Theorem ltaddpr 9856

Description: The sum of two positive reals is greater than one of them. Proposition 9-3.5(iii) of [Gleason] p. 123. (Contributed by NM, 26-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
ltaddpr	|- ( ( A e. P. /\ B e. P. ) -> A <P ( A +P. B ) )

Proof of Theorem ltaddpr

Dummy variables x y z w v are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prn0 9811	. . . . 5 |- ( B e. P. -> B =/= (/) )
2		n0 3931	. . . . 5 |- ( B =/= (/) <-> E. y y e. B )
3	1, 2	sylib 208	. . . 4 |- ( B e. P. -> E. y y e. B )
4	3	adantl 482	. . 3 |- ( ( A e. P. /\ B e. P. ) -> E. y y e. B )
5		addclpr 9840	. . . . . . . . . . . 12 |- ( ( A e. P. /\ B e. P. ) -> ( A +P. B ) e. P. )
6	5	adantr 481	. . . . . . . . . . 11 |- ( ( ( A e. P. /\ B e. P. ) /\ ( x e. A /\ y e. B ) ) -> ( A +P. B ) e. P. )
7		df-plp 9805	. . . . . . . . . . . . 13 |- +P. = ( w e. P. , v e. P. |-> { x | E. y e. w E. z e. v x = ( y +Q z ) } )
8		addclnq 9767	. . . . . . . . . . . . 13 |- ( ( y e. Q. /\ z e. Q. ) -> ( y +Q z ) e. Q. )
9	7, 8	genpprecl 9823	. . . . . . . . . . . 12 |- ( ( A e. P. /\ B e. P. ) -> ( ( x e. A /\ y e. B ) -> ( x +Q y ) e. ( A +P. B ) ) )
10	9	imp 445	. . . . . . . . . . 11 |- ( ( ( A e. P. /\ B e. P. ) /\ ( x e. A /\ y e. B ) ) -> ( x +Q y ) e. ( A +P. B ) )
11		elprnq 9813	. . . . . . . . . . . . 13 |- ( ( ( A +P. B ) e. P. /\ ( x +Q y ) e. ( A +P. B ) ) -> ( x +Q y ) e. Q. )
12		addnqf 9770	. . . . . . . . . . . . . . 15 |- +Q : ( Q. X. Q. ) --> Q.
13	12	fdmi 6052	. . . . . . . . . . . . . 14 |- dom +Q = ( Q. X. Q. )
14		0nnq 9746	. . . . . . . . . . . . . 14 |- -. (/) e. Q.
15	13, 14	ndmovrcl 6820	. . . . . . . . . . . . 13 |- ( ( x +Q y ) e. Q. -> ( x e. Q. /\ y e. Q. ) )
16		ltaddnq 9796	. . . . . . . . . . . . 13 |- ( ( x e. Q. /\ y e. Q. ) -> x <Q ( x +Q y ) )
17	11, 15, 16	3syl 18	. . . . . . . . . . . 12 |- ( ( ( A +P. B ) e. P. /\ ( x +Q y ) e. ( A +P. B ) ) -> x <Q ( x +Q y ) )
18		prcdnq 9815	. . . . . . . . . . . 12 |- ( ( ( A +P. B ) e. P. /\ ( x +Q y ) e. ( A +P. B ) ) -> ( x <Q ( x +Q y ) -> x e. ( A +P. B ) ) )
19	17, 18	mpd 15	. . . . . . . . . . 11 |- ( ( ( A +P. B ) e. P. /\ ( x +Q y ) e. ( A +P. B ) ) -> x e. ( A +P. B ) )
20	6, 10, 19	syl2anc 693	. . . . . . . . . 10 |- ( ( ( A e. P. /\ B e. P. ) /\ ( x e. A /\ y e. B ) ) -> x e. ( A +P. B ) )
21	20	exp32 631	. . . . . . . . 9 |- ( ( A e. P. /\ B e. P. ) -> ( x e. A -> ( y e. B -> x e. ( A +P. B ) ) ) )
22	21	com23 86	. . . . . . . 8 |- ( ( A e. P. /\ B e. P. ) -> ( y e. B -> ( x e. A -> x e. ( A +P. B ) ) ) )
23	22	alrimdv 1857	. . . . . . 7 |- ( ( A e. P. /\ B e. P. ) -> ( y e. B -> A. x ( x e. A -> x e. ( A +P. B ) ) ) )
24		dfss2 3591	. . . . . . 7 |- ( A C_ ( A +P. B ) <-> A. x ( x e. A -> x e. ( A +P. B ) ) )
