Theorem ltaddpr 7681

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 7559	. . . 4 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
2		prml 7561	. . . 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 7559	. . . . 5 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
6		prarloc 7587	. . . . 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 7566	. . . . . . . . . . 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 7552	. . . . . . . . . . . . 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 7459	. . . . . . . . . . . . 13 |- ( ( g e. Q. /\ h e. Q. ) -> ( g +Q h ) e. Q. )
20	18, 19	genpprecll 7598	. . . . . . . . . . . 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 7621	. . . . . . . . . . 11 |- ( ( A e. P. /\ B e. P. ) -> ( A +P. B ) e. P. )
24		prop 7559	. . . . . . . . . . . 12 |- ( ( A +P. B ) e. P. -> <. ( 1st ` ( A +P. B ) ) , ( 2nd ` ( A +P. B ) ) >. e. P. )
25		prcdnql 7568	. . . . . . . . . . . 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 2546	. . . . . . 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 1247	. . . . . 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 7590	. . . . . . . 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 2622	. . 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 2619	1 |- ( ( A e. P. /\ B e. P. ) -> A <P ( A +P. B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   e. wcel 2167   E.wrex 2476   <.cop 3626   class class class wbr 4034   ` cfv 5259  (class class class)co 5925   1stc1st 6205   2ndc2nd 6206   Q.cnq 7364   +Q cplq 7366   <Q cltq 7369   P.cnp 7375   +P. cpp 7377   <P cltp 7379

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-eprel 4325  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-irdg 6437  df-1o 6483  df-2o 6484  df-oadd 6487  df-omul 6488  df-er 6601  df-ec 6603  df-qs 6607  df-ni 7388  df-pli 7389  df-mi 7390  df-lti 7391  df-plpq 7428  df-mpq 7429  df-enq 7431  df-nqqs 7432  df-plqqs 7433  df-mqqs 7434  df-1nqqs 7435  df-rq 7436  df-ltnqqs 7437  df-enq0 7508  df-nq0 7509  df-0nq0 7510  df-plq0 7511  df-mq0 7512  df-inp 7550  df-iplp 7552  df-iltp 7554

This theorem is referenced by:  ltexprlemrl  7694  ltaprlem  7702  ltaprg  7703  prplnqu  7704  ltmprr  7726  caucvgprprlemnkltj  7773  caucvgprprlemnkeqj  7774  caucvgprprlemnbj  7777  0lt1sr  7849  recexgt0sr  7857  mulgt0sr  7862  archsr  7866  prsrpos  7869  mappsrprg  7888  pitoregt0  7933