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Theorem cauappcvgprlem1 6900
Description: Lemma for cauappcvgpr 6903. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 23-Jun-2020.)
Hypotheses
Ref Expression
cauappcvgpr.f (𝜑𝐹:QQ)
cauappcvgpr.app (𝜑 → ∀𝑝Q𝑞Q ((𝐹𝑝) <Q ((𝐹𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹𝑞) <Q ((𝐹𝑝) +Q (𝑝 +Q 𝑞))))
cauappcvgpr.bnd (𝜑 → ∀𝑝Q 𝐴 <Q (𝐹𝑝))
cauappcvgpr.lim 𝐿 = ⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢}⟩
cauappcvgprlem.q (𝜑𝑄Q)
cauappcvgprlem.r (𝜑𝑅Q)
Assertion
Ref Expression
cauappcvgprlem1 (𝜑 → ⟨{𝑙𝑙 <Q (𝐹𝑄)}, {𝑢 ∣ (𝐹𝑄) <Q 𝑢}⟩<P (𝐿 +P ⟨{𝑙𝑙 <Q (𝑄 +Q 𝑅)}, {𝑢 ∣ (𝑄 +Q 𝑅) <Q 𝑢}⟩))
Distinct variable groups:   𝐴,𝑝   𝐿,𝑝,𝑞   𝜑,𝑝,𝑞   𝐹,𝑝,𝑞,𝑙,𝑢   𝑄,𝑝,𝑞,𝑙,𝑢   𝑅,𝑝,𝑞,𝑙,𝑢
Allowed substitution hints:   𝜑(𝑢,𝑙)   𝐴(𝑢,𝑞,𝑙)   𝐿(𝑢,𝑙)

Proof of Theorem cauappcvgprlem1
Dummy variables 𝑓 𝑔 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cauappcvgprlem.r . . . . 5 (𝜑𝑅Q)
2 halfnqq 6651 . . . . 5 (𝑅Q → ∃𝑥Q (𝑥 +Q 𝑥) = 𝑅)
31, 2syl 14 . . . 4 (𝜑 → ∃𝑥Q (𝑥 +Q 𝑥) = 𝑅)
4 simprl 498 . . . . 5 ((𝜑 ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → 𝑥Q)
5 cauappcvgpr.app . . . . . . . . . . 11 (𝜑 → ∀𝑝Q𝑞Q ((𝐹𝑝) <Q ((𝐹𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹𝑞) <Q ((𝐹𝑝) +Q (𝑝 +Q 𝑞))))
65adantr 270 . . . . . . . . . 10 ((𝜑 ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → ∀𝑝Q𝑞Q ((𝐹𝑝) <Q ((𝐹𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹𝑞) <Q ((𝐹𝑝) +Q (𝑝 +Q 𝑞))))
7 cauappcvgprlem.q . . . . . . . . . . . 12 (𝜑𝑄Q)
87adantr 270 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → 𝑄Q)
9 fveq2 5203 . . . . . . . . . . . . . 14 (𝑝 = 𝑄 → (𝐹𝑝) = (𝐹𝑄))
10 oveq1 5544 . . . . . . . . . . . . . . 15 (𝑝 = 𝑄 → (𝑝 +Q 𝑞) = (𝑄 +Q 𝑞))
1110oveq2d 5553 . . . . . . . . . . . . . 14 (𝑝 = 𝑄 → ((𝐹𝑞) +Q (𝑝 +Q 𝑞)) = ((𝐹𝑞) +Q (𝑄 +Q 𝑞)))
129, 11breq12d 3800 . . . . . . . . . . . . 13 (𝑝 = 𝑄 → ((𝐹𝑝) <Q ((𝐹𝑞) +Q (𝑝 +Q 𝑞)) ↔ (𝐹𝑄) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑞))))
139, 10oveq12d 5555 . . . . . . . . . . . . . 14 (𝑝 = 𝑄 → ((𝐹𝑝) +Q (𝑝 +Q 𝑞)) = ((𝐹𝑄) +Q (𝑄 +Q 𝑞)))
1413breq2d 3799 . . . . . . . . . . . . 13 (𝑝 = 𝑄 → ((𝐹𝑞) <Q ((𝐹𝑝) +Q (𝑝 +Q 𝑞)) ↔ (𝐹𝑞) <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑞))))
