ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  xnn0dcle Unicode version
Theorem xnn0dcle 10204
Description: Decidability of <_ for extended nonnegative integers. (Contributed by Jim Kingdon, 13-Oct-2024.)
Assertion
Ref Expression
xnn0dcle |- ( ( A e. NN0* /\ B e. NN0* ) -> DECID A <_ B )
Proof of Theorem xnn0dcle
StepHypRef Expression
1 simpr 110 . . . . 5 |- ( ( ( ( A e. NN0* /\ B e. NN0* ) /\ B e. NN0 ) /\ A e. NN0 ) -> A e. NN0 )
21nn0zd 9766 . . . 4 |- ( ( ( ( A e. NN0* /\ B e. NN0* ) /\ B e. NN0 ) /\ A e. NN0 ) -> A e. ZZ )
3 simplr 533 . . . . 5 |- ( ( ( ( A e. NN0* /\ B e. NN0* ) /\ B e. NN0 ) /\ A e. NN0 ) -> B e. NN0 )
43nn0zd 9766 . . . 4 |- ( ( ( ( A e. NN0* /\ B e. NN0* ) /\ B e. NN0 ) /\ A e. NN0 ) -> B e. ZZ )
5 zdcle 9721 . . . 4 |- ( ( A e. ZZ /\ B e. ZZ ) -> DECID A <_ B )
62, 4, 5syl2anc 415 . . 3 |- ( ( ( ( A e. NN0* /\ B e. NN0* ) /\ B e. NN0 ) /\ A e. NN0 ) -> DECID A <_ B )
7 simpr 110 . . . . . 6 |- ( ( ( ( A e. NN0* /\ B e. NN0* ) /\ B e. NN0 ) /\ A = +oo ) -> A = +oo )
8 simplr 533 . . . . . . . . 9 |- ( ( ( ( A e. NN0* /\ B e. NN0* ) /\ B e. NN0 ) /\ A = +oo ) -> B e. NN0 )
98nn0red 9621 . . . . . . . 8 |- ( ( ( ( A e. NN0* /\ B e. NN0* ) /\ B e. NN0 ) /\ A = +oo ) -> B e. RR )
109ltpnfd 10183 . . . . . . 7 |- ( ( ( ( A e. NN0* /\ B e. NN0* ) /\ B e. NN0 ) /\ A = +oo ) -> B < +oo )
11 pnfxr 8378 . . . . . . . . 9 |- +oo e. RR*
129rexrd 8375 . . . . . . . . 9 |- ( ( ( ( A e. NN0* /\ B e. NN0* ) /\ B e. NN0 ) /\ A = +oo ) -> B e. RR* )
13 xrlenlt 8390 . . . . . . . . 9 |- ( ( +oo e. RR* /\ B e. RR* ) -> ( +oo <_ B <-> -. B < +oo ) )
1411, 12, 13sylancr 418 . . . . . . . 8 |- ( ( ( ( A e. NN0* /\ B e. NN0* ) /\ B e. NN0 ) /\ A = +oo ) -> ( +oo <_ B <-> -. B < +oo ) )
1514biimpd 144 . . . . . . 7 |- ( ( ( ( A e. NN0* /\ B e. NN0* ) /\ B e. NN0 ) /\ A = +oo ) -> ( +oo <_ B -> -. B < +oo ) )
1610, 15mt2d 634 . . . . . 6 |- ( ( ( ( A e. NN0* /\ B e. NN0* ) /\ B e. NN0 ) /\ A = +oo ) -> -. +oo <_ B )
177, 16eqnbrtrd 4148 . . . . 5 |- ( ( ( ( A e. NN0* /\ B e. NN0* ) /\ B e. NN0 ) /\ A = +oo ) -> -. A <_ B )
1817olcd 746 . . . 4 |- ( ( ( ( A e. NN0* /\ B e. NN0* ) /\ B e. NN0 ) /\ A = +oo ) -> ( A <_ B \/ -. A <_ B ) )
19 df-dc 847 . . . 4 |- (DECID A <_ B <-> ( A <_ B \/ -. A <_ B ) )
2018, 19sylibr 134 . . 3 |- ( ( ( ( A e. NN0* /\ B e. NN0* ) /\ B e. NN0 ) /\ A = +oo ) -> DECID A <_ B )
21 elxnn0 9632 . . . . 5 |- ( A e. NN0* <-> ( A e. NN0 \/ A = +oo ) )
2221biimpi 120 . . . 4 |- ( A e. NN0* -> ( A e. NN0 \/ A = +oo ) )
2322ad2antrr 492 . . 3 |- ( ( ( A e. NN0* /\ B e. NN0* ) /\ B e. NN0 ) -> ( A e. NN0 \/ A = +oo ) )
246, 20, 23mpjaodan 810 . 2 |- ( ( ( A e. NN0* /\ B e. NN0* ) /\ B e. NN0 ) -> DECID A <_ B )
25 xnn0xr 9635 . . . . . . 7 |- ( A e. NN0* -> A e. RR* )
2625ad2antrr 492 . . . . . 6 |- ( ( ( A e. NN0* /\ B e. NN0* ) /\ B = +oo ) -> A e. RR* )
27 pnfge 10191 . . . . . 6 |- ( A e. RR* -> A <_ +oo )
2826, 27syl 14 . . . . 5 |- ( ( ( A e. NN0* /\ B e. NN0* ) /\ B = +oo ) -> A <_ +oo )
29 simpr 110 . . . . 5 |- ( ( ( A e. NN0* /\ B e. NN0* ) /\ B = +oo ) -> B = +oo )
3028, 29breqtrrd 4158 . . . 4 |- ( ( ( A e. NN0* /\ B e. NN0* ) /\ B = +oo ) -> A <_ B )
3130orcd 745 . . 3 |- ( ( ( A e. NN0* /\ B e. NN0* ) /\ B = +oo ) -> ( A <_ B \/ -. A <_ B ) )
3231, 19sylibr 134 . 2 |- ( ( ( A e. NN0* /\ B e. NN0* ) /\ B = +oo ) -> DECID A <_ B )
33 elxnn0 9632 . . . 4 |- ( B e. NN0* <-> ( B e. NN0 \/ B = +oo ) )
3433biimpi 120 . . 3 |- ( B e. NN0* -> ( B e. NN0 \/ B = +oo ) )
3534adantl 277 . 2 |- ( ( A e. NN0* /\ B e. NN0* ) -> ( B e. NN0 \/ B = +oo ) )
3624, 32, 35mpjaodan 810 1 |- ( ( A e. NN0* /\ B e. NN0* ) -> DECID A <_ B )
Colors of variables:    wff set class
This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 720  DECID wdc 846   = wceq 1402   e. wcel 2209   class class class wbr 4130   +oocpnf 8357   RR*cxr 8359   < clt 8360   <_ cle 8361   NN0cn0 9563  NN0*cxnn0 9630   ZZcz 9644
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-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  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-addcom 8279  ax-addass 8281  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-ltadd 8295
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-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-opab 4193  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-iota 5337  df-fun 5379  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-inn 9305  df-n0 9564  df-xnn0 9631  df-z 9645
This theorem is used by:  pcgcd  13108
  Copyright terms: Public domain W3C validator