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Theorem difgtsumgt 9311
Description: If the difference of a real number and a nonnegative integer is greater than another real number, the sum of the real number and the nonnegative integer is also greater than the other real number. (Contributed by AV, 13-Aug-2021.)
Assertion
Ref Expression
difgtsumgt  |-  ( ( A  e.  RR  /\  B  e.  NN0  /\  C  e.  RR )  ->  ( C  <  ( A  -  B )  ->  C  <  ( A  +  B
) ) )

Proof of Theorem difgtsumgt
StepHypRef Expression
1 recn 7935 . . . . . . 7  |-  ( A  e.  RR  ->  A  e.  CC )
2 nn0cn 9175 . . . . . . 7  |-  ( B  e.  NN0  ->  B  e.  CC )
31, 2anim12i 338 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  NN0 )  -> 
( A  e.  CC  /\  B  e.  CC ) )
433adant3 1017 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  NN0  /\  C  e.  RR )  ->  ( A  e.  CC  /\  B  e.  CC ) )
5 negsub 8195 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  -u B )  =  ( A  -  B ) )
64, 5syl 14 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  NN0  /\  C  e.  RR )  ->  ( A  +  -u B )  =  ( A  -  B ) )
76eqcomd 2183 . . 3  |-  ( ( A  e.  RR  /\  B  e.  NN0  /\  C  e.  RR )  ->  ( A  -  B )  =  ( A  +  -u B ) )
87breq2d 4012 . 2  |-  ( ( A  e.  RR  /\  B  e.  NN0  /\  C  e.  RR )  ->  ( C  <  ( A  -  B )  <->  C  <  ( A  +  -u B
) ) )
9 simp3 999 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  NN0  /\  C  e.  RR )  ->  C  e.  RR )
10 simp1 997 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  NN0  /\  C  e.  RR )  ->  A  e.  RR )
11 nn0re 9174 . . . . . . 7  |-  ( B  e.  NN0  ->  B  e.  RR )
1211renegcld 8327 . . . . . 6  |-  ( B  e.  NN0  ->  -u B  e.  RR )
13123ad2ant2 1019 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  NN0  /\  C  e.  RR )  ->  -u B  e.  RR )
1410, 13readdcld 7977 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  NN0  /\  C  e.  RR )  ->  ( A  +  -u B )  e.  RR )
15113ad2ant2 1019 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  NN0  /\  C  e.  RR )  ->  B  e.  RR )
1610, 15readdcld 7977 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  NN0  /\  C  e.  RR )  ->  ( A  +  B )  e.  RR )
179, 14, 163jca 1177 . . 3  |-  ( ( A  e.  RR  /\  B  e.  NN0  /\  C  e.  RR )  ->  ( C  e.  RR  /\  ( A  +  -u B )  e.  RR  /\  ( A  +  B )  e.  RR ) )
18 nn0negleid 9310 . . . . 5  |-  ( B  e.  NN0  ->  -u B  <_  B )
19183ad2ant2 1019 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  NN0  /\  C  e.  RR )  ->  -u B  <_  B )
2013, 15, 10, 19leadd2dd 8507 . . 3  |-  ( ( A  e.  RR  /\  B  e.  NN0  /\  C  e.  RR )  ->  ( A  +  -u B )  <_  ( A  +  B ) )
2117, 20lelttrdi 8373 . 2  |-  ( ( A  e.  RR  /\  B  e.  NN0  /\  C  e.  RR )  ->  ( C  <  ( A  +  -u B )  ->  C  <  ( A  +  B
) ) )
228, 21sylbid 150 1  |-  ( ( A  e.  RR  /\  B  e.  NN0  /\  C  e.  RR )  ->  ( C  <  ( A  -  B )  ->  C  <  ( A  +  B
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    /\ w3a 978    = wceq 1353    e. wcel 2148   class class class wbr 4000  (class class class)co 5869   CCcc 7800   RRcr 7801    + caddc 7805    < clt 7982    <_ cle 7983    - cmin 8118   -ucneg 8119   NN0cn0 9165
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 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-addcom 7902  ax-addass 7904  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-0id 7910  ax-rnegex 7911  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-ltadd 7918
This theorem depends on definitions:  df-bi 117  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 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-br 4001  df-opab 4062  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-iota 5174  df-fun 5214  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-inn 8909  df-n0 9166
This theorem is referenced by:  difsqpwdvds  12320
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