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Theorem cauappcvgprlem1 6630
Description: Lemma for cauappcvgpr 6633. 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 (φ𝑝 Q 𝑞 Q ((𝐹𝑝) <Q ((𝐹𝑞) +Q (𝑝 +Q 𝑞)) (𝐹𝑞) <Q ((𝐹𝑝) +Q (𝑝 +Q 𝑞))))
cauappcvgpr.bnd (φ𝑝 Q A <Q (𝐹𝑝))
cauappcvgpr.lim 𝐿 = ⟨{𝑙 Q𝑞 Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {u Q𝑞 Q ((𝐹𝑞) +Q 𝑞) <Q u}⟩
cauappcvgprlem.q (φ𝑄 Q)
cauappcvgprlem.r (φ𝑅 Q)
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 (φ𝑅 Q)
2 halfnqq 6393 . . . . 5 (𝑅 Qx Q (x +Q x) = 𝑅)
31, 2syl 14 . . . 4 (φx Q (x +Q x) = 𝑅)
4 simprl 483 . . . . 5 ((φ (x Q (x +Q x) = 𝑅)) → x Q)
5 cauappcvgpr.app . . . . . . . . . . 11 (φ𝑝 Q 𝑞 Q ((𝐹𝑝) <Q ((𝐹𝑞) +Q (𝑝 +Q 𝑞)) (𝐹𝑞) <Q ((𝐹𝑝) +Q (𝑝 +Q 𝑞))))
65adantr 261 . . . . . . . . . 10 ((φ (x Q (x +Q x) = 𝑅)) → 𝑝 Q 𝑞 Q ((𝐹𝑝) <Q ((𝐹𝑞) +Q (𝑝 +Q 𝑞)) (𝐹𝑞) <Q ((𝐹𝑝) +Q (𝑝 +Q 𝑞))))
7 cauappcvgprlem.q . . . . . . . . . . . 12 (φ𝑄 Q)
87adantr 261 . . . . . . . . . . 11 ((φ (x Q (x +Q x) = 𝑅)) → 𝑄 Q)
9 fveq2 5121 . . . . . . . . . . . . . 14 (𝑝 = 𝑄 → (𝐹𝑝) = (𝐹𝑄))
10 oveq1 5462 . . . . . . . . . . . . . . 15 (𝑝 = 𝑄 → (𝑝 +Q 𝑞) = (𝑄 +Q 𝑞))
1110oveq2d 5471 . . . . . . . . . . . . . 14 (𝑝 = 𝑄 → ((𝐹𝑞) +Q (𝑝 +Q 𝑞)) = ((𝐹𝑞) +Q (𝑄 +Q 𝑞)))
129, 11breq12d 3768 . . . . . . . . . . . . 13 (𝑝 = 𝑄 → ((𝐹𝑝) <Q ((𝐹𝑞) +Q (𝑝 +Q 𝑞)) ↔ (𝐹𝑄) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑞))))
139, 10oveq12d 5473 . . . . . . . . . . . . . 14 (𝑝 = 𝑄 → ((𝐹𝑝) +Q (𝑝 +Q 𝑞)) = ((𝐹𝑄) +Q (𝑄 +Q 𝑞)))
1413breq2d 3767 . . . . . . . . . . . . 13 (𝑝 = 𝑄 → ((𝐹𝑞) <Q ((𝐹𝑝) +Q (𝑝 +Q 𝑞)) ↔ (𝐹𝑞) <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑞))))
1512, 14anbi12d 442 . . . . . . . . . . . 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 5121 . . . . . . . . . . . . . . 15 (𝑞 = x → (𝐹𝑞) = (𝐹x))
17 oveq2 5463 . . . . . . . . . . . . . . 15 (𝑞 = x → (𝑄 +Q 𝑞) = (𝑄 +Q x))
1816, 17oveq12d 5473 . . . . . . . . . . . . . 14 (𝑞 = x → ((𝐹𝑞) +Q (𝑄 +Q 𝑞)) = ((𝐹x) +Q (𝑄 +Q x)))
1918breq2d 3767 . . . . . . . . . . . . 13 (𝑞 = x → ((𝐹𝑄) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑞)) ↔ (𝐹𝑄) <Q ((𝐹x) +Q (𝑄 +Q x))))
2017oveq2d 5471 . . . . . . . . . . . . . 14 (𝑞 = x → ((𝐹𝑄) +Q (𝑄 +Q 𝑞)) = ((𝐹𝑄) +Q (𝑄 +Q x)))
2116, 20breq12d 3768 . . . . . . . . . . . . 13 (𝑞 = x → ((𝐹𝑞) <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑞)) ↔ (𝐹x) <Q ((𝐹𝑄) +Q (𝑄 +Q x))))
