ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  cauappcvgprlem2 GIF version

Theorem cauappcvgprlem2 7650
Description: Lemma for cauappcvgpr 7652. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 23-Jun-2020.)
Hypotheses
Ref Expression
cauappcvgpr.f (𝜑𝐹:QQ)
cauappcvgpr.app (𝜑 → ∀𝑝Q𝑞Q ((𝐹𝑝) <Q ((𝐹𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹𝑞) <Q ((𝐹𝑝) +Q (𝑝 +Q 𝑞))))
cauappcvgpr.bnd (𝜑 → ∀𝑝Q 𝐴 <Q (𝐹𝑝))
cauappcvgpr.lim 𝐿 = ⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢}⟩
cauappcvgprlem.q (𝜑𝑄Q)
cauappcvgprlem.r (𝜑𝑅Q)
Assertion
Ref Expression
cauappcvgprlem2 (𝜑𝐿<P ⟨{𝑙𝑙 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))}, {𝑢 ∣ ((𝐹𝑄) +Q (𝑄 +Q 𝑅)) <Q 𝑢}⟩)
Distinct variable groups:   𝐴,𝑝   𝐿,𝑝,𝑞   𝜑,𝑝,𝑞   𝐹,𝑝,𝑞,𝑙,𝑢   𝑄,𝑝,𝑞,𝑙,𝑢   𝑅,𝑝,𝑞,𝑙,𝑢
Allowed substitution hints:   𝜑(𝑢,𝑙)   𝐴(𝑢,𝑞,𝑙)   𝐿(𝑢,𝑙)

Proof of Theorem cauappcvgprlem2
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 cauappcvgprlem.q . . . . 5 (𝜑𝑄Q)
2 cauappcvgprlem.r . . . . 5 (𝜑𝑅Q)
3 ltaddnq 7397 . . . . 5 ((𝑄Q𝑅Q) → 𝑄 <Q (𝑄 +Q 𝑅))
41, 2, 3syl2anc 411 . . . 4 (𝜑𝑄 <Q (𝑄 +Q 𝑅))
5 cauappcvgpr.f . . . . 5 (𝜑𝐹:QQ)
65, 1ffvelcdmd 5648 . . . 4 (𝜑 → (𝐹𝑄) ∈ Q)
7 ltanqi 7392 . . . 4 ((𝑄 <Q (𝑄 +Q 𝑅) ∧ (𝐹𝑄) ∈ Q) → ((𝐹𝑄) +Q 𝑄) <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅)))
84, 6, 7syl2anc 411 . . 3 (𝜑 → ((𝐹𝑄) +Q 𝑄) <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅)))
9 ltbtwnnqq 7405 . . 3 (((𝐹𝑄) +Q 𝑄) <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅)) ↔ ∃𝑥Q (((𝐹𝑄) +Q 𝑄) <Q 𝑥𝑥 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))))
108, 9sylib 122 . 2 (𝜑 → ∃𝑥Q (((𝐹𝑄) +Q 𝑄) <Q 𝑥𝑥 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))))
11 simprl 529 . . . 4 ((𝜑 ∧ (𝑥Q ∧ (((𝐹𝑄) +Q 𝑄) <Q 𝑥𝑥 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))))) → 𝑥Q)
121adantr 276 . . . . . 6 ((𝜑 ∧ (𝑥Q ∧ (((𝐹𝑄) +Q 𝑄) <Q 𝑥𝑥 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))))) → 𝑄Q)
13 simprrl 539 . . . . . 6 ((𝜑 ∧ (𝑥Q ∧ (((𝐹𝑄) +Q 𝑄) <Q 𝑥𝑥 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))))) → ((𝐹𝑄) +Q 𝑄) <Q 𝑥)
14 fveq2 5511 . . . . . . . . 9 (𝑞 = 𝑄 → (𝐹𝑞) = (𝐹𝑄))
15 id 19 . . . . . . . . 9 (𝑞 = 𝑄𝑞 = 𝑄)
1614, 15oveq12d 5887 . . . . . . . 8 (𝑞 = 𝑄 → ((𝐹𝑞) +Q 𝑞) = ((𝐹𝑄) +Q 𝑄))
1716breq1d 4010 . . . . . . 7 (𝑞 = 𝑄 → (((𝐹𝑞) +Q 𝑞) <Q 𝑥 ↔ ((𝐹𝑄) +Q 𝑄) <Q 𝑥))
1817rspcev 2841 . . . . . 6 ((𝑄Q ∧ ((𝐹𝑄) +Q 𝑄) <Q 𝑥) → ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑥)
1912, 13, 18syl2anc 411 . . . . 5 ((𝜑 ∧ (𝑥Q ∧ (((𝐹𝑄) +Q 𝑄) <Q 𝑥𝑥 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))))) → ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑥)
20 breq2 4004 . . . . . . 7 (𝑢 = 𝑥 → (((𝐹𝑞) +Q 𝑞) <Q 𝑢 ↔ ((𝐹𝑞) +Q 𝑞) <Q 𝑥))
2120rexbidv 2478 . . . . . 6 (𝑢 = 𝑥 → (∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢 ↔ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑥))
