Theorem axltadd 8091

Description: Ordering property of addition on reals. Axiom for real and complex numbers, derived from set theory. (This restates ax-pre-ltadd 7990 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

Step	Hyp	Ref	Expression
1		ax-pre-ltadd 7990	. 2 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A <RR B -> ( C + A ) <RR ( C + B ) ) )
2		ltxrlt 8087	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( A < B <-> A <RR B ) )
3	2	3adant3 1019	. 2 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < B <-> A <RR B ) )
4		readdcl 8000	. . . . 5 |- ( ( C e. RR /\ A e. RR ) -> ( C + A ) e. RR )
5		readdcl 8000	. . . . 5 |- ( ( C e. RR /\ B e. RR ) -> ( C + B ) e. RR )
6		ltxrlt 8087	. . . . 5 |- ( ( ( C + A ) e. RR /\ ( C + B ) e. RR ) -> ( ( C + A ) < ( C + B ) <-> ( C + A ) <RR ( C + B ) ) )
7	4, 5, 6	syl2an 289	. . . 4 |- ( ( ( C e. RR /\ A e. RR ) /\ ( C e. RR /\ B e. RR ) ) -> ( ( C + A ) < ( C + B ) <-> ( C + A ) <RR ( C + B ) ) )
8	7	3impdi 1304	. . 3 |- ( ( C e. RR /\ A e. RR /\ B e. RR ) -> ( ( C + A ) < ( C + B ) <-> ( C + A ) <RR ( C + B ) ) )
9	8	3coml 1212	. 2 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( C + A ) < ( C + B ) <-> ( C + A ) <RR ( C + B ) ) )
10	1, 3, 9	3imtr4d 203	1 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < B -> ( C + A ) < ( C + B ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 980   e. wcel 2164   class class class wbr 4030  (class class class)co 5919   RRcr 7873   + caddc 7877   <RR cltrr 7878   < clt 8056

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-cnex 7965  ax-resscn 7966  ax-addrcl 7971  ax-pre-ltadd 7990

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-xp 4666  df-pnf 8058  df-mnf 8059  df-ltxr 8061

This theorem is referenced by:  ltadd2  8440  nnge1  9007  ltoddhalfle  12037