Theorem 2lgsoddprmlem3d 15702

Description: Lemma 4 for 2lgsoddprmlem3 15703. (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 9153	. . 3 |- 6 e. CC
2		8cn 9157	. . 3 |- 8 e. CC
3		8re 9156	. . . 4 |- 8 e. RR
4		8pos 9174	. . . 4 |- 0 < 8
5	3, 4	gt0ap0ii 8736	. . 3 |- 8 # 0
6	1, 2, 5	divcanap4i 8867	. 2 |- ( ( 6 x. 8 ) / 8 ) = 6
7	1, 2	mulcli 8112	. . . 4 |- ( 6 x. 8 ) e. CC
8		ax-1cn 8053	. . . 4 |- 1 e. CC
9		4p3e7 9216	. . . . . . 7 |- ( 4 + 3 ) = 7
10	9	eqcomi 2211	. . . . . 6 |- 7 = ( 4 + 3 )
11	10	oveq1i 5977	. . . . 5 |- ( 7 ^ 2 ) = ( ( 4 + 3 ) ^ 2 )
12		4cn 9149	. . . . . . 7 |- 4 e. CC
13		3cn 9146	. . . . . . 7 |- 3 e. CC
14	12, 13	binom2i 10830	. . . . . 6 |- ( ( 4 + 3 ) ^ 2 ) = ( ( ( 4 ^ 2 ) + ( 2 x. ( 4 x. 3 ) ) ) + ( 3 ^ 2 ) )
15		sq4e2t8 10819	. . . . . . . . . 10 |- ( 4 ^ 2 ) = ( 2 x. 8 )
16		2cn 9142	. . . . . . . . . . . . 13 |- 2 e. CC
17		4t2e8 9230	. . . . . . . . . . . . 13 |- ( 4 x. 2 ) = 8
18	12, 16, 17	mulcomli 8114	. . . . . . . . . . . 12 |- ( 2 x. 4 ) = 8
19	18	oveq1i 5977	. . . . . . . . . . 11 |- ( ( 2 x. 4 ) x. 3 ) = ( 8 x. 3 )
20	16, 12, 13	mulassi 8116	. . . . . . . . . . 11 |- ( ( 2 x. 4 ) x. 3 ) = ( 2 x. ( 4 x. 3 ) )
21	2, 13	mulcomi 8113	. . . . . . . . . . 11 |- ( 8 x. 3 ) = ( 3 x. 8 )
22	19, 20, 21	3eqtr3i 2236	. . . . . . . . . 10 |- ( 2 x. ( 4 x. 3 ) ) = ( 3 x. 8 )
23	15, 22	oveq12i 5979	. . . . . . . . 9 |- ( ( 4 ^ 2 ) + ( 2 x. ( 4 x. 3 ) ) ) = ( ( 2 x. 8 ) + ( 3 x. 8 ) )
24	16, 13, 2	adddiri 8118	. . . . . . . . 9 |- ( ( 2 + 3 ) x. 8 ) = ( ( 2 x. 8 ) + ( 3 x. 8 ) )
25		3p2e5 9213	. . . . . . . . . . 11 |- ( 3 + 2 ) = 5
26	13, 16, 25	addcomli 8252	. . . . . . . . . 10 |- ( 2 + 3 ) = 5
27	26	oveq1i 5977	. . . . . . . . 9 |- ( ( 2 + 3 ) x. 8 ) = ( 5 x. 8 )
28	23, 24, 27	3eqtr2i 2234	. . . . . . . 8 |- ( ( 4 ^ 2 ) + ( 2 x. ( 4 x. 3 ) ) ) = ( 5 x. 8 )
29		sq3 10818	. . . . . . . . 9 |- ( 3 ^ 2 ) = 9
30		df-9 9137	. . . . . . . . 9 |- 9 = ( 8 + 1 )
31	29, 30	eqtri 2228	. . . . . . . 8 |- ( 3 ^ 2 ) = ( 8 + 1 )
32	28, 31	oveq12i 5979	. . . . . . 7 |- ( ( ( 4 ^ 2 ) + ( 2 x. ( 4 x. 3 ) ) ) + ( 3 ^ 2 ) ) = ( ( 5 x. 8 ) + ( 8 + 1 ) )
33		5cn 9151	. . . . . . . . 9 |- 5 e. CC
34	33, 2	mulcli 8112	. . . . . . . 8 |- ( 5 x. 8 ) e. CC
