Theorem 2lgsoddprmlem3d 15783

Description: Lemma 4 for 2lgsoddprmlem3 15784. (Contributed by AV, 20-Jul-2021.)

Assertion

Ref	Expression
2lgsoddprmlem3d	|- ( ( ( 7 ^ 2 ) - 1 ) / 8 ) = ( 2 x. 3 )

Proof of Theorem 2lgsoddprmlem3d

Step	Hyp	Ref	Expression
1		6cn 9188	. . 3 |- 6 e. CC
2		8cn 9192	. . 3 |- 8 e. CC
3		8re 9191	. . . 4 |- 8 e. RR
4		8pos 9209	. . . 4 |- 0 < 8
5	3, 4	gt0ap0ii 8771	. . 3 |- 8 # 0
6	1, 2, 5	divcanap4i 8902	. 2 |- ( ( 6 x. 8 ) / 8 ) = 6
7	1, 2	mulcli 8147	. . . 4 |- ( 6 x. 8 ) e. CC
8		ax-1cn 8088	. . . 4 |- 1 e. CC
9		4p3e7 9251	. . . . . . 7 |- ( 4 + 3 ) = 7
10	9	eqcomi 2233	. . . . . 6 |- 7 = ( 4 + 3 )
11	10	oveq1i 6010	. . . . 5 |- ( 7 ^ 2 ) = ( ( 4 + 3 ) ^ 2 )
12		4cn 9184	. . . . . . 7 |- 4 e. CC
13		3cn 9181	. . . . . . 7 |- 3 e. CC
14	12, 13	binom2i 10865	. . . . . 6 |- ( ( 4 + 3 ) ^ 2 ) = ( ( ( 4 ^ 2 ) + ( 2 x. ( 4 x. 3 ) ) ) + ( 3 ^ 2 ) )
15		sq4e2t8 10854	. . . . . . . . . 10 |- ( 4 ^ 2 ) = ( 2 x. 8 )
16		2cn 9177	. . . . . . . . . . . . 13 |- 2 e. CC
17		4t2e8 9265	. . . . . . . . . . . . 13 |- ( 4 x. 2 ) = 8
18	12, 16, 17	mulcomli 8149	. . . . . . . . . . . 12 |- ( 2 x. 4 ) = 8
19	18	oveq1i 6010	. . . . . . . . . . 11 |- ( ( 2 x. 4 ) x. 3 ) = ( 8 x. 3 )
20	16, 12, 13	mulassi 8151	. . . . . . . . . . 11 |- ( ( 2 x. 4 ) x. 3 ) = ( 2 x. ( 4 x. 3 ) )
21	2, 13	mulcomi 8148	. . . . . . . . . . 11 |- ( 8 x. 3 ) = ( 3 x. 8 )
22	19, 20, 21	3eqtr3i 2258	. . . . . . . . . 10 |- ( 2 x. ( 4 x. 3 ) ) = ( 3 x. 8 )
23	15, 22	oveq12i 6012	. . . . . . . . 9 |- ( ( 4 ^ 2 ) + ( 2 x. ( 4 x. 3 ) ) ) = ( ( 2 x. 8 ) + ( 3 x. 8 ) )
24	16, 13, 2	adddiri 8153	. . . . . . . . 9 |- ( ( 2 + 3 ) x. 8 ) = ( ( 2 x. 8 ) + ( 3 x. 8 ) )
25		3p2e5 9248	. . . . . . . . . . 11 |- ( 3 + 2 ) = 5
26	13, 16, 25	addcomli 8287	. . . . . . . . . 10 |- ( 2 + 3 ) = 5
27	26	oveq1i 6010	. . . . . . . . 9 |- ( ( 2 + 3 ) x. 8 ) = ( 5 x. 8 )
28	23, 24, 27	3eqtr2i 2256	. . . . . . . 8 |- ( ( 4 ^ 2 ) + ( 2 x. ( 4 x. 3 ) ) ) = ( 5 x. 8 )
29		sq3 10853	. . . . . . . . 9 |- ( 3 ^ 2 ) = 9
30		df-9 9172	. . . . . . . . 9 |- 9 = ( 8 + 1 )
31	29, 30	eqtri 2250	. . . . . . . 8 |- ( 3 ^ 2 ) = ( 8 + 1 )
32	28, 31	oveq12i 6012	. . . . . . 7 |- ( ( ( 4 ^ 2 ) + ( 2 x. ( 4 x. 3 ) ) ) + ( 3 ^ 2 ) ) = ( ( 5 x. 8 ) + ( 8 + 1 ) )
33		5cn 9186	. . . . . . . . 9 |- 5 e. CC
34	33, 2	mulcli 8147	. . . . . . . 8 |- ( 5 x. 8 ) e. CC
35	34, 2, 8	addassi 8150	. . . . . . 7 |- ( ( ( 5 x. 8 ) + 8 ) + 1 ) = ( ( 5 x. 8 ) + ( 8 + 1 ) )
36		df-6 9169	. . . . . . . . . . 11 |- 6 = ( 5 + 1 )
37	36	oveq1i 6010	. . . . . . . . . 10 |- ( 6 x. 8 ) = ( ( 5 + 1 ) x. 8 )
38	33	a1i 9	. . . . . . . . . . . 12 |- ( 8 e. CC -> 5 e. CC )
39		id 19	. . . . . . . . . . . 12 |- ( 8 e. CC -> 8 e. CC )
40	38, 39	adddirp1d 8169	. . . . . . . . . . 11 |- ( 8 e. CC -> ( ( 5 + 1 ) x. 8 ) = ( ( 5 x. 8 ) + 8 ) )
41	2, 40	ax-mp 5	. . . . . . . . . 10 |- ( ( 5 + 1 ) x. 8 ) = ( ( 5 x. 8 ) + 8 )
42	37, 41	eqtri 2250	. . . . . . . . 9 |- ( 6 x. 8 ) = ( ( 5 x. 8 ) + 8 )
43	42	eqcomi 2233	. . . . . . . 8 |- ( ( 5 x. 8 ) + 8 ) = ( 6 x. 8 )
44	43	oveq1i 6010	. . . . . . 7 |- ( ( ( 5 x. 8 ) + 8 ) + 1 ) = ( ( 6 x. 8 ) + 1 )
45	32, 35, 44	3eqtr2i 2256	. . . . . 6 |- ( ( ( 4 ^ 2 ) + ( 2 x. ( 4 x. 3 ) ) ) + ( 3 ^ 2 ) ) = ( ( 6 x. 8 ) + 1 )
46	14, 45	eqtri 2250	. . . . 5 |- ( ( 4 + 3 ) ^ 2 ) = ( ( 6 x. 8 ) + 1 )
47	11, 46	eqtri 2250	. . . 4 |- ( 7 ^ 2 ) = ( ( 6 x. 8 ) + 1 )
48	7, 8, 47	mvrraddi 8359	. . 3 |- ( ( 7 ^ 2 ) - 1 ) = ( 6 x. 8 )
49	48	oveq1i 6010	. 2 |- ( ( ( 7 ^ 2 ) - 1 ) / 8 ) = ( ( 6 x. 8 ) / 8 )
50		3t2e6 9263	. . 3 |- ( 3 x. 2 ) = 6
51	13, 16, 50	mulcomli 8149	. 2 |- ( 2 x. 3 ) = 6
52	6, 49, 51	3eqtr4i 2260	1 |- ( ( ( 7 ^ 2 ) - 1 ) / 8 ) = ( 2 x. 3 )

