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Theorem xnn0xaddcl 12066
Description: The extended nonnegative integers are closed under extended addition. (Contributed by AV, 10-Dec-2020.)
Assertion
Ref Expression
xnn0xaddcl |- ( ( A e. NN0* /\ B e. NN0* ) -> ( A +e B ) e. NN0* )
Proof of Theorem xnn0xaddcl
StepHypRef Expression
1 nn0addcl 11328 . . . . 5 |- ( ( A e. NN0 /\ B e. NN0 ) -> ( A + B ) e. NN0 )
21nn0xnn0d 11372 . . . 4 |- ( ( A e. NN0 /\ B e. NN0 ) -> ( A + B ) e. NN0* )
3 nn0re 11301 . . . . 5 |- ( A e. NN0 -> A e. RR )
4 nn0re 11301 . . . . 5 |- ( B e. NN0 -> B e. RR )
5 rexadd 12063 . . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( A +e B ) = ( A + B ) )
65eleq1d 2686 . . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( ( A +e B ) e. NN0* <-> ( A + B ) e. NN0* ) )
73, 4, 6syl2an 494 . . . 4 |- ( ( A e. NN0 /\ B e. NN0 ) -> ( ( A +e B ) e. NN0* <-> ( A + B ) e. NN0* ) )
82, 7mpbird 247 . . 3 |- ( ( A e. NN0 /\ B e. NN0 ) -> ( A +e B ) e. NN0* )
98a1d 25 . 2 |- ( ( A e. NN0 /\ B e. NN0 ) -> ( ( A e. NN0* /\ B e. NN0* ) -> ( A +e B ) e. NN0* ) )
10 ianor 509 . . 3 |- ( -. ( A e. NN0 /\ B e. NN0 ) <-> ( -. A e. NN0 \/ -. B e. NN0 ) )
11 xnn0nnn0pnf 11376 . . . . . . . . . 10 |- ( ( A e. NN0* /\ -. A e. NN0 ) -> A = +oo )
12 oveq1 6657 . . . . . . . . . . . 12 |- ( A = +oo -> ( A +e B ) = ( +oo +e B ) )
13 xnn0xrnemnf 11375 . . . . . . . . . . . . 13 |- ( B e. NN0* -> ( B e. RR* /\ B =/= -oo ) )
14 xaddpnf2 12058 . . . . . . . . . . . . 13 |- ( ( B e. RR* /\ B =/= -oo ) -> ( +oo +e B ) = +oo )
1513, 14syl 17 . . . . . . . . . . . 12 |- ( B e. NN0* -> ( +oo +e B ) = +oo )
1612, 15sylan9eq 2676 . . . . . . . . . . 11 |- ( ( A = +oo /\ B e. NN0* ) -> ( A +e B ) = +oo )
1716ex 450 . . . . . . . . . 10 |- ( A = +oo -> ( B e. NN0* -> ( A +e B ) = +oo ) )
1811, 17syl 17 . . . . . . . . 9 |- ( ( A e. NN0* /\ -. A e. NN0 ) -> ( B e. NN0* -> ( A +e B ) = +oo ) )
1918expcom 451 . . . . . . . 8 |- ( -. A e. NN0 -> ( A e. NN0* -> ( B e. NN0* -> ( A +e B ) = +oo ) ) )
2019impd 447 . . . . . . 7 |- ( -. A e. NN0 -> ( ( A e. NN0* /\ B e. NN0* ) -> ( A +e B ) = +oo ) )
21 xnn0nnn0pnf 11376 . . . . . . . . . . 11 |- ( ( B e. NN0* /\ -. B e. NN0 ) -> B = +oo )
22 oveq2 6658 . . . . . . . . . . . . 13 |- ( B = +oo -> ( A +e B ) = ( A +e +oo ) )
23 xnn0xrnemnf 11375 . . . . . . . . . . . . . 14 |- ( A e. NN0* -> ( A e. RR* /\ A =/= -oo ) )
24 xaddpnf1 12057 . . . . . . . . . . . . . 14 |- ( ( A e. RR* /\ A =/= -oo ) -> ( A +e +oo ) = +oo )
2523, 24syl 17 . . . . . . . . . . . . 13 |- ( A e. NN0* -> ( A +e +oo ) = +oo )
2622, 25sylan9eq 2676 . . . . . . . . . . . 12 |- ( ( B = +oo /\ A e. NN0* ) -> ( A +e B ) = +oo )
2726ex 450 . . . . . . . . . . 11 |- ( B = +oo -> ( A e. NN0* -> ( A +e B ) = +oo ) )
2821, 27syl 17 . . . . . . . . . 10 |- ( ( B e. NN0* /\ -. B e. NN0 ) -> ( A e. NN0* -> ( A +e B ) = +oo ) )
2928expcom 451 . . . . . . . . 9 |- ( -. B e. NN0 -> ( B e. NN0* -> ( A e. NN0* -> ( A +e B ) = +oo ) ) )
3029com23 86 . . . . . . . 8 |- ( -. B e. NN0 -> ( A e. NN0* -> ( B e. NN0* -> ( A +e B ) = +oo ) ) )
3130impd 447 . . . . . . 7 |- ( -. B e. NN0 -> ( ( A e. NN0* /\ B e. NN0* ) -> ( A +e B ) = +oo ) )
3220, 31jaoi 394 . . . . . 6 |- ( ( -. A e. NN0 \/ -. B e. NN0 ) -> ( ( A e. NN0* /\ B e. NN0* ) -> ( A +e B ) = +oo ) )
3332imp 445 . . . . 5 |- ( ( ( -. A e. NN0 \/ -. B e. NN0 ) /\ ( A e. NN0* /\ B e. NN0* ) ) -> ( A +e B ) = +oo )
34 pnf0xnn0 11370 . . . . 5 |- +oo e. NN0*
3533, 34syl6eqel 2709 . . . 4 |- ( ( ( -. A e. NN0 \/ -. B e. NN0 ) /\ ( A e. NN0* /\ B e. NN0* ) ) -> ( A +e B ) e. NN0* )
3635ex 450 . . 3 |- ( ( -. A e. NN0 \/ -. B e. NN0 ) -> ( ( A e. NN0* /\ B e. NN0* ) -> ( A +e B ) e. NN0* ) )
3710, 36sylbi 207 . 2 |- ( -. ( A e. NN0 /\ B e. NN0 ) -> ( ( A e. NN0* /\ B e. NN0* ) -> ( A +e B ) e. NN0* ) )
389, 37pm2.61i 176 1 |- ( ( A e. NN0* /\ B e. NN0* ) -> ( A +e B ) e. NN0* )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794  (class class class)co 6650   RRcr 9935   + caddc 9939   +oocpnf 10071   -oocmnf 10072   RR*cxr 10073   NN0cn0 11292  NN0*cxnn0 11363   +ecxad 11944
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
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-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-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  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-nn 11021  df-n0 11293  df-xnn0 11364  df-xadd 11947
This theorem is referenced by:  vtxdgf  26367  vtxdginducedm1  26439
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