Theorem ltaddpr 7538

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 7416	. . . 4 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
2		prml 7418	. . . 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 275	. 2 |- ( ( A e. P. /\ B e. P. ) -> E. p e. Q. p e. ( 1st ` B ) )
5		prop 7416	. . . . 5 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
6		prarloc 7444	. . . . 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 281	. . . 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 501	. . 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 7423	. . . . . . . . . . 11 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ q e. ( 2nd ` A ) ) -> q e. Q. )
10	5, 9	sylan 281	. . . . . . . . . 10 |- ( ( A e. P. /\ q e. ( 2nd ` A ) ) -> q e. Q. )
11	10	adantlr 469	. . . . . . . . 9 |- ( ( ( A e. P. /\ B e. P. ) /\ q e. ( 2nd ` A ) ) -> q e. Q. )
12	11	ad2ant2rl 503	. . . . . . . 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 274	. . . . . . 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 526	. . . . . . 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 521	. . . . . . . . . . . . 13 |- ( ( ( p e. Q. /\ p e. ( 1st ` B ) ) /\ ( r e. ( 1st ` A ) /\ q e. ( 2nd ` A ) ) ) -> r e. ( 1st ` A ) )
16		simplr 520	. . . . . . . . . . . . 13 |- ( ( ( p e. Q. /\ p e. ( 1st ` B ) ) /\ ( r e. ( 1st ` A ) /\ q e. ( 2nd ` A ) ) ) -> p e. ( 1st ` B ) )
17	15, 16	jca 304	. . . . . . . . . . . 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 7409	. . . . . . . . . . . . 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 7316	. . . . . . . . . . . . 13 |- ( ( g e. Q. /\ h e. Q. ) -> ( g +Q h ) e. Q. )
20	18, 19	genpprecll 7455	. . . . . . . . . . . 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 442	. . . . . . . . . 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 7478	. . . . . . . . . . 11 |- ( ( A e. P. /\ B e. P. ) -> ( A +P. B ) e. P. )
24		prop 7416	. . . . . . . . . . . 12 |- ( ( A +P. B ) e. P. -> <. ( 1st ` ( A +P. B ) ) , ( 2nd ` ( A +P. B ) ) >. e. P. )
25		prcdnql 7425	. . . . . . . . . . . 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 281	. . . . . . . . . . 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 281	. . . . . . . . . 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 398	. . . . . . . 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 123	. . . . . . 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 2515	. . . . . . 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 1226	. . . . . 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 7447	. . . . . . . 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 280	. . . . . . 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 484	. . . . . 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 166	. . . . 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 114	. . . 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 2591	. . 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 2588	1 |- ( ( A e. P. /\ B e. P. ) -> A <P ( A +P. B ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   e. wcel 2136   E.wrex 2445   <.cop 3579   class class class wbr 3982   ` cfv 5188  (class class class)co 5842   1stc1st 6106   2ndc2nd 6107   Q.cnq 7221   +Q cplq 7223   <Q cltq 7226   P.cnp 7232   +P. cpp 7234   <P cltp 7236

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-eprel 4267  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-irdg 6338  df-1o 6384  df-2o 6385  df-oadd 6388  df-omul 6389  df-er 6501  df-ec 6503  df-qs 6507  df-ni 7245  df-pli 7246  df-mi 7247  df-lti 7248  df-plpq 7285  df-mpq 7286  df-enq 7288  df-nqqs 7289  df-plqqs 7290  df-mqqs 7291  df-1nqqs 7292  df-rq 7293  df-ltnqqs 7294  df-enq0 7365  df-nq0 7366  df-0nq0 7367  df-plq0 7368  df-mq0 7369  df-inp 7407  df-iplp 7409  df-iltp 7411

This theorem is referenced by:  ltexprlemrl  7551  ltaprlem  7559  ltaprg  7560  prplnqu  7561  ltmprr  7583  caucvgprprlemnkltj  7630  caucvgprprlemnkeqj  7631  caucvgprprlemnbj  7634  0lt1sr  7706  recexgt0sr  7714  mulgt0sr  7719  archsr  7723  prsrpos  7726  mappsrprg  7745  pitoregt0  7790