Theorem ltaddpr 5112

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 5065	. . . . 5 |- (B e. P. -> B =/= (/))
2		ne0 2278	. . . . 5 |- (B =/= (/) <-> E.y y e. B)
3	1, 2	sylib 198	. . . 4 |- (B e. P. -> E.y y e. B)
4	3	adantl 388	. . 3 |- ((A e. P. /\ B e. P.) -> E.y y e. B)
5		elprpq 5067	. . . . . . . . . . . . 13 |- (((A +P. B) e. P. /\ (x +Q y) e. (A +P. B)) -> (x +Q y) e. Q.)
6		visset 1804	. . . . . . . . . . . . . 14 |- y e. V
7		dmaddpq 5031	. . . . . . . . . . . . . 14 |- dom +Q = (Q. X. Q.)
8		0npq 5022	. . . . . . . . . . . . . 14 |- -. (/) e. Q.
9	6, 7, 8	ndmoprrcl 4032	. . . . . . . . . . . . 13 |- ((x +Q y) e. Q. -> (x e. Q. /\ y e. Q.))
10		visset 1804	. . . . . . . . . . . . . 14 |- x e. V
11	10, 6	ltaddpq 5051	. . . . . . . . . . . . 13 |- ((x e. Q. /\ y e. Q.) -> x <Q (x +Q y))
12	5, 9, 11	3syl 20	. . . . . . . . . . . 12 |- (((A +P. B) e. P. /\ (x +Q y) e. (A +P. B)) -> x <Q (x +Q y))
13		prcdpq 5069	. . . . . . . . . . . 12 |- (((A +P. B) e. P. /\ (x +Q y) e. (A +P. B)) -> (x <Q (x +Q y) -> x e. (A +P. B)))
14	12, 13	mpd 26	. . . . . . . . . . 11 |- (((A +P. B) e. P. /\ (x +Q y) e. (A +P. B)) -> x e. (A +P. B))
15		addclpr 5092	. . . . . . . . . . . 12 |- ((A e. P. /\ B e. P.) -> (A +P. B) e. P.)
16	15	adantr 389	. . . . . . . . . . 11 |- (((A e. P. /\ B e. P.) /\ (x e. A /\ y e. B)) -> (A +P. B) e. P.)
17		df-plp 5060	. . . . . . . . . . . . 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)})}
18	17	genpprecl 5076	. . . . . . . . . . . 12 |- ((A e. P. /\ B e. P.) -> ((x e. A /\ y e. B) -> (x +Q y) e. (A +P. B)))
19	18	imp 350	. . . . . . . . . . 11 |- (((A e. P. /\ B e. P.) /\ (x e. A /\ y e. B)) -> (x +Q y) e. (A +P. B))
20	14, 16, 19	sylanc 471	. . . . . . . . . 10 |- (((A e. P. /\ B e. P.) /\ (x e. A /\ y e. B)) -> x e. (A +P. B))
21	20	exp32 377	. . . . . . . . 9 |- ((A e. P. /\ B e. P.) -> (x e. A -> (y e. B -> x e. (A +P. B))))
22	21	com23 32	. . . . . . . 8 |- ((A e. P. /\ B e. P.) -> (y e. B -> (x e. A -> x e. (A +P. B))))
23	22	19.21adv 1283	. . . . . . 7 |- ((A e. P. /\ B e. P.) -> (y e. B -> A.x(x e. A -> x e. (A +P. B))))
24		dfss2 2048	. . . . . . 7 |- (A (_ (A +P. B) <-> A.x(x e. A -> x e. (A +P. B)))
25	23, 24	syl6ibr 213	. . . . . 6 |- ((A e. P. /\ B e. P.) -> (y e. B -> A (_ (A +P. B)))
26		eleq2 1527	. . . . . . . . . . . . . . . . . . 19 |- (A = (A +P. B) -> ((x +Q y) e. A <-> (x +Q y) e. (A +P. B)))
27	26	biimprcd 156	. . . . . . . . . . . . . . . . . 18 |- ((x +Q y) e. (A +P. B) -> (A = (A +P. B) -> (x +Q y) e. A))
28	27	con3d 95	. . . . . . . . . . . . . . . . 17 |- ((x +Q y) e. (A +P. B) -> (-. (x +Q y) e. A -> -. A = (A +P. B)))
