Theorem difgtsumgt 9389

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 8007	. . . . . . 7 |- ( A e. RR -> A e. CC )
2		nn0cn 9253	. . . . . . 7 |- ( B e. NN0 -> B e. CC )
3	1, 2	anim12i 338	. . . . . 6 |- ( ( A e. RR /\ B e. NN0 ) -> ( A e. CC /\ B e. CC ) )
4	3	3adant3 1019	. . . . 5 |- ( ( A e. RR /\ B e. NN0 /\ C e. RR ) -> ( A e. CC /\ B e. CC ) )
5		negsub 8269	. . . . 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 2199	. . 3 |- ( ( A e. RR /\ B e. NN0 /\ C e. RR ) -> ( A - B ) = ( A + -u B ) )
8	7	breq2d 4042	. 2 |- ( ( A e. RR /\ B e. NN0 /\ C e. RR ) -> ( C < ( A - B ) <-> C < ( A + -u B ) ) )
9		simp3 1001	. . . 4 |- ( ( A e. RR /\ B e. NN0 /\ C e. RR ) -> C e. RR )
10		simp1 999	. . . . 5 |- ( ( A e. RR /\ B e. NN0 /\ C e. RR ) -> A e. RR )
11		nn0re 9252	. . . . . . 7 |- ( B e. NN0 -> B e. RR )
12	11	renegcld 8401	. . . . . 6 |- ( B e. NN0 -> -u B e. RR )
13	12	3ad2ant2 1021	. . . . 5 |- ( ( A e. RR /\ B e. NN0 /\ C e. RR ) -> -u B e. RR )
14	10, 13	readdcld 8051	. . . 4 |- ( ( A e. RR /\ B e. NN0 /\ C e. RR ) -> ( A + -u B ) e. RR )
15	11	3ad2ant2 1021	. . . . 5 |- ( ( A e. RR /\ B e. NN0 /\ C e. RR ) -> B e. RR )
16	10, 15	readdcld 8051	. . . 4 |- ( ( A e. RR /\ B e. NN0 /\ C e. RR ) -> ( A + B ) e. RR )
17	9, 14, 16	3jca 1179	. . 3 |- ( ( A e. RR /\ B e. NN0 /\ C e. RR ) -> ( C e. RR /\ ( A + -u B ) e. RR /\ ( A + B ) e. RR ) )
18		nn0negleid 9388	. . . . 5 |- ( B e. NN0 -> -u B <_ B )
19	18	3ad2ant2 1021	. . . 4 |- ( ( A e. RR /\ B e. NN0 /\ C e. RR ) -> -u B <_ B )
20	13, 15, 10, 19	leadd2dd 8581	. . 3 |- ( ( A e. RR /\ B e. NN0 /\ C e. RR ) -> ( A + -u B ) <_ ( A + B ) )
21	17, 20	lelttrdi 8447	. 2 |- ( ( A e. RR /\ B e. NN0 /\ C e. RR ) -> ( C < ( A + -u B ) -> C < ( A + B ) ) )
22	8, 21	sylbid 150	1 |- ( ( A e. RR /\ B e. NN0 /\ C e. RR ) -> ( C < ( A - B ) -> C < ( A + B ) ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   = wceq 1364   e. wcel 2164   class class class wbr 4030  (class class class)co 5919   CCcc 7872   RRcr 7873   + caddc 7877   < clt 8056   <_ cle 8057   - cmin 8192   -ucneg 8193   NN0cn0 9243

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-addcom 7974  ax-addass 7976  ax-distr 7978  ax-i2m1 7979  ax-0lt1 7980  ax-0id 7982  ax-rnegex 7983  ax-cnre 7985  ax-pre-ltirr 7986  ax-pre-ltwlin 7987  ax-pre-lttrn 7988  ax-pre-ltadd 7990

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-br 4031  df-opab 4092  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-iota 5216  df-fun 5257  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-pnf 8058  df-mnf 8059  df-xr 8060  df-ltxr 8061  df-le 8062  df-sub 8194  df-neg 8195  df-inn 8985  df-n0 9244

This theorem is referenced by:  difsqpwdvds  12479