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Theorem axltadd 8893
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 8810 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 8810 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <RR  B  ->  ( C  +  A )  <RR  ( C  +  B
) ) )
2 ltxrlt 8890 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <->  A 
<RR  B ) )
323adant3 977 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <  B  <->  A  <RR  B ) )
4 readdcl 8817 . . . . 5  |-  ( ( C  e.  RR  /\  A  e.  RR )  ->  ( C  +  A
)  e.  RR )
5 readdcl 8817 . . . . 5  |-  ( ( C  e.  RR  /\  B  e.  RR )  ->  ( C  +  B
)  e.  RR )
6 ltxrlt 8890 . . . . 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 1239 . . 3  |-  ( ( C  e.  RR  /\  A  e.  RR  /\  B  e.  RR )  ->  (
( C  +  A
)  <  ( C  +  B )  <->  ( C  +  A )  <RR  ( C  +  B ) ) )
983coml 1160 . 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 936    e. wcel 1687   class class class wbr 4026  (class class class)co 5821   RRcr 8733    + caddc 8737    <RR cltrr 8738    < clt 8864
This theorem is referenced by:  ltadd2  8921  ltadd2i  8947  nnge1  9769  stoweidlem11  27161
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1638  ax-8 1646  ax-13 1689  ax-14 1691  ax-6 1706  ax-7 1711  ax-11 1718  ax-12 1870  ax-ext 2267  ax-sep 4144  ax-nul 4152  ax-pow 4189  ax-pr 4215  ax-un 4513  ax-resscn 8791  ax-addrcl 8795  ax-pre-ltadd 8810
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1531  df-nf 1534  df-sb 1633  df-eu 2150  df-mo 2151  df-clab 2273  df-cleq 2279  df-clel 2282  df-nfc 2411  df-ne 2451  df-nel 2452  df-ral 2551  df-rex 2552  df-rab 2555  df-v 2793  df-sbc 2995  df-csb 3085  df-dif 3158  df-un 3160  df-in 3162  df-ss 3169  df-nul 3459  df-if 3569  df-pw 3630  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3831  df-br 4027  df-opab 4081  df-mpt 4082  df-id 4310  df-xp 4696  df-rel 4697  df-cnv 4698  df-co 4699  df-dm 4700  df-rn 4701  df-res 4702  df-ima 4703  df-fun 5225  df-fn 5226  df-f 5227  df-f1 5228  df-fo 5229  df-f1o 5230  df-fv 5231  df-er 6657  df-en 6861  df-dom 6862  df-sdom 6863  df-pnf 8866  df-mnf 8867  df-ltxr 8869
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