Theorem ltaddpr 9317

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 9272	. . . . 5 |- ( B e. P. -> B =/= (/) )
2		n0 3757	. . . . 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 9301	. . . . . . . . . . . 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 9266	. . . . . . . . . . . . 13 |- +P. = ( w e. P. , v e. P. |-> { x | E. y e. w E. z e. v x = ( y +Q z ) } )
8		addclnq 9228	. . . . . . . . . . . . 13 |- ( ( y e. Q. /\ z e. Q. ) -> ( y +Q z ) e. Q. )
9	7, 8	genpprecl 9284	. . . . . . . . . . . 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 9274	. . . . . . . . . . . . 13 |- ( ( ( A +P. B ) e. P. /\ ( x +Q y ) e. ( A +P. B ) ) -> ( x +Q y ) e. Q. )
12		addnqf 9231	. . . . . . . . . . . . . . 15 |- +Q : ( Q. X. Q. ) --> Q.
13	12	fdmi 5675	. . . . . . . . . . . . . 14 |- dom +Q = ( Q. X. Q. )
14		0nnq 9207	. . . . . . . . . . . . . 14 |- -. (/) e. Q.
15	13, 14	ndmovrcl 6362	. . . . . . . . . . . . 13 |- ( ( x +Q y ) e. Q. -> ( x e. Q. /\ y e. Q. ) )
16		ltaddnq 9257	. . . . . . . . . . . . 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 9276	. . . . . . . . . . . 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 1688	. . . . . . 7 |- ( ( A e. P. /\ B e. P. ) -> ( y e. B -> A. x ( x e. A -> x e. ( A +P. B ) ) ) )
24		dfss2 3456	. . . . . . 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 3081	. . . . . . . . 9 |- y e. _V
27	26	prlem934 9316	. . . . . . . 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 2527	. . . . . . . . . . . . 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 2946	. . . . . . 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 3552	. . . . 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 1691	. . 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 9313	. . 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 1368   = wceq 1370   E.wex 1587   e. wcel 1758   =/= wne 2648   E.wrex 2800   C_ wss 3439   C. wpss 3440   (/)c0 3748   class class class wbr 4403   X. cxp 4949  (class class class)co 6203   Q.cnq 9133   +Q cplq 9136   <Q cltq 9139   P.cnp 9140   +P. cpp 9142   <P cltp 9144

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-inf2 7961

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-1o 7033  df-oadd 7037  df-omul 7038  df-er 7214  df-ni 9155  df-pli 9156  df-mi 9157  df-lti 9158  df-plpq 9191  df-mpq 9192  df-ltpq 9193  df-enq 9194  df-nq 9195  df-erq 9196  df-plq 9197  df-mq 9198  df-1nq 9199  df-rq 9200  df-ltnq 9201  df-np 9264  df-plp 9266  df-ltp 9268

This theorem is referenced by:  ltaddpr2  9318  ltexprlem7  9325  ltaprlem  9327  0lt1sr  9376  mappsrpr  9389