Theorem axltadd 8359

Description: Ordering property of addition on reals. Axiom for real and complex numbers, derived from set theory. (This restates ax-pre-ltadd 8259 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 8259	. 2 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A <RR B -> ( C + A ) <RR ( C + B ) ) )
2		ltxrlt 8355	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( A < B <-> A <RR B ) )
3	2	3adant3 1044	. 2 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < B <-> A <RR B ) )
4		readdcl 8269	. . . . 5 |- ( ( C e. RR /\ A e. RR ) -> ( C + A ) e. RR )
5		readdcl 8269	. . . . 5 |- ( ( C e. RR /\ B e. RR ) -> ( C + B ) e. RR )
6		ltxrlt 8355	. . . . 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 1330	. . 3 |- ( ( C e. RR /\ A e. RR /\ B e. RR ) -> ( ( C + A ) < ( C + B ) <-> ( C + A ) <RR ( C + B ) ) )
9	8	3coml 1237	. 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 1005   e. wcel 2205   class class class wbr 4114  (class class class)co 6058   RRcr 8142   + caddc 8146   <RR cltrr 8147   < clt 8324

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-addrcl 8240  ax-pre-ltadd 8259

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-xp 4760  df-pnf 8326  df-mnf 8327  df-ltxr 8329

This theorem is referenced by:  ltadd2  8710  nnge1  9277  ltoddhalfle  12604