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Theorem ltexprlemru 7074
Description: Lemma for ltexpri 7075. 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 6967 . . . . . . . 8 <P ⊆ (P × P)
21brel 4448 . . . . . . 7 (𝐴<P 𝐵 → (𝐴P𝐵P))
32simprd 112 . . . . . 6 (𝐴<P 𝐵𝐵P)
4 prop 6937 . . . . . 6 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
53, 4syl 14 . . . . 5 (𝐴<P 𝐵 → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
6 prnminu 6951 . . . . 5 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑤 ∈ (2nd𝐵)) → ∃𝑡 ∈ (2nd𝐵)𝑡 <Q 𝑤)
75, 6sylan 277 . . . 4 ((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) → ∃𝑡 ∈ (2nd𝐵)𝑡 <Q 𝑤)
8 simprr 499 . . . . . 6 (((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) → 𝑡 <Q 𝑤)
9 elprnqu 6944 . . . . . . . . 9 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑡 ∈ (2nd𝐵)) → 𝑡Q)
105, 9sylan 277 . . . . . . . 8 ((𝐴<P 𝐵𝑡 ∈ (2nd𝐵)) → 𝑡Q)
1110ad2ant2r 493 . . . . . . 7 (((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) → 𝑡Q)
12 elprnqu 6944 . . . . . . . . 9 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑤 ∈ (2nd𝐵)) → 𝑤Q)
135, 12sylan 277 . . . . . . . 8 ((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) → 𝑤Q)
1413adantr 270 . . . . . . 7 (((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) → 𝑤Q)
15 ltexnqq 6870 . . . . . . 7 ((𝑡Q𝑤Q) → (𝑡 <Q 𝑤 ↔ ∃𝑣Q (𝑡 +Q 𝑣) = 𝑤))
1611, 14, 15syl2anc 403 . . . . . 6 (((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) → (𝑡 <Q 𝑤 ↔ ∃𝑣Q (𝑡 +Q 𝑣) = 𝑤))
178, 16mpbid 145 . . . . 5 (((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) → ∃𝑣Q (𝑡 +Q 𝑣) = 𝑤)
182simpld 110 . . . . . . . . . 10 (𝐴<P 𝐵𝐴P)
19 prop 6937 . . . . . . . . . 10 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
2018, 19syl 14 . . . . . . . . 9 (𝐴<P 𝐵 → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
21 prarloc 6965 . . . . . . . . 9 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑣Q) → ∃𝑧 ∈ (1st𝐴)∃𝑢 ∈ (2nd𝐴)𝑢 <Q (𝑧 +Q 𝑣))
2220, 21sylan 277 . . . . . . . 8 ((𝐴<P 𝐵𝑣Q) → ∃𝑧 ∈ (1st𝐴)∃𝑢 ∈ (2nd𝐴)𝑢 <Q (𝑧 +Q 𝑣))
2322adantlr 461 . . . . . . 7 (((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ 𝑣Q) → ∃𝑧 ∈ (1st𝐴)∃𝑢 ∈ (2nd𝐴)𝑢 <Q (𝑧 +Q 𝑣))
2423ad2ant2r 493 . . . . . 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 500 . . . . . . . . . . . . 13 ((((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣Q ∧ (𝑡 +Q 𝑣) = 𝑤)) → 𝐴<P 𝐵)
2625ad2antrr 472 . . . . . . . . . . . 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 6968 . . . . . . . . . . . . . 14 ((𝐴P𝐵P) → (𝐴<P 𝐵 ↔ ∃𝑞Q (𝑞 ∈ (2nd𝐴) ∧ 𝑞 ∈ (1st𝐵))))
2827biimpd 142 . . . . . . . . . . . . 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 270 . . . . . . . . . . . . . 14 (((((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) → 𝐴<P 𝐵)
3231ad2antrr 472 . . . . . . . . . . . . 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 502 . . . . . . . . . . . . . 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 270 . . . . . . . . . . . . 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 506 . . . . . . . . . . . . 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 6949 . . . . . . . . . . . . . 14 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑧 ∈ (1st𝐴) ∧ 𝑞 ∈ (2nd𝐴)) → 𝑧 <Q 𝑞)
3720, 36syl3an1 1203 . . . . . . . . . . . . 13 ((𝐴<P 𝐵𝑧 ∈ (1st𝐴) ∧ 𝑞 ∈ (2nd𝐴)) → 𝑧 <Q 𝑞)
3832, 34, 35, 37syl3anc 1170 . . . . . . . . . . . 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 507 . . . . . . . . . . . . 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 502 . . . . . . . . . . . . . . 15 ((((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣Q ∧ (𝑡 +Q 𝑣) = 𝑤)) → 𝑡 ∈ (2nd𝐵))
4140adantr 270 . . . . . . . . . . . . . 14 (((((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) → 𝑡 ∈ (2nd𝐵))
