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Theorem caucvgsrlemfv 7563
Description: Lemma for caucvgsr 7574. 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 5387 . . . . . . . . 9 (𝑥 = 𝐴 → (𝐹𝑥) = (𝐹𝐴))
43eqeq1d 2124 . . . . . . . 8 (𝑥 = 𝐴 → ((𝐹𝑥) = [⟨(𝑦 +P 1P), 1P⟩] ~R ↔ (𝐹𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ))
54riotabidv 5698 . . . . . . 7 (𝑥 = 𝐴 → (𝑦P (𝐹𝑥) = [⟨(𝑦 +P 1P), 1P⟩] ~R ) = (𝑦P (𝐹𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ))
65adantl 273 . . . . . 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 7561 . . . . . 6 ((𝜑𝐴N) → (𝑦P (𝐹𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ) ∈ P)
112, 6, 7, 10fvmptd 5468 . . . . 5 ((𝜑𝐴N) → (𝐺𝐴) = (𝑦P (𝐹𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ))
1211oveq1d 5755 . . . 4 ((𝜑𝐴N) → ((𝐺𝐴) +P 1P) = ((𝑦P (𝐹𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ) +P 1P))
1312opeq1d 3679 . . 3 ((𝜑𝐴N) → ⟨((𝐺𝐴) +P 1P), 1P⟩ = ⟨((𝑦P (𝐹𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ) +P 1P), 1P⟩)
1413eceq1d 6431 . 2 ((𝜑𝐴N) → [⟨((𝐺𝐴) +P 1P), 1P⟩] ~R = [⟨((𝑦P (𝐹𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ) +P 1P), 1P⟩] ~R )
15 eqcom 2117 . . . . . . 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 5713 . . . . 5 (𝑦P (𝐹𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ) = (𝑦P [⟨(𝑦 +P 1P), 1P⟩] ~R = (𝐹𝐴))
1817oveq1i 5750 . . . 4 ((𝑦P (𝐹𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ) +P 1P) = ((𝑦P [⟨(𝑦 +P 1P), 1P⟩] ~R = (𝐹𝐴)) +P 1P)
1918opeq1i 3676 . . 3 ⟨((𝑦P (𝐹𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ) +P 1P), 1P⟩ = ⟨((𝑦P [⟨(𝑦 +P 1P), 1P⟩] ~R = (𝐹𝐴)) +P 1P), 1P
20 eceq1 6430 . . 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 5521 . . 3 ((𝜑𝐴N) → (𝐹𝐴) ∈ R)
23 0lt1sr 7537 . . . 4 0R <R 1R
24 fveq2 5387 . . . . . . 7 (𝑚 = 𝐴 → (𝐹𝑚) = (𝐹𝐴))
2524breq2d 3909 . . . . . 6 (𝑚 = 𝐴 → (1R <R (𝐹𝑚) ↔ 1R <R (𝐹𝐴)))
2625rspcv 2757 . . . . 5 (𝐴N → (∀𝑚N 1R <R (𝐹𝑚) → 1R <R (𝐹𝐴)))
279, 26mpan9 277 . . . 4 ((𝜑𝐴N) → 1R <R (𝐹𝐴))
28 ltsosr 7536 . . . . 5 <R Or R
29 ltrelsr 7510 . . . . 5 <R ⊆ (R × R)
3028, 29sotri 4902 . . . 4 ((0R <R 1R ∧ 1R <R (𝐹𝐴)) → 0R <R (𝐹𝐴))
3123, 27, 30sylancr 408 . . 3 ((𝜑𝐴N) → 0R <R (𝐹𝐴))
32 prsrriota 7560 . . 3 (((𝐹𝐴) ∈ R ∧ 0R <R (𝐹𝐴)) → [⟨((𝑦P [⟨(𝑦 +P 1P), 1P⟩] ~R = (𝐹𝐴)) +P 1P), 1P⟩] ~R = (𝐹𝐴))
3322, 31, 32syl2anc 406 . 2 ((𝜑𝐴N) → [⟨((𝑦P [⟨(𝑦 +P 1P), 1P⟩] ~R = (𝐹𝐴)) +P 1P), 1P⟩] ~R = (𝐹𝐴))
3414, 21, 333eqtrd 2152 1 ((𝜑𝐴N) → [⟨((𝐺𝐴) +P 1P), 1P⟩] ~R = (𝐹𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1314  wcel 1463  {cab 2101  wral 2391  cop 3498   class class class wbr 3897  cmpt 3957  wf 5087  cfv 5091  crio 5695  (class class class)co 5740  1oc1o 6272  [cec 6393  Ncnpi 7044   <N clti 7047   ~Q ceq 7051  *Qcrq 7056   <Q cltq 7057  Pcnp 7063  1Pc1p 7064   +P cpp 7065   ~R cer 7068  Rcnr 7069  0Rc0r 7070  1Rc1r 7071   +R cplr 7073   <R cltr 7075
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-reu 2398  df-rmo 2399  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-eprel 4179  df-id 4183  df-po 4186  df-iso 4187  df-iord 4256  df-on 4258  df-suc 4261  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-riota 5696  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-recs 6168  df-irdg 6233  df-1o 6279  df-2o 6280  df-oadd 6283  df-omul 6284  df-er 6395  df-ec 6397  df-qs 6401  df-ni 7076  df-pli 7077  df-mi 7078  df-lti 7079  df-plpq 7116  df-mpq 7117  df-enq 7119  df-nqqs 7120  df-plqqs 7121  df-mqqs 7122  df-1nqqs 7123  df-rq 7124  df-ltnqqs 7125  df-enq0 7196  df-nq0 7197  df-0nq0 7198  df-plq0 7199  df-mq0 7200  df-inp 7238  df-i1p 7239  df-iplp 7240  df-iltp 7242  df-enr 7498  df-nr 7499  df-ltr 7502  df-0r 7503  df-1r 7504
This theorem is referenced by:  caucvgsrlemcau  7565  caucvgsrlembound  7566  caucvgsrlemgt1  7567
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