Theorem pwbdvdseulemle 12962

Description: Lemma for pwbdvdseu 12963. Powers of a base which do and do not divide a natural number. (Contributed by Jim Kingdon, 17-Nov-2021.) (Revised by Jim Kingdon, 18-Aug-2026.)

Hypotheses

Ref	Expression
pwbdvdseulemle.n	|- ( ph -> N e. NN )
pwbdvdseulemle.a	|- ( ph -> A e. NN0 )
pwbdvdseulemle.b	|- ( ph -> B e. NN0 )
pwbdvdseulemle.p	|- ( ph -> P e. NN )
pwbdvdseulemle.2a	|- ( ph -> ( P ^ A ) || N )
pwbdvdseulemle.n2b	|- ( ph -> -. ( P ^ ( B + 1 ) ) || N )

Assertion

Ref	Expression
pwbdvdseulemle	|- ( ph -> A <_ B )

Proof of Theorem pwbdvdseulemle

Step	Hyp	Ref	Expression
1		pwbdvdseulemle.a	. . 3 |- ( ph -> A e. NN0 )
2	1	nn0red 9625	. 2 |- ( ph -> A e. RR )
3		pwbdvdseulemle.b	. . 3 |- ( ph -> B e. NN0 )
4	3	nn0red 9625	. 2 |- ( ph -> B e. RR )
5		pwbdvdseulemle.n2b	. . 3 |- ( ph -> -. ( P ^ ( B + 1 ) ) || N )
6		pwbdvdseulemle.p	. . . . . . . 8 |- ( ph -> P e. NN )
7	6	nncnd 9320	. . . . . . 7 |- ( ph -> P e. CC )
8	7	adantr 276	. . . . . 6 |- ( ( ph /\ B < A ) -> P e. CC )
9	3	adantr 276	. . . . . . . 8 |- ( ( ph /\ B < A ) -> B e. NN0 )
10		peano2nn0 9607	. . . . . . . 8 |- ( B e. NN0 -> ( B + 1 ) e. NN0 )
11	9, 10	syl 14	. . . . . . 7 |- ( ( ph /\ B < A ) -> ( B + 1 ) e. NN0 )
12	1	adantr 276	. . . . . . 7 |- ( ( ph /\ B < A ) -> A e. NN0 )
13		simpr 110	. . . . . . . 8 |- ( ( ph /\ B < A ) -> B < A )
14		nn0ltp1le 9711	. . . . . . . . 9 |- ( ( B e. NN0 /\ A e. NN0 ) -> ( B < A <-> ( B + 1 ) <_ A ) )
15	3, 12, 14	syl2an2r 603	. . . . . . . 8 |- ( ( ph /\ B < A ) -> ( B < A <-> ( B + 1 ) <_ A ) )
16	13, 15	mpbid 147	. . . . . . 7 |- ( ( ph /\ B < A ) -> ( B + 1 ) <_ A )
17		nn0sub2 9722	. . . . . . 7 |- ( ( ( B + 1 ) e. NN0 /\ A e. NN0 /\ ( B + 1 ) <_ A ) -> ( A - ( B + 1 ) ) e. NN0 )
18	11, 12, 16, 17	syl3anc 1278	. . . . . 6 |- ( ( ph /\ B < A ) -> ( A - ( B + 1 ) ) e. NN0 )
19	8, 18, 11	expaddd 11126	. . . . 5 |- ( ( ph /\ B < A ) -> ( P ^ ( ( B + 1 ) + ( A - ( B + 1 ) ) ) ) = ( ( P ^ ( B + 1 ) ) x. ( P ^ ( A - ( B + 1 ) ) ) ) )
20	11	nn0cnd 9626	. . . . . . . 8 |- ( ( ph /\ B < A ) -> ( B + 1 ) e. CC )
21	12	nn0cnd 9626	. . . . . . . 8 |- ( ( ph /\ B < A ) -> A e. CC )
22	20, 21	pncan3d 8641	. . . . . . 7 |- ( ( ph /\ B < A ) -> ( ( B + 1 ) + ( A - ( B + 1 ) ) ) = A )
23	22	oveq2d 6101	. . . . . 6 |- ( ( ph /\ B < A ) -> ( P ^ ( ( B + 1 ) + ( A - ( B + 1 ) ) ) ) = ( P ^ A ) )
