Theorem dpexpp1 29616

Description: Add one zero to the mantisse, and a one to the exponent in a scientific notation. (Contributed by Thierry Arnoux, 16-Dec-2021.)

Hypotheses

Ref	Expression
dpexpp1.a	|- A e. NN0
dpexpp1.b	|- B e. RR+
dpexpp1.1	|- ( P + 1 ) = Q
dpexpp1.p	|- P e. ZZ
dpexpp1.q	|- Q e. ZZ

Assertion

Ref	Expression
dpexpp1	|- ( ( A period B ) x. (; 1 0 ^ P ) ) = ( ( 0 period_ A B ) x. (; 1 0 ^ Q ) )

Proof of Theorem dpexpp1

Step	Hyp	Ref	Expression
1		0re 10040	. . . . . 6 |- 0 e. RR
2		10pos 11515	. . . . . 6 |- 0 < ; 1 0
3	1, 2	gtneii 10149	. . . . 5 |- ; 1 0 =/= 0
4		dpexpp1.a	. . . . . . . . . 10 |- A e. NN0
5		dpexpp1.b	. . . . . . . . . 10 |- B e. RR+
6	4, 5	rpdp2cl 29589	. . . . . . . . 9 |- _ A B e. RR+
7		rpre 11839	. . . . . . . . 9 |- (_ A B e. RR+ -> _ A B e. RR )
8	6, 7	ax-mp 5	. . . . . . . 8 |- _ A B e. RR
9	8	recni 10052	. . . . . . 7 |- _ A B e. CC
10		10re 11517	. . . . . . . . . . 11 |- ; 1 0 e. RR
11	10, 2	pm3.2i 471	. . . . . . . . . 10 |- (; 1 0 e. RR /\ 0 < ; 1 0 )
12		elrp 11834	. . . . . . . . . 10 |- (; 1 0 e. RR+ <-> (; 1 0 e. RR /\ 0 < ; 1 0 ) )
13	11, 12	mpbir 221	. . . . . . . . 9 |- ; 1 0 e. RR+
14		dpexpp1.p	. . . . . . . . 9 |- P e. ZZ
15		rpexpcl 12879	. . . . . . . . 9 |- ( (; 1 0 e. RR+ /\ P e. ZZ ) -> (; 1 0 ^ P ) e. RR+ )
16	13, 14, 15	mp2an 708	. . . . . . . 8 |- (; 1 0 ^ P ) e. RR+
17		rpcn 11841	. . . . . . . 8 |- ( (; 1 0 ^ P ) e. RR+ -> (; 1 0 ^ P ) e. CC )
18	16, 17	ax-mp 5	. . . . . . 7 |- (; 1 0 ^ P ) e. CC
19	9, 18	mulcli 10045	. . . . . 6 |- (_ A B x. (; 1 0 ^ P ) ) e. CC
20		10nn0 11516	. . . . . . 7 |- ; 1 0 e. NN0
21	20	nn0cni 11304	. . . . . 6 |- ; 1 0 e. CC
22	19, 21	divcan1zi 10761	. . . . 5 |- (; 1 0 =/= 0 -> ( ( (_ A B x. (; 1 0 ^ P ) ) / ; 1 0 ) x. ; 1 0 ) = (_ A B x. (; 1 0 ^ P ) ) )
23	3, 22	ax-mp 5	. . . 4 |- ( ( (_ A B x. (; 1 0 ^ P ) ) / ; 1 0 ) x. ; 1 0 ) = (_ A B x. (; 1 0 ^ P ) )
24	21, 3	pm3.2i 471	. . . . . 6 |- (; 1 0 e. CC /\ ; 1 0 =/= 0 )
25		div23 10704	. . . . . 6 |- ( (_ A B e. CC /\ (; 1 0 ^ P ) e. CC /\ (; 1 0 e. CC /\ ; 1 0 =/= 0 ) ) -> ( (_ A B x. (; 1 0 ^ P ) ) / ; 1 0 ) = ( (_ A B / ; 1 0 ) x. (; 1 0 ^ P ) ) )
26	9, 18, 24, 25	mp3an 1424	. . . . 5 |- ( (_ A B x. (; 1 0 ^ P ) ) / ; 1 0 ) = ( (_ A B / ; 1 0 ) x. (; 1 0 ^ P ) )
