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Theorem ltaddpr 10536
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 10491 . . . . 5 (𝐵P𝐵 ≠ ∅)
2 n0 4235 . . . . 5 (𝐵 ≠ ∅ ↔ ∃𝑦 𝑦𝐵)
31, 2sylib 221 . . . 4 (𝐵P → ∃𝑦 𝑦𝐵)
43adantl 485 . . 3 ((𝐴P𝐵P) → ∃𝑦 𝑦𝐵)
5 addclpr 10520 . . . . . . . . . . 11 ((𝐴P𝐵P) → (𝐴 +P 𝐵) ∈ P)
6 df-plp 10485 . . . . . . . . . . . . 13 +P = (𝑤P, 𝑣P ↦ {𝑥 ∣ ∃𝑦𝑤𝑧𝑣 𝑥 = (𝑦 +Q 𝑧)})
7 addclnq 10447 . . . . . . . . . . . . 13 ((𝑦Q𝑧Q) → (𝑦 +Q 𝑧) ∈ Q)
86, 7genpprecl 10503 . . . . . . . . . . . 12 ((𝐴P𝐵P) → ((𝑥𝐴𝑦𝐵) → (𝑥 +Q 𝑦) ∈ (𝐴 +P 𝐵)))
98imp 410 . . . . . . . . . . 11 (((𝐴P𝐵P) ∧ (𝑥𝐴𝑦𝐵)) → (𝑥 +Q 𝑦) ∈ (𝐴 +P 𝐵))
10 elprnq 10493 . . . . . . . . . . . . 13 (((𝐴 +P 𝐵) ∈ P ∧ (𝑥 +Q 𝑦) ∈ (𝐴 +P 𝐵)) → (𝑥 +Q 𝑦) ∈ Q)
11 addnqf 10450 . . . . . . . . . . . . . . 15 +Q :(Q × Q)⟶Q
1211fdmi 6516 . . . . . . . . . . . . . 14 dom +Q = (Q × Q)
13 0nnq 10426 . . . . . . . . . . . . . 14 ¬ ∅ ∈ Q
1412, 13ndmovrcl 7352 . . . . . . . . . . . . 13 ((𝑥 +Q 𝑦) ∈ Q → (𝑥Q𝑦Q))
15 ltaddnq 10476 . . . . . . . . . . . . 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 10495 . . . . . . . . . . . 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 424 . . . . . . . . 9 ((𝐴P𝐵P) → (𝑥𝐴 → (𝑦𝐵𝑥 ∈ (𝐴 +P 𝐵))))
2120com23 86 . . . . . . . 8 ((𝐴P𝐵P) → (𝑦𝐵 → (𝑥𝐴𝑥 ∈ (𝐴 +P 𝐵))))
2221alrimdv 1936 . . . . . . 7 ((𝐴P𝐵P) → (𝑦𝐵 → ∀𝑥(𝑥𝐴𝑥 ∈ (𝐴 +P 𝐵))))
23 dfss2 3863 . . . . . . 7 (𝐴 ⊆ (𝐴 +P 𝐵) ↔ ∀𝑥(𝑥𝐴𝑥 ∈ (𝐴 +P 𝐵)))
2422, 23syl6ibr 255 . . . . . 6 ((𝐴P𝐵P) → (𝑦𝐵𝐴 ⊆ (𝐴 +P 𝐵)))
25 vex 3402 . . . . . . . . 9 𝑦 ∈ V
2625prlem934 10535 . . . . . . . 8 (𝐴P → ∃𝑥𝐴 ¬ (𝑥 +Q 𝑦) ∈ 𝐴)
2726adantr 484 . . . . . . 7 ((𝐴P𝐵P) → ∃𝑥𝐴 ¬ (𝑥 +Q 𝑦) ∈ 𝐴)
28 eleq2 2821 . . . . . . . . . . . . 13 (𝐴 = (𝐴 +P 𝐵) → ((𝑥 +Q 𝑦) ∈ 𝐴 ↔ (𝑥 +Q 𝑦) ∈ (𝐴 +P 𝐵)))
2928biimprcd 253 . . . . . . . . . . . 12 ((𝑥 +Q 𝑦) ∈ (𝐴 +P 𝐵) → (𝐴 = (𝐴 +P 𝐵) → (𝑥 +Q 𝑦) ∈ 𝐴))
3029con3d 155 . . . . . . . . . . 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 419 . . . . . . . . 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 3193 . . . . . . 7 ((𝐴P𝐵P) → (∃𝑥𝐴 ¬ (𝑥 +Q 𝑦) ∈ 𝐴 → (𝑦𝐵 → ¬ 𝐴 = (𝐴 +P 𝐵))))
3527, 34mpd 15 . . . . . 6 ((𝐴P𝐵P) → (𝑦𝐵 → ¬ 𝐴 = (𝐴 +P 𝐵)))
3624, 35jcad 516 . . . . 5 ((𝐴P𝐵P) → (𝑦𝐵 → (𝐴 ⊆ (𝐴 +P 𝐵) ∧ ¬ 𝐴 = (𝐴 +P 𝐵))))
37 dfpss2 3976 . . . . 5 (𝐴 ⊊ (𝐴 +P 𝐵) ↔ (𝐴 ⊆ (𝐴 +P 𝐵) ∧ ¬ 𝐴 = (𝐴 +P 𝐵)))
3836, 37syl6ibr 255 . . . 4 ((𝐴P𝐵P) → (𝑦𝐵𝐴 ⊊ (𝐴 +P 𝐵)))
3938exlimdv 1940 . . 3 ((𝐴P𝐵P) → (∃𝑦 𝑦𝐵𝐴 ⊊ (𝐴 +P 𝐵)))
404, 39mpd 15 . 2 ((𝐴P𝐵P) → 𝐴 ⊊ (𝐴 +P 𝐵))
41 ltprord 10532 . . 3 ((𝐴P ∧ (𝐴 +P 𝐵) ∈ P) → (𝐴<P (𝐴 +P 𝐵) ↔ 𝐴 ⊊ (𝐴 +P 𝐵)))
425, 41syldan 594 . 2 ((𝐴P𝐵P) → (𝐴<P (𝐴 +P 𝐵) ↔ 𝐴 ⊊ (𝐴 +P 𝐵)))
4340, 42mpbird 260 1 ((𝐴P𝐵P) → 𝐴<P (𝐴 +P 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wal 1540   = wceq 1542  wex 1786  wcel 2114  wne 2934  wrex 3054  wss 3843  wpss 3844  c0 4211   class class class wbr 5030   × cxp 5523  (class class class)co 7172  Qcnq 10354   +Q cplq 10357   <Q cltq 10360  Pcnp 10361   +P cpp 10363  <P cltp 10365
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7481  ax-inf2 9179
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rmo 3061  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-uni 4797  df-int 4837  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-we 5485  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6129  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-ov 7175  df-oprab 7176  df-mpo 7177  df-om 7602  df-1st 7716  df-2nd 7717  df-wrecs 7978  df-recs 8039  df-rdg 8077  df-1o 8133  df-oadd 8137  df-omul 8138  df-er 8322  df-ni 10374  df-pli 10375  df-mi 10376  df-lti 10377  df-plpq 10410  df-mpq 10411  df-ltpq 10412  df-enq 10413  df-nq 10414  df-erq 10415  df-plq 10416  df-mq 10417  df-1nq 10418  df-rq 10419  df-ltnq 10420  df-np 10483  df-plp 10485  df-ltp 10487
This theorem is referenced by:  ltaddpr2  10537  ltexprlem7  10544  ltaprlem  10546  0lt1sr  10597  mappsrpr  10610
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