Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  ltexprlemfu GIF version

Theorem ltexprlemfu 7431
 Description: Lemma for ltexpri 7433. One direction of our result for upper cuts. (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
ltexprlemfu (𝐴<P 𝐵 → (2nd ‘(𝐴 +P 𝐶)) ⊆ (2nd𝐵))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦

Proof of Theorem ltexprlemfu
Dummy variables 𝑧 𝑤 𝑢 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltrelpr 7325 . . . . . 6 <P ⊆ (P × P)
21brel 4591 . . . . 5 (𝐴<P 𝐵 → (𝐴P𝐵P))
32simpld 111 . . . 4 (𝐴<P 𝐵𝐴P)
4 ltexprlem.1 . . . . 5 𝐶 = ⟨{𝑥Q ∣ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st𝐵))}, {𝑥Q ∣ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd𝐵))}⟩
54ltexprlempr 7428 . . . 4 (𝐴<P 𝐵𝐶P)
6 df-iplp 7288 . . . . 5 +P = (𝑧P, 𝑦P ↦ ⟨{𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (1st𝑧) ∧ ∈ (1st𝑦) ∧ 𝑓 = (𝑔 +Q ))}, {𝑓Q ∣ ∃𝑔QQ (𝑔 ∈ (2nd𝑧) ∧ ∈ (2nd𝑦) ∧ 𝑓 = (𝑔 +Q ))}⟩)
7 addclnq 7195 . . . . 5 ((𝑔QQ) → (𝑔 +Q ) ∈ Q)
86, 7genpelvu 7333 . . . 4 ((𝐴P𝐶P) → (𝑧 ∈ (2nd ‘(𝐴 +P 𝐶)) ↔ ∃𝑤 ∈ (2nd𝐴)∃𝑢 ∈ (2nd𝐶)𝑧 = (𝑤 +Q 𝑢)))
93, 5, 8syl2anc 408 . . 3 (𝐴<P 𝐵 → (𝑧 ∈ (2nd ‘(𝐴 +P 𝐶)) ↔ ∃𝑤 ∈ (2nd𝐴)∃𝑢 ∈ (2nd𝐶)𝑧 = (𝑤 +Q 𝑢)))
10 simprr 521 . . . . . 6 ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (2nd𝐴) ∧ 𝑢 ∈ (2nd𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢))) → 𝑧 = (𝑤 +Q 𝑢))
114ltexprlemelu 7419 . . . . . . . . . . 11 (𝑢 ∈ (2nd𝐶) ↔ (𝑢Q ∧ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑢) ∈ (2nd𝐵))))
1211biimpi 119 . . . . . . . . . 10 (𝑢 ∈ (2nd𝐶) → (𝑢Q ∧ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑢) ∈ (2nd𝐵))))
1312ad2antlr 480 . . . . . . . . 9 (((𝑤 ∈ (2nd𝐴) ∧ 𝑢 ∈ (2nd𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢)) → (𝑢Q ∧ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑢) ∈ (2nd𝐵))))
1413simprd 113 . . . . . . . 8 (((𝑤 ∈ (2nd𝐴) ∧ 𝑢 ∈ (2nd𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢)) → ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑢) ∈ (2nd𝐵)))
1514adantl 275 . . . . . . 7 ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (2nd𝐴) ∧ 𝑢 ∈ (2nd𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢))) → ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑢) ∈ (2nd𝐵)))
16 prop 7295 . . . . . . . . . . . . . . 15 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
173, 16syl 14 . . . . . . . . . . . . . 14 (𝐴<P 𝐵 → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
18 prltlu 7307 . . . . . . . . . . . . . 14 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑦 ∈ (1st𝐴) ∧ 𝑤 ∈ (2nd𝐴)) → 𝑦 <Q 𝑤)
1917, 18syl3an1 1249 . . . . . . . . . . . . 13 ((𝐴<P 𝐵𝑦 ∈ (1st𝐴) ∧ 𝑤 ∈ (2nd𝐴)) → 𝑦 <Q 𝑤)
20193com23 1187 . . . . . . . . . . . 12 ((𝐴<P 𝐵𝑤 ∈ (2nd𝐴) ∧ 𝑦 ∈ (1st𝐴)) → 𝑦 <Q 𝑤)
21203adant2r 1211 . . . . . . . . . . 11 ((𝐴<P 𝐵 ∧ (𝑤 ∈ (2nd𝐴) ∧ 𝑢 ∈ (2nd𝐶)) ∧ 𝑦 ∈ (1st𝐴)) → 𝑦 <Q 𝑤)
22213adant2r 1211 . . . . . . . . . 10 ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (2nd𝐴) ∧ 𝑢 ∈ (2nd𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢)) ∧ 𝑦 ∈ (1st𝐴)) → 𝑦 <Q 𝑤)
