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Theorem ltexprlemrl 7266
 Description: Lemma for ltexpri 7269. Reverse directon 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 7161 . . . . . . . 8 <P ⊆ (P × P)
21brel 4519 . . . . . . 7 (𝐴<P 𝐵 → (𝐴P𝐵P))
32simprd 113 . . . . . 6 (𝐴<P 𝐵𝐵P)
4 prop 7131 . . . . . 6 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
53, 4syl 14 . . . . 5 (𝐴<P 𝐵 → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
6 prnmaddl 7146 . . . . 5 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑤 ∈ (1st𝐵)) → ∃𝑣Q (𝑤 +Q 𝑣) ∈ (1st𝐵))
75, 6sylan 278 . . . 4 ((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) → ∃𝑣Q (𝑤 +Q 𝑣) ∈ (1st𝐵))
82simpld 111 . . . . . . . 8 (𝐴<P 𝐵𝐴P)
9 prop 7131 . . . . . . . 8 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
108, 9syl 14 . . . . . . 7 (𝐴<P 𝐵 → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
11 prarloc 7159 . . . . . . 7 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑣Q) → ∃𝑧 ∈ (1st𝐴)∃𝑢 ∈ (2nd𝐴)𝑢 <Q (𝑧 +Q 𝑣))
1210, 11sylan 278 . . . . . 6 ((𝐴<P 𝐵𝑣Q) → ∃𝑧 ∈ (1st𝐴)∃𝑢 ∈ (2nd𝐴)𝑢 <Q (𝑧 +Q 𝑣))
1312ad2ant2r 494 . . . . 5 (((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) ∧ (𝑣Q ∧ (𝑤 +Q 𝑣) ∈ (1st𝐵))) → ∃𝑧 ∈ (1st𝐴)∃𝑢 ∈ (2nd𝐴)𝑢 <Q (𝑧 +Q 𝑣))
14 simplll 501 . . . . . . . . . . 11 ((((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) ∧ (𝑣Q ∧ (𝑤 +Q 𝑣) ∈ (1st𝐵))) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) → 𝐴<P 𝐵)
1514adantr 271 . . . . . . . . . 10 (((((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) ∧ (𝑣Q ∧ (𝑤 +Q 𝑣) ∈ (1st𝐵))) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) → 𝐴<P 𝐵)
16 simplrl 503 . . . . . . . . . 10 (((((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) ∧ (𝑣Q ∧ (𝑤 +Q 𝑣) ∈ (1st𝐵))) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) → 𝑧 ∈ (1st𝐴))
17 elprnql 7137 . . . . . . . . . . 11 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑧 ∈ (1st𝐴)) → 𝑧Q)
1810, 17sylan 278 . . . . . . . . . 10 ((𝐴<P 𝐵𝑧 ∈ (1st𝐴)) → 𝑧Q)
1915, 16, 18syl2anc 404 . . . . . . . . 9 (((((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) ∧ (𝑣Q ∧ (𝑤 +Q 𝑣) ∈ (1st𝐵))) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) → 𝑧Q)
20 elprnql 7137 . . . . . . . . . . 11 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑤 ∈ (1st𝐵)) → 𝑤Q)
215, 20sylan 278 . . . . . . . . . 10 ((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) → 𝑤Q)
2221ad3antrrr 477 . . . . . . . . 9 (((((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) ∧ (𝑣Q ∧ (𝑤 +Q 𝑣) ∈ (1st𝐵))) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) → 𝑤Q)
23 nqtri3or 7052 . . . . . . . . 9 ((𝑧Q𝑤Q) → (𝑧 <Q 𝑤𝑧 = 𝑤𝑤 <Q 𝑧))
2419, 22, 23syl2anc 404 . . . . . . . 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 7064 . . . . . . . . . . . . 13 ((𝑧Q𝑤Q) → (𝑧 <Q 𝑤 ↔ ∃𝑠Q (𝑧 +Q 𝑠) = 𝑤))
