Theorem ltaddpr 9459

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 9414	. . . . 5 |- ( B e. P. -> B =/= (/) )
2		n0 3741	. . . . 5 |- ( B =/= (/) <-> E. y y e. B )
3	1, 2	sylib 200	. . . 4 |- ( B e. P. -> E. y y e. B )
4	3	adantl 468	. . 3 |- ( ( A e. P. /\ B e. P. ) -> E. y y e. B )
5		addclpr 9443	. . . . . . . . . . . 12 |- ( ( A e. P. /\ B e. P. ) -> ( A +P. B ) e. P. )
6	5	adantr 467	. . . . . . . . . . 11 |- ( ( ( A e. P. /\ B e. P. ) /\ ( x e. A /\ y e. B ) ) -> ( A +P. B ) e. P. )
7		df-plp 9408	. . . . . . . . . . . . 13 |- +P. = ( w e. P. , v e. P. |-> { x | E. y e. w E. z e. v x = ( y +Q z ) } )
8		addclnq 9370	. . . . . . . . . . . . 13 |- ( ( y e. Q. /\ z e. Q. ) -> ( y +Q z ) e. Q. )
9	7, 8	genpprecl 9426	. . . . . . . . . . . 12 |- ( ( A e. P. /\ B e. P. ) -> ( ( x e. A /\ y e. B ) -> ( x +Q y ) e. ( A +P. B ) ) )
10	9	imp 431	. . . . . . . . . . 11 |- ( ( ( A e. P. /\ B e. P. ) /\ ( x e. A /\ y e. B ) ) -> ( x +Q y ) e. ( A +P. B ) )
11		elprnq 9416	. . . . . . . . . . . . 13 |- ( ( ( A +P. B ) e. P. /\ ( x +Q y ) e. ( A +P. B ) ) -> ( x +Q y ) e. Q. )
12		addnqf 9373	. . . . . . . . . . . . . . 15 |- +Q : ( Q. X. Q. ) --> Q.
13	12	fdmi 5734	. . . . . . . . . . . . . 14 |- dom +Q = ( Q. X. Q. )
14		0nnq 9349	. . . . . . . . . . . . . 14 |- -. (/) e. Q.
15	13, 14	ndmovrcl 6455	. . . . . . . . . . . . 13 |- ( ( x +Q y ) e. Q. -> ( x e. Q. /\ y e. Q. ) )
16		ltaddnq 9399	. . . . . . . . . . . . 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 9418	. . . . . . . . . . . 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 667	. . . . . . . . . 10 |- ( ( ( A e. P. /\ B e. P. ) /\ ( x e. A /\ y e. B ) ) -> x e. ( A +P. B ) )
21	20	exp32 610	. . . . . . . . 9 |- ( ( A e. P. /\ B e. P. ) -> ( x e. A -> ( y e. B -> x e. ( A +P. B ) ) ) )
22	21	com23 81	. . . . . . . 8 |- ( ( A e. P. /\ B e. P. ) -> ( y e. B -> ( x e. A -> x e. ( A +P. B ) ) ) )
23	22	alrimdv 1775	. . . . . . 7 |- ( ( A e. P. /\ B e. P. ) -> ( y e. B -> A. x ( x e. A -> x e. ( A +P. B ) ) ) )
24		dfss2 3421	. . . . . . 7 |- ( A C_ ( A +P. B ) <-> A. x ( x e. A -> x e. ( A +P. B ) ) )
25	23, 24	syl6ibr 231	. . . . . 6 |- ( ( A e. P. /\ B e. P. ) -> ( y e. B -> A C_ ( A +P. B ) ) )
26		vex 3048	. . . . . . . . 9 |- y e. _V
27	26	prlem934 9458	. . . . . . . 8 |- ( A e. P. -> E. x e. A -. ( x +Q y ) e. A )
28	27	adantr 467	. . . . . . 7 |- ( ( A e. P. /\ B e. P. ) -> E. x e. A -. ( x +Q y ) e. A )
29		eleq2 2518	. . . . . . . . . . . . 13 |- ( A = ( A +P. B ) -> ( ( x +Q y ) e. A <-> ( x +Q y ) e. ( A +P. B ) ) )
30	29	biimprcd 229	. . . . . . . . . . . 12 |- ( ( x +Q y ) e. ( A +P. B ) -> ( A = ( A +P. B ) -> ( x +Q y ) e. A ) )
31	30	con3d 139	. . . . . . . . . . 11 |- ( ( x +Q y ) e. ( A +P. B ) -> ( -. ( x +Q y ) e. A -> -. A = ( A +P. B ) ) )
32	9, 31	syl6 34	. . . . . . . . . 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 438	. . . . . . . . 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 86	. . . . . . . 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 2877	. . . . . . 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 536	. . . . 5 |- ( ( A e. P. /\ B e. P. ) -> ( y e. B -> ( A C_ ( A +P. B ) /\ -. A = ( A +P. B ) ) ) )
38		dfpss2 3518	. . . . 5 |- ( A C. ( A +P. B ) <-> ( A C_ ( A +P. B ) /\ -. A = ( A +P. B ) ) )
39	37, 38	syl6ibr 231	. . . 4 |- ( ( A e. P. /\ B e. P. ) -> ( y e. B -> A C. ( A +P. B ) ) )
40	39	exlimdv 1779	. . 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 9455	. . 3 |- ( ( A e. P. /\ ( A +P. B ) e. P. ) -> ( A <P ( A +P. B ) <-> A C. ( A +P. B ) ) )
43	5, 42	syldan 473	. 2 |- ( ( A e. P. /\ B e. P. ) -> ( A <P ( A +P. B ) <-> A C. ( A +P. B ) ) )
44	41, 43	mpbird 236	1 |- ( ( A e. P. /\ B e. P. ) -> A <P ( A +P. B ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 188   /\ wa 371   A.wal 1442   = wceq 1444   E.wex 1663   e. wcel 1887   =/= wne 2622   E.wrex 2738   C_ wss 3404   C. wpss 3405   (/)c0 3731   class class class wbr 4402   X. cxp 4832  (class class class)co 6290   Q.cnq 9277   +Q cplq 9280   <Q cltq 9283   P.cnp 9284   +P. cpp 9286   <P cltp 9288

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-omul 7187  df-er 7363  df-ni 9297  df-pli 9298  df-mi 9299  df-lti 9300  df-plpq 9333  df-mpq 9334  df-ltpq 9335  df-enq 9336  df-nq 9337  df-erq 9338  df-plq 9339  df-mq 9340  df-1nq 9341  df-rq 9342  df-ltnq 9343  df-np 9406  df-plp 9408  df-ltp 9410

This theorem is referenced by:  ltaddpr2  9460  ltexprlem7  9467  ltaprlem  9469  0lt1sr  9519  mappsrpr  9532