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Theorem axltadd 9609
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 9518 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 9518 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <RR  B  ->  ( C  +  A )  <RR  ( C  +  B
) ) )
2 ltxrlt 9606 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <->  A 
<RR  B ) )
323adant3 1017 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <  B  <->  A  <RR  B ) )
4 readdcl 9525 . . . . 5  |-  ( ( C  e.  RR  /\  A  e.  RR )  ->  ( C  +  A
)  e.  RR )
5 readdcl 9525 . . . . 5  |-  ( ( C  e.  RR  /\  B  e.  RR )  ->  ( C  +  B
)  e.  RR )
6 ltxrlt 9606 . . . . 5  |-  ( ( ( C  +  A
)  e.  RR  /\  ( C  +  B
)  e.  RR )  ->  ( ( C  +  A )  < 
( C  +  B
)  <->  ( C  +  A )  <RR  ( C  +  B ) ) )
74, 5, 6syl2an 475 . . . 4  |-  ( ( ( C  e.  RR  /\  A  e.  RR )  /\  ( C  e.  RR  /\  B  e.  RR ) )  -> 
( ( C  +  A )  <  ( C  +  B )  <->  ( C  +  A ) 
<RR  ( C  +  B
) ) )
873impdi 1285 . . 3  |-  ( ( C  e.  RR  /\  A  e.  RR  /\  B  e.  RR )  ->  (
( C  +  A
)  <  ( C  +  B )  <->  ( C  +  A )  <RR  ( C  +  B ) ) )
983coml 1204 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( C  +  A
)  <  ( C  +  B )  <->  ( C  +  A )  <RR  ( C  +  B ) ) )
101, 3, 93imtr4d 268 1  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <  B  ->  ( C  +  A )  <  ( C  +  B
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 974    e. wcel 1842   class class class wbr 4394  (class class class)co 6234   RRcr 9441    + caddc 9445    <RR cltrr 9446    < clt 9578
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530  ax-resscn 9499  ax-addrcl 9503  ax-pre-ltadd 9518
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-er 7268  df-en 7475  df-dom 7476  df-sdom 7477  df-pnf 9580  df-mnf 9581  df-ltxr 9583
This theorem is referenced by:  ltadd2  9640  ltadd2iOLD  9668  nnge1  10522
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