Theorem ltaddpr 7784

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.)

Assertion

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

Proof of Theorem ltaddpr

Dummy variables f g h x y p q r are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prop 7662	. . . 4 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
2		prml 7664	. . . 4 |- ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. -> E. p e. Q. p e. ( 1st ` B ) )
3	1, 2	syl 14	. . 3 |- ( B e. P. -> E. p e. Q. p e. ( 1st ` B ) )
4	3	adantl 277	. 2 |- ( ( A e. P. /\ B e. P. ) -> E. p e. Q. p e. ( 1st ` B ) )
5		prop 7662	. . . . 5 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
6		prarloc 7690	. . . . 5 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ p e. Q. ) -> E. r e. ( 1st ` A ) E. q e. ( 2nd ` A ) q <Q ( r +Q p ) )
7	5, 6	sylan 283	. . . 4 |- ( ( A e. P. /\ p e. Q. ) -> E. r e. ( 1st ` A ) E. q e. ( 2nd ` A ) q <Q ( r +Q p ) )
8	7	ad2ant2r 509	. . 3 |- ( ( ( A e. P. /\ B e. P. ) /\ ( p e. Q. /\ p e. ( 1st ` B ) ) ) -> E. r e. ( 1st ` A ) E. q e. ( 2nd ` A ) q <Q ( r +Q p ) )
9		elprnqu 7669	. . . . . . . . . . 11 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ q e. ( 2nd ` A ) ) -> q e. Q. )
10	5, 9	sylan 283	. . . . . . . . . 10 |- ( ( A e. P. /\ q e. ( 2nd ` A ) ) -> q e. Q. )
11	10	adantlr 477	. . . . . . . . 9 |- ( ( ( A e. P. /\ B e. P. ) /\ q e. ( 2nd ` A ) ) -> q e. Q. )
12	11	ad2ant2rl 511	. . . . . . . 8 |- ( ( ( ( A e. P. /\ B e. P. ) /\ ( p e. Q. /\ p e. ( 1st ` B ) ) ) /\ ( r e. ( 1st ` A ) /\ q e. ( 2nd ` A ) ) ) -> q e. Q. )
13	12	adantr 276	. . . . . . 7 |- ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( p e. Q. /\ p e. ( 1st ` B ) ) ) /\ ( r e. ( 1st ` A ) /\ q e. ( 2nd ` A ) ) ) /\ q <Q ( r +Q p ) ) -> q e. Q. )
14		simplrr 536	. . . . . . 7 |- ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( p e. Q. /\ p e. ( 1st ` B ) ) ) /\ ( r e. ( 1st ` A ) /\ q e. ( 2nd ` A ) ) ) /\ q <Q ( r +Q p ) ) -> q e. ( 2nd ` A ) )
15		simprl 529	. . . . . . . . . . . . 13 |- ( ( ( p e. Q. /\ p e. ( 1st ` B ) ) /\ ( r e. ( 1st ` A ) /\ q e. ( 2nd ` A ) ) ) -> r e. ( 1st ` A ) )
16		simplr 528	. . . . . . . . . . . . 13 |- ( ( ( p e. Q. /\ p e. ( 1st ` B ) ) /\ ( r e. ( 1st ` A ) /\ q e. ( 2nd ` A ) ) ) -> p e. ( 1st ` B ) )
17	15, 16	jca 306	. . . . . . . . . . . 12 |- ( ( ( p e. Q. /\ p e. ( 1st ` B ) ) /\ ( r e. ( 1st ` A ) /\ q e. ( 2nd ` A ) ) ) -> ( r e. ( 1st ` A ) /\ p e. ( 1st ` B ) ) )
