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Theorem ltexprlemrl 7070
Description: Lemma for ltexpri 7073. 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 6965 . . . . . . . 8 <P ⊆ (P × P)
21brel 4446 . . . . . . 7 (𝐴<P 𝐵 → (𝐴P𝐵P))
32simprd 112 . . . . . 6 (𝐴<P 𝐵𝐵P)
4 prop 6935 . . . . . 6 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
53, 4syl 14 . . . . 5 (𝐴<P 𝐵 → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
6 prnmaddl 6950 . . . . 5 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑤 ∈ (1st𝐵)) → ∃𝑣Q (𝑤 +Q 𝑣) ∈ (1st𝐵))
75, 6sylan 277 . . . 4 ((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) → ∃𝑣Q (𝑤 +Q 𝑣) ∈ (1st𝐵))
82simpld 110 . . . . . . . 8 (𝐴<P 𝐵𝐴P)
9 prop 6935 . . . . . . . 8 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
108, 9syl 14 . . . . . . 7 (𝐴<P 𝐵 → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
11 prarloc 6963 . . . . . . 7 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑣Q) → ∃𝑧 ∈ (1st𝐴)∃𝑢 ∈ (2nd𝐴)𝑢 <Q (𝑧 +Q 𝑣))
1210, 11sylan 277 . . . . . 6 ((𝐴<P 𝐵𝑣Q) → ∃𝑧 ∈ (1st𝐴)∃𝑢 ∈ (2nd𝐴)𝑢 <Q (𝑧 +Q 𝑣))
1312ad2ant2r 493 . . . . 5 (((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) ∧ (𝑣Q ∧ (𝑤 +Q 𝑣) ∈ (1st𝐵))) → ∃𝑧 ∈ (1st𝐴)∃𝑢 ∈ (2nd𝐴)𝑢 <Q (𝑧 +Q 𝑣))
14 simplll 500 . . . . . . . . . . 11 ((((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) ∧ (𝑣Q ∧ (𝑤 +Q 𝑣) ∈ (1st𝐵))) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) → 𝐴<P 𝐵)
1514adantr 270 . . . . . . . . . 10 (((((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) ∧ (𝑣Q ∧ (𝑤 +Q 𝑣) ∈ (1st𝐵))) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) → 𝐴<P 𝐵)
16 simplrl 502 . . . . . . . . . 10 (((((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) ∧ (𝑣Q ∧ (𝑤 +Q 𝑣) ∈ (1st𝐵))) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) → 𝑧 ∈ (1st𝐴))
17 elprnql 6941 . . . . . . . . . . 11 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑧 ∈ (1st𝐴)) → 𝑧Q)
1810, 17sylan 277 . . . . . . . . . 10 ((𝐴<P 𝐵𝑧 ∈ (1st𝐴)) → 𝑧Q)
1915, 16, 18syl2anc 403 . . . . . . . . 9 (((((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) ∧ (𝑣Q ∧ (𝑤 +Q 𝑣) ∈ (1st𝐵))) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) → 𝑧Q)
20 elprnql 6941 . . . . . . . . . . 11 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑤 ∈ (1st𝐵)) → 𝑤Q)
215, 20sylan 277 . . . . . . . . . 10 ((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) → 𝑤Q)
2221ad3antrrr 476 . . . . . . . . 9 (((((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) ∧ (𝑣Q ∧ (𝑤 +Q 𝑣) ∈ (1st𝐵))) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) → 𝑤Q)
23 nqtri3or 6856 . . . . . . . . 9 ((𝑧Q𝑤Q) → (𝑧 <Q 𝑤𝑧 = 𝑤𝑤 <Q 𝑧))
2419, 22, 23syl2anc 403 . . . . . . . 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 6868 . . . . . . . . . . . . 13 ((𝑧Q𝑤Q) → (𝑧 <Q 𝑤 ↔ ∃𝑠Q (𝑧 +Q 𝑠) = 𝑤))
2619, 22, 25syl2anc 403 . . . . . . . . . . . 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 290 . . . . . . . . . . 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 499 . . . . . . . . . . . 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 472 . . . . . . . . . . . . 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 498 . . . . . . . . . . . . . 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 108 . . . . . . . . . . . . . . . . 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 503 . . . . . . . . . . . . . . . . . 18 (((((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) ∧ (𝑣Q ∧ (𝑤 +Q 𝑣) ∈ (1st𝐵))) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) → 𝑢 ∈ (2nd𝐴))
33 prcunqu 6945 . . . . . . . . . . . . . . . . . . 19 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑢 ∈ (2nd𝐴)) → (𝑢 <Q (𝑧 +Q 𝑣) → (𝑧 +Q 𝑣) ∈ (2nd𝐴)))
