Theorem o1dif 14360

Description: If the difference of two functions is eventually bounded, eventual boundedness of either one implies the other. (Contributed by Mario Carneiro, 26-May-2016.)

Hypotheses

Ref	Expression
o1dif.1	|- ( ( ph /\ x e. A ) -> B e. CC )
o1dif.2	|- ( ( ph /\ x e. A ) -> C e. CC )
o1dif.3	|- ( ph -> ( x e. A |-> ( B - C ) ) e. O(1) )

Assertion

Ref	Expression
o1dif	|- ( ph -> ( ( x e. A |-> B ) e. O(1) <-> ( x e. A |-> C ) e. O(1) ) )

Distinct variable groups:   x, A   ph, x

Allowed substitution hints:   B( x)   C( x)

Proof of Theorem o1dif

Step	Hyp	Ref	Expression
1		o1dif.3	. . . 4 |- ( ph -> ( x e. A |-> ( B - C ) ) e. O(1) )
2		o1sub 14346	. . . . 5 |- ( ( ( x e. A |-> B ) e. O(1) /\ ( x e. A |-> ( B - C ) ) e. O(1) ) -> ( ( x e. A |-> B ) oF - ( x e. A |-> ( B - C ) ) ) e. O(1) )
3	2	expcom 451	. . . 4 |- ( ( x e. A |-> ( B - C ) ) e. O(1) -> ( ( x e. A |-> B ) e. O(1) -> ( ( x e. A |-> B ) oF - ( x e. A |-> ( B - C ) ) ) e. O(1) ) )
4	1, 3	syl 17	. . 3 |- ( ph -> ( ( x e. A |-> B ) e. O(1) -> ( ( x e. A |-> B ) oF - ( x e. A |-> ( B - C ) ) ) e. O(1) ) )
5		o1dif.1	. . . . . . . . . . 11 |- ( ( ph /\ x e. A ) -> B e. CC )
6		o1dif.2	. . . . . . . . . . 11 |- ( ( ph /\ x e. A ) -> C e. CC )
7	5, 6	subcld 10392	. . . . . . . . . 10 |- ( ( ph /\ x e. A ) -> ( B - C ) e. CC )
8	7	ralrimiva 2966	. . . . . . . . 9 |- ( ph -> A. x e. A ( B - C ) e. CC )
9		dmmptg 5632	. . . . . . . . 9 |- ( A. x e. A ( B - C ) e. CC -> dom ( x e. A |-> ( B - C ) ) = A )
10	8, 9	syl 17	. . . . . . . 8 |- ( ph -> dom ( x e. A |-> ( B - C ) ) = A )
11		o1dm 14261	. . . . . . . . 9 |- ( ( x e. A |-> ( B - C ) ) e. O(1) -> dom ( x e. A |-> ( B - C ) ) C_ RR )
12	1, 11	syl 17	. . . . . . . 8 |- ( ph -> dom ( x e. A |-> ( B - C ) ) C_ RR )
13	10, 12	eqsstr3d 3640	. . . . . . 7 |- ( ph -> A C_ RR )
14		reex 10027	. . . . . . . 8 |- RR e. _V
15	14	ssex 4802	. . . . . . 7 |- ( A C_ RR -> A e. _V )
16	13, 15	syl 17	. . . . . 6 |- ( ph -> A e. _V )
17		eqidd 2623	. . . . . 6 |- ( ph -> ( x e. A |-> B ) = ( x e. A |-> B ) )
18		eqidd 2623	. . . . . 6 |- ( ph -> ( x e. A |-> ( B - C ) ) = ( x e. A |-> ( B - C ) ) )
19	16, 5, 7, 17, 18	offval2 6914	. . . . 5 |- ( ph -> ( ( x e. A |-> B ) oF - ( x e. A |-> ( B - C ) ) ) = ( x e. A |-> ( B - ( B - C ) ) ) )
20	5, 6	nncand 10397	. . . . . 6 |- ( ( ph /\ x e. A ) -> ( B - ( B - C ) ) = C )
21	20	mpteq2dva 4744	. . . . 5 |- ( ph -> ( x e. A |-> ( B - ( B - C ) ) ) = ( x e. A |-> C ) )
