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Theorem ltexprlemru 7413
 Description: Lemma for ltexpri 7414. 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
ltexprlemru (𝐴<P 𝐵 → (2nd𝐵) ⊆ (2nd ‘(𝐴 +P 𝐶)))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦

Proof of Theorem ltexprlemru
Dummy variables 𝑧 𝑤 𝑢 𝑣 𝑓 𝑔 𝑞 𝑠 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltrelpr 7306 . . . . . . . 8 <P ⊆ (P × P)
21brel 4586 . . . . . . 7 (𝐴<P 𝐵 → (𝐴P𝐵P))
32simprd 113 . . . . . 6 (𝐴<P 𝐵𝐵P)
4 prop 7276 . . . . . 6 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
53, 4syl 14 . . . . 5 (𝐴<P 𝐵 → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
6 prnminu 7290 . . . . 5 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑤 ∈ (2nd𝐵)) → ∃𝑡 ∈ (2nd𝐵)𝑡 <Q 𝑤)
75, 6sylan 281 . . . 4 ((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) → ∃𝑡 ∈ (2nd𝐵)𝑡 <Q 𝑤)
8 simprr 521 . . . . . 6 (((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) → 𝑡 <Q 𝑤)
9 elprnqu 7283 . . . . . . . . 9 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑡 ∈ (2nd𝐵)) → 𝑡Q)
105, 9sylan 281 . . . . . . . 8 ((𝐴<P 𝐵𝑡 ∈ (2nd𝐵)) → 𝑡Q)
1110ad2ant2r 500 . . . . . . 7 (((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) → 𝑡Q)
12 elprnqu 7283 . . . . . . . . 9 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑤 ∈ (2nd𝐵)) → 𝑤Q)
135, 12sylan 281 . . . . . . . 8 ((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) → 𝑤Q)
1413adantr 274 . . . . . . 7 (((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) → 𝑤Q)
15 ltexnqq 7209 . . . . . . 7 ((𝑡Q𝑤Q) → (𝑡 <Q 𝑤 ↔ ∃𝑣Q (𝑡 +Q 𝑣) = 𝑤))
1611, 14, 15syl2anc 408 . . . . . 6 (((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) → (𝑡 <Q 𝑤 ↔ ∃𝑣Q (𝑡 +Q 𝑣) = 𝑤))
178, 16mpbid 146 . . . . 5 (((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) → ∃𝑣Q (𝑡 +Q 𝑣) = 𝑤)
182simpld 111 . . . . . . . . . 10 (𝐴<P 𝐵𝐴P)
19 prop 7276 . . . . . . . . . 10 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
2018, 19syl 14 . . . . . . . . 9 (𝐴<P 𝐵 → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
21 prarloc 7304 . . . . . . . . 9 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑣Q) → ∃𝑧 ∈ (1st𝐴)∃𝑢 ∈ (2nd𝐴)𝑢 <Q (𝑧 +Q 𝑣))
2220, 21sylan 281 . . . . . . . 8 ((𝐴<P 𝐵𝑣Q) → ∃𝑧 ∈ (1st𝐴)∃𝑢 ∈ (2nd𝐴)𝑢 <Q (𝑧 +Q 𝑣))
2322adantlr 468 . . . . . . 7 (((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ 𝑣Q) → ∃𝑧 ∈ (1st𝐴)∃𝑢 ∈ (2nd𝐴)𝑢 <Q (𝑧 +Q 𝑣))
2423ad2ant2r 500 . . . . . 6 ((((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣Q ∧ (𝑡 +Q 𝑣) = 𝑤)) → ∃𝑧 ∈ (1st𝐴)∃𝑢 ∈ (2nd𝐴)𝑢 <Q (𝑧 +Q 𝑣))
25 simplll 522 . . . . . . . . . . . . 13 ((((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣Q ∧ (𝑡 +Q 𝑣) = 𝑤)) → 𝐴<P 𝐵)
2625ad2antrr 479 . . . . . . . . . . . 12 ((((((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) → 𝐴<P 𝐵)
27 ltdfpr 7307 . . . . . . . . . . . . . 14 ((𝐴P𝐵P) → (𝐴<P 𝐵 ↔ ∃𝑞Q (𝑞 ∈ (2nd𝐴) ∧ 𝑞 ∈ (1st𝐵))))