18		df-iplp 7655	. . . . . . . . . . . . 13 |- +P. = ( x e. P. , y e. P. |-> <. { f e. Q. | E. g e. Q. E. h e. Q. ( g e. ( 1st ` x ) /\ h e. ( 1st ` y ) /\ f = ( g +Q h ) ) } , { f e. Q. | E. g e. Q. E. h e. Q. ( g e. ( 2nd ` x ) /\ h e. ( 2nd ` y ) /\ f = ( g +Q h ) ) } >. )
19		addclnq 7562	. . . . . . . . . . . . 13 |- ( ( g e. Q. /\ h e. Q. ) -> ( g +Q h ) e. Q. )
20	18, 19	genpprecll 7701	. . . . . . . . . . . 12 |- ( ( A e. P. /\ B e. P. ) -> ( ( r e. ( 1st ` A ) /\ p e. ( 1st ` B ) ) -> ( r +Q p ) e. ( 1st ` ( A +P. B ) ) ) )
21	17, 20	syl5 32	. . . . . . . . . . 11 |- ( ( A e. P. /\ B e. P. ) -> ( ( ( p e. Q. /\ p e. ( 1st ` B ) ) /\ ( r e. ( 1st ` A ) /\ q e. ( 2nd ` A ) ) ) -> ( r +Q p ) e. ( 1st ` ( A +P. B ) ) ) )
22	21	imdistani 445	. . . . . . . . . 10 |- ( ( ( A e. P. /\ B e. P. ) /\ ( ( p e. Q. /\ p e. ( 1st ` B ) ) /\ ( r e. ( 1st ` A ) /\ q e. ( 2nd ` A ) ) ) ) -> ( ( A e. P. /\ B e. P. ) /\ ( r +Q p ) e. ( 1st ` ( A +P. B ) ) ) )
23		addclpr 7724	. . . . . . . . . . 11 |- ( ( A e. P. /\ B e. P. ) -> ( A +P. B ) e. P. )
24		prop 7662	. . . . . . . . . . . 12 |- ( ( A +P. B ) e. P. -> <. ( 1st ` ( A +P. B ) ) , ( 2nd ` ( A +P. B ) ) >. e. P. )
25		prcdnql 7671	. . . . . . . . . . . 12 |- ( ( <. ( 1st ` ( A +P. B ) ) , ( 2nd ` ( A +P. B ) ) >. e. P. /\ ( r +Q p ) e. ( 1st ` ( A +P. B ) ) ) -> ( q <Q ( r +Q p ) -> q e. ( 1st ` ( A +P. B ) ) ) )
26	24, 25	sylan 283	. . . . . . . . . . 11 |- ( ( ( A +P. B ) e. P. /\ ( r +Q p ) e. ( 1st ` ( A +P. B ) ) ) -> ( q <Q ( r +Q p ) -> q e. ( 1st ` ( A +P. B ) ) ) )
27	23, 26	sylan 283	. . . . . . . . . 10 |- ( ( ( A e. P. /\ B e. P. ) /\ ( r +Q p ) e. ( 1st ` ( A +P. B ) ) ) -> ( q <Q ( r +Q p ) -> q e. ( 1st ` ( A +P. B ) ) ) )
28	22, 27	syl 14	. . . . . . . . 9 |- ( ( ( A e. P. /\ B e. P. ) /\ ( ( p e. Q. /\ p e. ( 1st ` B ) ) /\ ( r e. ( 1st ` A ) /\ q e. ( 2nd ` A ) ) ) ) -> ( q <Q ( r +Q p ) -> q e. ( 1st ` ( A +P. B ) ) ) )
29	28	anassrs 400	. . . . . . . 8 |- ( ( ( ( A e. P. /\ B e. P. ) /\ ( p e. Q. /\ p e. ( 1st ` B ) ) ) /\ ( r e. ( 1st ` A ) /\ q e. ( 2nd ` A ) ) ) -> ( q <Q ( r +Q p ) -> q e. ( 1st ` ( A +P. B ) ) ) )
30	29	imp 124	. . . . . . 7 |- ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( p e. Q. /\ p e. ( 1st ` B ) ) ) /\ ( r e. ( 1st ` A ) /\ q e. ( 2nd ` A ) ) ) /\ q <Q ( r +Q p ) ) -> q e. ( 1st ` ( A +P. B ) ) )
31		rspe 2579	. . . . . . 7 |- ( ( q e. Q. /\ ( q e. ( 2nd ` A ) /\ q e. ( 1st ` ( A +P. B ) ) ) ) -> E. q e. Q. ( q e. ( 2nd ` A ) /\ q e. ( 1st ` ( A +P. B ) ) ) )
32	13, 14, 30, 31	syl12anc 1269	. . . . . 6 |- ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( p e. Q. /\ p e. ( 1st ` B ) ) ) /\ ( r e. ( 1st ` A ) /\ q e. ( 2nd ` A ) ) ) /\ q <Q ( r +Q p ) ) -> E. q e. Q. ( q e. ( 2nd ` A ) /\ q e. ( 1st ` ( A +P. B ) ) ) )
