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Theorem caucvgsrlemfv 6873
Description: Lemma for caucvgsr 6884. 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 5178 . . . . . . . . 9 (𝑥 = 𝐴 → (𝐹𝑥) = (𝐹𝐴))
43eqeq1d 2048 . . . . . . . 8 (𝑥 = 𝐴 → ((𝐹𝑥) = [⟨(𝑦 +P 1P), 1P⟩] ~R ↔ (𝐹𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ))
54riotabidv 5470 . . . . . . 7 (𝑥 = 𝐴 → (𝑦P (𝐹𝑥) = [⟨(𝑦 +P 1P), 1P⟩] ~R ) = (𝑦P (𝐹𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ))
65adantl 262 . . . . . 6 (((𝜑𝐴N) ∧ 𝑥 = 𝐴) → (𝑦P (𝐹𝑥) = [⟨(𝑦 +P 1P), 1P⟩] ~R ) = (𝑦P (𝐹𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ))
7 simpr 103 . . . . . 6 ((𝜑𝐴N) → 𝐴N)
8 caucvgsr.f . . . . . . 7 (𝜑𝐹:NR)
9 caucvgsrlemgt1.gt1 . . . . . . 7 (𝜑 → ∀𝑚N 1R <R (𝐹𝑚))
108, 9caucvgsrlemcl 6871 . . . . . 6 ((𝜑𝐴N) → (𝑦P (𝐹𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ) ∈ P)
112, 6, 7, 10fvmptd 5253 . . . . 5 ((𝜑𝐴N) → (𝐺𝐴) = (𝑦P (𝐹𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ))
1211oveq1d 5527 . . . 4 ((𝜑𝐴N) → ((𝐺𝐴) +P 1P) = ((𝑦P (𝐹𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ) +P 1P))
1312opeq1d 3555 . . 3 ((𝜑𝐴N) → ⟨((𝐺𝐴) +P 1P), 1P⟩ = ⟨((𝑦P (𝐹𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ) +P 1P), 1P⟩)
1413eceq1d 6142 . 2 ((𝜑𝐴N) → [⟨((𝐺𝐴) +P 1P), 1P⟩] ~R = [⟨((𝑦P (𝐹𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ) +P 1P), 1P⟩] ~R )
15 eqcom 2042 . . . . . . 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 5485 . . . . 5 (𝑦P (𝐹𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ) = (𝑦P [⟨(𝑦 +P 1P), 1P⟩] ~R = (𝐹𝐴))
1817oveq1i 5522 . . . 4 ((𝑦P (𝐹𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ) +P 1P) = ((𝑦P [⟨(𝑦 +P 1P), 1P⟩] ~R = (𝐹𝐴)) +P 1P)
1918opeq1i 3552 . . 3 ⟨((𝑦P (𝐹𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ) +P 1P), 1P⟩ = ⟨((𝑦P [⟨(𝑦 +P 1P), 1P⟩] ~R = (𝐹𝐴)) +P 1P), 1P
20 eceq1 6141 . . 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 5302 . . 3 ((𝜑𝐴N) → (𝐹𝐴) ∈ R)
23 0lt1sr 6848 . . . 4 0R <R 1R
24 fveq2 5178 . . . . . . 7 (𝑚 = 𝐴 → (𝐹𝑚) = (𝐹𝐴))
2524breq2d 3776 . . . . . 6 (𝑚 = 𝐴 → (1R <R (𝐹𝑚) ↔ 1R <R (𝐹𝐴)))
2625rspcv 2652 . . . . 5 (𝐴N → (∀𝑚N 1R <R (𝐹𝑚) → 1R <R (𝐹𝐴)))
279, 26mpan9 265 . . . 4 ((𝜑𝐴N) → 1R <R (𝐹𝐴))
28 ltsosr 6847 . . . . 5 <R Or R
29 ltrelsr 6821 . . . . 5 <R ⊆ (R × R)
3028, 29sotri 4720 . . . 4 ((0R <R 1R ∧ 1R <R (𝐹𝐴)) → 0R <R (𝐹𝐴))
3123, 27, 30sylancr 393 . . 3 ((𝜑𝐴N) → 0R <R (𝐹𝐴))
32 prsrriota 6870 . . 3 (((𝐹𝐴) ∈ R ∧ 0R <R (𝐹𝐴)) → [⟨((𝑦P [⟨(𝑦 +P 1P), 1P⟩] ~R = (𝐹𝐴)) +P 1P), 1P⟩] ~R = (𝐹𝐴))
3322, 31, 32syl2anc 391 . 2 ((𝜑𝐴N) → [⟨((𝑦P [⟨(𝑦 +P 1P), 1P⟩] ~R = (𝐹𝐴)) +P 1P), 1P⟩] ~R = (𝐹𝐴))
3414, 21, 333eqtrd 2076 1 ((𝜑𝐴N) → [⟨((𝐺𝐴) +P 1P), 1P⟩] ~R = (𝐹𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wb 98   = wceq 1243  wcel 1393  {cab 2026  wral 2306  cop 3378   class class class wbr 3764  cmpt 3818  wf 4898  cfv 4902  crio 5467  (class class class)co 5512  1𝑜c1o 5994  [cec 6104  Ncnpi 6368   <N clti 6371   ~Q ceq 6375  *Qcrq 6380   <Q cltq 6381  Pcnp 6387  1Pc1p 6388   +P cpp 6389   ~R cer 6392  Rcnr 6393  0Rc0r 6394  1Rc1r 6395   +R cplr 6397   <R cltr 6399
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-iltp 6566  df-enr 6809  df-nr 6810  df-ltr 6813  df-0r 6814  df-1r 6815
This theorem is referenced by:  caucvgsrlemcau  6875  caucvgsrlembound  6876  caucvgsrlemgt1  6877
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