35	34, 2, 8	addassi 8115	. . . . . . 7 |- ( ( ( 5 x. 8 ) + 8 ) + 1 ) = ( ( 5 x. 8 ) + ( 8 + 1 ) )
36		df-6 9134	. . . . . . . . . . 11 |- 6 = ( 5 + 1 )
37	36	oveq1i 5977	. . . . . . . . . 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 8134	. . . . . . . . . . 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 2228	. . . . . . . . 9 |- ( 6 x. 8 ) = ( ( 5 x. 8 ) + 8 )
43	42	eqcomi 2211	. . . . . . . 8 |- ( ( 5 x. 8 ) + 8 ) = ( 6 x. 8 )
44	43	oveq1i 5977	. . . . . . 7 |- ( ( ( 5 x. 8 ) + 8 ) + 1 ) = ( ( 6 x. 8 ) + 1 )
45	32, 35, 44	3eqtr2i 2234	. . . . . 6 |- ( ( ( 4 ^ 2 ) + ( 2 x. ( 4 x. 3 ) ) ) + ( 3 ^ 2 ) ) = ( ( 6 x. 8 ) + 1 )
46	14, 45	eqtri 2228	. . . . 5 |- ( ( 4 + 3 ) ^ 2 ) = ( ( 6 x. 8 ) + 1 )
47	11, 46	eqtri 2228	. . . 4 |- ( 7 ^ 2 ) = ( ( 6 x. 8 ) + 1 )
48	7, 8, 47	mvrraddi 8324	. . 3 |- ( ( 7 ^ 2 ) - 1 ) = ( 6 x. 8 )
49	48	oveq1i 5977	. 2 |- ( ( ( 7 ^ 2 ) - 1 ) / 8 ) = ( ( 6 x. 8 ) / 8 )
50		3t2e6 9228	. . 3 |- ( 3 x. 2 ) = 6
51	13, 16, 50	mulcomli 8114	. 2 |- ( 2 x. 3 ) = 6
52	6, 49, 51	3eqtr4i 2238	1 |- ( ( ( 7 ^ 2 ) - 1 ) / 8 ) = ( 2 x. 3 )

Syntax hints:   = wceq 1373   e. wcel 2178  (class class class)co 5967   CCcc 7958   1c1 7961   + caddc 7963   x. cmul 7965   - cmin 8278   / cdiv 8780   2c2 9122   3c3 9123   4c4 9124   5c5 9125   6c6 9126   7c7 9127   8c8 9128   9c9 9129   ^cexp 10720

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-mulrcl 8059  ax-addcom 8060  ax-mulcom 8061  ax-addass 8062  ax-mulass 8063  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-1rid 8067  ax-0id 8068  ax-rnegex 8069  ax-precex 8070  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-apti 8075  ax-pre-ltadd 8076  ax-pre-mulgt0 8077  ax-pre-mulext 8078

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-if 3580  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-id 4358  df-po 4361  df-iso 4362  df-iord 4431  df-on 4433  df-ilim 4434  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-recs 6414  df-frec 6500  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-reap 8683  df-ap 8690  df-div 8781  df-inn 9072  df-2 9130  df-3 9131  df-4 9132  df-5 9133  df-6 9134  df-7 9135  df-8 9136  df-9 9137  df-n0 9331  df-z 9408  df-uz 9684  df-seqfrec 10630  df-exp 10721

This theorem is referenced by:  2lgsoddprmlem3  15703