Theorem caucvgprlemdisj 7800

Description: Lemma for caucvgpr 7808. 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 5961	. . . . . . . . . . . 12 ⊢ (𝑙 = 𝑠 → (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) = (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )))
2	1	breq1d 4058	. . . . . . . . . . 11 ⊢ (𝑙 = 𝑠 → ((𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗) ↔ (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)))
3	2	rexbidv 2508	. . . . . . . . . 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 5589	. . . . . . . . . . 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 7489	. . . . . . . . . . . . 13 ⊢ Q ∈ V
7	6	rabex 4193	. . . . . . . . . . . 12 ⊢ {𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)} ∈ V
8	6	rabex 4193	. . . . . . . . . . . 12 ⊢ {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢} ∈ V
9	7, 8	op1st 6242	. . . . . . . . . . 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 2227	. . . . . . . . . 10 ⊢ (1st ‘𝐿) = {𝑙 ∈ Q ∣ ∃𝑗 ∈ N (𝑙 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗)}
11	3, 10	elrab2 2934	. . . . . . . . 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 3822	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑘 → ⟨𝑗, 1o⟩ = ⟨𝑘, 1o⟩)
14	13	eceq1d 6666	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑘 → [⟨𝑗, 1o⟩] ~Q = [⟨𝑘, 1o⟩] ~Q )
15	14	fveq2d 5590	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑘 → (*Q‘[⟨𝑗, 1o⟩] ~Q ) = (*Q‘[⟨𝑘, 1o⟩] ~Q ))
16	15	oveq2d 5970	. . . . . . . . . 10 ⊢ (𝑗 = 𝑘 → (𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) = (𝑠 +Q (*Q‘[⟨𝑘, 1o⟩] ~Q )))
17		fveq2 5586	. . . . . . . . . 10 ⊢ (𝑗 = 𝑘 → (𝐹‘𝑗) = (𝐹‘𝑘))
18	16, 17	breq12d 4061	. . . . . . . . 9 ⊢ (𝑗 = 𝑘 → ((𝑠 +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q (𝐹‘𝑗) ↔ (𝑠 +Q (*Q‘[⟨𝑘, 1o⟩] ~Q )) <Q (𝐹‘𝑘)))
19	18	cbvrexv 2740	. . . . . . . 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 4052	. . . . . . . . . 10 ⊢ (𝑢 = 𝑠 → (((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢 ↔ ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑠))
22	21	rexbidv 2508	. . . . . . . . 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 5589	. . . . . . . . . 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 6243	. . . . . . . . . 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 2227	. . . . . . . . 9 ⊢ (2nd ‘𝐿) = {𝑢 ∈ Q ∣ ∃𝑗 ∈ N ((𝐹‘𝑗) +Q (*Q‘[⟨𝑗, 1o⟩] ~Q )) <Q 𝑢}
26	22, 25	elrab2 2934	. . . . . . . 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 2677	. . . . . 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 529	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑠 ∈ (1st ‘𝐿) ∧ 𝑠 ∈ (2nd ‘𝐿))) ∧ (𝑘 ∈ N ∧ 𝑗 ∈ N)) → 𝑘 ∈ N)
37		simprr 531	. . . . . . 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 7792	. . . . . 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 620	. . . . 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 2632	. . . 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 1392	. 2 ⊢ (𝜑 → ¬ (𝑠 ∈ (1st ‘𝐿) ∧ 𝑠 ∈ (2nd ‘𝐿)))
46	45	ralrimivw 2581	1 ⊢ (𝜑 → ∀𝑠 ∈ Q ¬ (𝑠 ∈ (1st ‘𝐿) ∧ 𝑠 ∈ (2nd ‘𝐿)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   = wceq 1373  ⊥wfal 1378   ∈ wcel 2177  ∀wral 2485  ∃wrex 2486  {crab 2489  ⟨cop 3638   class class class wbr 4048  ⟶wf 5273  ‘cfv 5277  (class class class)co 5954  1^st c1st 6234  2^nd c2nd 6235  1_oc1o 6505  [cec 6628  Ncnpi 7398   <_N clti 7401   ~_Q ceq 7405  Qcnq 7406   +_Q cplq 7408  *_Qcrq 7410   <_Q cltq 7411

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4164  ax-sep 4167  ax-nul 4175  ax-pow 4223  ax-pr 4258  ax-un 4485  ax-setind 4590  ax-iinf 4641

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3001  df-csb 3096  df-dif 3170  df-un 3172  df-in 3174  df-ss 3181  df-nul 3463  df-pw 3620  df-sn 3641  df-pr 3642  df-op 3644  df-uni 3854  df-int 3889  df-iun 3932  df-br 4049  df-opab 4111  df-mpt 4112  df-tr 4148  df-eprel 4341  df-id 4345  df-po 4348  df-iso 4349  df-iord 4418  df-on 4420  df-suc 4423  df-iom 4644  df-xp 4686  df-rel 4687  df-cnv 4688  df-co 4689  df-dm 4690  df-rn 4691  df-res 4692  df-ima 4693  df-iota 5238  df-fun 5279  df-fn 5280  df-f 5281  df-f1 5282  df-fo 5283  df-f1o 5284  df-fv 5285  df-ov 5957  df-oprab 5958  df-mpo 5959  df-1st 6236  df-2nd 6237  df-recs 6401  df-irdg 6466  df-1o 6512  df-oadd 6516  df-omul 6517  df-er 6630  df-ec 6632  df-qs 6636  df-ni 7430  df-pli 7431  df-mi 7432  df-lti 7433  df-plpq 7470  df-mpq 7471  df-enq 7473  df-nqqs 7474  df-plqqs 7475  df-mqqs 7476  df-1nqqs 7477  df-rq 7478  df-ltnqqs 7479

This theorem is referenced by:  caucvgprlemcl  7802