Theorem caucvgsrlemfv 7886

Description: Lemma for caucvgsr 7897. 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 5570	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴))
4	3	eqeq1d 2213	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → ((𝐹‘𝑥) = [⟨(𝑦 +P 1P), 1P⟩] ~R ↔ (𝐹‘𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ))
5	4	riotabidv 5891	. . . . . . 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 7884	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ N) → (℩𝑦 ∈ P (𝐹‘𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ) ∈ P)
11	2, 6, 7, 10	fvmptd 5654	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ N) → (𝐺‘𝐴) = (℩𝑦 ∈ P (𝐹‘𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ))
12	11	oveq1d 5949	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ N) → ((𝐺‘𝐴) +P 1P) = ((℩𝑦 ∈ P (𝐹‘𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ) +P 1P))
13	12	opeq1d 3824	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ N) → ⟨((𝐺‘𝐴) +P 1P), 1P⟩ = ⟨((℩𝑦 ∈ P (𝐹‘𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ) +P 1P), 1P⟩)
14	13	eceq1d 6646	. 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 2206	. . . . . . 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 5907	. . . . 5 ⊢ (℩𝑦 ∈ P (𝐹‘𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ) = (℩𝑦 ∈ P [⟨(𝑦 +P 1P), 1P⟩] ~R = (𝐹‘𝐴))
18	17	oveq1i 5944	. . . 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 3821	. . 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 6645	. . 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 5709	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ N) → (𝐹‘𝐴) ∈ R)
23		0lt1sr 7860	. . . 4 ⊢ 0R <R 1R
24		fveq2 5570	. . . . . . 7 ⊢ (𝑚 = 𝐴 → (𝐹‘𝑚) = (𝐹‘𝐴))
25	24	breq2d 4055	. . . . . 6 ⊢ (𝑚 = 𝐴 → (1R <R (𝐹‘𝑚) ↔ 1R <R (𝐹‘𝐴)))
26	25	rspcv 2872	. . . . 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 7859	. . . . 5 ⊢ <R Or R
29		ltrelsr 7833	. . . . 5 ⊢ <R ⊆ (R × R)
30	28, 29	sotri 5075	. . . 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 7883	. . 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 2241	1 ⊢ ((𝜑 ∧ 𝐴 ∈ N) → [⟨((𝐺‘𝐴) +P 1P), 1P⟩] ~R = (𝐹‘𝐴))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1372   ∈ wcel 2175  {cab 2190  ∀wral 2483  ⟨cop 3635   class class class wbr 4043   ↦ cmpt 4104  ⟶wf 5264  ‘cfv 5268  ℩crio 5888  (class class class)co 5934  1_oc1o 6485  [cec 6608  Ncnpi 7367   <_N clti 7370   ~_Q ceq 7374  *_Qcrq 7379   <_Q cltq 7380  Pcnp 7386  1_Pc1p 7387   +_P cpp 7388   ~_R cer 7391  Rcnr 7392  0_Rc0r 7393  1_Rc1r 7394   +_R cplr 7396   <_R cltr 7398

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-nul 4169  ax-pow 4217  ax-pr 4252  ax-un 4478  ax-setind 4583  ax-iinf 4634

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-ral 2488  df-rex 2489  df-reu 2490  df-rmo 2491  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-tr 4142  df-eprel 4334  df-id 4338  df-po 4341  df-iso 4342  df-iord 4411  df-on 4413  df-suc 4416  df-iom 4637  df-xp 4679  df-rel 4680  df-cnv 4681  df-co 4682  df-dm 4683  df-rn 4684  df-res 4685  df-ima 4686  df-iota 5229  df-fun 5270  df-fn 5271  df-f 5272  df-f1 5273  df-fo 5274  df-f1o 5275  df-fv 5276  df-riota 5889  df-ov 5937  df-oprab 5938  df-mpo 5939  df-1st 6216  df-2nd 6217  df-recs 6381  df-irdg 6446  df-1o 6492  df-2o 6493  df-oadd 6496  df-omul 6497  df-er 6610  df-ec 6612  df-qs 6616  df-ni 7399  df-pli 7400  df-mi 7401  df-lti 7402  df-plpq 7439  df-mpq 7440  df-enq 7442  df-nqqs 7443  df-plqqs 7444  df-mqqs 7445  df-1nqqs 7446  df-rq 7447  df-ltnqqs 7448  df-enq0 7519  df-nq0 7520  df-0nq0 7521  df-plq0 7522  df-mq0 7523  df-inp 7561  df-i1p 7562  df-iplp 7563  df-iltp 7565  df-enr 7821  df-nr 7822  df-ltr 7825  df-0r 7826  df-1r 7827

This theorem is referenced by:  caucvgsrlemcau  7888  caucvgsrlembound  7889  caucvgsrlemgt1  7890