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Theorem pell14qrexpclnn0 37430
Description: Lemma for pell14qrexpcl 37431. (Contributed by Stefan O'Rear, 18-Sep-2014.)
Assertion
Ref Expression
pell14qrexpclnn0 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) /\ B e. NN0 ) -> ( A ^ B ) e. (Pell14QR ` D ) )
Proof of Theorem pell14qrexpclnn0
Dummy variables a b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 6658 . . . . . 6 |- ( a = 0 -> ( A ^ a ) = ( A ^ 0 ) )
21eleq1d 2686 . . . . 5 |- ( a = 0 -> ( ( A ^ a ) e. (Pell14QR ` D ) <-> ( A ^ 0 ) e. (Pell14QR ` D ) ) )
32imbi2d 330 . . . 4 |- ( a = 0 -> ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) -> ( A ^ a ) e. (Pell14QR ` D ) ) <-> ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) -> ( A ^ 0 ) e. (Pell14QR ` D ) ) ) )
4 oveq2 6658 . . . . . 6 |- ( a = b -> ( A ^ a ) = ( A ^ b ) )
54eleq1d 2686 . . . . 5 |- ( a = b -> ( ( A ^ a ) e. (Pell14QR ` D ) <-> ( A ^ b ) e. (Pell14QR ` D ) ) )
65imbi2d 330 . . . 4 |- ( a = b -> ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) -> ( A ^ a ) e. (Pell14QR ` D ) ) <-> ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) -> ( A ^ b ) e. (Pell14QR ` D ) ) ) )
7 oveq2 6658 . . . . . 6 |- ( a = ( b + 1 ) -> ( A ^ a ) = ( A ^ ( b + 1 ) ) )
87eleq1d 2686 . . . . 5 |- ( a = ( b + 1 ) -> ( ( A ^ a ) e. (Pell14QR ` D ) <-> ( A ^ ( b + 1 ) ) e. (Pell14QR ` D ) ) )
98imbi2d 330 . . . 4 |- ( a = ( b + 1 ) -> ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) -> ( A ^ a ) e. (Pell14QR ` D ) ) <-> ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) -> ( A ^ ( b + 1 ) ) e. (Pell14QR ` D ) ) ) )
10 oveq2 6658 . . . . . 6 |- ( a = B -> ( A ^ a ) = ( A ^ B ) )
1110eleq1d 2686 . . . . 5 |- ( a = B -> ( ( A ^ a ) e. (Pell14QR ` D ) <-> ( A ^ B ) e. (Pell14QR ` D ) ) )
1211imbi2d 330 . . . 4 |- ( a = B -> ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) -> ( A ^ a ) e. (Pell14QR ` D ) ) <-> ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) -> ( A ^ B ) e. (Pell14QR ` D ) ) ) )
13 pell14qrre 37421 . . . . . . . 8 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) -> A e. RR )
1413recnd 10068 . . . . . . 7 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) -> A e. CC )
1514exp0d 13002 . . . . . 6 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) -> ( A ^ 0 ) = 1 )
16 pell14qrne0 37422 . . . . . . 7 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) -> A =/= 0 )
1714, 16dividd 10799 . . . . . 6 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) -> ( A / A ) = 1 )
1815, 17eqtr4d 2659 . . . . 5 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) -> ( A ^ 0 ) = ( A / A ) )
19 pell14qrdivcl 37429 . . . . . 6 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) /\ A e. (Pell14QR ` D ) ) -> ( A / A ) e. (Pell14QR ` D ) )
20193anidm23 1385 . . . . 5 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) -> ( A / A ) e. (Pell14QR ` D ) )
2118, 20eqeltrd 2701 . . . 4 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) -> ( A ^ 0 ) e. (Pell14QR ` D ) )
22143ad2ant2 1083 . . . . . . . 8 |- ( ( b e. NN0 /\ ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) /\ ( A ^ b ) e. (Pell14QR ` D ) ) -> A e. CC )
23 simp1 1061 . . . . . . . 8 |- ( ( b e. NN0 /\ ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) /\ ( A ^ b ) e. (Pell14QR ` D ) ) -> b e. NN0 )
2422, 23expp1d 13009 . . . . . . 7 |- ( ( b e. NN0 /\ ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) /\ ( A ^ b ) e. (Pell14QR ` D ) ) -> ( A ^ ( b + 1 ) ) = ( ( A ^ b ) x. A ) )
25 simp2l 1087 . . . . . . . 8 |- ( ( b e. NN0 /\ ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) /\ ( A ^ b ) e. (Pell14QR ` D ) ) -> D e. ( NN \ ◻NN ) )
26 simp3 1063 . . . . . . . 8 |- ( ( b e. NN0 /\ ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) /\ ( A ^ b ) e. (Pell14QR ` D ) ) -> ( A ^ b ) e. (Pell14QR ` D ) )
27 simp2r 1088 . . . . . . . 8 |- ( ( b e. NN0 /\ ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) /\ ( A ^ b ) e. (Pell14QR ` D ) ) -> A e. (Pell14QR ` D ) )
28 pell14qrmulcl 37427 . . . . . . . 8 |- ( ( D e. ( NN \ ◻NN ) /\ ( A ^ b ) e. (Pell14QR ` D ) /\ A e. (Pell14QR ` D ) ) -> ( ( A ^ b ) x. A ) e. (Pell14QR ` D ) )
2925, 26, 27, 28syl3anc 1326 . . . . . . 7 |- ( ( b e. NN0 /\ ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) /\ ( A ^ b ) e. (Pell14QR ` D ) ) -> ( ( A ^ b ) x. A ) e. (Pell14QR ` D ) )
3024, 29eqeltrd 2701 . . . . . 6 |- ( ( b e. NN0 /\ ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) /\ ( A ^ b ) e. (Pell14QR ` D ) ) -> ( A ^ ( b + 1 ) ) e. (Pell14QR ` D ) )
31303exp 1264 . . . . 5 |- ( b e. NN0 -> ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) -> ( ( A ^ b ) e. (Pell14QR ` D ) -> ( A ^ ( b + 1 ) ) e. (Pell14QR ` D ) ) ) )
3231a2d 29 . . . 4 |- ( b e. NN0 -> ( ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) -> ( A ^ b ) e. (Pell14QR ` D ) ) -> ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) -> ( A ^ ( b + 1 ) ) e. (Pell14QR ` D ) ) ) )
333, 6, 9, 12, 21, 32nn0ind 11472 . . 3 |- ( B e. NN0 -> ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) ) -> ( A ^ B ) e. (Pell14QR ` D ) ) )
3433expdcom 455 . 2 |- ( D e. ( NN \ ◻NN ) -> ( A e. (Pell14QR ` D ) -> ( B e. NN0 -> ( A ^ B ) e. (Pell14QR ` D ) ) ) )
35343imp 1256 1 |- ( ( D e. ( NN \ ◻NN ) /\ A e. (Pell14QR ` D ) /\ B e. NN0 ) -> ( A ^ B ) e. (Pell14QR ` D ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   \ cdif 3571   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   / cdiv 10684   NNcn 11020   NN0cn0 11292   ^cexp 12860  ◻NNcsquarenn 37400  Pell14QRcpell14qr 37403
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-pell14qr 37407  df-pell1234qr 37408
This theorem is referenced by:  pell14qrexpcl  37431
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