Theorem caucvgsrlemfv 7858

Description: Lemma for caucvgsr 7869. 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 5558	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴))
4	3	eqeq1d 2205	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → ((𝐹‘𝑥) = [⟨(𝑦 +P 1P), 1P⟩] ~R ↔ (𝐹‘𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ))
5	4	riotabidv 5879	. . . . . . 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 7856	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ N) → (℩𝑦 ∈ P (𝐹‘𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ) ∈ P)
11	2, 6, 7, 10	fvmptd 5642	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ N) → (𝐺‘𝐴) = (℩𝑦 ∈ P (𝐹‘𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ))
12	11	oveq1d 5937	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ N) → ((𝐺‘𝐴) +P 1P) = ((℩𝑦 ∈ P (𝐹‘𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ) +P 1P))
13	12	opeq1d 3814	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ N) → ⟨((𝐺‘𝐴) +P 1P), 1P⟩ = ⟨((℩𝑦 ∈ P (𝐹‘𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ) +P 1P), 1P⟩)
14	13	eceq1d 6628	. 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 2198	. . . . . . 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 5895	. . . . 5 ⊢ (℩𝑦 ∈ P (𝐹‘𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ) = (℩𝑦 ∈ P [⟨(𝑦 +P 1P), 1P⟩] ~R = (𝐹‘𝐴))
18	17	oveq1i 5932	. . . 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 3811	. . 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 6627	. . 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 5697	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ N) → (𝐹‘𝐴) ∈ R)
23		0lt1sr 7832	. . . 4 ⊢ 0R <R 1R
24		fveq2 5558	. . . . . . 7 ⊢ (𝑚 = 𝐴 → (𝐹‘𝑚) = (𝐹‘𝐴))
25	24	breq2d 4045	. . . . . 6 ⊢ (𝑚 = 𝐴 → (1R <R (𝐹‘𝑚) ↔ 1R <R (𝐹‘𝐴)))
26	25	rspcv 2864	. . . . 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 7831	. . . . 5 ⊢ <R Or R
29		ltrelsr 7805	. . . . 5 ⊢ <R ⊆ (R × R)
30	28, 29	sotri 5065	. . . 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 7855	. . 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 2233	1 ⊢ ((𝜑 ∧ 𝐴 ∈ N) → [⟨((𝐺‘𝐴) +P 1P), 1P⟩] ~R = (𝐹‘𝐴))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364   ∈ wcel 2167  {cab 2182  ∀wral 2475  ⟨cop 3625   class class class wbr 4033   ↦ cmpt 4094  ⟶wf 5254  ‘cfv 5258  ℩crio 5876  (class class class)co 5922  1_oc1o 6467  [cec 6590  Ncnpi 7339   <_N clti 7342   ~_Q ceq 7346  *_Qcrq 7351   <_Q cltq 7352  Pcnp 7358  1_Pc1p 7359   +_P cpp 7360   ~_R cer 7363  Rcnr 7364  0_Rc0r 7365  1_Rc1r 7366   +_R cplr 7368   <_R cltr 7370

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-eprel 4324  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-irdg 6428  df-1o 6474  df-2o 6475  df-oadd 6478  df-omul 6479  df-er 6592  df-ec 6594  df-qs 6598  df-ni 7371  df-pli 7372  df-mi 7373  df-lti 7374  df-plpq 7411  df-mpq 7412  df-enq 7414  df-nqqs 7415  df-plqqs 7416  df-mqqs 7417  df-1nqqs 7418  df-rq 7419  df-ltnqqs 7420  df-enq0 7491  df-nq0 7492  df-0nq0 7493  df-plq0 7494  df-mq0 7495  df-inp 7533  df-i1p 7534  df-iplp 7535  df-iltp 7537  df-enr 7793  df-nr 7794  df-ltr 7797  df-0r 7798  df-1r 7799

This theorem is referenced by:  caucvgsrlemcau  7860  caucvgsrlembound  7861  caucvgsrlemgt1  7862