29	18, 28	syl6 22	. . . . . . . . . . . . . . . 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 375	. . . . . . . . . . . . . . 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 36	. . . . . . . . . . . . . 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 361	. . . . . . . . . . . . 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 1209	. . . . . . . . . . . 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 5111	. . . . . . . . . . . 12 |- ((A e. P. /\ y e. Q.) -> E.x(x e. A /\ -. (x +Q y) e. A))
35	33, 34	syl5 21	. . . . . . . . . . 11 |- ((A e. P. /\ B e. P.) -> ((A e. P. /\ y e. Q.) -> (y e. B -> -. A = (A +P. B))))
36		elprpq 5067	. . . . . . . . . . 11 |- ((B e. P. /\ y e. B) -> y e. Q.)
37	35, 36	sylan2i 465	. . . . . . . . . 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 381	. . . . . . . . 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 361	. . . . . . . 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 64	. . . . . . 7 |- ((A e. P. /\ B e. P.) -> (y e. B -> (y e. B -> -. A = (A +P. B))))
41	40	pm2.43d 65	. . . . . 6 |- ((A e. P. /\ B e. P.) -> (y e. B -> -. A = (A +P. B)))
42	25, 41	jcad 598	. . . . 5 |- ((A e. P. /\ B e. P.) -> (y e. B -> (A (_ (A +P. B) /\ -. A = (A +P. B))))
43		dfpss2 2123	. . . . 5 |- (A (. (A +P. B) <-> (A (_ (A +P. B) /\ -. A = (A +P. B)))
44	42, 43	syl6ibr 213	. . . 4 |- ((A e. P. /\ B e. P.) -> (y e. B -> A (. (A +P. B)))
45	44	19.23adv 1209	. . 3 |- ((A e. P. /\ B e. P.) -> (E.y y e. B -> A (. (A +P. B)))
46	4, 45	mpd 26	. 2 |- ((A e. P. /\ B e. P.) -> A (. (A +P. B))
47		ltprord 5106	. . 3 |- ((A e. P. /\ (A +P. B) e. P.) -> (A <P (A +P. B) <-> A (. (A +P. B)))
48	15, 47	syldan 467	. 2 |- ((A e. P. /\ B e. P.) -> (A <P (A +P. B) <-> A (. (A +P. B)))
49	46, 48	mpbird 196	1 |- ((A e. P. /\ B e. P.) -> A <P (A +P. B))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223  A.wal 951   = wceq 953   e. wcel 955  E.wex 977   =/= wne 1577   (_ wss 2037   (. wpss 2038  (/)c0 2270   class class class wbr 2609  (class class class)co 3948  Q.cnq 4951   +Q cplq 4953   <Q cltq 4956  P.cnp 4957   +P. cpp 4959   <P cltp 4961

This theorem is referenced by:  ltaddpr2 5113  ltexprlem7 5120  ltaprlem 5122  0lt1sr 5176  mappsrpr 5190

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-inf2 4597

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-reu 1643  df-rab 1644  df-v 1803  df-sbc 1932  df-csb 1992  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-pss 2045  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-int 2524  df-iun 2558  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-om 3122  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-fv 3188  df-rdg 3917  df-opr 3950  df-oprab 3951  df-1st 4063  df-2nd 4064  df-1o 4117  df-oadd 4119  df-omul 4120  df-er 4245  df-ec 4247  df-qs 4250  df-ni 4972  df-pli 4973  df-mi 4974  df-lti 4975  df-plpq 5007  df-mpq 5008  df-enq 5009  df-nq 5010  df-plq 5011  df-mq 5012  df-rq 5013  df-ltq 5014  df-1q 5015  df-np 5058  df-plp 5060  df-ltp 5062

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