Syntax hints:   = wceq 1395   e. wcel 2200  (class class class)co 6000   CCcc 7993   1c1 7996   + caddc 7998   x. cmul 8000   - cmin 8313   / cdiv 8815   2c2 9157   3c3 9158   4c4 9159   5c5 9160   6c6 9161   7c7 9162   8c8 9163   9c9 9164   ^cexp 10755

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 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-mulrcl 8094  ax-addcom 8095  ax-mulcom 8096  ax-addass 8097  ax-mulass 8098  ax-distr 8099  ax-i2m1 8100  ax-0lt1 8101  ax-1rid 8102  ax-0id 8103  ax-rnegex 8104  ax-precex 8105  ax-cnre 8106  ax-pre-ltirr 8107  ax-pre-ltwlin 8108  ax-pre-lttrn 8109  ax-pre-apti 8110  ax-pre-ltadd 8111  ax-pre-mulgt0 8112  ax-pre-mulext 8113

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 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4383  df-po 4386  df-iso 4387  df-iord 4456  df-on 4458  df-ilim 4459  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-recs 6449  df-frec 6535  df-pnf 8179  df-mnf 8180  df-xr 8181  df-ltxr 8182  df-le 8183  df-sub 8315  df-neg 8316  df-reap 8718  df-ap 8725  df-div 8816  df-inn 9107  df-2 9165  df-3 9166  df-4 9167  df-5 9168  df-6 9169  df-7 9170  df-8 9171  df-9 9172  df-n0 9366  df-z 9443  df-uz 9719  df-seqfrec 10665  df-exp 10756

This theorem is referenced by:  2lgsoddprmlem3  15784