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

Theorem ltexprlemrl 7425
Description: Lemma for ltexpri 7428. Reverse direction of our result for lower 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
ltexprlemrl (𝐴<P 𝐵 → (1st𝐵) ⊆ (1st ‘(𝐴 +P 𝐶)))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦

Proof of Theorem ltexprlemrl
Dummy variables 𝑧 𝑤 𝑢 𝑣 𝑓 𝑔 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltrelpr 7320 . . . . . . . 8 <P ⊆ (P × P)
21brel 4591 . . . . . . 7 (𝐴<P 𝐵 → (𝐴P𝐵P))
32simprd 113 . . . . . 6 (𝐴<P 𝐵𝐵P)
4 prop 7290 . . . . . 6 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
53, 4syl 14 . . . . 5 (𝐴<P 𝐵 → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
6 prnmaddl 7305 . . . . 5 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑤 ∈ (1st𝐵)) → ∃𝑣Q (𝑤 +Q 𝑣) ∈ (1st𝐵))
75, 6sylan 281 . . . 4 ((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) → ∃𝑣Q (𝑤 +Q 𝑣) ∈ (1st𝐵))
82simpld 111 . . . . . . . 8 (𝐴<P 𝐵𝐴P)
9 prop 7290 . . . . . . . 8 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
108, 9syl 14 . . . . . . 7 (𝐴<P 𝐵 → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
11 prarloc 7318 . . . . . . 7 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑣Q) → ∃𝑧 ∈ (1st𝐴)∃𝑢 ∈ (2nd𝐴)𝑢 <Q (𝑧 +Q 𝑣))
1210, 11sylan 281 . . . . . 6 ((𝐴<P 𝐵𝑣Q) → ∃𝑧 ∈ (1st𝐴)∃𝑢 ∈ (2nd𝐴)𝑢 <Q (𝑧 +Q 𝑣))
1312ad2ant2r 500 . . . . 5 (((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) ∧ (𝑣Q ∧ (𝑤 +Q 𝑣) ∈ (1st𝐵))) → ∃𝑧 ∈ (1st𝐴)∃𝑢 ∈ (2nd𝐴)𝑢 <Q (𝑧 +Q 𝑣))
14 simplll 522 . . . . . . . . . . 11 ((((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) ∧ (𝑣Q ∧ (𝑤 +Q 𝑣) ∈ (1st𝐵))) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) → 𝐴<P 𝐵)
1514adantr 274 . . . . . . . . . 10 (((((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) ∧ (𝑣Q ∧ (𝑤 +Q 𝑣) ∈ (1st𝐵))) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) → 𝐴<P 𝐵)
16 simplrl 524 . . . . . . . . . 10 (((((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) ∧ (𝑣Q ∧ (𝑤 +Q 𝑣) ∈ (1st𝐵))) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) → 𝑧 ∈ (1st𝐴))
17 elprnql 7296 . . . . . . . . . . 11 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑧 ∈ (1st𝐴)) → 𝑧Q)
1810, 17sylan 281 . . . . . . . . . 10 ((𝐴<P 𝐵𝑧 ∈ (1st𝐴)) → 𝑧Q)
1915, 16, 18syl2anc 408 . . . . . . . . 9 (((((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) ∧ (𝑣Q ∧ (𝑤 +Q 𝑣) ∈ (1st𝐵))) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) → 𝑧Q)
