Theorem 2lgsoddprmlem3d 16434

Description: Lemma 4 for 2lgsoddprmlem3 16435. (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 9395	. . 3 |- 6 e. CC
2		8cn 9399	. . 3 |- 8 e. CC
3		8re 9398	. . . 4 |- 8 e. RR
4		8pos 9416	. . . 4 |- 0 < 8
5	3, 4	gt0ap0ii 8964	. . 3 |- 8 # 0
6	1, 2, 5	divcanap4i 9097	. 2 |- ( ( 6 x. 8 ) / 8 ) = 6
7	1, 2	mulcli 8337	. . . 4 |- ( 6 x. 8 ) e. CC
8		ax-1cn 8278	. . . 4 |- 1 e. CC
9		4p3e7 9458	. . . . . . 7 |- ( 4 + 3 ) = 7
10	9	eqcomi 2242	. . . . . 6 |- 7 = ( 4 + 3 )
11	10	oveq1i 6098	. . . . 5 |- ( 7 ^ 2 ) = ( ( 4 + 3 ) ^ 2 )
12		4cn 9391	. . . . . . 7 |- 4 e. CC
13		3cn 9388	. . . . . . 7 |- 3 e. CC
14	12, 13	binom2i 11106	. . . . . 6 |- ( ( 4 + 3 ) ^ 2 ) = ( ( ( 4 ^ 2 ) + ( 2 x. ( 4 x. 3 ) ) ) + ( 3 ^ 2 ) )
15		sq4e2t8 11095	. . . . . . . . . 10 |- ( 4 ^ 2 ) = ( 2 x. 8 )
16		2cn 9384	. . . . . . . . . . . . 13 |- 2 e. CC
17		4t2e8 9473	. . . . . . . . . . . . 13 |- ( 4 x. 2 ) = 8
18	12, 16, 17	mulcomli 8339	. . . . . . . . . . . 12 |- ( 2 x. 4 ) = 8
19	18	oveq1i 6098	. . . . . . . . . . 11 |- ( ( 2 x. 4 ) x. 3 ) = ( 8 x. 3 )
20	16, 12, 13	mulassi 8341	. . . . . . . . . . 11 |- ( ( 2 x. 4 ) x. 3 ) = ( 2 x. ( 4 x. 3 ) )
21	2, 13	mulcomi 8338	. . . . . . . . . . 11 |- ( 8 x. 3 ) = ( 3 x. 8 )
22	19, 20, 21	3eqtr3i 2267	. . . . . . . . . 10 |- ( 2 x. ( 4 x. 3 ) ) = ( 3 x. 8 )
23	15, 22	oveq12i 6100	. . . . . . . . 9 |- ( ( 4 ^ 2 ) + ( 2 x. ( 4 x. 3 ) ) ) = ( ( 2 x. 8 ) + ( 3 x. 8 ) )
24	16, 13, 2	adddiri 8343	. . . . . . . . 9 |- ( ( 2 + 3 ) x. 8 ) = ( ( 2 x. 8 ) + ( 3 x. 8 ) )
25		3p2e5 9455	. . . . . . . . . . 11 |- ( 3 + 2 ) = 5
26	13, 16, 25	addcomli 8478	. . . . . . . . . 10 |- ( 2 + 3 ) = 5
27	26	oveq1i 6098	. . . . . . . . 9 |- ( ( 2 + 3 ) x. 8 ) = ( 5 x. 8 )
28	23, 24, 27	3eqtr2i 2265	. . . . . . . 8 |- ( ( 4 ^ 2 ) + ( 2 x. ( 4 x. 3 ) ) ) = ( 5 x. 8 )
29		sq3 11094	. . . . . . . . 9 |- ( 3 ^ 2 ) = 9
30		df-9 9379	. . . . . . . . 9 |- 9 = ( 8 + 1 )
31	29, 30	eqtri 2259	. . . . . . . 8 |- ( 3 ^ 2 ) = ( 8 + 1 )
32	28, 31	oveq12i 6100	. . . . . . 7 |- ( ( ( 4 ^ 2 ) + ( 2 x. ( 4 x. 3 ) ) ) + ( 3 ^ 2 ) ) = ( ( 5 x. 8 ) + ( 8 + 1 ) )
33		5cn 9393	. . . . . . . . 9 |- 5 e. CC
34	33, 2	mulcli 8337	. . . . . . . 8 |- ( 5 x. 8 ) e. CC
