Theorem 2lgsoddprmlem3d 15912

Description: Lemma 4 for 2lgsoddprmlem3 15913. (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 9267	. . 3 |- 6 e. CC
2		8cn 9271	. . 3 |- 8 e. CC
3		8re 9270	. . . 4 |- 8 e. RR
4		8pos 9288	. . . 4 |- 0 < 8
5	3, 4	gt0ap0ii 8850	. . 3 |- 8 # 0
6	1, 2, 5	divcanap4i 8981	. 2 |- ( ( 6 x. 8 ) / 8 ) = 6
7	1, 2	mulcli 8227	. . . 4 |- ( 6 x. 8 ) e. CC
8		ax-1cn 8168	. . . 4 |- 1 e. CC
9		4p3e7 9330	. . . . . . 7 |- ( 4 + 3 ) = 7
10	9	eqcomi 2235	. . . . . 6 |- 7 = ( 4 + 3 )
11	10	oveq1i 6038	. . . . 5 |- ( 7 ^ 2 ) = ( ( 4 + 3 ) ^ 2 )
12		4cn 9263	. . . . . . 7 |- 4 e. CC
13		3cn 9260	. . . . . . 7 |- 3 e. CC
14	12, 13	binom2i 10956	. . . . . 6 |- ( ( 4 + 3 ) ^ 2 ) = ( ( ( 4 ^ 2 ) + ( 2 x. ( 4 x. 3 ) ) ) + ( 3 ^ 2 ) )
15		sq4e2t8 10945	. . . . . . . . . 10 |- ( 4 ^ 2 ) = ( 2 x. 8 )
16		2cn 9256	. . . . . . . . . . . . 13 |- 2 e. CC
17		4t2e8 9344	. . . . . . . . . . . . 13 |- ( 4 x. 2 ) = 8
18	12, 16, 17	mulcomli 8229	. . . . . . . . . . . 12 |- ( 2 x. 4 ) = 8
19	18	oveq1i 6038	. . . . . . . . . . 11 |- ( ( 2 x. 4 ) x. 3 ) = ( 8 x. 3 )
20	16, 12, 13	mulassi 8231	. . . . . . . . . . 11 |- ( ( 2 x. 4 ) x. 3 ) = ( 2 x. ( 4 x. 3 ) )
21	2, 13	mulcomi 8228	. . . . . . . . . . 11 |- ( 8 x. 3 ) = ( 3 x. 8 )
22	19, 20, 21	3eqtr3i 2260	. . . . . . . . . 10 |- ( 2 x. ( 4 x. 3 ) ) = ( 3 x. 8 )
23	15, 22	oveq12i 6040	. . . . . . . . 9 |- ( ( 4 ^ 2 ) + ( 2 x. ( 4 x. 3 ) ) ) = ( ( 2 x. 8 ) + ( 3 x. 8 ) )
24	16, 13, 2	adddiri 8233	. . . . . . . . 9 |- ( ( 2 + 3 ) x. 8 ) = ( ( 2 x. 8 ) + ( 3 x. 8 ) )
25		3p2e5 9327	. . . . . . . . . . 11 |- ( 3 + 2 ) = 5
26	13, 16, 25	addcomli 8366	. . . . . . . . . 10 |- ( 2 + 3 ) = 5
27	26	oveq1i 6038	. . . . . . . . 9 |- ( ( 2 + 3 ) x. 8 ) = ( 5 x. 8 )
28	23, 24, 27	3eqtr2i 2258	. . . . . . . 8 |- ( ( 4 ^ 2 ) + ( 2 x. ( 4 x. 3 ) ) ) = ( 5 x. 8 )
29		sq3 10944	. . . . . . . . 9 |- ( 3 ^ 2 ) = 9
30		df-9 9251	. . . . . . . . 9 |- 9 = ( 8 + 1 )
31	29, 30	eqtri 2252	. . . . . . . 8 |- ( 3 ^ 2 ) = ( 8 + 1 )
32	28, 31	oveq12i 6040	. . . . . . 7 |- ( ( ( 4 ^ 2 ) + ( 2 x. ( 4 x. 3 ) ) ) + ( 3 ^ 2 ) ) = ( ( 5 x. 8 ) + ( 8 + 1 ) )
33		5cn 9265	. . . . . . . . 9 |- 5 e. CC
34	33, 2	mulcli 8227	. . . . . . . 8 |- ( 5 x. 8 ) e. CC
