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Theorem ltaddpr 9712
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 9667 . . . . 5 (𝐵P𝐵 ≠ ∅)
2 n0 3889 . . . . 5 (𝐵 ≠ ∅ ↔ ∃𝑦 𝑦𝐵)
31, 2sylib 206 . . . 4 (𝐵P → ∃𝑦 𝑦𝐵)
43adantl 480 . . 3 ((𝐴P𝐵P) → ∃𝑦 𝑦𝐵)
5 addclpr 9696 . . . . . . . . . . . 12 ((𝐴P𝐵P) → (𝐴 +P 𝐵) ∈ P)
65adantr 479 . . . . . . . . . . 11 (((𝐴P𝐵P) ∧ (𝑥𝐴𝑦𝐵)) → (𝐴 +P 𝐵) ∈ P)
7 df-plp 9661 . . . . . . . . . . . . 13 +P = (𝑤P, 𝑣P ↦ {𝑥 ∣ ∃𝑦𝑤𝑧𝑣 𝑥 = (𝑦 +Q 𝑧)})
8 addclnq 9623 . . . . . . . . . . . . 13 ((𝑦Q𝑧Q) → (𝑦 +Q 𝑧) ∈ Q)
97, 8genpprecl 9679 . . . . . . . . . . . 12 ((𝐴P𝐵P) → ((𝑥𝐴𝑦𝐵) → (𝑥 +Q 𝑦) ∈ (𝐴 +P 𝐵)))
109imp 443 . . . . . . . . . . 11 (((𝐴P𝐵P) ∧ (𝑥𝐴𝑦𝐵)) → (𝑥 +Q 𝑦) ∈ (𝐴 +P 𝐵))
11 elprnq 9669 . . . . . . . . . . . . 13 (((𝐴 +P 𝐵) ∈ P ∧ (𝑥 +Q 𝑦) ∈ (𝐴 +P 𝐵)) → (𝑥 +Q 𝑦) ∈ Q)
12 addnqf 9626 . . . . . . . . . . . . . . 15 +Q :(Q × Q)⟶Q
1312fdmi 5951 . . . . . . . . . . . . . 14 dom +Q = (Q × Q)
14 0nnq 9602 . . . . . . . . . . . . . 14 ¬ ∅ ∈ Q
1513, 14ndmovrcl 6695 . . . . . . . . . . . . 13 ((𝑥 +Q 𝑦) ∈ Q → (𝑥Q𝑦Q))
16 ltaddnq 9652 . . . . . . . . . . . . 13 ((𝑥Q𝑦Q) → 𝑥 <Q (𝑥 +Q 𝑦))
1711, 15, 163syl 18 . . . . . . . . . . . 12 (((𝐴 +P 𝐵) ∈ P ∧ (𝑥 +Q 𝑦) ∈ (𝐴 +P 𝐵)) → 𝑥 <Q (𝑥 +Q 𝑦))
18 prcdnq 9671 . . . . . . . . . . . 12 (((𝐴 +P 𝐵) ∈ P ∧ (𝑥 +Q 𝑦) ∈ (𝐴 +P 𝐵)) → (𝑥 <Q (𝑥 +Q 𝑦) → 𝑥 ∈ (𝐴 +P 𝐵)))
1917, 18mpd 15 . . . . . . . . . . 11 (((𝐴 +P 𝐵) ∈ P ∧ (𝑥 +Q 𝑦) ∈ (𝐴 +P 𝐵)) → 𝑥 ∈ (𝐴 +P 𝐵))
206, 10, 19syl2anc 690 . . . . . . . . . 10 (((𝐴P𝐵P) ∧ (𝑥𝐴𝑦𝐵)) → 𝑥 ∈ (𝐴 +P 𝐵))
2120exp32 628 . . . . . . . . 9 ((𝐴P𝐵P) → (𝑥𝐴 → (𝑦𝐵𝑥 ∈ (𝐴 +P 𝐵))))
2221com23 83 . . . . . . . 8 ((𝐴P𝐵P) → (𝑦𝐵 → (𝑥𝐴𝑥 ∈ (𝐴 +P 𝐵))))
2322alrimdv 1843 . . . . . . 7 ((𝐴P𝐵P) → (𝑦𝐵 → ∀𝑥(𝑥𝐴𝑥 ∈ (𝐴 +P 𝐵))))
24 dfss2 3556 . . . . . . 7 (𝐴 ⊆ (𝐴 +P 𝐵) ↔ ∀𝑥(𝑥𝐴𝑥 ∈ (𝐴 +P 𝐵)))
2523, 24syl6ibr 240 . . . . . 6 ((𝐴P𝐵P) → (𝑦𝐵𝐴 ⊆ (𝐴 +P 𝐵)))
26 vex 3175 . . . . . . . . 9 𝑦 ∈ V
2726prlem934 9711 . . . . . . . 8 (𝐴P → ∃𝑥𝐴 ¬ (𝑥 +Q 𝑦) ∈ 𝐴)
2827adantr 479 . . . . . . 7 ((𝐴P𝐵P) → ∃𝑥𝐴 ¬ (𝑥 +Q 𝑦) ∈ 𝐴)
29 eleq2 2676 . . . . . . . . . . . . 13 (𝐴 = (𝐴 +P 𝐵) → ((𝑥 +Q 𝑦) ∈ 𝐴 ↔ (𝑥 +Q 𝑦) ∈ (𝐴 +P 𝐵)))