2219, 21anbi12d 442 . . . . . . . . . . . 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 2656 . . . . . . . . . . 11 ((𝑄 Q x Q) → (𝑝 Q 𝑞 Q ((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 391 . . . . . . . . . 10 ((φ (x Q (x +Q x) = 𝑅)) → (𝑝 Q 𝑞 Q ((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 (x Q (x +Q x) = 𝑅)) → ((𝐹𝑄) <Q ((𝐹x) +Q (𝑄 +Q x)) (𝐹x) <Q ((𝐹𝑄) +Q (𝑄 +Q x))))
2625simpld 105 . . . . . . . 8 ((φ (x Q (x +Q x) = 𝑅)) → (𝐹𝑄) <Q ((𝐹x) +Q (𝑄 +Q x)))
27 cauappcvgpr.f . . . . . . . . . . 11 (φ𝐹:QQ)
2827adantr 261 . . . . . . . . . 10 ((φ (x Q (x +Q x) = 𝑅)) → 𝐹:QQ)
2928, 4ffvelrnd 5246 . . . . . . . . 9 ((φ (x Q (x +Q x) = 𝑅)) → (𝐹x) Q)
30 addassnqg 6366 . . . . . . . . 9 (((𝐹x) Q 𝑄 Q x Q) → (((𝐹x) +Q 𝑄) +Q x) = ((𝐹x) +Q (𝑄 +Q x)))
3129, 8, 4, 30syl3anc 1134 . . . . . . . 8 ((φ (x Q (x +Q x) = 𝑅)) → (((𝐹x) +Q 𝑄) +Q x) = ((𝐹x) +Q (𝑄 +Q x)))
3226, 31breqtrrd 3781 . . . . . . 7 ((φ (x Q (x +Q x) = 𝑅)) → (𝐹𝑄) <Q (((𝐹x) +Q 𝑄) +Q x))
33 ltanqg 6384 . . . . . . . . 9 ((f Q g Q Q) → (f <Q g ↔ ( +Q f) <Q ( +Q g)))
3433adantl 262 . . . . . . . 8 (((φ (x Q (x +Q x) = 𝑅)) (f Q g Q Q)) → (f <Q g ↔ ( +Q f) <Q ( +Q g)))
3527, 7ffvelrnd 5246 . . . . . . . . 9 (φ → (𝐹𝑄) Q)
3635adantr 261 . . . . . . . 8 ((φ (x Q (x +Q x) = 𝑅)) → (𝐹𝑄) Q)
37 addclnq 6359 . . . . . . . . . 10 (((𝐹x) Q 𝑄 Q) → ((𝐹x) +Q 𝑄) Q)
3829, 8, 37syl2anc 391 . . . . . . . . 9 ((φ (x Q (x +Q x) = 𝑅)) → ((𝐹x) +Q 𝑄) Q)
39 addclnq 6359 . . . . . . . . 9 ((((𝐹x) +Q 𝑄) Q x Q) → (((𝐹x) +Q 𝑄) +Q x) Q)
4038, 4, 39syl2anc 391 . . . . . . . 8 ((φ (x Q (x +Q x) = 𝑅)) → (((𝐹x) +Q 𝑄) +Q x) Q)
41 addcomnqg 6365 . . . . . . . . 9 ((f Q g Q) → (f +Q g) = (g +Q f))
4241adantl 262 . . . . . . . 8 (((φ (x Q (x +Q x) = 𝑅)) (f Q g Q)) → (f +Q g) = (g +Q f))
4334, 36, 40, 4, 42caovord2d 5612 . . . . . . 7 ((φ (x Q (x +Q x) = 𝑅)) → ((𝐹𝑄) <Q (((𝐹x) +Q 𝑄) +Q x) ↔ ((𝐹𝑄) +Q x) <Q ((((𝐹x) +Q 𝑄) +Q x) +Q x)))
4432, 43mpbid 135 . . . . . 6 ((φ (x Q (x +Q x) = 𝑅)) → ((𝐹𝑄) +Q x) <Q ((((𝐹x) +Q 𝑄) +Q x) +Q x))
45 addassnqg 6366 . . . . . . . 8 ((((𝐹x) +Q 𝑄) Q x Q x Q) → ((((𝐹x) +Q 𝑄) +Q x) +Q x) = (((𝐹x) +Q 𝑄) +Q (x +Q x)))
4638, 4, 4, 45syl3anc 1134 . . . . . . 7 ((φ (x Q (x +Q x) = 𝑅)) → ((((𝐹x) +Q 𝑄) +Q x) +Q x) = (((𝐹x) +Q 𝑄) +Q (x +Q x)))
47 simprr 484 . . . . . . . 8 ((φ (x Q (x +Q x) = 𝑅)) → (x +Q x) = 𝑅)
4847oveq2d 5471 . . . . . . 7 ((φ (x Q (x +Q x) = 𝑅)) → (((𝐹x) +Q 𝑄) +Q (x +Q x)) = (((𝐹x) +Q 𝑄) +Q 𝑅))
491adantr 261 . . . . . . . 8 ((φ (x Q (x +Q x) = 𝑅)) → 𝑅 Q)
50 addassnqg 6366 . . . . . . . 8 (((𝐹x) Q 𝑄 Q 𝑅 Q) → (((𝐹x) +Q 𝑄) +Q 𝑅) = ((𝐹x) +Q (𝑄 +Q 𝑅)))
5129, 8, 49, 50syl3anc 1134 . . . . . . 7 ((φ (x Q (x +Q x) = 𝑅)) → (((𝐹x) +Q 𝑄) +Q 𝑅) = ((𝐹x) +Q (𝑄 +Q 𝑅)))
