Theorem axltadd 8096

Description: Ordering property of addition on reals. Axiom for real and complex numbers, derived from set theory. (This restates ax-pre-ltadd 7995 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 7995	. 2 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A <RR B -> ( C + A ) <RR ( C + B ) ) )
2		ltxrlt 8092	. . 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 8005	. . . . 5 |- ( ( C e. RR /\ A e. RR ) -> ( C + A ) e. RR )
5		readdcl 8005	. . . . 5 |- ( ( C e. RR /\ B e. RR ) -> ( C + B ) e. RR )
6		ltxrlt 8092	. . . . 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 2167   class class class wbr 4033  (class class class)co 5922   RRcr 7878   + caddc 7882   <RR cltrr 7883   < clt 8061

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-addrcl 7976  ax-pre-ltadd 7995

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-xp 4669  df-pnf 8063  df-mnf 8064  df-ltxr 8066

This theorem is referenced by:  ltadd2  8446  nnge1  9013  ltoddhalfle  12058