Theorem caucvgprlemdisj 7894

Description: Lemma for caucvgpr 7902. The putative limit is disjoint. (Contributed by Jim Kingdon, 27-Sep-2020.)

Hypotheses

Ref	Expression
caucvgpr.f	⊢ (𝜑 → 𝐹:N⟶Q)
caucvgpr.cau	⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <Q ((𝐹‘𝑘) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )) ∧ (𝐹‘𝑘) <Q ((𝐹‘𝑛) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )))))
caucvgpr.bnd	⊢ (𝜑 → ∀𝑗 ∈ N 𝐴 <Q (𝐹‘𝑗))
caucvgpr.lim	⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)}, {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}⟩

Assertion

Ref	Expression
caucvgprlemdisj	⊢ (𝜑 → ∀𝑠 ∈ Q ¬ (𝑠 ∈ (1st ‘𝐿) ∧ 𝑠 ∈ (2nd ‘𝐿)))

Distinct variable groups:   𝐴,𝑗   𝑗,𝐹,𝑘   𝐹,𝑙,𝑗   𝑢,𝐹,𝑗   𝑛,𝐹   𝑗,𝐿,𝑘   𝜑,𝑗,𝑠,𝑘   𝑠,𝑙   𝑢,𝑠   𝑘,𝑛

Allowed substitution hints:   𝜑(𝑢,𝑛,𝑙)   𝐴(𝑢,𝑘,𝑛,𝑠,𝑙)   𝐹(𝑠)   𝐿(𝑢,𝑛,𝑠,𝑙)

