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Theorem cauappcvgprlem1 7431
Description: Lemma for cauappcvgpr 7434. 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 7182 . . . . 5 (𝑅Q → ∃𝑥Q (𝑥 +Q 𝑥) = 𝑅)
31, 2syl 14 . . . 4 (𝜑 → ∃𝑥Q (𝑥 +Q 𝑥) = 𝑅)
4 simprl 503 . . . . 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 272 . . . . . . . . . 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 272 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → 𝑄Q)
9 fveq2 5387 . . . . . . . . . . . . . 14 (𝑝 = 𝑄 → (𝐹𝑝) = (𝐹𝑄))
10 oveq1 5747 . . . . . . . . . . . . . . 15 (𝑝 = 𝑄 → (𝑝 +Q 𝑞) = (𝑄 +Q 𝑞))
1110oveq2d 5756 . . . . . . . . . . . . . 14 (𝑝 = 𝑄 → ((𝐹𝑞) +Q (𝑝 +Q 𝑞)) = ((𝐹𝑞) +Q (𝑄 +Q 𝑞)))
129, 11breq12d 3910 . . . . . . . . . . . . 13 (𝑝 = 𝑄 → ((𝐹𝑝) <Q ((𝐹𝑞) +Q (𝑝 +Q 𝑞)) ↔ (𝐹𝑄) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑞))))
139, 10oveq12d 5758 . . . . . . . . . . . . . 14 (𝑝 = 𝑄 → ((𝐹𝑝) +Q (𝑝 +Q 𝑞)) = ((𝐹𝑄) +Q (𝑄 +Q 𝑞)))
1413breq2d 3909 . . . . . . . . . . . . 13 (𝑝 = 𝑄 → ((𝐹𝑞) <Q ((𝐹𝑝) +Q (𝑝 +Q 𝑞)) ↔ (𝐹𝑞) <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑞))))
1512, 14anbi12d 462 . . . . . . . . . . . 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 5387 . . . . . . . . . . . . . . 15 (𝑞 = 𝑥 → (𝐹𝑞) = (𝐹𝑥))
17 oveq2 5748 . . . . . . . . . . . . . . 15 (𝑞 = 𝑥 → (𝑄 +Q 𝑞) = (𝑄 +Q 𝑥))
1816, 17oveq12d 5758 . . . . . . . . . . . . . 14 (𝑞 = 𝑥 → ((𝐹𝑞) +Q (𝑄 +Q 𝑞)) = ((𝐹𝑥) +Q (𝑄 +Q 𝑥)))
1918breq2d 3909 . . . . . . . . . . . . 13 (𝑞 = 𝑥 → ((𝐹𝑄) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑞)) ↔ (𝐹𝑄) <Q ((𝐹𝑥) +Q (𝑄 +Q 𝑥))))
2017oveq2d 5756 . . . . . . . . . . . . . 14 (𝑞 = 𝑥 → ((𝐹𝑄) +Q (𝑄 +Q 𝑞)) = ((𝐹𝑄) +Q (𝑄 +Q 𝑥)))
2116, 20breq12d 3910 . . . . . . . . . . . . 13 (𝑞 = 𝑥 → ((𝐹𝑞) <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑞)) ↔ (𝐹𝑥) <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑥))))
2219, 21anbi12d 462 . . . . . . . . . . . 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 2774 . . . . . . . . . . 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 406 . . . . . . . . . 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 111 . . . . . . . 8 ((𝜑 ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → (𝐹𝑄) <Q ((𝐹𝑥) +Q (𝑄 +Q 𝑥)))
27 cauappcvgpr.f . . . . . . . . . . 11 (𝜑𝐹:QQ)
2827adantr 272 . . . . . . . . . 10 ((𝜑 ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → 𝐹:QQ)
2928, 4ffvelrnd 5522 . . . . . . . . 9 ((𝜑 ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → (𝐹𝑥) ∈ Q)
30 addassnqg 7154 . . . . . . . . 9 (((𝐹𝑥) ∈ Q𝑄Q𝑥Q) → (((𝐹𝑥) +Q 𝑄) +Q 𝑥) = ((𝐹𝑥) +Q (𝑄 +Q 𝑥)))
3129, 8, 4, 30syl3anc 1199 . . . . . . . 8 ((𝜑 ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → (((𝐹𝑥) +Q 𝑄) +Q 𝑥) = ((𝐹𝑥) +Q (𝑄 +Q 𝑥)))
3226, 31breqtrrd 3924 . . . . . . 7 ((𝜑 ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → (𝐹𝑄) <Q (((𝐹𝑥) +Q 𝑄) +Q 𝑥))
33 ltanqg 7172 . . . . . . . . 9 ((𝑓Q𝑔QQ) → (𝑓 <Q 𝑔 ↔ ( +Q 𝑓) <Q ( +Q 𝑔)))
