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Theorem 2lgslem3d1 16341
Description: Lemma 4 for 2lgslem3 16342. (Contributed by AV, 15-Jul-2021.)
Hypothesis
Ref Expression
2lgslem2.n |- N = ( ( ( P - 1 ) / 2 ) - ( |_ ` ( P / 4 ) ) )
Assertion
Ref Expression
2lgslem3d1 |- ( ( P e. NN /\ ( P mod 8 ) = 7 ) -> ( N mod 2 ) = 0 )
Proof of Theorem 2lgslem3d1
Dummy variable k is distinct from all other variables.
StepHypRef Expression
1 nnnn0 9575 . . . 4 |- ( P e. NN -> P e. NN0 )
2 8nn 9477 . . . . 5 |- 8 e. NN
3 nnq 10043 . . . . 5 |- ( 8 e. NN -> 8 e. QQ )
42, 3mp1i 10 . . . 4 |- ( P e. NN -> 8 e. QQ )
52nngt0i 9337 . . . . 5 |- 0 < 8
65a1i 9 . . . 4 |- ( P e. NN -> 0 < 8 )
7 modqmuladdnn0 10819 . . . 4 |- ( ( P e. NN0 /\ 8 e. QQ /\ 0 < 8 ) -> ( ( P mod 8 ) = 7 -> E. k e. NN0 P = ( ( k x. 8 ) + 7 ) ) )
81, 4, 6, 7syl3anc 1278 . . 3 |- ( P e. NN -> ( ( P mod 8 ) = 7 -> E. k e. NN0 P = ( ( k x. 8 ) + 7 ) ) )
9 simpr 110 . . . . 5 |- ( ( P e. NN /\ k e. NN0 ) -> k e. NN0 )
10 nn0cn 9578 . . . . . . . . . . 11 |- ( k e. NN0 -> k e. CC )
11 8cn 9393 . . . . . . . . . . . 12 |- 8 e. CC
1211a1i 9 . . . . . . . . . . 11 |- ( k e. NN0 -> 8 e. CC )
1310, 12mulcomd 8348 . . . . . . . . . 10 |- ( k e. NN0 -> ( k x. 8 ) = ( 8 x. k ) )
1413adantl 277 . . . . . . . . 9 |- ( ( P e. NN /\ k e. NN0 ) -> ( k x. 8 ) = ( 8 x. k ) )
1514oveq1d 6100 . . . . . . . 8 |- ( ( P e. NN /\ k e. NN0 ) -> ( ( k x. 8 ) + 7 ) = ( ( 8 x. k ) + 7 ) )
1615eqeq2d 2250 . . . . . . 7 |- ( ( P e. NN /\ k e. NN0 ) -> ( P = ( ( k x. 8 ) + 7 ) <-> P = ( ( 8 x. k ) + 7 ) ) )
1716biimpa 296 . . . . . 6 |- ( ( ( P e. NN /\ k e. NN0 ) /\ P = ( ( k x. 8 ) + 7 ) ) -> P = ( ( 8 x. k ) + 7 ) )
18 2lgslem2.n . . . . . . 7 |- N = ( ( ( P - 1 ) / 2 ) - ( |_ ` ( P / 4 ) ) )
19182lgslem3d 16337 . . . . . 6 |- ( ( k e. NN0 /\ P = ( ( 8 x. k ) + 7 ) ) -> N = ( ( 2 x. k ) + 2 ) )
209, 17, 19syl2an2r 603 . . . . 5 |- ( ( ( P e. NN /\ k e. NN0 ) /\ P = ( ( k x. 8 ) + 7 ) ) -> N = ( ( 2 x. k ) + 2 ) )
21 oveq1 6092 . . . . . 6 |- ( N = ( ( 2 x. k ) + 2 ) -> ( N mod 2 ) = ( ( ( 2 x. k ) + 2 ) mod 2 ) )
22 2t1e2 9461 . . . . . . . . . . . 12 |- ( 2 x. 1 ) = 2
2322eqcomi 2242 . . . . . . . . . . 11 |- 2 = ( 2 x. 1 )
2423a1i 9 . . . . . . . . . 10 |- ( k e. NN0 -> 2 = ( 2 x. 1 ) )
2524oveq2d 6101 . . . . . . . . 9 |- ( k e. NN0 -> ( ( 2 x. k ) + 2 ) = ( ( 2 x. k ) + ( 2 x. 1 ) ) )
26 2cnd 9380 . . . . . . . . . 10 |- ( k e. NN0 -> 2 e. CC )
27 1cnd 8343 . . . . . . . . . 10 |- ( k e. NN0 -> 1 e. CC )
28 adddi 8312 . . . . . . . . . . 11 |- ( ( 2 e. CC /\ k e. CC /\ 1 e. CC ) -> ( 2 x. ( k + 1 ) ) = ( ( 2 x. k ) + ( 2 x. 1 ) ) )
2928eqcomd 2244 . . . . . . . . . 10 |- ( ( 2 e. CC /\ k e. CC /\ 1 e. CC ) -> ( ( 2 x. k ) + ( 2 x. 1 ) ) = ( 2 x. ( k + 1 ) ) )
3026, 10, 27, 29syl3anc 1278 . . . . . . . . 9 |- ( k e. NN0 -> ( ( 2 x. k ) + ( 2 x. 1 ) ) = ( 2 x. ( k + 1 ) ) )
3110, 27addcld 8346 . . . . . . . . . 10 |- ( k e. NN0 -> ( k + 1 ) e. CC )
3226, 31mulcomd 8348 . . . . . . . . 9 |- ( k e. NN0 -> ( 2 x. ( k + 1 ) ) = ( ( k + 1 ) x. 2 ) )
