Theorem difgtsumgt 9281

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 7907	. . . . . . 7 |- ( A e. RR -> A e. CC )
2		nn0cn 9145	. . . . . . 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 1012	. . . . 5 |- ( ( A e. RR /\ B e. NN0 /\ C e. RR ) -> ( A e. CC /\ B e. CC ) )
5		negsub 8167	. . . . 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 2176	. . 3 |- ( ( A e. RR /\ B e. NN0 /\ C e. RR ) -> ( A - B ) = ( A + -u B ) )
8	7	breq2d 4001	. 2 |- ( ( A e. RR /\ B e. NN0 /\ C e. RR ) -> ( C < ( A - B ) <-> C < ( A + -u B ) ) )
9		simp3 994	. . . 4 |- ( ( A e. RR /\ B e. NN0 /\ C e. RR ) -> C e. RR )
10		simp1 992	. . . . 5 |- ( ( A e. RR /\ B e. NN0 /\ C e. RR ) -> A e. RR )
11		nn0re 9144	. . . . . . 7 |- ( B e. NN0 -> B e. RR )
12	11	renegcld 8299	. . . . . 6 |- ( B e. NN0 -> -u B e. RR )
13	12	3ad2ant2 1014	. . . . 5 |- ( ( A e. RR /\ B e. NN0 /\ C e. RR ) -> -u B e. RR )
14	10, 13	readdcld 7949	. . . 4 |- ( ( A e. RR /\ B e. NN0 /\ C e. RR ) -> ( A + -u B ) e. RR )
15	11	3ad2ant2 1014	. . . . 5 |- ( ( A e. RR /\ B e. NN0 /\ C e. RR ) -> B e. RR )
16	10, 15	readdcld 7949	. . . 4 |- ( ( A e. RR /\ B e. NN0 /\ C e. RR ) -> ( A + B ) e. RR )
17	9, 14, 16	3jca 1172	. . 3 |- ( ( A e. RR /\ B e. NN0 /\ C e. RR ) -> ( C e. RR /\ ( A + -u B ) e. RR /\ ( A + B ) e. RR ) )
18		nn0negleid 9280	. . . . 5 |- ( B e. NN0 -> -u B <_ B )
19	18	3ad2ant2 1014	. . . 4 |- ( ( A e. RR /\ B e. NN0 /\ C e. RR ) -> -u B <_ B )
20	13, 15, 10, 19	leadd2dd 8479	. . 3 |- ( ( A e. RR /\ B e. NN0 /\ C e. RR ) -> ( A + -u B ) <_ ( A + B ) )
21	17, 20	lelttrdi 8345	. 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 973   = wceq 1348   e. wcel 2141   class class class wbr 3989  (class class class)co 5853   CCcc 7772   RRcr 7773   + caddc 7777   < clt 7954   <_ cle 7955   - cmin 8090   -ucneg 8091   NN0cn0 9135

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-addcom 7874  ax-addass 7876  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-0id 7882  ax-rnegex 7883  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-ltadd 7890

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-inn 8879  df-n0 9136

This theorem is referenced by:  difsqpwdvds  12291