Theorem axltadd 8248

Description: Ordering property of addition on reals. Axiom for real and complex numbers, derived from set theory. (This restates ax-pre-ltadd 8147 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 8147	. 2 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A <RR B -> ( C + A ) <RR ( C + B ) ) )
2		ltxrlt 8244	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( A < B <-> A <RR B ) )
3	2	3adant3 1043	. 2 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < B <-> A <RR B ) )
4		readdcl 8157	. . . . 5 |- ( ( C e. RR /\ A e. RR ) -> ( C + A ) e. RR )
5		readdcl 8157	. . . . 5 |- ( ( C e. RR /\ B e. RR ) -> ( C + B ) e. RR )
6		ltxrlt 8244	. . . . 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 1329	. . 3 |- ( ( C e. RR /\ A e. RR /\ B e. RR ) -> ( ( C + A ) < ( C + B ) <-> ( C + A ) <RR ( C + B ) ) )
9	8	3coml 1236	. 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 1004   e. wcel 2202   class class class wbr 4088  (class class class)co 6017   RRcr 8030   + caddc 8034   <RR cltrr 8035   < clt 8213

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-addrcl 8128  ax-pre-ltadd 8147

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-xp 4731  df-pnf 8215  df-mnf 8216  df-ltxr 8218

This theorem is referenced by:  ltadd2  8598  nnge1  9165  ltoddhalfle  12453