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Theorem ltexprlemru 7533
Description: Lemma for ltexpri 7534. 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 7426 . . . . . . . 8 <P ⊆ (P × P)
21brel 4639 . . . . . . 7 (𝐴<P 𝐵 → (𝐴P𝐵P))
32simprd 113 . . . . . 6 (𝐴<P 𝐵𝐵P)
4 prop 7396 . . . . . 6 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
53, 4syl 14 . . . . 5 (𝐴<P 𝐵 → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
6 prnminu 7410 . . . . 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 522 . . . . . 6 (((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) → 𝑡 <Q 𝑤)
9 elprnqu 7403 . . . . . . . . 9 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑡 ∈ (2nd𝐵)) → 𝑡Q)
105, 9sylan 281 . . . . . . . 8 ((𝐴<P 𝐵𝑡 ∈ (2nd𝐵)) → 𝑡Q)
1110ad2ant2r 501 . . . . . . 7 (((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) → 𝑡Q)
12 elprnqu 7403 . . . . . . . . 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 7329 . . . . . . 7 ((𝑡Q𝑤Q) → (𝑡 <Q 𝑤 ↔ ∃𝑣Q (𝑡 +Q 𝑣) = 𝑤))
1611, 14, 15syl2anc 409 . . . . . 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 7396 . . . . . . . . . 10 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
2018, 19syl 14 . . . . . . . . 9 (𝐴<P 𝐵 → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
21 prarloc 7424 . . . . . . . . 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 469 . . . . . . 7 (((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ 𝑣Q) → ∃𝑧 ∈ (1st𝐴)∃𝑢 ∈ (2nd𝐴)𝑢 <Q (𝑧 +Q 𝑣))
2423ad2ant2r 501 . . . . . 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 523 . . . . . . . . . . . . 13 ((((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣Q ∧ (𝑡 +Q 𝑣) = 𝑤)) → 𝐴<P 𝐵)
2625ad2antrr 480 . . . . . . . . . . . 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 7427 . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . 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 525 . . . . . . . . . . . . . 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 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 ∈ (2nd𝐴))
36 prltlu 7408 . . . . . . . . . . . . . 14 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑧 ∈ (1st𝐴) ∧ 𝑞 ∈ (2nd𝐴)) → 𝑧 <Q 𝑞)
3720, 36syl3an1 1253 . . . . . . . . . . . . 13 ((𝐴<P 𝐵𝑧 ∈ (1st𝐴) ∧ 𝑞 ∈ (2nd𝐴)) → 𝑧 <Q 𝑞)
3832, 34, 35, 37syl3anc 1220 . . . . . . . . . . . 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 530 . . . . . . . . . . . . 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 525 . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . 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 7408 . . . . . . . . . . . . . 14 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑞 ∈ (1st𝐵) ∧ 𝑡 ∈ (2nd𝐵)) → 𝑞 <Q 𝑡)
445, 43syl3an1 1253 . . . . . . . . . . . . 13 ((𝐴<P 𝐵𝑞 ∈ (1st𝐵) ∧ 𝑡 ∈ (2nd𝐵)) → 𝑞 <Q 𝑡)
4532, 39, 42, 44syl3anc 1220 . . . . . . . . . . . 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 7319 . . . . . . . . . . . . 13 <Q Or Q
47 ltrelnq 7286 . . . . . . . . . . . . 13 <Q ⊆ (Q × Q)
4846, 47sotri 4982 . . . . . . . . . . . 12 ((𝑧 <Q 𝑞𝑞 <Q 𝑡) → 𝑧 <Q 𝑡)
4938, 45, 48syl2anc 409 . . . . . . . . . . 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 2579 . . . . . . . . . 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 7330 . . . . . . . . . 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 526 . . . . . . . . . . . 12 (((((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) → (𝑡 +Q 𝑣) = 𝑤)
5453ad2antrr 480 . . . . . . . . . . 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 522 . . . . . . . . . . . 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 5832 . . . . . . . . . . . . 13 ((𝑧 +Q 𝑠) = 𝑡 → ((𝑧 +Q 𝑠) +Q 𝑣) = (𝑡 +Q 𝑣))
5756eqeq1d 2166 . . . . . . . . . . . 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 7402 . . . . . . . . . . . . . . . . 17 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑧 ∈ (1st𝐴)) → 𝑧Q)
6120, 60sylan 281 . . . . . . . . . . . . . . . 16 ((𝐴<P 𝐵𝑧 ∈ (1st𝐴)) → 𝑧Q)
6261adantlr 469 . . . . . . . . . . . . . . 15 (((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ 𝑧 ∈ (1st𝐴)) → 𝑧Q)
6362ad2ant2r 501 . . . . . . . . . . . . . 14 ((((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) → 𝑧Q)
6463adantlr 469 . . . . . . . . . . . . 13 (((((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) → 𝑧Q)
6564ad2antrr 480 . . . . . . . . . . . 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 525 . . . . . . . . . . . . 13 (((((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) ∧ (𝑣Q ∧ (𝑡 +Q 𝑣) = 𝑤)) ∧ (𝑧 ∈ (1st𝐴) ∧ 𝑢 ∈ (2nd𝐴))) → 𝑣Q)
6766ad2antrr 480 . . . . . . . . . . . 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 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)) → 𝑠Q)
