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Theorem ltexprlem7 9824
Description: Lemma for Proposition 9-3.5(iv) of [Gleason] p. 123. (Contributed by NM, 8-Apr-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
Hypothesis
Ref Expression
ltexprlem.1 𝐶 = {𝑥 ∣ ∃𝑦𝑦𝐴 ∧ (𝑦 +Q 𝑥) ∈ 𝐵)}
Assertion
Ref Expression
ltexprlem7 (((𝐴P𝐵P) ∧ 𝐴𝐵) → 𝐵 ⊆ (𝐴 +P 𝐶))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶
Allowed substitution hint:   𝐶(𝑦)

Proof of Theorem ltexprlem7
Dummy variables 𝑧 𝑤 𝑣 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltexprlem.1 . . . . . . . 8 𝐶 = {𝑥 ∣ ∃𝑦𝑦𝐴 ∧ (𝑦 +Q 𝑥) ∈ 𝐵)}
21ltexprlem5 9822 . . . . . . 7 ((𝐵P𝐴𝐵) → 𝐶P)
3 ltaddpr 9816 . . . . . . . . . . . . . 14 ((𝐴P𝐶P) → 𝐴<P (𝐴 +P 𝐶))
4 addclpr 9800 . . . . . . . . . . . . . . 15 ((𝐴P𝐶P) → (𝐴 +P 𝐶) ∈ P)
5 ltprord 9812 . . . . . . . . . . . . . . 15 ((𝐴P ∧ (𝐴 +P 𝐶) ∈ P) → (𝐴<P (𝐴 +P 𝐶) ↔ 𝐴 ⊊ (𝐴 +P 𝐶)))
64, 5syldan 487 . . . . . . . . . . . . . 14 ((𝐴P𝐶P) → (𝐴<P (𝐴 +P 𝐶) ↔ 𝐴 ⊊ (𝐴 +P 𝐶)))
73, 6mpbid 222 . . . . . . . . . . . . 13 ((𝐴P𝐶P) → 𝐴 ⊊ (𝐴 +P 𝐶))
87pssssd 3688 . . . . . . . . . . . 12 ((𝐴P𝐶P) → 𝐴 ⊆ (𝐴 +P 𝐶))
98sseld 3587 . . . . . . . . . . 11 ((𝐴P𝐶P) → (𝑤𝐴𝑤 ∈ (𝐴 +P 𝐶)))
1092a1d 26 . . . . . . . . . 10 ((𝐴P𝐶P) → (𝐵P → (𝑤𝐵 → (𝑤𝐴𝑤 ∈ (𝐴 +P 𝐶)))))
1110com4r 94 . . . . . . . . 9 (𝑤𝐴 → ((𝐴P𝐶P) → (𝐵P → (𝑤𝐵𝑤 ∈ (𝐴 +P 𝐶)))))
1211expd 452 . . . . . . . 8 (𝑤𝐴 → (𝐴P → (𝐶P → (𝐵P → (𝑤𝐵𝑤 ∈ (𝐴 +P 𝐶))))))
13 prnmadd 9779 . . . . . . . . . . . 12 ((𝐵P𝑤𝐵) → ∃𝑣(𝑤 +Q 𝑣) ∈ 𝐵)
1413ex 450 . . . . . . . . . . 11 (𝐵P → (𝑤𝐵 → ∃𝑣(𝑤 +Q 𝑣) ∈ 𝐵))
15 elprnq 9773 . . . . . . . . . . . . . . . 16 ((𝐵P ∧ (𝑤 +Q 𝑣) ∈ 𝐵) → (𝑤 +Q 𝑣) ∈ Q)
16 addnqf 9730 . . . . . . . . . . . . . . . . . 18 +Q :(Q × Q)⟶Q
1716fdmi 6019 . . . . . . . . . . . . . . . . 17 dom +Q = (Q × Q)
18 0nnq 9706 . . . . . . . . . . . . . . . . 17 ¬ ∅ ∈ Q
1917, 18ndmovrcl 6785 . . . . . . . . . . . . . . . 16 ((𝑤 +Q 𝑣) ∈ Q → (𝑤Q𝑣Q))
2015, 19syl 17 . . . . . . . . . . . . . . 15 ((𝐵P ∧ (𝑤 +Q 𝑣) ∈ 𝐵) → (𝑤Q𝑣Q))
2120simpld 475 . . . . . . . . . . . . . 14 ((𝐵P ∧ (𝑤 +Q 𝑣) ∈ 𝐵) → 𝑤Q)
