Theorem ltaddpr 7587

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 7465	. . . 4 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
2		prml 7467	. . . 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 7465	. . . . 5 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
6		prarloc 7493	. . . . 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 7472	. . . . . . . . . . 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 7458	. . . . . . . . . . . . 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 7365	. . . . . . . . . . . . 13 |- ( ( g e. Q. /\ h e. Q. ) -> ( g +Q h ) e. Q. )
20	18, 19	genpprecll 7504	. . . . . . . . . . . 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 7527	. . . . . . . . . . 11 |- ( ( A e. P. /\ B e. P. ) -> ( A +P. B ) e. P. )
24		prop 7465	. . . . . . . . . . . 12 |- ( ( A +P. B ) e. P. -> <. ( 1st ` ( A +P. B ) ) , ( 2nd ` ( A +P. B ) ) >. e. P. )
25		prcdnql 7474	. . . . . . . . . . . 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 2526	. . . . . . 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 1236	. . . . . 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 7496	. . . . . . . 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 2602	. . 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 2599	1 |- ( ( A e. P. /\ B e. P. ) -> A <P ( A +P. B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   e. wcel 2148   E.wrex 2456   <.cop 3594   class class class wbr 4000   ` cfv 5212  (class class class)co 5869   1stc1st 6133   2ndc2nd 6134   Q.cnq 7270   +Q cplq 7272   <Q cltq 7275   P.cnp 7281   +P. cpp 7283   <P cltp 7285

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-eprel 4286  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-irdg 6365  df-1o 6411  df-2o 6412  df-oadd 6415  df-omul 6416  df-er 6529  df-ec 6531  df-qs 6535  df-ni 7294  df-pli 7295  df-mi 7296  df-lti 7297  df-plpq 7334  df-mpq 7335  df-enq 7337  df-nqqs 7338  df-plqqs 7339  df-mqqs 7340  df-1nqqs 7341  df-rq 7342  df-ltnqqs 7343  df-enq0 7414  df-nq0 7415  df-0nq0 7416  df-plq0 7417  df-mq0 7418  df-inp 7456  df-iplp 7458  df-iltp 7460

This theorem is referenced by:  ltexprlemrl  7600  ltaprlem  7608  ltaprg  7609  prplnqu  7610  ltmprr  7632  caucvgprprlemnkltj  7679  caucvgprprlemnkeqj  7680  caucvgprprlemnbj  7683  0lt1sr  7755  recexgt0sr  7763  mulgt0sr  7768  archsr  7772  prsrpos  7775  mappsrprg  7794  pitoregt0  7839