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Theorem ltexprlemrl 7627
Description: Lemma for ltexpri 7630. 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 7522 . . . . . . . 8 <P ⊆ (P × P)
21brel 4693 . . . . . . 7 (𝐴<P 𝐵 → (𝐴P𝐵P))
32simprd 114 . . . . . 6 (𝐴<P 𝐵𝐵P)
4 prop 7492 . . . . . 6 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
53, 4syl 14 . . . . 5 (𝐴<P 𝐵 → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
6 prnmaddl 7507 . . . . 5 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑤 ∈ (1st𝐵)) → ∃𝑣Q (𝑤 +Q 𝑣) ∈ (1st𝐵))
75, 6sylan 283 . . . 4 ((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) → ∃𝑣Q (𝑤 +Q 𝑣) ∈ (1st𝐵))
82simpld 112 . . . . . . . 8 (𝐴<P 𝐵𝐴P)
9 prop 7492 . . . . . . . 8 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
108, 9syl 14 . . . . . . 7 (𝐴<P 𝐵 → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
11 prarloc 7520 . . . . . . 7 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑣Q) → ∃𝑧 ∈ (1st𝐴)∃𝑢 ∈ (2nd𝐴)𝑢 <Q (𝑧 +Q 𝑣))
1210, 11sylan 283 . . . . . 6 ((𝐴<P 𝐵𝑣Q) → ∃𝑧 ∈ (1st𝐴)∃𝑢 ∈ (2nd𝐴)𝑢 <Q (𝑧 +Q 𝑣))
1312ad2ant2r 509 . . . . 5 (((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) ∧ (𝑣Q ∧ (𝑤 +Q 𝑣) ∈ (1st𝐵))) → ∃𝑧 ∈ (1st𝐴)∃𝑢 ∈ (2nd𝐴)𝑢 <Q (𝑧 +Q 𝑣))
14 simplll 533 . . . . . . . . . . 11 ((((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) ∧ (𝑣Q ∧ (𝑤 +Q 𝑣) ∈ (1st𝐵))) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) → 𝐴<P 𝐵)
1514adantr 276 . . . . . . . . . 10 (((((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) ∧ (𝑣Q ∧ (𝑤 +Q 𝑣) ∈ (1st𝐵))) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) → 𝐴<P 𝐵)
16 simplrl 535 . . . . . . . . . 10 (((((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) ∧ (𝑣Q ∧ (𝑤 +Q 𝑣) ∈ (1st𝐵))) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) → 𝑧 ∈ (1st𝐴))
17 elprnql 7498 . . . . . . . . . . 11 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑧 ∈ (1st𝐴)) → 𝑧Q)
1810, 17sylan 283 . . . . . . . . . 10 ((𝐴<P 𝐵𝑧 ∈ (1st𝐴)) → 𝑧Q)
1915, 16, 18syl2anc 411 . . . . . . . . 9 (((((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) ∧ (𝑣Q ∧ (𝑤 +Q 𝑣) ∈ (1st𝐵))) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) → 𝑧Q)
20 elprnql 7498 . . . . . . . . . . 11 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑤 ∈ (1st𝐵)) → 𝑤Q)
215, 20sylan 283 . . . . . . . . . 10 ((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) → 𝑤Q)
2221ad3antrrr 492 . . . . . . . . 9 (((((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) ∧ (𝑣Q ∧ (𝑤 +Q 𝑣) ∈ (1st𝐵))) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) → 𝑤Q)
23 nqtri3or 7413 . . . . . . . . 9 ((𝑧Q𝑤Q) → (𝑧 <Q 𝑤𝑧 = 𝑤𝑤 <Q 𝑧))
2419, 22, 23syl2anc 411 . . . . . . . 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 7425 . . . . . . . . . . . . 13 ((𝑧Q𝑤Q) → (𝑧 <Q 𝑤 ↔ ∃𝑠Q (𝑧 +Q 𝑠) = 𝑤))
2619, 22, 25syl2anc 411 . . . . . . . . . . . 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 296 . . . . . . . . . . 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 531 . . . . . . . . . . . 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 488 . . . . . . . . . . . . 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 529 . . . . . . . . . . . . . 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 110 . . . . . . . . . . . . . . . . 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 536 . . . . . . . . . . . . . . . . . 18 (((((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) ∧ (𝑣Q ∧ (𝑤 +Q 𝑣) ∈ (1st𝐵))) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) → 𝑢 ∈ (2nd𝐴))