2827biimpd 143 . . . . . . . . . . . . 13 ((𝐴P𝐵P) → (𝐴<P 𝐵 → ∃𝑞Q (𝑞 ∈ (2nd𝐴) ∧ 𝑞 ∈ (1st𝐵))))
292, 28mpcom 36 . . . . . . . . . . . 12 (𝐴<P 𝐵 → ∃𝑞Q (𝑞 ∈ (2nd𝐴) ∧ 𝑞 ∈ (1st𝐵)))
3026, 29syl 14 . . . . . . . . . . 11 ((((((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) → ∃𝑞Q (𝑞 ∈ (2nd𝐴) ∧ 𝑞 ∈ (1st𝐵)))
3125adantr 274 . . . . . . . . . . . . . 14 (((((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) → 𝐴<P 𝐵)
3231ad2antrr 479 . . . . . . . . . . . . 13 (((((((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ (𝑞Q ∧ (𝑞 ∈ (2nd𝐴) ∧ 𝑞 ∈ (1st𝐵)))) → 𝐴<P 𝐵)
33 simplrl 524 . . . . . . . . . . . . . 14 ((((((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) → 𝑧 ∈ (1st𝐴))
3433adantr 274 . . . . . . . . . . . . 13 (((((((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ (𝑞Q ∧ (𝑞 ∈ (2nd𝐴) ∧ 𝑞 ∈ (1st𝐵)))) → 𝑧 ∈ (1st𝐴))
35 simprrl 528 . . . . . . . . . . . . 13 (((((((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ (𝑞Q ∧ (𝑞 ∈ (2nd𝐴) ∧ 𝑞 ∈ (1st𝐵)))) → 𝑞 ∈ (2nd𝐴))
36 prltlu 7288 . . . . . . . . . . . . . 14 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑧 ∈ (1st𝐴) ∧ 𝑞 ∈ (2nd𝐴)) → 𝑧 <Q 𝑞)
3720, 36syl3an1 1249 . . . . . . . . . . . . 13 ((𝐴<P 𝐵𝑧 ∈ (1st𝐴) ∧ 𝑞 ∈ (2nd𝐴)) → 𝑧 <Q 𝑞)
3832, 34, 35, 37syl3anc 1216 . . . . . . . . . . . 12 (((((((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ (𝑞Q ∧ (𝑞 ∈ (2nd𝐴) ∧ 𝑞 ∈ (1st𝐵)))) → 𝑧 <Q 𝑞)
39 simprrr 529 . . . . . . . . . . . . 13 (((((((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ (𝑞Q ∧ (𝑞 ∈ (2nd𝐴) ∧ 𝑞 ∈ (1st𝐵)))) → 𝑞 ∈ (1st𝐵))
40 simplrl 524 . . . . . . . . . . . . . . 15 ((((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣Q ∧ (𝑡 +Q 𝑣) = 𝑤)) → 𝑡 ∈ (2nd𝐵))
4140adantr 274 . . . . . . . . . . . . . 14 (((((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) → 𝑡 ∈ (2nd𝐵))
4241ad2antrr 479 . . . . . . . . . . . . 13 (((((((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ (𝑞Q ∧ (𝑞 ∈ (2nd𝐴) ∧ 𝑞 ∈ (1st𝐵)))) → 𝑡 ∈ (2nd𝐵))
43 prltlu 7288 . . . . . . . . . . . . . 14 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑞 ∈ (1st𝐵) ∧ 𝑡 ∈ (2nd𝐵)) → 𝑞 <Q 𝑡)
445, 43syl3an1 1249 . . . . . . . . . . . . 13 ((𝐴<P 𝐵𝑞 ∈ (1st𝐵) ∧ 𝑡 ∈ (2nd𝐵)) → 𝑞 <Q 𝑡)
4532, 39, 42, 44syl3anc 1216 . . . . . . . . . . . 12 (((((((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ (𝑞Q ∧ (𝑞 ∈ (2nd𝐴) ∧ 𝑞 ∈ (1st𝐵)))) → 𝑞 <Q 𝑡)
46 ltsonq 7199 . . . . . . . . . . . . 13 <Q Or Q
47 ltrelnq 7166 . . . . . . . . . . . . 13 <Q ⊆ (Q × Q)
4846, 47sotri 4929 . . . . . . . . . . . 12 ((𝑧 <Q 𝑞𝑞 <Q 𝑡) → 𝑧 <Q 𝑡)
4938, 45, 48syl2anc 408 . . . . . . . . . . 11 (((((((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ (𝑞Q ∧ (𝑞 ∈ (2nd𝐴) ∧ 𝑞 ∈ (1st𝐵)))) → 𝑧 <Q 𝑡)
5030, 49rexlimddv 2552 . . . . . . . . . 10 ((((((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) → 𝑧 <Q 𝑡)
51 ltexnqi 7210 . . . . . . . . . 10 (𝑧 <Q 𝑡 → ∃𝑠Q (𝑧 +Q 𝑠) = 𝑡)