Proof of Theorem caucvgprlemdisj

Step	Hyp	Ref	Expression
1		oveq1 6025	. . . . . . . . . . . 12 ⊢ (𝑙 = 𝑠 → (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) = (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )))
2	1	breq1d 4098	. . . . . . . . . . 11 ⊢ (𝑙 = 𝑠 → ((𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗) ↔ (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)))
3	2	rexbidv 2533	. . . . . . . . . 10 ⊢ (𝑙 = 𝑠 → (∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗) ↔ ∃𝑗 ∈ N (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)))
4		caucvgpr.lim	. . . . . . . . . . . 12 ⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)}, {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}⟩
5	4	fveq2i 5642	. . . . . . . . . . 11 ⊢ (1st ‘𝐿) = (1st ‘⟨{𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)}, {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}⟩)
6		nqex 7583	. . . . . . . . . . . . 13 ⊢ Q ∈ V
7	6	rabex 4234	. . . . . . . . . . . 12 ⊢ {𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)} ∈ V
8	6	rabex 4234	. . . . . . . . . . . 12 ⊢ {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢} ∈ V
9	7, 8	op1st 6309	. . . . . . . . . . 11 ⊢ (1st ‘⟨{𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)}, {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}⟩) = {𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)}
10	5, 9	eqtri 2252	. . . . . . . . . 10 ⊢ (1st ‘𝐿) = {𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)}
11	3, 10	elrab2 2965	. . . . . . . . 9 ⊢ (𝑠 ∈ (1st ‘𝐿) ↔ (𝑠 ∈ Q ∧ ∃𝑗 ∈ N (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)))
12	11	simprbi 275	. . . . . . . 8 ⊢ (𝑠 ∈ (1st ‘𝐿) → ∃𝑗 ∈ N (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗))
13		opeq1 3862	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑘 → ⟨𝑗, 1o⟩ = ⟨𝑘, 1o⟩)
14	13	eceq1d 6738	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑘 → [⟨𝑗, 1o⟩] ~Q = [⟨𝑘, 1o⟩] ~Q )
15	14	fveq2d 5643	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑘 → (*Q‘[⟨𝑗, 1o⟩] ~Q ) = (*Q‘[⟨𝑘, 1o⟩] ~Q ))
16	15	oveq2d 6034	. . . . . . . . . 10 ⊢ (𝑗 = 𝑘 → (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) = (𝑠 +Q (*Q‘[⟨𝑘, 1o⟩] ~Q )))
17		fveq2 5639	. . . . . . . . . 10 ⊢ (𝑗 = 𝑘 → (𝐹‘𝑗) = (𝐹‘𝑘))
18	16, 17	breq12d 4101	. . . . . . . . 9 ⊢ (𝑗 = 𝑘 → ((𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗) ↔ (𝑠 +Q (*Q‘[⟨𝑘, 1o⟩] ~Q )) <Q (𝐹‘𝑘)))
19	18	cbvrexv 2768	. . . . . . . 8 ⊢ (∃𝑗 ∈ N (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗) ↔ ∃𝑘 ∈ N (𝑠 +Q (*Q‘[⟨𝑘, 1o⟩] ~Q )) <Q (𝐹‘𝑘))
20	12, 19	sylib 122	. . . . . . 7 ⊢ (𝑠 ∈ (1st ‘𝐿) → ∃𝑘 ∈ N (𝑠 +Q (*Q‘[⟨𝑘, 1o⟩] ~Q )) <Q (𝐹‘𝑘))
21		breq2 4092	. . . . . . . . . 10 ⊢ (𝑢 = 𝑠 → (((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢 ↔ ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑠))
22	21	rexbidv 2533	. . . . . . . . 9 ⊢ (𝑢 = 𝑠 → (∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢 ↔ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑠))
23	4	fveq2i 5642	. . . . . . . . . 10 ⊢ (2nd ‘𝐿) = (2nd ‘⟨{𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)}, {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}⟩)
24	7, 8	op2nd 6310	. . . . . . . . . 10 ⊢ (2nd ‘⟨{𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)}, {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}⟩) = {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}
25	23, 24	eqtri 2252	. . . . . . . . 9 ⊢ (2nd ‘𝐿) = {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}
26	22, 25	elrab2 2965	. . . . . . . 8 ⊢ (𝑠 ∈ (2nd ‘𝐿) ↔ (𝑠 ∈ Q ∧ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑠))
27	26	simprbi 275	. . . . . . 7 ⊢ (𝑠 ∈ (2nd ‘𝐿) → ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑠)
