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Theorem axcaucvglemval 6969
Description: Lemma for axcaucvg 6972. Value of sequence when mapping to N and R. (Contributed by Jim Kingdon, 10-Jul-2021.)
Hypotheses
Ref Expression
axcaucvg.n 𝑁 = {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}
axcaucvg.f (𝜑𝐹:𝑁⟶ℝ)
axcaucvg.cau (𝜑 → ∀𝑛𝑁𝑘𝑁 (𝑛 < 𝑘 → ((𝐹𝑛) < ((𝐹𝑘) + (𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)) ∧ (𝐹𝑘) < ((𝐹𝑛) + (𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)))))
axcaucvg.g 𝐺 = (𝑗N ↦ (𝑧R (𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑗, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑗, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨𝑧, 0R⟩))
Assertion
Ref Expression
axcaucvglemval ((𝜑𝐽N) → (𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝐽, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨(𝐺𝐽), 0R⟩)
Distinct variable groups:   𝑗,𝐹,𝑧   𝑧,𝐺   𝑗,𝐽,𝑙,𝑢,𝑧   𝜑,𝑗   𝑦,𝑙,𝑢   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑢,𝑘,𝑛,𝑟,𝑙)   𝐹(𝑥,𝑦,𝑢,𝑘,𝑛,𝑟,𝑙)   𝐺(𝑥,𝑦,𝑢,𝑗,𝑘,𝑛,𝑟,𝑙)   𝐽(𝑥,𝑦,𝑘,𝑛,𝑟)   𝑁(𝑥,𝑦,𝑧,𝑢,𝑗,𝑘,𝑛,𝑟,𝑙)

Proof of Theorem axcaucvglemval
StepHypRef Expression
1 axcaucvg.g . . . . 5 𝐺 = (𝑗N ↦ (𝑧R (𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑗, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑗, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨𝑧, 0R⟩))
21a1i 9 . . . 4 ((𝜑𝐽N) → 𝐺 = (𝑗N ↦ (𝑧R (𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑗, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑗, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨𝑧, 0R⟩)))
3 opeq1 3549 . . . . . . . . . . . . . . . 16 (𝑗 = 𝐽 → ⟨𝑗, 1𝑜⟩ = ⟨𝐽, 1𝑜⟩)
43eceq1d 6142 . . . . . . . . . . . . . . 15 (𝑗 = 𝐽 → [⟨𝑗, 1𝑜⟩] ~Q = [⟨𝐽, 1𝑜⟩] ~Q )
54breq2d 3776 . . . . . . . . . . . . . 14 (𝑗 = 𝐽 → (𝑙 <Q [⟨𝑗, 1𝑜⟩] ~Q𝑙 <Q [⟨𝐽, 1𝑜⟩] ~Q ))
65abbidv 2155 . . . . . . . . . . . . 13 (𝑗 = 𝐽 → {𝑙𝑙 <Q [⟨𝑗, 1𝑜⟩] ~Q } = {𝑙𝑙 <Q [⟨𝐽, 1𝑜⟩] ~Q })
74breq1d 3774 . . . . . . . . . . . . . 14 (𝑗 = 𝐽 → ([⟨𝑗, 1𝑜⟩] ~Q <Q 𝑢 ↔ [⟨𝐽, 1𝑜⟩] ~Q <Q 𝑢))
87abbidv 2155 . . . . . . . . . . . . 13 (𝑗 = 𝐽 → {𝑢 ∣ [⟨𝑗, 1𝑜⟩] ~Q <Q 𝑢} = {𝑢 ∣ [⟨𝐽, 1𝑜⟩] ~Q <Q 𝑢})
96, 8opeq12d 3557 . . . . . . . . . . . 12 (𝑗 = 𝐽 → ⟨{𝑙𝑙 <Q [⟨𝑗, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑗, 1𝑜⟩] ~Q <Q 𝑢}⟩ = ⟨{𝑙𝑙 <Q [⟨𝐽, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1𝑜⟩] ~Q <Q 𝑢}⟩)
109oveq1d 5527 . . . . . . . . . . 11 (𝑗 = 𝐽 → (⟨{𝑙𝑙 <Q [⟨𝑗, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑗, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P) = (⟨{𝑙𝑙 <Q [⟨𝐽, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P))
1110opeq1d 3555 . . . . . . . . . 10 (𝑗 = 𝐽 → ⟨(⟨{𝑙𝑙 <Q [⟨𝑗, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑗, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩ = ⟨(⟨{𝑙𝑙 <Q [⟨𝐽, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩)
1211eceq1d 6142 . . . . . . . . 9 (𝑗 = 𝐽 → [⟨(⟨{𝑙𝑙 <Q [⟨𝑗, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑗, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R = [⟨(⟨{𝑙𝑙 <Q [⟨𝐽, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )
1312opeq1d 3555 . . . . . . . 8 (𝑗 = 𝐽 → ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑗, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑗, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝐽, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩)
1413fveq2d 5182 . . . . . . 7 (𝑗 = 𝐽 → (𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑗, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑗, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = (𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝐽, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩))
1514eqeq1d 2048 . . . . . 6 (𝑗 = 𝐽 → ((𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑗, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑗, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨𝑧, 0R⟩ ↔ (𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝐽, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨𝑧, 0R⟩))
1615riotabidv 5470 . . . . 5 (𝑗 = 𝐽 → (𝑧R (𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑗, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑗, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨𝑧, 0R⟩) = (𝑧R (𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝐽, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨𝑧, 0R⟩))
1716adantl 262 . . . 4 (((𝜑𝐽N) ∧ 𝑗 = 𝐽) → (𝑧R (𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑗, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑗, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨𝑧, 0R⟩) = (𝑧R (𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝐽, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨𝑧, 0R⟩))
18 simpr 103 . . . 4 ((𝜑𝐽N) → 𝐽N)
19 axcaucvg.n . . . . 5 𝑁 = {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}
