Theorem ltaddpr 6292

Description: The sum of two positive reals is greater than one of them. Proposition 9-3.5(iii) of [Gleason] p. 123.

Assertion

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

Proof of Theorem ltaddpr

Step	Hyp	Ref	Expression
1		prn0 6245	. . . . 5 |- (B e. P. -> B =/= (/))
2		n0 2884	. . . . 5 |- (B =/= (/) <-> E.y y e. B)
3	1, 2	sylib 215	. . . 4 |- (B e. P. -> E.y y e. B)
4	3	adantl 424	. . 3 |- ((A e. P. /\ B e. P.) -> E.y y e. B)
5		addclpr 6272	. . . . . . . . . . . 12 |- ((A e. P. /\ B e. P.) -> (A +P. B) e. P.)
6	5	adantr 425	. . . . . . . . . . 11 |- (((A e. P. /\ B e. P.) /\ (x e. A /\ y e. B)) -> (A +P. B) e. P.)
7		df-plp 6240	. . . . . . . . . . . . 13 |- +P. = {<.<.w, v>., u>. | ((w e. P. /\ v e. P.) /\ u = {x | E.y e. w E.z e. v x = (y +Q z)})}
8	7	genpprecl 6256	. . . . . . . . . . . 12 |- ((A e. P. /\ B e. P.) -> ((x e. A /\ y e. B) -> (x +Q y) e. (A +P. B)))
9	8	imp 377	. . . . . . . . . . 11 |- (((A e. P. /\ B e. P.) /\ (x e. A /\ y e. B)) -> (x +Q y) e. (A +P. B))
10		elprpq 6247	. . . . . . . . . . . . 13 |- (((A +P. B) e. P. /\ (x +Q y) e. (A +P. B)) -> (x +Q y) e. Q.)
11		visset 2295	. . . . . . . . . . . . . 14 |- y e. _V
12		dmaddpq 6211	. . . . . . . . . . . . . 14 |- dom +Q = (Q. X. Q.)
13		0npq 6202	. . . . . . . . . . . . . 14 |- -. (/) e. Q.
14	11, 12, 13	ndmoprrcl 4979	. . . . . . . . . . . . 13 |- ((x +Q y) e. Q. -> (x e. Q. /\ y e. Q.))
15		visset 2295	. . . . . . . . . . . . . 14 |- x e. _V
16	15, 11	ltaddpq 6231	. . . . . . . . . . . . 13 |- ((x e. Q. /\ y e. Q.) -> x <Q (x +Q y))
17	10, 14, 16	3syl 24	. . . . . . . . . . . 12 |- (((A +P. B) e. P. /\ (x +Q y) e. (A +P. B)) -> x <Q (x +Q y))
18		prcdpq 6249	. . . . . . . . . . . 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 29	. . . . . . . . . . 11 |- (((A +P. B) e. P. /\ (x +Q y) e. (A +P. B)) -> x e. (A +P. B))
20	6, 9, 19	syl11anc 524	. . . . . . . . . 10 |- (((A e. P. /\ B e. P.) /\ (x e. A /\ y e. B)) -> x e. (A +P. B))
21	20	exp32 408	. . . . . . . . 9 |- ((A e. P. /\ B e. P.) -> (x e. A -> (y e. B -> x e. (A +P. B))))
22	21	com23 36	. . . . . . . 8 |- ((A e. P. /\ B e. P.) -> (y e. B -> (x e. A -> x e. (A +P. B))))
23	22	19.21adv 1666	. . . . . . 7 |- ((A e. P. /\ B e. P.) -> (y e. B -> A.x(x e. A -> x e. (A +P. B))))
24		dfss2 2610	. . . . . . 7 |- (A C_ (A +P. B) <-> A.x(x e. A -> x e. (A +P. B)))
25	23, 24	syl6ibr 230	. . . . . 6 |- ((A e. P. /\ B e. P.) -> (y e. B -> A C_ (A +P. B)))
26		eleq2 1958	. . . . . . . . . . . . . . . . . . 19 |- (A = (A +P. B) -> ((x +Q y) e. A <-> (x +Q y) e. (A +P. B)))
27	26	biimprcd 173	. . . . . . . . . . . . . . . . . 18 |- ((x +Q y) e. (A +P. B) -> (A = (A +P. B) -> (x +Q y) e. A))
28	27	con3d 111	. . . . . . . . . . . . . . . . 17 |- ((x +Q y) e. (A +P. B) -> (-. (x +Q y) e. A -> -. A = (A +P. B)))
