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Theorem limsupbnd1OLD 13544
 Description: If a sequence is eventually at most , then the limsup is also at most . (The converse is only true if the less or equal is replaced by strictly less than; consider the sequence which is never less or equal to zero even though the limsup is.) (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by Mario Carneiro, 7-May-2016.) Obsolete version of limsupbnd1 13543 as of 12-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
limsupbnd.1
limsupbnd.2
limsupbnd.3
limsupbnd1.4
Assertion
Ref Expression
limsupbnd1OLD
Distinct variable groups:   ,,   ,,   ,,   ,,

Proof of Theorem limsupbnd1OLD
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 limsupbnd1.4 . 2
2 limsupbnd.1 . . . . . 6
32adantr 466 . . . . 5
4 limsupbnd.2 . . . . . 6
54adantr 466 . . . . 5
6 simpr 462 . . . . 5
7 limsupbnd.3 . . . . . 6
87adantr 466 . . . . 5
9 eqid 2422 . . . . . 6
109limsupgle 13534 . . . . 5
113, 5, 6, 8, 10syl211anc 1270 . . . 4
12 reex 9637 . . . . . . . . . . . 12
1312ssex 4568 . . . . . . . . . . 11
142, 13syl 17 . . . . . . . . . 10
15 xrex 11306 . . . . . . . . . . 11
1615a1i 11 . . . . . . . . . 10
17 fex2 6762 . . . . . . . . . 10
184, 14, 16, 17syl3anc 1264 . . . . . . . . 9
19 limsupclOLD 13529 . . . . . . . . 9
2018, 19syl 17 . . . . . . . 8
21 xrleid 11456 . . . . . . . 8
2220, 21syl 17 . . . . . . 7
239limsupleOLD 13536 . . . . . . . 8
242, 4, 20, 23syl3anc 1264 . . . . . . 7
2522, 24mpbid 213 . . . . . 6
2625r19.21bi 2791 . . . . 5
2720adantr 466 . . . . . 6
289limsupgf 13532 . . . . . . . 8
2928a1i 11 . . . . . . 7
3029ffvelrnda 6037 . . . . . 6
31 xrletr 11462 . . . . . 6
3227, 30, 8, 31syl3anc 1264 . . . . 5
3326, 32mpand 679 . . . 4
3411, 33sylbird 238 . . 3
3534rexlimdva 2914 . 2
361, 35mpd 15 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 187   wa 370   wcel 1872  wral 2771  wrex 2772  cvv 3080   cin 3435   wss 3436   class class class wbr 4423   cmpt 4482  cima 4856  wf 5597  cfv 5601  (class class class)co 6305  csup 7963  cr 9545   cpnf 9679  cxr 9681   clt 9682   cle 9683  cico 11644  clspold 13524 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597  ax-cnex 9602  ax-resscn 9603  ax-1cn 9604  ax-icn 9605  ax-addcl 9606  ax-addrcl 9607  ax-mulcl 9608  ax-mulrcl 9609  ax-mulcom 9610  ax-addass 9611  ax-mulass 9612  ax-distr 9613  ax-i2m1 9614  ax-1ne0 9615  ax-1rid 9616  ax-rnegex 9617  ax-rrecex 9618  ax-cnre 9619  ax-pre-lttri 9620  ax-pre-lttrn 9621  ax-pre-ltadd 9622  ax-pre-mulgt0 9623  ax-pre-sup 9624 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rmo 2779  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-po 4774  df-so 4775  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-er 7374  df-en 7581  df-dom 7582  df-sdom 7583  df-sup 7965  df-pnf 9684  df-mnf 9685  df-xr 9686  df-ltxr 9687  df-le 9688  df-sub 9869  df-neg 9870  df-ico 11648  df-limsupOLD 13526 This theorem is referenced by:  caucvgrlemOLD  13736  limsupreOLD  37662
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