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Theorem ltaddpr 11074
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 11029 . . . . 5 (𝐵P𝐵 ≠ ∅)
2 n0 4353 . . . . 5 (𝐵 ≠ ∅ ↔ ∃𝑦 𝑦𝐵)
31, 2sylib 218 . . . 4 (𝐵P → ∃𝑦 𝑦𝐵)
43adantl 481 . . 3 ((𝐴P𝐵P) → ∃𝑦 𝑦𝐵)
5 addclpr 11058 . . . . . . . . . . 11 ((𝐴P𝐵P) → (𝐴 +P 𝐵) ∈ P)
6 df-plp 11023 . . . . . . . . . . . . 13 +P = (𝑤P, 𝑣P ↦ {𝑥 ∣ ∃𝑦𝑤𝑧𝑣 𝑥 = (𝑦 +Q 𝑧)})
7 addclnq 10985 . . . . . . . . . . . . 13 ((𝑦Q𝑧Q) → (𝑦 +Q 𝑧) ∈ Q)
86, 7genpprecl 11041 . . . . . . . . . . . 12 ((𝐴P𝐵P) → ((𝑥𝐴𝑦𝐵) → (𝑥 +Q 𝑦) ∈ (𝐴 +P 𝐵)))
98imp 406 . . . . . . . . . . 11 (((𝐴P𝐵P) ∧ (𝑥𝐴𝑦𝐵)) → (𝑥 +Q 𝑦) ∈ (𝐴 +P 𝐵))
10 elprnq 11031 . . . . . . . . . . . . 13 (((𝐴 +P 𝐵) ∈ P ∧ (𝑥 +Q 𝑦) ∈ (𝐴 +P 𝐵)) → (𝑥 +Q 𝑦) ∈ Q)
11 addnqf 10988 . . . . . . . . . . . . . . 15 +Q :(Q × Q)⟶Q
1211fdmi 6747 . . . . . . . . . . . . . 14 dom +Q = (Q × Q)
13 0nnq 10964 . . . . . . . . . . . . . 14 ¬ ∅ ∈ Q
1412, 13ndmovrcl 7619 . . . . . . . . . . . . 13 ((𝑥 +Q 𝑦) ∈ Q → (𝑥Q𝑦Q))
15 ltaddnq 11014 . . . . . . . . . . . . 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 11033 . . . . . . . . . . . 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 685 . . . . . . . . . 10 (((𝐴P𝐵P) ∧ (𝑥𝐴𝑦𝐵)) → 𝑥 ∈ (𝐴 +P 𝐵))
2019exp32 420 . . . . . . . . 9 ((𝐴P𝐵P) → (𝑥𝐴 → (𝑦𝐵𝑥 ∈ (𝐴 +P 𝐵))))
2120com23 86 . . . . . . . 8 ((𝐴P𝐵P) → (𝑦𝐵 → (𝑥𝐴𝑥 ∈ (𝐴 +P 𝐵))))
2221alrimdv 1929 . . . . . . 7 ((𝐴P𝐵P) → (𝑦𝐵 → ∀𝑥(𝑥𝐴𝑥 ∈ (𝐴 +P 𝐵))))
23 df-ss 3968 . . . . . . 7 (𝐴 ⊆ (𝐴 +P 𝐵) ↔ ∀𝑥(𝑥𝐴𝑥 ∈ (𝐴 +P 𝐵)))
2422, 23imbitrrdi 252 . . . . . 6 ((𝐴P𝐵P) → (𝑦𝐵𝐴 ⊆ (𝐴 +P 𝐵)))
25 vex 3484 . . . . . . . . 9 𝑦 ∈ V
2625prlem934 11073 . . . . . . . 8 (𝐴P → ∃𝑥𝐴 ¬ (𝑥 +Q 𝑦) ∈ 𝐴)
2726adantr 480 . . . . . . 7 ((𝐴P𝐵P) → ∃𝑥𝐴 ¬ (𝑥 +Q 𝑦) ∈ 𝐴)
28 eleq2 2830 . . . . . . . . . . . . 13 (𝐴 = (𝐴 +P 𝐵) → ((𝑥 +Q 𝑦) ∈ 𝐴 ↔ (𝑥 +Q 𝑦) ∈ (𝐴 +P 𝐵)))
2928biimprcd 250 . . . . . . . . . . . 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 415 . . . . . . . . 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 3153 . . . . . . 7 ((𝐴P𝐵P) → (∃𝑥𝐴 ¬ (𝑥 +Q 𝑦) ∈ 𝐴 → (𝑦𝐵 → ¬ 𝐴 = (𝐴 +P 𝐵))))
3527, 34mpd 15 . . . . . 6 ((𝐴P𝐵P) → (𝑦𝐵 → ¬ 𝐴 = (𝐴 +P 𝐵)))
3624, 35jcad 512 . . . . 5 ((𝐴P𝐵P) → (𝑦𝐵 → (𝐴 ⊆ (𝐴 +P 𝐵) ∧ ¬ 𝐴 = (𝐴 +P 𝐵))))
37 dfpss2 4088 . . . . 5 (𝐴 ⊊ (𝐴 +P 𝐵) ↔ (𝐴 ⊆ (𝐴 +P 𝐵) ∧ ¬ 𝐴 = (𝐴 +P 𝐵)))
3836, 37imbitrrdi 252 . . . 4 ((𝐴P𝐵P) → (𝑦𝐵𝐴 ⊊ (𝐴 +P 𝐵)))
3938exlimdv 1933 . . 3 ((𝐴P𝐵P) → (∃𝑦 𝑦𝐵𝐴 ⊊ (𝐴 +P 𝐵)))
404, 39mpd 15 . 2 ((𝐴P𝐵P) → 𝐴 ⊊ (𝐴 +P 𝐵))
41 ltprord 11070 . . 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 257 1 ((𝐴P𝐵P) → 𝐴<P (𝐴 +P 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wal 1538   = wceq 1540  wex 1779  wcel 2108  wne 2940  wrex 3070  wss 3951  wpss 3952  c0 4333   class class class wbr 5143   × cxp 5683  (class class class)co 7431  Qcnq 10892   +Q cplq 10895   <Q cltq 10898  Pcnp 10899   +P cpp 10901  <P cltp 10903
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-oadd 8510  df-omul 8511  df-er 8745  df-ni 10912  df-pli 10913  df-mi 10914  df-lti 10915  df-plpq 10948  df-mpq 10949  df-ltpq 10950  df-enq 10951  df-nq 10952  df-erq 10953  df-plq 10954  df-mq 10955  df-1nq 10956  df-rq 10957  df-ltnq 10958  df-np 11021  df-plp 11023  df-ltp 11025
This theorem is referenced by:  ltaddpr2  11075  ltexprlem7  11082  ltaprlem  11084  0lt1sr  11135  mappsrpr  11148
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