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Theorem caucvgsrlemfv 6932
 Description: Lemma for caucvgsr 6943. Coercing sequence value from a positive real to a signed real. (Contributed by Jim Kingdon, 29-Jun-2021.)
Hypotheses
Ref Expression
caucvgsr.f (𝜑𝐹:NR)
caucvgsr.cau (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <R ((𝐹𝑘) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐹𝑘) <R ((𝐹𝑛) +R [⟨(⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))))
caucvgsrlemgt1.gt1 (𝜑 → ∀𝑚N 1R <R (𝐹𝑚))
caucvgsrlemf.xfr 𝐺 = (𝑥N ↦ (𝑦P (𝐹𝑥) = [⟨(𝑦 +P 1P), 1P⟩] ~R ))
Assertion
Ref Expression
caucvgsrlemfv ((𝜑𝐴N) → [⟨((𝐺𝐴) +P 1P), 1P⟩] ~R = (𝐹𝐴))
Distinct variable groups:   𝐴,𝑚   𝑥,𝐴,𝑦   𝑚,𝐹   𝑥,𝐹,𝑦   𝜑,𝑥
Allowed substitution hints:   𝜑(𝑦,𝑢,𝑘,𝑚,𝑛,𝑙)   𝐴(𝑢,𝑘,𝑛,𝑙)   𝐹(𝑢,𝑘,𝑛,𝑙)   𝐺(𝑥,𝑦,𝑢,𝑘,𝑚,𝑛,𝑙)

Proof of Theorem caucvgsrlemfv
StepHypRef Expression
1 caucvgsrlemf.xfr . . . . . . 7 𝐺 = (𝑥N ↦ (𝑦P (𝐹𝑥) = [⟨(𝑦 +P 1P), 1P⟩] ~R ))
21a1i 9 . . . . . 6 ((𝜑𝐴N) → 𝐺 = (𝑥N ↦ (𝑦P (𝐹𝑥) = [⟨(𝑦 +P 1P), 1P⟩] ~R )))
3 fveq2 5205 . . . . . . . . 9 (𝑥 = 𝐴 → (𝐹𝑥) = (𝐹𝐴))
43eqeq1d 2064 . . . . . . . 8 (𝑥 = 𝐴 → ((𝐹𝑥) = [⟨(𝑦 +P 1P), 1P⟩] ~R ↔ (𝐹𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ))
54riotabidv 5497 . . . . . . 7 (𝑥 = 𝐴 → (𝑦P (𝐹𝑥) = [⟨(𝑦 +P 1P), 1P⟩] ~R ) = (𝑦P (𝐹𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ))
65adantl 266 . . . . . 6 (((𝜑𝐴N) ∧ 𝑥 = 𝐴) → (𝑦P (𝐹𝑥) = [⟨(𝑦 +P 1P), 1P⟩] ~R ) = (𝑦P (𝐹𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ))
7 simpr 107 . . . . . 6 ((𝜑𝐴N) → 𝐴N)
8 caucvgsr.f . . . . . . 7 (𝜑𝐹:NR)
9 caucvgsrlemgt1.gt1 . . . . . . 7 (𝜑 → ∀𝑚N 1R <R (𝐹𝑚))
108, 9caucvgsrlemcl 6930 . . . . . 6 ((𝜑𝐴N) → (𝑦P (𝐹𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ) ∈ P)
112, 6, 7, 10fvmptd 5280 . . . . 5 ((𝜑𝐴N) → (𝐺𝐴) = (𝑦P (𝐹𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ))
1211oveq1d 5554 . . . 4 ((𝜑𝐴N) → ((𝐺𝐴) +P 1P) = ((𝑦P (𝐹𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ) +P 1P))
1312opeq1d 3582 . . 3 ((𝜑𝐴N) → ⟨((𝐺𝐴) +P 1P), 1P⟩ = ⟨((𝑦P (𝐹𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ) +P 1P), 1P⟩)
1413eceq1d 6172 . 2 ((𝜑𝐴N) → [⟨((𝐺𝐴) +P 1P), 1P⟩] ~R = [⟨((𝑦P (𝐹𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ) +P 1P), 1P⟩] ~R )