69 addcomnqg 7302 . . . . . . . . . . . . 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 7303 . . . . . . . . . . . . 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 6002 . . . . . . . . . . 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 526 . . . . . . . . . . . . . . 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 7406 . . . . . . . . . . . . . . . 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 409 . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . 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 2234 . . . . . . . . . . . . . 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 2220 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑧 → (𝑦 ∈ (1st𝐴) ↔ 𝑧 ∈ (1st𝐴)))
85 oveq1 5832 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑧 → (𝑦 +Q 𝑠) = (𝑧 +Q 𝑠))
8685eleq1d 2226 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑧 → ((𝑦 +Q 𝑠) ∈ (2nd𝐵) ↔ (𝑧 +Q 𝑠) ∈ (2nd𝐵)))
8784, 86anbi12d 465 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑧 → ((𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑠) ∈ (2nd𝐵)) ↔ (𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑠) ∈ (2nd𝐵))))
8887spcegv 2800 . . . . . . . . . . . . . . 15 (𝑧 ∈ (1st𝐴) → ((𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑠) ∈ (2nd𝐵)) → ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑠) ∈ (2nd𝐵))))
8988anabsi5 569 . . . . . . . . . . . . . 14 ((𝑧 ∈ (1st𝐴) ∧ (𝑧 +Q 𝑠) ∈ (2nd𝐵)) → ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑠) ∈ (2nd𝐵)))
9081, 83, 89syl2anc 409 . . . . . . . . . . . . 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 7520 . . . . . . . . . . . . 13 (𝑠 ∈ (2nd𝐶) ↔ (𝑠Q ∧ ∃𝑦(𝑦 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑠) ∈ (2nd𝐵))))
9368, 90, 92sylanbrc 414 . . . . . . . . . . . 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 480 . . . . . . . . . . . . . 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 7529 . . . . . . . . . . . . . 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 7389 . . . . . . . . . . . . . 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 7296 . . . . . . . . . . . . . 14 ((𝑓Q𝑣Q) → (𝑓 +Q 𝑣) ∈ Q)
10098, 99genppreclu 7436 . . . . . . . . . . . . 13 ((𝐴P𝐶P) → (((𝑧 +Q 𝑣) ∈ (2nd𝐴) ∧ 𝑠 ∈ (2nd𝐶)) → ((𝑧 +Q 𝑣) +Q 𝑠) ∈ (2nd ‘(𝐴 +P 𝐶))))
10195, 97, 100syl2anc 409 . . . . . . . . . . . 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 430 . . . . . . . . . . 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 2235 . . . . . . . . . 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 2235 . . . . . . . . 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 2579 . . . . . . . 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 2582 . . . . . 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 2579 . . . 4 (((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) ∧ (𝑡 ∈ (2nd𝐵) ∧ 𝑡 <Q 𝑤)) → 𝑤 ∈ (2nd ‘(𝐴 +P 𝐶)))
1107, 109rexlimddv 2579 . . 3 ((𝐴<P 𝐵𝑤 ∈ (2nd𝐵)) → 𝑤 ∈ (2nd ‘(𝐴 +P 𝐶)))
111110ex 114 . 2 (𝐴<P 𝐵 → (𝑤 ∈ (2nd𝐵) → 𝑤 ∈ (2nd ‘(𝐴 +P 𝐶))))
112111ssrdv 3134 1 (𝐴<P 𝐵 → (2nd𝐵) ⊆ (2nd ‘(𝐴 +P 𝐶)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  w3a 963   = wceq 1335  wex 1472  wcel 2128  wrex 2436  {crab 2439  wss 3102  cop 3563   class class class wbr 3966  cfv 5171  (class class class)co 5825  1st c1st 6087  2nd c2nd 6088  Qcnq 7201   +Q cplq 7203   <Q cltq 7206  Pcnp 7212   +P cpp 7214  <P cltp 7216
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4080  ax-sep 4083  ax-nul 4091  ax-pow 4136  ax-pr 4170  ax-un 4394  ax-setind 4497  ax-iinf 4548
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3774  df-int 3809  df-iun 3852  df-br 3967  df-opab 4027  df-mpt 4028  df-tr 4064  df-eprel 4250  df-id 4254  df-po 4257  df-iso 4258  df-iord 4327  df-on 4329  df-suc 4332  df-iom 4551  df-xp 4593  df-rel 4594  df-cnv 4595  df-co 4596  df-dm 4597  df-rn 4598  df-res 4599  df-ima 4600  df-iota 5136  df-fun 5173  df-fn 5174  df-f 5175  df-f1 5176  df-fo 5177  df-f1o 5178  df-fv 5179  df-ov 5828  df-oprab 5829  df-mpo 5830  df-1st 6089  df-2nd 6090  df-recs 6253  df-irdg 6318  df-1o 6364  df-2o 6365  df-oadd 6368  df-omul 6369  df-er 6481  df-ec 6483  df-qs 6487  df-ni 7225  df-pli 7226  df-mi 7227  df-lti 7228  df-plpq 7265  df-mpq 7266  df-enq 7268  df-nqqs 7269  df-plqqs 7270  df-mqqs 7271  df-1nqqs 7272  df-rq 7273  df-ltnqqs 7274  df-enq0 7345  df-nq0 7346  df-0nq0 7347  df-plq0 7348  df-mq0 7349  df-inp 7387  df-iplp 7389  df-iltp 7391
This theorem is referenced by:  ltexpri  7534
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