5250, 51syl 14 . . . . . . . . 9 ((((((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) → ∃𝑠Q (𝑧 +Q 𝑠) = 𝑡)
53 simplrr 525 . . . . . . . . . . . 12 (((((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) → (𝑡 +Q 𝑣) = 𝑤)
5453ad2antrr 479 . . . . . . . . . . 11 (((((((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ (𝑠Q ∧ (𝑧 +Q 𝑠) = 𝑡)) → (𝑡 +Q 𝑣) = 𝑤)
55 simprr 521 . . . . . . . . . . . 12 (((((((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ (𝑠Q ∧ (𝑧 +Q 𝑠) = 𝑡)) → (𝑧 +Q 𝑠) = 𝑡)
56 oveq1 5774 . . . . . . . . . . . . 13 ((𝑧 +Q 𝑠) = 𝑡 → ((𝑧 +Q 𝑠) +Q 𝑣) = (𝑡 +Q 𝑣))
5756eqeq1d 2146 . . . . . . . . . . . 12 ((𝑧 +Q 𝑠) = 𝑡 → (((𝑧 +Q 𝑠) +Q 𝑣) = 𝑤 ↔ (𝑡 +Q 𝑣) = 𝑤))
5855, 57syl 14 . . . . . . . . . . 11 (((((((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ (𝑠Q ∧ (𝑧 +Q 𝑠) = 𝑡)) → (((𝑧 +Q 𝑠) +Q 𝑣) = 𝑤 ↔ (𝑡 +Q 𝑣) = 𝑤))
5954, 58mpbird 166 . . . . . . . . . 10 (((((((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ (𝑠Q ∧ (𝑧 +Q 𝑠) = 𝑡)) → ((𝑧 +Q 𝑠) +Q 𝑣) = 𝑤)
60 elprnql 7282 . . . . . . . . . . . . . . . . 17 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑧 ∈ (1st𝐴)) → 𝑧Q)
6120, 60sylan 281 . . . . . . . . . . . . . . . 16 ((𝐴<P 𝐵𝑧 ∈ (1st𝐴)) → 𝑧Q)
6261adantlr 468 . . . . . . . . . . . . . . 15 (((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ 𝑧 ∈ (1st𝐴)) → 𝑧Q)
6362ad2ant2r 500 . . . . . . . . . . . . . 14 ((((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) → 𝑧Q)
6463adantlr 468 . . . . . . . . . . . . 13 (((((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) → 𝑧Q)
6564ad2antrr 479 . . . . . . . . . . . 12 (((((((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ (𝑠Q ∧ (𝑧 +Q 𝑠) = 𝑡)) → 𝑧Q)
66 simplrl 524 . . . . . . . . . . . . 13 (((((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) → 𝑣Q)
6766ad2antrr 479 . . . . . . . . . . . 12 (((((((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ (𝑠Q ∧ (𝑧 +Q 𝑠) = 𝑡)) → 𝑣Q)
68 simprl 520 . . . . . . . . . . . 12 (((((((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ (𝑠Q ∧ (𝑧 +Q 𝑠) = 𝑡)) → 𝑠Q)
69 addcomnqg 7182 . . . . . . . . . . . . 13 ((𝑓Q𝑔Q) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
7069adantl 275 . . . . . . . . . . . 12 ((((((((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ (𝑠Q ∧ (𝑧 +Q 𝑠) = 𝑡)) ∧ (𝑓Q𝑔Q)) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
71 addassnqg 7183 . . . . . . . . . . . . 13 ((𝑓Q𝑔QQ) → ((𝑓 +Q 𝑔) +Q ) = (𝑓 +Q (𝑔 +Q )))
7271adantl 275 . . . . . . . . . . . 12 ((((((((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ (𝑠Q ∧ (𝑧 +Q 𝑠) = 𝑡)) ∧ (𝑓Q𝑔QQ)) → ((𝑓 +Q 𝑔) +Q ) = (𝑓 +Q (𝑔 +Q )))
7365, 67, 68, 70, 72caov32d 5944 . . . . . . . . . . 11 (((((((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ (𝑠Q ∧ (𝑧 +Q 𝑠) = 𝑡)) → ((𝑧 +Q 𝑣) +Q 𝑠) = ((𝑧 +Q 𝑠) +Q 𝑣))