2619, 22, 25syl2anc 404 . . . . . . . . . . . 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 291 . . . . . . . . . . 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 500 . . . . . . . . . . . 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 473 . . . . . . . . . . . . 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 499 . . . . . . . . . . . . . 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 504 . . . . . . . . . . . . . . . . . 18 (((((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) ∧ (𝑣Q ∧ (𝑤 +Q 𝑣) ∈ (1st𝐵))) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) → 𝑢 ∈ (2nd𝐴))
33 prcunqu 7141 . . . . . . . . . . . . . . . . . . 19 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑢 ∈ (2nd𝐴)) → (𝑢 <Q (𝑧 +Q 𝑣) → (𝑧 +Q 𝑣) ∈ (2nd𝐴)))
3410, 33sylan 278 . . . . . . . . . . . . . . . . . 18 ((𝐴<P 𝐵𝑢 ∈ (2nd𝐴)) → (𝑢 <Q (𝑧 +Q 𝑣) → (𝑧 +Q 𝑣) ∈ (2nd𝐴)))
3515, 32, 34syl2anc 404 . . . . . . . . . . . . . . . . 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 473 . . . . . . . . . . . . . . 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 473 . . . . . . . . . . . . . . . . 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 503 . . . . . . . . . . . . . . . . . 18 ((((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) ∧ (𝑣Q ∧ (𝑤 +Q 𝑣) ∈ (1st𝐵))) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) → 𝑣Q)
4039ad3antrrr 477 . . . . . . . . . . . . . . . . 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 7037 . . . . . . . . . . . . . . . . . 18 ((𝑓Q𝑔Q) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
4241adantl 272 . . . . . . . . . . . . . . . . 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 7038 . . . . . . . . . . . . . . . . . 18 ((𝑓Q𝑔QQ) → ((𝑓 +Q 𝑔) +Q ) = (𝑓 +Q (𝑔 +Q )))
4443adantl 272 . . . . . . . . . . . . . . . . 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 5863 . . . . . . . . . . . . . . . 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 504 . . . . . . . . . . . . . . . . . 18 ((((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) ∧ (𝑣Q ∧ (𝑤 +Q 𝑣) ∈ (1st𝐵))) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) → (𝑤 +Q 𝑣) ∈ (1st𝐵))
4746ad3antrrr 477 . . . . . . . . . . . . . . . . 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 5697 . . . . . . . . . . . . . . . . . . 19 ((𝑧 +Q 𝑠) = 𝑤 → ((𝑧 +Q 𝑠) +Q 𝑣) = (𝑤 +Q 𝑣))
4948eleq1d 2163 . . . . . . . . . . . . . . . . . 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 2171 . . . . . . . . . . . . . . 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 2157 . . . . . . . . . . . . . . . . . 18 (𝑦 = (𝑧 +Q 𝑣) → (𝑦 ∈ (2nd𝐴) ↔ (𝑧 +Q 𝑣) ∈ (2nd𝐴)))
54 oveq1 5697 . . . . . . . . . . . . . . . . . . 19 (𝑦 = (𝑧 +Q 𝑣) → (𝑦 +Q 𝑠) = ((𝑧 +Q 𝑣) +Q 𝑠))
5554eleq1d 2163 . . . . . . . . . . . . . . . . . 18 (𝑦 = (𝑧 +Q 𝑣) → ((𝑦 +Q 𝑠) ∈ (1st𝐵) ↔ ((𝑧 +Q 𝑣) +Q 𝑠) ∈ (1st𝐵)))
5653, 55anbi12d 458 . . . . . . . . . . . . . . . . 17 (𝑦 = (𝑧 +Q 𝑣) → ((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑠) ∈ (1st𝐵)) ↔ ((𝑧 +Q 𝑣) ∈ (2nd𝐴) ∧ ((𝑧 +Q 𝑣) +Q 𝑠) ∈ (1st𝐵))))
