Theorem caucvgsrlemfv 7807

Description: Lemma for caucvgsr 7818. Coercing sequence value from a positive real to a signed real. (Contributed by Jim Kingdon, 29-Jun-2021.)

Hypotheses

Ref	Expression
caucvgsr.f	⊢ (𝜑 → 𝐹:N⟶R)
caucvgsr.cau	⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <R ((𝐹‘𝑘) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ) ∧ (𝐹‘𝑘) <R ((𝐹‘𝑛) +R [⟨(⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ))))
caucvgsrlemgt1.gt1	⊢ (𝜑 → ∀𝑚 ∈ N 1R <R (𝐹‘𝑚))
caucvgsrlemf.xfr	⊢ 𝐺 = (𝑥 ∈ N ↦ (℩𝑦 ∈ P (𝐹‘𝑥) = [⟨(𝑦 +P 1P), 1P⟩] ~R ))

Assertion

Ref	Expression
caucvgsrlemfv	⊢ ((𝜑 ∧ 𝐴 ∈ N) → [⟨((𝐺‘𝐴) +P 1P), 1P⟩] ~R = (𝐹‘𝐴))

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

Step	Hyp	Ref	Expression
1		caucvgsrlemf.xfr	. . . . . . 7 ⊢ 𝐺 = (𝑥 ∈ N ↦ (℩𝑦 ∈ P (𝐹‘𝑥) = [⟨(𝑦 +P 1P), 1P⟩] ~R ))
2	1	a1i 9	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ N) → 𝐺 = (𝑥 ∈ N ↦ (℩𝑦 ∈ P (𝐹‘𝑥) = [⟨(𝑦 +P 1P), 1P⟩] ~R )))
3		fveq2 5529	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴))
4	3	eqeq1d 2197	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → ((𝐹‘𝑥) = [⟨(𝑦 +P 1P), 1P⟩] ~R ↔ (𝐹‘𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ))
5	4	riotabidv 5848	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (℩𝑦 ∈ P (𝐹‘𝑥) = [⟨(𝑦 +P 1P), 1P⟩] ~R ) = (℩𝑦 ∈ P (𝐹‘𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ))
6	5	adantl 277	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ∈ N) ∧ 𝑥 = 𝐴) → (℩𝑦 ∈ P (𝐹‘𝑥) = [⟨(𝑦 +P 1P), 1P⟩] ~R ) = (℩𝑦 ∈ P (𝐹‘𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ))
7		simpr 110	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ N) → 𝐴 ∈ N)
8		caucvgsr.f	. . . . . . 7 ⊢ (𝜑 → 𝐹:N⟶R)
9		caucvgsrlemgt1.gt1	. . . . . . 7 ⊢ (𝜑 → ∀𝑚 ∈ N 1R <R (𝐹‘𝑚))
10	8, 9	caucvgsrlemcl 7805	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ N) → (℩𝑦 ∈ P (𝐹‘𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ) ∈ P)
11	2, 6, 7, 10	fvmptd 5612	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ N) → (𝐺‘𝐴) = (℩𝑦 ∈ P (𝐹‘𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ))
12	11	oveq1d 5905	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ N) → ((𝐺‘𝐴) +P 1P) = ((℩𝑦 ∈ P (𝐹‘𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ) +P 1P))
13	12	opeq1d 3798	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ N) → ⟨((𝐺‘𝐴) +P 1P), 1P⟩ = ⟨((℩𝑦 ∈ P (𝐹‘𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ) +P 1P), 1P⟩)
14	13	eceq1d 6588	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ N) → [⟨((𝐺‘𝐴) +P 1P), 1P⟩] ~R = [⟨((℩𝑦 ∈ P (𝐹‘𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ) +P 1P), 1P⟩] ~R )
15		eqcom 2190	. . . . . . 7 ⊢ ((𝐹‘𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ↔ [⟨(𝑦 +P 1P), 1P⟩] ~R = (𝐹‘𝐴))
16	15	a1i 9	. . . . . 6 ⊢ (𝑦 ∈ P → ((𝐹‘𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ↔ [⟨(𝑦 +P 1P), 1P⟩] ~R = (𝐹‘𝐴)))
17	16	riotabiia 5863	. . . . 5 ⊢ (℩𝑦 ∈ P (𝐹‘𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ) = (℩𝑦 ∈ P [⟨(𝑦 +P 1P), 1P⟩] ~R = (𝐹‘𝐴))
18	17	oveq1i 5900	. . . 4 ⊢ ((℩𝑦 ∈ P (𝐹‘𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ) +P 1P) = ((℩𝑦 ∈ P [⟨(𝑦 +P 1P), 1P⟩] ~R = (𝐹‘𝐴)) +P 1P)
19	18	opeq1i 3795	. . 3 ⊢ ⟨((℩𝑦 ∈ P (𝐹‘𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ) +P 1P), 1P⟩ = ⟨((℩𝑦 ∈ P [⟨(𝑦 +P 1P), 1P⟩] ~R = (𝐹‘𝐴)) +P 1P), 1P⟩
20		eceq1 6587	. . 3 ⊢ (⟨((℩𝑦 ∈ P (𝐹‘𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ) +P 1P), 1P⟩ = ⟨((℩𝑦 ∈ P [⟨(𝑦 +P 1P), 1P⟩] ~R = (𝐹‘𝐴)) +P 1P), 1P⟩ → [⟨((℩𝑦 ∈ P (𝐹‘𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ) +P 1P), 1P⟩] ~R = [⟨((℩𝑦 ∈ P [⟨(𝑦 +P 1P), 1P⟩] ~R = (𝐹‘𝐴)) +P 1P), 1P⟩] ~R )
21	19, 20	mp1i 10	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ N) → [⟨((℩𝑦 ∈ P (𝐹‘𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ) +P 1P), 1P⟩] ~R = [⟨((℩𝑦 ∈ P [⟨(𝑦 +P 1P), 1P⟩] ~R = (𝐹‘𝐴)) +P 1P), 1P⟩] ~R )
22	8	ffvelcdmda 5666	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ N) → (𝐹‘𝐴) ∈ R)
23		0lt1sr 7781	. . . 4 ⊢ 0R <R 1R
24		fveq2 5529	. . . . . . 7 ⊢ (𝑚 = 𝐴 → (𝐹‘𝑚) = (𝐹‘𝐴))
25	24	breq2d 4029	. . . . . 6 ⊢ (𝑚 = 𝐴 → (1R <R (𝐹‘𝑚) ↔ 1R <R (𝐹‘𝐴)))
26	25	rspcv 2851	. . . . 5 ⊢ (𝐴 ∈ N → (∀𝑚 ∈ N 1R <R (𝐹‘𝑚) → 1R <R (𝐹‘𝐴)))
27	9, 26	mpan9 281	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ N) → 1R <R (𝐹‘𝐴))
28		ltsosr 7780	. . . . 5 ⊢ <R Or R
29		ltrelsr 7754	. . . . 5 ⊢ <R ⊆ (R × R)
30	28, 29	sotri 5038	. . . 4 ⊢ ((0R <R 1R ∧ 1R <R (𝐹‘𝐴)) → 0R <R (𝐹‘𝐴))
31	23, 27, 30	sylancr 414	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ N) → 0R <R (𝐹‘𝐴))
32		prsrriota 7804	. . 3 ⊢ (((𝐹‘𝐴) ∈ R ∧ 0R <R (𝐹‘𝐴)) → [⟨((℩𝑦 ∈ P [⟨(𝑦 +P 1P), 1P⟩] ~R = (𝐹‘𝐴)) +P 1P), 1P⟩] ~R = (𝐹‘𝐴))
33	22, 31, 32	syl2anc 411	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ N) → [⟨((℩𝑦 ∈ P [⟨(𝑦 +P 1P), 1P⟩] ~R = (𝐹‘𝐴)) +P 1P), 1P⟩] ~R = (𝐹‘𝐴))
34	14, 21, 33	3eqtrd 2225	1 ⊢ ((𝜑 ∧ 𝐴 ∈ N) → [⟨((𝐺‘𝐴) +P 1P), 1P⟩] ~R = (𝐹‘𝐴))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1363   ∈ wcel 2159  {cab 2174  ∀wral 2467  ⟨cop 3609   class class class wbr 4017   ↦ cmpt 4078  ⟶wf 5226  ‘cfv 5230  ℩crio 5845  (class class class)co 5890  1_oc1o 6427  [cec 6550  Ncnpi 7288   <_N clti 7291   ~_Q ceq 7295  *_Qcrq 7300   <_Q cltq 7301  Pcnp 7307  1_Pc1p 7308   +_P cpp 7309   ~_R cer 7312  Rcnr 7313  0_Rc0r 7314  1_Rc1r 7315   +_R cplr 7317   <_R cltr 7319