23223adant3r 1213 . . . . . . . . 9 ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (2nd𝐴) ∧ 𝑢 ∈ (2nd𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢)) ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑢) ∈ (2nd𝐵))) → 𝑦 <Q 𝑤)
24 ltanqg 7220 . . . . . . . . . . . 12 ((𝑓Q𝑔QQ) → (𝑓 <Q 𝑔 ↔ ( +Q 𝑓) <Q ( +Q 𝑔)))
2524adantl 275 . . . . . . . . . . 11 (((𝐴<P 𝐵 ∧ ((𝑤 ∈ (2nd𝐴) ∧ 𝑢 ∈ (2nd𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢)) ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑢) ∈ (2nd𝐵))) ∧ (𝑓Q𝑔QQ)) → (𝑓 <Q 𝑔 ↔ ( +Q 𝑓) <Q ( +Q 𝑔)))
26 elprnql 7301 . . . . . . . . . . . . . 14 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑦 ∈ (1st𝐴)) → 𝑦Q)
2717, 26sylan 281 . . . . . . . . . . . . 13 ((𝐴<P 𝐵𝑦 ∈ (1st𝐴)) → 𝑦Q)
2827adantrr 470 . . . . . . . . . . . 12 ((𝐴<P 𝐵 ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑢) ∈ (2nd𝐵))) → 𝑦Q)
29283adant2 1000 . . . . . . . . . . 11 ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (2nd𝐴) ∧ 𝑢 ∈ (2nd𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢)) ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑢) ∈ (2nd𝐵))) → 𝑦Q)
30 elprnqu 7302 . . . . . . . . . . . . . . 15 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑤 ∈ (2nd𝐴)) → 𝑤Q)
3117, 30sylan 281 . . . . . . . . . . . . . 14 ((𝐴<P 𝐵𝑤 ∈ (2nd𝐴)) → 𝑤Q)
3231adantrr 470 . . . . . . . . . . . . 13 ((𝐴<P 𝐵 ∧ (𝑤 ∈ (2nd𝐴) ∧ 𝑢 ∈ (2nd𝐶))) → 𝑤Q)
3332adantrr 470 . . . . . . . . . . . 12 ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (2nd𝐴) ∧ 𝑢 ∈ (2nd𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢))) → 𝑤Q)
34333adant3 1001 . . . . . . . . . . 11 ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (2nd𝐴) ∧ 𝑢 ∈ (2nd𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢)) ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑢) ∈ (2nd𝐵))) → 𝑤Q)
35 prop 7295 . . . . . . . . . . . . . . . 16 (𝐶P → ⟨(1st𝐶), (2nd𝐶)⟩ ∈ P)
365, 35syl 14 . . . . . . . . . . . . . . 15 (𝐴<P 𝐵 → ⟨(1st𝐶), (2nd𝐶)⟩ ∈ P)
37 elprnqu 7302 . . . . . . . . . . . . . . 15 ((⟨(1st𝐶), (2nd𝐶)⟩ ∈ P𝑢 ∈ (2nd𝐶)) → 𝑢Q)
3836, 37sylan 281 . . . . . . . . . . . . . 14 ((𝐴<P 𝐵𝑢 ∈ (2nd𝐶)) → 𝑢Q)
3938adantrl 469 . . . . . . . . . . . . 13 ((𝐴<P 𝐵 ∧ (𝑤 ∈ (2nd𝐴) ∧ 𝑢 ∈ (2nd𝐶))) → 𝑢Q)
4039adantrr 470 . . . . . . . . . . . 12 ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (2nd𝐴) ∧ 𝑢 ∈ (2nd𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢))) → 𝑢Q)
41403adant3 1001 . . . . . . . . . . 11 ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (2nd𝐴) ∧ 𝑢 ∈ (2nd𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢)) ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑢) ∈ (2nd𝐵))) → 𝑢Q)
42 addcomnqg 7201 . . . . . . . . . . . 12 ((𝑓Q𝑔Q) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
4342adantl 275 . . . . . . . . . . 11 (((𝐴<P 𝐵 ∧ ((𝑤 ∈ (2nd𝐴) ∧ 𝑢 ∈ (2nd𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢)) ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑢) ∈ (2nd𝐵))) ∧ (𝑓Q𝑔Q)) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
