Theorem 2lgsoddprmlem1 15948

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 6048	. . . . 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 6056	. . 3 |- ( ( A e. ZZ /\ B e. ZZ /\ N = ( ( 8 x. A ) + B ) ) -> ( ( N ^ 2 ) - 1 ) = ( ( ( ( 8 x. A ) + B ) ^ 2 ) - 1 ) )
4	3	oveq1d 6056	. 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 9568	. . . . 5 |- ( A e. ZZ -> A e. CC )
6	5	adantr 276	. . . 4 |- ( ( A e. ZZ /\ B e. ZZ ) -> A e. CC )
7		zcn 9568	. . . . 5 |- ( B e. ZZ -> B e. CC )
8	7	adantl 277	. . . 4 |- ( ( A e. ZZ /\ B e. ZZ ) -> B e. CC )
9		1cnd 8278	. . . 4 |- ( ( A e. ZZ /\ B e. ZZ ) -> 1 e. CC )
10		8cn 9311	. . . . . 6 |- 8 e. CC
11		8re 9310	. . . . . . 7 |- 8 e. RR
12		8pos 9328	. . . . . . 7 |- 0 < 8
13	11, 12	gt0ap0ii 8890	. . . . . 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 11059	. . . 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 2265	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 2203   class class class wbr 4102  (class class class)co 6041   CCcc 8113   0cc0 8115   1c1 8116   + caddc 8118   x. cmul 8120   - cmin 8432   # cap 8843   / cdiv 8934   2c2 9276   8c8 9282   ZZcz 9563   ^cexp 10886

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 2205  ax-14 2206  ax-ext 2214  ax-coll 4218  ax-sep 4221  ax-nul 4229  ax-pow 4279  ax-pr 4314  ax-un 4545  ax-setind 4650  ax-iinf 4701  ax-cnex 8206  ax-resscn 8207  ax-1cn 8208  ax-1re 8209  ax-icn 8210  ax-addcl 8211  ax-addrcl 8212  ax-mulcl 8213  ax-mulrcl 8214  ax-addcom 8215  ax-mulcom 8216  ax-addass 8217  ax-mulass 8218  ax-distr 8219  ax-i2m1 8220  ax-0lt1 8221  ax-1rid 8222  ax-0id 8223  ax-rnegex 8224  ax-precex 8225  ax-cnre 8226  ax-pre-ltirr 8227  ax-pre-ltwlin 8228  ax-pre-lttrn 8229  ax-pre-apti 8230  ax-pre-ltadd 8231  ax-pre-mulgt0 8232  ax-pre-mulext 8233

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 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3506  df-if 3617  df-pw 3667  df-sn 3688  df-pr 3689  df-op 3691  df-uni 3908  df-int 3943  df-iun 3986  df-br 4103  df-opab 4165  df-mpt 4166  df-tr 4202  df-id 4405  df-po 4408  df-iso 4409  df-iord 4478  df-on 4480  df-ilim 4481  df-suc 4483  df-iom 4704  df-xp 4746  df-rel 4747  df-cnv 4748  df-co 4749  df-dm 4750  df-rn 4751  df-res 4752  df-ima 4753  df-iota 5303  df-fun 5345  df-fn 5346  df-f 5347  df-f1 5348  df-fo 5349  df-f1o 5350  df-fv 5351  df-riota 5994  df-ov 6044  df-oprab 6045  df-mpo 6046  df-1st 6325  df-2nd 6326  df-recs 6527  df-frec 6613  df-pnf 8298  df-mnf 8299  df-xr 8300  df-ltxr 8301  df-le 8302  df-sub 8434  df-neg 8435  df-reap 8837  df-ap 8844  df-div 8935  df-inn 9226  df-2 9284  df-3 9285  df-4 9286  df-5 9287  df-6 9288  df-7 9289  df-8 9290  df-n0 9485  df-z 9564  df-uz 9840  df-seqfrec 10796  df-exp 10887

This theorem is referenced by:  2lgsoddprmlem2  15949