3325, 30, 323eqtrd 2275 . . . . . . . 8 |- ( k e. NN0 -> ( ( 2 x. k ) + 2 ) = ( ( k + 1 ) x. 2 ) )
3433oveq1d 6100 . . . . . . 7 |- ( k e. NN0 -> ( ( ( 2 x. k ) + 2 ) mod 2 ) = ( ( ( k + 1 ) x. 2 ) mod 2 ) )
35 peano2nn0 9608 . . . . . . . . 9 |- ( k e. NN0 -> ( k + 1 ) e. NN0 )
3635nn0zd 9771 . . . . . . . 8 |- ( k e. NN0 -> ( k + 1 ) e. ZZ )
37 2nn 9471 . . . . . . . . 9 |- 2 e. NN
38 nnq 10043 . . . . . . . . 9 |- ( 2 e. NN -> 2 e. QQ )
3937, 38mp1i 10 . . . . . . . 8 |- ( k e. NN0 -> 2 e. QQ )
4037nngt0i 9337 . . . . . . . . 9 |- 0 < 2
4140a1i 9 . . . . . . . 8 |- ( k e. NN0 -> 0 < 2 )
42 mulqmod0 10781 . . . . . . . 8 |- ( ( ( k + 1 ) e. ZZ /\ 2 e. QQ /\ 0 < 2 ) -> ( ( ( k + 1 ) x. 2 ) mod 2 ) = 0 )
4336, 39, 41, 42syl3anc 1278 . . . . . . 7 |- ( k e. NN0 -> ( ( ( k + 1 ) x. 2 ) mod 2 ) = 0 )
4434, 43eqtrd 2271 . . . . . 6 |- ( k e. NN0 -> ( ( ( 2 x. k ) + 2 ) mod 2 ) = 0 )
4521, 44sylan9eqr 2293 . . . . 5 |- ( ( k e. NN0 /\ N = ( ( 2 x. k ) + 2 ) ) -> ( N mod 2 ) = 0 )
469, 20, 45syl2an2r 603 . . . 4 |- ( ( ( P e. NN /\ k e. NN0 ) /\ P = ( ( k x. 8 ) + 7 ) ) -> ( N mod 2 ) = 0 )
4746rexlimdva2 2671 . . 3 |- ( P e. NN -> ( E. k e. NN0 P = ( ( k x. 8 ) + 7 ) -> ( N mod 2 ) = 0 ) )
488, 47syld 45 . 2 |- ( P e. NN -> ( ( P mod 8 ) = 7 -> ( N mod 2 ) = 0 ) )
4948imp 124 1 |- ( ( P e. NN /\ ( P mod 8 ) = 7 ) -> ( N mod 2 ) = 0 )
Colors of variables:    wff set class
This proof depends on syntax axioms:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209   E.wrex 2529   class class class wbr 4130   ` cfv 5377  (class class class)co 6085   CCcc 8178   0cc0 8180   1c1 8181   + caddc 8183   x. cmul 8185   < clt 8361   - cmin 8499   / cdiv 9005   NNcn 9307   2c2 9358   4c4 9360   7c7 9363   8c8 9364   NN0cn0 9568   ZZcz 9649   QQcq 10029   |_cfl 10714   mod cmo 10773
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-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8271  ax-resscn 8272  ax-1cn 8273  ax-1re 8274  ax-icn 8275  ax-addcl 8276  ax-addrcl 8277  ax-mulcl 8278  ax-mulrcl 8279  ax-addcom 8280  ax-mulcom 8281  ax-addass 8282  ax-mulass 8283  ax-distr 8284  ax-i2m1 8285  ax-0lt1 8286  ax-1rid 8287  ax-0id 8288  ax-rnegex 8289  ax-precex 8290  ax-cnre 8291  ax-pre-ltirr 8292  ax-pre-ltwlin 8293  ax-pre-lttrn 8294  ax-pre-apti 8295  ax-pre-ltadd 8296  ax-pre-mulgt0 8297  ax-pre-mulext 8298  ax-arch 8299
This proof depends on definitions:  df-bi 117  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-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-po 4441  df-iso 4442  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-pnf 8363  df-mnf 8364  df-xr 8365  df-ltxr 8366  df-le 8367  df-sub 8501  df-neg 8502  df-reap 8906  df-ap 8913  df-div 9006  df-inn 9308  df-2 9366  df-3 9367  df-4 9368  df-5 9369  df-6 9370  df-7 9371  df-8 9372  df-n0 9569  df-z 9650  df-q 10030  df-rp 10066  df-ico 10307  df-fl 10716  df-mod 10774
This theorem is used by:  2lgslem3  16342
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