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Theorem axltadd 8772
Description: Ordering property of addition on reals. Axiom 20 of 22 for real and complex numbers, derived from ZF set theory. (This restates ax-pre-ltadd 8690 with ordering on the extended reals.) (Contributed by NM, 13-Oct-2005.)
Assertion
Ref Expression
axltadd  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <  B  ->  ( C  +  A )  <  ( C  +  B
) ) )

Proof of Theorem axltadd
StepHypRef Expression
1 ax-pre-ltadd 8690 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <RR  B  ->  ( C  +  A )  <RR  ( C  +  B
) ) )
2 ltxrlt 8769 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <->  A 
<RR  B ) )
323adant3 980 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <  B  <->  A  <RR  B ) )
4 readdcl 8697 . . . . 5  |-  ( ( C  e.  RR  /\  A  e.  RR )  ->  ( C  +  A
)  e.  RR )
5 readdcl 8697 . . . . 5  |-  ( ( C  e.  RR  /\  B  e.  RR )  ->  ( C  +  B
)  e.  RR )
6 ltxrlt 8769 . . . . 5  |-  ( ( ( C  +  A
)  e.  RR  /\  ( C  +  B
)  e.  RR )  ->  ( ( C  +  A )  < 
( C  +  B
)  <->  ( C  +  A )  <RR  ( C  +  B ) ) )
74, 5, 6syl2an 465 . . . 4  |-  ( ( ( C  e.  RR  /\  A  e.  RR )  /\  ( C  e.  RR  /\  B  e.  RR ) )  -> 
( ( C  +  A )  <  ( C  +  B )  <->  ( C  +  A ) 
<RR  ( C  +  B
) ) )
873impdi 1242 . . 3  |-  ( ( C  e.  RR  /\  A  e.  RR  /\  B  e.  RR )  ->  (
( C  +  A
)  <  ( C  +  B )  <->  ( C  +  A )  <RR  ( C  +  B ) ) )
983coml 1163 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( C  +  A
)  <  ( C  +  B )  <->  ( C  +  A )  <RR  ( C  +  B ) ) )
101, 3, 93imtr4d 261 1  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <  B  ->  ( C  +  A )  <  ( C  +  B
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    /\ w3a 939    e. wcel 1621   class class class wbr 3917  (class class class)co 5707   RRcr 8613    + caddc 8617    <RR cltrr 8618    < clt 8744
This theorem is referenced by:  ltadd2  8800  ltadd2i  8826  nnge1  9620
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4035  ax-nul 4043  ax-pow 4079  ax-pr 4105  ax-un 4400  ax-resscn 8671  ax-addrcl 8675  ax-pre-ltadd 8690
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2511  df-rex 2512  df-rab 2514  df-v 2727  df-sbc 2920  df-csb 3007  df-dif 3078  df-un 3080  df-in 3082  df-ss 3086  df-nul 3360  df-if 3468  df-pw 3529  df-sn 3547  df-pr 3548  df-op 3550  df-uni 3725  df-br 3918  df-opab 3972  df-mpt 3973  df-id 4199  df-xp 4591  df-rel 4592  df-cnv 4593  df-co 4594  df-dm 4595  df-rn 4596  df-res 4597  df-ima 4598  df-fun 4599  df-fn 4600  df-f 4601  df-f1 4602  df-fo 4603  df-f1o 4604  df-fv 4605  df-er 6543  df-en 6747  df-dom 6748  df-sdom 6749  df-pnf 8746  df-mnf 8747  df-ltxr 8749
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