3410, 33sylan 277 . . . . . . . . . . . . . . . . . 18 ((𝐴<P 𝐵𝑢 ∈ (2nd𝐴)) → (𝑢 <Q (𝑧 +Q 𝑣) → (𝑧 +Q 𝑣) ∈ (2nd𝐴)))
3515, 32, 34syl2anc 403 . . . . . . . . . . . . . . . . 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 472 . . . . . . . . . . . . . . 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 472 . . . . . . . . . . . . . . . . 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 502 . . . . . . . . . . . . . . . . . 18 ((((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) ∧ (𝑣Q ∧ (𝑤 +Q 𝑣) ∈ (1st𝐵))) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) → 𝑣Q)
4039ad3antrrr 476 . . . . . . . . . . . . . . . . 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 6841 . . . . . . . . . . . . . . . . . 18 ((𝑓Q𝑔Q) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
4241adantl 271 . . . . . . . . . . . . . . . . 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 6842 . . . . . . . . . . . . . . . . . 18 ((𝑓Q𝑔QQ) → ((𝑓 +Q 𝑔) +Q ) = (𝑓 +Q (𝑔 +Q )))
4443adantl 271 . . . . . . . . . . . . . . . . 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 5758 . . . . . . . . . . . . . . . 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 503 . . . . . . . . . . . . . . . . . 18 ((((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) ∧ (𝑣Q ∧ (𝑤 +Q 𝑣) ∈ (1st𝐵))) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) → (𝑤 +Q 𝑣) ∈ (1st𝐵))
4746ad3antrrr 476 . . . . . . . . . . . . . . . . 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 5596 . . . . . . . . . . . . . . . . . . 19 ((𝑧 +Q 𝑠) = 𝑤 → ((𝑧 +Q 𝑠) +Q 𝑣) = (𝑤 +Q 𝑣))
4948eleq1d 2151 . . . . . . . . . . . . . . . . . 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 165 . . . . . . . . . . . . . . . 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 2159 . . . . . . . . . . . . . . 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 2145 . . . . . . . . . . . . . . . . . 18 (𝑦 = (𝑧 +Q 𝑣) → (𝑦 ∈ (2nd𝐴) ↔ (𝑧 +Q 𝑣) ∈ (2nd𝐴)))
54 oveq1 5596 . . . . . . . . . . . . . . . . . . 19 (𝑦 = (𝑧 +Q 𝑣) → (𝑦 +Q 𝑠) = ((𝑧 +Q 𝑣) +Q 𝑠))
5554eleq1d 2151 . . . . . . . . . . . . . . . . . 18 (𝑦 = (𝑧 +Q 𝑣) → ((𝑦 +Q 𝑠) ∈ (1st𝐵) ↔ ((𝑧 +Q 𝑣) +Q 𝑠) ∈ (1st𝐵)))
5653, 55anbi12d 457 . . . . . . . . . . . . . . . . 17 (𝑦 = (𝑧 +Q 𝑣) → ((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑠) ∈ (1st𝐵)) ↔ ((𝑧 +Q 𝑣) ∈ (2nd𝐴) ∧ ((𝑧 +Q 𝑣) +Q 𝑠) ∈ (1st𝐵))))
5756spcegv 2697 . . . . . . . . . . . . . . . 16 ((𝑧 +Q 𝑣) ∈ (2nd𝐴) → (((𝑧 +Q 𝑣) ∈ (2nd𝐴) ∧ ((𝑧 +Q 𝑣) +Q 𝑠) ∈ (1st𝐵)) → ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑠) ∈ (1st𝐵))))
5857anabsi5 544 . . . . . . . . . . . . . . 15 (((𝑧 +Q 𝑣) ∈ (2nd𝐴) ∧ ((𝑧 +Q 𝑣) +Q 𝑠) ∈ (1st𝐵)) → ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑠) ∈ (1st𝐵)))
5937, 52, 58syl2anc 403 . . . . . . . . . . . . . 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 7058 . . . . . . . . . . . . . 14 (𝑠 ∈ (1st𝐶) ↔ (𝑠Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑠) ∈ (1st𝐵))))
6230, 59, 61sylanbrc 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)) → 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 472 . . . . . . . . . . . . . 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 7068 . . . . . . . . . . . . . . . 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 472 . . . . . . . . . . . . . 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 6928 . . . . . . . . . . . . . . 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 6835 . . . . . . . . . . . . . . 15 ((𝑓Q𝑣Q) → (𝑓 +Q 𝑣) ∈ Q)
7068, 69genpprecll 6974 . . . . . . . . . . . . . 14 ((𝐴P𝐶P) → ((𝑧 ∈ (1st𝐴) ∧ 𝑠 ∈ (1st𝐶)) → (𝑧 +Q 𝑠) ∈ (1st ‘(𝐴 +P 𝐶))))
