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Theorem axltadd 8776
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 8693 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 8693 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <RR  B  ->  ( C  +  A )  <RR  ( C  +  B
) ) )
2 ltxrlt 8773 . . 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 8700 . . . . 5  |-  ( ( C  e.  RR  /\  A  e.  RR )  ->  ( C  +  A
)  e.  RR )
5 readdcl 8700 . . . . 5  |-  ( ( C  e.  RR  /\  B  e.  RR )  ->  ( C  +  B
)  e.  RR )
6 ltxrlt 8773 . . . . 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 3920  (class class class)co 5710   RRcr 8616    + caddc 8620    <RR cltrr 8621    < clt 8747
This theorem is referenced by:  ltadd2  8804  ltadd2i  8830  nnge1  9652
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 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-resscn 8674  ax-addrcl 8678  ax-pre-ltadd 8693
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 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-er 6546  df-en 6750  df-dom 6751  df-sdom 6752  df-pnf 8749  df-mnf 8750  df-ltxr 8752
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