1512, 14anbi12d 457 . . . . . . . . . . . 12 (𝑝 = 𝑄 → (((𝐹𝑝) <Q ((𝐹𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹𝑞) <Q ((𝐹𝑝) +Q (𝑝 +Q 𝑞))) ↔ ((𝐹𝑄) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑞)) ∧ (𝐹𝑞) <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑞)))))
16 fveq2 5203 . . . . . . . . . . . . . . 15 (𝑞 = 𝑥 → (𝐹𝑞) = (𝐹𝑥))
17 oveq2 5545 . . . . . . . . . . . . . . 15 (𝑞 = 𝑥 → (𝑄 +Q 𝑞) = (𝑄 +Q 𝑥))
1816, 17oveq12d 5555 . . . . . . . . . . . . . 14 (𝑞 = 𝑥 → ((𝐹𝑞) +Q (𝑄 +Q 𝑞)) = ((𝐹𝑥) +Q (𝑄 +Q 𝑥)))
1918breq2d 3799 . . . . . . . . . . . . 13 (𝑞 = 𝑥 → ((𝐹𝑄) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑞)) ↔ (𝐹𝑄) <Q ((𝐹𝑥) +Q (𝑄 +Q 𝑥))))
2017oveq2d 5553 . . . . . . . . . . . . . 14 (𝑞 = 𝑥 → ((𝐹𝑄) +Q (𝑄 +Q 𝑞)) = ((𝐹𝑄) +Q (𝑄 +Q 𝑥)))
2116, 20breq12d 3800 . . . . . . . . . . . . 13 (𝑞 = 𝑥 → ((𝐹𝑞) <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑞)) ↔ (𝐹𝑥) <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑥))))
2219, 21anbi12d 457 . . . . . . . . . . . 12 (𝑞 = 𝑥 → (((𝐹𝑄) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑞)) ∧ (𝐹𝑞) <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑞))) ↔ ((𝐹𝑄) <Q ((𝐹𝑥) +Q (𝑄 +Q 𝑥)) ∧ (𝐹𝑥) <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑥)))))
2315, 22rspc2v 2714 . . . . . . . . . . 11 ((𝑄Q𝑥Q) → (∀𝑝Q𝑞Q ((𝐹𝑝) <Q ((𝐹𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹𝑞) <Q ((𝐹𝑝) +Q (𝑝 +Q 𝑞))) → ((𝐹𝑄) <Q ((𝐹𝑥) +Q (𝑄 +Q 𝑥)) ∧ (𝐹𝑥) <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑥)))))
248, 4, 23syl2anc 403 . . . . . . . . . 10 ((𝜑 ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → (∀𝑝Q𝑞Q ((𝐹𝑝) <Q ((𝐹𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹𝑞) <Q ((𝐹𝑝) +Q (𝑝 +Q 𝑞))) → ((𝐹𝑄) <Q ((𝐹𝑥) +Q (𝑄 +Q 𝑥)) ∧ (𝐹𝑥) <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑥)))))
256, 24mpd 13 . . . . . . . . 9 ((𝜑 ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → ((𝐹𝑄) <Q ((𝐹𝑥) +Q (𝑄 +Q 𝑥)) ∧ (𝐹𝑥) <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑥))))
2625simpld 110 . . . . . . . 8 ((𝜑 ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → (𝐹𝑄) <Q ((𝐹𝑥) +Q (𝑄 +Q 𝑥)))
27 cauappcvgpr.f . . . . . . . . . . 11 (𝜑𝐹:QQ)
2827adantr 270 . . . . . . . . . 10 ((𝜑 ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → 𝐹:QQ)
