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Theorem ltexprlempr 7430
Description: Our constructed difference is a positive real. Lemma for ltexpri 7435. (Contributed by Jim Kingdon, 17-Dec-2019.)
Hypothesis
Ref Expression
ltexprlem.1 𝐶 = ⟨{𝑥Q ∣ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st𝐵))}, {𝑥Q ∣ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd𝐵))}⟩
Assertion
Ref Expression
ltexprlempr (𝐴<P 𝐵𝐶P)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦

Proof of Theorem ltexprlempr
Dummy variables 𝑞 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltexprlem.1 . . . 4 𝐶 = ⟨{𝑥Q ∣ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st𝐵))}, {𝑥Q ∣ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd𝐵))}⟩
21ltexprlemm 7422 . . 3 (𝐴<P 𝐵 → (∃𝑞Q 𝑞 ∈ (1st𝐶) ∧ ∃𝑟Q 𝑟 ∈ (2nd𝐶)))
3 ssrab2 3182 . . . . . 6 {𝑥Q ∣ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st𝐵))} ⊆ Q
4 nqex 7185 . . . . . . 7 Q ∈ V
54elpw2 4082 . . . . . 6 ({𝑥Q ∣ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st𝐵))} ∈ 𝒫 Q ↔ {𝑥Q ∣ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st𝐵))} ⊆ Q)
63, 5mpbir 145 . . . . 5 {𝑥Q ∣ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st𝐵))} ∈ 𝒫 Q
7 ssrab2 3182 . . . . . 6 {𝑥Q ∣ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd𝐵))} ⊆ Q
84elpw2 4082 . . . . . 6 ({𝑥Q ∣ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd𝐵))} ∈ 𝒫 Q ↔ {𝑥Q ∣ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd𝐵))} ⊆ Q)
97, 8mpbir 145 . . . . 5 {𝑥Q ∣ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd𝐵))} ∈ 𝒫 Q
10 opelxpi 4571 . . . . 5 (({𝑥Q ∣ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st𝐵))} ∈ 𝒫 Q ∧ {𝑥Q ∣ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd𝐵))} ∈ 𝒫 Q) → ⟨{𝑥Q ∣ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st𝐵))}, {𝑥Q ∣ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd𝐵))}⟩ ∈ (𝒫 Q × 𝒫 Q))
116, 9, 10mp2an 422 . . . 4 ⟨{𝑥Q ∣ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st𝐵))}, {𝑥Q ∣ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd𝐵))}⟩ ∈ (𝒫 Q × 𝒫 Q)
121, 11eqeltri 2212 . . 3 𝐶 ∈ (𝒫 Q × 𝒫 Q)
132, 12jctil 310 . 2 (𝐴<P 𝐵 → (𝐶 ∈ (𝒫 Q × 𝒫 Q) ∧ (∃𝑞Q 𝑞 ∈ (1st𝐶) ∧ ∃𝑟Q 𝑟 ∈ (2nd𝐶))))
141ltexprlemrnd 7427 . . 3 (𝐴<P 𝐵 → (∀𝑞Q (𝑞 ∈ (1st𝐶) ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st𝐶))) ∧ ∀𝑟Q (𝑟 ∈ (2nd𝐶) ↔ ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd𝐶)))))
151ltexprlemdisj 7428 . . 3 (𝐴<P 𝐵 → ∀𝑞Q ¬ (𝑞 ∈ (1st𝐶) ∧ 𝑞 ∈ (2nd𝐶)))
161ltexprlemloc 7429 . . 3 (𝐴<P 𝐵 → ∀𝑞Q𝑟Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st𝐶) ∨ 𝑟 ∈ (2nd𝐶))))
1714, 15, 163jca 1161 . 2 (𝐴<P 𝐵 → ((∀𝑞Q (𝑞 ∈ (1st𝐶) ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st𝐶))) ∧ ∀𝑟Q (𝑟 ∈ (2nd𝐶) ↔ ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd𝐶)))) ∧ ∀𝑞Q ¬ (𝑞 ∈ (1st𝐶) ∧ 𝑞 ∈ (2nd𝐶)) ∧ ∀𝑞Q𝑟Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st𝐶) ∨ 𝑟 ∈ (2nd𝐶)))))
18 elnp1st2nd 7298 . 2 (𝐶P ↔ ((𝐶 ∈ (𝒫 Q × 𝒫 Q) ∧ (∃𝑞Q 𝑞 ∈ (1st𝐶) ∧ ∃𝑟Q 𝑟 ∈ (2nd𝐶))) ∧ ((∀𝑞Q (𝑞 ∈ (1st𝐶) ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ (1st𝐶))) ∧ ∀𝑟Q (𝑟 ∈ (2nd𝐶) ↔ ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ (2nd𝐶)))) ∧ ∀𝑞Q ¬ (𝑞 ∈ (1st𝐶) ∧ 𝑞 ∈ (2nd𝐶)) ∧ ∀𝑞Q𝑟Q (𝑞 <Q 𝑟 → (𝑞 ∈ (1st𝐶) ∨ 𝑟 ∈ (2nd𝐶))))))
1913, 17, 18sylanbrc 413 1 (𝐴<P 𝐵𝐶P)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  wo 697  w3a 962   = wceq 1331  wex 1468  wcel 1480  wral 2416  wrex 2417  {crab 2420  wss 3071  𝒫 cpw 3510  cop 3530   class class class wbr 3929   × cxp 4537  cfv 5123  (class class class)co 5774  1st c1st 6036  2nd c2nd 6037  Qcnq 7102   +Q cplq 7104   <Q cltq 7107  Pcnp 7113  <P cltp 7117
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-eprel 4211  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-2o 6314  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7126  df-pli 7127  df-mi 7128  df-lti 7129  df-plpq 7166  df-mpq 7167  df-enq 7169  df-nqqs 7170  df-plqqs 7171  df-mqqs 7172  df-1nqqs 7173  df-rq 7174  df-ltnqqs 7175  df-enq0 7246  df-nq0 7247  df-0nq0 7248  df-plq0 7249  df-mq0 7250  df-inp 7288  df-iltp 7292
This theorem is referenced by:  ltexprlemfl  7431  ltexprlemrl  7432  ltexprlemfu  7433  ltexprlemru  7434  ltexpri  7435
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