24		pwbdvdseulemle.2a	. . . . . . 7 |- ( ph -> ( P ^ A ) || N )
25	24	adantr 276	. . . . . 6 |- ( ( ph /\ B < A ) -> ( P ^ A ) || N )
26	23, 25	eqbrtrd 4152	. . . . 5 |- ( ( ph /\ B < A ) -> ( P ^ ( ( B + 1 ) + ( A - ( B + 1 ) ) ) ) || N )
27	19, 26	eqbrtrrd 4154	. . . 4 |- ( ( ph /\ B < A ) -> ( ( P ^ ( B + 1 ) ) x. ( P ^ ( A - ( B + 1 ) ) ) ) || N )
28	6	adantr 276	. . . . . . 7 |- ( ( ph /\ B < A ) -> P e. NN )
29	28, 11	nnexpcld 11146	. . . . . 6 |- ( ( ph /\ B < A ) -> ( P ^ ( B + 1 ) ) e. NN )
30	29	nnzd 9771	. . . . 5 |- ( ( ph /\ B < A ) -> ( P ^ ( B + 1 ) ) e. ZZ )
31	28, 18	nnexpcld 11146	. . . . . 6 |- ( ( ph /\ B < A ) -> ( P ^ ( A - ( B + 1 ) ) ) e. NN )
32	31	nnzd 9771	. . . . 5 |- ( ( ph /\ B < A ) -> ( P ^ ( A - ( B + 1 ) ) ) e. ZZ )
33		pwbdvdseulemle.n	. . . . . . 7 |- ( ph -> N e. NN )
34	33	adantr 276	. . . . . 6 |- ( ( ph /\ B < A ) -> N e. NN )
35	34	nnzd 9771	. . . . 5 |- ( ( ph /\ B < A ) -> N e. ZZ )
36		muldvds1 12599	. . . . 5 |- ( ( ( P ^ ( B + 1 ) ) e. ZZ /\ ( P ^ ( A - ( B + 1 ) ) ) e. ZZ /\ N e. ZZ ) -> ( ( ( P ^ ( B + 1 ) ) x. ( P ^ ( A - ( B + 1 ) ) ) ) || N -> ( P ^ ( B + 1 ) ) || N ) )
37	30, 32, 35, 36	syl3anc 1278	. . . 4 |- ( ( ph /\ B < A ) -> ( ( ( P ^ ( B + 1 ) ) x. ( P ^ ( A - ( B + 1 ) ) ) ) || N -> ( P ^ ( B + 1 ) ) || N ) )
38	27, 37	mpd 13	. . 3 |- ( ( ph /\ B < A ) -> ( P ^ ( B + 1 ) ) || N )
39	5, 38	mtand 675	. 2 |- ( ph -> -. B < A )
40	2, 4, 39	nltled 8448	1 |- ( ph -> A <_ B )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   e. wcel 2209   class class class wbr 4130  (class class class)co 6085   CCcc 8177   1c1 8180   + caddc 8182   x. cmul 8184   < clt 8360   <_ cle 8361   - cmin 8498   NNcn 9306   NN0cn0 9567   ZZcz 9648   ^cexp 10988   || cdvds 12570

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 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297

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 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  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-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8500  df-neg 8501  df-reap 8905  df-ap 8912  df-div 9005  df-inn 9307  df-n0 9568  df-z 9649  df-uz 9931  df-seqfrec 10898  df-exp 10989  df-dvds 12571

This theorem is used by:  pwbdvdseu  12963