20 elprnql 7296 . . . . . . . . . . 11 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑤 ∈ (1st𝐵)) → 𝑤Q)
215, 20sylan 281 . . . . . . . . . 10 ((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) → 𝑤Q)
2221ad3antrrr 483 . . . . . . . . 9 (((((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) ∧ (𝑣Q ∧ (𝑤 +Q 𝑣) ∈ (1st𝐵))) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) → 𝑤Q)
23 nqtri3or 7211 . . . . . . . . 9 ((𝑧Q𝑤Q) → (𝑧 <Q 𝑤𝑧 = 𝑤𝑤 <Q 𝑧))
2419, 22, 23syl2anc 408 . . . . . . . 8 (((((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) ∧ (𝑣Q ∧ (𝑤 +Q 𝑣) ∈ (1st𝐵))) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) → (𝑧 <Q 𝑤𝑧 = 𝑤𝑤 <Q 𝑧))
25 ltexnqq 7223 . . . . . . . . . . . . 13 ((𝑧Q𝑤Q) → (𝑧 <Q 𝑤 ↔ ∃𝑠Q (𝑧 +Q 𝑠) = 𝑤))
2619, 22, 25syl2anc 408 . . . . . . . . . . . 12 (((((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) ∧ (𝑣Q ∧ (𝑤 +Q 𝑣) ∈ (1st𝐵))) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) → (𝑧 <Q 𝑤 ↔ ∃𝑠Q (𝑧 +Q 𝑠) = 𝑤))
2726biimpa 294 . . . . . . . . . . 11 ((((((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) ∧ (𝑣Q ∧ (𝑤 +Q 𝑣) ∈ (1st𝐵))) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ 𝑧 <Q 𝑤) → ∃𝑠Q (𝑧 +Q 𝑠) = 𝑤)
28 simprr 521 . . . . . . . . . . . 12 (((((((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) ∧ (𝑣Q ∧ (𝑤 +Q 𝑣) ∈ (1st𝐵))) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ 𝑧 <Q 𝑤) ∧ (𝑠Q ∧ (𝑧 +Q 𝑠) = 𝑤)) → (𝑧 +Q 𝑠) = 𝑤)
2916ad2antrr 479 . . . . . . . . . . . . 13 (((((((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) ∧ (𝑣Q ∧ (𝑤 +Q 𝑣) ∈ (1st𝐵))) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ 𝑧 <Q 𝑤) ∧ (𝑠Q ∧ (𝑧 +Q 𝑠) = 𝑤)) → 𝑧 ∈ (1st𝐴))
30 simprl 520 . . . . . . . . . . . . . 14 (((((((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) ∧ (𝑣Q ∧ (𝑤 +Q 𝑣) ∈ (1st𝐵))) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ 𝑧 <Q 𝑤) ∧ (𝑠Q ∧ (𝑧 +Q 𝑠) = 𝑤)) → 𝑠Q)
31 simpr 109 . . . . . . . . . . . . . . . . 17 (((((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) ∧ (𝑣Q ∧ (𝑤 +Q 𝑣) ∈ (1st𝐵))) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) → 𝑢 <Q (𝑧 +Q 𝑣))
32 simplrr 525 . . . . . . . . . . . . . . . . . 18 (((((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) ∧ (𝑣Q ∧ (𝑤 +Q 𝑣) ∈ (1st𝐵))) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) → 𝑢 ∈ (2nd𝐴))
33 prcunqu 7300 . . . . . . . . . . . . . . . . . . 19 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑢 ∈ (2nd𝐴)) → (𝑢 <Q (𝑧 +Q 𝑣) → (𝑧 +Q 𝑣) ∈ (2nd𝐴)))