25	23, 24	syl6ibr 242	. . . . . 6 |- ( ( A e. P. /\ B e. P. ) -> ( y e. B -> A C_ ( A +P. B ) ) )
26		vex 3203	. . . . . . . . 9 |- y e. _V
27	26	prlem934 9855	. . . . . . . 8 |- ( A e. P. -> E. x e. A -. ( x +Q y ) e. A )
28	27	adantr 481	. . . . . . 7 |- ( ( A e. P. /\ B e. P. ) -> E. x e. A -. ( x +Q y ) e. A )
29		eleq2 2690	. . . . . . . . . . . . 13 |- ( A = ( A +P. B ) -> ( ( x +Q y ) e. A <-> ( x +Q y ) e. ( A +P. B ) ) )
30	29	biimprcd 240	. . . . . . . . . . . 12 |- ( ( x +Q y ) e. ( A +P. B ) -> ( A = ( A +P. B ) -> ( x +Q y ) e. A ) )
31	30	con3d 148	. . . . . . . . . . 11 |- ( ( x +Q y ) e. ( A +P. B ) -> ( -. ( x +Q y ) e. A -> -. A = ( A +P. B ) ) )
32	9, 31	syl6 35	. . . . . . . . . 10 |- ( ( A e. P. /\ B e. P. ) -> ( ( x e. A /\ y e. B ) -> ( -. ( x +Q y ) e. A -> -. A = ( A +P. B ) ) ) )
33	32	expd 452	. . . . . . . . 9 |- ( ( A e. P. /\ B e. P. ) -> ( x e. A -> ( y e. B -> ( -. ( x +Q y ) e. A -> -. A = ( A +P. B ) ) ) ) )
34	33	com34 91	. . . . . . . 8 |- ( ( A e. P. /\ B e. P. ) -> ( x e. A -> ( -. ( x +Q y ) e. A -> ( y e. B -> -. A = ( A +P. B ) ) ) ) )
35	34	rexlimdv 3030	. . . . . . 7 |- ( ( A e. P. /\ B e. P. ) -> ( E. x e. A -. ( x +Q y ) e. A -> ( y e. B -> -. A = ( A +P. B ) ) ) )
36	28, 35	mpd 15	. . . . . 6 |- ( ( A e. P. /\ B e. P. ) -> ( y e. B -> -. A = ( A +P. B ) ) )
37	25, 36	jcad 555	. . . . 5 |- ( ( A e. P. /\ B e. P. ) -> ( y e. B -> ( A C_ ( A +P. B ) /\ -. A = ( A +P. B ) ) ) )
38		dfpss2 3692	. . . . 5 |- ( A C. ( A +P. B ) <-> ( A C_ ( A +P. B ) /\ -. A = ( A +P. B ) ) )
39	37, 38	syl6ibr 242	. . . 4 |- ( ( A e. P. /\ B e. P. ) -> ( y e. B -> A C. ( A +P. B ) ) )
40	39	exlimdv 1861	. . 3 |- ( ( A e. P. /\ B e. P. ) -> ( E. y y e. B -> A C. ( A +P. B ) ) )
41	4, 40	mpd 15	. 2 |- ( ( A e. P. /\ B e. P. ) -> A C. ( A +P. B ) )
42		ltprord 9852	. . 3 |- ( ( A e. P. /\ ( A +P. B ) e. P. ) -> ( A <P ( A +P. B ) <-> A C. ( A +P. B ) ) )
43	5, 42	syldan 487	. 2 |- ( ( A e. P. /\ B e. P. ) -> ( A <P ( A +P. B ) <-> A C. ( A +P. B ) ) )
44	41, 43	mpbird 247	1 |- ( ( A e. P. /\ B e. P. ) -> A <P ( A +P. B ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   E.wrex 2913   C_ wss 3574   C. wpss 3575   (/)c0 3915   class class class wbr 4653   X. cxp 5112  (class class class)co 6650   Q.cnq 9674   +Q cplq 9677   <Q cltq 9680   P.cnp 9681   +P. cpp 9683   <P cltp 9685

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-omul 7565  df-er 7742  df-ni 9694  df-pli 9695  df-mi 9696  df-lti 9697  df-plpq 9730  df-mpq 9731  df-ltpq 9732  df-enq 9733  df-nq 9734  df-erq 9735  df-plq 9736  df-mq 9737  df-1nq 9738  df-rq 9739  df-ltnq 9740  df-np 9803  df-plp 9805  df-ltp 9807

This theorem is referenced by:  ltaddpr2  9857  ltexprlem7  9864  ltaprlem  9866  0lt1sr  9916  mappsrpr  9929