27	26	oveq1i 6660	. . . 4 |- ( ( (_ A B x. (; 1 0 ^ P ) ) / ; 1 0 ) x. ; 1 0 ) = ( ( (_ A B / ; 1 0 ) x. (; 1 0 ^ P ) ) x. ; 1 0 )
28	23, 27	eqtr3i 2646	. . 3 |- (_ A B x. (; 1 0 ^ P ) ) = ( ( (_ A B / ; 1 0 ) x. (; 1 0 ^ P ) ) x. ; 1 0 )
29	9, 21, 3	divcli 10767	. . . 4 |- (_ A B / ; 1 0 ) e. CC
30	29, 18, 21	mulassi 10049	. . 3 |- ( ( (_ A B / ; 1 0 ) x. (; 1 0 ^ P ) ) x. ; 1 0 ) = ( (_ A B / ; 1 0 ) x. ( (; 1 0 ^ P ) x. ; 1 0 ) )
31		expp1z 12909	. . . . . 6 |- ( (; 1 0 e. CC /\ ; 1 0 =/= 0 /\ P e. ZZ ) -> (; 1 0 ^ ( P + 1 ) ) = ( (; 1 0 ^ P ) x. ; 1 0 ) )
32	21, 3, 14, 31	mp3an 1424	. . . . 5 |- (; 1 0 ^ ( P + 1 ) ) = ( (; 1 0 ^ P ) x. ; 1 0 )
33		dpexpp1.1	. . . . . 6 |- ( P + 1 ) = Q
34	33	oveq2i 6661	. . . . 5 |- (; 1 0 ^ ( P + 1 ) ) = (; 1 0 ^ Q )
35	32, 34	eqtr3i 2646	. . . 4 |- ( (; 1 0 ^ P ) x. ; 1 0 ) = (; 1 0 ^ Q )
36	35	oveq2i 6661	. . 3 |- ( (_ A B / ; 1 0 ) x. ( (; 1 0 ^ P ) x. ; 1 0 ) ) = ( (_ A B / ; 1 0 ) x. (; 1 0 ^ Q ) )
37	28, 30, 36	3eqtri 2648	. 2 |- (_ A B x. (; 1 0 ^ P ) ) = ( (_ A B / ; 1 0 ) x. (; 1 0 ^ Q ) )
38	4, 5	dpval3rp 29608	. . 3 |- ( A period B ) = _ A B
39	38	oveq1i 6660	. 2 |- ( ( A period B ) x. (; 1 0 ^ P ) ) = (_ A B x. (; 1 0 ^ P ) )
40		0nn0 11307	. . . . 5 |- 0 e. NN0
41	40, 6	dpval3rp 29608	. . . 4 |- ( 0 period_ A B ) = _ 0_ A B
42	6	dp20h 29586	. . . 4 |- _ 0_ A B = (_ A B / ; 1 0 )
43	41, 42	eqtri 2644	. . 3 |- ( 0 period_ A B ) = (_ A B / ; 1 0 )
44	43	oveq1i 6660	. 2 |- ( ( 0 period_ A B ) x. (; 1 0 ^ Q ) ) = ( (_ A B / ; 1 0 ) x. (; 1 0 ^ Q ) )
45	37, 39, 44	3eqtr4i 2654	1 |- ( ( A period B ) x. (; 1 0 ^ P ) ) = ( ( 0 period_ A B ) x. (; 1 0 ^ Q ) )

Syntax hints:   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   class class class wbr 4653  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   < clt 10074   / cdiv 10684   NN0cn0 11292   ZZcz 11377  ;cdc 11493   RR+crp 11832   ^cexp 12860  _cdp2 29577   periodcdp 29595

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

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-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-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-rp 11833  df-seq 12802  df-exp 12861  df-dp2 29578  df-dp 29596

This theorem is referenced by:  0dp2dp  29617  hgt750lemd  30726  hgt750lem  30729