5756spcegv 2721 . . . . . . . . . . . . . . . 16 ((𝑧 +Q 𝑣) ∈ (2nd𝐴) → (((𝑧 +Q 𝑣) ∈ (2nd𝐴) ∧ ((𝑧 +Q 𝑣) +Q 𝑠) ∈ (1st𝐵)) → ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑠) ∈ (1st𝐵))))
5857anabsi5 547 . . . . . . . . . . . . . . 15 (((𝑧 +Q 𝑣) ∈ (2nd𝐴) ∧ ((𝑧 +Q 𝑣) +Q 𝑠) ∈ (1st𝐵)) → ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑠) ∈ (1st𝐵)))
5937, 52, 58syl2anc 404 . . . . . . . . . . . . . 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 7254 . . . . . . . . . . . . . 14 (𝑠 ∈ (1st𝐶) ↔ (𝑠Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑠) ∈ (1st𝐵))))
6230, 59, 61sylanbrc 409 . . . . . . . . . . . . 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 473 . . . . . . . . . . . . . 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 7264 . . . . . . . . . . . . . . . 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 473 . . . . . . . . . . . . . 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 7124 . . . . . . . . . . . . . . 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 7031 . . . . . . . . . . . . . . 15 ((𝑓Q𝑣Q) → (𝑓 +Q 𝑣) ∈ Q)
7068, 69genpprecll 7170 . . . . . . . . . . . . . 14 ((𝐴P𝐶P) → ((𝑧 ∈ (1st𝐴) ∧ 𝑠 ∈ (1st𝐶)) → (𝑧 +Q 𝑠) ∈ (1st ‘(𝐴 +P 𝐶))))
7164, 67, 70syl2anc 404 . . . . . . . . . . . . 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 425 . . . . . . . . . . . 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 2172 . . . . . . . . . . 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 2507 . . . . . . . . . 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 473 . . . . . . . . . . 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 271 . . . . . . . . . . . 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 2172 . . . . . . . . . . 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 7253 . . . . . . . . . . . . 13 ((𝐴P𝐶P) → 𝐴<P (𝐴 +P 𝐶))
818, 65, 80syl2anc 404 . . . . . . . . . . . 12 (𝐴<P 𝐵𝐴<P (𝐴 +P 𝐶))
82 ltprordil 7245 . . . . . . . . . . . . 13 (𝐴<P (𝐴 +P 𝐶) → (1st𝐴) ⊆ (1st ‘(𝐴 +P 𝐶)))
8382sseld 3038 . . . . . . . . . . . 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 7140 . . . . . . . . . . . 12 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑧 ∈ (1st𝐴)) → (𝑤 <Q 𝑧𝑤 ∈ (1st𝐴)))
8810, 87sylan 278 . . . . . . . . . . 11 ((𝐴<P 𝐵𝑧 ∈ (1st𝐴)) → (𝑤 <Q 𝑧𝑤 ∈ (1st𝐴)))
8915, 16, 88syl2anc 404 . . . . . . . . . 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 1247 . . . . . . . 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 2510 . . . . 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 2507 . . 3 ((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) → 𝑤 ∈ (1st ‘(𝐴 +P 𝐶)))
9796ex 114 . 2 (𝐴<P 𝐵 → (𝑤 ∈ (1st𝐵) → 𝑤 ∈ (1st ‘(𝐴 +P 𝐶))))
9897ssrdv 3045 1 (𝐴<P 𝐵 → (1st𝐵) ⊆ (1st ‘(𝐴 +P 𝐶)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∨ w3o 926   ∧ w3a 927   = wceq 1296  ∃wex 1433   ∈ wcel 1445  ∃wrex 2371  {crab 2374   ⊆ wss 3013  ⟨cop 3469   class class class wbr 3867  ‘cfv 5049  (class class class)co 5690  1st c1st 5947  2nd c2nd 5948  Qcnq 6936   +Q cplq 6938
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