2928, 4ffvelrnd 5329 . . . . . . . . 9 ((𝜑 ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → (𝐹𝑥) ∈ Q)
30 addassnqg 6623 . . . . . . . . 9 (((𝐹𝑥) ∈ Q𝑄Q𝑥Q) → (((𝐹𝑥) +Q 𝑄) +Q 𝑥) = ((𝐹𝑥) +Q (𝑄 +Q 𝑥)))
3129, 8, 4, 30syl3anc 1170 . . . . . . . 8 ((𝜑 ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → (((𝐹𝑥) +Q 𝑄) +Q 𝑥) = ((𝐹𝑥) +Q (𝑄 +Q 𝑥)))
3226, 31breqtrrd 3813 . . . . . . 7 ((𝜑 ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → (𝐹𝑄) <Q (((𝐹𝑥) +Q 𝑄) +Q 𝑥))
33 ltanqg 6641 . . . . . . . . 9 ((𝑓Q𝑔QQ) → (𝑓 <Q 𝑔 ↔ ( +Q 𝑓) <Q ( +Q 𝑔)))
3433adantl 271 . . . . . . . 8 (((𝜑 ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑅)) ∧ (𝑓Q𝑔QQ)) → (𝑓 <Q 𝑔 ↔ ( +Q 𝑓) <Q ( +Q 𝑔)))
3527, 7ffvelrnd 5329 . . . . . . . . 9 (𝜑 → (𝐹𝑄) ∈ Q)
3635adantr 270 . . . . . . . 8 ((𝜑 ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → (𝐹𝑄) ∈ Q)
37 addclnq 6616 . . . . . . . . . 10 (((𝐹𝑥) ∈ Q𝑄Q) → ((𝐹𝑥) +Q 𝑄) ∈ Q)
3829, 8, 37syl2anc 403 . . . . . . . . 9 ((𝜑 ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → ((𝐹𝑥) +Q 𝑄) ∈ Q)
39 addclnq 6616 . . . . . . . . 9 ((((𝐹𝑥) +Q 𝑄) ∈ Q𝑥Q) → (((𝐹𝑥) +Q 𝑄) +Q 𝑥) ∈ Q)
4038, 4, 39syl2anc 403 . . . . . . . 8 ((𝜑 ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → (((𝐹𝑥) +Q 𝑄) +Q 𝑥) ∈ Q)
41 addcomnqg 6622 . . . . . . . . 9 ((𝑓Q𝑔Q) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
4241adantl 271 . . . . . . . 8 (((𝜑 ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑅)) ∧ (𝑓Q𝑔Q)) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
4334, 36, 40, 4, 42caovord2d 5695 . . . . . . 7 ((𝜑 ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → ((𝐹𝑄) <Q (((𝐹𝑥) +Q 𝑄) +Q 𝑥) ↔ ((𝐹𝑄) +Q 𝑥) <Q ((((𝐹𝑥) +Q 𝑄) +Q 𝑥) +Q 𝑥)))
4432, 43mpbid 145 . . . . . 6 ((𝜑 ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → ((𝐹𝑄) +Q 𝑥) <Q ((((𝐹𝑥) +Q 𝑄) +Q 𝑥) +Q 𝑥))
45 addassnqg 6623 . . . . . . . 8 ((((𝐹𝑥) +Q 𝑄) ∈ Q𝑥Q𝑥Q) → ((((𝐹𝑥) +Q 𝑄) +Q 𝑥) +Q 𝑥) = (((𝐹𝑥) +Q 𝑄) +Q (𝑥 +Q 𝑥)))
4638, 4, 4, 45syl3anc 1170 . . . . . . 7 ((𝜑 ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → ((((𝐹𝑥) +Q 𝑄) +Q 𝑥) +Q 𝑥) = (((𝐹𝑥) +Q 𝑄) +Q (𝑥 +Q 𝑥)))
47 simprr 499 . . . . . . . 8 ((𝜑 ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → (𝑥 +Q 𝑥) = 𝑅)