3410, 33sylan 281 . . . . . . . . . . . . . . . . . 18 ((𝐴<P 𝐵𝑢 ∈ (2nd𝐴)) → (𝑢 <Q (𝑧 +Q 𝑣) → (𝑧 +Q 𝑣) ∈ (2nd𝐴)))
3515, 32, 34syl2anc 408 . . . . . . . . . . . . . . . . 17 (((((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) ∧ (𝑣Q ∧ (𝑤 +Q 𝑣) ∈ (1st𝐵))) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) → (𝑢 <Q (𝑧 +Q 𝑣) → (𝑧 +Q 𝑣) ∈ (2nd𝐴)))
3631, 35mpd 13 . . . . . . . . . . . . . . . 16 (((((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) ∧ (𝑣Q ∧ (𝑤 +Q 𝑣) ∈ (1st𝐵))) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) → (𝑧 +Q 𝑣) ∈ (2nd𝐴))
3736ad2antrr 479 . . . . . . . . . . . . . . 15 (((((((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) ∧ (𝑣Q ∧ (𝑤 +Q 𝑣) ∈ (1st𝐵))) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ 𝑧 <Q 𝑤) ∧ (𝑠Q ∧ (𝑧 +Q 𝑠) = 𝑤)) → (𝑧 +Q 𝑣) ∈ (2nd𝐴))
3819ad2antrr 479 . . . . . . . . . . . . . . . . 17 (((((((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) ∧ (𝑣Q ∧ (𝑤 +Q 𝑣) ∈ (1st𝐵))) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ 𝑧 <Q 𝑤) ∧ (𝑠Q ∧ (𝑧 +Q 𝑠) = 𝑤)) → 𝑧Q)
39 simplrl 524 . . . . . . . . . . . . . . . . . 18 ((((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) ∧ (𝑣Q ∧ (𝑤 +Q 𝑣) ∈ (1st𝐵))) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) → 𝑣Q)
4039ad3antrrr 483 . . . . . . . . . . . . . . . . 17 (((((((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) ∧ (𝑣Q ∧ (𝑤 +Q 𝑣) ∈ (1st𝐵))) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ 𝑧 <Q 𝑤) ∧ (𝑠Q ∧ (𝑧 +Q 𝑠) = 𝑤)) → 𝑣Q)
41 addcomnqg 7196 . . . . . . . . . . . . . . . . . 18 ((𝑓Q𝑔Q) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
4241adantl 275 . . . . . . . . . . . . . . . . 17 ((((((((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) ∧ (𝑣Q ∧ (𝑤 +Q 𝑣) ∈ (1st𝐵))) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ 𝑧 <Q 𝑤) ∧ (𝑠Q ∧ (𝑧 +Q 𝑠) = 𝑤)) ∧ (𝑓Q𝑔Q)) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
43 addassnqg 7197 . . . . . . . . . . . . . . . . . 18 ((𝑓Q𝑔QQ) → ((𝑓 +Q 𝑔) +Q ) = (𝑓 +Q (𝑔 +Q )))
4443adantl 275 . . . . . . . . . . . . . . . . 17 ((((((((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) ∧ (𝑣Q ∧ (𝑤 +Q 𝑣) ∈ (1st𝐵))) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ 𝑧 <Q 𝑤) ∧ (𝑠Q ∧ (𝑧 +Q 𝑠) = 𝑤)) ∧ (𝑓Q𝑔QQ)) → ((𝑓 +Q 𝑔) +Q ) = (𝑓 +Q (𝑔 +Q )))
4538, 40, 30, 42, 44caov32d 5951 . . . . . . . . . . . . . . . 16 (((((((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) ∧ (𝑣Q ∧ (𝑤 +Q 𝑣) ∈ (1st𝐵))) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ 𝑧 <Q 𝑤) ∧ (𝑠Q ∧ (𝑧 +Q 𝑠) = 𝑤)) → ((𝑧 +Q 𝑣) +Q 𝑠) = ((𝑧 +Q 𝑠) +Q 𝑣))
