Theorem caucvgprprlemcbv 7649

Description: Lemma for caucvgprpr 7674. Change bound variables in Cauchy condition. (Contributed by Jim Kingdon, 12-Feb-2021.)

Hypotheses

Ref	Expression
caucvgprpr.f	⊢ (𝜑 → 𝐹:N⟶P)
caucvgprpr.cau	⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛)<P ((𝐹‘𝑘) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹‘𝑘)<P ((𝐹‘𝑛) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩))))

Assertion

Ref	Expression
caucvgprprlemcbv	⊢ (𝜑 → ∀𝑎 ∈ N ∀𝑏 ∈ N (𝑎 <N 𝑏 → ((𝐹‘𝑎)<P ((𝐹‘𝑏) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹‘𝑏)<P ((𝐹‘𝑎) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑢}⟩))))

Distinct variable groups:   𝐹,𝑎,𝑏,𝑘   𝑛,𝐹,𝑎,𝑘   𝑎,𝑙,𝑏,𝑘   𝑢,𝑎,𝑏,𝑘   𝑛,𝑙   𝑢,𝑛

Allowed substitution hints:   𝜑(𝑢,𝑘,𝑛,𝑎,𝑏,𝑙)   𝐹(𝑢,𝑙)

Proof of Theorem caucvgprprlemcbv

Step	Hyp	Ref	Expression
1		caucvgprpr.cau	. 2 ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛)<P ((𝐹‘𝑘) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹‘𝑘)<P ((𝐹‘𝑛) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩))))
2		breq1 3992	. . . 4 ⊢ (𝑛 = 𝑎 → (𝑛 <N 𝑘 ↔ 𝑎 <N 𝑘))
3		fveq2 5496	. . . . . 6 ⊢ (𝑛 = 𝑎 → (𝐹‘𝑛) = (𝐹‘𝑎))
4		opeq1 3765	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑎 → ⟨𝑛, 1o⟩ = ⟨𝑎, 1o⟩)
5	4	eceq1d 6549	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑎 → [⟨𝑛, 1o⟩] ~Q = [⟨𝑎, 1o⟩] ~Q )
6	5	fveq2d 5500	. . . . . . . . . 10 ⊢ (𝑛 = 𝑎 → (*Q‘[⟨𝑛, 1o⟩] ~Q ) = (*Q‘[⟨𝑎, 1o⟩] ~Q ))
7	6	breq2d 4001	. . . . . . . . 9 ⊢ (𝑛 = 𝑎 → (𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q ) ↔ 𝑙 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )))
8	7	abbidv 2288	. . . . . . . 8 ⊢ (𝑛 = 𝑎 → {𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )} = {𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )})
9	6	breq1d 3999	. . . . . . . . 9 ⊢ (𝑛 = 𝑎 → ((*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢 ↔ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑢))
10	9	abbidv 2288	. . . . . . . 8 ⊢ (𝑛 = 𝑎 → {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢} = {𝑢 ∣ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑢})
11	8, 10	opeq12d 3773	. . . . . . 7 ⊢ (𝑛 = 𝑎 → ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩ = ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑢}⟩)
12	11	oveq2d 5869	. . . . . 6 ⊢ (𝑛 = 𝑎 → ((𝐹‘𝑘) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩) = ((𝐹‘𝑘) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑢}⟩))
13	3, 12	breq12d 4002	. . . . 5 ⊢ (𝑛 = 𝑎 → ((𝐹‘𝑛)<P ((𝐹‘𝑘) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩) ↔ (𝐹‘𝑎)<P ((𝐹‘𝑘) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑢}⟩)))
14	3, 11	oveq12d 5871	. . . . . 6 ⊢ (𝑛 = 𝑎 → ((𝐹‘𝑛) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩) = ((𝐹‘𝑎) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑢}⟩))
15	14	breq2d 4001	. . . . 5 ⊢ (𝑛 = 𝑎 → ((𝐹‘𝑘)<P ((𝐹‘𝑛) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩) ↔ (𝐹‘𝑘)<P ((𝐹‘𝑎) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑢}⟩)))