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2161  ax-14 2162  ax-ext 2170  ax-coll 4132  ax-sep 4135  ax-nul 4143  ax-pow 4188  ax-pr 4223  ax-un 4447  ax-setind 4550  ax-iinf 4601

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2040  df-mo 2041  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-ne 2360  df-ral 2472  df-rex 2473  df-reu 2474  df-rmo 2475  df-rab 2476  df-v 2753  df-sbc 2977  df-csb 3072  df-dif 3145  df-un 3147  df-in 3149  df-ss 3156  df-nul 3437  df-pw 3591  df-sn 3612  df-pr 3613  df-op 3615  df-uni 3824  df-int 3859  df-iun 3902  df-br 4018  df-opab 4079  df-mpt 4080  df-tr 4116  df-eprel 4303  df-id 4307  df-po 4310  df-iso 4311  df-iord 4380  df-on 4382  df-suc 4385  df-iom 4604  df-xp 4646  df-rel 4647  df-cnv 4648  df-co 4649  df-dm 4650  df-rn 4651  df-res 4652  df-ima 4653  df-iota 5192  df-fun 5232  df-fn 5233  df-f 5234  df-f1 5235  df-fo 5236  df-f1o 5237  df-fv 5238  df-riota 5846  df-ov 5893  df-oprab 5894  df-mpo 5895  df-1st 6158  df-2nd 6159  df-recs 6323  df-irdg 6388  df-1o 6434  df-2o 6435  df-oadd 6438  df-omul 6439  df-er 6552  df-ec 6554  df-qs 6558  df-ni 7320  df-pli 7321  df-mi 7322  df-lti 7323  df-plpq 7360  df-mpq 7361  df-enq 7363  df-nqqs 7364  df-plqqs 7365  df-mqqs 7366  df-1nqqs 7367  df-rq 7368  df-ltnqqs 7369  df-enq0 7440  df-nq0 7441  df-0nq0 7442  df-plq0 7443  df-mq0 7444  df-inp 7482  df-i1p 7483  df-iplp 7484  df-iltp 7486  df-enr 7742  df-nr 7743  df-ltr 7746  df-0r 7747  df-1r 7748

This theorem is referenced by:  caucvgsrlemcau  7809  caucvgsrlembound  7810  caucvgsrlemgt1  7811