33 prcunqu 7502 . . . . . . . . . . . . . . . . . . 19 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑢 ∈ (2nd𝐴)) → (𝑢 <Q (𝑧 +Q 𝑣) → (𝑧 +Q 𝑣) ∈ (2nd𝐴)))
3410, 33sylan 283 . . . . . . . . . . . . . . . . . 18 ((𝐴<P 𝐵𝑢 ∈ (2nd𝐴)) → (𝑢 <Q (𝑧 +Q 𝑣) → (𝑧 +Q 𝑣) ∈ (2nd𝐴)))
3515, 32, 34syl2anc 411 . . . . . . . . . . . . . . . . 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 488 . . . . . . . . . . . . . . 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 488 . . . . . . . . . . . . . . . . 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 535 . . . . . . . . . . . . . . . . . 18 ((((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) ∧ (𝑣Q ∧ (𝑤 +Q 𝑣) ∈ (1st𝐵))) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) → 𝑣Q)
4039ad3antrrr 492 . . . . . . . . . . . . . . . . 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 7398 . . . . . . . . . . . . . . . . . 18 ((𝑓Q𝑔Q) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
4241adantl 277 . . . . . . . . . . . . . . . . 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 7399 . . . . . . . . . . . . . . . . . 18 ((𝑓Q𝑔QQ) → ((𝑓 +Q 𝑔) +Q ) = (𝑓 +Q (𝑔 +Q )))
4443adantl 277 . . . . . . . . . . . . . . . . 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 6072 . . . . . . . . . . . . . . . 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 536 . . . . . . . . . . . . . . . . . 18 ((((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) ∧ (𝑣Q ∧ (𝑤 +Q 𝑣) ∈ (1st𝐵))) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) → (𝑤 +Q 𝑣) ∈ (1st𝐵))
4746ad3antrrr 492 . . . . . . . . . . . . . . . . 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 5898 . . . . . . . . . . . . . . . . . . 19 ((𝑧 +Q 𝑠) = 𝑤 → ((𝑧 +Q 𝑠) +Q 𝑣) = (𝑤 +Q 𝑣))
4948eleq1d 2258 . . . . . . . . . . . . . . . . . 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 167 . . . . . . . . . . . . . . . 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 2266 . . . . . . . . . . . . . . 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 2252 . . . . . . . . . . . . . . . . . 18 (𝑦 = (𝑧 +Q 𝑣) → (𝑦 ∈ (2nd𝐴) ↔ (𝑧 +Q 𝑣) ∈ (2nd𝐴)))
54 oveq1 5898 . . . . . . . . . . . . . . . . . . 19 (𝑦 = (𝑧 +Q 𝑣) → (𝑦 +Q 𝑠) = ((𝑧 +Q 𝑣) +Q 𝑠))
5554eleq1d 2258 . . . . . . . . . . . . . . . . . 18 (𝑦 = (𝑧 +Q 𝑣) → ((𝑦 +Q 𝑠) ∈ (1st𝐵) ↔ ((𝑧 +Q 𝑣) +Q 𝑠) ∈ (1st𝐵)))
5653, 55anbi12d 473 . . . . . . . . . . . . . . . . 17 (𝑦 = (𝑧 +Q 𝑣) → ((𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑠) ∈ (1st𝐵)) ↔ ((𝑧 +Q 𝑣) ∈ (2nd𝐴) ∧ ((𝑧 +Q 𝑣) +Q 𝑠) ∈ (1st𝐵))))
5756spcegv 2840 . . . . . . . . . . . . . . . 16 ((𝑧 +Q 𝑣) ∈ (2nd𝐴) → (((𝑧 +Q 𝑣) ∈ (2nd𝐴) ∧ ((𝑧 +Q 𝑣) +Q 𝑠) ∈ (1st𝐵)) → ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑠) ∈ (1st𝐵))))
5857anabsi5 579 . . . . . . . . . . . . . . 15 (((𝑧 +Q 𝑣) ∈ (2nd𝐴) ∧ ((𝑧 +Q 𝑣) +Q 𝑠) ∈ (1st𝐵)) → ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑠) ∈ (1st𝐵)))
5937, 52, 58syl2anc 411 . . . . . . . . . . . . . 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 7615 . . . . . . . . . . . . . 14 (𝑠 ∈ (1st𝐶) ↔ (𝑠Q ∧ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑠) ∈ (1st𝐵))))
6230, 59, 61sylanbrc 417 . . . . . . . . . . . . 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 488 . . . . . . . . . . . . . 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 7625 . . . . . . . . . . . . . . . 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 488 . . . . . . . . . . . . . 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 7485 . . . . . . . . . . . . . . 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 7392 . . . . . . . . . . . . . . 15 ((𝑓Q𝑣Q) → (𝑓 +Q 𝑣) ∈ Q)