4241ad2antrr 472 . . . . . . . . . . . . 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 6949 . . . . . . . . . . . . . 14 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑞 ∈ (1st𝐵) ∧ 𝑡 ∈ (2nd𝐵)) → 𝑞 <Q 𝑡)
445, 43syl3an1 1203 . . . . . . . . . . . . 13 ((𝐴<P 𝐵𝑞 ∈ (1st𝐵) ∧ 𝑡 ∈ (2nd𝐵)) → 𝑞 <Q 𝑡)
4532, 39, 42, 44syl3anc 1170 . . . . . . . . . . . 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 6860 . . . . . . . . . . . . 13 <Q Or Q
47 ltrelnq 6827 . . . . . . . . . . . . 13 <Q ⊆ (Q × Q)
4846, 47sotri 4782 . . . . . . . . . . . 12 ((𝑧 <Q 𝑞𝑞 <Q 𝑡) → 𝑧 <Q 𝑡)
4938, 45, 48syl2anc 403 . . . . . . . . . . 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 2487 . . . . . . . . . 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 6871 . . . . . . . . . 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 503 . . . . . . . . . . . 12 (((((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) → (𝑡 +Q 𝑣) = 𝑤)
5453ad2antrr 472 . . . . . . . . . . 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 499 . . . . . . . . . . . 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 5598 . . . . . . . . . . . . 13 ((𝑧 +Q 𝑠) = 𝑡 → ((𝑧 +Q 𝑠) +Q 𝑣) = (𝑡 +Q 𝑣))
5756eqeq1d 2091 . . . . . . . . . . . 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 165 . . . . . . . . . 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 6943 . . . . . . . . . . . . . . . . 17 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑧 ∈ (1st𝐴)) → 𝑧Q)
6120, 60sylan 277 . . . . . . . . . . . . . . . 16 ((𝐴<P 𝐵𝑧 ∈ (1st𝐴)) → 𝑧Q)
6261adantlr 461 . . . . . . . . . . . . . . 15 (((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ 𝑧 ∈ (1st𝐴)) → 𝑧Q)
6362ad2ant2r 493 . . . . . . . . . . . . . 14 ((((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) → 𝑧Q)
6463adantlr 461 . . . . . . . . . . . . 13 (((((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) → 𝑧Q)
6564ad2antrr 472 . . . . . . . . . . . 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 502 . . . . . . . . . . . . 13 (((((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) → 𝑣Q)
6766ad2antrr 472 . . . . . . . . . . . 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 498 . . . . . . . . . . . 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 6843 . . . . . . . . . . . . 13 ((𝑓Q𝑔Q) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
7069adantl 271 . . . . . . . . . . . 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 6844 . . . . . . . . . . . . 13 ((𝑓Q𝑔QQ) → ((𝑓 +Q 𝑔) +Q ) = (𝑓 +Q (𝑔 +Q )))
7271adantl 271 . . . . . . . . . . . 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 5760 . . . . . . . . . . 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 108 . . . . . . . . . . . . . 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 503 . . . . . . . . . . . . . . 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 6947 . . . . . . . . . . . . . . . 16 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑢 ∈ (2nd𝐴)) → (𝑢 <Q (𝑧 +Q 𝑣) → (𝑧 +Q 𝑣) ∈ (2nd𝐴)))
7720, 76sylan 277 . . . . . . . . . . . . . . 15 ((𝐴<P 𝐵𝑢 ∈ (2nd𝐴)) → (𝑢 <Q (𝑧 +Q 𝑣) → (𝑧 +Q 𝑣) ∈ (2nd𝐴)))
7826, 75, 77syl2anc 403 . . . . . . . . . . . . . 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 270 . . . . . . . . . . . 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 270 . . . . . . . . . . . . . 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 472 . . . . . . . . . . . . . . 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 2159 . . . . . . . . . . . . . 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 2145 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑧 → (𝑦 ∈ (1st𝐴) ↔ 𝑧 ∈ (1st𝐴)))
85 oveq1 5598 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑧 → (𝑦 +Q 𝑠) = (𝑧 +Q 𝑠))
