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Theorem cauappcvgprlem1 7621
Description: Lemma for cauappcvgpr 7624. 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 7372 . . . . 5 (𝑅Q → ∃𝑥Q (𝑥 +Q 𝑥) = 𝑅)
31, 2syl 14 . . . 4 (𝜑 → ∃𝑥Q (𝑥 +Q 𝑥) = 𝑅)
4 simprl 526 . . . . 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 274 . . . . . . . . . 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 274 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → 𝑄Q)
9 fveq2 5496 . . . . . . . . . . . . . 14 (𝑝 = 𝑄 → (𝐹𝑝) = (𝐹𝑄))
10 oveq1 5860 . . . . . . . . . . . . . . 15 (𝑝 = 𝑄 → (𝑝 +Q 𝑞) = (𝑄 +Q 𝑞))
1110oveq2d 5869 . . . . . . . . . . . . . 14 (𝑝 = 𝑄 → ((𝐹𝑞) +Q (𝑝 +Q 𝑞)) = ((𝐹𝑞) +Q (𝑄 +Q 𝑞)))
129, 11breq12d 4002 . . . . . . . . . . . . 13 (𝑝 = 𝑄 → ((𝐹𝑝) <Q ((𝐹𝑞) +Q (𝑝 +Q 𝑞)) ↔ (𝐹𝑄) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑞))))
139, 10oveq12d 5871 . . . . . . . . . . . . . 14 (𝑝 = 𝑄 → ((𝐹𝑝) +Q (𝑝 +Q 𝑞)) = ((𝐹𝑄) +Q (𝑄 +Q 𝑞)))
1413breq2d 4001 . . . . . . . . . . . . 13 (𝑝 = 𝑄 → ((𝐹𝑞) <Q ((𝐹𝑝) +Q (𝑝 +Q 𝑞)) ↔ (𝐹𝑞) <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑞))))
1512, 14anbi12d 470 . . . . . . . . . . . 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 5496 . . . . . . . . . . . . . . 15 (𝑞 = 𝑥 → (𝐹𝑞) = (𝐹𝑥))
17 oveq2 5861 . . . . . . . . . . . . . . 15 (𝑞 = 𝑥 → (𝑄 +Q 𝑞) = (𝑄 +Q 𝑥))
1816, 17oveq12d 5871 . . . . . . . . . . . . . 14 (𝑞 = 𝑥 → ((𝐹𝑞) +Q (𝑄 +Q 𝑞)) = ((𝐹𝑥) +Q (𝑄 +Q 𝑥)))
1918breq2d 4001 . . . . . . . . . . . . 13 (𝑞 = 𝑥 → ((𝐹𝑄) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑞)) ↔ (𝐹𝑄) <Q ((𝐹𝑥) +Q (𝑄 +Q 𝑥))))
2017oveq2d 5869 . . . . . . . . . . . . . 14 (𝑞 = 𝑥 → ((𝐹𝑄) +Q (𝑄 +Q 𝑞)) = ((𝐹𝑄) +Q (𝑄 +Q 𝑥)))
2116, 20breq12d 4002 . . . . . . . . . . . . 13 (𝑞 = 𝑥 → ((𝐹𝑞) <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑞)) ↔ (𝐹𝑥) <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑥))))
2219, 21anbi12d 470 . . . . . . . . . . . 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 2847 . . . . . . . . . . 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 409 . . . . . . . . . 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 274 . . . . . . . . . 10 ((𝜑 ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → 𝐹:QQ)
2928, 4ffvelrnd 5632 . . . . . . . . 9 ((𝜑 ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → (𝐹𝑥) ∈ Q)
30 addassnqg 7344 . . . . . . . . 9 (((𝐹𝑥) ∈ Q𝑄Q𝑥Q) → (((𝐹𝑥) +Q 𝑄) +Q 𝑥) = ((𝐹𝑥) +Q (𝑄 +Q 𝑥)))
3129, 8, 4, 30syl3anc 1233 . . . . . . . 8 ((𝜑 ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → (((𝐹𝑥) +Q 𝑄) +Q 𝑥) = ((𝐹𝑥) +Q (𝑄 +Q 𝑥)))
3226, 31breqtrrd 4017 . . . . . . 7 ((𝜑 ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → (𝐹𝑄) <Q (((𝐹𝑥) +Q 𝑄) +Q 𝑥))