5246, 48, 513eqtrd 2073 . . . . . 6 ((φ (x Q (x +Q x) = 𝑅)) → ((((𝐹x) +Q 𝑄) +Q x) +Q x) = ((𝐹x) +Q (𝑄 +Q 𝑅)))
5344, 52breqtrd 3779 . . . . 5 ((φ (x Q (x +Q x) = 𝑅)) → ((𝐹𝑄) +Q x) <Q ((𝐹x) +Q (𝑄 +Q 𝑅)))
54 oveq2 5463 . . . . . . 7 (𝑞 = x → ((𝐹𝑄) +Q 𝑞) = ((𝐹𝑄) +Q x))
5516oveq1d 5470 . . . . . . 7 (𝑞 = x → ((𝐹𝑞) +Q (𝑄 +Q 𝑅)) = ((𝐹x) +Q (𝑄 +Q 𝑅)))
5654, 55breq12d 3768 . . . . . 6 (𝑞 = x → (((𝐹𝑄) +Q 𝑞) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑅)) ↔ ((𝐹𝑄) +Q x) <Q ((𝐹x) +Q (𝑄 +Q 𝑅))))
5756rspcev 2650 . . . . 5 ((x Q ((𝐹𝑄) +Q x) <Q ((𝐹x) +Q (𝑄 +Q 𝑅))) → 𝑞 Q ((𝐹𝑄) +Q 𝑞) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑅)))
584, 53, 57syl2anc 391 . . . 4 ((φ (x Q (x +Q x) = 𝑅)) → 𝑞 Q ((𝐹𝑄) +Q 𝑞) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑅)))
593, 58rexlimddv 2431 . . 3 (φ𝑞 Q ((𝐹𝑄) +Q 𝑞) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑅)))
60 cauappcvgpr.bnd . . . . . . . 8 (φ𝑝 Q A <Q (𝐹𝑝))
61 cauappcvgpr.lim . . . . . . . 8 𝐿 = ⟨{𝑙 Q𝑞 Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {u Q𝑞 Q ((𝐹𝑞) +Q 𝑞) <Q u}⟩
62 addclnq 6359 . . . . . . . . 9 ((𝑄 Q 𝑅 Q) → (𝑄 +Q 𝑅) Q)
637, 1, 62syl2anc 391 . . . . . . . 8 (φ → (𝑄 +Q 𝑅) Q)
6427, 5, 60, 61, 63cauappcvgprlemladd 6629 . . . . . . 7 (φ → (𝐿 +P ⟨{𝑙𝑙 <Q (𝑄 +Q 𝑅)}, {u ∣ (𝑄 +Q 𝑅) <Q u}⟩) = ⟨{𝑙 Q𝑞 Q (𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑅))}, {u Q𝑞 Q (((𝐹𝑞) +Q 𝑞) +Q (𝑄 +Q 𝑅)) <Q u}⟩)
6564fveq2d 5125 . . . . . 6 (φ → (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q (𝑄 +Q 𝑅)}, {u ∣ (𝑄 +Q 𝑅) <Q u}⟩)) = (1st ‘⟨{𝑙 Q𝑞 Q (𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑅))}, {u Q𝑞 Q (((𝐹𝑞) +Q 𝑞) +Q (𝑄 +Q 𝑅)) <Q u}⟩))
66 nqex 6347 . . . . . . . 8 Q V
6766rabex 3892 . . . . . . 7 {𝑙 Q𝑞 Q (𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑅))} V
6866rabex 3892 . . . . . . 7 {u Q𝑞 Q (((𝐹𝑞) +Q 𝑞) +Q (𝑄 +Q 𝑅)) <Q u} V
6967, 68op1st 5715 . . . . . 6 (1st ‘⟨{𝑙 Q𝑞 Q (𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑅))}, {u Q𝑞 Q (((𝐹𝑞) +Q 𝑞) +Q (𝑄 +Q 𝑅)) <Q u}⟩) = {𝑙 Q𝑞 Q (𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑅))}
7065, 69syl6eq 2085 . . . . 5 (φ → (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q (𝑄 +Q 𝑅)}, {u ∣ (𝑄 +Q 𝑅) <Q u}⟩)) = {𝑙 Q𝑞 Q (𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑅))})
7170eleq2d 2104 . . . 4 (φ → ((𝐹𝑄) (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q (𝑄 +Q 𝑅)}, {u ∣ (𝑄 +Q 𝑅) <Q u}⟩)) ↔ (𝐹𝑄) {𝑙 Q𝑞 Q (𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑅))}))
72 oveq1 5462 . . . . . . . 8 (𝑙 = (𝐹𝑄) → (𝑙 +Q 𝑞) = ((𝐹𝑄) +Q 𝑞))
