Theorem cauappcvgprlemcl 7773

Description: Lemma for cauappcvgpr 7782. The putative limit is a positive real. (Contributed by Jim Kingdon, 20-Jun-2020.)

Hypotheses

Ref	Expression
cauappcvgpr.f	⊢ (𝜑 → 𝐹:Q⟶Q)
cauappcvgpr.app	⊢ (𝜑 → ∀𝑝 ∈ Q ∀𝑞 ∈ Q ((𝐹‘𝑝) <Q ((𝐹‘𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹‘𝑞) <Q ((𝐹‘𝑝) +Q (𝑝 +Q 𝑞))))
cauappcvgpr.bnd	⊢ (𝜑 → ∀𝑝 ∈ Q 𝐴 <Q (𝐹‘𝑝))
cauappcvgpr.lim	⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢}⟩

Assertion

Ref	Expression
cauappcvgprlemcl	⊢ (𝜑 → 𝐿 ∈ P)

Distinct variable groups:   𝐴,𝑝   𝐿,𝑝,𝑞   𝜑,𝑝,𝑞   𝐹,𝑙,𝑢,𝑝,𝑞

Allowed substitution hints:   𝜑(𝑢,𝑙)   𝐴(𝑢,𝑞,𝑙)   𝐿(𝑢,𝑙)

Proof of Theorem cauappcvgprlemcl

Dummy variables 𝑟 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cauappcvgpr.f	. . . 4 ⊢ (𝜑 → 𝐹:Q⟶Q)
2		cauappcvgpr.app	. . . 4 ⊢ (𝜑 → ∀𝑝 ∈ Q ∀𝑞 ∈ Q ((𝐹‘𝑝) <Q ((𝐹‘𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹‘𝑞) <Q ((𝐹‘𝑝) +Q (𝑝 +Q 𝑞))))
3		cauappcvgpr.bnd	. . . 4 ⊢ (𝜑 → ∀𝑝 ∈ Q 𝐴 <Q (𝐹‘𝑝))
4		cauappcvgpr.lim	. . . 4 ⊢ 𝐿 = ⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢}⟩
5	1, 2, 3, 4	cauappcvgprlemm 7765	. . 3 ⊢ (𝜑 → (∃𝑠 ∈ Q 𝑠 ∈ (1st ‘𝐿) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐿)))
6		ssrab2 3279	. . . . . 6 ⊢ {𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)} ⊆ Q
7		nqex 7483	. . . . . . 7 ⊢ Q ∈ V
8	7	elpw2 4205	. . . . . 6 ⊢ ({𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)} ∈ 𝒫 Q ↔ {𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)} ⊆ Q)
9	6, 8	mpbir 146	. . . . 5 ⊢ {𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)} ∈ 𝒫 Q
10		ssrab2 3279	. . . . . 6 ⊢ {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢} ⊆ Q
11	7	elpw2 4205	. . . . . 6 ⊢ ({𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢} ∈ 𝒫 Q ↔ {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢} ⊆ Q)
12	10, 11	mpbir 146	. . . . 5 ⊢ {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢} ∈ 𝒫 Q
13		opelxpi 4711	. . . . 5 ⊢ (({𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)} ∈ 𝒫 Q ∧ {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢} ∈ 𝒫 Q) → ⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢}⟩ ∈ (𝒫 Q × 𝒫 Q))
14	9, 12, 13	mp2an 426	. . . 4 ⊢ ⟨{𝑙 ∈ Q ∣ ∃𝑞 ∈ Q (𝑙 +Q 𝑞) <Q (𝐹‘𝑞)}, {𝑢 ∈ Q ∣ ∃𝑞 ∈ Q ((𝐹‘𝑞) +Q 𝑞) <Q 𝑢}⟩ ∈ (𝒫 Q × 𝒫 Q)
15	4, 14	eqeltri 2279	. . 3 ⊢ 𝐿 ∈ (𝒫 Q × 𝒫 Q)
16	5, 15	jctil 312	. 2 ⊢ (𝜑 → (𝐿 ∈ (𝒫 Q × 𝒫 Q) ∧ (∃𝑠 ∈ Q 𝑠 ∈ (1st ‘𝐿) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐿))))
17	1, 2, 3, 4	cauappcvgprlemrnd 7770	. . 3 ⊢ (𝜑 → (∀𝑠 ∈ Q (𝑠 ∈ (1st ‘𝐿) ↔ ∃𝑟 ∈ Q (𝑠 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐿))) ∧ ∀𝑟 ∈ Q (𝑟 ∈ (2nd ‘𝐿) ↔ ∃𝑠 ∈ Q (𝑠 <Q 𝑟 ∧ 𝑠 ∈ (2nd ‘𝐿)))))
18	1, 2, 3, 4	cauappcvgprlemdisj 7771	. . 3 ⊢ (𝜑 → ∀𝑠 ∈ Q ¬ (𝑠 ∈ (1st ‘𝐿) ∧ 𝑠 ∈ (2nd ‘𝐿)))
19	1, 2, 3, 4	cauappcvgprlemloc 7772	. . 3 ⊢ (𝜑 → ∀𝑠 ∈ Q ∀𝑟 ∈ Q (𝑠 <Q 𝑟 → (𝑠 ∈ (1st ‘𝐿) ∨ 𝑟 ∈ (2nd ‘𝐿))))
20	17, 18, 19	3jca 1180	. 2 ⊢ (𝜑 → ((∀𝑠 ∈ Q (𝑠 ∈ (1st ‘𝐿) ↔ ∃𝑟 ∈ Q (𝑠 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐿))) ∧ ∀𝑟 ∈ Q (𝑟 ∈ (2nd ‘𝐿) ↔ ∃𝑠 ∈ Q (𝑠 <Q 𝑟 ∧ 𝑠 ∈ (2nd ‘𝐿)))) ∧ ∀𝑠 ∈ Q ¬ (𝑠 ∈ (1st ‘𝐿) ∧ 𝑠 ∈ (2nd ‘𝐿)) ∧ ∀𝑠 ∈ Q ∀𝑟 ∈ Q (𝑠 <Q 𝑟 → (𝑠 ∈ (1st ‘𝐿) ∨ 𝑟 ∈ (2nd ‘𝐿)))))
21		elnp1st2nd 7596	. 2 ⊢ (𝐿 ∈ P ↔ ((𝐿 ∈ (𝒫 Q × 𝒫 Q) ∧ (∃𝑠 ∈ Q 𝑠 ∈ (1st ‘𝐿) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2nd ‘𝐿))) ∧ ((∀𝑠 ∈ Q (𝑠 ∈ (1st ‘𝐿) ↔ ∃𝑟 ∈ Q (𝑠 <Q 𝑟 ∧ 𝑟 ∈ (1st ‘𝐿))) ∧ ∀𝑟 ∈ Q (𝑟 ∈ (2nd ‘𝐿) ↔ ∃𝑠 ∈ Q (𝑠 <Q 𝑟 ∧ 𝑠 ∈ (2nd ‘𝐿)))) ∧ ∀𝑠 ∈ Q ¬ (𝑠 ∈ (1st ‘𝐿) ∧ 𝑠 ∈ (2nd ‘𝐿)) ∧ ∀𝑠 ∈ Q ∀𝑟 ∈ Q (𝑠 <Q 𝑟 → (𝑠 ∈ (1st ‘𝐿) ∨ 𝑟 ∈ (2nd ‘𝐿))))))
22	16, 20, 21	sylanbrc 417	1 ⊢ (𝜑 → 𝐿 ∈ P)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 710   ∧ w3a 981   = wceq 1373   ∈ wcel 2177  ∀wral 2485  ∃wrex 2486  {crab 2489   ⊆ wss 3167  𝒫 cpw 3617  ⟨cop 3637   class class class wbr 4047   × cxp 4677  ⟶wf 5272  ‘cfv 5276  (class class class)co 5951  1^st c1st 6231  2^nd c2nd 6232  Qcnq 7400   +_Q cplq 7402   <_Q cltq 7405  Pcnp 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 4163  ax-sep 4166  ax-nul 4174  ax-pow 4222  ax-pr 4257  ax-un 4484  ax-setind 4589  ax-iinf 4640