33 ltanqg 7362 . . . . . . . . 9 ((𝑓Q𝑔QQ) → (𝑓 <Q 𝑔 ↔ ( +Q 𝑓) <Q ( +Q 𝑔)))
3433adantl 275 . . . . . . . 8 (((𝜑 ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑅)) ∧ (𝑓Q𝑔QQ)) → (𝑓 <Q 𝑔 ↔ ( +Q 𝑓) <Q ( +Q 𝑔)))
3527, 7ffvelrnd 5632 . . . . . . . . 9 (𝜑 → (𝐹𝑄) ∈ Q)
3635adantr 274 . . . . . . . 8 ((𝜑 ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → (𝐹𝑄) ∈ Q)
37 addclnq 7337 . . . . . . . . . 10 (((𝐹𝑥) ∈ Q𝑄Q) → ((𝐹𝑥) +Q 𝑄) ∈ Q)
3829, 8, 37syl2anc 409 . . . . . . . . 9 ((𝜑 ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → ((𝐹𝑥) +Q 𝑄) ∈ Q)
39 addclnq 7337 . . . . . . . . 9 ((((𝐹𝑥) +Q 𝑄) ∈ Q𝑥Q) → (((𝐹𝑥) +Q 𝑄) +Q 𝑥) ∈ Q)
4038, 4, 39syl2anc 409 . . . . . . . 8 ((𝜑 ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → (((𝐹𝑥) +Q 𝑄) +Q 𝑥) ∈ Q)
41 addcomnqg 7343 . . . . . . . . 9 ((𝑓Q𝑔Q) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
4241adantl 275 . . . . . . . 8 (((𝜑 ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑅)) ∧ (𝑓Q𝑔Q)) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
4334, 36, 40, 4, 42caovord2d 6022 . . . . . . 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 7344 . . . . . . . 8 ((((𝐹𝑥) +Q 𝑄) ∈ Q𝑥Q𝑥Q) → ((((𝐹𝑥) +Q 𝑄) +Q 𝑥) +Q 𝑥) = (((𝐹𝑥) +Q 𝑄) +Q (𝑥 +Q 𝑥)))
4638, 4, 4, 45syl3anc 1233 . . . . . . 7 ((𝜑 ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → ((((𝐹𝑥) +Q 𝑄) +Q 𝑥) +Q 𝑥) = (((𝐹𝑥) +Q 𝑄) +Q (𝑥 +Q 𝑥)))
47 simprr 527 . . . . . . . 8 ((𝜑 ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → (𝑥 +Q 𝑥) = 𝑅)
4847oveq2d 5869 . . . . . . 7 ((𝜑 ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → (((𝐹𝑥) +Q 𝑄) +Q (𝑥 +Q 𝑥)) = (((𝐹𝑥) +Q 𝑄) +Q 𝑅))
491adantr 274 . . . . . . . 8 ((𝜑 ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → 𝑅Q)
50 addassnqg 7344 . . . . . . . 8 (((𝐹𝑥) ∈ Q𝑄Q𝑅Q) → (((𝐹𝑥) +Q 𝑄) +Q 𝑅) = ((𝐹𝑥) +Q (𝑄 +Q 𝑅)))
5129, 8, 49, 50syl3anc 1233 . . . . . . 7 ((𝜑 ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → (((𝐹𝑥) +Q 𝑄) +Q 𝑅) = ((𝐹𝑥) +Q (𝑄 +Q 𝑅)))
5246, 48, 513eqtrd 2207 . . . . . 6 ((𝜑 ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → ((((𝐹𝑥) +Q 𝑄) +Q 𝑥) +Q 𝑥) = ((𝐹𝑥) +Q (𝑄 +Q 𝑅)))
5344, 52breqtrd 4015 . . . . 5 ((𝜑 ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → ((𝐹𝑄) +Q 𝑥) <Q ((𝐹𝑥) +Q (𝑄 +Q 𝑅)))
54 oveq2 5861 . . . . . . 7 (𝑞 = 𝑥 → ((𝐹𝑄) +Q 𝑞) = ((𝐹𝑄) +Q 𝑥))
5516oveq1d 5868 . . . . . . 7 (𝑞 = 𝑥 → ((𝐹𝑞) +Q (𝑄 +Q 𝑅)) = ((𝐹𝑥) +Q (𝑄 +Q 𝑅)))
5654, 55breq12d 4002 . . . . . 6 (𝑞 = 𝑥 → (((𝐹𝑄) +Q 𝑞) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑅)) ↔ ((𝐹𝑄) +Q 𝑥) <Q ((𝐹𝑥) +Q (𝑄 +Q 𝑅))))
5756rspcev 2834 . . . . 5 ((𝑥Q ∧ ((𝐹𝑄) +Q 𝑥) <Q ((𝐹𝑥) +Q (𝑄 +Q 𝑅))) → ∃𝑞Q ((𝐹𝑄) +Q 𝑞) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑅)))