4847oveq2d 5553 . . . . . . 7 ((𝜑 ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → (((𝐹𝑥) +Q 𝑄) +Q (𝑥 +Q 𝑥)) = (((𝐹𝑥) +Q 𝑄) +Q 𝑅))
491adantr 270 . . . . . . . 8 ((𝜑 ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → 𝑅Q)
50 addassnqg 6623 . . . . . . . 8 (((𝐹𝑥) ∈ Q𝑄Q𝑅Q) → (((𝐹𝑥) +Q 𝑄) +Q 𝑅) = ((𝐹𝑥) +Q (𝑄 +Q 𝑅)))
5129, 8, 49, 50syl3anc 1170 . . . . . . 7 ((𝜑 ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → (((𝐹𝑥) +Q 𝑄) +Q 𝑅) = ((𝐹𝑥) +Q (𝑄 +Q 𝑅)))
5246, 48, 513eqtrd 2118 . . . . . 6 ((𝜑 ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → ((((𝐹𝑥) +Q 𝑄) +Q 𝑥) +Q 𝑥) = ((𝐹𝑥) +Q (𝑄 +Q 𝑅)))
5344, 52breqtrd 3811 . . . . 5 ((𝜑 ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → ((𝐹𝑄) +Q 𝑥) <Q ((𝐹𝑥) +Q (𝑄 +Q 𝑅)))
54 oveq2 5545 . . . . . . 7 (𝑞 = 𝑥 → ((𝐹𝑄) +Q 𝑞) = ((𝐹𝑄) +Q 𝑥))
5516oveq1d 5552 . . . . . . 7 (𝑞 = 𝑥 → ((𝐹𝑞) +Q (𝑄 +Q 𝑅)) = ((𝐹𝑥) +Q (𝑄 +Q 𝑅)))
5654, 55breq12d 3800 . . . . . 6 (𝑞 = 𝑥 → (((𝐹𝑄) +Q 𝑞) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑅)) ↔ ((𝐹𝑄) +Q 𝑥) <Q ((𝐹𝑥) +Q (𝑄 +Q 𝑅))))
5756rspcev 2702 . . . . 5 ((𝑥Q ∧ ((𝐹𝑄) +Q 𝑥) <Q ((𝐹𝑥) +Q (𝑄 +Q 𝑅))) → ∃𝑞Q ((𝐹𝑄) +Q 𝑞) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑅)))
584, 53, 57syl2anc 403 . . . 4 ((𝜑 ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → ∃𝑞Q ((𝐹𝑄) +Q 𝑞) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑅)))
593, 58rexlimddv 2482 . . 3 (𝜑 → ∃𝑞Q ((𝐹𝑄) +Q 𝑞) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑅)))
60 cauappcvgpr.bnd . . . . . . . 8 (𝜑 → ∀𝑝Q 𝐴 <Q (𝐹𝑝))
61 cauappcvgpr.lim . . . . . . . 8 𝐿 = ⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢}⟩
62 addclnq 6616 . . . . . . . . 9 ((𝑄Q𝑅Q) → (𝑄 +Q 𝑅) ∈ Q)
637, 1, 62syl2anc 403 . . . . . . . 8 (𝜑 → (𝑄 +Q 𝑅) ∈ Q)
6427, 5, 60, 61, 63cauappcvgprlemladd 6899 . . . . . . 7 (𝜑 → (𝐿 +P ⟨{𝑙𝑙 <Q (𝑄 +Q 𝑅)}, {𝑢 ∣ (𝑄 +Q 𝑅) <Q 𝑢}⟩) = ⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑅))}, {𝑢Q ∣ ∃𝑞Q (((𝐹𝑞) +Q 𝑞) +Q (𝑄 +Q 𝑅)) <Q 𝑢}⟩)
6564fveq2d 5207 . . . . . 6 (𝜑 → (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q (𝑄 +Q 𝑅)}, {𝑢 ∣ (𝑄 +Q 𝑅) <Q 𝑢}⟩)) = (1st ‘⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑅))}, {𝑢Q ∣ ∃𝑞Q (((𝐹𝑞) +Q 𝑞) +Q (𝑄 +Q 𝑅)) <Q 𝑢}⟩))
66 nqex 6604 . . . . . . . 8 Q ∈ V
6766rabex 3924 . . . . . . 7 {𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑅))} ∈ V
