Theorem ltaddpr 9195

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 9150	. . . . 5 |- ( B e. P. -> B =/= (/) )
2		n0 3641	. . . . 5 |- ( B =/= (/) <-> E. y y e. B )
3	1, 2	sylib 196	. . . 4 |- ( B e. P. -> E. y y e. B )
4	3	adantl 466	. . 3 |- ( ( A e. P. /\ B e. P. ) -> E. y y e. B )
5		addclpr 9179	. . . . . . . . . . . 12 |- ( ( A e. P. /\ B e. P. ) -> ( A +P. B ) e. P. )
6	5	adantr 465	. . . . . . . . . . 11 |- ( ( ( A e. P. /\ B e. P. ) /\ ( x e. A /\ y e. B ) ) -> ( A +P. B ) e. P. )
7		df-plp 9144	. . . . . . . . . . . . 13 |- +P. = ( w e. P. , v e. P. |-> { x | E. y e. w E. z e. v x = ( y +Q z ) } )
8		addclnq 9106	. . . . . . . . . . . . 13 |- ( ( y e. Q. /\ z e. Q. ) -> ( y +Q z ) e. Q. )
9	7, 8	genpprecl 9162	. . . . . . . . . . . 12 |- ( ( A e. P. /\ B e. P. ) -> ( ( x e. A /\ y e. B ) -> ( x +Q y ) e. ( A +P. B ) ) )
10	9	imp 429	. . . . . . . . . . 11 |- ( ( ( A e. P. /\ B e. P. ) /\ ( x e. A /\ y e. B ) ) -> ( x +Q y ) e. ( A +P. B ) )
11		elprnq 9152	. . . . . . . . . . . . 13 |- ( ( ( A +P. B ) e. P. /\ ( x +Q y ) e. ( A +P. B ) ) -> ( x +Q y ) e. Q. )
12		addnqf 9109	. . . . . . . . . . . . . . 15 |- +Q : ( Q. X. Q. ) --> Q.
13	12	fdmi 5559	. . . . . . . . . . . . . 14 |- dom +Q = ( Q. X. Q. )
14		0nnq 9085	. . . . . . . . . . . . . 14 |- -. (/) e. Q.
15	13, 14	ndmovrcl 6244	. . . . . . . . . . . . 13 |- ( ( x +Q y ) e. Q. -> ( x e. Q. /\ y e. Q. ) )
16		ltaddnq 9135	. . . . . . . . . . . . 13 |- ( ( x e. Q. /\ y e. Q. ) -> x <Q ( x +Q y ) )
17	11, 15, 16	3syl 20	. . . . . . . . . . . 12 |- ( ( ( A +P. B ) e. P. /\ ( x +Q y ) e. ( A +P. B ) ) -> x <Q ( x +Q y ) )
18		prcdnq 9154	. . . . . . . . . . . 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 661	. . . . . . . . . 10 |- ( ( ( A e. P. /\ B e. P. ) /\ ( x e. A /\ y e. B ) ) -> x e. ( A +P. B ) )
21	20	exp32 605	. . . . . . . . 9 |- ( ( A e. P. /\ B e. P. ) -> ( x e. A -> ( y e. B -> x e. ( A +P. B ) ) ) )
22	21	com23 78	. . . . . . . 8 |- ( ( A e. P. /\ B e. P. ) -> ( y e. B -> ( x e. A -> x e. ( A +P. B ) ) ) )
23	22	alrimdv 1687	. . . . . . 7 |- ( ( A e. P. /\ B e. P. ) -> ( y e. B -> A. x ( x e. A -> x e. ( A +P. B ) ) ) )
24		dfss2 3340	. . . . . . 7 |- ( A C_ ( A +P. B ) <-> A. x ( x e. A -> x e. ( A +P. B ) ) )
25	23, 24	syl6ibr 227	. . . . . 6 |- ( ( A e. P. /\ B e. P. ) -> ( y e. B -> A C_ ( A +P. B ) ) )
26		vex 2970	. . . . . . . . 9 |- y e. _V
27	26	prlem934 9194	. . . . . . . 8 |- ( A e. P. -> E. x e. A -. ( x +Q y ) e. A )
28	27	adantr 465	. . . . . . 7 |- ( ( A e. P. /\ B e. P. ) -> E. x e. A -. ( x +Q y ) e. A )
29		eleq2 2499	. . . . . . . . . . . . 13 |- ( A = ( A +P. B ) -> ( ( x +Q y ) e. A <-> ( x +Q y ) e. ( A +P. B ) ) )
30	29	biimprcd 225	. . . . . . . . . . . 12 |- ( ( x +Q y ) e. ( A +P. B ) -> ( A = ( A +P. B ) -> ( x +Q y ) e. A ) )
31	30	con3d 133	. . . . . . . . . . 11 |- ( ( x +Q y ) e. ( A +P. B ) -> ( -. ( x +Q y ) e. A -> -. A = ( A +P. B ) ) )
32	9, 31	syl6 33	. . . . . . . . . 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 436	. . . . . . . . 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 83	. . . . . . . 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 2835	. . . . . . 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 533	. . . . 5 |- ( ( A e. P. /\ B e. P. ) -> ( y e. B -> ( A C_ ( A +P. B ) /\ -. A = ( A +P. B ) ) ) )
38		dfpss2 3436	. . . . 5 |- ( A C. ( A +P. B ) <-> ( A C_ ( A +P. B ) /\ -. A = ( A +P. B ) ) )
39	37, 38	syl6ibr 227	. . . 4 |- ( ( A e. P. /\ B e. P. ) -> ( y e. B -> A C. ( A +P. B ) ) )
40	39	exlimdv 1690	. . 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 9191	. . 3 |- ( ( A e. P. /\ ( A +P. B ) e. P. ) -> ( A <P ( A +P. B ) <-> A C. ( A +P. B ) ) )
43	5, 42	syldan 470	. 2 |- ( ( A e. P. /\ B e. P. ) -> ( A <P ( A +P. B ) <-> A C. ( A +P. B ) ) )
44	41, 43	mpbird 232	1 |- ( ( A e. P. /\ B e. P. ) -> A <P ( A +P. B ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   A.wal 1367   = wceq 1369   E.wex 1586   e. wcel 1756   =/= wne 2601   E.wrex 2711   C_ wss 3323   C. wpss 3324   (/)c0 3632   class class class wbr 4287   X. cxp 4833  (class class class)co 6086   Q.cnq 9011   +Q cplq 9014   <Q cltq 9017   P.cnp 9018   +P. cpp 9020   <P cltp 9022

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-1o 6912  df-oadd 6916  df-omul 6917  df-er 7093  df-ni 9033  df-pli 9034  df-mi 9035  df-lti 9036  df-plpq 9069  df-mpq 9070  df-ltpq 9071  df-enq 9072  df-nq 9073  df-erq 9074  df-plq 9075  df-mq 9076  df-1nq 9077  df-rq 9078  df-ltnq 9079  df-np 9142  df-plp 9144  df-ltp 9146

This theorem is referenced by:  ltaddpr2  9196  ltexprlem7  9203  ltaprlem  9205  0lt1sr  9254  mappsrpr  9267