15 eqcom 2058 . . . . . . 7 ((𝐹𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ↔ [⟨(𝑦 +P 1P), 1P⟩] ~R = (𝐹𝐴))
1615a1i 9 . . . . . 6 (𝑦P → ((𝐹𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ↔ [⟨(𝑦 +P 1P), 1P⟩] ~R = (𝐹𝐴)))
1716riotabiia 5512 . . . . 5 (𝑦P (𝐹𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ) = (𝑦P [⟨(𝑦 +P 1P), 1P⟩] ~R = (𝐹𝐴))
1817oveq1i 5549 . . . 4 ((𝑦P (𝐹𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ) +P 1P) = ((𝑦P [⟨(𝑦 +P 1P), 1P⟩] ~R = (𝐹𝐴)) +P 1P)
1918opeq1i 3579 . . 3 ⟨((𝑦P (𝐹𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ) +P 1P), 1P⟩ = ⟨((𝑦P [⟨(𝑦 +P 1P), 1P⟩] ~R = (𝐹𝐴)) +P 1P), 1P
20 eceq1 6171 . . 3 (⟨((𝑦P (𝐹𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ) +P 1P), 1P⟩ = ⟨((𝑦P [⟨(𝑦 +P 1P), 1P⟩] ~R = (𝐹𝐴)) +P 1P), 1P⟩ → [⟨((𝑦P (𝐹𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ) +P 1P), 1P⟩] ~R = [⟨((𝑦P [⟨(𝑦 +P 1P), 1P⟩] ~R = (𝐹𝐴)) +P 1P), 1P⟩] ~R )
2119, 20mp1i 10 . 2 ((𝜑𝐴N) → [⟨((𝑦P (𝐹𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ) +P 1P), 1P⟩] ~R = [⟨((𝑦P [⟨(𝑦 +P 1P), 1P⟩] ~R = (𝐹𝐴)) +P 1P), 1P⟩] ~R )
228ffvelrnda 5329 . . 3 ((𝜑𝐴N) → (𝐹𝐴) ∈ R)
23 0lt1sr 6907 . . . 4 0R <R 1R
24 fveq2 5205 . . . . . . 7 (𝑚 = 𝐴 → (𝐹𝑚) = (𝐹𝐴))
2524breq2d 3803 . . . . . 6 (𝑚 = 𝐴 → (1R <R (𝐹𝑚) ↔ 1R <R (𝐹𝐴)))
2625rspcv 2669 . . . . 5 (𝐴N → (∀𝑚N 1R <R (𝐹𝑚) → 1R <R (𝐹𝐴)))
279, 26mpan9 269 . . . 4 ((𝜑𝐴N) → 1R <R (𝐹𝐴))
28 ltsosr 6906 . . . . 5 <R Or R
29 ltrelsr 6880 . . . . 5 <R ⊆ (R × R)
3028, 29sotri 4747 . . . 4 ((0R <R 1R ∧ 1R <R (𝐹𝐴)) → 0R <R (𝐹𝐴))
3123, 27, 30sylancr 399 . . 3 ((𝜑𝐴N) → 0R <R (𝐹𝐴))
32 prsrriota 6929 . . 3 (((𝐹𝐴) ∈ R ∧ 0R <R (𝐹𝐴)) → [⟨((𝑦P [⟨(𝑦 +P 1P), 1P⟩] ~R = (𝐹𝐴)) +P 1P), 1P⟩] ~R = (𝐹𝐴))
3322, 31, 32syl2anc 397 . 2 ((𝜑𝐴N) → [⟨((𝑦P [⟨(𝑦 +P 1P), 1P⟩] ~R = (𝐹𝐴)) +P 1P), 1P⟩] ~R = (𝐹𝐴))
3414, 21, 333eqtrd 2092 1 ((𝜑𝐴N) → [⟨((𝐺𝐴) +P 1P), 1P⟩] ~R = (𝐹𝐴))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 101   ↔ wb 102   = wceq 1259   ∈ wcel 1409  {cab 2042  ∀wral 2323  ⟨cop 3405   class class class wbr 3791   ↦ cmpt 3845  ⟶wf 4925  ‘cfv 4929  ℩crio 5494  (class class class)co 5539  1𝑜c1o 6024  [cec 6134  Ncnpi 6427
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