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Theorem ltexprlemru 7602
Description: Lemma for ltexpri 7603. 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 7495 . . . . . . . 8 <P ⊆ (P × P)
21brel 4675 . . . . . . 7 (𝐴<P 𝐵 → (𝐴P𝐵P))
32simprd 114 . . . . . 6 (𝐴<P 𝐵𝐵P)
4 prop 7465 . . . . . 6 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
53, 4syl 14 . . . . 5 (𝐴<P 𝐵 → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
6 prnminu 7479 . . . . 5 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑤 ∈ (2nd𝐵)) → ∃𝑡 ∈ (2nd𝐵)𝑡 <Q 𝑤)
75, 6sylan 283 . . . 4 ((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) → ∃𝑡 ∈ (2nd𝐵)𝑡 <Q 𝑤)
8 simprr 531 . . . . . 6 (((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) → 𝑡 <Q 𝑤)
9 elprnqu 7472 . . . . . . . . 9 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑡 ∈ (2nd𝐵)) → 𝑡Q)
105, 9sylan 283 . . . . . . . 8 ((𝐴<P 𝐵𝑡 ∈ (2nd𝐵)) → 𝑡Q)
1110ad2ant2r 509 . . . . . . 7 (((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) → 𝑡Q)
12 elprnqu 7472 . . . . . . . . 9 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑤 ∈ (2nd𝐵)) → 𝑤Q)
135, 12sylan 283 . . . . . . . 8 ((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) → 𝑤Q)
1413adantr 276 . . . . . . 7 (((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) → 𝑤Q)
15 ltexnqq 7398 . . . . . . 7 ((𝑡Q𝑤Q) → (𝑡 <Q 𝑤 ↔ ∃𝑣Q (𝑡 +Q 𝑣) = 𝑤))
1611, 14, 15syl2anc 411 . . . . . 6 (((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) → (𝑡 <Q 𝑤 ↔ ∃𝑣Q (𝑡 +Q 𝑣) = 𝑤))
178, 16mpbid 147 . . . . 5 (((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) → ∃𝑣Q (𝑡 +Q 𝑣) = 𝑤)
182simpld 112 . . . . . . . . . 10 (𝐴<P 𝐵𝐴P)
19 prop 7465 . . . . . . . . . 10 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
2018, 19syl 14 . . . . . . . . 9 (𝐴<P 𝐵 → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
21 prarloc 7493 . . . . . . . . 9 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑣Q) → ∃𝑧 ∈ (1st𝐴)∃𝑢 ∈ (2nd𝐴)𝑢 <Q (𝑧 +Q 𝑣))
2220, 21sylan 283 . . . . . . . 8 ((𝐴<P 𝐵𝑣Q) → ∃𝑧 ∈ (1st𝐴)∃𝑢 ∈ (2nd𝐴)𝑢 <Q (𝑧 +Q 𝑣))
2322adantlr 477 . . . . . . 7 (((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ 𝑣Q) → ∃𝑧 ∈ (1st𝐴)∃𝑢 ∈ (2nd𝐴)𝑢 <Q (𝑧 +Q 𝑣))
2423ad2ant2r 509 . . . . . 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 533 . . . . . . . . . . . . 13 ((((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣Q ∧ (𝑡 +Q 𝑣) = 𝑤)) → 𝐴<P 𝐵)
2625ad2antrr 488 . . . . . . . . . . . 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 7496 . . . . . . . . . . . . . 14 ((𝐴P𝐵P) → (𝐴<P 𝐵 ↔ ∃𝑞Q (𝑞 ∈ (2nd𝐴) ∧ 𝑞 ∈ (1st𝐵))))
2827biimpd 144 . . . . . . . . . . . . 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 276 . . . . . . . . . . . . . 14 (((((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) → 𝐴<P 𝐵)
3231ad2antrr 488 . . . . . . . . . . . . 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 535 . . . . . . . . . . . . . 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 276 . . . . . . . . . . . . 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 539 . . . . . . . . . . . . 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 7477 . . . . . . . . . . . . . 14 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑧 ∈ (1st𝐴) ∧ 𝑞 ∈ (2nd𝐴)) → 𝑧 <Q 𝑞)
3720, 36syl3an1 1271 . . . . . . . . . . . . 13 ((𝐴<P 𝐵𝑧 ∈ (1st𝐴) ∧ 𝑞 ∈ (2nd𝐴)) → 𝑧 <Q 𝑞)
3832, 34, 35, 37syl3anc 1238 . . . . . . . . . . . 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 540 . . . . . . . . . . . . 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 535 . . . . . . . . . . . . . . 15 ((((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣Q ∧ (𝑡 +Q 𝑣) = 𝑤)) → 𝑡 ∈ (2nd𝐵))