16	13, 15	anbi12d 470	. . . 4 ⊢ (𝑛 = 𝑎 → (((𝐹‘𝑛)<P ((𝐹‘𝑘) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹‘𝑘)<P ((𝐹‘𝑛) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩)) ↔ ((𝐹‘𝑎)<P ((𝐹‘𝑘) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹‘𝑘)<P ((𝐹‘𝑎) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑢}⟩))))
17	2, 16	imbi12d 233	. . 3 ⊢ (𝑛 = 𝑎 → ((𝑛 <N 𝑘 → ((𝐹‘𝑛)<P ((𝐹‘𝑘) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹‘𝑘)<P ((𝐹‘𝑛) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩))) ↔ (𝑎 <N 𝑘 → ((𝐹‘𝑎)<P ((𝐹‘𝑘) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹‘𝑘)<P ((𝐹‘𝑎) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑢}⟩)))))
18		breq2 3993	. . . 4 ⊢ (𝑘 = 𝑏 → (𝑎 <N 𝑘 ↔ 𝑎 <N 𝑏))
19		fveq2 5496	. . . . . . 7 ⊢ (𝑘 = 𝑏 → (𝐹‘𝑘) = (𝐹‘𝑏))
20	19	oveq1d 5868	. . . . . 6 ⊢ (𝑘 = 𝑏 → ((𝐹‘𝑘) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑢}⟩) = ((𝐹‘𝑏) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑢}⟩))
21	20	breq2d 4001	. . . . 5 ⊢ (𝑘 = 𝑏 → ((𝐹‘𝑎)<P ((𝐹‘𝑘) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑢}⟩) ↔ (𝐹‘𝑎)<P ((𝐹‘𝑏) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑢}⟩)))
22	19	breq1d 3999	. . . . 5 ⊢ (𝑘 = 𝑏 → ((𝐹‘𝑘)<P ((𝐹‘𝑎) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑢}⟩) ↔ (𝐹‘𝑏)<P ((𝐹‘𝑎) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑢}⟩)))
23	21, 22	anbi12d 470	. . . 4 ⊢ (𝑘 = 𝑏 → (((𝐹‘𝑎)<P ((𝐹‘𝑘) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹‘𝑘)<P ((𝐹‘𝑎) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑢}⟩)) ↔ ((𝐹‘𝑎)<P ((𝐹‘𝑏) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹‘𝑏)<P ((𝐹‘𝑎) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑢}⟩))))
24	18, 23	imbi12d 233	. . 3 ⊢ (𝑘 = 𝑏 → ((𝑎 <N 𝑘 → ((𝐹‘𝑎)<P ((𝐹‘𝑘) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹‘𝑘)<P ((𝐹‘𝑎) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑢}⟩))) ↔ (𝑎 <N 𝑏 → ((𝐹‘𝑎)<P ((𝐹‘𝑏) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹‘𝑏)<P ((𝐹‘𝑎) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑢}⟩)))))
25	17, 24	cbvral2v 2709	. 2 ⊢ (∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛)<P ((𝐹‘𝑘) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹‘𝑘)<P ((𝐹‘𝑛) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩))) ↔ ∀𝑎 ∈ N ∀𝑏 ∈ N (𝑎 <N 𝑏 → ((𝐹‘𝑎)<P ((𝐹‘𝑏) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹‘𝑏)<P ((𝐹‘𝑎) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑢}⟩))))
26	1, 25	sylib 121	1 ⊢ (𝜑 → ∀𝑎 ∈ N ∀𝑏 ∈ N (𝑎 <N 𝑏 → ((𝐹‘𝑎)<P ((𝐹‘𝑏) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹‘𝑏)<P ((𝐹‘𝑎) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑢}⟩))))

Syntax hints:   → wi 4   ∧ wa 103  {cab 2156  ∀wral 2448  ⟨cop 3586   class class class wbr 3989  ⟶wf 5194  ‘cfv 5198  (class class class)co 5853  1_oc1o 6388  [cec 6511  Ncnpi 7234   <_N clti 7237   ~_Q ceq 7241  *_Qcrq 7246   <_Q cltq 7247  Pcnp 7253   +_P cpp 7255  <_P cltp 7257

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-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-xp 4617  df-cnv 4619  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fv 5206  df-ov 5856  df-ec 6515

This theorem is referenced by:  caucvgprprlemval  7650