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Theorem ltaddpr 7519
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 ((𝐴P𝐵P) → 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 7397 . . . 4 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
2 prml 7399 . . . 4 (⟨(1st𝐵), (2nd𝐵)⟩ ∈ P → ∃𝑝Q 𝑝 ∈ (1st𝐵))
31, 2syl 14 . . 3 (𝐵P → ∃𝑝Q 𝑝 ∈ (1st𝐵))
43adantl 275 . 2 ((𝐴P𝐵P) → ∃𝑝Q 𝑝 ∈ (1st𝐵))
5 prop 7397 . . . . 5 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
6 prarloc 7425 . . . . 5 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑝Q) → ∃𝑟 ∈ (1st𝐴)∃𝑞 ∈ (2nd𝐴)𝑞 <Q (𝑟 +Q 𝑝))
75, 6sylan 281 . . . 4 ((𝐴P𝑝Q) → ∃𝑟 ∈ (1st𝐴)∃𝑞 ∈ (2nd𝐴)𝑞 <Q (𝑟 +Q 𝑝))
87ad2ant2r 501 . . 3 (((𝐴P𝐵P) ∧ (𝑝Q𝑝 ∈ (1st𝐵))) → ∃𝑟 ∈ (1st𝐴)∃𝑞 ∈ (2nd𝐴)𝑞 <Q (𝑟 +Q 𝑝))
9 elprnqu 7404 . . . . . . . . . . 11 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑞 ∈ (2nd𝐴)) → 𝑞Q)
105, 9sylan 281 . . . . . . . . . 10 ((𝐴P𝑞 ∈ (2nd𝐴)) → 𝑞Q)
1110adantlr 469 . . . . . . . . 9 (((𝐴P𝐵P) ∧ 𝑞 ∈ (2nd𝐴)) → 𝑞Q)
1211ad2ant2rl 503 . . . . . . . 8 ((((𝐴P𝐵P) ∧ (𝑝Q𝑝 ∈ (1st𝐵))) ∧ (𝑟 ∈ (1st𝐴) ∧ 𝑞 ∈ (2nd𝐴))) → 𝑞Q)
1312adantr 274 . . . . . . 7 (((((𝐴P𝐵P) ∧ (𝑝Q𝑝 ∈ (1st𝐵))) ∧ (𝑟 ∈ (1st𝐴) ∧ 𝑞 ∈ (2nd𝐴))) ∧ 𝑞 <Q (𝑟 +Q 𝑝)) → 𝑞Q)
14 simplrr 526 . . . . . . 7 (((((𝐴P𝐵P) ∧ (𝑝Q𝑝 ∈ (1st𝐵))) ∧ (𝑟 ∈ (1st𝐴) ∧ 𝑞 ∈ (2nd𝐴))) ∧ 𝑞 <Q (𝑟 +Q 𝑝)) → 𝑞 ∈ (2nd𝐴))
15 simprl 521 . . . . . . . . . . . . 13 (((𝑝Q𝑝 ∈ (1st𝐵)) ∧ (𝑟 ∈ (1st𝐴) ∧ 𝑞 ∈ (2nd𝐴))) → 𝑟 ∈ (1st𝐴))
16 simplr 520 . . . . . . . . . . . . 13 (((𝑝Q𝑝 ∈ (1st𝐵)) ∧ (𝑟 ∈ (1st𝐴) ∧ 𝑞 ∈ (2nd𝐴))) → 𝑝 ∈ (1st𝐵))
1715, 16jca 304 . . . . . . . . . . . 12 (((𝑝Q𝑝 ∈ (1st𝐵)) ∧ (𝑟 ∈ (1st𝐴) ∧ 𝑞 ∈ (2nd𝐴))) → (𝑟 ∈ (1st𝐴) ∧ 𝑝 ∈ (1st𝐵)))
18 df-iplp 7390 . . . . . . . . . . . . 13 +P = (𝑥P, 𝑦P ↦ ⟨{𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (1st𝑥) ∧ ∈ (1st𝑦) ∧ 𝑓 = (𝑔 +Q ))}, {𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (2nd𝑥) ∧ ∈ (2nd𝑦) ∧ 𝑓 = (𝑔 +Q ))}⟩)
19 addclnq 7297 . . . . . . . . . . . . 13 ((𝑔QQ) → (𝑔 +Q ) ∈ Q)
2018, 19genpprecll 7436 . . . . . . . . . . . 12 ((𝐴P𝐵P) → ((𝑟 ∈ (1st𝐴) ∧ 𝑝 ∈ (1st𝐵)) → (𝑟 +Q 𝑝) ∈ (1st ‘(𝐴 +P 𝐵))))