3433adantl 273 . . . . . . . 8 (((𝜑 ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑅)) ∧ (𝑓Q𝑔QQ)) → (𝑓 <Q 𝑔 ↔ ( +Q 𝑓) <Q ( +Q 𝑔)))
3527, 7ffvelrnd 5522 . . . . . . . . 9 (𝜑 → (𝐹𝑄) ∈ Q)
3635adantr 272 . . . . . . . 8 ((𝜑 ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → (𝐹𝑄) ∈ Q)
37 addclnq 7147 . . . . . . . . . 10 (((𝐹𝑥) ∈ Q𝑄Q) → ((𝐹𝑥) +Q 𝑄) ∈ Q)
3829, 8, 37syl2anc 406 . . . . . . . . 9 ((𝜑 ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → ((𝐹𝑥) +Q 𝑄) ∈ Q)
39 addclnq 7147 . . . . . . . . 9 ((((𝐹𝑥) +Q 𝑄) ∈ Q𝑥Q) → (((𝐹𝑥) +Q 𝑄) +Q 𝑥) ∈ Q)
4038, 4, 39syl2anc 406 . . . . . . . 8 ((𝜑 ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → (((𝐹𝑥) +Q 𝑄) +Q 𝑥) ∈ Q)
41 addcomnqg 7153 . . . . . . . . 9 ((𝑓Q𝑔Q) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
4241adantl 273 . . . . . . . 8 (((𝜑 ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑅)) ∧ (𝑓Q𝑔Q)) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
4334, 36, 40, 4, 42caovord2d 5906 . . . . . . 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 146 . . . . . 6 ((𝜑 ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → ((𝐹𝑄) +Q 𝑥) <Q ((((𝐹𝑥) +Q 𝑄) +Q 𝑥) +Q 𝑥))
45 addassnqg 7154 . . . . . . . 8 ((((𝐹𝑥) +Q 𝑄) ∈ Q𝑥Q𝑥Q) → ((((𝐹𝑥) +Q 𝑄) +Q 𝑥) +Q 𝑥) = (((𝐹𝑥) +Q 𝑄) +Q (𝑥 +Q 𝑥)))
4638, 4, 4, 45syl3anc 1199 . . . . . . 7 ((𝜑 ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → ((((𝐹𝑥) +Q 𝑄) +Q 𝑥) +Q 𝑥) = (((𝐹𝑥) +Q 𝑄) +Q (𝑥 +Q 𝑥)))
47 simprr 504 . . . . . . . 8 ((𝜑 ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → (𝑥 +Q 𝑥) = 𝑅)
4847oveq2d 5756 . . . . . . 7 ((𝜑 ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → (((𝐹𝑥) +Q 𝑄) +Q (𝑥 +Q 𝑥)) = (((𝐹𝑥) +Q 𝑄) +Q 𝑅))
491adantr 272 . . . . . . . 8 ((𝜑 ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → 𝑅Q)
50 addassnqg 7154 . . . . . . . 8 (((𝐹𝑥) ∈ Q𝑄Q𝑅Q) → (((𝐹𝑥) +Q 𝑄) +Q 𝑅) = ((𝐹𝑥) +Q (𝑄 +Q 𝑅)))
5129, 8, 49, 50syl3anc 1199 . . . . . . 7 ((𝜑 ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → (((𝐹𝑥) +Q 𝑄) +Q 𝑅) = ((𝐹𝑥) +Q (𝑄 +Q 𝑅)))
5246, 48, 513eqtrd 2152 . . . . . 6 ((𝜑 ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → ((((𝐹𝑥) +Q 𝑄) +Q 𝑥) +Q 𝑥) = ((𝐹𝑥) +Q (𝑄 +Q 𝑅)))
5344, 52breqtrd 3922 . . . . 5 ((𝜑 ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → ((𝐹𝑄) +Q 𝑥) <Q ((𝐹𝑥) +Q (𝑄 +Q 𝑅)))
54 oveq2 5748 . . . . . . 7 (𝑞 = 𝑥 → ((𝐹𝑄) +Q 𝑞) = ((𝐹𝑄) +Q 𝑥))
5516oveq1d 5755 . . . . . . 7 (𝑞 = 𝑥 → ((𝐹𝑞) +Q (𝑄 +Q 𝑅)) = ((𝐹𝑥) +Q (𝑄 +Q 𝑅)))
5654, 55breq12d 3910 . . . . . 6 (𝑞 = 𝑥 → (((𝐹𝑄) +Q 𝑞) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑅)) ↔ ((𝐹𝑄) +Q 𝑥) <Q ((𝐹𝑥) +Q (𝑄 +Q 𝑅))))
5756rspcev 2761 . . . . 5 ((𝑥Q ∧ ((𝐹𝑄) +Q 𝑥) <Q ((𝐹𝑥) +Q (𝑄 +Q 𝑅))) → ∃𝑞Q ((𝐹𝑄) +Q 𝑞) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑅)))
584, 53, 57syl2anc 406 . . . 4 ((𝜑 ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → ∃𝑞Q ((𝐹𝑄) +Q 𝑞) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑅)))
