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Theorem xnn0dcle 9804
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 9375 . . . 4 |- ( ( ( ( A e. NN0* /\ B e. NN0* ) /\ B e. NN0 ) /\ A e. NN0 ) -> A e. ZZ )
3 simplr 528 . . . . 5 |- ( ( ( ( A e. NN0* /\ B e. NN0* ) /\ B e. NN0 ) /\ A e. NN0 ) -> B e. NN0 )
43nn0zd 9375 . . . 4 |- ( ( ( ( A e. NN0* /\ B e. NN0* ) /\ B e. NN0 ) /\ A e. NN0 ) -> B e. ZZ )
5 zdcle 9331 . . . 4 |- ( ( A e. ZZ /\ B e. ZZ ) -> DECID A <_ B )
62, 4, 5syl2anc 411 . . 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 528 . . . . . . . . 9 |- ( ( ( ( A e. NN0* /\ B e. NN0* ) /\ B e. NN0 ) /\ A = +oo ) -> B e. NN0 )
98nn0red 9232 . . . . . . . 8 |- ( ( ( ( A e. NN0* /\ B e. NN0* ) /\ B e. NN0 ) /\ A = +oo ) -> B e. RR )
109ltpnfd 9783 . . . . . . 7 |- ( ( ( ( A e. NN0* /\ B e. NN0* ) /\ B e. NN0 ) /\ A = +oo ) -> B < +oo )
11 pnfxr 8012 . . . . . . . . 9 |- +oo e. RR*
129rexrd 8009 . . . . . . . . 9 |- ( ( ( ( A e. NN0* /\ B e. NN0* ) /\ B e. NN0 ) /\ A = +oo ) -> B e. RR* )
13 xrlenlt 8024 . . . . . . . . 9 |- ( ( +oo e. RR* /\ B e. RR* ) -> ( +oo <_ B <-> -. B < +oo ) )
1411, 12, 13sylancr 414 . . . . . . . 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 625 . . . . . 6 |- ( ( ( ( A e. NN0* /\ B e. NN0* ) /\ B e. NN0 ) /\ A = +oo ) -> -. +oo <_ B )
177, 16eqnbrtrd 4023 . . . . 5 |- ( ( ( ( A e. NN0* /\ B e. NN0* ) /\ B e. NN0 ) /\ A = +oo ) -> -. A <_ B )
1817olcd 734 . . . 4 |- ( ( ( ( A e. NN0* /\ B e. NN0* ) /\ B e. NN0 ) /\ A = +oo ) -> ( A <_ B \/ -. A <_ B ) )
19 df-dc 835 . . . 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 9243 . . . . 5 |- ( A e. NN0* <-> ( A e. NN0 \/ A = +oo ) )
2221biimpi 120 . . . 4 |- ( A e. NN0* -> ( A e. NN0 \/ A = +oo ) )
2322ad2antrr 488 . . 3 |- ( ( ( A e. NN0* /\ B e. NN0* ) /\ B e. NN0 ) -> ( A e. NN0 \/ A = +oo ) )
246, 20, 23mpjaodan 798 . 2 |- ( ( ( A e. NN0* /\ B e. NN0* ) /\ B e. NN0 ) -> DECID A <_ B )
25 xnn0xr 9246 . . . . . . 7 |- ( A e. NN0* -> A e. RR* )
2625ad2antrr 488 . . . . . 6 |- ( ( ( A e. NN0* /\ B e. NN0* ) /\ B = +oo ) -> A e. RR* )
27 pnfge 9791 . . . . . 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 4033 . . . 4 |- ( ( ( A e. NN0* /\ B e. NN0* ) /\ B = +oo ) -> A <_ B )
3130orcd 733 . . 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 9243 . . . 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 798 1 |- ( ( A e. NN0* /\ B e. NN0* ) -> DECID A <_ B )
Colors of variables: wff set class
Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 708  DECID wdc 834   = wceq 1353   e. wcel 2148   class class class wbr 4005   +oocpnf 7991   RR*cxr 7993   < clt 7994   <_ cle 7995   NN0cn0 9178  NN0*cxnn0 9241   ZZcz 9255
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-addcom 7913  ax-addass 7915  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-0id 7921  ax-rnegex 7922  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-ltadd 7929
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-br 4006  df-opab 4067  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-iota 5180  df-fun 5220  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-inn 8922  df-n0 9179  df-xnn0 9242  df-z 9256
This theorem is referenced by:  pcgcd  12330
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