Theorem difgtsumgt 9251

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 7877	. . . . . . 7 |- ( A e. RR -> A e. CC )
2		nn0cn 9115	. . . . . . 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 1006	. . . . 5 |- ( ( A e. RR /\ B e. NN0 /\ C e. RR ) -> ( A e. CC /\ B e. CC ) )
5		negsub 8137	. . . . 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 2170	. . 3 |- ( ( A e. RR /\ B e. NN0 /\ C e. RR ) -> ( A - B ) = ( A + -u B ) )
8	7	breq2d 3988	. 2 |- ( ( A e. RR /\ B e. NN0 /\ C e. RR ) -> ( C < ( A - B ) <-> C < ( A + -u B ) ) )
9		simp3 988	. . . 4 |- ( ( A e. RR /\ B e. NN0 /\ C e. RR ) -> C e. RR )
10		simp1 986	. . . . 5 |- ( ( A e. RR /\ B e. NN0 /\ C e. RR ) -> A e. RR )
11		nn0re 9114	. . . . . . 7 |- ( B e. NN0 -> B e. RR )
12	11	renegcld 8269	. . . . . 6 |- ( B e. NN0 -> -u B e. RR )
13	12	3ad2ant2 1008	. . . . 5 |- ( ( A e. RR /\ B e. NN0 /\ C e. RR ) -> -u B e. RR )
14	10, 13	readdcld 7919	. . . 4 |- ( ( A e. RR /\ B e. NN0 /\ C e. RR ) -> ( A + -u B ) e. RR )
15	11	3ad2ant2 1008	. . . . 5 |- ( ( A e. RR /\ B e. NN0 /\ C e. RR ) -> B e. RR )
16	10, 15	readdcld 7919	. . . 4 |- ( ( A e. RR /\ B e. NN0 /\ C e. RR ) -> ( A + B ) e. RR )
17	9, 14, 16	3jca 1166	. . 3 |- ( ( A e. RR /\ B e. NN0 /\ C e. RR ) -> ( C e. RR /\ ( A + -u B ) e. RR /\ ( A + B ) e. RR ) )
18		nn0negleid 9250	. . . . 5 |- ( B e. NN0 -> -u B <_ B )
19	18	3ad2ant2 1008	. . . 4 |- ( ( A e. RR /\ B e. NN0 /\ C e. RR ) -> -u B <_ B )
20	13, 15, 10, 19	leadd2dd 8449	. . 3 |- ( ( A e. RR /\ B e. NN0 /\ C e. RR ) -> ( A + -u B ) <_ ( A + B ) )
21	17, 20	lelttrdi 8315	. 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 967   = wceq 1342   e. wcel 2135   class class class wbr 3976  (class class class)co 5836   CCcc 7742   RRcr 7743   + caddc 7747   < clt 7924   <_ cle 7925   - cmin 8060   -ucneg 8061   NN0cn0 9105

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 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-sep 4094  ax-pow 4147  ax-pr 4181  ax-un 4405  ax-setind 4508  ax-cnex 7835  ax-resscn 7836  ax-1cn 7837  ax-1re 7838  ax-icn 7839  ax-addcl 7840  ax-addrcl 7841  ax-mulcl 7842  ax-addcom 7844  ax-addass 7846  ax-distr 7848  ax-i2m1 7849  ax-0lt1 7850  ax-0id 7852  ax-rnegex 7853  ax-cnre 7855  ax-pre-ltirr 7856  ax-pre-ltwlin 7857  ax-pre-lttrn 7858  ax-pre-ltadd 7860

This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-nel 2430  df-ral 2447  df-rex 2448  df-reu 2449  df-rab 2451  df-v 2723  df-sbc 2947  df-dif 3113  df-un 3115  df-in 3117  df-ss 3124  df-pw 3555  df-sn 3576  df-pr 3577  df-op 3579  df-uni 3784  df-int 3819  df-br 3977  df-opab 4038  df-id 4265  df-xp 4604  df-rel 4605  df-cnv 4606  df-co 4607  df-dm 4608  df-iota 5147  df-fun 5184  df-fv 5190  df-riota 5792  df-ov 5839  df-oprab 5840  df-mpo 5841  df-pnf 7926  df-mnf 7927  df-xr 7928  df-ltxr 7929  df-le 7930  df-sub 8062  df-neg 8063  df-inn 8849  df-n0 9106

This theorem is referenced by:  difsqpwdvds  12246