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Theorem ltaddpr 10790
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.) (New usage is discouraged.)
Assertion
Ref Expression
ltaddpr ((𝐴P𝐵P) → 𝐴<P (𝐴 +P 𝐵))

Proof of Theorem ltaddpr
Dummy variables 𝑥 𝑦 𝑧 𝑤 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prn0 10745 . . . . 5 (𝐵P𝐵 ≠ ∅)
2 n0 4280 . . . . 5 (𝐵 ≠ ∅ ↔ ∃𝑦 𝑦𝐵)
31, 2sylib 217 . . . 4 (𝐵P → ∃𝑦 𝑦𝐵)
43adantl 482 . . 3 ((𝐴P𝐵P) → ∃𝑦 𝑦𝐵)
5 addclpr 10774 . . . . . . . . . . 11 ((𝐴P𝐵P) → (𝐴 +P 𝐵) ∈ P)
6 df-plp 10739 . . . . . . . . . . . . 13 +P = (𝑤P, 𝑣P ↦ {𝑥 ∣ ∃𝑦𝑤𝑧𝑣 𝑥 = (𝑦 +Q 𝑧)})
7 addclnq 10701 . . . . . . . . . . . . 13 ((𝑦Q𝑧Q) → (𝑦 +Q 𝑧) ∈ Q)
86, 7genpprecl 10757 . . . . . . . . . . . 12 ((𝐴P𝐵P) → ((𝑥𝐴𝑦𝐵) → (𝑥 +Q 𝑦) ∈ (𝐴 +P 𝐵)))
98imp 407 . . . . . . . . . . 11 (((𝐴P𝐵P) ∧ (𝑥𝐴𝑦𝐵)) → (𝑥 +Q 𝑦) ∈ (𝐴 +P 𝐵))
10 elprnq 10747 . . . . . . . . . . . . 13 (((𝐴 +P 𝐵) ∈ P ∧ (𝑥 +Q 𝑦) ∈ (𝐴 +P 𝐵)) → (𝑥 +Q 𝑦) ∈ Q)
11 addnqf 10704 . . . . . . . . . . . . . . 15 +Q :(Q × Q)⟶Q
1211fdmi 6612 . . . . . . . . . . . . . 14 dom +Q = (Q × Q)
13 0nnq 10680 . . . . . . . . . . . . . 14 ¬ ∅ ∈ Q
1412, 13ndmovrcl 7458 . . . . . . . . . . . . 13 ((𝑥 +Q 𝑦) ∈ Q → (𝑥Q𝑦Q))
15 ltaddnq 10730 . . . . . . . . . . . . 13 ((𝑥Q𝑦Q) → 𝑥 <Q (𝑥 +Q 𝑦))
1610, 14, 153syl 18 . . . . . . . . . . . 12 (((𝐴 +P 𝐵) ∈ P ∧ (𝑥 +Q 𝑦) ∈ (𝐴 +P 𝐵)) → 𝑥 <Q (𝑥 +Q 𝑦))
17 prcdnq 10749 . . . . . . . . . . . 12 (((𝐴 +P 𝐵) ∈ P ∧ (𝑥 +Q 𝑦) ∈ (𝐴 +P 𝐵)) → (𝑥 <Q (𝑥 +Q 𝑦) → 𝑥 ∈ (𝐴 +P 𝐵)))
1816, 17mpd 15 . . . . . . . . . . 11 (((𝐴 +P 𝐵) ∈ P ∧ (𝑥 +Q 𝑦) ∈ (𝐴 +P 𝐵)) → 𝑥 ∈ (𝐴 +P 𝐵))
195, 9, 18syl2an2r 682 . . . . . . . . . 10 (((𝐴P𝐵P) ∧ (𝑥𝐴𝑦𝐵)) → 𝑥 ∈ (𝐴 +P 𝐵))
2019exp32 421 . . . . . . . . 9 ((𝐴P𝐵P) → (𝑥𝐴 → (𝑦𝐵𝑥 ∈ (𝐴 +P 𝐵))))
2120com23 86 . . . . . . . 8 ((𝐴P𝐵P) → (𝑦𝐵 → (𝑥𝐴𝑥 ∈ (𝐴 +P 𝐵))))
2221alrimdv 1932 . . . . . . 7 ((𝐴P𝐵P) → (𝑦𝐵 → ∀𝑥(𝑥𝐴𝑥 ∈ (𝐴 +P 𝐵))))
23 dfss2 3907 . . . . . . 7 (𝐴 ⊆ (𝐴 +P 𝐵) ↔ ∀𝑥(𝑥𝐴𝑥 ∈ (𝐴 +P 𝐵)))
2422, 23syl6ibr 251 . . . . . 6 ((𝐴P𝐵P) → (𝑦𝐵𝐴 ⊆ (𝐴 +P 𝐵)))
25 vex 3436 . . . . . . . . 9 𝑦 ∈ V
2625prlem934 10789 . . . . . . . 8 (𝐴P → ∃𝑥𝐴 ¬ (𝑥 +Q 𝑦) ∈ 𝐴)
2726adantr 481 . . . . . . 7 ((𝐴P𝐵P) → ∃𝑥𝐴 ¬ (𝑥 +Q 𝑦) ∈ 𝐴)
