Theorem ltaddpr 7954

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 7832	. . . 4 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
2		prml 7834	. . . 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 7832	. . . . 5 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
6		prarloc 7860	. . . . 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 513	. . 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 7839	. . . . . . . . . . 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 481	. . . . . . . . 9 |- ( ( ( A e. P. /\ B e. P. ) /\ q e. ( 2nd ` A ) ) -> q e. Q. )
12	11	ad2ant2rl 515	. . . . . . . 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 542	. . . . . . 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 535	. . . . . . . . . . . . 13 |- ( ( ( p e. Q. /\ p e. ( 1st ` B ) ) /\ ( r e. ( 1st ` A ) /\ q e. ( 2nd ` A ) ) ) -> r e. ( 1st ` A ) )
16		simplr 533	. . . . . . . . . . . . 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 7825	. . . . . . . . . . . . 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 7732	. . . . . . . . . . . . 13 |- ( ( g e. Q. /\ h e. Q. ) -> ( g +Q h ) e. Q. )
20	18, 19	genpprecll 7871	. . . . . . . . . . . 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 449	. . . . . . . . . 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 7894	. . . . . . . . . . 11 |- ( ( A e. P. /\ B e. P. ) -> ( A +P. B ) e. P. )
24		prop 7832	. . . . . . . . . . . 12 |- ( ( A +P. B ) e. P. -> <. ( 1st ` ( A +P. B ) ) , ( 2nd ` ( A +P. B ) ) >. e. P. )
25		prcdnql 7841	. . . . . . . . . . . 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 404	. . . . . . . 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 2599	. . . . . . 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 1276	. . . . . 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 7863	. . . . . . . 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 496	. . . . . 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 2676	. . 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 2673	1 |- ( ( A e. P. /\ B e. P. ) -> A <P ( A +P. B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   e. wcel 2209   E.wrex 2529   <.cop 3708   class class class wbr 4125   ` cfv 5372  (class class class)co 6075   1stc1st 6362   2ndc2nd 6363   Q.cnq 7637   +Q cplq 7639   <Q cltq 7642   P.cnp 7648   +P. cpp 7650   <P cltp 7652

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-eprel 4429  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-irdg 6631  df-1o 6677  df-2o 6678  df-oadd 6681  df-omul 6682  df-er 6797  df-ec 6799  df-qs 6803  df-ni 7661  df-pli 7662  df-mi 7663  df-lti 7664  df-plpq 7701  df-mpq 7702  df-enq 7704  df-nqqs 7705  df-plqqs 7706  df-mqqs 7707  df-1nqqs 7708  df-rq 7709  df-ltnqqs 7710  df-enq0 7781  df-nq0 7782  df-0nq0 7783  df-plq0 7784  df-mq0 7785  df-inp 7823  df-iplp 7825  df-iltp 7827

This theorem is referenced by:  ltexprlemrl  7967  ltaprlem  7975  ltaprg  7976  prplnqu  7977  ltmprr  7999  caucvgprprlemnkltj  8046  caucvgprprlemnkeqj  8047  caucvgprprlemnbj  8050  0lt1sr  8122  recexgt0sr  8130  mulgt0sr  8135  archsr  8139  prsrpos  8142  mappsrprg  8161  pitoregt0  8206