Theorem 2lgsoddprmlem3d 15377

Description: Lemma 4 for 2lgsoddprmlem3 15378. (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 9075	. . 3 |- 6 e. CC
2		8cn 9079	. . 3 |- 8 e. CC
3		8re 9078	. . . 4 |- 8 e. RR
4		8pos 9096	. . . 4 |- 0 < 8
5	3, 4	gt0ap0ii 8658	. . 3 |- 8 # 0
6	1, 2, 5	divcanap4i 8789	. 2 |- ( ( 6 x. 8 ) / 8 ) = 6
7	1, 2	mulcli 8034	. . . 4 |- ( 6 x. 8 ) e. CC
8		ax-1cn 7975	. . . 4 |- 1 e. CC
9		4p3e7 9138	. . . . . . 7 |- ( 4 + 3 ) = 7
10	9	eqcomi 2200	. . . . . 6 |- 7 = ( 4 + 3 )
11	10	oveq1i 5933	. . . . 5 |- ( 7 ^ 2 ) = ( ( 4 + 3 ) ^ 2 )
12		4cn 9071	. . . . . . 7 |- 4 e. CC
13		3cn 9068	. . . . . . 7 |- 3 e. CC
14	12, 13	binom2i 10743	. . . . . 6 |- ( ( 4 + 3 ) ^ 2 ) = ( ( ( 4 ^ 2 ) + ( 2 x. ( 4 x. 3 ) ) ) + ( 3 ^ 2 ) )
15		sq4e2t8 10732	. . . . . . . . . 10 |- ( 4 ^ 2 ) = ( 2 x. 8 )
16		2cn 9064	. . . . . . . . . . . . 13 |- 2 e. CC
17		4t2e8 9152	. . . . . . . . . . . . 13 |- ( 4 x. 2 ) = 8
18	12, 16, 17	mulcomli 8036	. . . . . . . . . . . 12 |- ( 2 x. 4 ) = 8
19	18	oveq1i 5933	. . . . . . . . . . 11 |- ( ( 2 x. 4 ) x. 3 ) = ( 8 x. 3 )
20	16, 12, 13	mulassi 8038	. . . . . . . . . . 11 |- ( ( 2 x. 4 ) x. 3 ) = ( 2 x. ( 4 x. 3 ) )
21	2, 13	mulcomi 8035	. . . . . . . . . . 11 |- ( 8 x. 3 ) = ( 3 x. 8 )
22	19, 20, 21	3eqtr3i 2225	. . . . . . . . . 10 |- ( 2 x. ( 4 x. 3 ) ) = ( 3 x. 8 )
23	15, 22	oveq12i 5935	. . . . . . . . 9 |- ( ( 4 ^ 2 ) + ( 2 x. ( 4 x. 3 ) ) ) = ( ( 2 x. 8 ) + ( 3 x. 8 ) )
24	16, 13, 2	adddiri 8040	. . . . . . . . 9 |- ( ( 2 + 3 ) x. 8 ) = ( ( 2 x. 8 ) + ( 3 x. 8 ) )
25		3p2e5 9135	. . . . . . . . . . 11 |- ( 3 + 2 ) = 5
26	13, 16, 25	addcomli 8174	. . . . . . . . . 10 |- ( 2 + 3 ) = 5
27	26	oveq1i 5933	. . . . . . . . 9 |- ( ( 2 + 3 ) x. 8 ) = ( 5 x. 8 )
28	23, 24, 27	3eqtr2i 2223	. . . . . . . 8 |- ( ( 4 ^ 2 ) + ( 2 x. ( 4 x. 3 ) ) ) = ( 5 x. 8 )
29		sq3 10731	. . . . . . . . 9 |- ( 3 ^ 2 ) = 9
30		df-9 9059	. . . . . . . . 9 |- 9 = ( 8 + 1 )
31	29, 30	eqtri 2217	. . . . . . . 8 |- ( 3 ^ 2 ) = ( 8 + 1 )
32	28, 31	oveq12i 5935	. . . . . . 7 |- ( ( ( 4 ^ 2 ) + ( 2 x. ( 4 x. 3 ) ) ) + ( 3 ^ 2 ) ) = ( ( 5 x. 8 ) + ( 8 + 1 ) )
33		5cn 9073	. . . . . . . . 9 |- 5 e. CC
34	33, 2	mulcli 8034	. . . . . . . 8 |- ( 5 x. 8 ) e. CC
35	34, 2, 8	addassi 8037	. . . . . . 7 |- ( ( ( 5 x. 8 ) + 8 ) + 1 ) = ( ( 5 x. 8 ) + ( 8 + 1 ) )
36		df-6 9056	. . . . . . . . . . 11 |- 6 = ( 5 + 1 )
37	36	oveq1i 5933	. . . . . . . . . 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 8056	. . . . . . . . . . 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 2217	. . . . . . . . 9 |- ( 6 x. 8 ) = ( ( 5 x. 8 ) + 8 )
43	42	eqcomi 2200	. . . . . . . 8 |- ( ( 5 x. 8 ) + 8 ) = ( 6 x. 8 )
44	43	oveq1i 5933	. . . . . . 7 |- ( ( ( 5 x. 8 ) + 8 ) + 1 ) = ( ( 6 x. 8 ) + 1 )
45	32, 35, 44	3eqtr2i 2223	. . . . . 6 |- ( ( ( 4 ^ 2 ) + ( 2 x. ( 4 x. 3 ) ) ) + ( 3 ^ 2 ) ) = ( ( 6 x. 8 ) + 1 )
46	14, 45	eqtri 2217	. . . . 5 |- ( ( 4 + 3 ) ^ 2 ) = ( ( 6 x. 8 ) + 1 )
47	11, 46	eqtri 2217	. . . 4 |- ( 7 ^ 2 ) = ( ( 6 x. 8 ) + 1 )
48	7, 8, 47	mvrraddi 8246	. . 3 |- ( ( 7 ^ 2 ) - 1 ) = ( 6 x. 8 )
49	48	oveq1i 5933	. 2 |- ( ( ( 7 ^ 2 ) - 1 ) / 8 ) = ( ( 6 x. 8 ) / 8 )
50		3t2e6 9150	. . 3 |- ( 3 x. 2 ) = 6
51	13, 16, 50	mulcomli 8036	. 2 |- ( 2 x. 3 ) = 6
52	6, 49, 51	3eqtr4i 2227	1 |- ( ( ( 7 ^ 2 ) - 1 ) / 8 ) = ( 2 x. 3 )