6866rabex 3924 . . . . . . 7 {𝑢Q ∣ ∃𝑞Q (((𝐹𝑞) +Q 𝑞) +Q (𝑄 +Q 𝑅)) <Q 𝑢} ∈ V
6967, 68op1st 5798 . . . . . 6 (1st ‘⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑅))}, {𝑢Q ∣ ∃𝑞Q (((𝐹𝑞) +Q 𝑞) +Q (𝑄 +Q 𝑅)) <Q 𝑢}⟩) = {𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑅))}
7065, 69syl6eq 2130 . . . . 5 (𝜑 → (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q (𝑄 +Q 𝑅)}, {𝑢 ∣ (𝑄 +Q 𝑅) <Q 𝑢}⟩)) = {𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑅))})
7170eleq2d 2149 . . . 4 (𝜑 → ((𝐹𝑄) ∈ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q (𝑄 +Q 𝑅)}, {𝑢 ∣ (𝑄 +Q 𝑅) <Q 𝑢}⟩)) ↔ (𝐹𝑄) ∈ {𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑅))}))
72 oveq1 5544 . . . . . . . 8 (𝑙 = (𝐹𝑄) → (𝑙 +Q 𝑞) = ((𝐹𝑄) +Q 𝑞))
7372breq1d 3797 . . . . . . 7 (𝑙 = (𝐹𝑄) → ((𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑅)) ↔ ((𝐹𝑄) +Q 𝑞) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑅))))
7473rexbidv 2370 . . . . . 6 (𝑙 = (𝐹𝑄) → (∃𝑞Q (𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑅)) ↔ ∃𝑞Q ((𝐹𝑄) +Q 𝑞) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑅))))
7574elrab3 2751 . . . . 5 ((𝐹𝑄) ∈ Q → ((𝐹𝑄) ∈ {𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑅))} ↔ ∃𝑞Q ((𝐹𝑄) +Q 𝑞) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑅))))
7635, 75syl 14 . . . 4 (𝜑 → ((𝐹𝑄) ∈ {𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑅))} ↔ ∃𝑞Q ((𝐹𝑄) +Q 𝑞) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑅))))
7771, 76bitrd 186 . . 3 (𝜑 → ((𝐹𝑄) ∈ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q (𝑄 +Q 𝑅)}, {𝑢 ∣ (𝑄 +Q 𝑅) <Q 𝑢}⟩)) ↔ ∃𝑞Q ((𝐹𝑄) +Q 𝑞) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑅))))
7859, 77mpbird 165 . 2 (𝜑 → (𝐹𝑄) ∈ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q (𝑄 +Q 𝑅)}, {𝑢 ∣ (𝑄 +Q 𝑅) <Q 𝑢}⟩)))
7927, 5, 60, 61cauappcvgprlemcl 6894 . . . 4 (𝜑𝐿P)
80 nqprlu 6788 . . . . 5 ((𝑄 +Q 𝑅) ∈ Q → ⟨{𝑙𝑙 <Q (𝑄 +Q 𝑅)}, {𝑢 ∣ (𝑄 +Q 𝑅) <Q 𝑢}⟩ ∈ P)
8163, 80syl 14 . . . 4 (𝜑 → ⟨{𝑙𝑙 <Q (𝑄 +Q 𝑅)}, {𝑢 ∣ (𝑄 +Q 𝑅) <Q 𝑢}⟩ ∈ P)
82 addclpr 6778 . . . 4 ((𝐿P ∧ ⟨{𝑙𝑙 <Q (𝑄 +Q 𝑅)}, {𝑢 ∣ (𝑄 +Q 𝑅) <Q 𝑢}⟩ ∈ P) → (𝐿 +P ⟨{𝑙𝑙 <Q (𝑄 +Q 𝑅)}, {𝑢 ∣ (𝑄 +Q 𝑅) <Q 𝑢}⟩) ∈ P)
8379, 81, 82syl2anc 403 . . 3 (𝜑 → (𝐿 +P ⟨{𝑙𝑙 <Q (𝑄 +Q 𝑅)}, {𝑢 ∣ (𝑄 +Q 𝑅) <Q 𝑢}⟩) ∈ P)
