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Theorem ltaddpr 6570
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 P B P) → A<P (A +P B))

Proof of Theorem ltaddpr
Dummy variables f g x y 𝑝 𝑞 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prop 6457 . . . 4 (B P → ⟨(1stB), (2ndB)⟩ P)
2 prml 6459 . . . 4 (⟨(1stB), (2ndB)⟩ P𝑝 Q 𝑝 (1stB))
31, 2syl 14 . . 3 (B P𝑝 Q 𝑝 (1stB))
43adantl 262 . 2 ((A P B P) → 𝑝 Q 𝑝 (1stB))
5 prop 6457 . . . . 5 (A P → ⟨(1stA), (2ndA)⟩ P)
6 prarloc 6485 . . . . 5 ((⟨(1stA), (2ndA)⟩ P 𝑝 Q) → 𝑟 (1stA)𝑞 (2ndA)𝑞 <Q (𝑟 +Q 𝑝))
75, 6sylan 267 . . . 4 ((A P 𝑝 Q) → 𝑟 (1stA)𝑞 (2ndA)𝑞 <Q (𝑟 +Q 𝑝))
87ad2ant2r 478 . . 3 (((A P B P) (𝑝 Q 𝑝 (1stB))) → 𝑟 (1stA)𝑞 (2ndA)𝑞 <Q (𝑟 +Q 𝑝))
9 elprnqu 6464 . . . . . . . . . . 11 ((⟨(1stA), (2ndA)⟩ P 𝑞 (2ndA)) → 𝑞 Q)
105, 9sylan 267 . . . . . . . . . 10 ((A P 𝑞 (2ndA)) → 𝑞 Q)
1110adantlr 446 . . . . . . . . 9 (((A P B P) 𝑞 (2ndA)) → 𝑞 Q)
1211ad2ant2rl 480 . . . . . . . 8 ((((A P B P) (𝑝 Q 𝑝 (1stB))) (𝑟 (1stA) 𝑞 (2ndA))) → 𝑞 Q)
1312adantr 261 . . . . . . 7 (((((A P B P) (𝑝 Q 𝑝 (1stB))) (𝑟 (1stA) 𝑞 (2ndA))) 𝑞 <Q (𝑟 +Q 𝑝)) → 𝑞 Q)
14 simplrr 488 . . . . . . 7 (((((A P B P) (𝑝 Q 𝑝 (1stB))) (𝑟 (1stA) 𝑞 (2ndA))) 𝑞 <Q (𝑟 +Q 𝑝)) → 𝑞 (2ndA))
15 simprl 483 . . . . . . . . . . . . 13 (((𝑝 Q 𝑝 (1stB)) (𝑟 (1stA) 𝑞 (2ndA))) → 𝑟 (1stA))
16 simplr 482 . . . . . . . . . . . . 13 (((𝑝 Q 𝑝 (1stB)) (𝑟 (1stA) 𝑞 (2ndA))) → 𝑝 (1stB))
1715, 16jca 290 . . . . . . . . . . . 12 (((𝑝 Q 𝑝 (1stB)) (𝑟 (1stA) 𝑞 (2ndA))) → (𝑟 (1stA) 𝑝 (1stB)))
18 df-iplp 6450 . . . . . . . . . . . . 13 +P = (x P, y P ↦ ⟨{f Qg Q Q (g (1stx) (1sty) f = (g +Q ))}, {f Qg Q Q (g (2ndx) (2ndy) f = (g +Q ))}⟩)
19 addclnq 6359 . . . . . . . . . . . . 13 ((g Q Q) → (g +Q ) Q)
2018, 19genpprecll 6496 . . . . . . . . . . . 12 ((A P B P) → ((𝑟 (1stA) 𝑝 (1stB)) → (𝑟 +Q 𝑝) (1st ‘(A +P B))))
2117, 20syl5 28 . . . . . . . . . . 11 ((A P B P) → (((𝑝 Q 𝑝 (1stB)) (𝑟 (1stA) 𝑞 (2ndA))) → (𝑟 +Q 𝑝) (1st ‘(A +P B))))
2221imdistani 419 . . . . . . . . . 10 (((A P B P) ((𝑝 Q 𝑝 (1stB)) (𝑟 (1stA) 𝑞 (2ndA)))) → ((A P B P) (𝑟 +Q 𝑝) (1st ‘(A +P B))))
23 addclpr 6519 . . . . . . . . . . 11 ((A P B P) → (A +P B) P)
24 prop 6457 . . . . . . . . . . . 12 ((A +P B) P → ⟨(1st ‘(A +P B)), (2nd ‘(A +P B))⟩ P)
25 prcdnql 6466 . . . . . . . . . . . 12 ((⟨(1st ‘(A +P B)), (2nd ‘(A +P B))⟩ P (𝑟 +Q 𝑝) (1st ‘(A +P B))) → (𝑞 <Q (𝑟 +Q 𝑝) → 𝑞 (1st ‘(A +P B))))
2624, 25sylan 267 . . . . . . . . . . 11 (((A +P B) P (𝑟 +Q 𝑝) (1st ‘(A +P B))) → (𝑞 <Q (𝑟 +Q 𝑝) → 𝑞 (1st ‘(A +P B))))
2723, 26sylan 267 . . . . . . . . . 10 (((A P B P) (𝑟 +Q 𝑝) (1st ‘(A +P B))) → (𝑞 <Q (𝑟 +Q 𝑝) → 𝑞 (1st ‘(A +P B))))