3029biimprcd 238 . . . . . . . . . . . 12 ((𝑥 +Q 𝑦) ∈ (𝐴 +P 𝐵) → (𝐴 = (𝐴 +P 𝐵) → (𝑥 +Q 𝑦) ∈ 𝐴))
3130con3d 146 . . . . . . . . . . 11 ((𝑥 +Q 𝑦) ∈ (𝐴 +P 𝐵) → (¬ (𝑥 +Q 𝑦) ∈ 𝐴 → ¬ 𝐴 = (𝐴 +P 𝐵)))
329, 31syl6 34 . . . . . . . . . 10 ((𝐴P𝐵P) → ((𝑥𝐴𝑦𝐵) → (¬ (𝑥 +Q 𝑦) ∈ 𝐴 → ¬ 𝐴 = (𝐴 +P 𝐵))))
3332expd 450 . . . . . . . . 9 ((𝐴P𝐵P) → (𝑥𝐴 → (𝑦𝐵 → (¬ (𝑥 +Q 𝑦) ∈ 𝐴 → ¬ 𝐴 = (𝐴 +P 𝐵)))))
3433com34 88 . . . . . . . 8 ((𝐴P𝐵P) → (𝑥𝐴 → (¬ (𝑥 +Q 𝑦) ∈ 𝐴 → (𝑦𝐵 → ¬ 𝐴 = (𝐴 +P 𝐵)))))
3534rexlimdv 3011 . . . . . . 7 ((𝐴P𝐵P) → (∃𝑥𝐴 ¬ (𝑥 +Q 𝑦) ∈ 𝐴 → (𝑦𝐵 → ¬ 𝐴 = (𝐴 +P 𝐵))))
3628, 35mpd 15 . . . . . 6 ((𝐴P𝐵P) → (𝑦𝐵 → ¬ 𝐴 = (𝐴 +P 𝐵)))
3725, 36jcad 553 . . . . 5 ((𝐴P𝐵P) → (𝑦𝐵 → (𝐴 ⊆ (𝐴 +P 𝐵) ∧ ¬ 𝐴 = (𝐴 +P 𝐵))))
38 dfpss2 3653 . . . . 5 (𝐴 ⊊ (𝐴 +P 𝐵) ↔ (𝐴 ⊆ (𝐴 +P 𝐵) ∧ ¬ 𝐴 = (𝐴 +P 𝐵)))
3937, 38syl6ibr 240 . . . 4 ((𝐴P𝐵P) → (𝑦𝐵𝐴 ⊊ (𝐴 +P 𝐵)))
4039exlimdv 1847 . . 3 ((𝐴P𝐵P) → (∃𝑦 𝑦𝐵𝐴 ⊊ (𝐴 +P 𝐵)))
414, 40mpd 15 . 2 ((𝐴P𝐵P) → 𝐴 ⊊ (𝐴 +P 𝐵))
42 ltprord 9708 . . 3 ((𝐴P ∧ (𝐴 +P 𝐵) ∈ P) → (𝐴<P (𝐴 +P 𝐵) ↔ 𝐴 ⊊ (𝐴 +P 𝐵)))
435, 42syldan 485 . 2 ((𝐴P𝐵P) → (𝐴<P (𝐴 +P 𝐵) ↔ 𝐴 ⊊ (𝐴 +P 𝐵)))
4441, 43mpbird 245 1 ((𝐴P𝐵P) → 𝐴<P (𝐴 +P 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382  wal 1472   = wceq 1474  wex 1694  wcel 1976  wne 2779  wrex 2896  wss 3539  wpss 3540  c0 3873   class class class wbr 4577   × cxp 5026  (class class class)co 6527  Qcnq 9530   +Q cplq 9533   <Q cltq 9536  Pcnp 9537   +P cpp 9539  <P cltp 9541
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-inf2 8398
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6935  df-1st 7036  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-1o 7424  df-oadd 7428  df-omul 7429  df-er 7606  df-ni 9550  df-pli 9551  df-mi 9552  df-lti 9553  df-plpq 9586  df-mpq 9587  df-ltpq 9588  df-enq 9589  df-nq 9590  df-erq 9591  df-plq 9592  df-mq 9593  df-1nq 9594  df-rq 9595  df-ltnq 9596  df-np 9659  df-plp 9661  df-ltp 9663
This theorem is referenced by:  ltaddpr2  9713  ltexprlem7  9720  ltaprlem  9722  0lt1sr  9772  mappsrpr  9785
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