84 nqprl 6792 . . 3 (((𝐹𝑄) ∈ Q ∧ (𝐿 +P ⟨{𝑙𝑙 <Q (𝑄 +Q 𝑅)}, {𝑢 ∣ (𝑄 +Q 𝑅) <Q 𝑢}⟩) ∈ P) → ((𝐹𝑄) ∈ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q (𝑄 +Q 𝑅)}, {𝑢 ∣ (𝑄 +Q 𝑅) <Q 𝑢}⟩)) ↔ ⟨{𝑙𝑙 <Q (𝐹𝑄)}, {𝑢 ∣ (𝐹𝑄) <Q 𝑢}⟩<P (𝐿 +P ⟨{𝑙𝑙 <Q (𝑄 +Q 𝑅)}, {𝑢 ∣ (𝑄 +Q 𝑅) <Q 𝑢}⟩)))
8535, 83, 84syl2anc 403 . 2 (𝜑 → ((𝐹𝑄) ∈ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q (𝑄 +Q 𝑅)}, {𝑢 ∣ (𝑄 +Q 𝑅) <Q 𝑢}⟩)) ↔ ⟨{𝑙𝑙 <Q (𝐹𝑄)}, {𝑢 ∣ (𝐹𝑄) <Q 𝑢}⟩<P (𝐿 +P ⟨{𝑙𝑙 <Q (𝑄 +Q 𝑅)}, {𝑢 ∣ (𝑄 +Q 𝑅) <Q 𝑢}⟩)))
8678, 85mpbid 145 1 (𝜑 → ⟨{𝑙𝑙 <Q (𝐹𝑄)}, {𝑢 ∣ (𝐹𝑄) <Q 𝑢}⟩<P (𝐿 +P ⟨{𝑙𝑙 <Q (𝑄 +Q 𝑅)}, {𝑢 ∣ (𝑄 +Q 𝑅) <Q 𝑢}⟩))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  w3a 920   = wceq 1285  wcel 1434  {cab 2068  wral 2349  wrex 2350  {crab 2353  cop 3403   class class class wbr 3787  wf 4922  cfv 4926  (class class class)co 5537  1st c1st 5790  Qcnq 6521   +Q cplq 6523   <Q cltq 6526  Pcnp 6532   +P cpp 6534  <P cltp 6536
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 2064  ax-coll 3895  ax-sep 3898  ax-nul 3906  ax-pow 3950  ax-pr 3966  ax-un 4190  ax-setind 4282  ax-iinf 4331
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 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3253  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3604  df-int 3639  df-iun 3682  df-br 3788  df-opab 3842  df-mpt 3843  df-tr 3878  df-eprel 4046  df-id 4050  df-po 4053  df-iso 4054  df-iord 4123  df-on 4125  df-suc 4128  df-iom 4334  df-xp 4371  df-rel 4372  df-cnv 4373  df-co 4374  df-dm 4375  df-rn 4376  df-res 4377  df-ima 4378  df-iota 4891  df-fun 4928  df-fn 4929  df-f 4930  df-f1 4931  df-fo 4932  df-f1o 4933  df-fv 4934  df-ov 5540  df-oprab 5541  df-mpt2 5542  df-1st 5792  df-2nd 5793  df-recs 5948  df-irdg 6013  df-1o 6059  df-2o 6060  df-oadd 6063  df-omul 6064  df-er 6165  df-ec 6167  df-qs 6171  df-ni 6545  df-pli 6546  df-mi 6547  df-lti 6548  df-plpq 6585  df-mpq 6586  df-enq 6588  df-nqqs 6589  df-plqqs 6590  df-mqqs 6591  df-1nqqs 6592  df-rq 6593  df-ltnqqs 6594  df-enq0 6665  df-nq0 6666  df-0nq0 6667  df-plq0 6668  df-mq0 6669  df-inp 6707  df-iplp 6709  df-iltp 6711
This theorem is referenced by:  cauappcvgprlemlim  6902
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