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Theorem ltaddpr 9816
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 9771 . . . . 5 (𝐵P𝐵 ≠ ∅)
2 n0 3913 . . . . 5 (𝐵 ≠ ∅ ↔ ∃𝑦 𝑦𝐵)
31, 2sylib 208 . . . 4 (𝐵P → ∃𝑦 𝑦𝐵)
43adantl 482 . . 3 ((𝐴P𝐵P) → ∃𝑦 𝑦𝐵)
5 addclpr 9800 . . . . . . . . . . . 12 ((𝐴P𝐵P) → (𝐴 +P 𝐵) ∈ P)
65adantr 481 . . . . . . . . . . 11 (((𝐴P𝐵P) ∧ (𝑥𝐴𝑦𝐵)) → (𝐴 +P 𝐵) ∈ P)
7 df-plp 9765 . . . . . . . . . . . . 13 +P = (𝑤P, 𝑣P ↦ {𝑥 ∣ ∃𝑦𝑤𝑧𝑣 𝑥 = (𝑦 +Q 𝑧)})
8 addclnq 9727 . . . . . . . . . . . . 13 ((𝑦Q𝑧Q) → (𝑦 +Q 𝑧) ∈ Q)
97, 8genpprecl 9783 . . . . . . . . . . . 12 ((𝐴P𝐵P) → ((𝑥𝐴𝑦𝐵) → (𝑥 +Q 𝑦) ∈ (𝐴 +P 𝐵)))
109imp 445 . . . . . . . . . . 11 (((𝐴P𝐵P) ∧ (𝑥𝐴𝑦𝐵)) → (𝑥 +Q 𝑦) ∈ (𝐴 +P 𝐵))
11 elprnq 9773 . . . . . . . . . . . . 13 (((𝐴 +P 𝐵) ∈ P ∧ (𝑥 +Q 𝑦) ∈ (𝐴 +P 𝐵)) → (𝑥 +Q 𝑦) ∈ Q)
12 addnqf 9730 . . . . . . . . . . . . . . 15 +Q :(Q × Q)⟶Q
1312fdmi 6019 . . . . . . . . . . . . . 14 dom +Q = (Q × Q)
14 0nnq 9706 . . . . . . . . . . . . . 14 ¬ ∅ ∈ Q
1513, 14ndmovrcl 6785 . . . . . . . . . . . . 13 ((𝑥 +Q 𝑦) ∈ Q → (𝑥Q𝑦Q))
16 ltaddnq 9756 . . . . . . . . . . . . 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 9775 . . . . . . . . . . . 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 692 . . . . . . . . . 10 (((𝐴P𝐵P) ∧ (𝑥𝐴𝑦𝐵)) → 𝑥 ∈ (𝐴 +P 𝐵))
2120exp32 630 . . . . . . . . 9 ((𝐴P𝐵P) → (𝑥𝐴 → (𝑦𝐵𝑥 ∈ (𝐴 +P 𝐵))))
2221com23 86 . . . . . . . 8 ((𝐴P𝐵P) → (𝑦𝐵 → (𝑥𝐴𝑥 ∈ (𝐴 +P 𝐵))))
2322alrimdv 1854 . . . . . . 7 ((𝐴P𝐵P) → (𝑦𝐵 → ∀𝑥(𝑥𝐴𝑥 ∈ (𝐴 +P 𝐵))))
24 dfss2 3577 . . . . . . 7 (𝐴 ⊆ (𝐴 +P 𝐵) ↔ ∀𝑥(𝑥𝐴𝑥 ∈ (𝐴 +P 𝐵)))
2523, 24syl6ibr 242 . . . . . 6 ((𝐴P𝐵P) → (𝑦𝐵𝐴 ⊆ (𝐴 +P 𝐵)))
26 vex 3193 . . . . . . . . 9 𝑦 ∈ V
2726prlem934 9815 . . . . . . . 8 (𝐴P → ∃𝑥𝐴 ¬ (𝑥 +Q 𝑦) ∈ 𝐴)
2827adantr 481 . . . . . . 7 ((𝐴P𝐵P) → ∃𝑥𝐴 ¬ (𝑥 +Q 𝑦) ∈ 𝐴)
29 eleq2 2687 . . . . . . . . . . . . 13 (𝐴 = (𝐴 +P 𝐵) → ((𝑥 +Q 𝑦) ∈ 𝐴 ↔ (𝑥 +Q 𝑦) ∈ (𝐴 +P 𝐵)))