35	34, 2, 8	addassi 8340	. . . . . . 7 |- ( ( ( 5 x. 8 ) + 8 ) + 1 ) = ( ( 5 x. 8 ) + ( 8 + 1 ) )
36		df-6 9376	. . . . . . . . . . 11 |- 6 = ( 5 + 1 )
37	36	oveq1i 6098	. . . . . . . . . 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 8358	. . . . . . . . . . 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 2259	. . . . . . . . 9 |- ( 6 x. 8 ) = ( ( 5 x. 8 ) + 8 )
43	42	eqcomi 2242	. . . . . . . 8 |- ( ( 5 x. 8 ) + 8 ) = ( 6 x. 8 )
44	43	oveq1i 6098	. . . . . . 7 |- ( ( ( 5 x. 8 ) + 8 ) + 1 ) = ( ( 6 x. 8 ) + 1 )
45	32, 35, 44	3eqtr2i 2265	. . . . . 6 |- ( ( ( 4 ^ 2 ) + ( 2 x. ( 4 x. 3 ) ) ) + ( 3 ^ 2 ) ) = ( ( 6 x. 8 ) + 1 )
46	14, 45	eqtri 2259	. . . . 5 |- ( ( 4 + 3 ) ^ 2 ) = ( ( 6 x. 8 ) + 1 )
47	11, 46	eqtri 2259	. . . 4 |- ( 7 ^ 2 ) = ( ( 6 x. 8 ) + 1 )
48	7, 8, 47	mvrraddi 8550	. . 3 |- ( ( 7 ^ 2 ) - 1 ) = ( 6 x. 8 )
49	48	oveq1i 6098	. 2 |- ( ( ( 7 ^ 2 ) - 1 ) / 8 ) = ( ( 6 x. 8 ) / 8 )
50		3t2e6 9470	. . 3 |- ( 3 x. 2 ) = 6
51	13, 16, 50	mulcomli 8339	. 2 |- ( 2 x. 3 ) = 6
52	6, 49, 51	3eqtr4i 2269	1 |- ( ( ( 7 ^ 2 ) - 1 ) / 8 ) = ( 2 x. 3 )

This proof depends on syntax axioms:   = wceq 1402   e. wcel 2209  (class class class)co 6088   CCcc 8183   1c1 8186   + caddc 8188   x. cmul 8190   - cmin 8504   / cdiv 9010   2c2 9364   3c3 9365   4c4 9366   5c5 9367   6c6 9368   7c7 9369   8c8 9370   9c9 9371   ^cexp 10996

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-mulrcl 8284  ax-addcom 8285  ax-mulcom 8286  ax-addass 8287  ax-mulass 8288  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-1rid 8292  ax-0id 8293  ax-rnegex 8294  ax-precex 8295  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-apti 8300  ax-pre-ltadd 8301  ax-pre-mulgt0 8302  ax-pre-mulext 8303

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-po 4443  df-iso 4444  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-frec 6666  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-reap 8911  df-ap 8918  df-div 9011  df-inn 9314  df-2 9372  df-3 9373  df-4 9374  df-5 9375  df-6 9376  df-7 9377  df-8 9378  df-9 9379  df-n0 9575  df-z 9656  df-uz 9938  df-seqfrec 10906  df-exp 10997

This theorem is used by:  2lgsoddprmlem3  16435