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 3000  df-csb 3095  df-dif 3169  df-un 3171  df-in 3173  df-ss 3180  df-nul 3462  df-pw 3619  df-sn 3640  df-pr 3641  df-op 3643  df-uni 3853  df-int 3888  df-iun 3931  df-br 4048  df-opab 4110  df-mpt 4111  df-tr 4147  df-eprel 4340  df-id 4344  df-po 4347  df-iso 4348  df-iord 4417  df-on 4419  df-suc 4422  df-iom 4643  df-xp 4685  df-rel 4686  df-cnv 4687  df-co 4688  df-dm 4689  df-rn 4690  df-res 4691  df-ima 4692  df-iota 5237  df-fun 5278  df-fn 5279  df-f 5280  df-f1 5281  df-fo 5282  df-f1o 5283  df-fv 5284  df-ov 5954  df-oprab 5955  df-mpo 5956  df-1st 6233  df-2nd 6234  df-recs 6398  df-irdg 6463  df-1o 6509  df-oadd 6513  df-omul 6514  df-er 6627  df-ec 6629  df-qs 6633  df-ni 7424  df-pli 7425  df-mi 7426  df-lti 7427  df-plpq 7464  df-mpq 7465  df-enq 7467  df-nqqs 7468  df-plqqs 7469  df-mqqs 7470  df-1nqqs 7471  df-rq 7472  df-ltnqqs 7473  df-inp 7586

This theorem is referenced by:  cauappcvgprlemladdfu  7774  cauappcvgprlemladdfl  7775  cauappcvgprlemladdru  7776  cauappcvgprlemladdrl  7777  cauappcvgprlemladd  7778  cauappcvgprlem1  7779  cauappcvgprlem2  7780  cauappcvgpr  7782