46 simplrr 525 . . . . . . . . . . . . . . . . . 18 ((((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) ∧ (𝑣Q ∧ (𝑤 +Q 𝑣) ∈ (1st𝐵))) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) → (𝑤 +Q 𝑣) ∈ (1st𝐵))
4746ad3antrrr 483 . . . . . . . . . . . . . . . . 17 (((((((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) ∧ (𝑣Q ∧ (𝑤 +Q 𝑣) ∈ (1st𝐵))) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ 𝑧 <Q 𝑤) ∧ (𝑠Q ∧ (𝑧 +Q 𝑠) = 𝑤)) → (𝑤 +Q 𝑣) ∈ (1st𝐵))
48 oveq1 5781 . . . . . . . . . . . . . . . . . . 19 ((𝑧 +Q 𝑠) = 𝑤 → ((𝑧 +Q 𝑠) +Q 𝑣) = (𝑤 +Q 𝑣))
4948eleq1d 2208 . . . . . . . . . . . . . . . . . 18 ((𝑧 +Q 𝑠) = 𝑤 → (((𝑧 +Q 𝑠) +Q 𝑣) ∈ (1st𝐵) ↔ (𝑤 +Q 𝑣) ∈ (1st𝐵)))
5028, 49syl 14 . . . . . . . . . . . . . . . . 17 (((((((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) ∧ (𝑣Q ∧ (𝑤 +Q 𝑣) ∈ (1st𝐵))) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ 𝑧 <Q 𝑤) ∧ (𝑠Q ∧ (𝑧 +Q 𝑠) = 𝑤)) → (((𝑧 +Q 𝑠) +Q 𝑣) ∈ (1st𝐵) ↔ (𝑤 +Q 𝑣) ∈ (1st𝐵)))
5147, 50mpbird 166 . . . . . . . . . . . . . . . 16 (((((((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) ∧ (𝑣Q ∧ (𝑤 +Q 𝑣) ∈ (1st𝐵))) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ 𝑧 <Q 𝑤) ∧ (𝑠Q ∧ (𝑧 +Q 𝑠) = 𝑤)) → ((𝑧 +Q 𝑠) +Q 𝑣) ∈ (1st𝐵))
5245, 51eqeltrd 2216 . . . . . . . . . . . . . . 15 (((((((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) ∧ (𝑣Q ∧ (𝑤 +Q 𝑣) ∈ (1st𝐵))) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ 𝑧 <Q 𝑤) ∧ (𝑠Q ∧ (𝑧 +Q 𝑠) = 𝑤)) → ((𝑧 +Q 𝑣) +Q 𝑠) ∈ (1st𝐵))
53 eleq1 2202 . . . . . . . . . . . . . . . . . 18 (𝑦 = (𝑧 +Q 𝑣) → (𝑦 ∈ (2nd𝐴) ↔ (𝑧 +Q 𝑣) ∈ (2nd𝐴)))
54 oveq1 5781 . . . . . . . . . . . . . . . . . . 19 (𝑦 = (𝑧 +Q 𝑣) → (𝑦 +Q 𝑠) = ((𝑧 +Q 𝑣) +Q 𝑠))
5554eleq1d 2208 . . . . . . . . . . . . . . . . . 18 (𝑦 = (𝑧 +Q 𝑣) → ((𝑦 +Q 𝑠) ∈ (1st𝐵) ↔ ((𝑧 +Q 𝑣) +Q 𝑠) ∈ (1st𝐵)))
5653, 55anbi12d 464 . . . . . . . . . . . . . . . . 17 (𝑦 = (𝑧 +Q 𝑣) → ((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑠) ∈ (1st𝐵)) ↔ ((𝑧 +Q 𝑣) ∈ (2nd𝐴) ∧ ((𝑧 +Q 𝑣) +Q 𝑠) ∈ (1st𝐵))))
5756spcegv 2774 . . . . . . . . . . . . . . . 16 ((𝑧 +Q 𝑣) ∈ (2nd𝐴) → (((𝑧 +Q 𝑣) ∈ (2nd𝐴) ∧ ((𝑧 +Q 𝑣) +Q 𝑠) ∈ (1st𝐵)) → ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑠) ∈ (1st𝐵))))
5857anabsi5 568 . . . . . . . . . . . . . . 15 (((𝑧 +Q 𝑣) ∈ (2nd𝐴) ∧ ((𝑧 +Q 𝑣) +Q 𝑠) ∈ (1st𝐵)) → ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑠) ∈ (1st𝐵)))