74 simpr 109 . . . . . . . . . . . . . 14 ((((((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) → 𝑢 <Q (𝑧 +Q 𝑣))
75 simplrr 525 . . . . . . . . . . . . . . 15 ((((((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) → 𝑢 ∈ (2nd𝐴))
76 prcunqu 7286 . . . . . . . . . . . . . . . 16 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑢 ∈ (2nd𝐴)) → (𝑢 <Q (𝑧 +Q 𝑣) → (𝑧 +Q 𝑣) ∈ (2nd𝐴)))
7720, 76sylan 281 . . . . . . . . . . . . . . 15 ((𝐴<P 𝐵𝑢 ∈ (2nd𝐴)) → (𝑢 <Q (𝑧 +Q 𝑣) → (𝑧 +Q 𝑣) ∈ (2nd𝐴)))
7826, 75, 77syl2anc 408 . . . . . . . . . . . . . 14 ((((((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) → (𝑢 <Q (𝑧 +Q 𝑣) → (𝑧 +Q 𝑣) ∈ (2nd𝐴)))
7974, 78mpd 13 . . . . . . . . . . . . 13 ((((((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) → (𝑧 +Q 𝑣) ∈ (2nd𝐴))
8079adantr 274 . . . . . . . . . . . 12 (((((((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ (𝑠Q ∧ (𝑧 +Q 𝑠) = 𝑡)) → (𝑧 +Q 𝑣) ∈ (2nd𝐴))
8133adantr 274 . . . . . . . . . . . . . 14 (((((((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ (𝑠Q ∧ (𝑧 +Q 𝑠) = 𝑡)) → 𝑧 ∈ (1st𝐴))
8241ad2antrr 479 . . . . . . . . . . . . . . 15 (((((((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ (𝑠Q ∧ (𝑧 +Q 𝑠) = 𝑡)) → 𝑡 ∈ (2nd𝐵))
8355, 82eqeltrd 2214 . . . . . . . . . . . . . 14 (((((((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ (𝑠Q ∧ (𝑧 +Q 𝑠) = 𝑡)) → (𝑧 +Q 𝑠) ∈ (2nd𝐵))
84 eleq1 2200 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑧 → (𝑦 ∈ (1st𝐴) ↔ 𝑧 ∈ (1st𝐴)))
85 oveq1 5774 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑧 → (𝑦 +Q 𝑠) = (𝑧 +Q 𝑠))
8685eleq1d 2206 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑧 → ((𝑦 +Q 𝑠) ∈ (2nd𝐵) ↔ (𝑧 +Q 𝑠) ∈ (2nd𝐵)))
8784, 86anbi12d 464 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑧 → ((𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑠) ∈ (2nd𝐵)) ↔ (𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑠) ∈ (2nd𝐵))))
8887spcegv 2769 . . . . . . . . . . . . . . 15 (𝑧 ∈ (1st𝐴) → ((𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑠) ∈ (2nd𝐵)) → ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑠) ∈ (2nd𝐵))))
8988anabsi5 568 . . . . . . . . . . . . . 14 ((𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑠) ∈ (2nd𝐵)) → ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑠) ∈ (2nd𝐵)))
9081, 83, 89syl2anc 408 . . . . . . . . . . . . 13 (((((((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ (𝑠Q ∧ (𝑧 +Q 𝑠) = 𝑡)) → ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑠) ∈ (2nd𝐵)))
91 ltexprlem.1 . . . . . . . . . . . . . 14 𝐶 = ⟨{𝑥Q ∣ ∃𝑦(𝑦 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑥) ∈ (1st𝐵))}, {𝑥Q ∣ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑥) ∈ (2nd𝐵))}⟩
9291ltexprlemelu 7400 . . . . . . . . . . . . 13 (𝑠 ∈ (2nd𝐶) ↔ (𝑠Q ∧ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑠) ∈ (2nd𝐵))))
