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Theorem ltexpri 9619
Description: Proposition 9-3.5(iv) of [Gleason] p. 123. (Contributed by NM, 13-May-1996.) (Revised by Mario Carneiro, 14-Jun-2013.) (New usage is discouraged.)
Assertion
Ref Expression
ltexpri (𝐴<P 𝐵 → ∃𝑥P (𝐴 +P 𝑥) = 𝐵)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem ltexpri
Dummy variables 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltrelpr 9574 . . 3 <P ⊆ (P × P)
21brel 4984 . 2 (𝐴<P 𝐵 → (𝐴P𝐵P))
3 ltprord 9606 . . 3 ((𝐴P𝐵P) → (𝐴<P 𝐵𝐴𝐵))
4 oveq2 6433 . . . . . . . . . . 11 (𝑦 = 𝑧 → (𝑤 +Q 𝑦) = (𝑤 +Q 𝑧))
54eleq1d 2576 . . . . . . . . . 10 (𝑦 = 𝑧 → ((𝑤 +Q 𝑦) ∈ 𝐵 ↔ (𝑤 +Q 𝑧) ∈ 𝐵))
65anbi2d 735 . . . . . . . . 9 (𝑦 = 𝑧 → ((¬ 𝑤𝐴 ∧ (𝑤 +Q 𝑦) ∈ 𝐵) ↔ (¬ 𝑤𝐴 ∧ (𝑤 +Q 𝑧) ∈ 𝐵)))
76exbidv 1803 . . . . . . . 8 (𝑦 = 𝑧 → (∃𝑤𝑤𝐴 ∧ (𝑤 +Q 𝑦) ∈ 𝐵) ↔ ∃𝑤𝑤𝐴 ∧ (𝑤 +Q 𝑧) ∈ 𝐵)))
87cbvabv 2638 . . . . . . 7 {𝑦 ∣ ∃𝑤𝑤𝐴 ∧ (𝑤 +Q 𝑦) ∈ 𝐵)} = {𝑧 ∣ ∃𝑤𝑤𝐴 ∧ (𝑤 +Q 𝑧) ∈ 𝐵)}
98ltexprlem5 9616 . . . . . 6 ((𝐵P𝐴𝐵) → {𝑦 ∣ ∃𝑤𝑤𝐴 ∧ (𝑤 +Q 𝑦) ∈ 𝐵)} ∈ P)
109adantll 745 . . . . 5 (((𝐴P𝐵P) ∧ 𝐴𝐵) → {𝑦 ∣ ∃𝑤𝑤𝐴 ∧ (𝑤 +Q 𝑦) ∈ 𝐵)} ∈ P)
118ltexprlem6 9617 . . . . . 6 (((𝐴P𝐵P) ∧ 𝐴𝐵) → (𝐴 +P {𝑦 ∣ ∃𝑤𝑤𝐴 ∧ (𝑤 +Q 𝑦) ∈ 𝐵)}) ⊆ 𝐵)
128ltexprlem7 9618 . . . . . 6 (((𝐴P𝐵P) ∧ 𝐴𝐵) → 𝐵 ⊆ (𝐴 +P {𝑦 ∣ ∃𝑤𝑤𝐴 ∧ (𝑤 +Q 𝑦) ∈ 𝐵)}))
1311, 12eqssd 3489 . . . . 5 (((𝐴P𝐵P) ∧ 𝐴𝐵) → (𝐴 +P {𝑦 ∣ ∃𝑤𝑤𝐴 ∧ (𝑤 +Q 𝑦) ∈ 𝐵)}) = 𝐵)
14 oveq2 6433 . . . . . . 7 (𝑥 = {𝑦 ∣ ∃𝑤𝑤𝐴 ∧ (𝑤 +Q 𝑦) ∈ 𝐵)} → (𝐴 +P 𝑥) = (𝐴 +P {𝑦 ∣ ∃𝑤𝑤𝐴 ∧ (𝑤 +Q 𝑦) ∈ 𝐵)}))
1514eqeq1d 2516 . . . . . 6 (𝑥 = {𝑦 ∣ ∃𝑤𝑤𝐴 ∧ (𝑤 +Q 𝑦) ∈ 𝐵)} → ((𝐴 +P 𝑥) = 𝐵 ↔ (𝐴 +P {𝑦 ∣ ∃𝑤𝑤𝐴 ∧ (𝑤 +Q 𝑦) ∈ 𝐵)}) = 𝐵))
1615rspcev 3186 . . . . 5 (({𝑦 ∣ ∃𝑤𝑤𝐴 ∧ (𝑤 +Q 𝑦) ∈ 𝐵)} ∈ P ∧ (𝐴 +P {𝑦 ∣ ∃𝑤𝑤𝐴 ∧ (𝑤 +Q 𝑦) ∈ 𝐵)}) = 𝐵) → ∃𝑥P (𝐴 +P 𝑥) = 𝐵)
1710, 13, 16syl2anc 690 . . . 4 (((𝐴P𝐵P) ∧ 𝐴𝐵) → ∃𝑥P (𝐴 +P 𝑥) = 𝐵)
1817ex 448 . . 3 ((𝐴P𝐵P) → (𝐴𝐵 → ∃𝑥P (𝐴 +P 𝑥) = 𝐵))
193, 18sylbid 228 . 2 ((𝐴P𝐵P) → (𝐴<P 𝐵 → ∃𝑥P (𝐴 +P 𝑥) = 𝐵))
202, 19mpcom 37 1 (𝐴<P 𝐵 → ∃𝑥P (𝐴 +P 𝑥) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382   = wceq 1474  wex 1694  wcel 1938  {cab 2500  wrex 2801  wpss 3445   class class class wbr 4481  (class class class)co 6425   +Q cplq 9431  Pcnp 9435   +P cpp 9437  <P cltp 9439
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-8 1940  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-sep 4607  ax-nul 4616  ax-pow 4668  ax-pr 4732  ax-un 6722  ax-inf2 8296
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 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ne 2686  df-ral 2805  df-rex 2806  df-reu 2807  df-rmo 2808  df-rab 2809  df-v 3079  df-sbc 3307  df-csb 3404  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-pss 3460  df-nul 3778  df-if 3940  df-pw 4013  df-sn 4029  df-pr 4031  df-tp 4033  df-op 4035  df-uni 4271  df-int 4309  df-iun 4355  df-br 4482  df-opab 4542  df-mpt 4543  df-tr 4579  df-eprel 4843  df-id 4847  df-po 4853  df-so 4854  df-fr 4891  df-we 4893  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-rn 4943  df-res 4944  df-ima 4945  df-pred 5487  df-ord 5533  df-on 5534  df-lim 5535  df-suc 5536  df-iota 5653  df-fun 5691  df-fn 5692  df-f 5693  df-f1 5694  df-fo 5695  df-f1o 5696  df-fv 5697  df-ov 6428  df-oprab 6429  df-mpt2 6430  df-om 6833  df-1st 6933  df-2nd 6934  df-wrecs 7168  df-recs 7230  df-rdg 7268  df-1o 7322  df-oadd 7326  df-omul 7327  df-er 7504  df-ni 9448  df-pli 9449  df-mi 9450  df-lti 9451  df-plpq 9484  df-mpq 9485  df-ltpq 9486  df-enq 9487  df-nq 9488  df-erq 9489  df-plq 9490  df-mq 9491  df-1nq 9492  df-rq 9493  df-ltnq 9494  df-np 9557  df-plp 9559  df-ltp 9561
This theorem is referenced by:  ltaprlem  9620  recexsrlem  9678  mulgt0sr  9680  map2psrpr  9685
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