5937, 52, 58syl2anc 408 . . . . . . . . . . . . . 14 (((((((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) ∧ (𝑣Q ∧ (𝑤 +Q 𝑣) ∈ (1st𝐵))) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ 𝑧 <Q 𝑤) ∧ (𝑠Q ∧ (𝑧 +Q 𝑠) = 𝑤)) → ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑠) ∈ (1st𝐵)))
60 ltexprlem.1 . . . . . . . . . . . . . . 15 𝐶 = ⟨{𝑥Q ∣ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st𝐵))}, {𝑥Q ∣ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd𝐵))}⟩
6160ltexprlemell 7413 . . . . . . . . . . . . . 14 (𝑠 ∈ (1st𝐶) ↔ (𝑠Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑠) ∈ (1st𝐵))))
6230, 59, 61sylanbrc 413 . . . . . . . . . . . . 13 (((((((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) ∧ (𝑣Q ∧ (𝑤 +Q 𝑣) ∈ (1st𝐵))) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ 𝑧 <Q 𝑤) ∧ (𝑠Q ∧ (𝑧 +Q 𝑠) = 𝑤)) → 𝑠 ∈ (1st𝐶))
6315, 8syl 14 . . . . . . . . . . . . . . 15 (((((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) ∧ (𝑣Q ∧ (𝑤 +Q 𝑣) ∈ (1st𝐵))) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) → 𝐴P)
6463ad2antrr 479 . . . . . . . . . . . . . 14 (((((((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) ∧ (𝑣Q ∧ (𝑤 +Q 𝑣) ∈ (1st𝐵))) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ 𝑧 <Q 𝑤) ∧ (𝑠Q ∧ (𝑧 +Q 𝑠) = 𝑤)) → 𝐴P)
6560ltexprlempr 7423 . . . . . . . . . . . . . . . 16 (𝐴<P 𝐵𝐶P)
6615, 65syl 14 . . . . . . . . . . . . . . 15 (((((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) ∧ (𝑣Q ∧ (𝑤 +Q 𝑣) ∈ (1st𝐵))) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) → 𝐶P)
6766ad2antrr 479 . . . . . . . . . . . . . 14 (((((((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) ∧ (𝑣Q ∧ (𝑤 +Q 𝑣) ∈ (1st𝐵))) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ 𝑧 <Q 𝑤) ∧ (𝑠Q ∧ (𝑧 +Q 𝑠) = 𝑤)) → 𝐶P)
68 df-iplp 7283 . . . . . . . . . . . . . . 15 +P = (𝑥P, 𝑤P ↦ ⟨{𝑧Q ∣ ∃𝑓Q𝑣Q (𝑓 ∈ (1st𝑥) ∧ 𝑣 ∈ (1st𝑤) ∧ 𝑧 = (𝑓 +Q 𝑣))}, {𝑧Q ∣ ∃𝑓Q𝑣Q (𝑓 ∈ (2nd𝑥) ∧ 𝑣 ∈ (2nd𝑤) ∧ 𝑧 = (𝑓 +Q 𝑣))}⟩)
69 addclnq 7190 . . . . . . . . . . . . . . 15 ((𝑓Q𝑣Q) → (𝑓 +Q 𝑣) ∈ Q)
7068, 69genpprecll 7329 . . . . . . . . . . . . . 14 ((𝐴P𝐶P) → ((𝑧 ∈ (1st𝐴) ∧ 𝑠 ∈ (1st𝐶)) → (𝑧 +Q 𝑠) ∈ (1st ‘(𝐴 +P 𝐶))))
7164, 67, 70syl2anc 408 . . . . . . . . . . . . 13 (((((((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) ∧ (𝑣Q ∧ (𝑤 +Q 𝑣) ∈ (1st𝐵))) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ 𝑧 <Q 𝑤) ∧ (𝑠Q ∧ (𝑧 +Q 𝑠) = 𝑤)) → ((𝑧 ∈ (1st𝐴) ∧ 𝑠 ∈ (1st𝐶)) → (𝑧 +Q 𝑠) ∈ (1st ‘(𝐴 +P 𝐶))))