22 vex 3193 . . . . . . . . . . . . . . . . . . 19 𝑣 ∈ V
2322prlem934 9815 . . . . . . . . . . . . . . . . . 18 (𝐴P → ∃𝑧𝐴 ¬ (𝑧 +Q 𝑣) ∈ 𝐴)
2423adantr 481 . . . . . . . . . . . . . . . . 17 ((𝐴P𝐶P) → ∃𝑧𝐴 ¬ (𝑧 +Q 𝑣) ∈ 𝐴)
25 prub 9776 . . . . . . . . . . . . . . . . . . . . 21 (((𝐴P𝑧𝐴) ∧ 𝑤Q) → (¬ 𝑤𝐴𝑧 <Q 𝑤))
26 ltexnq 9757 . . . . . . . . . . . . . . . . . . . . . 22 (𝑤Q → (𝑧 <Q 𝑤 ↔ ∃𝑥(𝑧 +Q 𝑥) = 𝑤))
2726adantl 482 . . . . . . . . . . . . . . . . . . . . 21 (((𝐴P𝑧𝐴) ∧ 𝑤Q) → (𝑧 <Q 𝑤 ↔ ∃𝑥(𝑧 +Q 𝑥) = 𝑤))
2825, 27sylibd 229 . . . . . . . . . . . . . . . . . . . 20 (((𝐴P𝑧𝐴) ∧ 𝑤Q) → (¬ 𝑤𝐴 → ∃𝑥(𝑧 +Q 𝑥) = 𝑤))
2928ex 450 . . . . . . . . . . . . . . . . . . 19 ((𝐴P𝑧𝐴) → (𝑤Q → (¬ 𝑤𝐴 → ∃𝑥(𝑧 +Q 𝑥) = 𝑤)))
3029ad2ant2r 782 . . . . . . . . . . . . . . . . . 18 (((𝐴P𝐶P) ∧ (𝑧𝐴 ∧ ¬ (𝑧 +Q 𝑣) ∈ 𝐴)) → (𝑤Q → (¬ 𝑤𝐴 → ∃𝑥(𝑧 +Q 𝑥) = 𝑤)))
31 vex 3193 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 𝑧 ∈ V
32 vex 3193 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 𝑥 ∈ V
33 addcomnq 9733 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓)
34 addassnq 9740 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑓 +Q 𝑔) +Q ) = (𝑓 +Q (𝑔 +Q ))
3531, 22, 32, 33, 34caov32 6826 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑧 +Q 𝑣) +Q 𝑥) = ((𝑧 +Q 𝑥) +Q 𝑣)
36 oveq1 6622 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑧 +Q 𝑥) = 𝑤 → ((𝑧 +Q 𝑥) +Q 𝑣) = (𝑤 +Q 𝑣))
3735, 36syl5eq 2667 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑧 +Q 𝑥) = 𝑤 → ((𝑧 +Q 𝑣) +Q 𝑥) = (𝑤 +Q 𝑣))
3837eleq1d 2683 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑧 +Q 𝑥) = 𝑤 → (((𝑧 +Q 𝑣) +Q 𝑥) ∈ 𝐵 ↔ (𝑤 +Q 𝑣) ∈ 𝐵))
3938biimpar 502 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑧 +Q 𝑥) = 𝑤 ∧ (𝑤 +Q 𝑣) ∈ 𝐵) → ((𝑧 +Q 𝑣) +Q 𝑥) ∈ 𝐵)
40 ovex 6643 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑧 +Q 𝑣) ∈ V
41 eleq1 2686 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑦 = (𝑧 +Q 𝑣) → (𝑦𝐴 ↔ (𝑧 +Q 𝑣) ∈ 𝐴))
4241notbid 308 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑦 = (𝑧 +Q 𝑣) → (¬ 𝑦𝐴 ↔ ¬ (𝑧 +Q 𝑣) ∈ 𝐴))