7068, 69genpprecll 7531 . . . . . . . . . . . . . 14 ((𝐴P𝐶P) → ((𝑧 ∈ (1st𝐴) ∧ 𝑠 ∈ (1st𝐶)) → (𝑧 +Q 𝑠) ∈ (1st ‘(𝐴 +P 𝐶))))
7164, 67, 70syl2anc 411 . . . . . . . . . . . . 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 433 . . . . . . . . . . . 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 2267 . . . . . . . . . . 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 2612 . . . . . . . . . 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 115 . . . . . . . . 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 488 . . . . . . . . . . 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 110 . . . . . . . . . . . 12 ((((((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) ∧ (𝑣Q ∧ (𝑤 +Q 𝑣) ∈ (1st𝐵))) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ 𝑧 = 𝑤) → 𝑧 = 𝑤)
7816adantr 276 . . . . . . . . . . . 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 2267 . . . . . . . . . . 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 7614 . . . . . . . . . . . . 13 ((𝐴P𝐶P) → 𝐴<P (𝐴 +P 𝐶))
818, 65, 80syl2anc 411 . . . . . . . . . . . 12 (𝐴<P 𝐵𝐴<P (𝐴 +P 𝐶))
82 ltprordil 7606 . . . . . . . . . . . . 13 (𝐴<P (𝐴 +P 𝐶) → (1st𝐴) ⊆ (1st ‘(𝐴 +P 𝐶)))
8382sseld 3169 . . . . . . . . . . . 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 115 . . . . . . . . 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 7501 . . . . . . . . . . . 12 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑧 ∈ (1st𝐴)) → (𝑤 <Q 𝑧𝑤 ∈ (1st𝐴)))
8810, 87sylan 283 . . . . . . . . . . 11 ((𝐴<P 𝐵𝑧 ∈ (1st𝐴)) → (𝑤 <Q 𝑧𝑤 ∈ (1st𝐴)))
8915, 16, 88syl2anc 411 . . . . . . . . . 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 1315 . . . . . . . 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 115 . . . . . 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 2615 . . . . 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 2612 . . 3 ((𝐴<P 𝐵𝑤 ∈ (1st𝐵)) → 𝑤 ∈ (1st ‘(𝐴 +P 𝐶)))
9796ex 115 . 2 (𝐴<P 𝐵 → (𝑤 ∈ (1st𝐵) → 𝑤 ∈ (1st ‘(𝐴 +P 𝐶))))
9897ssrdv 3176 1 (𝐴<P 𝐵 → (1st𝐵) ⊆ (1st ‘(𝐴 +P 𝐶)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  w3o 979  w3a 980   = wceq 1364  wex 1503  wcel 2160  wrex 2469  {crab 2472  wss 3144  cop 3610   class class class wbr 4018  cfv 5231  (class class class)co 5891  1st c1st 6157  2nd c2nd 6158  Qcnq 7297   +Q cplq 7299   <Q cltq 7302  Pcnp 7308   +P cpp 7310  <P cltp 7312
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-iinf 4602
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-eprel 4304  df-id 4308  df-po 4311  df-iso 4312  df-iord 4381  df-on 4383  df-suc 4386  df-iom 4605  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-f1 5236  df-fo 5237  df-f1o 5238  df-fv 5239  df-ov 5894  df-oprab 5895  df-mpo 5896  df-1st 6159  df-2nd 6160  df-recs 6324  df-irdg 6389  df-1o 6435  df-2o 6436  df-oadd 6439  df-omul 6440  df-er 6553  df-ec 6555  df-qs 6559  df-ni 7321  df-pli 7322  df-mi 7323  df-lti 7324  df-plpq 7361  df-mpq 7362  df-enq 7364  df-nqqs 7365  df-plqqs 7366  df-mqqs 7367  df-1nqqs 7368  df-rq 7369  df-ltnqqs 7370  df-enq0 7441  df-nq0 7442  df-0nq0 7443  df-plq0 7444  df-mq0 7445  df-inp 7483  df-iplp 7485  df-iltp 7487
This theorem is referenced by:  ltexpri  7630
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