584, 53, 57syl2anc 409 . . . 4 ((𝜑 ∧ (𝑥Q ∧ (𝑥 +Q 𝑥) = 𝑅)) → ∃𝑞Q ((𝐹𝑄) +Q 𝑞) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑅)))
593, 58rexlimddv 2592 . . 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 7337 . . . . . . . . 9 ((𝑄Q𝑅Q) → (𝑄 +Q 𝑅) ∈ Q)
637, 1, 62syl2anc 409 . . . . . . . 8 (𝜑 → (𝑄 +Q 𝑅) ∈ Q)
6427, 5, 60, 61, 63cauappcvgprlemladd 7620 . . . . . . 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 5500 . . . . . 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 7325 . . . . . . . 8 Q ∈ V
6766rabex 4133 . . . . . . 7 {𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑅))} ∈ V
6866rabex 4133 . . . . . . 7 {𝑢Q ∣ ∃𝑞Q (((𝐹𝑞) +Q 𝑞) +Q (𝑄 +Q 𝑅)) <Q 𝑢} ∈ V
6967, 68op1st 6125 . . . . . 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, 69eqtrdi 2219 . . . . 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 2240 . . . 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 5860 . . . . . . . 8 (𝑙 = (𝐹𝑄) → (𝑙 +Q 𝑞) = ((𝐹𝑄) +Q 𝑞))
7372breq1d 3999 . . . . . . 7 (𝑙 = (𝐹𝑄) → ((𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑅)) ↔ ((𝐹𝑄) +Q 𝑞) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑅))))
7473rexbidv 2471 . . . . . 6 (𝑙 = (𝐹𝑄) → (∃𝑞Q (𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑅)) ↔ ∃𝑞Q ((𝐹𝑄) +Q 𝑞) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑅))))
7574elrab3 2887 . . . . 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 7615 . . . 4 (𝜑𝐿P)
80 nqprlu 7509 . . . . 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 7499 . . . 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 409 . . 3 (𝜑 → (𝐿 +P ⟨{𝑙𝑙 <Q (𝑄 +Q 𝑅)}, {𝑢 ∣ (𝑄 +Q 𝑅) <Q 𝑢}⟩) ∈ P)
84 nqprl 7513 . . 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 409 . 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 973   = wceq 1348  wcel 2141  {cab 2156  wral 2448  wrex 2449  {crab 2452  cop 3586   class class class wbr 3989  wf 5194  cfv 5198  (class class class)co 5853  1st c1st 6117  Qcnq 7242   +Q cplq 7244   <Q cltq 7247  Pcnp 7253   +P cpp 7255  <P cltp 7257
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-eprel 4274  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-1o 6395  df-2o 6396  df-oadd 6399  df-omul 6400  df-er 6513  df-ec 6515  df-qs 6519  df-ni 7266  df-pli 7267  df-mi 7268  df-lti 7269  df-plpq 7306  df-mpq 7307  df-enq 7309  df-nqqs 7310  df-plqqs 7311  df-mqqs 7312  df-1nqqs 7313  df-rq 7314  df-ltnqqs 7315  df-enq0 7386  df-nq0 7387  df-0nq0 7388  df-plq0 7389  df-mq0 7390  df-inp 7428  df-iplp 7430  df-iltp 7432
This theorem is referenced by:  cauappcvgprlemlim  7623
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