22 cauappcvgpr.lim . . . . . . . 8 𝐿 = ⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢}⟩
2322fveq2i 5514 . . . . . . 7 (2nd𝐿) = (2nd ‘⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢}⟩)
24 nqex 7353 . . . . . . . . 9 Q ∈ V
2524rabex 4144 . . . . . . . 8 {𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)} ∈ V
2624rabex 4144 . . . . . . . 8 {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢} ∈ V
2725, 26op2nd 6142 . . . . . . 7 (2nd ‘⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢}⟩) = {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢}
2823, 27eqtri 2198 . . . . . 6 (2nd𝐿) = {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢}
2921, 28elrab2 2896 . . . . 5 (𝑥 ∈ (2nd𝐿) ↔ (𝑥Q ∧ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑥))
3011, 19, 29sylanbrc 417 . . . 4 ((𝜑 ∧ (𝑥Q ∧ (((𝐹𝑄) +Q 𝑄) <Q 𝑥𝑥 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))))) → 𝑥 ∈ (2nd𝐿))
31 simprrr 540 . . . . . 6 ((𝜑 ∧ (𝑥Q ∧ (((𝐹𝑄) +Q 𝑄) <Q 𝑥𝑥 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))))) → 𝑥 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅)))
32 vex 2740 . . . . . . 7 𝑥 ∈ V
33 breq1 4003 . . . . . . 7 (𝑙 = 𝑥 → (𝑙 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅)) ↔ 𝑥 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))))
3432, 33elab 2881 . . . . . 6 (𝑥 ∈ {𝑙𝑙 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))} ↔ 𝑥 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅)))
3531, 34sylibr 134 . . . . 5 ((𝜑 ∧ (𝑥Q ∧ (((𝐹𝑄) +Q 𝑄) <Q 𝑥𝑥 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))))) → 𝑥 ∈ {𝑙𝑙 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))})
36 ltnqex 7539 . . . . . 6 {𝑙𝑙 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))} ∈ V
37 gtnqex 7540 . . . . . 6 {𝑢 ∣ ((𝐹𝑄) +Q (𝑄 +Q 𝑅)) <Q 𝑢} ∈ V
3836, 37op1st 6141 . . . . 5 (1st ‘⟨{𝑙𝑙 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))}, {𝑢 ∣ ((𝐹𝑄) +Q (𝑄 +Q 𝑅)) <Q 𝑢}⟩) = {𝑙𝑙 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))}
3935, 38eleqtrrdi 2271 . . . 4 ((𝜑 ∧ (𝑥Q ∧ (((𝐹𝑄) +Q 𝑄) <Q 𝑥𝑥 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))))) → 𝑥 ∈ (1st ‘⟨{𝑙𝑙 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))}, {𝑢 ∣ ((𝐹𝑄) +Q (𝑄 +Q 𝑅)) <Q 𝑢}⟩))
40 rspe 2526 . . . 4 ((𝑥Q ∧ (𝑥 ∈ (2nd𝐿) ∧ 𝑥 ∈ (1st ‘⟨{𝑙𝑙 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))}, {𝑢 ∣ ((𝐹𝑄) +Q (𝑄 +Q 𝑅)) <Q 𝑢}⟩))) → ∃𝑥Q (𝑥 ∈ (2nd𝐿) ∧ 𝑥 ∈ (1st ‘⟨{𝑙𝑙 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))}, {𝑢 ∣ ((𝐹𝑄) +Q (𝑄 +Q 𝑅)) <Q 𝑢}⟩)))
4111, 30, 39, 40syl12anc 1236 . . 3 ((𝜑 ∧ (𝑥Q ∧ (((𝐹𝑄) +Q 𝑄) <Q 𝑥𝑥 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))))) → ∃𝑥Q (𝑥 ∈ (2nd𝐿) ∧ 𝑥 ∈ (1st ‘⟨{𝑙𝑙 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))}, {𝑢 ∣ ((𝐹𝑄) +Q (𝑄 +Q 𝑅)) <Q 𝑢}⟩)))
42 cauappcvgpr.app . . . . . 6 (𝜑 → ∀𝑝Q𝑞Q ((𝐹𝑝) <Q ((𝐹𝑞) +Q (𝑝 +Q 𝑞)) ∧ (𝐹𝑞) <Q ((𝐹𝑝) +Q (𝑝 +Q 𝑞))))