43 oveq1 6622 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑦 = (𝑧 +Q 𝑣) → (𝑦 +Q 𝑥) = ((𝑧 +Q 𝑣) +Q 𝑥))
4443eleq1d 2683 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑦 = (𝑧 +Q 𝑣) → ((𝑦 +Q 𝑥) ∈ 𝐵 ↔ ((𝑧 +Q 𝑣) +Q 𝑥) ∈ 𝐵))
4542, 44anbi12d 746 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑦 = (𝑧 +Q 𝑣) → ((¬ 𝑦𝐴 ∧ (𝑦 +Q 𝑥) ∈ 𝐵) ↔ (¬ (𝑧 +Q 𝑣) ∈ 𝐴 ∧ ((𝑧 +Q 𝑣) +Q 𝑥) ∈ 𝐵)))
4640, 45spcev 3290 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((¬ (𝑧 +Q 𝑣) ∈ 𝐴 ∧ ((𝑧 +Q 𝑣) +Q 𝑥) ∈ 𝐵) → ∃𝑦𝑦𝐴 ∧ (𝑦 +Q 𝑥) ∈ 𝐵))
471abeq2i 2732 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥𝐶 ↔ ∃𝑦𝑦𝐴 ∧ (𝑦 +Q 𝑥) ∈ 𝐵))
4846, 47sylibr 224 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((¬ (𝑧 +Q 𝑣) ∈ 𝐴 ∧ ((𝑧 +Q 𝑣) +Q 𝑥) ∈ 𝐵) → 𝑥𝐶)
4939, 48sylan2 491 . . . . . . . . . . . . . . . . . . . . . . . 24 ((¬ (𝑧 +Q 𝑣) ∈ 𝐴 ∧ ((𝑧 +Q 𝑥) = 𝑤 ∧ (𝑤 +Q 𝑣) ∈ 𝐵)) → 𝑥𝐶)
50 df-plp 9765 . . . . . . . . . . . . . . . . . . . . . . . . 25 +P = (𝑥P, 𝑤P ↦ {𝑧 ∣ ∃𝑓𝑥𝑣𝑤 𝑧 = (𝑓 +Q 𝑣)})
51 addclnq 9727 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑓Q𝑣Q) → (𝑓 +Q 𝑣) ∈ Q)
5250, 51genpprecl 9783 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐴P𝐶P) → ((𝑧𝐴𝑥𝐶) → (𝑧 +Q 𝑥) ∈ (𝐴 +P 𝐶)))
5349, 52sylan2i 686 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴P𝐶P) → ((𝑧𝐴 ∧ (¬ (𝑧 +Q 𝑣) ∈ 𝐴 ∧ ((𝑧 +Q 𝑥) = 𝑤 ∧ (𝑤 +Q 𝑣) ∈ 𝐵))) → (𝑧 +Q 𝑥) ∈ (𝐴 +P 𝐶)))
5453exp4d 636 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐴P𝐶P) → (𝑧𝐴 → (¬ (𝑧 +Q 𝑣) ∈ 𝐴 → (((𝑧 +Q 𝑥) = 𝑤 ∧ (𝑤 +Q 𝑣) ∈ 𝐵) → (𝑧 +Q 𝑥) ∈ (𝐴 +P 𝐶)))))
5554imp42 619 . . . . . . . . . . . . . . . . . . . . 21 ((((𝐴P𝐶P) ∧ (𝑧𝐴 ∧ ¬ (𝑧 +Q 𝑣) ∈ 𝐴)) ∧ ((𝑧 +Q 𝑥) = 𝑤 ∧ (𝑤 +Q 𝑣) ∈ 𝐵)) → (𝑧 +Q 𝑥) ∈ (𝐴 +P 𝐶))
56 eleq1 2686 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑧 +Q 𝑥) = 𝑤 → ((𝑧 +Q 𝑥) ∈ (𝐴 +P 𝐶) ↔ 𝑤 ∈ (𝐴 +P 𝐶)))
5756ad2antrl 763 . . . . . . . . . . . . . . . . . . . . 21 ((((𝐴P𝐶P) ∧ (𝑧𝐴 ∧ ¬ (𝑧 +Q 𝑣) ∈ 𝐴)) ∧ ((𝑧 +Q 𝑥) = 𝑤 ∧ (𝑤 +Q 𝑣) ∈ 𝐵)) → ((𝑧 +Q 𝑥) ∈ (𝐴 +P 𝐶) ↔ 𝑤 ∈ (𝐴 +P 𝐶)))