7164, 67, 70syl2anc 403 . . . . . . . . . . . . 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 424 . . . . . . . . . . . 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 2160 . . . . . . . . . . 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 2487 . . . . . . . . . 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 113 . . . . . . . . 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 472 . . . . . . . . . . 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 108 . . . . . . . . . . . 12 ((((((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) ∧ (𝑣Q ∧ (𝑤 +Q 𝑣) ∈ (1st𝐵))) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ 𝑧 = 𝑤) → 𝑧 = 𝑤)
7816adantr 270 . . . . . . . . . . . 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 2160 . . . . . . . . . . 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 7057 . . . . . . . . . . . . 13 ((𝐴P𝐶P) → 𝐴<P (𝐴 +P 𝐶))
818, 65, 80syl2anc 403 . . . . . . . . . . . 12 (𝐴<P 𝐵𝐴<P (𝐴 +P 𝐶))
82 ltprordil 7049 . . . . . . . . . . . . 13 (𝐴<P (𝐴 +P 𝐶) → (1st𝐴) ⊆ (1st ‘(𝐴 +P 𝐶)))
8382sseld 3009 . . . . . . . . . . . 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 61 . . . . . . . . . 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 113 . . . . . . . . 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 6944 . . . . . . . . . . . 12 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑧 ∈ (1st𝐴)) → (𝑤 <Q 𝑧𝑤 ∈ (1st𝐴)))
8810, 87sylan 277 . . . . . . . . . . 11 ((𝐴<P 𝐵𝑧 ∈ (1st𝐴)) → (𝑤 <Q 𝑧𝑤 ∈ (1st𝐴)))
8915, 16, 88syl2anc 403 . . . . . . . . . 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 57 . . . . . . . . 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 1236 . . . . . . . 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 113 . . . . . 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 2490 . . . . 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 2487 . . 3 ((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) → 𝑤 ∈ (1st ‘(𝐴 +P 𝐶)))
9796ex 113 . 2 (𝐴<P 𝐵 → (𝑤 ∈ (1st𝐵) → 𝑤 ∈ (1st ‘(𝐴 +P 𝐶))))
9897ssrdv 3016 1 (𝐴<P 𝐵 → (1st𝐵) ⊆ (1st ‘(𝐴 +P 𝐶)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  w3o 919  w3a 920   = wceq 1285  wex 1422  wcel 1434  wrex 2354  {crab 2357  wss 2984  cop 3425   class class class wbr 3811  cfv 4967  (class class class)co 5589  1st c1st 5842  2nd c2nd 5843  Qcnq 6740   +Q cplq 6742   <Q cltq 6745  Pcnp 6751   +P cpp 6753  <P cltp 6755
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3919  ax-sep 3922  ax-nul 3930  ax-pow 3974  ax-pr 3999  ax-un 4223  ax-setind 4315  ax-iinf 4365
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2614  df-sbc 2827  df-csb 2920  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-int 3663  df-iun 3706  df-br 3812  df-opab 3866  df-mpt 3867  df-tr 3902  df-eprel 4079  df-id 4083  df-po 4086  df-iso 4087  df-iord 4156  df-on 4158  df-suc 4161  df-iom 4368  df-xp 4405  df-rel 4406  df-cnv 4407  df-co 4408  df-dm 4409  df-rn 4410  df-res 4411  df-ima 4412  df-iota 4932  df-fun 4969  df-fn 4970  df-f 4971  df-f1 4972  df-fo 4973  df-f1o 4974  df-fv 4975  df-ov 5592  df-oprab 5593  df-mpt2 5594  df-1st 5844  df-2nd 5845  df-recs 6000  df-irdg 6065  df-1o 6111  df-2o 6112  df-oadd 6115  df-omul 6116  df-er 6220  df-ec 6222  df-qs 6226  df-ni 6764  df-pli 6765  df-mi 6766  df-lti 6767  df-plpq 6804  df-mpq 6805  df-enq 6807  df-nqqs 6808  df-plqqs 6809  df-mqqs 6810  df-1nqqs 6811  df-rq 6812  df-ltnqqs 6813  df-enq0 6884  df-nq0 6885  df-0nq0 6886  df-plq0 6887  df-mq0 6888  df-inp 6926  df-iplp 6928  df-iltp 6930
This theorem is referenced by:  ltexpri  7073
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