43 cauappcvgpr.bnd . . . . . 6 (𝜑 → ∀𝑝Q 𝐴 <Q (𝐹𝑝))
445, 42, 43, 22cauappcvgprlemcl 7643 . . . . 5 (𝜑𝐿P)
4544adantr 276 . . . 4 ((𝜑 ∧ (𝑥Q ∧ (((𝐹𝑄) +Q 𝑄) <Q 𝑥𝑥 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))))) → 𝐿P)
46 addclnq 7365 . . . . . . . 8 ((𝑄Q𝑅Q) → (𝑄 +Q 𝑅) ∈ Q)
471, 2, 46syl2anc 411 . . . . . . 7 (𝜑 → (𝑄 +Q 𝑅) ∈ Q)
48 addclnq 7365 . . . . . . 7 (((𝐹𝑄) ∈ Q ∧ (𝑄 +Q 𝑅) ∈ Q) → ((𝐹𝑄) +Q (𝑄 +Q 𝑅)) ∈ Q)
496, 47, 48syl2anc 411 . . . . . 6 (𝜑 → ((𝐹𝑄) +Q (𝑄 +Q 𝑅)) ∈ Q)
50 nqprlu 7537 . . . . . 6 (((𝐹𝑄) +Q (𝑄 +Q 𝑅)) ∈ Q → ⟨{𝑙𝑙 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))}, {𝑢 ∣ ((𝐹𝑄) +Q (𝑄 +Q 𝑅)) <Q 𝑢}⟩ ∈ P)
5149, 50syl 14 . . . . 5 (𝜑 → ⟨{𝑙𝑙 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))}, {𝑢 ∣ ((𝐹𝑄) +Q (𝑄 +Q 𝑅)) <Q 𝑢}⟩ ∈ P)
5251adantr 276 . . . 4 ((𝜑 ∧ (𝑥Q ∧ (((𝐹𝑄) +Q 𝑄) <Q 𝑥𝑥 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))))) → ⟨{𝑙𝑙 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))}, {𝑢 ∣ ((𝐹𝑄) +Q (𝑄 +Q 𝑅)) <Q 𝑢}⟩ ∈ P)
53 ltdfpr 7496 . . . 4 ((𝐿P ∧ ⟨{𝑙𝑙 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))}, {𝑢 ∣ ((𝐹𝑄) +Q (𝑄 +Q 𝑅)) <Q 𝑢}⟩ ∈ P) → (𝐿<P ⟨{𝑙𝑙 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))}, {𝑢 ∣ ((𝐹𝑄) +Q (𝑄 +Q 𝑅)) <Q 𝑢}⟩ ↔ ∃𝑥Q (𝑥 ∈ (2nd𝐿) ∧ 𝑥 ∈ (1st ‘⟨{𝑙𝑙 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))}, {𝑢 ∣ ((𝐹𝑄) +Q (𝑄 +Q 𝑅)) <Q 𝑢}⟩))))
5445, 52, 53syl2anc 411 . . 3 ((𝜑 ∧ (𝑥Q ∧ (((𝐹𝑄) +Q 𝑄) <Q 𝑥𝑥 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))))) → (𝐿<P ⟨{𝑙𝑙 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))}, {𝑢 ∣ ((𝐹𝑄) +Q (𝑄 +Q 𝑅)) <Q 𝑢}⟩ ↔ ∃𝑥Q (𝑥 ∈ (2nd𝐿) ∧ 𝑥 ∈ (1st ‘⟨{𝑙𝑙 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))}, {𝑢 ∣ ((𝐹𝑄) +Q (𝑄 +Q 𝑅)) <Q 𝑢}⟩))))
5541, 54mpbird 167 . 2 ((𝜑 ∧ (𝑥Q ∧ (((𝐹𝑄) +Q 𝑄) <Q 𝑥𝑥 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))))) → 𝐿<P ⟨{𝑙𝑙 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))}, {𝑢 ∣ ((𝐹𝑄) +Q (𝑄 +Q 𝑅)) <Q 𝑢}⟩)
5610, 55rexlimddv 2599 1 (𝜑𝐿<P ⟨{𝑙𝑙 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))}, {𝑢 ∣ ((𝐹𝑄) +Q (𝑄 +Q 𝑅)) <Q 𝑢}⟩)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1353  wcel 2148  {cab 2163  wral 2455  wrex 2456  {crab 2459  cop 3594   class class class wbr 4000  wf 5208  cfv 5212  (class class class)co 5869  1st c1st 6133  2nd c2nd 6134  Qcnq 7270   +Q cplq 7272   <Q cltq 7275  Pcnp 7281  <P cltp 7285
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-eprel 4286  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-irdg 6365  df-1o 6411  df-oadd 6415  df-omul 6416  df-er 6529  df-ec 6531  df-qs 6535  df-ni 7294  df-pli 7295  df-mi 7296  df-lti 7297  df-plpq 7334  df-mpq 7335  df-enq 7337  df-nqqs 7338  df-plqqs 7339  df-mqqs 7340  df-1nqqs 7341  df-rq 7342  df-ltnqqs 7343  df-inp 7456  df-iltp 7460
This theorem is referenced by:  cauappcvgprlemlim  7651
  Copyright terms: Public domain W3C validator