7229, 62, 71mp2and 429 . . . . . . . . . . . 12 (((((((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) ∧ (𝑣Q ∧ (𝑤 +Q 𝑣) ∈ (1st𝐵))) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ 𝑧 <Q 𝑤) ∧ (𝑠Q ∧ (𝑧 +Q 𝑠) = 𝑤)) → (𝑧 +Q 𝑠) ∈ (1st ‘(𝐴 +P 𝐶)))
7328, 72eqeltrrd 2217 . . . . . . . . . . 11 (((((((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) ∧ (𝑣Q ∧ (𝑤 +Q 𝑣) ∈ (1st𝐵))) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ 𝑧 <Q 𝑤) ∧ (𝑠Q ∧ (𝑧 +Q 𝑠) = 𝑤)) → 𝑤 ∈ (1st ‘(𝐴 +P 𝐶)))
7427, 73rexlimddv 2554 . . . . . . . . . 10 ((((((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) ∧ (𝑣Q ∧ (𝑤 +Q 𝑣) ∈ (1st𝐵))) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ 𝑧 <Q 𝑤) → 𝑤 ∈ (1st ‘(𝐴 +P 𝐶)))
7574ex 114 . . . . . . . . 9 (((((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) ∧ (𝑣Q ∧ (𝑤 +Q 𝑣) ∈ (1st𝐵))) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) → (𝑧 <Q 𝑤𝑤 ∈ (1st ‘(𝐴 +P 𝐶))))
7614ad2antrr 479 . . . . . . . . . . 11 ((((((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) ∧ (𝑣Q ∧ (𝑤 +Q 𝑣) ∈ (1st𝐵))) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ 𝑧 = 𝑤) → 𝐴<P 𝐵)
77 simpr 109 . . . . . . . . . . . 12 ((((((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) ∧ (𝑣Q ∧ (𝑤 +Q 𝑣) ∈ (1st𝐵))) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ 𝑧 = 𝑤) → 𝑧 = 𝑤)
7816adantr 274 . . . . . . . . . . . 12 ((((((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) ∧ (𝑣Q ∧ (𝑤 +Q 𝑣) ∈ (1st𝐵))) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ 𝑧 = 𝑤) → 𝑧 ∈ (1st𝐴))
7977, 78eqeltrrd 2217 . . . . . . . . . . 11 ((((((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) ∧ (𝑣Q ∧ (𝑤 +Q 𝑣) ∈ (1st𝐵))) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ 𝑧 = 𝑤) → 𝑤 ∈ (1st𝐴))
80 ltaddpr 7412 . . . . . . . . . . . . 13 ((𝐴P𝐶P) → 𝐴<P (𝐴 +P 𝐶))
818, 65, 80syl2anc 408 . . . . . . . . . . . 12 (𝐴<P 𝐵𝐴<P (𝐴 +P 𝐶))
82 ltprordil 7404 . . . . . . . . . . . . 13 (𝐴<P (𝐴 +P 𝐶) → (1st𝐴) ⊆ (1st ‘(𝐴 +P 𝐶)))
8382sseld 3096 . . . . . . . . . . . 12 (𝐴<P (𝐴 +P 𝐶) → (𝑤 ∈ (1st𝐴) → 𝑤 ∈ (1st ‘(𝐴 +P 𝐶))))
8481, 83syl 14 . . . . . . . . . . 11 (𝐴<P 𝐵 → (𝑤 ∈ (1st𝐴) → 𝑤 ∈ (1st ‘(𝐴 +P 𝐶))))
8576, 79, 84sylc 62 . . . . . . . . . 10 ((((((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) ∧ (𝑣Q ∧ (𝑤 +Q 𝑣) ∈ (1st𝐵))) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ 𝑧 = 𝑤) → 𝑤 ∈ (1st ‘(𝐴 +P 𝐶)))
8685ex 114 . . . . . . . . 9 (((((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) ∧ (𝑣Q ∧ (𝑤 +Q 𝑣) ∈ (1st𝐵))) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) → (𝑧 = 𝑤𝑤 ∈ (1st ‘(𝐴 +P 𝐶))))