35	34, 2, 8	addassi 8230	. . . . . . 7 |- ( ( ( 5 x. 8 ) + 8 ) + 1 ) = ( ( 5 x. 8 ) + ( 8 + 1 ) )
36		df-6 9248	. . . . . . . . . . 11 |- 6 = ( 5 + 1 )
37	36	oveq1i 6038	. . . . . . . . . 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 8248	. . . . . . . . . . 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 2252	. . . . . . . . 9 |- ( 6 x. 8 ) = ( ( 5 x. 8 ) + 8 )
43	42	eqcomi 2235	. . . . . . . 8 |- ( ( 5 x. 8 ) + 8 ) = ( 6 x. 8 )
44	43	oveq1i 6038	. . . . . . 7 |- ( ( ( 5 x. 8 ) + 8 ) + 1 ) = ( ( 6 x. 8 ) + 1 )
45	32, 35, 44	3eqtr2i 2258	. . . . . 6 |- ( ( ( 4 ^ 2 ) + ( 2 x. ( 4 x. 3 ) ) ) + ( 3 ^ 2 ) ) = ( ( 6 x. 8 ) + 1 )
46	14, 45	eqtri 2252	. . . . 5 |- ( ( 4 + 3 ) ^ 2 ) = ( ( 6 x. 8 ) + 1 )
47	11, 46	eqtri 2252	. . . 4 |- ( 7 ^ 2 ) = ( ( 6 x. 8 ) + 1 )
48	7, 8, 47	mvrraddi 8438	. . 3 |- ( ( 7 ^ 2 ) - 1 ) = ( 6 x. 8 )
49	48	oveq1i 6038	. 2 |- ( ( ( 7 ^ 2 ) - 1 ) / 8 ) = ( ( 6 x. 8 ) / 8 )
50		3t2e6 9342	. . 3 |- ( 3 x. 2 ) = 6
51	13, 16, 50	mulcomli 8229	. 2 |- ( 2 x. 3 ) = 6
52	6, 49, 51	3eqtr4i 2262	1 |- ( ( ( 7 ^ 2 ) - 1 ) / 8 ) = ( 2 x. 3 )

Syntax hints:   = wceq 1398   e. wcel 2202  (class class class)co 6028   CCcc 8073   1c1 8076   + caddc 8078   x. cmul 8080   - cmin 8392   / cdiv 8894   2c2 9236   3c3 9237   4c4 9238   5c5 9239   6c6 9240   7c7 9241   8c8 9242   9c9 9243   ^cexp 10846

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-mulrcl 8174  ax-addcom 8175  ax-mulcom 8176  ax-addass 8177  ax-mulass 8178  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-1rid 8182  ax-0id 8183  ax-rnegex 8184  ax-precex 8185  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-apti 8190  ax-pre-ltadd 8191  ax-pre-mulgt0 8192  ax-pre-mulext 8193

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 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-frec 6600  df-pnf 8258  df-mnf 8259  df-xr 8260  df-ltxr 8261  df-le 8262  df-sub 8394  df-neg 8395  df-reap 8797  df-ap 8804  df-div 8895  df-inn 9186  df-2 9244  df-3 9245  df-4 9246  df-5 9247  df-6 9248  df-7 9249  df-8 9250  df-9 9251  df-n0 9445  df-z 9524  df-uz 9800  df-seqfrec 10756  df-exp 10847

This theorem is referenced by:  2lgsoddprmlem3  15913