5855, 57mpbid 222 . . . . . . . . . . . . . . . . . . . 20 ((((𝐴P𝐶P) ∧ (𝑧𝐴 ∧ ¬ (𝑧 +Q 𝑣) ∈ 𝐴)) ∧ ((𝑧 +Q 𝑥) = 𝑤 ∧ (𝑤 +Q 𝑣) ∈ 𝐵)) → 𝑤 ∈ (𝐴 +P 𝐶))
5958exp32 630 . . . . . . . . . . . . . . . . . . 19 (((𝐴P𝐶P) ∧ (𝑧𝐴 ∧ ¬ (𝑧 +Q 𝑣) ∈ 𝐴)) → ((𝑧 +Q 𝑥) = 𝑤 → ((𝑤 +Q 𝑣) ∈ 𝐵𝑤 ∈ (𝐴 +P 𝐶))))
6059exlimdv 1858 . . . . . . . . . . . . . . . . . 18 (((𝐴P𝐶P) ∧ (𝑧𝐴 ∧ ¬ (𝑧 +Q 𝑣) ∈ 𝐴)) → (∃𝑥(𝑧 +Q 𝑥) = 𝑤 → ((𝑤 +Q 𝑣) ∈ 𝐵𝑤 ∈ (𝐴 +P 𝐶))))
6130, 60syl6d 75 . . . . . . . . . . . . . . . . 17 (((𝐴P𝐶P) ∧ (𝑧𝐴 ∧ ¬ (𝑧 +Q 𝑣) ∈ 𝐴)) → (𝑤Q → (¬ 𝑤𝐴 → ((𝑤 +Q 𝑣) ∈ 𝐵𝑤 ∈ (𝐴 +P 𝐶)))))
6224, 61rexlimddv 3030 . . . . . . . . . . . . . . . 16 ((𝐴P𝐶P) → (𝑤Q → (¬ 𝑤𝐴 → ((𝑤 +Q 𝑣) ∈ 𝐵𝑤 ∈ (𝐴 +P 𝐶)))))
6362com14 96 . . . . . . . . . . . . . . 15 ((𝑤 +Q 𝑣) ∈ 𝐵 → (𝑤Q → (¬ 𝑤𝐴 → ((𝐴P𝐶P) → 𝑤 ∈ (𝐴 +P 𝐶)))))
6463adantl 482 . . . . . . . . . . . . . 14 ((𝐵P ∧ (𝑤 +Q 𝑣) ∈ 𝐵) → (𝑤Q → (¬ 𝑤𝐴 → ((𝐴P𝐶P) → 𝑤 ∈ (𝐴 +P 𝐶)))))
6521, 64mpd 15 . . . . . . . . . . . . 13 ((𝐵P ∧ (𝑤 +Q 𝑣) ∈ 𝐵) → (¬ 𝑤𝐴 → ((𝐴P𝐶P) → 𝑤 ∈ (𝐴 +P 𝐶))))
6665ex 450 . . . . . . . . . . . 12 (𝐵P → ((𝑤 +Q 𝑣) ∈ 𝐵 → (¬ 𝑤𝐴 → ((𝐴P𝐶P) → 𝑤 ∈ (𝐴 +P 𝐶)))))
6766exlimdv 1858 . . . . . . . . . . 11 (𝐵P → (∃𝑣(𝑤 +Q 𝑣) ∈ 𝐵 → (¬ 𝑤𝐴 → ((𝐴P𝐶P) → 𝑤 ∈ (𝐴 +P 𝐶)))))
6814, 67syld 47 . . . . . . . . . 10 (𝐵P → (𝑤𝐵 → (¬ 𝑤𝐴 → ((𝐴P𝐶P) → 𝑤 ∈ (𝐴 +P 𝐶)))))
6968com4t 93 . . . . . . . . 9 𝑤𝐴 → ((𝐴P𝐶P) → (𝐵P → (𝑤𝐵𝑤 ∈ (𝐴 +P 𝐶)))))
7069expd 452 . . . . . . . 8 𝑤𝐴 → (𝐴P → (𝐶P → (𝐵P → (𝑤𝐵𝑤 ∈ (𝐴 +P 𝐶))))))
7112, 70pm2.61i 176 . . . . . . 7 (𝐴P → (𝐶P → (𝐵P → (𝑤𝐵𝑤 ∈ (𝐴 +P 𝐶)))))
722, 71syl5 34 . . . . . 6 (𝐴P → ((𝐵P𝐴𝐵) → (𝐵P → (𝑤𝐵𝑤 ∈ (𝐴 +P 𝐶)))))
7372expd 452 . . . . 5 (𝐴P → (𝐵P → (𝐴𝐵 → (𝐵P → (𝑤𝐵𝑤 ∈ (𝐴 +P 𝐶))))))
7473com34 91 . . . 4 (𝐴P → (𝐵P → (𝐵P → (𝐴𝐵 → (𝑤𝐵𝑤 ∈ (𝐴 +P 𝐶))))))
7574pm2.43d 53 . . 3 (𝐴P → (𝐵P → (𝐴𝐵 → (𝑤𝐵𝑤 ∈ (𝐴 +P 𝐶)))))
7675imp31 448 . 2 (((𝐴P𝐵P) ∧ 𝐴𝐵) → (𝑤𝐵𝑤 ∈ (𝐴 +P 𝐶)))
7776ssrdv 3594 1 (((𝐴P𝐵P) ∧ 𝐴𝐵) → 𝐵 ⊆ (𝐴 +P 𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384   = wceq 1480  wex 1701  wcel 1987  {cab 2607  wrex 2909  wss 3560  wpss 3561   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:  ltexpri  9825
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