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Theorem ltexprlem6 9901
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
ltexprlem6 (((𝐴P𝐵P) ∧ 𝐴𝐵) → (𝐴 +P 𝐶) ⊆ 𝐵)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶
Allowed substitution hint:   𝐶(𝑦)

Proof of Theorem ltexprlem6
Dummy variables 𝑧 𝑤 𝑣 𝑓 𝑔 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltexprlem.1 . . . . . 6 𝐶 = {𝑥 ∣ ∃𝑦𝑦𝐴 ∧ (𝑦 +Q 𝑥) ∈ 𝐵)}
21ltexprlem5 9900 . . . . 5 ((𝐵P𝐴𝐵) → 𝐶P)
3 df-plp 9843 . . . . . 6 +P = (𝑧P, 𝑦P ↦ {𝑓 ∣ ∃𝑔𝑧𝑦 𝑓 = (𝑔 +Q )})
4 addclnq 9805 . . . . . 6 ((𝑔QQ) → (𝑔 +Q ) ∈ Q)
53, 4genpelv 9860 . . . . 5 ((𝐴P𝐶P) → (𝑧 ∈ (𝐴 +P 𝐶) ↔ ∃𝑤𝐴𝑥𝐶 𝑧 = (𝑤 +Q 𝑥)))
62, 5sylan2 490 . . . 4 ((𝐴P ∧ (𝐵P𝐴𝐵)) → (𝑧 ∈ (𝐴 +P 𝐶) ↔ ∃𝑤𝐴𝑥𝐶 𝑧 = (𝑤 +Q 𝑥)))
71abeq2i 2764 . . . . . . . . . . . 12 (𝑥𝐶 ↔ ∃𝑦𝑦𝐴 ∧ (𝑦 +Q 𝑥) ∈ 𝐵))
8 elprnq 9851 . . . . . . . . . . . . . . . . . . 19 ((𝐵P ∧ (𝑦 +Q 𝑥) ∈ 𝐵) → (𝑦 +Q 𝑥) ∈ Q)
9 addnqf 9808 . . . . . . . . . . . . . . . . . . . . . 22 +Q :(Q × Q)⟶Q
109fdmi 6090 . . . . . . . . . . . . . . . . . . . . 21 dom +Q = (Q × Q)
11 0nnq 9784 . . . . . . . . . . . . . . . . . . . . 21 ¬ ∅ ∈ Q
1210, 11ndmovrcl 6862 . . . . . . . . . . . . . . . . . . . 20 ((𝑦 +Q 𝑥) ∈ Q → (𝑦Q𝑥Q))
1312simpld 474 . . . . . . . . . . . . . . . . . . 19 ((𝑦 +Q 𝑥) ∈ Q𝑦Q)
148, 13syl 17 . . . . . . . . . . . . . . . . . 18 ((𝐵P ∧ (𝑦 +Q 𝑥) ∈ 𝐵) → 𝑦Q)
15 prub 9854 . . . . . . . . . . . . . . . . . 18 (((𝐴P𝑤𝐴) ∧ 𝑦Q) → (¬ 𝑦𝐴𝑤 <Q 𝑦))
1614, 15sylan2 490 . . . . . . . . . . . . . . . . 17 (((𝐴P𝑤𝐴) ∧ (𝐵P ∧ (𝑦 +Q 𝑥) ∈ 𝐵)) → (¬ 𝑦𝐴𝑤 <Q 𝑦))
1712simprd 478 . . . . . . . . . . . . . . . . . . . 20 ((𝑦 +Q 𝑥) ∈ Q𝑥Q)
18 vex 3234 . . . . . . . . . . . . . . . . . . . . 21 𝑤 ∈ V
19 vex 3234 . . . . . . . . . . . . . . . . . . . . 21 𝑦 ∈ V
20 ltanq 9831 . . . . . . . . . . . . . . . . . . . . 21 (𝑢Q → (𝑧 <Q 𝑣 ↔ (𝑢 +Q 𝑧) <Q (𝑢 +Q 𝑣)))
21 vex 3234 . . . . . . . . . . . . . . . . . . . . 21 𝑥 ∈ V
22 addcomnq 9811 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 +Q 𝑣) = (𝑣 +Q 𝑧)
2318, 19, 20, 21, 22caovord2 6888 . . . . . . . . . . . . . . . . . . . 20 (𝑥Q → (𝑤 <Q 𝑦 ↔ (𝑤 +Q 𝑥) <Q (𝑦 +Q 𝑥)))
248, 17, 233syl 18 . . . . . . . . . . . . . . . . . . 19 ((𝐵P ∧ (𝑦 +Q 𝑥) ∈ 𝐵) → (𝑤 <Q 𝑦 ↔ (𝑤 +Q 𝑥) <Q (𝑦 +Q 𝑥)))
25 prcdnq 9853 . . . . . . . . . . . . . . . . . . 19 ((𝐵P ∧ (𝑦 +Q 𝑥) ∈ 𝐵) → ((𝑤 +Q 𝑥) <Q (𝑦 +Q 𝑥) → (𝑤 +Q 𝑥) ∈ 𝐵))
