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Theorem le2sub 8380
Description: Subtracting both sides of two 'less than or equal to' relations. (Contributed by Mario Carneiro, 14-Apr-2016.)
Assertion
Ref Expression
le2sub |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( ( A <_ C /\ D <_ B ) -> ( A - B ) <_ ( C - D ) ) )
Proof of Theorem le2sub
StepHypRef Expression
1 simpll 524 . . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> A e. RR )
2 simprl 526 . . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> C e. RR )
3 simplr 525 . . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> B e. RR )
4 lesub1 8375 . . . 4 |- ( ( A e. RR /\ C e. RR /\ B e. RR ) -> ( A <_ C <-> ( A - B ) <_ ( C - B ) ) )
51, 2, 3, 4syl3anc 1233 . . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( A <_ C <-> ( A - B ) <_ ( C - B ) ) )
6 simprr 527 . . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> D e. RR )
7 lesub2 8376 . . . 4 |- ( ( D e. RR /\ B e. RR /\ C e. RR ) -> ( D <_ B <-> ( C - B ) <_ ( C - D ) ) )
86, 3, 2, 7syl3anc 1233 . . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( D <_ B <-> ( C - B ) <_ ( C - D ) ) )
95, 8anbi12d 470 . 2 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( ( A <_ C /\ D <_ B ) <-> ( ( A - B ) <_ ( C - B ) /\ ( C - B ) <_ ( C - D ) ) ) )
10 resubcl 8183 . . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( A - B ) e. RR )
1110adantr 274 . . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( A - B ) e. RR )
122, 3resubcld 8300 . . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( C - B ) e. RR )
13 resubcl 8183 . . . 4 |- ( ( C e. RR /\ D e. RR ) -> ( C - D ) e. RR )
1413adantl 275 . . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( C - D ) e. RR )
15 letr 8002 . . 3 |- ( ( ( A - B ) e. RR /\ ( C - B ) e. RR /\ ( C - D ) e. RR ) -> ( ( ( A - B ) <_ ( C - B ) /\ ( C - B ) <_ ( C - D ) ) -> ( A - B ) <_ ( C - D ) ) )
1611, 12, 14, 15syl3anc 1233 . 2 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( ( ( A - B ) <_ ( C - B ) /\ ( C - B ) <_ ( C - D ) ) -> ( A - B ) <_ ( C - D ) ) )
179, 16sylbid 149 1 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( ( A <_ C /\ D <_ B ) -> ( A - B ) <_ ( C - D ) ) )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   e. wcel 2141   class class class wbr 3989  (class class class)co 5853   RRcr 7773   <_ cle 7955   - cmin 8090
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-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-addcom 7874  ax-addass 7876  ax-distr 7878  ax-i2m1 7879  ax-0id 7882  ax-rnegex 7883  ax-cnre 7885  ax-pre-ltwlin 7887  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-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
This theorem is referenced by:  le2subd  8483
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