28	20, 27	anim12i 338	. . . . . 6 ⊢ ((𝑠 ∈ (1st ‘𝐿) ∧ 𝑠 ∈ (2nd ‘𝐿)) → (∃𝑘 ∈ N (𝑠 +Q (*Q‘[⟨𝑘, 1o⟩] ~Q )) <Q (𝐹‘𝑘) ∧ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑠))
29		reeanv 2703	. . . . . 6 ⊢ (∃𝑘 ∈ N ∃𝑗 ∈ N ((𝑠 +Q (*Q‘[⟨𝑘, 1o⟩] ~Q )) <Q (𝐹‘𝑘) ∧ ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑠) ↔ (∃𝑘 ∈ N (𝑠 +Q (*Q‘[⟨𝑘, 1o⟩] ~Q )) <Q (𝐹‘𝑘) ∧ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑠))
30	28, 29	sylibr 134	. . . . 5 ⊢ ((𝑠 ∈ (1st ‘𝐿) ∧ 𝑠 ∈ (2nd ‘𝐿)) → ∃𝑘 ∈ N ∃𝑗 ∈ N ((𝑠 +Q (*Q‘[⟨𝑘, 1o⟩] ~Q )) <Q (𝐹‘𝑘) ∧ ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑠))
31	30	adantl 277	. . . 4 ⊢ ((𝜑 ∧ (𝑠 ∈ (1st ‘𝐿) ∧ 𝑠 ∈ (2nd ‘𝐿))) → ∃𝑘 ∈ N ∃𝑗 ∈ N ((𝑠 +Q (*Q‘[⟨𝑘, 1o⟩] ~Q )) <Q (𝐹‘𝑘) ∧ ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑠))
32		caucvgpr.f	. . . . . . . 8 ⊢ (𝜑 → 𝐹:N⟶Q)
33	32	ad2antrr 488	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑠 ∈ (1st ‘𝐿) ∧ 𝑠 ∈ (2nd ‘𝐿))) ∧ (𝑘 ∈ N ∧ 𝑗 ∈ N)) → 𝐹:N⟶Q)
34		caucvgpr.cau	. . . . . . . 8 ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <Q ((𝐹‘𝑘) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )) ∧ (𝐹‘𝑘) <Q ((𝐹‘𝑛) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )))))
35	34	ad2antrr 488	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑠 ∈ (1st ‘𝐿) ∧ 𝑠 ∈ (2nd ‘𝐿))) ∧ (𝑘 ∈ N ∧ 𝑗 ∈ N)) → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <Q ((𝐹‘𝑘) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )) ∧ (𝐹‘𝑘) <Q ((𝐹‘𝑛) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )))))
36		simprl 531	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑠 ∈ (1st ‘𝐿) ∧ 𝑠 ∈ (2nd ‘𝐿))) ∧ (𝑘 ∈ N ∧ 𝑗 ∈ N)) → 𝑘 ∈ N)
37		simprr 533	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑠 ∈ (1st ‘𝐿) ∧ 𝑠 ∈ (2nd ‘𝐿))) ∧ (𝑘 ∈ N ∧ 𝑗 ∈ N)) → 𝑗 ∈ N)
38	11	simplbi 274	. . . . . . . . 9 ⊢ (𝑠 ∈ (1st ‘𝐿) → 𝑠 ∈ Q)
39	38	ad2antrl 490	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑠 ∈ (1st ‘𝐿) ∧ 𝑠 ∈ (2nd ‘𝐿))) → 𝑠 ∈ Q)
40	39	adantr 276	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑠 ∈ (1st ‘𝐿) ∧ 𝑠 ∈ (2nd ‘𝐿))) ∧ (𝑘 ∈ N ∧ 𝑗 ∈ N)) → 𝑠 ∈ Q)
41	33, 35, 36, 37, 40	caucvgprlemnkj 7886	. . . . . 6 ⊢ (((𝜑 ∧ (𝑠 ∈ (1st ‘𝐿) ∧ 𝑠 ∈ (2nd ‘𝐿))) ∧ (𝑘 ∈ N ∧ 𝑗 ∈ N)) → ¬ ((𝑠 +Q (*Q‘[⟨𝑘, 1o⟩] ~Q )) <Q (𝐹‘𝑘) ∧ ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑠))
42	41	pm2.21d 624	. . . . 5 ⊢ (((𝜑 ∧ (𝑠 ∈ (1st ‘𝐿) ∧ 𝑠 ∈ (2nd ‘𝐿))) ∧ (𝑘 ∈ N ∧ 𝑗 ∈ N)) → (((𝑠 +Q (*Q‘[⟨𝑘, 1o⟩] ~Q )) <Q (𝐹‘𝑘) ∧ ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑠) → ⊥))
43	42	rexlimdvva 2658	. . . 4 ⊢ ((𝜑 ∧ (𝑠 ∈ (1st ‘𝐿) ∧ 𝑠 ∈ (2nd ‘𝐿))) → (∃𝑘 ∈ N ∃𝑗 ∈ N ((𝑠 +Q (*Q‘[⟨𝑘, 1o⟩] ~Q )) <Q (𝐹‘𝑘) ∧ ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑠) → ⊥))
44	31, 43	mpd 13	. . 3 ⊢ ((𝜑 ∧ (𝑠 ∈ (1st ‘𝐿) ∧ 𝑠 ∈ (2nd ‘𝐿))) → ⊥)
45	44	inegd 1416	. 2 ⊢ (𝜑 → ¬ (𝑠 ∈ (1st ‘𝐿) ∧ 𝑠 ∈ (2nd ‘𝐿)))
46	45	ralrimivw 2606	1 ⊢ (𝜑 → ∀𝑠 ∈ Q ¬ (𝑠 ∈ (1st ‘𝐿) ∧ 𝑠 ∈ (2nd ‘𝐿)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   = wceq 1397  ⊥wfal 1402   ∈ wcel 2202  ∀wral 2510  ∃wrex 2511  {crab 2514  ⟨cop 3672   class class class wbr 4088  ⟶wf 5322  ‘cfv 5326  (class class class)co 6018  1^st c1st 6301  2^nd c2nd 6302  1_oc1o 6575  [cec 6700  Ncnpi 7492   <_N clti 7495   ~_Q ceq 7499  Qcnq 7500   +_Q cplq 7502  *_Qcrq 7504   <_Q cltq 7505

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-eprel 4386  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-irdg 6536  df-1o 6582  df-oadd 6586  df-omul 6587  df-er 6702  df-ec 6704  df-qs 6708  df-ni 7524  df-pli 7525  df-mi 7526  df-lti 7527  df-plpq 7564  df-mpq 7565  df-enq 7567  df-nqqs 7568  df-plqqs 7569  df-mqqs 7570  df-1nqqs 7571  df-rq 7572  df-ltnqqs 7573

This theorem is referenced by:  caucvgprlemcl  7896