4425, 29, 34, 41, 43caovord2d 5940 . . . . . . . . . 10 ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (2nd𝐴) ∧ 𝑢 ∈ (2nd𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢)) ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑢) ∈ (2nd𝐵))) → (𝑦 <Q 𝑤 ↔ (𝑦 +Q 𝑢) <Q (𝑤 +Q 𝑢)))
452simprd 113 . . . . . . . . . . . . . 14 (𝐴<P 𝐵𝐵P)
46 prop 7295 . . . . . . . . . . . . . 14 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
4745, 46syl 14 . . . . . . . . . . . . 13 (𝐴<P 𝐵 → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
48 prcunqu 7305 . . . . . . . . . . . . 13 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P ∧ (𝑦 +Q 𝑢) ∈ (2nd𝐵)) → ((𝑦 +Q 𝑢) <Q (𝑤 +Q 𝑢) → (𝑤 +Q 𝑢) ∈ (2nd𝐵)))
4947, 48sylan 281 . . . . . . . . . . . 12 ((𝐴<P 𝐵 ∧ (𝑦 +Q 𝑢) ∈ (2nd𝐵)) → ((𝑦 +Q 𝑢) <Q (𝑤 +Q 𝑢) → (𝑤 +Q 𝑢) ∈ (2nd𝐵)))
5049adantrl 469 . . . . . . . . . . 11 ((𝐴<P 𝐵 ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑢) ∈ (2nd𝐵))) → ((𝑦 +Q 𝑢) <Q (𝑤 +Q 𝑢) → (𝑤 +Q 𝑢) ∈ (2nd𝐵)))
51503adant2 1000 . . . . . . . . . 10 ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (2nd𝐴) ∧ 𝑢 ∈ (2nd𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢)) ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑢) ∈ (2nd𝐵))) → ((𝑦 +Q 𝑢) <Q (𝑤 +Q 𝑢) → (𝑤 +Q 𝑢) ∈ (2nd𝐵)))
5244, 51sylbid 149 . . . . . . . . 9 ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (2nd𝐴) ∧ 𝑢 ∈ (2nd𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢)) ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑢) ∈ (2nd𝐵))) → (𝑦 <Q 𝑤 → (𝑤 +Q 𝑢) ∈ (2nd𝐵)))
5323, 52mpd 13 . . . . . . . 8 ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (2nd𝐴) ∧ 𝑢 ∈ (2nd𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢)) ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑢) ∈ (2nd𝐵))) → (𝑤 +Q 𝑢) ∈ (2nd𝐵))
54533expa 1181 . . . . . . 7 (((𝐴<P 𝐵 ∧ ((𝑤 ∈ (2nd𝐴) ∧ 𝑢 ∈ (2nd𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢))) ∧ (𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑢) ∈ (2nd𝐵))) → (𝑤 +Q 𝑢) ∈ (2nd𝐵))
5515, 54exlimddv 1870 . . . . . 6 ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (2nd𝐴) ∧ 𝑢 ∈ (2nd𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢))) → (𝑤 +Q 𝑢) ∈ (2nd𝐵))
5610, 55eqeltrd 2216 . . . . 5 ((𝐴<P 𝐵 ∧ ((𝑤 ∈ (2nd𝐴) ∧ 𝑢 ∈ (2nd𝐶)) ∧ 𝑧 = (𝑤 +Q 𝑢))) → 𝑧 ∈ (2nd𝐵))
5756expr 372 . . . 4 ((𝐴<P 𝐵 ∧ (𝑤 ∈ (2nd𝐴) ∧ 𝑢 ∈ (2nd𝐶))) → (𝑧 = (𝑤 +Q 𝑢) → 𝑧 ∈ (2nd𝐵)))
5857rexlimdvva 2557 . . 3 (𝐴<P 𝐵 → (∃𝑤 ∈ (2nd𝐴)∃𝑢 ∈ (2nd𝐶)𝑧 = (𝑤 +Q 𝑢) → 𝑧 ∈ (2nd𝐵)))
599, 58sylbid 149 . 2 (𝐴<P 𝐵 → (𝑧 ∈ (2nd ‘(𝐴 +P 𝐶)) → 𝑧 ∈ (2nd𝐵)))
6059ssrdv 3103 1 (𝐴<P 𝐵 → (2nd ‘(𝐴 +P 𝐶)) ⊆ (2nd𝐵))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∧ w3a 962   = wceq 1331  ∃wex 1468   ∈ wcel 1480  ∃wrex 2417  {crab 2420   ⊆ wss 3071  ⟨cop 3530   class class class wbr 3929  ‘cfv 5123  (class class class)co 5774  1st c1st 6036  2nd c2nd 6037  Qcnq 7100   +Q cplq 7102
 Copyright terms: Public domain W3C validator