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Theorem xleadd1 10826
Description: Weakened version of xleadd1a 10824 under which the reverse implication is true. (Contributed by Mario Carneiro, 20-Aug-2015.)
Assertion
Ref Expression
xleadd1  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  ->  ( A  <_  B  <->  ( A + e C )  <_ 
( B + e C ) ) )

Proof of Theorem xleadd1
StepHypRef Expression
1 rexr 9122 . . 3  |-  ( C  e.  RR  ->  C  e.  RR* )
2 xleadd1a 10824 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  A  <_  B )  ->  ( A + e C )  <_  ( B + e C ) )
32ex 424 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  ( A  <_  B  ->  ( A + e C )  <_  ( B + e C ) ) )
41, 3syl3an3 1219 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  ->  ( A  <_  B  ->  ( A + e C )  <_  ( B + e C ) ) )
5 simp1 957 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  ->  A  e.  RR* )
613ad2ant3 980 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  ->  C  e.  RR* )
7 xaddcl 10815 . . . . 5  |-  ( ( A  e.  RR*  /\  C  e.  RR* )  ->  ( A + e C )  e.  RR* )
85, 6, 7syl2anc 643 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  ->  ( A + e C )  e.  RR* )
9 simp2 958 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  ->  B  e.  RR* )
10 xaddcl 10815 . . . . 5  |-  ( ( B  e.  RR*  /\  C  e.  RR* )  ->  ( B + e C )  e.  RR* )
119, 6, 10syl2anc 643 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  ->  ( B + e C )  e.  RR* )
12 xnegcl 10791 . . . . 5  |-  ( C  e.  RR*  ->  - e C  e.  RR* )
136, 12syl 16 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  ->  - e C  e.  RR* )
14 xleadd1a 10824 . . . . 5  |-  ( ( ( ( A + e C )  e.  RR*  /\  ( B + e C )  e.  RR*  /\  - e C  e.  RR* )  /\  ( A + e C )  <_  ( B + e C ) )  ->  ( ( A + e C ) + e  - e C )  <_  (
( B + e C ) + e  - e C ) )
1514ex 424 . . . 4  |-  ( ( ( A + e C )  e.  RR*  /\  ( B + e C )  e.  RR*  /\  - e C  e.  RR* )  ->  ( ( A + e C )  <_  ( B + e C )  ->  (
( A + e C ) + e  - e C )  <_ 
( ( B + e C ) + e  - e C ) ) )
168, 11, 13, 15syl3anc 1184 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  ->  (
( A + e C )  <_  ( B + e C )  ->  ( ( A + e C ) + e  - e C )  <_  (
( B + e C ) + e  - e C ) ) )
17 xpncan 10822 . . . . 5  |-  ( ( A  e.  RR*  /\  C  e.  RR )  ->  (
( A + e C ) + e  - e C )  =  A )
18173adant2 976 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  ->  (
( A + e C ) + e  - e C )  =  A )
19 xpncan 10822 . . . . 5  |-  ( ( B  e.  RR*  /\  C  e.  RR )  ->  (
( B + e C ) + e  - e C )  =  B )
20193adant1 975 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  ->  (
( B + e C ) + e  - e C )  =  B )
2118, 20breq12d 4217 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  ->  (
( ( A + e C ) + e  - e C )  <_ 
( ( B + e C ) + e  - e C )  <->  A  <_  B ) )
2216, 21sylibd 206 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  ->  (
( A + e C )  <_  ( B + e C )  ->  A  <_  B
) )
234, 22impbid 184 1  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR )  ->  ( A  <_  B  <->  ( A + e C )  <_ 
( B + e C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ w3a 936    = wceq 1652    e. wcel 1725   class class class wbr 4204  (class class class)co 6073   RRcr 8981   RR*cxr 9111    <_ cle 9113    - ecxne 10699   + ecxad 10700
This theorem is referenced by:  xltadd1  10827  xsubge0  10832  xlesubadd  10834
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-po 4495  df-so 4496  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-xneg 10702  df-xadd 10703
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