3029biimprcd 240 . . . . . . . . . . . 12 ((𝑥 +Q 𝑦) ∈ (𝐴 +P 𝐵) → (𝐴 = (𝐴 +P 𝐵) → (𝑥 +Q 𝑦) ∈ 𝐴))
3130con3d 148 . . . . . . . . . . 11 ((𝑥 +Q 𝑦) ∈ (𝐴 +P 𝐵) → (¬ (𝑥 +Q 𝑦) ∈ 𝐴 → ¬ 𝐴 = (𝐴 +P 𝐵)))
329, 31syl6 35 . . . . . . . . . 10 ((𝐴P𝐵P) → ((𝑥𝐴𝑦𝐵) → (¬ (𝑥 +Q 𝑦) ∈ 𝐴 → ¬ 𝐴 = (𝐴 +P 𝐵))))
3332expd 452 . . . . . . . . 9 ((𝐴P𝐵P) → (𝑥𝐴 → (𝑦𝐵 → (¬ (𝑥 +Q 𝑦) ∈ 𝐴 → ¬ 𝐴 = (𝐴 +P 𝐵)))))
3433com34 91 . . . . . . . 8 ((𝐴P𝐵P) → (𝑥𝐴 → (¬ (𝑥 +Q 𝑦) ∈ 𝐴 → (𝑦𝐵 → ¬ 𝐴 = (𝐴 +P 𝐵)))))
3534rexlimdv 3025 . . . . . . 7 ((𝐴P𝐵P) → (∃𝑥𝐴 ¬ (𝑥 +Q 𝑦) ∈ 𝐴 → (𝑦𝐵 → ¬ 𝐴 = (𝐴 +P 𝐵))))
3628, 35mpd 15 . . . . . 6 ((𝐴P𝐵P) → (𝑦𝐵 → ¬ 𝐴 = (𝐴 +P 𝐵)))
3725, 36jcad 555 . . . . 5 ((𝐴P𝐵P) → (𝑦𝐵 → (𝐴 ⊆ (𝐴 +P 𝐵) ∧ ¬ 𝐴 = (𝐴 +P 𝐵))))
38 dfpss2 3676 . . . . 5 (𝐴 ⊊ (𝐴 +P 𝐵) ↔ (𝐴 ⊆ (𝐴 +P 𝐵) ∧ ¬ 𝐴 = (𝐴 +P 𝐵)))
3937, 38syl6ibr 242 . . . 4 ((𝐴P𝐵P) → (𝑦𝐵𝐴 ⊊ (𝐴 +P 𝐵)))
4039exlimdv 1858 . . 3 ((𝐴P𝐵P) → (∃𝑦 𝑦𝐵𝐴 ⊊ (𝐴 +P 𝐵)))
414, 40mpd 15 . 2 ((𝐴P𝐵P) → 𝐴 ⊊ (𝐴 +P 𝐵))
42 ltprord 9812 . . 3 ((𝐴P ∧ (𝐴 +P 𝐵) ∈ P) → (𝐴<P (𝐴 +P 𝐵) ↔ 𝐴 ⊊ (𝐴 +P 𝐵)))
435, 42syldan 487 . 2 ((𝐴P𝐵P) → (𝐴<P (𝐴 +P 𝐵) ↔ 𝐴 ⊊ (𝐴 +P 𝐵)))
4441, 43mpbird 247 1 ((𝐴P𝐵P) → 𝐴<P (𝐴 +P 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  wal 1478   = wceq 1480  wex 1701  wcel 1987  wne 2790  wrex 2909  wss 3560  wpss 3561  c0 3897   class class class wbr 4623   × cxp 5082  (class class class)co 6615  Qcnq 9634   +Q cplq 9637   <Q cltq 9640  Pcnp 9641   +P cpp 9643  <P cltp 9645
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-inf2 8498
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-oadd 7524  df-omul 7525  df-er 7702  df-ni 9654  df-pli 9655  df-mi 9656  df-lti 9657  df-plpq 9690  df-mpq 9691  df-ltpq 9692  df-enq 9693  df-nq 9694  df-erq 9695  df-plq 9696  df-mq 9697  df-1nq 9698  df-rq 9699  df-ltnq 9700  df-np 9763  df-plp 9765  df-ltp 9767
This theorem is referenced by:  ltaddpr2  9817  ltexprlem7  9824  ltaprlem  9826  0lt1sr  9876  mappsrpr  9889
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