87 prcdnql 7299 . . . . . . . . . . . 12 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑧 ∈ (1st𝐴)) → (𝑤 <Q 𝑧𝑤 ∈ (1st𝐴)))
8810, 87sylan 281 . . . . . . . . . . 11 ((𝐴<P 𝐵𝑧 ∈ (1st𝐴)) → (𝑤 <Q 𝑧𝑤 ∈ (1st𝐴)))
8915, 16, 88syl2anc 408 . . . . . . . . . 10 (((((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) ∧ (𝑣Q ∧ (𝑤 +Q 𝑣) ∈ (1st𝐵))) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) → (𝑤 <Q 𝑧𝑤 ∈ (1st𝐴)))
9015, 89, 84sylsyld 58 . . . . . . . . 9 (((((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) ∧ (𝑣Q ∧ (𝑤 +Q 𝑣) ∈ (1st𝐵))) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) → (𝑤 <Q 𝑧𝑤 ∈ (1st ‘(𝐴 +P 𝐶))))
9175, 86, 903jaod 1282 . . . . . . . 8 (((((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) ∧ (𝑣Q ∧ (𝑤 +Q 𝑣) ∈ (1st𝐵))) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) → ((𝑧 <Q 𝑤𝑧 = 𝑤𝑤 <Q 𝑧) → 𝑤 ∈ (1st ‘(𝐴 +P 𝐶))))
9224, 91mpd 13 . . . . . . 7 (((((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) ∧ (𝑣Q ∧ (𝑤 +Q 𝑣) ∈ (1st𝐵))) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) → 𝑤 ∈ (1st ‘(𝐴 +P 𝐶)))
9392ex 114 . . . . . 6 ((((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) ∧ (𝑣Q ∧ (𝑤 +Q 𝑣) ∈ (1st𝐵))) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) → (𝑢 <Q (𝑧 +Q 𝑣) → 𝑤 ∈ (1st ‘(𝐴 +P 𝐶))))
9493rexlimdvva 2557 . . . . 5 (((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) ∧ (𝑣Q ∧ (𝑤 +Q 𝑣) ∈ (1st𝐵))) → (∃𝑧 ∈ (1st𝐴)∃𝑢 ∈ (2nd𝐴)𝑢 <Q (𝑧 +Q 𝑣) → 𝑤 ∈ (1st ‘(𝐴 +P 𝐶))))
9513, 94mpd 13 . . . 4 (((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) ∧ (𝑣Q ∧ (𝑤 +Q 𝑣) ∈ (1st𝐵))) → 𝑤 ∈ (1st ‘(𝐴 +P 𝐶)))
967, 95rexlimddv 2554 . . 3 ((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) → 𝑤 ∈ (1st ‘(𝐴 +P 𝐶)))
9796ex 114 . 2 (𝐴<P 𝐵 → (𝑤 ∈ (1st𝐵) → 𝑤 ∈ (1st ‘(𝐴 +P 𝐶))))
9897ssrdv 3103 1 (𝐴<P 𝐵 → (1st𝐵) ⊆ (1st ‘(𝐴 +P 𝐶)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  w3o 961  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 7095   +Q cplq 7097   <Q cltq 7100  Pcnp 7106   +P cpp 7108  <P cltp 7110
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 7119  df-pli 7120  df-mi 7121  df-lti 7122  df-plpq 7159  df-mpq 7160  df-enq 7162  df-nqqs 7163  df-plqqs 7164  df-mqqs 7165  df-1nqqs 7166  df-rq 7167  df-ltnqqs 7168  df-enq0 7239  df-nq0 7240  df-0nq0 7241  df-plq0 7242  df-mq0 7243  df-inp 7281  df-iplp 7283  df-iltp 7285
This theorem is referenced by:  ltexpri  7428
  Copyright terms: Public domain W3C validator