . . . 6 |- 8 # 0
14	10, 13	pm3.2i 272	. . . . 5 |- ( 8 e. CC /\ 8 # 0 )
15	14	a1i 9	. . . 4 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( 8 e. CC /\ 8 # 0 ) )
16		mulsubdivbinom2ap 10933	. . . 4 |- ( ( ( A e. CC /\ B e. CC /\ 1 e. CC ) /\ ( 8 e. CC /\ 8 # 0 ) ) -> ( ( ( ( ( 8 x. A ) + B ) ^ 2 ) - 1 ) / 8 ) = ( ( ( 8 x. ( A ^ 2 ) ) + ( 2 x. ( A x. B ) ) ) + ( ( ( B ^ 2 ) - 1 ) / 8 ) ) )
17	6, 8, 9, 15, 16	syl31anc 1274	. . 3 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( ( ( ( 8 x. A ) + B ) ^ 2 ) - 1 ) / 8 ) = ( ( ( 8 x. ( A ^ 2 ) ) + ( 2 x. ( A x. B ) ) ) + ( ( ( B ^ 2 ) - 1 ) / 8 ) ) )
18	17	3adant3 1041	. 2 |- ( ( A e. ZZ /\ B e. ZZ /\ N = ( ( 8 x. A ) + B ) ) -> ( ( ( ( ( 8 x. A ) + B ) ^ 2 ) - 1 ) / 8 ) = ( ( ( 8 x. ( A ^ 2 ) ) + ( 2 x. ( A x. B ) ) ) + ( ( ( B ^ 2 ) - 1 ) / 8 ) ) )
19	4, 18	eqtrd 2262	1 |- ( ( A e. ZZ /\ B e. ZZ /\ N = ( ( 8 x. A ) + B ) ) -> ( ( ( N ^ 2 ) - 1 ) / 8 ) = ( ( ( 8 x. ( A ^ 2 ) ) + ( 2 x. ( A x. B ) ) ) + ( ( ( B ^ 2 ) - 1 ) / 8 ) ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1002   = wceq 1395   e. wcel 2200   class class class wbr 4083  (class class class)co 6001   CCcc 7997   0cc0 7999   1c1 8000   + caddc 8002   x. cmul 8004   - cmin 8317   # cap 8728   / cdiv 8819   2c2 9161   8c8 9167   ZZcz 9446   ^cexp 10760

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-mulrcl 8098  ax-addcom 8099  ax-mulcom 8100  ax-addass 8101  ax-mulass 8102  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-1rid 8106  ax-0id 8107  ax-rnegex 8108  ax-precex 8109  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-apti 8114  ax-pre-ltadd 8115  ax-pre-mulgt0 8116  ax-pre-mulext 8117

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-recs 6451  df-frec 6537  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-reap 8722  df-ap 8729  df-div 8820  df-inn 9111  df-2 9169  df-3 9170  df-4 9171  df-5 9172  df-6 9173  df-7 9174  df-8 9175  df-n0 9370  df-z 9447  df-uz 9723  df-seqfrec 10670  df-exp 10761

This theorem is referenced by:  2lgsoddprmlem2  15785