28 eleq2 2827 . . . . . . . . . . . . 13 (𝐴 = (𝐴 +P 𝐵) → ((𝑥 +Q 𝑦) ∈ 𝐴 ↔ (𝑥 +Q 𝑦) ∈ (𝐴 +P 𝐵)))
2928biimprcd 249 . . . . . . . . . . . 12 ((𝑥 +Q 𝑦) ∈ (𝐴 +P 𝐵) → (𝐴 = (𝐴 +P 𝐵) → (𝑥 +Q 𝑦) ∈ 𝐴))
3029con3d 152 . . . . . . . . . . 11 ((𝑥 +Q 𝑦) ∈ (𝐴 +P 𝐵) → (¬ (𝑥 +Q 𝑦) ∈ 𝐴 → ¬ 𝐴 = (𝐴 +P 𝐵)))
318, 30syl6 35 . . . . . . . . . 10 ((𝐴P𝐵P) → ((𝑥𝐴𝑦𝐵) → (¬ (𝑥 +Q 𝑦) ∈ 𝐴 → ¬ 𝐴 = (𝐴 +P 𝐵))))
3231expd 416 . . . . . . . . 9 ((𝐴P𝐵P) → (𝑥𝐴 → (𝑦𝐵 → (¬ (𝑥 +Q 𝑦) ∈ 𝐴 → ¬ 𝐴 = (𝐴 +P 𝐵)))))
3332com34 91 . . . . . . . 8 ((𝐴P𝐵P) → (𝑥𝐴 → (¬ (𝑥 +Q 𝑦) ∈ 𝐴 → (𝑦𝐵 → ¬ 𝐴 = (𝐴 +P 𝐵)))))
3433rexlimdv 3212 . . . . . . 7 ((𝐴P𝐵P) → (∃𝑥𝐴 ¬ (𝑥 +Q 𝑦) ∈ 𝐴 → (𝑦𝐵 → ¬ 𝐴 = (𝐴 +P 𝐵))))
3527, 34mpd 15 . . . . . 6 ((𝐴P𝐵P) → (𝑦𝐵 → ¬ 𝐴 = (𝐴 +P 𝐵)))
3624, 35jcad 513 . . . . 5 ((𝐴P𝐵P) → (𝑦𝐵 → (𝐴 ⊆ (𝐴 +P 𝐵) ∧ ¬ 𝐴 = (𝐴 +P 𝐵))))
37 dfpss2 4020 . . . . 5 (𝐴 ⊊ (𝐴 +P 𝐵) ↔ (𝐴 ⊆ (𝐴 +P 𝐵) ∧ ¬ 𝐴 = (𝐴 +P 𝐵)))
3836, 37syl6ibr 251 . . . 4 ((𝐴P𝐵P) → (𝑦𝐵𝐴 ⊊ (𝐴 +P 𝐵)))
3938exlimdv 1936 . . 3 ((𝐴P𝐵P) → (∃𝑦 𝑦𝐵𝐴 ⊊ (𝐴 +P 𝐵)))
404, 39mpd 15 . 2 ((𝐴P𝐵P) → 𝐴 ⊊ (𝐴 +P 𝐵))
41 ltprord 10786 . . 3 ((𝐴P ∧ (𝐴 +P 𝐵) ∈ P) → (𝐴<P (𝐴 +P 𝐵) ↔ 𝐴 ⊊ (𝐴 +P 𝐵)))
425, 41syldan 591 . 2 ((𝐴P𝐵P) → (𝐴<P (𝐴 +P 𝐵) ↔ 𝐴 ⊊ (𝐴 +P 𝐵)))
4340, 42mpbird 256 1 ((𝐴P𝐵P) → 𝐴<P (𝐴 +P 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  wal 1537   = wceq 1539  wex 1782  wcel 2106  wne 2943  wrex 3065  wss 3887  wpss 3888  c0 4256   class class class wbr 5074   × cxp 5587  (class class class)co 7275  Qcnq 10608   +Q cplq 10611   <Q cltq 10614  Pcnp 10615   +P cpp 10617  <P cltp 10619
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-inf2 9399
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-oadd 8301  df-omul 8302  df-er 8498  df-ni 10628  df-pli 10629  df-mi 10630  df-lti 10631  df-plpq 10664  df-mpq 10665  df-ltpq 10666  df-enq 10667  df-nq 10668  df-erq 10669  df-plq 10670  df-mq 10671  df-1nq 10672  df-rq 10673  df-ltnq 10674  df-np 10737  df-plp 10739  df-ltp 10741
This theorem is referenced by:  ltaddpr2  10791  ltexprlem7  10798  ltaprlem  10800  0lt1sr  10851  mappsrpr  10864
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