Theorem ltadd2 8522

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 8172	. 2 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < B -> ( C + A ) < ( C + B ) ) )
2		ax-rnegex 8064	. . . 4 |- ( C e. RR -> E. x e. RR ( C + x ) = 0 )
3	2	3ad2ant3 1023	. . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> E. x e. RR ( C + x ) = 0 )
4		simpl3 1005	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> C e. RR )
5		simpl1 1003	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> A e. RR )
6	4, 5	readdcld 8132	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> ( C + A ) e. RR )
7		simpl2 1004	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> B e. RR )
8	4, 7	readdcld 8132	. . . . . 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 8172	. . . . . 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 1250	. . . . 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 8131	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> x e. CC )
13	4	recnd 8131	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> C e. CC )
14	5	recnd 8131	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> A e. CC )
15	12, 13, 14	addassd 8125	. . . . . 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 8131	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> B e. CC )
17	12, 13, 16	addassd 8125	. . . . . 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 4067	. . . . 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 8239	. . . . . . . . . 10 |- ( ( C e. CC /\ x e. CC ) -> ( C + x ) = ( x + C ) )
22	21	eqeq1d 2215	. . . . . . . . 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 5977	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> ( ( x + C ) + A ) = ( 0 + A ) )
26	14	addlidd 8252	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> ( 0 + A ) = A )
27	25, 26	eqtrd 2239	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> ( ( x + C ) + A ) = A )
28	24	oveq1d 5977	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> ( ( x + C ) + B ) = ( 0 + B ) )
29	16	addlidd 8252	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> ( 0 + B ) = B )
30	28, 29	eqtrd 2239	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( x e. RR /\ ( C + x ) = 0 ) ) -> ( ( x + C ) + B ) = B )
31	27, 30	breq12d 4067	. . . 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 2629	. 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 981   = wceq 1373   e. wcel 2177   E.wrex 2486   class class class wbr 4054  (class class class)co 5962   CCcc 7953   RRcr 7954   0cc0 7955   + caddc 7958   < clt 8137

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4173  ax-pow 4229  ax-pr 4264  ax-un 4493  ax-setind 4598  ax-cnex 8046  ax-resscn 8047  ax-1cn 8048  ax-icn 8050  ax-addcl 8051  ax-addrcl 8052  ax-mulcl 8053  ax-addcom 8055  ax-addass 8057  ax-i2m1 8060  ax-0id 8063  ax-rnegex 8064  ax-pre-ltadd 8071

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-rab 2494  df-v 2775  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3860  df-br 4055  df-opab 4117  df-xp 4694  df-iota 5246  df-fv 5293  df-ov 5965  df-pnf 8139  df-mnf 8140  df-ltxr 8142

This theorem is referenced by:  ltadd2i  8523  ltadd2d  8524  ltaddneg  8527  ltadd1  8532  ltaddpos  8555  ltsub2  8562  ltaddsublt  8674  avglt1  9306  flqbi2  10466