7372breq1d 3765 . . . . . . 7 (𝑙 = (𝐹𝑄) → ((𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑅)) ↔ ((𝐹𝑄) +Q 𝑞) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑅))))
7473rexbidv 2321 . . . . . 6 (𝑙 = (𝐹𝑄) → (𝑞 Q (𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑅)) ↔ 𝑞 Q ((𝐹𝑄) +Q 𝑞) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑅))))
7574elrab3 2693 . . . . 5 ((𝐹𝑄) Q → ((𝐹𝑄) {𝑙 Q𝑞 Q (𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑅))} ↔ 𝑞 Q ((𝐹𝑄) +Q 𝑞) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑅))))
7635, 75syl 14 . . . 4 (φ → ((𝐹𝑄) {𝑙 Q𝑞 Q (𝑙 +Q 𝑞) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑅))} ↔ 𝑞 Q ((𝐹𝑄) +Q 𝑞) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑅))))
7771, 76bitrd 177 . . 3 (φ → ((𝐹𝑄) (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q (𝑄 +Q 𝑅)}, {u ∣ (𝑄 +Q 𝑅) <Q u}⟩)) ↔ 𝑞 Q ((𝐹𝑄) +Q 𝑞) <Q ((𝐹𝑞) +Q (𝑄 +Q 𝑅))))
7859, 77mpbird 156 . 2 (φ → (𝐹𝑄) (1st ‘(𝐿 +P ⟨{𝑙𝑙 <Q (𝑄 +Q 𝑅)}, {u ∣ (𝑄 +Q 𝑅) <Q u}⟩)))
7927, 5, 60, 61cauappcvgprlemcl 6624 . . . 4 (φ𝐿 P)
80 nqprlu 6529 . . . . 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 6519 . . . 4 ((𝐿 P ⟨{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 391 . . 3 (φ → (𝐿 +P ⟨{𝑙𝑙 <Q (𝑄 +Q 𝑅)}, {u ∣ (𝑄 +Q 𝑅) <Q u}⟩) P)
84 nqprl 6532 . . 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 391 . 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 135 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 97  wb 98   w3a 884   = wceq 1242   wcel 1390  {cab 2023  wral 2300  wrex 2301  {crab 2304  cop 3370   class class class wbr 3755  wf 4841  cfv 4845  (class class class)co 5455  1st c1st 5707  Qcnq 6264   +Q cplq 6266   <Q cltq 6269  Pcnp 6275   +P cpp 6277  <P cltp 6279
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254
This theorem depends on definitions:  df-bi 110  df-dc 742  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-eprel 4017  df-id 4021  df-po 4024  df-iso 4025  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-1o 5940  df-2o 5941  df-oadd 5944  df-omul 5945  df-er 6042  df-ec 6044  df-qs 6048  df-ni 6288  df-pli 6289  df-mi 6290  df-lti 6291  df-plpq 6328  df-mpq 6329  df-enq 6331  df-nqqs 6332  df-plqqs 6333  df-mqqs 6334  df-1nqqs 6335  df-rq 6336  df-ltnqqs 6337  df-enq0 6406  df-nq0 6407  df-0nq0 6408  df-plq0 6409  df-mq0 6410  df-inp 6448  df-iplp 6450  df-iltp 6452
This theorem is referenced by:  cauappcvgprlemlim  6632
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