Theorem 2lgsoddprmlem1 15995

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 6059	. . . . 5 |- ( N = ( ( 8 x. A ) + B ) -> ( N ^ 2 ) = ( ( ( 8 x. A ) + B ) ^ 2 ) )
2	1	3ad2ant3 1047	. . . 4 |- ( ( A e. ZZ /\ B e. ZZ /\ N = ( ( 8 x. A ) + B ) ) -> ( N ^ 2 ) = ( ( ( 8 x. A ) + B ) ^ 2 ) )
3	2	oveq1d 6067	. . 3 |- ( ( A e. ZZ /\ B e. ZZ /\ N = ( ( 8 x. A ) + B ) ) -> ( ( N ^ 2 ) - 1 ) = ( ( ( ( 8 x. A ) + B ) ^ 2 ) - 1 ) )
4	3	oveq1d 6067	. 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 9584	. . . . 5 |- ( A e. ZZ -> A e. CC )
6	5	adantr 276	. . . 4 |- ( ( A e. ZZ /\ B e. ZZ ) -> A e. CC )
7		zcn 9584	. . . . 5 |- ( B e. ZZ -> B e. CC )
8	7	adantl 277	. . . 4 |- ( ( A e. ZZ /\ B e. ZZ ) -> B e. CC )
9		1cnd 8292	. . . 4 |- ( ( A e. ZZ /\ B e. ZZ ) -> 1 e. CC )
10		8cn 9325	. . . . . 6 |- 8 e. CC
11		8re 9324	. . . . . . 7 |- 8 e. RR
12		8pos 9342	. . . . . . 7 |- 0 < 8
13	11, 12	gt0ap0ii 8904	. . . . . 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 11077	. . . 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 1277	. . 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 1044	. 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 2267	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 1005   = wceq 1398   e. wcel 2205   class class class wbr 4111  (class class class)co 6052   CCcc 8127   0cc0 8129   1c1 8130   + caddc 8132   x. cmul 8134   - cmin 8446   # cap 8857   / cdiv 8948   2c2 9290   8c8 9296   ZZcz 9579   ^cexp 10904

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-iinf 4712  ax-cnex 8220  ax-resscn 8221  ax-1cn 8222  ax-1re 8223  ax-icn 8224  ax-addcl 8225  ax-addrcl 8226  ax-mulcl 8227  ax-mulrcl 8228  ax-addcom 8229  ax-mulcom 8230  ax-addass 8231  ax-mulass 8232  ax-distr 8233  ax-i2m1 8234  ax-0lt1 8235  ax-1rid 8236  ax-0id 8237  ax-rnegex 8238  ax-precex 8239  ax-cnre 8240  ax-pre-ltirr 8241  ax-pre-ltwlin 8242  ax-pre-lttrn 8243  ax-pre-apti 8244  ax-pre-ltadd 8245  ax-pre-mulgt0 8246  ax-pre-mulext 8247

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-if 3623  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-id 4416  df-po 4419  df-iso 4420  df-iord 4489  df-on 4491  df-ilim 4492  df-suc 4494  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-recs 6538  df-frec 6624  df-pnf 8312  df-mnf 8313  df-xr 8314  df-ltxr 8315  df-le 8316  df-sub 8448  df-neg 8449  df-reap 8851  df-ap 8858  df-div 8949  df-inn 9240  df-2 9298  df-3 9299  df-4 9300  df-5 9301  df-6 9302  df-7 9303  df-8 9304  df-n0 9499  df-z 9580  df-uz 9857  df-seqfrec 10814  df-exp 10905

This theorem is referenced by:  2lgsoddprmlem2  15996