8685eleq1d 2151 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑧 → ((𝑦 +Q 𝑠) ∈ (2nd𝐵) ↔ (𝑧 +Q 𝑠) ∈ (2nd𝐵)))
8784, 86anbi12d 457 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑧 → ((𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑠) ∈ (2nd𝐵)) ↔ (𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑠) ∈ (2nd𝐵))))
8887spcegv 2697 . . . . . . . . . . . . . . 15 (𝑧 ∈ (1st𝐴) → ((𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑠) ∈ (2nd𝐵)) → ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑠) ∈ (2nd𝐵))))
8988anabsi5 544 . . . . . . . . . . . . . 14 ((𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑠) ∈ (2nd𝐵)) → ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑠) ∈ (2nd𝐵)))
9081, 83, 89syl2anc 403 . . . . . . . . . . . . 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 7061 . . . . . . . . . . . . 13 (𝑠 ∈ (2nd𝐶) ↔ (𝑠Q ∧ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑠) ∈ (2nd𝐵))))
9368, 90, 92sylanbrc 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)) → 𝑠 ∈ (2nd𝐶))
9431ad2antrr 472 . . . . . . . . . . . . . 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 7070 . . . . . . . . . . . . . 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 6930 . . . . . . . . . . . . . 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 6837 . . . . . . . . . . . . . 14 ((𝑓Q𝑣Q) → (𝑓 +Q 𝑣) ∈ Q)
10098, 99genppreclu 6977 . . . . . . . . . . . . 13 ((𝐴P𝐶P) → (((𝑧 +Q 𝑣) ∈ (2nd𝐴) ∧ 𝑠 ∈ (2nd𝐶)) → ((𝑧 +Q 𝑣) +Q 𝑠) ∈ (2nd ‘(𝐴 +P 𝐶))))
10195, 97, 100syl2anc 403 . . . . . . . . . . . 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 424 . . . . . . . . . . 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 2160 . . . . . . . . . 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 2160 . . . . . . . . 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 2487 . . . . . . . 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 113 . . . . . . 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 2490 . . . . . 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 2487 . . . 4 (((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) → 𝑤 ∈ (2nd ‘(𝐴 +P 𝐶)))
1107, 109rexlimddv 2487 . . 3 ((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) → 𝑤 ∈ (2nd ‘(𝐴 +P 𝐶)))
111110ex 113 . 2 (𝐴<P 𝐵 → (𝑤 ∈ (2nd𝐵) → 𝑤 ∈ (2nd ‘(𝐴 +P 𝐶))))
112111ssrdv 3016 1 (𝐴<P 𝐵 → (2nd𝐵) ⊆ (2nd ‘(𝐴 +P 𝐶)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  w3a 920   = wceq 1285  wex 1422  wcel 1434  wrex 2354  {crab 2357  wss 2984  cop 3425   class class class wbr 3811  cfv 4969  (class class class)co 5591  1st c1st 5844  2nd c2nd 5845  Qcnq 6742   +Q cplq 6744   <Q cltq 6747  Pcnp 6753   +P cpp 6755  <P cltp 6757
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 4000  ax-un 4224  ax-setind 4316  ax-iinf 4366
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 4080  df-id 4084  df-po 4087  df-iso 4088  df-iord 4157  df-on 4159  df-suc 4162  df-iom 4369  df-xp 4407  df-rel 4408  df-cnv 4409  df-co 4410  df-dm 4411  df-rn 4412  df-res 4413  df-ima 4414  df-iota 4934  df-fun 4971  df-fn 4972  df-f 4973  df-f1 4974  df-fo 4975  df-f1o 4976  df-fv 4977  df-ov 5594  df-oprab 5595  df-mpt2 5596  df-1st 5846  df-2nd 5847  df-recs 6002  df-irdg 6067  df-1o 6113  df-2o 6114  df-oadd 6117  df-omul 6118  df-er 6222  df-ec 6224  df-qs 6228  df-ni 6766  df-pli 6767  df-mi 6768  df-lti 6769  df-plpq 6806  df-mpq 6807  df-enq 6809  df-nqqs 6810  df-plqqs 6811  df-mqqs 6812  df-1nqqs 6813  df-rq 6814  df-ltnqqs 6815  df-enq0 6886  df-nq0 6887  df-0nq0 6888  df-plq0 6889  df-mq0 6890  df-inp 6928  df-iplp 6930  df-iltp 6932
This theorem is referenced by:  ltexpri  7075
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