4140adantr 276 . . . . . . . . . . . . . 14 (((((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) → 𝑡 ∈ (2nd𝐵))
4241ad2antrr 488 . . . . . . . . . . . . 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 7477 . . . . . . . . . . . . . 14 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑞 ∈ (1st𝐵) ∧ 𝑡 ∈ (2nd𝐵)) → 𝑞 <Q 𝑡)
445, 43syl3an1 1271 . . . . . . . . . . . . 13 ((𝐴<P 𝐵𝑞 ∈ (1st𝐵) ∧ 𝑡 ∈ (2nd𝐵)) → 𝑞 <Q 𝑡)
4532, 39, 42, 44syl3anc 1238 . . . . . . . . . . . 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 7388 . . . . . . . . . . . . 13 <Q Or Q
47 ltrelnq 7355 . . . . . . . . . . . . 13 <Q ⊆ (Q × Q)
4846, 47sotri 5020 . . . . . . . . . . . 12 ((𝑧 <Q 𝑞𝑞 <Q 𝑡) → 𝑧 <Q 𝑡)
4938, 45, 48syl2anc 411 . . . . . . . . . . 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 2599 . . . . . . . . . 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 7399 . . . . . . . . . 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 536 . . . . . . . . . . . 12 (((((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) → (𝑡 +Q 𝑣) = 𝑤)
5453ad2antrr 488 . . . . . . . . . . 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 531 . . . . . . . . . . . 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 5876 . . . . . . . . . . . . 13 ((𝑧 +Q 𝑠) = 𝑡 → ((𝑧 +Q 𝑠) +Q 𝑣) = (𝑡 +Q 𝑣))
5756eqeq1d 2186 . . . . . . . . . . . 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 167 . . . . . . . . . 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 7471 . . . . . . . . . . . . . . . . 17 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑧 ∈ (1st𝐴)) → 𝑧Q)
6120, 60sylan 283 . . . . . . . . . . . . . . . 16 ((𝐴<P 𝐵𝑧 ∈ (1st𝐴)) → 𝑧Q)
6261adantlr 477 . . . . . . . . . . . . . . 15 (((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ 𝑧 ∈ (1st𝐴)) → 𝑧Q)
6362ad2ant2r 509 . . . . . . . . . . . . . 14 ((((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) → 𝑧Q)
6463adantlr 477 . . . . . . . . . . . . 13 (((((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) → 𝑧Q)
6564ad2antrr 488 . . . . . . . . . . . 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 535 . . . . . . . . . . . . 13 (((((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) → 𝑣Q)
6766ad2antrr 488 . . . . . . . . . . . 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 529 . . . . . . . . . . . 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 7371 . . . . . . . . . . . . 13 ((𝑓Q𝑔Q) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
7069adantl 277 . . . . . . . . . . . 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 7372 . . . . . . . . . . . . 13 ((𝑓Q𝑔QQ) → ((𝑓 +Q 𝑔) +Q ) = (𝑓 +Q (𝑔 +Q )))
7271adantl 277 . . . . . . . . . . . 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 6049 . . . . . . . . . . 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 110 . . . . . . . . . . . . . 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 536 . . . . . . . . . . . . . . 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 7475 . . . . . . . . . . . . . . . 16 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑢 ∈ (2nd𝐴)) → (𝑢 <Q (𝑧 +Q 𝑣) → (𝑧 +Q 𝑣) ∈ (2nd𝐴)))
7720, 76sylan 283 . . . . . . . . . . . . . . 15 ((𝐴<P 𝐵𝑢 ∈ (2nd𝐴)) → (𝑢 <Q (𝑧 +Q 𝑣) → (𝑧 +Q 𝑣) ∈ (2nd𝐴)))
7826, 75, 77syl2anc 411 . . . . . . . . . . . . . 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 276 . . . . . . . . . . . 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 276 . . . . . . . . . . . . . 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 488 . . . . . . . . . . . . . . 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 2254 . . . . . . . . . . . . . 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 2240 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑧 → (𝑦 ∈ (1st𝐴) ↔ 𝑧 ∈ (1st𝐴)))