2822, 27syl 14 . . . . . . . . 9 (((A P B P) ((𝑝 Q 𝑝 (1stB)) (𝑟 (1stA) 𝑞 (2ndA)))) → (𝑞 <Q (𝑟 +Q 𝑝) → 𝑞 (1st ‘(A +P B))))
2928anassrs 380 . . . . . . . 8 ((((A P B P) (𝑝 Q 𝑝 (1stB))) (𝑟 (1stA) 𝑞 (2ndA))) → (𝑞 <Q (𝑟 +Q 𝑝) → 𝑞 (1st ‘(A +P B))))
3029imp 115 . . . . . . 7 (((((A P B P) (𝑝 Q 𝑝 (1stB))) (𝑟 (1stA) 𝑞 (2ndA))) 𝑞 <Q (𝑟 +Q 𝑝)) → 𝑞 (1st ‘(A +P B)))
31 rspe 2364 . . . . . . 7 ((𝑞 Q (𝑞 (2ndA) 𝑞 (1st ‘(A +P B)))) → 𝑞 Q (𝑞 (2ndA) 𝑞 (1st ‘(A +P B))))
3213, 14, 30, 31syl12anc 1132 . . . . . 6 (((((A P B P) (𝑝 Q 𝑝 (1stB))) (𝑟 (1stA) 𝑞 (2ndA))) 𝑞 <Q (𝑟 +Q 𝑝)) → 𝑞 Q (𝑞 (2ndA) 𝑞 (1st ‘(A +P B))))
33 ltdfpr 6488 . . . . . . . 8 ((A P (A +P B) P) → (A<P (A +P B) ↔ 𝑞 Q (𝑞 (2ndA) 𝑞 (1st ‘(A +P B)))))
3423, 33syldan 266 . . . . . . 7 ((A P B P) → (A<P (A +P B) ↔ 𝑞 Q (𝑞 (2ndA) 𝑞 (1st ‘(A +P B)))))
3534ad3antrrr 461 . . . . . 6 (((((A P B P) (𝑝 Q 𝑝 (1stB))) (𝑟 (1stA) 𝑞 (2ndA))) 𝑞 <Q (𝑟 +Q 𝑝)) → (A<P (A +P B) ↔ 𝑞 Q (𝑞 (2ndA) 𝑞 (1st ‘(A +P B)))))
3632, 35mpbird 156 . . . . 5 (((((A P B P) (𝑝 Q 𝑝 (1stB))) (𝑟 (1stA) 𝑞 (2ndA))) 𝑞 <Q (𝑟 +Q 𝑝)) → A<P (A +P B))
3736ex 108 . . . 4 ((((A P B P) (𝑝 Q 𝑝 (1stB))) (𝑟 (1stA) 𝑞 (2ndA))) → (𝑞 <Q (𝑟 +Q 𝑝) → A<P (A +P B)))
3837rexlimdvva 2434 . . 3 (((A P B P) (𝑝 Q 𝑝 (1stB))) → (𝑟 (1stA)𝑞 (2ndA)𝑞 <Q (𝑟 +Q 𝑝) → A<P (A +P B)))
398, 38mpd 13 . 2 (((A P B P) (𝑝 Q 𝑝 (1stB))) → A<P (A +P B))
404, 39rexlimddv 2431 1 ((A P B P) → A<P (A +P B))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   wcel 1390  wrex 2301  cop 3370   class class class wbr 3755  cfv 4845  (class class class)co 5455  1st c1st 5707  2nd c2nd 5708  Qcnq 6264   +Q cplq 6266   <Q cltq 6269  Pcnp 6275   +P cpp 6277  <P cltp 6279
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254
This theorem depends on definitions:  df-bi 110  df-dc 742  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-eprel 4017  df-id 4021  df-po 4024  df-iso 4025  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-1o 5940  df-2o 5941  df-oadd 5944  df-omul 5945  df-er 6042  df-ec 6044  df-qs 6048  df-ni 6288  df-pli 6289  df-mi 6290  df-lti 6291  df-plpq 6328  df-mpq 6329  df-enq 6331  df-nqqs 6332  df-plqqs 6333  df-mqqs 6334  df-1nqqs 6335  df-rq 6336  df-ltnqqs 6337  df-enq0 6406  df-nq0 6407  df-0nq0 6408  df-plq0 6409  df-mq0 6410  df-inp 6448  df-iplp 6450  df-iltp 6452
This theorem is referenced by:  ltexprlemrl  6583  ltaprlem  6590  ltaprg  6591  ltmprr  6613  0lt1sr  6673  recexgt0sr  6681  mulgt0sr  6684  archsr  6688
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