2117, 20syl5 32 . . . . . . . . . . 11 ((𝐴P𝐵P) → (((𝑝Q𝑝 ∈ (1st𝐵)) ∧ (𝑟 ∈ (1st𝐴) ∧ 𝑞 ∈ (2nd𝐴))) → (𝑟 +Q 𝑝) ∈ (1st ‘(𝐴 +P 𝐵))))
2221imdistani 442 . . . . . . . . . 10 (((𝐴P𝐵P) ∧ ((𝑝Q𝑝 ∈ (1st𝐵)) ∧ (𝑟 ∈ (1st𝐴) ∧ 𝑞 ∈ (2nd𝐴)))) → ((𝐴P𝐵P) ∧ (𝑟 +Q 𝑝) ∈ (1st ‘(𝐴 +P 𝐵))))
23 addclpr 7459 . . . . . . . . . . 11 ((𝐴P𝐵P) → (𝐴 +P 𝐵) ∈ P)
24 prop 7397 . . . . . . . . . . . 12 ((𝐴 +P 𝐵) ∈ P → ⟨(1st ‘(𝐴 +P 𝐵)), (2nd ‘(𝐴 +P 𝐵))⟩ ∈ P)
25 prcdnql 7406 . . . . . . . . . . . 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 281 . . . . . . . . . . 11 (((𝐴 +P 𝐵) ∈ P ∧ (𝑟 +Q 𝑝) ∈ (1st ‘(𝐴 +P 𝐵))) → (𝑞 <Q (𝑟 +Q 𝑝) → 𝑞 ∈ (1st ‘(𝐴 +P 𝐵))))
2723, 26sylan 281 . . . . . . . . . 10 (((𝐴P𝐵P) ∧ (𝑟 +Q 𝑝) ∈ (1st ‘(𝐴 +P 𝐵))) → (𝑞 <Q (𝑟 +Q 𝑝) → 𝑞 ∈ (1st ‘(𝐴 +P 𝐵))))
2822, 27syl 14 . . . . . . . . 9 (((𝐴P𝐵P) ∧ ((𝑝Q𝑝 ∈ (1st𝐵)) ∧ (𝑟 ∈ (1st𝐴) ∧ 𝑞 ∈ (2nd𝐴)))) → (𝑞 <Q (𝑟 +Q 𝑝) → 𝑞 ∈ (1st ‘(𝐴 +P 𝐵))))
2928anassrs 398 . . . . . . . 8 ((((𝐴P𝐵P) ∧ (𝑝Q𝑝 ∈ (1st𝐵))) ∧ (𝑟 ∈ (1st𝐴) ∧ 𝑞 ∈ (2nd𝐴))) → (𝑞 <Q (𝑟 +Q 𝑝) → 𝑞 ∈ (1st ‘(𝐴 +P 𝐵))))
3029imp 123 . . . . . . 7 (((((𝐴P𝐵P) ∧ (𝑝Q𝑝 ∈ (1st𝐵))) ∧ (𝑟 ∈ (1st𝐴) ∧ 𝑞 ∈ (2nd𝐴))) ∧ 𝑞 <Q (𝑟 +Q 𝑝)) → 𝑞 ∈ (1st ‘(𝐴 +P 𝐵)))
31 rspe 2506 . . . . . . 7 ((𝑞Q ∧ (𝑞 ∈ (2nd𝐴) ∧ 𝑞 ∈ (1st ‘(𝐴 +P 𝐵)))) → ∃𝑞Q (𝑞 ∈ (2nd𝐴) ∧ 𝑞 ∈ (1st ‘(𝐴 +P 𝐵))))
3213, 14, 30, 31syl12anc 1218 . . . . . 6 (((((𝐴P𝐵P) ∧ (𝑝Q𝑝 ∈ (1st𝐵))) ∧ (𝑟 ∈ (1st𝐴) ∧ 𝑞 ∈ (2nd𝐴))) ∧ 𝑞 <Q (𝑟 +Q 𝑝)) → ∃𝑞Q (𝑞 ∈ (2nd𝐴) ∧ 𝑞 ∈ (1st ‘(𝐴 +P 𝐵))))
33 ltdfpr 7428 . . . . . . . 8 ((𝐴P ∧ (𝐴 +P 𝐵) ∈ P) → (𝐴<P (𝐴 +P 𝐵) ↔ ∃𝑞Q (𝑞 ∈ (2nd𝐴) ∧ 𝑞 ∈ (1st ‘(𝐴 +P 𝐵)))))
3423, 33syldan 280 . . . . . . 7 ((𝐴P𝐵P) → (𝐴<P (𝐴 +P 𝐵) ↔ ∃𝑞Q (𝑞 ∈ (2nd𝐴) ∧ 𝑞 ∈ (1st ‘(𝐴 +P 𝐵)))))
3534ad3antrrr 484 . . . . . 6 (((((𝐴P𝐵P) ∧ (𝑝Q𝑝 ∈ (1st𝐵))) ∧ (𝑟 ∈ (1st𝐴) ∧ 𝑞 ∈ (2nd𝐴))) ∧ 𝑞 <Q (𝑟 +Q 𝑝)) → (𝐴<P (𝐴 +P 𝐵) ↔ ∃𝑞Q (𝑞 ∈ (2nd𝐴) ∧ 𝑞 ∈ (1st ‘(𝐴 +P 𝐵)))))
3632, 35mpbird 166 . . . . 5 (((((𝐴P𝐵P) ∧ (𝑝Q𝑝 ∈ (1st𝐵))) ∧ (𝑟 ∈ (1st𝐴) ∧ 𝑞 ∈ (2nd𝐴))) ∧ 𝑞 <Q (𝑟 +Q 𝑝)) → 𝐴<P (𝐴 +P 𝐵))
