Theorem ltadd2 8446

Description: Addition to both sides of 'less than'. (Contributed by NM, 12-Nov-1999.) (Revised by Mario Carneiro, 27-May-2016.)

Assertion

Ref	Expression
ltadd2	|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < B <-> ( C + A ) < ( C + B ) ) )

Proof of Theorem ltadd2

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		axltadd 8096	. 2 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < B -> ( C + A ) < ( C + B ) ) )
2		ax-rnegex 7988	. . . 4 |- ( C e. RR -> E. x e. RR ( C + x ) = 0 )
3	2	3ad2ant3 1022	. . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> E. x e. RR ( C + x ) = 0 )
4		simpl3 1004	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> C e. RR )
5		simpl1 1002	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> A e. RR )
6	4, 5	readdcld 8056	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> ( C + A ) e. RR )
7		simpl2 1003	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> B e. RR )
8	4, 7	readdcld 8056	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> ( C + B ) e. RR )
9		simprl 529	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> x e. RR )
10		axltadd 8096	. . . . . 6 |- ( ( ( C + A ) e. RR /\ ( C + B ) e. RR /\ x e. RR ) -> ( ( C + A ) < ( C + B ) -> ( x + ( C + A ) ) < ( x + ( C + B ) ) ) )
11	6, 8, 9, 10	syl3anc 1249	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> ( ( C + A ) < ( C + B ) -> ( x + ( C + A ) ) < ( x + ( C + B ) ) ) )
12	9	recnd 8055	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> x e. CC )
13	4	recnd 8055	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> C e. CC )
14	5	recnd 8055	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> A e. CC )
15	12, 13, 14	addassd 8049	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> ( ( x + C ) + A ) = ( x + ( C + A ) ) )
16	7	recnd 8055	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> B e. CC )
17	12, 13, 16	addassd 8049	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> ( ( x + C ) + B ) = ( x + ( C + B ) ) )
18	15, 17	breq12d 4046	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> ( ( ( x + C ) + A ) < ( ( x + C ) + B ) <-> ( x + ( C + A ) ) < ( x + ( C + B ) ) ) )
19	11, 18	sylibrd 169	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> ( ( C + A ) < ( C + B ) -> ( ( x + C ) + A ) < ( ( x + C ) + B ) ) )
20		simprr 531	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> ( C + x ) = 0 )
21		addcom 8163	. . . . . . . . . 10 |- ( ( C e. CC /\ x e. CC ) -> ( C + x ) = ( x + C ) )
22	21	eqeq1d 2205	. . . . . . . . 9 |- ( ( C e. CC /\ x e. CC ) -> ( ( C + x ) = 0 <-> ( x + C ) = 0 ) )
23	13, 12, 22	syl2anc 411	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> ( ( C + x ) = 0 <-> ( x + C ) = 0 ) )
24	20, 23	mpbid 147	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> ( x + C ) = 0 )
25	24	oveq1d 5937	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> ( ( x + C ) + A ) = ( 0 + A ) )
26	14	addlidd 8176	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> ( 0 + A ) = A )
27	25, 26	eqtrd 2229	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> ( ( x + C ) + A ) = A )
28	24	oveq1d 5937	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> ( ( x + C ) + B ) = ( 0 + B ) )
29	16	addlidd 8176	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> ( 0 + B ) = B )
30	28, 29	eqtrd 2229	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> ( ( x + C ) + B ) = B )
31	27, 30	breq12d 4046	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> ( ( ( x + C ) + A ) < ( ( x + C ) + B ) <-> A < B ) )
32	19, 31	sylibd 149	. . 3 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> ( ( C + A ) < ( C + B ) -> A < B ) )
33	3, 32	rexlimddv 2619	. 2 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( C + A ) < ( C + B ) -> A < B ) )
34	1, 33	impbid 129	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   = wceq 1364   e. wcel 2167   E.wrex 2476   class class class wbr 4033  (class class class)co 5922   CCcc 7877   RRcr 7878   0cc0 7879   + caddc 7882   < 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-1cn 7972  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-i2m1 7984  ax-0id 7987  ax-rnegex 7988  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-iota 5219  df-fv 5266  df-ov 5925  df-pnf 8063  df-mnf 8064  df-ltxr 8066

This theorem is referenced by:  ltadd2i  8447  ltadd2d  8448  ltaddneg  8451  ltadd1  8456  ltaddpos  8479  ltsub2  8486  ltaddsublt  8598  avglt1  9230  flqbi2  10381