593, 58rexlimddv 2529 . . 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 7147 . . . . . . . . 9 ((𝑄Q𝑅Q) → (𝑄 +Q 𝑅) ∈ Q)
637, 1, 62syl2anc 406 . . . . . . . 8 (𝜑 → (𝑄 +Q 𝑅) ∈ Q)
6427, 5, 60, 61, 63cauappcvgprlemladd 7430 . . . . . . 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 5391 . . . . . 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 7135 . . . . . . . 8 Q ∈ V
6766rabex 4040 . . . . . . 7 {𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑅))} ∈ V
6866rabex 4040 . . . . . . 7 {𝑢Q ∣ ∃𝑞Q (((𝐹𝑞) +Q 𝑞) +Q (𝑄 +Q 𝑅)) <Q 𝑢} ∈ V
6967, 68op1st 6010 . . . . . 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 2164 . . . . 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 2185 . . . 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 5747 . . . . . . . 8 (𝑙 = (𝐹𝑄) → (𝑙 +Q 𝑞) = ((𝐹𝑄) +Q 𝑞))
7372breq1d 3907 . . . . . . 7 (𝑙 = (𝐹𝑄) → ((𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑅)) ↔ ((𝐹𝑄) +Q 𝑞) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑅))))
7473rexbidv 2413 . . . . . 6 (𝑙 = (𝐹𝑄) → (∃𝑞Q (𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑅)) ↔ ∃𝑞Q ((𝐹𝑄) +Q 𝑞) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑅))))
7574elrab3 2812 . . . . 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 187 . . 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 166 . 2 (𝜑 → (𝐹𝑄) ∈ (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q (𝑄 +Q 𝑅)}, {𝑢 ∣ (𝑄 +Q 𝑅) <Q 𝑢}⟩)))
7927, 5, 60, 61cauappcvgprlemcl 7425 . . . 4 (𝜑𝐿P)
80 nqprlu 7319 . . . . 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 7309 . . . 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 406 . . 3 (𝜑 → (𝐿 +P ⟨{𝑙𝑙 <Q (𝑄 +Q 𝑅)}, {𝑢 ∣ (𝑄 +Q 𝑅) <Q 𝑢}⟩) ∈ P)
84 nqprl 7323 . . 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 406 . 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 146 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 103  wb 104  w3a 945   = wceq 1314  wcel 1463  {cab 2101  wral 2391  wrex 2392  {crab 2395  cop 3498   class class class wbr 3897  wf 5087  cfv 5091  (class class class)co 5740  1st c1st 6002  Qcnq 7052   +Q cplq 7054   <Q cltq 7057  Pcnp 7063   +P cpp 7065  <P cltp 7067
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-eprel 4179  df-id 4183  df-po 4186  df-iso 4187  df-iord 4256  df-on 4258  df-suc 4261  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-recs 6168  df-irdg 6233  df-1o 6279  df-2o 6280  df-oadd 6283  df-omul 6284  df-er 6395  df-ec 6397  df-qs 6401  df-ni 7076  df-pli 7077  df-mi 7078  df-lti 7079  df-plpq 7116  df-mpq 7117  df-enq 7119  df-nqqs 7120  df-plqqs 7121  df-mqqs 7122  df-1nqqs 7123  df-rq 7124  df-ltnqqs 7125  df-enq0 7196  df-nq0 7197  df-0nq0 7198  df-plq0 7199  df-mq0 7200  df-inp 7238  df-iplp 7240  df-iltp 7242
This theorem is referenced by:  cauappcvgprlemlim  7433
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