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Theorem ltaddpr 7412
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 7290 . . . 4 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
2 prml 7292 . . . 4 (⟨(1st𝐵), (2nd𝐵)⟩ ∈ P → ∃𝑝Q 𝑝 ∈ (1st𝐵))
31, 2syl 14 . . 3 (𝐵P → ∃𝑝Q 𝑝 ∈ (1st𝐵))
43adantl 275 . 2 ((𝐴P𝐵P) → ∃𝑝Q 𝑝 ∈ (1st𝐵))
5 prop 7290 . . . . 5 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
6 prarloc 7318 . . . . 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 500 . . 3 (((𝐴P𝐵P) ∧ (𝑝Q𝑝 ∈ (1st𝐵))) → ∃𝑟 ∈ (1st𝐴)∃𝑞 ∈ (2nd𝐴)𝑞 <Q (𝑟 +Q 𝑝))
9 elprnqu 7297 . . . . . . . . . . 11 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑞 ∈ (2nd𝐴)) → 𝑞Q)
105, 9sylan 281 . . . . . . . . . 10 ((𝐴P𝑞 ∈ (2nd𝐴)) → 𝑞Q)
1110adantlr 468 . . . . . . . . 9 (((𝐴P𝐵P) ∧ 𝑞 ∈ (2nd𝐴)) → 𝑞Q)
1211ad2ant2rl 502 . . . . . . . 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 525 . . . . . . 7 (((((𝐴P𝐵P) ∧ (𝑝Q𝑝 ∈ (1st𝐵))) ∧ (𝑟 ∈ (1st𝐴) ∧ 𝑞 ∈ (2nd𝐴))) ∧ 𝑞 <Q (𝑟 +Q 𝑝)) → 𝑞 ∈ (2nd𝐴))
15 simprl 520 . . . . . . . . . . . . 13 (((𝑝Q𝑝 ∈ (1st𝐵)) ∧ (𝑟 ∈ (1st𝐴) ∧ 𝑞 ∈ (2nd𝐴))) → 𝑟 ∈ (1st𝐴))
16 simplr 519 . . . . . . . . . . . . 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 7283 . . . . . . . . . . . . 13 +P = (𝑥P, 𝑦P ↦ ⟨{𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (1st𝑥) ∧ ∈ (1st𝑦) ∧ 𝑓 = (𝑔 +Q ))}, {𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (2nd𝑥) ∧ ∈ (2nd𝑦) ∧ 𝑓 = (𝑔 +Q ))}⟩)
19 addclnq 7190 . . . . . . . . . . . . 13 ((𝑔QQ) → (𝑔 +Q ) ∈ Q)
2018, 19genpprecll 7329 . . . . . . . . . . . 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 441 . . . . . . . . . 10 (((𝐴P𝐵P) ∧ ((𝑝Q𝑝 ∈ (1st𝐵)) ∧ (𝑟 ∈ (1st𝐴) ∧ 𝑞 ∈ (2nd𝐴)))) → ((𝐴P𝐵P) ∧ (𝑟 +Q 𝑝) ∈ (1st ‘(𝐴 +P 𝐵))))
23 addclpr 7352 . . . . . . . . . . 11 ((𝐴P𝐵P) → (𝐴 +P 𝐵) ∈ P)
24 prop 7290 . . . . . . . . . . . 12 ((𝐴 +P 𝐵) ∈ P → ⟨(1st ‘(𝐴 +P 𝐵)), (2nd ‘(𝐴 +P 𝐵))⟩ ∈ P)
25 prcdnql 7299 . . . . . . . . . . . 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 397 . . . . . . . 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 2481 . . . . . . 7 ((𝑞Q ∧ (𝑞 ∈ (2nd𝐴) ∧ 𝑞 ∈ (1st ‘(𝐴 +P 𝐵)))) → ∃𝑞Q (𝑞 ∈ (2nd𝐴) ∧ 𝑞 ∈ (1st ‘(𝐴 +P 𝐵))))
3213, 14, 30, 31syl12anc 1214 . . . . . 6 (((((𝐴P𝐵P) ∧ (𝑝Q𝑝 ∈ (1st𝐵))) ∧ (𝑟 ∈ (1st𝐴) ∧ 𝑞 ∈ (2nd𝐴))) ∧ 𝑞 <Q (𝑟 +Q 𝑝)) → ∃𝑞Q (𝑞 ∈ (2nd𝐴) ∧ 𝑞 ∈ (1st ‘(𝐴 +P 𝐵))))
33 ltdfpr 7321 . . . . . . . 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 483 . . . . . 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 2557 . . 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 2554 1 ((𝐴P𝐵P) → 𝐴<P (𝐴 +P 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wcel 1480  wrex 2417  cop 3530   class class class wbr 3929  cfv 5123  (class class class)co 5774  1st c1st 6036  2nd c2nd 6037  Qcnq 7095   +Q cplq 7097   <Q cltq 7100  Pcnp 7106   +P cpp 7108  <P cltp 7110
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-eprel 4211  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-2o 6314  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7119  df-pli 7120  df-mi 7121  df-lti 7122  df-plpq 7159  df-mpq 7160  df-enq 7162  df-nqqs 7163  df-plqqs 7164  df-mqqs 7165  df-1nqqs 7166  df-rq 7167  df-ltnqqs 7168  df-enq0 7239  df-nq0 7240  df-0nq0 7241  df-plq0 7242  df-mq0 7243  df-inp 7281  df-iplp 7283  df-iltp 7285
This theorem is referenced by:  ltexprlemrl  7425  ltaprlem  7433  ltaprg  7434  prplnqu  7435  ltmprr  7457  caucvgprprlemnkltj  7504  caucvgprprlemnkeqj  7505  caucvgprprlemnbj  7508  0lt1sr  7580  recexgt0sr  7588  mulgt0sr  7593  archsr  7597  prsrpos  7600  mappsrprg  7619  pitoregt0  7664
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