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Theorem caucvgsrlemfv 7713
Description: Lemma for caucvgsr 7724. 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 5470 . . . . . . . . 9 (𝑥 = 𝐴 → (𝐹𝑥) = (𝐹𝐴))
43eqeq1d 2166 . . . . . . . 8 (𝑥 = 𝐴 → ((𝐹𝑥) = [⟨(𝑦 +P 1P), 1P⟩] ~R ↔ (𝐹𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ))
54riotabidv 5784 . . . . . . 7 (𝑥 = 𝐴 → (𝑦P (𝐹𝑥) = [⟨(𝑦 +P 1P), 1P⟩] ~R ) = (𝑦P (𝐹𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ))
65adantl 275 . . . . . 6 (((𝜑𝐴N) ∧ 𝑥 = 𝐴) → (𝑦P (𝐹𝑥) = [⟨(𝑦 +P 1P), 1P⟩] ~R ) = (𝑦P (𝐹𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ))
7 simpr 109 . . . . . 6 ((𝜑𝐴N) → 𝐴N)
8 caucvgsr.f . . . . . . 7 (𝜑𝐹:NR)
9 caucvgsrlemgt1.gt1 . . . . . . 7 (𝜑 → ∀𝑚N 1R <R (𝐹𝑚))
108, 9caucvgsrlemcl 7711 . . . . . 6 ((𝜑𝐴N) → (𝑦P (𝐹𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ) ∈ P)
112, 6, 7, 10fvmptd 5551 . . . . 5 ((𝜑𝐴N) → (𝐺𝐴) = (𝑦P (𝐹𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ))
1211oveq1d 5841 . . . 4 ((𝜑𝐴N) → ((𝐺𝐴) +P 1P) = ((𝑦P (𝐹𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ) +P 1P))
1312opeq1d 3749 . . 3 ((𝜑𝐴N) → ⟨((𝐺𝐴) +P 1P), 1P⟩ = ⟨((𝑦P (𝐹𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ) +P 1P), 1P⟩)
1413eceq1d 6518 . 2 ((𝜑𝐴N) → [⟨((𝐺𝐴) +P 1P), 1P⟩] ~R = [⟨((𝑦P (𝐹𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ) +P 1P), 1P⟩] ~R )
15 eqcom 2159 . . . . . . 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 5799 . . . . 5 (𝑦P (𝐹𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ) = (𝑦P [⟨(𝑦 +P 1P), 1P⟩] ~R = (𝐹𝐴))
1817oveq1i 5836 . . . 4 ((𝑦P (𝐹𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ) +P 1P) = ((𝑦P [⟨(𝑦 +P 1P), 1P⟩] ~R = (𝐹𝐴)) +P 1P)
1918opeq1i 3746 . . 3 ⟨((𝑦P (𝐹𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ) +P 1P), 1P⟩ = ⟨((𝑦P [⟨(𝑦 +P 1P), 1P⟩] ~R = (𝐹𝐴)) +P 1P), 1P
20 eceq1 6517 . . 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 5604 . . 3 ((𝜑𝐴N) → (𝐹𝐴) ∈ R)
23 0lt1sr 7687 . . . 4 0R <R 1R
24 fveq2 5470 . . . . . . 7 (𝑚 = 𝐴 → (𝐹𝑚) = (𝐹𝐴))
2524breq2d 3979 . . . . . 6 (𝑚 = 𝐴 → (1R <R (𝐹𝑚) ↔ 1R <R (𝐹𝐴)))
2625rspcv 2812 . . . . 5 (𝐴N → (∀𝑚N 1R <R (𝐹𝑚) → 1R <R (𝐹𝐴)))
279, 26mpan9 279 . . . 4 ((𝜑𝐴N) → 1R <R (𝐹𝐴))
28 ltsosr 7686 . . . . 5 <R Or R
29 ltrelsr 7660 . . . . 5 <R ⊆ (R × R)
3028, 29sotri 4983 . . . 4 ((0R <R 1R ∧ 1R <R (𝐹𝐴)) → 0R <R (𝐹𝐴))
3123, 27, 30sylancr 411 . . 3 ((𝜑𝐴N) → 0R <R (𝐹𝐴))
32 prsrriota 7710 . . 3 (((𝐹𝐴) ∈ R ∧ 0R <R (𝐹𝐴)) → [⟨((𝑦P [⟨(𝑦 +P 1P), 1P⟩] ~R = (𝐹𝐴)) +P 1P), 1P⟩] ~R = (𝐹𝐴))
3322, 31, 32syl2anc 409 . 2 ((𝜑𝐴N) → [⟨((𝑦P [⟨(𝑦 +P 1P), 1P⟩] ~R = (𝐹𝐴)) +P 1P), 1P⟩] ~R = (𝐹𝐴))
3414, 21, 333eqtrd 2194 1 ((𝜑𝐴N) → [⟨((𝐺𝐴) +P 1P), 1P⟩] ~R = (𝐹𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1335  wcel 2128  {cab 2143  wral 2435  cop 3564   class class class wbr 3967  cmpt 4027  wf 5168  cfv 5172  crio 5781  (class class class)co 5826  1oc1o 6358  [cec 6480  Ncnpi 7194   <N clti 7197   ~Q ceq 7201  *Qcrq 7206   <Q cltq 7207  Pcnp 7213  1Pc1p 7214   +P cpp 7215   ~R cer 7218  Rcnr 7219  0Rc0r 7220  1Rc1r 7221   +R cplr 7223   <R cltr 7225
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4081  ax-sep 4084  ax-nul 4092  ax-pow 4137  ax-pr 4171  ax-un 4395  ax-setind 4498  ax-iinf 4549
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3396  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-int 3810  df-iun 3853  df-br 3968  df-opab 4028  df-mpt 4029  df-tr 4065  df-eprel 4251  df-id 4255  df-po 4258  df-iso 4259  df-iord 4328  df-on 4330  df-suc 4333  df-iom 4552  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-iota 5137  df-fun 5174  df-fn 5175  df-f 5176  df-f1 5177  df-fo 5178  df-f1o 5179  df-fv 5180  df-riota 5782  df-ov 5829  df-oprab 5830  df-mpo 5831  df-1st 6090  df-2nd 6091  df-recs 6254  df-irdg 6319  df-1o 6365  df-2o 6366  df-oadd 6369  df-omul 6370  df-er 6482  df-ec 6484  df-qs 6488  df-ni 7226  df-pli 7227  df-mi 7228  df-lti 7229  df-plpq 7266  df-mpq 7267  df-enq 7269  df-nqqs 7270  df-plqqs 7271  df-mqqs 7272  df-1nqqs 7273  df-rq 7274  df-ltnqqs 7275  df-enq0 7346  df-nq0 7347  df-0nq0 7348  df-plq0 7349  df-mq0 7350  df-inp 7388  df-i1p 7389  df-iplp 7390  df-iltp 7392  df-enr 7648  df-nr 7649  df-ltr 7652  df-0r 7653  df-1r 7654
This theorem is referenced by:  caucvgsrlemcau  7715  caucvgsrlembound  7716  caucvgsrlemgt1  7717
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