Theorem caucvgprlemdisj 8005

Description: Lemma for caucvgpr 8013. 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 6065	. . . . . . . . . . . 12 ⊢ (𝑙 = 𝑠 → (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) = (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )))
2	1	breq1d 4124	. . . . . . . . . . 11 ⊢ (𝑙 = 𝑠 → ((𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗) ↔ (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)))
3	2	rexbidv 2545	. . . . . . . . . 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 5678	. . . . . . . . . . 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 7694	. . . . . . . . . . . . 13 ⊢ Q ∈ V
7	6	rabex 4261	. . . . . . . . . . . 12 ⊢ {𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)} ∈ V
8	6	rabex 4261	. . . . . . . . . . . 12 ⊢ {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢} ∈ V
9	7, 8	op1st 6353	. . . . . . . . . . 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 2255	. . . . . . . . . 10 ⊢ (1st ‘𝐿) = {𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)}
11	3, 10	elrab2 2979	. . . . . . . . 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 3888	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑘 → ⟨𝑗, 1o⟩ = ⟨𝑘, 1o⟩)
14	13	eceq1d 6816	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑘 → [⟨𝑗, 1o⟩] ~Q = [⟨𝑘, 1o⟩] ~Q )
15	14	fveq2d 5679	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑘 → (*Q‘[⟨𝑗, 1o⟩] ~Q ) = (*Q‘[⟨𝑘, 1o⟩] ~Q ))
16	15	oveq2d 6074	. . . . . . . . . 10 ⊢ (𝑗 = 𝑘 → (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) = (𝑠 +Q (*Q‘[⟨𝑘, 1o⟩] ~Q )))
17		fveq2 5675	. . . . . . . . . 10 ⊢ (𝑗 = 𝑘 → (𝐹‘𝑗) = (𝐹‘𝑘))
18	16, 17	breq12d 4127	. . . . . . . . 9 ⊢ (𝑗 = 𝑘 → ((𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗) ↔ (𝑠 +Q (*Q‘[⟨𝑘, 1o⟩] ~Q )) <Q (𝐹‘𝑘)))
19	18	cbvrexv 2781	. . . . . . . 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 4118	. . . . . . . . . 10 ⊢ (𝑢 = 𝑠 → (((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢 ↔ ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑠))
22	21	rexbidv 2545	. . . . . . . . 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 5678	. . . . . . . . . 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 6354	. . . . . . . . . 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 2255	. . . . . . . . 9 ⊢ (2nd ‘𝐿) = {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}
26	22, 25	elrab2 2979	. . . . . . . 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 2715	. . . . . 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 7997	. . . . . 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 2670	. . . 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 1417	. 2 ⊢ (𝜑 → ¬ (𝑠 ∈ (1st ‘𝐿) ∧ 𝑠 ∈ (2nd ‘𝐿)))
46	45	ralrimivw 2618	1 ⊢ (𝜑 → ∀𝑠 ∈ Q ¬ (𝑠 ∈ (1st ‘𝐿) ∧ 𝑠 ∈ (2nd ‘𝐿)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   = wceq 1398  ⊥wfal 1403   ∈ wcel 2205  ∀wral 2522  ∃wrex 2523  {crab 2526  ⟨cop 3697   class class class wbr 4114  ⟶wf 5353  ‘cfv 5357  (class class class)co 6058  1^st c1st 6345  2^nd c2nd 6346  1_oc1o 6653  [cec 6778  Ncnpi 7603   <_N clti 7606   ~_Q ceq 7610  Qcnq 7611   +_Q cplq 7613  *_Qcrq 7615   <_Q cltq 7616

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-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-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-irdg 6614  df-1o 6660  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

This theorem is referenced by:  caucvgprlemcl  8007