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Theorem axltadd 8896
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 8813 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 8813 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <RR  B  ->  ( C  +  A )  <RR  ( C  +  B
) ) )
2 ltxrlt 8893 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <->  A 
<RR  B ) )
323adant3 975 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <  B  <->  A  <RR  B ) )
4 readdcl 8820 . . . . 5  |-  ( ( C  e.  RR  /\  A  e.  RR )  ->  ( C  +  A
)  e.  RR )
5 readdcl 8820 . . . . 5  |-  ( ( C  e.  RR  /\  B  e.  RR )  ->  ( C  +  B
)  e.  RR )
6 ltxrlt 8893 . . . . 5  |-  ( ( ( C  +  A
)  e.  RR  /\  ( C  +  B
)  e.  RR )  ->  ( ( C  +  A )  < 
( C  +  B
)  <->  ( C  +  A )  <RR  ( C  +  B ) ) )
74, 5, 6syl2an 463 . . . 4  |-  ( ( ( C  e.  RR  /\  A  e.  RR )  /\  ( C  e.  RR  /\  B  e.  RR ) )  -> 
( ( C  +  A )  <  ( C  +  B )  <->  ( C  +  A ) 
<RR  ( C  +  B
) ) )
873impdi 1237 . . 3  |-  ( ( C  e.  RR  /\  A  e.  RR  /\  B  e.  RR )  ->  (
( C  +  A
)  <  ( C  +  B )  <->  ( C  +  A )  <RR  ( C  +  B ) ) )
983coml 1158 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( C  +  A
)  <  ( C  +  B )  <->  ( C  +  A )  <RR  ( C  +  B ) ) )
101, 3, 93imtr4d 259 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 4    <-> wb 176    /\ wa 358    /\ w3a 934    e. wcel 1684   class class class wbr 4023  (class class class)co 5858   RRcr 8736    + caddc 8740    <RR cltrr 8741    < clt 8867
This theorem is referenced by:  ltadd2  8924  ltadd2i  8950  nnge1  9772  stoweidlem11  27760
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8794  ax-addrcl 8798  ax-pre-ltadd 8813
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-ltxr 8872
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