Theorem ltaddpr 9477

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 9432	. . . . 5 |- ( B e. P. -> B =/= (/) )
2		n0 3732	. . . . 5 |- ( B =/= (/) <-> E. y y e. B )
3	1, 2	sylib 201	. . . 4 |- ( B e. P. -> E. y y e. B )
4	3	adantl 473	. . 3 |- ( ( A e. P. /\ B e. P. ) -> E. y y e. B )
5		addclpr 9461	. . . . . . . . . . . 12 |- ( ( A e. P. /\ B e. P. ) -> ( A +P. B ) e. P. )
6	5	adantr 472	. . . . . . . . . . 11 |- ( ( ( A e. P. /\ B e. P. ) /\ ( x e. A /\ y e. B ) ) -> ( A +P. B ) e. P. )
7		df-plp 9426	. . . . . . . . . . . . 13 |- +P. = ( w e. P. , v e. P. |-> { x | E. y e. w E. z e. v x = ( y +Q z ) } )
8		addclnq 9388	. . . . . . . . . . . . 13 |- ( ( y e. Q. /\ z e. Q. ) -> ( y +Q z ) e. Q. )
9	7, 8	genpprecl 9444	. . . . . . . . . . . 12 |- ( ( A e. P. /\ B e. P. ) -> ( ( x e. A /\ y e. B ) -> ( x +Q y ) e. ( A +P. B ) ) )
10	9	imp 436	. . . . . . . . . . 11 |- ( ( ( A e. P. /\ B e. P. ) /\ ( x e. A /\ y e. B ) ) -> ( x +Q y ) e. ( A +P. B ) )
11		elprnq 9434	. . . . . . . . . . . . 13 |- ( ( ( A +P. B ) e. P. /\ ( x +Q y ) e. ( A +P. B ) ) -> ( x +Q y ) e. Q. )
12		addnqf 9391	. . . . . . . . . . . . . . 15 |- +Q : ( Q. X. Q. ) --> Q.
13	12	fdmi 5746	. . . . . . . . . . . . . 14 |- dom +Q = ( Q. X. Q. )
14		0nnq 9367	. . . . . . . . . . . . . 14 |- -. (/) e. Q.
15	13, 14	ndmovrcl 6474	. . . . . . . . . . . . 13 |- ( ( x +Q y ) e. Q. -> ( x e. Q. /\ y e. Q. ) )
16		ltaddnq 9417	. . . . . . . . . . . . 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 9436	. . . . . . . . . . . 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 673	. . . . . . . . . 10 |- ( ( ( A e. P. /\ B e. P. ) /\ ( x e. A /\ y e. B ) ) -> x e. ( A +P. B ) )
21	20	exp32 616	. . . . . . . . 9 |- ( ( A e. P. /\ B e. P. ) -> ( x e. A -> ( y e. B -> x e. ( A +P. B ) ) ) )
22	21	com23 80	. . . . . . . 8 |- ( ( A e. P. /\ B e. P. ) -> ( y e. B -> ( x e. A -> x e. ( A +P. B ) ) ) )
23	22	alrimdv 1783	. . . . . . 7 |- ( ( A e. P. /\ B e. P. ) -> ( y e. B -> A. x ( x e. A -> x e. ( A +P. B ) ) ) )
24		dfss2 3407	. . . . . . 7 |- ( A C_ ( A +P. B ) <-> A. x ( x e. A -> x e. ( A +P. B ) ) )
25	23, 24	syl6ibr 235	. . . . . 6 |- ( ( A e. P. /\ B e. P. ) -> ( y e. B -> A C_ ( A +P. B ) ) )
26		vex 3034	. . . . . . . . 9 |- y e. _V
27	26	prlem934 9476	. . . . . . . 8 |- ( A e. P. -> E. x e. A -. ( x +Q y ) e. A )
28	27	adantr 472	. . . . . . 7 |- ( ( A e. P. /\ B e. P. ) -> E. x e. A -. ( x +Q y ) e. A )
29		eleq2 2538	. . . . . . . . . . . . 13 |- ( A = ( A +P. B ) -> ( ( x +Q y ) e. A <-> ( x +Q y ) e. ( A +P. B ) ) )
30	29	biimprcd 233	. . . . . . . . . . . 12 |- ( ( x +Q y ) e. ( A +P. B ) -> ( A = ( A +P. B ) -> ( x +Q y ) e. A ) )
31	30	con3d 140	. . . . . . . . . . 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 443	. . . . . . . . 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 85	. . . . . . . 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 2870	. . . . . . 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 542	. . . . 5 |- ( ( A e. P. /\ B e. P. ) -> ( y e. B -> ( A C_ ( A +P. B ) /\ -. A = ( A +P. B ) ) ) )
38		dfpss2 3504	. . . . 5 |- ( A C. ( A +P. B ) <-> ( A C_ ( A +P. B ) /\ -. A = ( A +P. B ) ) )
39	37, 38	syl6ibr 235	. . . 4 |- ( ( A e. P. /\ B e. P. ) -> ( y e. B -> A C. ( A +P. B ) ) )
40	39	exlimdv 1787	. . 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 9473	. . 3 |- ( ( A e. P. /\ ( A +P. B ) e. P. ) -> ( A <P ( A +P. B ) <-> A C. ( A +P. B ) ) )
43	5, 42	syldan 478	. 2 |- ( ( A e. P. /\ B e. P. ) -> ( A <P ( A +P. B ) <-> A C. ( A +P. B ) ) )
44	41, 43	mpbird 240	1 |- ( ( A e. P. /\ B e. P. ) -> A <P ( A +P. B ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 189   /\ wa 376   A.wal 1450   = wceq 1452   E.wex 1671   e. wcel 1904   =/= wne 2641   E.wrex 2757   C_ wss 3390   C. wpss 3391   (/)c0 3722   class class class wbr 4395   X. cxp 4837  (class class class)co 6308   Q.cnq 9295   +Q cplq 9298   <Q cltq 9301   P.cnp 9302   +P. cpp 9304   <P cltp 9306

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-omul 7205  df-er 7381  df-ni 9315  df-pli 9316  df-mi 9317  df-lti 9318  df-plpq 9351  df-mpq 9352  df-ltpq 9353  df-enq 9354  df-nq 9355  df-erq 9356  df-plq 9357  df-mq 9358  df-1nq 9359  df-rq 9360  df-ltnq 9361  df-np 9424  df-plp 9426  df-ltp 9428

This theorem is referenced by:  ltaddpr2  9478  ltexprlem7  9485  ltaprlem  9487  0lt1sr  9537  mappsrpr  9550