Theorem caucvgsrlemfv 8122

Description: Lemma for caucvgsr 8133. 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 5675	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴))
4	3	eqeq1d 2243	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → ((𝐹‘𝑥) = [⟨(𝑦 +P 1P), 1P⟩] ~R ↔ (𝐹‘𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ))
5	4	riotabidv 6013	. . . . . . 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 8120	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ N) → (℩𝑦 ∈ P (𝐹‘𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ) ∈ P)
11	2, 6, 7, 10	fvmptd 5763	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ N) → (𝐺‘𝐴) = (℩𝑦 ∈ P (𝐹‘𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ))
12	11	oveq1d 6073	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ N) → ((𝐺‘𝐴) +P 1P) = ((℩𝑦 ∈ P (𝐹‘𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ) +P 1P))
13	12	opeq1d 3894	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ N) → ⟨((𝐺‘𝐴) +P 1P), 1P⟩ = ⟨((℩𝑦 ∈ P (𝐹‘𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ) +P 1P), 1P⟩)
14	13	eceq1d 6816	. 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 2236	. . . . . . 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 6030	. . . . 5 ⊢ (℩𝑦 ∈ P (𝐹‘𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ) = (℩𝑦 ∈ P [⟨(𝑦 +P 1P), 1P⟩] ~R = (𝐹‘𝐴))
18	17	oveq1i 6068	. . . 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 3891	. . 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 6815	. . 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 5817	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ N) → (𝐹‘𝐴) ∈ R)
23		0lt1sr 8096	. . . 4 ⊢ 0R <R 1R
24		fveq2 5675	. . . . . . 7 ⊢ (𝑚 = 𝐴 → (𝐹‘𝑚) = (𝐹‘𝐴))
25	24	breq2d 4126	. . . . . 6 ⊢ (𝑚 = 𝐴 → (1R <R (𝐹‘𝑚) ↔ 1R <R (𝐹‘𝐴)))
26	25	rspcv 2919	. . . . 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 8095	. . . . 5 ⊢ <R Or R
29		ltrelsr 8069	. . . . 5 ⊢ <R ⊆ (R × R)
30	28, 29	sotri 5163	. . . 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 8119	. . 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 2271	1 ⊢ ((𝜑 ∧ 𝐴 ∈ N) → [⟨((𝐺‘𝐴) +P 1P), 1P⟩] ~R = (𝐹‘𝐴))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398   ∈ wcel 2205  {cab 2220  ∀wral 2522  ⟨cop 3697   class class class wbr 4114   ↦ cmpt 4176  ⟶wf 5353  ‘cfv 5357  ℩crio 6010  (class class class)co 6058  1_oc1o 6653  [cec 6778  Ncnpi 7603   <_N clti 7606   ~_Q ceq 7610  *_Qcrq 7615   <_Q cltq 7616  Pcnp 7622  1_Pc1p 7623   +_P cpp 7624   ~_R cer 7627  Rcnr 7628  0_Rc0r 7629  1_Rc1r 7630   +_R cplr 7632   <_R cltr 7634

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-eprel 4415  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-irdg 6614  df-1o 6660  df-2o 6661  df-oadd 6664  df-omul 6665  df-er 6780  df-ec 6782  df-qs 6786  df-ni 7635  df-pli 7636  df-mi 7637  df-lti 7638  df-plpq 7675  df-mpq 7676  df-enq 7678  df-nqqs 7679  df-plqqs 7680  df-mqqs 7681  df-1nqqs 7682  df-rq 7683  df-ltnqqs 7684  df-enq0 7755  df-nq0 7756  df-0nq0 7757  df-plq0 7758  df-mq0 7759  df-inp 7797  df-i1p 7798  df-iplp 7799  df-iltp 7801  df-enr 8057  df-nr 8058  df-ltr 8061  df-0r 8062  df-1r 8063

This theorem is referenced by:  caucvgsrlemcau  8124  caucvgsrlembound  8125  caucvgsrlemgt1  8126