20 axcaucvg.f . . . . 5 (𝜑𝐹:𝑁⟶ℝ)
2119, 20axcaucvglemcl 6967 . . . 4 ((𝜑𝐽N) → (𝑧R (𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝐽, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨𝑧, 0R⟩) ∈ R)
222, 17, 18, 21fvmptd 5253 . . 3 ((𝜑𝐽N) → (𝐺𝐽) = (𝑧R (𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝐽, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨𝑧, 0R⟩))
2322eqcomd 2045 . 2 ((𝜑𝐽N) → (𝑧R (𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝐽, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨𝑧, 0R⟩) = (𝐺𝐽))
2422, 21eqeltrd 2114 . . 3 ((𝜑𝐽N) → (𝐺𝐽) ∈ R)
2520adantr 261 . . . . . 6 ((𝜑𝐽N) → 𝐹:𝑁⟶ℝ)
26 pitonn 6922 . . . . . . . 8 (𝐽N → ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝐽, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)})
2726, 19syl6eleqr 2131 . . . . . . 7 (𝐽N → ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝐽, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑁)
2827adantl 262 . . . . . 6 ((𝜑𝐽N) → ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝐽, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑁)
2925, 28ffvelrnd 5303 . . . . 5 ((𝜑𝐽N) → (𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝐽, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) ∈ ℝ)
30 elrealeu 6904 . . . . 5 ((𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝐽, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) ∈ ℝ ↔ ∃!𝑧R𝑧, 0R⟩ = (𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝐽, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩))
3129, 30sylib 127 . . . 4 ((𝜑𝐽N) → ∃!𝑧R𝑧, 0R⟩ = (𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝐽, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩))
32 eqcom 2042 . . . . 5 (⟨𝑧, 0R⟩ = (𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝐽, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) ↔ (𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝐽, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨𝑧, 0R⟩)
3332reubii 2495 . . . 4 (∃!𝑧R𝑧, 0R⟩ = (𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝐽, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) ↔ ∃!𝑧R (𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝐽, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨𝑧, 0R⟩)
3431, 33sylib 127 . . 3 ((𝜑𝐽N) → ∃!𝑧R (𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝐽, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨𝑧, 0R⟩)
35 opeq1 3549 . . . . 5 (𝑧 = (𝐺𝐽) → ⟨𝑧, 0R⟩ = ⟨(𝐺𝐽), 0R⟩)
3635eqeq2d 2051 . . . 4 (𝑧 = (𝐺𝐽) → ((𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝐽, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨𝑧, 0R⟩ ↔ (𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝐽, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨(𝐺𝐽), 0R⟩))
3736riota2 5490 . . 3 (((𝐺𝐽) ∈ R ∧ ∃!𝑧R (𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝐽, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨𝑧, 0R⟩) → ((𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝐽, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨(𝐺𝐽), 0R⟩ ↔ (𝑧R (𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝐽, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨𝑧, 0R⟩) = (𝐺𝐽)))
3824, 34, 37syl2anc 391 . 2 ((𝜑𝐽N) → ((𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝐽, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨(𝐺𝐽), 0R⟩ ↔ (𝑧R (𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝐽, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨𝑧, 0R⟩) = (𝐺𝐽)))
3923, 38mpbird 156 1 ((𝜑𝐽N) → (𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝐽, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨(𝐺𝐽), 0R⟩)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wb 98   = wceq 1243  wcel 1393  {cab 2026  wral 2306  ∃!wreu 2308  cop 3378   cint 3615   class class class wbr 3764  cmpt 3818  wf 4898  cfv 4902  crio 5467  (class class class)co 5512  1𝑜c1o 5994  [cec 6104  Ncnpi 6368   ~Q ceq 6375   <Q cltq 6381  1Pc1p 6388   +P cpp 6389   ~R cer 6392  Rcnr 6393  0Rc0r 6394  cr 6886  1c1 6888   + caddc 6890   < cltrr 6891   · cmul 6892
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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rmo 2314  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-riota 5468  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6400  df-pli 6401  df-mi 6402  df-lti 6403  df-plpq 6440  df-mpq 6441  df-enq 6443  df-nqqs 6444  df-plqqs 6445  df-mqqs 6446  df-1nqqs 6447  df-rq 6448  df-ltnqqs 6449  df-enq0 6520  df-nq0 6521  df-0nq0 6522  df-plq0 6523  df-mq0 6524  df-inp 6562  df-i1p 6563  df-iplp 6564  df-enr 6809  df-nr 6810  df-plr 6811  df-0r 6814  df-1r 6815  df-c 6893  df-1 6895  df-r 6897  df-add 6898
This theorem is referenced by:  axcaucvglemcau  6970  axcaucvglemres  6971
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