3736ex 114 . . . 4 ((((𝐴P𝐵P) ∧ (𝑝Q𝑝 ∈ (1st𝐵))) ∧ (𝑟 ∈ (1st𝐴) ∧ 𝑞 ∈ (2nd𝐴))) → (𝑞 <Q (𝑟 +Q 𝑝) → 𝐴<P (𝐴 +P 𝐵)))
3837rexlimdvva 2582 . . 3 (((𝐴P𝐵P) ∧ (𝑝Q𝑝 ∈ (1st𝐵))) → (∃𝑟 ∈ (1st𝐴)∃𝑞 ∈ (2nd𝐴)𝑞 <Q (𝑟 +Q 𝑝) → 𝐴<P (𝐴 +P 𝐵)))
398, 38mpd 13 . 2 (((𝐴P𝐵P) ∧ (𝑝Q𝑝 ∈ (1st𝐵))) → 𝐴<P (𝐴 +P 𝐵))
404, 39rexlimddv 2579 1 ((𝐴P𝐵P) → 𝐴<P (𝐴 +P 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wcel 2128  wrex 2436  cop 3564   class class class wbr 3967  cfv 5172  (class class class)co 5826  1st c1st 6088  2nd c2nd 6089  Qcnq 7202   +Q cplq 7204   <Q cltq 7207  Pcnp 7213   +P cpp 7215  <P cltp 7217
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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4081  ax-sep 4084  ax-nul 4092  ax-pow 4137  ax-pr 4171  ax-un 4395  ax-setind 4498  ax-iinf 4549
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3396  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-int 3810  df-iun 3853  df-br 3968  df-opab 4028  df-mpt 4029  df-tr 4065  df-eprel 4251  df-id 4255  df-po 4258  df-iso 4259  df-iord 4328  df-on 4330  df-suc 4333  df-iom 4552  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-iota 5137  df-fun 5174  df-fn 5175  df-f 5176  df-f1 5177  df-fo 5178  df-f1o 5179  df-fv 5180  df-ov 5829  df-oprab 5830  df-mpo 5831  df-1st 6090  df-2nd 6091  df-recs 6254  df-irdg 6319  df-1o 6365  df-2o 6366  df-oadd 6369  df-omul 6370  df-er 6482  df-ec 6484  df-qs 6488  df-ni 7226  df-pli 7227  df-mi 7228  df-lti 7229  df-plpq 7266  df-mpq 7267  df-enq 7269  df-nqqs 7270  df-plqqs 7271  df-mqqs 7272  df-1nqqs 7273  df-rq 7274  df-ltnqqs 7275  df-enq0 7346  df-nq0 7347  df-0nq0 7348  df-plq0 7349  df-mq0 7350  df-inp 7388  df-iplp 7390  df-iltp 7392
This theorem is referenced by:  ltexprlemrl  7532  ltaprlem  7540  ltaprg  7541  prplnqu  7542  ltmprr  7564  caucvgprprlemnkltj  7611  caucvgprprlemnkeqj  7612  caucvgprprlemnbj  7615  0lt1sr  7687  recexgt0sr  7695  mulgt0sr  7700  archsr  7704  prsrpos  7707  mappsrprg  7726  pitoregt0  7771
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