85 oveq1 5876 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑧 → (𝑦 +Q 𝑠) = (𝑧 +Q 𝑠))
8685eleq1d 2246 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑧 → ((𝑦 +Q 𝑠) ∈ (2nd𝐵) ↔ (𝑧 +Q 𝑠) ∈ (2nd𝐵)))
8784, 86anbi12d 473 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑧 → ((𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑠) ∈ (2nd𝐵)) ↔ (𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑠) ∈ (2nd𝐵))))
8887spcegv 2825 . . . . . . . . . . . . . . 15 (𝑧 ∈ (1st𝐴) → ((𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑠) ∈ (2nd𝐵)) → ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑠) ∈ (2nd𝐵))))
8988anabsi5 579 . . . . . . . . . . . . . 14 ((𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑠) ∈ (2nd𝐵)) → ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑠) ∈ (2nd𝐵)))
9081, 83, 89syl2anc 411 . . . . . . . . . . . . 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 7589 . . . . . . . . . . . . 13 (𝑠 ∈ (2nd𝐶) ↔ (𝑠Q ∧ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑠) ∈ (2nd𝐵))))
9368, 90, 92sylanbrc 417 . . . . . . . . . . . 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 488 . . . . . . . . . . . . . 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 7598 . . . . . . . . . . . . . 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 7458 . . . . . . . . . . . . . 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 7365 . . . . . . . . . . . . . 14 ((𝑓Q𝑣Q) → (𝑓 +Q 𝑣) ∈ Q)
10098, 99genppreclu 7505 . . . . . . . . . . . . 13 ((𝐴P𝐶P) → (((𝑧 +Q 𝑣) ∈ (2nd𝐴) ∧ 𝑠 ∈ (2nd𝐶)) → ((𝑧 +Q 𝑣) +Q 𝑠) ∈ (2nd ‘(𝐴 +P 𝐶))))
10195, 97, 100syl2anc 411 . . . . . . . . . . . 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 433 . . . . . . . . . . 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 2255 . . . . . . . . . 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 2255 . . . . . . . . 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 2599 . . . . . . . 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 115 . . . . . . 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 2602 . . . . . 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 2599 . . . 4 (((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) → 𝑤 ∈ (2nd ‘(𝐴 +P 𝐶)))
1107, 109rexlimddv 2599 . . 3 ((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) → 𝑤 ∈ (2nd ‘(𝐴 +P 𝐶)))
111110ex 115 . 2 (𝐴<P 𝐵 → (𝑤 ∈ (2nd𝐵) → 𝑤 ∈ (2nd ‘(𝐴 +P 𝐶))))
112111ssrdv 3161 1 (𝐴<P 𝐵 → (2nd𝐵) ⊆ (2nd ‘(𝐴 +P 𝐶)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  w3a 978   = wceq 1353  wex 1492  wcel 2148  wrex 2456  {crab 2459  wss 3129  cop 3594   class class class wbr 4000  cfv 5212  (class class class)co 5869  1st c1st 6133  2nd c2nd 6134  Qcnq 7270   +Q cplq 7272   <Q cltq 7275  Pcnp 7281   +P cpp 7283  <P cltp 7285
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-eprel 4286  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-irdg 6365  df-1o 6411  df-2o 6412  df-oadd 6415  df-omul 6416  df-er 6529  df-ec 6531  df-qs 6535  df-ni 7294  df-pli 7295  df-mi 7296  df-lti 7297  df-plpq 7334  df-mpq 7335  df-enq 7337  df-nqqs 7338  df-plqqs 7339  df-mqqs 7340  df-1nqqs 7341  df-rq 7342  df-ltnqqs 7343  df-enq0 7414  df-nq0 7415  df-0nq0 7416  df-plq0 7417  df-mq0 7418  df-inp 7456  df-iplp 7458  df-iltp 7460
This theorem is referenced by:  ltexpri  7603
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