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Theorem nn0sumltlt 31880
Description: If the sum of two nonnegative integers is less than a third integer, then one of the summands is already less than this third integer. (Contributed by AV, 19-Oct-2019.)
Assertion
Ref Expression
nn0sumltlt  |-  ( ( a  e.  NN0  /\  b  e.  NN0  /\  c  e.  NN0 )  ->  (
( a  +  b )  <  c  -> 
b  <  c )
)

Proof of Theorem nn0sumltlt
StepHypRef Expression
1 nn0re 10795 . . 3  |-  ( a  e.  NN0  ->  a  e.  RR )
2 nn0re 10795 . . 3  |-  ( b  e.  NN0  ->  b  e.  RR )
3 nn0re 10795 . . 3  |-  ( c  e.  NN0  ->  c  e.  RR )
4 ltaddsub2 10018 . . 3  |-  ( ( a  e.  RR  /\  b  e.  RR  /\  c  e.  RR )  ->  (
( a  +  b )  <  c  <->  b  <  ( c  -  a ) ) )
51, 2, 3, 4syl3an 1265 . 2  |-  ( ( a  e.  NN0  /\  b  e.  NN0  /\  c  e.  NN0 )  ->  (
( a  +  b )  <  c  <->  b  <  ( c  -  a ) ) )
6 nn0ge0 10812 . . . . 5  |-  ( a  e.  NN0  ->  0  <_ 
a )
763ad2ant1 1012 . . . 4  |-  ( ( a  e.  NN0  /\  b  e.  NN0  /\  c  e.  NN0 )  ->  0  <_  a )
81, 3anim12ci 567 . . . . . 6  |-  ( ( a  e.  NN0  /\  c  e.  NN0 )  -> 
( c  e.  RR  /\  a  e.  RR ) )
983adant2 1010 . . . . 5  |-  ( ( a  e.  NN0  /\  b  e.  NN0  /\  c  e.  NN0 )  ->  (
c  e.  RR  /\  a  e.  RR )
)
10 subge02 10059 . . . . . 6  |-  ( ( c  e.  RR  /\  a  e.  RR )  ->  ( 0  <_  a  <->  ( c  -  a )  <_  c ) )
1110bicomd 201 . . . . 5  |-  ( ( c  e.  RR  /\  a  e.  RR )  ->  ( ( c  -  a )  <_  c  <->  0  <_  a ) )
129, 11syl 16 . . . 4  |-  ( ( a  e.  NN0  /\  b  e.  NN0  /\  c  e.  NN0 )  ->  (
( c  -  a
)  <_  c  <->  0  <_  a ) )
137, 12mpbird 232 . . 3  |-  ( ( a  e.  NN0  /\  b  e.  NN0  /\  c  e.  NN0 )  ->  (
c  -  a )  <_  c )
1423ad2ant2 1013 . . . 4  |-  ( ( a  e.  NN0  /\  b  e.  NN0  /\  c  e.  NN0 )  ->  b  e.  RR )
15 nn0resubcl 31754 . . . . . 6  |-  ( ( c  e.  NN0  /\  a  e.  NN0 )  -> 
( c  -  a
)  e.  RR )
1615ancoms 453 . . . . 5  |-  ( ( a  e.  NN0  /\  c  e.  NN0 )  -> 
( c  -  a
)  e.  RR )
17163adant2 1010 . . . 4  |-  ( ( a  e.  NN0  /\  b  e.  NN0  /\  c  e.  NN0 )  ->  (
c  -  a )  e.  RR )
1833ad2ant3 1014 . . . 4  |-  ( ( a  e.  NN0  /\  b  e.  NN0  /\  c  e.  NN0 )  ->  c  e.  RR )
19 ltletr 9667 . . . 4  |-  ( ( b  e.  RR  /\  ( c  -  a
)  e.  RR  /\  c  e.  RR )  ->  ( ( b  < 
( c  -  a
)  /\  ( c  -  a )  <_ 
c )  ->  b  <  c ) )
2014, 17, 18, 19syl3anc 1223 . . 3  |-  ( ( a  e.  NN0  /\  b  e.  NN0  /\  c  e.  NN0 )  ->  (
( b  <  (
c  -  a )  /\  ( c  -  a )  <_  c
)  ->  b  <  c ) )
2113, 20mpan2d 674 . 2  |-  ( ( a  e.  NN0  /\  b  e.  NN0  /\  c  e.  NN0 )  ->  (
b  <  ( c  -  a )  -> 
b  <  c )
)
225, 21sylbid 215 1  |-  ( ( a  e.  NN0  /\  b  e.  NN0  /\  c  e.  NN0 )  ->  (
( a  +  b )  <  c  -> 
b  <  c )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 968    e. wcel 1762   class class class wbr 4442  (class class class)co 6277   RRcr 9482   0cc0 9483    + caddc 9486    < clt 9619    <_ cle 9620    - cmin 9796   NN0cn0 10786
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6674  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7509  df-dom 7510  df-sdom 7511  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-nn 10528  df-n0 10787
This theorem is referenced by:  ply1mulgsumlem1  31936
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