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Theorem axcaucvglemcl 6969
 Description: Lemma for axcaucvg 6974. Mapping to N and R. (Contributed by Jim Kingdon, 10-Jul-2021.)
Hypotheses
Ref Expression
axcaucvglemcl.n 𝑁 = {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}
axcaucvglemcl.f (𝜑𝐹:𝑁⟶ℝ)
Assertion
Ref Expression
axcaucvglemcl ((𝜑𝐽N) → (𝑧R (𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝐽, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨𝑧, 0R⟩) ∈ R)
Distinct variable groups:   𝑧,𝐹   𝐽,𝑙,𝑢,𝑧   𝑦,𝑙,𝑢   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑢,𝑙)   𝐹(𝑥,𝑦,𝑢,𝑙)   𝐽(𝑥,𝑦)   𝑁(𝑥,𝑦,𝑧,𝑢,𝑙)

Proof of Theorem axcaucvglemcl
StepHypRef Expression
1 pitonn 6924 . . . . . 6 (𝐽N → ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝐽, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)})
2 axcaucvglemcl.n . . . . . 6 𝑁 = {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦𝑥 (𝑦 + 1) ∈ 𝑥)}
31, 2syl6eleqr 2131 . . . . 5 (𝐽N → ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝐽, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑁)
4 axcaucvglemcl.f . . . . . 6 (𝜑𝐹:𝑁⟶ℝ)
54ffvelrnda 5302 . . . . 5 ((𝜑 ∧ ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝐽, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑁) → (𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝐽, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) ∈ ℝ)
63, 5sylan2 270 . . . 4 ((𝜑𝐽N) → (𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝐽, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) ∈ ℝ)
7 elrealeu 6906 . . . 4 ((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⟩))
86, 7sylib 127 . . 3 ((𝜑𝐽N) → ∃!𝑧R𝑧, 0R⟩ = (𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝐽, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩))
9 eqcom 2042 . . . 4 (⟨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⟩)
109reubii 2495 . . 3 (∃!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⟩)
118, 10sylib 127 . 2 ((𝜑𝐽N) → ∃!𝑧R (𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝐽, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨𝑧, 0R⟩)
12 riotacl 5482 . 2 (∃!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⟩) ∈ R)
1311, 12syl 14 1 ((𝜑𝐽N) → (𝑧R (𝐹‘⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝐽, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1𝑜⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨𝑧, 0R⟩) ∈ R)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1243   ∈ wcel 1393  {cab 2026  ∀wral 2306  ∃!wreu 2308  ⟨cop 3378  ∩ cint 3615   class class class wbr 3764  ⟶wf 4898  ‘cfv 4902  ℩crio 5467  (class class class)co 5512  1𝑜c1o 5994  [cec 6104  Ncnpi 6370   ~Q ceq 6377
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