Syntax hints:   = wceq 1364   e. wcel 2167  (class class class)co 5923   CCcc 7880   1c1 7883   + caddc 7885   x. cmul 7887   - cmin 8200   / cdiv 8702   2c2 9044   3c3 9045   4c4 9046   5c5 9047   6c6 9048   7c7 9049   8c8 9050   9c9 9051   ^cexp 10633

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7973  ax-resscn 7974  ax-1cn 7975  ax-1re 7976  ax-icn 7977  ax-addcl 7978  ax-addrcl 7979  ax-mulcl 7980  ax-mulrcl 7981  ax-addcom 7982  ax-mulcom 7983  ax-addass 7984  ax-mulass 7985  ax-distr 7986  ax-i2m1 7987  ax-0lt1 7988  ax-1rid 7989  ax-0id 7990  ax-rnegex 7991  ax-precex 7992  ax-cnre 7993  ax-pre-ltirr 7994  ax-pre-ltwlin 7995  ax-pre-lttrn 7996  ax-pre-apti 7997  ax-pre-ltadd 7998  ax-pre-mulgt0 7999  ax-pre-mulext 8000

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5878  df-ov 5926  df-oprab 5927  df-mpo 5928  df-1st 6200  df-2nd 6201  df-recs 6365  df-frec 6451  df-pnf 8066  df-mnf 8067  df-xr 8068  df-ltxr 8069  df-le 8070  df-sub 8202  df-neg 8203  df-reap 8605  df-ap 8612  df-div 8703  df-inn 8994  df-2 9052  df-3 9053  df-4 9054  df-5 9055  df-6 9056  df-7 9057  df-8 9058  df-9 9059  df-n0 9253  df-z 9330  df-uz 9605  df-seqfrec 10543  df-exp 10634

This theorem is referenced by:  2lgsoddprmlem3  15378