Theorem ltadd2 8327

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 7978	. 2 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < B -> ( C + A ) < ( C + B ) ) )
2		ax-rnegex 7872	. . . 4 |- ( C e. RR -> E. x e. RR ( C + x ) = 0 )
3	2	3ad2ant3 1015	. . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> E. x e. RR ( C + x ) = 0 )
4		simpl3 997	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> C e. RR )
5		simpl1 995	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> A e. RR )
6	4, 5	readdcld 7938	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> ( C + A ) e. RR )
7		simpl2 996	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> B e. RR )
8	4, 7	readdcld 7938	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> ( C + B ) e. RR )
9		simprl 526	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> x e. RR )
10		axltadd 7978	. . . . . 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 1233	. . . . 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 7937	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> x e. CC )
13	4	recnd 7937	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> C e. CC )
14	5	recnd 7937	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> A e. CC )
15	12, 13, 14	addassd 7931	. . . . . 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 7937	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> B e. CC )
17	12, 13, 16	addassd 7931	. . . . . 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 4000	. . . . 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 168	. . . 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 527	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> ( C + x ) = 0 )
21		addcom 8045	. . . . . . . . . 10 |- ( ( C e. CC /\ x e. CC ) -> ( C + x ) = ( x + C ) )
22	21	eqeq1d 2179	. . . . . . . . 9 |- ( ( C e. CC /\ x e. CC ) -> ( ( C + x ) = 0 <-> ( x + C ) = 0 ) )
23	13, 12, 22	syl2anc 409	. . . . . . . 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 146	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> ( x + C ) = 0 )
25	24	oveq1d 5866	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> ( ( x + C ) + A ) = ( 0 + A ) )
26	14	addid2d 8058	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> ( 0 + A ) = A )
27	25, 26	eqtrd 2203	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> ( ( x + C ) + A ) = A )
28	24	oveq1d 5866	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> ( ( x + C ) + B ) = ( 0 + B ) )
29	16	addid2d 8058	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> ( 0 + B ) = B )
30	28, 29	eqtrd 2203	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> ( ( x + C ) + B ) = B )
31	27, 30	breq12d 4000	. . . 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 148	. . 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 2592	. 2 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( C + A ) < ( C + B ) -> A < B ) )
34	1, 33	impbid 128	1 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < B <-> ( C + A ) < ( C + B ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 973   = wceq 1348   e. wcel 2141   E.wrex 2449   class class class wbr 3987  (class class class)co 5851   CCcc 7761   RRcr 7762   0cc0 7763   + caddc 7766   < clt 7943

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-cnex 7854  ax-resscn 7855  ax-1cn 7856  ax-icn 7858  ax-addcl 7859  ax-addrcl 7860  ax-mulcl 7861  ax-addcom 7863  ax-addass 7865  ax-i2m1 7868  ax-0id 7871  ax-rnegex 7872  ax-pre-ltadd 7879

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-opab 4049  df-xp 4615  df-iota 5158  df-fv 5204  df-ov 5854  df-pnf 7945  df-mnf 7946  df-ltxr 7948

This theorem is referenced by:  ltadd2i  8328  ltadd2d  8329  ltaddneg  8332  ltadd1  8337  ltaddpos  8360  ltsub2  8367  ltaddsublt  8479  avglt1  9105  flqbi2  10236