2624, 25sylbid 230 . . . . . . . . . . . . . . . . . 18 ((𝐵P ∧ (𝑦 +Q 𝑥) ∈ 𝐵) → (𝑤 <Q 𝑦 → (𝑤 +Q 𝑥) ∈ 𝐵))
2726adantl 481 . . . . . . . . . . . . . . . . 17 (((𝐴P𝑤𝐴) ∧ (𝐵P ∧ (𝑦 +Q 𝑥) ∈ 𝐵)) → (𝑤 <Q 𝑦 → (𝑤 +Q 𝑥) ∈ 𝐵))
2816, 27syld 47 . . . . . . . . . . . . . . . 16 (((𝐴P𝑤𝐴) ∧ (𝐵P ∧ (𝑦 +Q 𝑥) ∈ 𝐵)) → (¬ 𝑦𝐴 → (𝑤 +Q 𝑥) ∈ 𝐵))
2928exp32 630 . . . . . . . . . . . . . . 15 ((𝐴P𝑤𝐴) → (𝐵P → ((𝑦 +Q 𝑥) ∈ 𝐵 → (¬ 𝑦𝐴 → (𝑤 +Q 𝑥) ∈ 𝐵))))
3029com34 91 . . . . . . . . . . . . . 14 ((𝐴P𝑤𝐴) → (𝐵P → (¬ 𝑦𝐴 → ((𝑦 +Q 𝑥) ∈ 𝐵 → (𝑤 +Q 𝑥) ∈ 𝐵))))
3130imp4b 612 . . . . . . . . . . . . 13 (((𝐴P𝑤𝐴) ∧ 𝐵P) → ((¬ 𝑦𝐴 ∧ (𝑦 +Q 𝑥) ∈ 𝐵) → (𝑤 +Q 𝑥) ∈ 𝐵))
3231exlimdv 1901 . . . . . . . . . . . 12 (((𝐴P𝑤𝐴) ∧ 𝐵P) → (∃𝑦𝑦𝐴 ∧ (𝑦 +Q 𝑥) ∈ 𝐵) → (𝑤 +Q 𝑥) ∈ 𝐵))
337, 32syl5bi 232 . . . . . . . . . . 11 (((𝐴P𝑤𝐴) ∧ 𝐵P) → (𝑥𝐶 → (𝑤 +Q 𝑥) ∈ 𝐵))
3433exp31 629 . . . . . . . . . 10 (𝐴P → (𝑤𝐴 → (𝐵P → (𝑥𝐶 → (𝑤 +Q 𝑥) ∈ 𝐵))))
3534com23 86 . . . . . . . . 9 (𝐴P → (𝐵P → (𝑤𝐴 → (𝑥𝐶 → (𝑤 +Q 𝑥) ∈ 𝐵))))
3635imp43 620 . . . . . . . 8 (((𝐴P𝐵P) ∧ (𝑤𝐴𝑥𝐶)) → (𝑤 +Q 𝑥) ∈ 𝐵)
37 eleq1 2718 . . . . . . . . 9 (𝑧 = (𝑤 +Q 𝑥) → (𝑧𝐵 ↔ (𝑤 +Q 𝑥) ∈ 𝐵))
3837biimparc 503 . . . . . . . 8 (((𝑤 +Q 𝑥) ∈ 𝐵𝑧 = (𝑤 +Q 𝑥)) → 𝑧𝐵)
3936, 38sylan 487 . . . . . . 7 ((((𝐴P𝐵P) ∧ (𝑤𝐴𝑥𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑥)) → 𝑧𝐵)
4039exp31 629 . . . . . 6 ((𝐴P𝐵P) → ((𝑤𝐴𝑥𝐶) → (𝑧 = (𝑤 +Q 𝑥) → 𝑧𝐵)))
4140rexlimdvv 3066 . . . . 5 ((𝐴P𝐵P) → (∃𝑤𝐴𝑥𝐶 𝑧 = (𝑤 +Q 𝑥) → 𝑧𝐵))
4241adantrr 753 . . . 4 ((𝐴P ∧ (𝐵P𝐴𝐵)) → (∃𝑤𝐴𝑥𝐶 𝑧 = (𝑤 +Q 𝑥) → 𝑧𝐵))
436, 42sylbid 230 . . 3 ((𝐴P ∧ (𝐵P𝐴𝐵)) → (𝑧 ∈ (𝐴 +P 𝐶) → 𝑧𝐵))
4443ssrdv 3642 . 2 ((𝐴P ∧ (𝐵P𝐴𝐵)) → (𝐴 +P 𝐶) ⊆ 𝐵)
4544anassrs 681 1 (((𝐴P𝐵P) ∧ 𝐴𝐵) → (𝐴 +P 𝐶) ⊆ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383   = wceq 1523  wex 1744  wcel 2030  {cab 2637  wrex 2942  wss 3607  wpss 3608   class class class wbr 4685   × cxp 5141  (class class class)co 6690  Qcnq 9712   +Q cplq 9715   <Q cltq 9718  Pcnp 9719   +P cpp 9721
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-omul 7610  df-er 7787  df-ni 9732  df-pli 9733  df-mi 9734  df-lti 9735  df-plpq 9768  df-mpq 9769  df-ltpq 9770  df-enq 9771  df-nq 9772  df-erq 9773  df-plq 9774  df-mq 9775  df-1nq 9776  df-ltnq 9778  df-np 9841  df-plp 9843
This theorem is referenced by:  ltexpri  9903
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