33		ltdfpr 7693	. . . . . . . 8 |- ( ( A e. P. /\ ( A +P. B ) e. P. ) -> ( A <P ( A +P. B ) <-> E. q e. Q. ( q e. ( 2nd ` A ) /\ q e. ( 1st ` ( A +P. B ) ) ) ) )
34	23, 33	syldan 282	. . . . . . 7 |- ( ( A e. P. /\ B e. P. ) -> ( A <P ( A +P. B ) <-> E. q e. Q. ( q e. ( 2nd ` A ) /\ q e. ( 1st ` ( A +P. B ) ) ) ) )
35	34	ad3antrrr 492	. . . . . 6 |- ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( p e. Q. /\ p e. ( 1st ` B ) ) ) /\ ( r e. ( 1st ` A ) /\ q e. ( 2nd ` A ) ) ) /\ q <Q ( r +Q p ) ) -> ( A <P ( A +P. B ) <-> E. q e. Q. ( q e. ( 2nd ` A ) /\ q e. ( 1st ` ( A +P. B ) ) ) ) )
36	32, 35	mpbird 167	. . . . 5 |- ( ( ( ( ( A e. P. /\ B e. P. ) /\ ( p e. Q. /\ p e. ( 1st ` B ) ) ) /\ ( r e. ( 1st ` A ) /\ q e. ( 2nd ` A ) ) ) /\ q <Q ( r +Q p ) ) -> A <P ( A +P. B ) )
37	36	ex 115	. . . 4 |- ( ( ( ( A e. P. /\ B e. P. ) /\ ( p e. Q. /\ p e. ( 1st ` B ) ) ) /\ ( r e. ( 1st ` A ) /\ q e. ( 2nd ` A ) ) ) -> ( q <Q ( r +Q p ) -> A <P ( A +P. B ) ) )
38	37	rexlimdvva 2656	. . 3 |- ( ( ( A e. P. /\ B e. P. ) /\ ( p e. Q. /\ p e. ( 1st ` B ) ) ) -> ( E. r e. ( 1st ` A ) E. q e. ( 2nd ` A ) q <Q ( r +Q p ) -> A <P ( A +P. B ) ) )
39	8, 38	mpd 13	. 2 |- ( ( ( A e. P. /\ B e. P. ) /\ ( p e. Q. /\ p e. ( 1st ` B ) ) ) -> A <P ( A +P. B ) )
40	4, 39	rexlimddv 2653	1 |- ( ( A e. P. /\ B e. P. ) -> A <P ( A +P. B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   e. wcel 2200   E.wrex 2509   <.cop 3669   class class class wbr 4083   ` cfv 5318  (class class class)co 6001   1stc1st 6284   2ndc2nd 6285   Q.cnq 7467   +Q cplq 7469   <Q cltq 7472   P.cnp 7478   +P. cpp 7480   <P cltp 7482

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-eprel 4380  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-recs 6451  df-irdg 6516  df-1o 6562  df-2o 6563  df-oadd 6566  df-omul 6567  df-er 6680  df-ec 6682  df-qs 6686  df-ni 7491  df-pli 7492  df-mi 7493  df-lti 7494  df-plpq 7531  df-mpq 7532  df-enq 7534  df-nqqs 7535  df-plqqs 7536  df-mqqs 7537  df-1nqqs 7538  df-rq 7539  df-ltnqqs 7540  df-enq0 7611  df-nq0 7612  df-0nq0 7613  df-plq0 7614  df-mq0 7615  df-inp 7653  df-iplp 7655  df-iltp 7657

This theorem is referenced by:  ltexprlemrl  7797  ltaprlem  7805  ltaprg  7806  prplnqu  7807  ltmprr  7829  caucvgprprlemnkltj  7876  caucvgprprlemnkeqj  7877  caucvgprprlemnbj  7880  0lt1sr  7952  recexgt0sr  7960  mulgt0sr  7965  archsr  7969  prsrpos  7972  mappsrprg  7991  pitoregt0  8036