22	19, 21	eqtrd 2656	. . . 4 |- ( ph -> ( ( x e. A |-> B ) oF - ( x e. A |-> ( B - C ) ) ) = ( x e. A |-> C ) )
23	22	eleq1d 2686	. . 3 |- ( ph -> ( ( ( x e. A |-> B ) oF - ( x e. A |-> ( B - C ) ) ) e. O(1) <-> ( x e. A |-> C ) e. O(1) ) )
24	4, 23	sylibd 229	. 2 |- ( ph -> ( ( x e. A |-> B ) e. O(1) -> ( x e. A |-> C ) e. O(1) ) )
25		o1add 14344	. . . . 5 |- ( ( ( x e. A |-> ( B - C ) ) e. O(1) /\ ( x e. A |-> C ) e. O(1) ) -> ( ( x e. A |-> ( B - C ) ) oF + ( x e. A |-> C ) ) e. O(1) )
26	25	ex 450	. . . 4 |- ( ( x e. A |-> ( B - C ) ) e. O(1) -> ( ( x e. A |-> C ) e. O(1) -> ( ( x e. A |-> ( B - C ) ) oF + ( x e. A |-> C ) ) e. O(1) ) )
27	1, 26	syl 17	. . 3 |- ( ph -> ( ( x e. A |-> C ) e. O(1) -> ( ( x e. A |-> ( B - C ) ) oF + ( x e. A |-> C ) ) e. O(1) ) )
28		eqidd 2623	. . . . . 6 |- ( ph -> ( x e. A |-> C ) = ( x e. A |-> C ) )
29	16, 7, 6, 18, 28	offval2 6914	. . . . 5 |- ( ph -> ( ( x e. A |-> ( B - C ) ) oF + ( x e. A |-> C ) ) = ( x e. A |-> ( ( B - C ) + C ) ) )
30	5, 6	npcand 10396	. . . . . 6 |- ( ( ph /\ x e. A ) -> ( ( B - C ) + C ) = B )
31	30	mpteq2dva 4744	. . . . 5 |- ( ph -> ( x e. A |-> ( ( B - C ) + C ) ) = ( x e. A |-> B ) )
32	29, 31	eqtrd 2656	. . . 4 |- ( ph -> ( ( x e. A |-> ( B - C ) ) oF + ( x e. A |-> C ) ) = ( x e. A |-> B ) )
33	32	eleq1d 2686	. . 3 |- ( ph -> ( ( ( x e. A |-> ( B - C ) ) oF + ( x e. A |-> C ) ) e. O(1) <-> ( x e. A |-> B ) e. O(1) ) )
34	27, 33	sylibd 229	. 2 |- ( ph -> ( ( x e. A |-> C ) e. O(1) -> ( x e. A |-> B ) e. O(1) ) )
35	24, 34	impbid 202	1 |- ( ph -> ( ( x e. A |-> B ) e. O(1) <-> ( x e. A |-> C ) e. O(1) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   C_ wss 3574   |-> cmpt 4729   dom cdm 5114  (class class class)co 6650   oFcof 6895   CCcc 9934   RRcr 9935   + caddc 9939   - cmin 10266   O(1)co1 14217

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-rep 4771  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-of 6897  df-om 7066  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-pm 7860  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-ico 12181  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-o1 14221

This theorem is referenced by:  dchrmusum2  25183  dchrvmasumiflem2  25191  dchrisum0lem2a  25206  dchrisum0lem2  25207  rplogsum  25216  dirith2  25217  mulogsumlem  25220  mulogsum  25221  vmalogdivsum2  25227  vmalogdivsum  25228  2vmadivsumlem  25229  selberg3lem1  25246  selberg4lem1  25249  selberg4  25250  pntrlog2bndlem4  25269