9368, 90, 92sylanbrc 413 . . . . . . . . . . . 12 (((((((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ (𝑠Q ∧ (𝑧 +Q 𝑠) = 𝑡)) → 𝑠 ∈ (2nd𝐶))
9431ad2antrr 479 . . . . . . . . . . . . . 14 (((((((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ (𝑠Q ∧ (𝑧 +Q 𝑠) = 𝑡)) → 𝐴<P 𝐵)
9594, 18syl 14 . . . . . . . . . . . . 13 (((((((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ (𝑠Q ∧ (𝑧 +Q 𝑠) = 𝑡)) → 𝐴P)
9691ltexprlempr 7409 . . . . . . . . . . . . . 14 (𝐴<P 𝐵𝐶P)
9794, 96syl 14 . . . . . . . . . . . . 13 (((((((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ (𝑠Q ∧ (𝑧 +Q 𝑠) = 𝑡)) → 𝐶P)
98 df-iplp 7269 . . . . . . . . . . . . . 14 +P = (𝑥P, 𝑤P ↦ ⟨{𝑧Q ∣ ∃𝑓Q𝑣Q (𝑓 ∈ (1st𝑥) ∧ 𝑣 ∈ (1st𝑤) ∧ 𝑧 = (𝑓 +Q 𝑣))}, {𝑧Q ∣ ∃𝑓Q𝑣Q (𝑓 ∈ (2nd𝑥) ∧ 𝑣 ∈ (2nd𝑤) ∧ 𝑧 = (𝑓 +Q 𝑣))}⟩)
99 addclnq 7176 . . . . . . . . . . . . . 14 ((𝑓Q𝑣Q) → (𝑓 +Q 𝑣) ∈ Q)
10098, 99genppreclu 7316 . . . . . . . . . . . . 13 ((𝐴P𝐶P) → (((𝑧 +Q 𝑣) ∈ (2nd𝐴) ∧ 𝑠 ∈ (2nd𝐶)) → ((𝑧 +Q 𝑣) +Q 𝑠) ∈ (2nd ‘(𝐴 +P 𝐶))))
10195, 97, 100syl2anc 408 . . . . . . . . . . . 12 (((((((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ (𝑠Q ∧ (𝑧 +Q 𝑠) = 𝑡)) → (((𝑧 +Q 𝑣) ∈ (2nd𝐴) ∧ 𝑠 ∈ (2nd𝐶)) → ((𝑧 +Q 𝑣) +Q 𝑠) ∈ (2nd ‘(𝐴 +P 𝐶))))
10280, 93, 101mp2and 429 . . . . . . . . . . 11 (((((((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ (𝑠Q ∧ (𝑧 +Q 𝑠) = 𝑡)) → ((𝑧 +Q 𝑣) +Q 𝑠) ∈ (2nd ‘(𝐴 +P 𝐶)))
10373, 102eqeltrrd 2215 . . . . . . . . . 10 (((((((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ (𝑠Q ∧ (𝑧 +Q 𝑠) = 𝑡)) → ((𝑧 +Q 𝑠) +Q 𝑣) ∈ (2nd ‘(𝐴 +P 𝐶)))
10459, 103eqeltrrd 2215 . . . . . . . . 9 (((((((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) ∧ (𝑠Q ∧ (𝑧 +Q 𝑠) = 𝑡)) → 𝑤 ∈ (2nd ‘(𝐴 +P 𝐶)))
10552, 104rexlimddv 2552 . . . . . . . 8 ((((((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) ∧ 𝑢 <Q (𝑧 +Q 𝑣)) → 𝑤 ∈ (2nd ‘(𝐴 +P 𝐶)))
106105ex 114 . . . . . . 7 (((((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) → (𝑢 <Q (𝑧 +Q 𝑣) → 𝑤 ∈ (2nd ‘(𝐴 +P 𝐶))))
107106rexlimdvva 2555 . . . . . 6 ((((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣Q ∧ (𝑡 +Q 𝑣) = 𝑤)) → (∃𝑧 ∈ (1st𝐴)∃𝑢 ∈ (2nd𝐴)𝑢 <Q (𝑧 +Q 𝑣) → 𝑤 ∈ (2nd ‘(𝐴 +P 𝐶))))
10824, 107mpd 13 . . . . 5 ((((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣Q ∧ (𝑡 +Q 𝑣) = 𝑤)) → 𝑤 ∈ (2nd ‘(𝐴 +P 𝐶)))
10917, 108rexlimddv 2552 . . . 4 (((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) → 𝑤 ∈ (2nd ‘(𝐴 +P 𝐶)))
1107, 109rexlimddv 2552 . . 3 ((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) → 𝑤 ∈ (2nd ‘(𝐴 +P 𝐶)))
111110ex 114 . 2 (𝐴<P 𝐵 → (𝑤 ∈ (2nd𝐵) → 𝑤 ∈ (2nd ‘(𝐴 +P 𝐶))))
112111ssrdv 3098 1 (𝐴<P 𝐵 → (2nd𝐵) ⊆ (2nd ‘(𝐴 +P 𝐶)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∧ w3a 962   = wceq 1331  ∃wex 1468   ∈ wcel 1480  ∃wrex 2415  {crab 2418   ⊆ wss 3066  ⟨cop 3525   class class class wbr 3924  ‘cfv 5118  (class class class)co 5767  1st c1st 6029  2nd c2nd 6030  Qcnq 7081   +Q cplq 7083
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