Theorem difgtsumgt 9260

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

Step	Hyp	Ref	Expression
1		recn 7886	. . . . . . 7 |- ( A e. RR -> A e. CC )
2		nn0cn 9124	. . . . . . 7 |- ( B e. NN0 -> B e. CC )
3	1, 2	anim12i 336	. . . . . 6 |- ( ( A e. RR /\ B e. NN0 ) -> ( A e. CC /\ B e. CC ) )
4	3	3adant3 1007	. . . . 5 |- ( ( A e. RR /\ B e. NN0 /\ C e. RR ) -> ( A e. CC /\ B e. CC ) )
5		negsub 8146	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( A + -u B ) = ( A - B ) )
6	4, 5	syl 14	. . . 4 |- ( ( A e. RR /\ B e. NN0 /\ C e. RR ) -> ( A + -u B ) = ( A - B ) )
7	6	eqcomd 2171	. . 3 |- ( ( A e. RR /\ B e. NN0 /\ C e. RR ) -> ( A - B ) = ( A + -u B ) )
8	7	breq2d 3994	. 2 |- ( ( A e. RR /\ B e. NN0 /\ C e. RR ) -> ( C < ( A - B ) <-> C < ( A + -u B ) ) )
9		simp3 989	. . . 4 |- ( ( A e. RR /\ B e. NN0 /\ C e. RR ) -> C e. RR )
10		simp1 987	. . . . 5 |- ( ( A e. RR /\ B e. NN0 /\ C e. RR ) -> A e. RR )
11		nn0re 9123	. . . . . . 7 |- ( B e. NN0 -> B e. RR )
12	11	renegcld 8278	. . . . . 6 |- ( B e. NN0 -> -u B e. RR )
13	12	3ad2ant2 1009	. . . . 5 |- ( ( A e. RR /\ B e. NN0 /\ C e. RR ) -> -u B e. RR )
14	10, 13	readdcld 7928	. . . 4 |- ( ( A e. RR /\ B e. NN0 /\ C e. RR ) -> ( A + -u B ) e. RR )
15	11	3ad2ant2 1009	. . . . 5 |- ( ( A e. RR /\ B e. NN0 /\ C e. RR ) -> B e. RR )
16	10, 15	readdcld 7928	. . . 4 |- ( ( A e. RR /\ B e. NN0 /\ C e. RR ) -> ( A + B ) e. RR )
17	9, 14, 16	3jca 1167	. . 3 |- ( ( A e. RR /\ B e. NN0 /\ C e. RR ) -> ( C e. RR /\ ( A + -u B ) e. RR /\ ( A + B ) e. RR ) )
18		nn0negleid 9259	. . . . 5 |- ( B e. NN0 -> -u B <_ B )
19	18	3ad2ant2 1009	. . . 4 |- ( ( A e. RR /\ B e. NN0 /\ C e. RR ) -> -u B <_ B )
20	13, 15, 10, 19	leadd2dd 8458	. . 3 |- ( ( A e. RR /\ B e. NN0 /\ C e. RR ) -> ( A + -u B ) <_ ( A + B ) )
21	17, 20	lelttrdi 8324	. 2 |- ( ( A e. RR /\ B e. NN0 /\ C e. RR ) -> ( C < ( A + -u B ) -> C < ( A + B ) ) )
22	8, 21	sylbid 149	1 |- ( ( A e. RR /\ B e. NN0 /\ C e. RR ) -> ( C < ( A - B ) -> C < ( A + B ) ) )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 968   = wceq 1343   e. wcel 2136   class class class wbr 3982  (class class class)co 5842   CCcc 7751   RRcr 7752   + caddc 7756   < clt 7933   <_ cle 7934   - cmin 8069   -ucneg 8070   NN0cn0 9114

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-addcom 7853  ax-addass 7855  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-0id 7861  ax-rnegex 7862  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-ltadd 7869

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-iota 5153  df-fun 5190  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-inn 8858  df-n0 9115

This theorem is referenced by:  difsqpwdvds  12269