29	8, 28	syl6 25	. . . . . . . . . . . . . . . 16 |- ((A e. P. /\ B e. P.) -> ((x e. A /\ y e. B) -> (-. (x +Q y) e. A -> -. A = (A +P. B))))
30	29	exp3a 405	. . . . . . . . . . . . . . 15 |- ((A e. P. /\ B e. P.) -> (x e. A -> (y e. B -> (-. (x +Q y) e. A -> -. A = (A +P. B)))))
31	30	com34 40	. . . . . . . . . . . . . 14 |- ((A e. P. /\ B e. P.) -> (x e. A -> (-. (x +Q y) e. A -> (y e. B -> -. A = (A +P. B)))))
32	31	imp3a 388	. . . . . . . . . . . . 13 |- ((A e. P. /\ B e. P.) -> ((x e. A /\ -. (x +Q y) e. A) -> (y e. B -> -. A = (A +P. B))))
33	32	19.23adv 1584	. . . . . . . . . . . 12 |- ((A e. P. /\ B e. P.) -> (E.x(x e. A /\ -. (x +Q y) e. A) -> (y e. B -> -. A = (A +P. B))))
34		prlem934 6291	. . . . . . . . . . . 12 |- ((A e. P. /\ y e. Q.) -> E.x(x e. A /\ -. (x +Q y) e. A))
35	33, 34	syl5 20	. . . . . . . . . . 11 |- ((A e. P. /\ B e. P.) -> ((A e. P. /\ y e. Q.) -> (y e. B -> -. A = (A +P. B))))
36		elprpq 6247	. . . . . . . . . . 11 |- ((B e. P. /\ y e. B) -> y e. Q.)
37	35, 36	sylan2i 514	. . . . . . . . . 10 |- ((A e. P. /\ B e. P.) -> ((A e. P. /\ (B e. P. /\ y e. B)) -> (y e. B -> -. A = (A +P. B))))
38	37	exp4d 412	. . . . . . . . 9 |- ((A e. P. /\ B e. P.) -> (A e. P. -> (B e. P. -> (y e. B -> (y e. B -> -. A = (A +P. B))))))
39	38	imp3a 388	. . . . . . . 8 |- ((A e. P. /\ B e. P.) -> ((A e. P. /\ B e. P.) -> (y e. B -> (y e. B -> -. A = (A +P. B)))))
40	39	pm2.43i 78	. . . . . . 7 |- ((A e. P. /\ B e. P.) -> (y e. B -> (y e. B -> -. A = (A +P. B))))
41	40	pm2.43d 79	. . . . . 6 |- ((A e. P. /\ B e. P.) -> (y e. B -> -. A = (A +P. B)))
42	25, 41	jcad 661	. . . . 5 |- ((A e. P. /\ B e. P.) -> (y e. B -> (A C_ (A +P. B) /\ -. A = (A +P. B))))
43		dfpss2 2694	. . . . 5 |- (A C. (A +P. B) <-> (A C_ (A +P. B) /\ -. A = (A +P. B)))
44	42, 43	syl6ibr 230	. . . 4 |- ((A e. P. /\ B e. P.) -> (y e. B -> A C. (A +P. B)))
45	44	19.23adv 1584	. . 3 |- ((A e. P. /\ B e. P.) -> (E.y y e. B -> A C. (A +P. B)))
46	4, 45	mpd 29	. 2 |- ((A e. P. /\ B e. P.) -> A C. (A +P. B))
47		ltprord 6286	. . 3 |- ((A e. P. /\ (A +P. B) e. P.) -> (A <P (A +P. B) <-> A C. (A +P. B)))
48	5, 47	syldan 516	. 2 |- ((A e. P. /\ B e. P.) -> (A <P (A +P. B) <-> A C. (A +P. B)))
49	46, 48	mpbird 213	1 |- ((A e. P. /\ B e. P.) -> A <P (A +P. B))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  E.wex 1326   =/= wne 2017   C_ wss 2593   C. wpss 2594  (/)c0 2875   class class class wbr 3338  (class class class)co 4884  Q.cnq 6131   +Q cplq 6133   <Q cltq 6136  P.cnp 6137   +P. cpp 6139   <P cltp 6141

This theorem is referenced by:  ltaddpr2 6293  ltexprlem7 6300  ltaprlem 6302  0lt1sr 6356  mappsrpr 6370

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-fv 4014  df-opr 4886  df-oprab 4887  df-1st 5020  df-2nd 5021  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-plp 6240  df-ltp 6242

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