Theorem caucvgsrlemfv 7732

Description: Lemma for caucvgsr 7743. 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 5486	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴))
4	3	eqeq1d 2174	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → ((𝐹‘𝑥) = [⟨(𝑦 +P 1P), 1P⟩] ~R ↔ (𝐹‘𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ))
5	4	riotabidv 5800	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (℩𝑦 ∈ P (𝐹‘𝑥) = [⟨(𝑦 +P 1P), 1P⟩] ~R ) = (℩𝑦 ∈ P (𝐹‘𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ))
6	5	adantl 275	. . . . . 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 109	. . . . . 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 7730	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ N) → (℩𝑦 ∈ P (𝐹‘𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ) ∈ P)
11	2, 6, 7, 10	fvmptd 5567	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ N) → (𝐺‘𝐴) = (℩𝑦 ∈ P (𝐹‘𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ))
12	11	oveq1d 5857	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ N) → ((𝐺‘𝐴) +P 1P) = ((℩𝑦 ∈ P (𝐹‘𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ) +P 1P))
13	12	opeq1d 3764	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ N) → ⟨((𝐺‘𝐴) +P 1P), 1P⟩ = ⟨((℩𝑦 ∈ P (𝐹‘𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ) +P 1P), 1P⟩)
14	13	eceq1d 6537	. 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 2167	. . . . . . 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 5815	. . . . 5 ⊢ (℩𝑦 ∈ P (𝐹‘𝐴) = [⟨(𝑦 +P 1P), 1P⟩] ~R ) = (℩𝑦 ∈ P [⟨(𝑦 +P 1P), 1P⟩] ~R = (𝐹‘𝐴))
18	17	oveq1i 5852	. . . 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 3761	. . 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 6536	. . 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	ffvelrnda 5620	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ N) → (𝐹‘𝐴) ∈ R)
23		0lt1sr 7706	. . . 4 ⊢ 0R <R 1R
24		fveq2 5486	. . . . . . 7 ⊢ (𝑚 = 𝐴 → (𝐹‘𝑚) = (𝐹‘𝐴))
25	24	breq2d 3994	. . . . . 6 ⊢ (𝑚 = 𝐴 → (1R <R (𝐹‘𝑚) ↔ 1R <R (𝐹‘𝐴)))
26	25	rspcv 2826	. . . . 5 ⊢ (𝐴 ∈ N → (∀𝑚 ∈ N 1R <R (𝐹‘𝑚) → 1R <R (𝐹‘𝐴)))
27	9, 26	mpan9 279	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ N) → 1R <R (𝐹‘𝐴))
28		ltsosr 7705	. . . . 5 ⊢ <R Or R
29		ltrelsr 7679	. . . . 5 ⊢ <R ⊆ (R × R)
30	28, 29	sotri 4999	. . . 4 ⊢ ((0R <R 1R ∧ 1R <R (𝐹‘𝐴)) → 0R <R (𝐹‘𝐴))
31	23, 27, 30	sylancr 411	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ N) → 0R <R (𝐹‘𝐴))
32		prsrriota 7729	. . 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 409	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ N) → [⟨((℩𝑦 ∈ P [⟨(𝑦 +P 1P), 1P⟩] ~R = (𝐹‘𝐴)) +P 1P), 1P⟩] ~R = (𝐹‘𝐴))
34	14, 21, 33	3eqtrd 2202	1 ⊢ ((𝜑 ∧ 𝐴 ∈ N) → [⟨((𝐺‘𝐴) +P 1P), 1P⟩] ~R = (𝐹‘𝐴))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1343   ∈ wcel 2136  {cab 2151  ∀wral 2444  ⟨cop 3579   class class class wbr 3982   ↦ cmpt 4043  ⟶wf 5184  ‘cfv 5188  ℩crio 5797  (class class class)co 5842  1_oc1o 6377  [cec 6499  Ncnpi 7213   <_N clti 7216   ~_Q ceq 7220  *_Qcrq 7225   <_Q cltq 7226  Pcnp 7232  1_Pc1p 7233   +_P cpp 7234   ~_R cer 7237  Rcnr 7238  0_Rc0r 7239  1_Rc1r 7240   +_R cplr 7242   <_R cltr 7244

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-eprel 4267  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-irdg 6338  df-1o 6384  df-2o 6385  df-oadd 6388  df-omul 6389  df-er 6501  df-ec 6503  df-qs 6507  df-ni 7245  df-pli 7246  df-mi 7247  df-lti 7248  df-plpq 7285  df-mpq 7286  df-enq 7288  df-nqqs 7289  df-plqqs 7290  df-mqqs 7291  df-1nqqs 7292  df-rq 7293  df-ltnqqs 7294  df-enq0 7365  df-nq0 7366  df-0nq0 7367  df-plq0 7368  df-mq0 7369  df-inp 7407  df-i1p 7408  df-iplp 7409  df-iltp 7411  df-enr 7667  df-nr 7668  df-ltr 7671  df-0r 7672  df-1r 7673

This theorem is referenced by:  caucvgsrlemcau  7734  caucvgsrlembound  7735  caucvgsrlemgt1  7736