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Theorem cauappcvgprlem2 7634
Description: Lemma for cauappcvgpr 7636. 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 7381 . . . . 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 5644 . . . 4 (𝜑 → (𝐹𝑄) ∈ Q)
7 ltanqi 7376 . . . 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 7389 . . 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 5507 . . . . . . . . 9 (𝑞 = 𝑄 → (𝐹𝑞) = (𝐹𝑄))
15 id 19 . . . . . . . . 9 (𝑞 = 𝑄𝑞 = 𝑄)
1614, 15oveq12d 5883 . . . . . . . 8 (𝑞 = 𝑄 → ((𝐹𝑞) +Q 𝑞) = ((𝐹𝑄) +Q 𝑄))
1716breq1d 4008 . . . . . . 7 (𝑞 = 𝑄 → (((𝐹𝑞) +Q 𝑞) <Q 𝑥 ↔ ((𝐹𝑄) +Q 𝑄) <Q 𝑥))
1817rspcev 2839 . . . . . 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 4002 . . . . . . 7 (𝑢 = 𝑥 → (((𝐹𝑞) +Q 𝑞) <Q 𝑢 ↔ ((𝐹𝑞) +Q 𝑞) <Q 𝑥))
2120rexbidv 2476 . . . . . 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 5510 . . . . . . 7 (2nd𝐿) = (2nd ‘⟨{𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)}, {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢}⟩)
24 nqex 7337 . . . . . . . . 9 Q ∈ V
2524rabex 4142 . . . . . . . 8 {𝑙Q ∣ ∃𝑞Q (𝑙 +Q 𝑞) <Q (𝐹𝑞)} ∈ V
2624rabex 4142 . . . . . . . 8 {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢} ∈ V
2725, 26op2nd 6138 . . . . . . 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 2196 . . . . . 6 (2nd𝐿) = {𝑢Q ∣ ∃𝑞Q ((𝐹𝑞) +Q 𝑞) <Q 𝑢}
2921, 28elrab2 2894 . . . . 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 2738 . . . . . . 7 𝑥 ∈ V
33 breq1 4001 . . . . . . 7 (𝑙 = 𝑥 → (𝑙 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅)) ↔ 𝑥 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))))
3432, 33elab 2879 . . . . . 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 7523 . . . . . 6 {𝑙𝑙 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))} ∈ V
37 gtnqex 7524 . . . . . 6 {𝑢 ∣ ((𝐹𝑄) +Q (𝑄 +Q 𝑅)) <Q 𝑢} ∈ V
3836, 37op1st 6137 . . . . 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 2269 . . . 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 2524 . . . 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 7627 . . . . 5 (𝜑𝐿P)
4544adantr 276 . . . 4 ((𝜑 ∧ (𝑥Q ∧ (((𝐹𝑄) +Q 𝑄) <Q 𝑥𝑥 <Q ((𝐹𝑄) +Q (𝑄 +Q 𝑅))))) → 𝐿P)
46 addclnq 7349 . . . . . . . 8 ((𝑄Q𝑅Q) → (𝑄 +Q 𝑅) ∈ Q)
471, 2, 46syl2anc 411 . . . . . . 7 (𝜑 → (𝑄 +Q 𝑅) ∈ Q)
48 addclnq 7349 . . . . . . 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 7521 . . . . . 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 7480 . . . 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 2597 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 2146  {cab 2161  wral 2453  wrex 2454  {crab 2457  cop 3592   class class class wbr 3998  wf 5204  cfv 5208  (class class class)co 5865  1st c1st 6129  2nd c2nd 6130  Qcnq 7254   +Q cplq 7256   <Q cltq 7259  Pcnp 7265  <P cltp 7269
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 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-iinf 4581
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 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-ral 2458  df-rex 2459  df-reu 2460  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-eprel 4283  df-id 4287  df-po 4290  df-iso 4291  df-iord 4360  df-on 4362  df-suc 4365  df-iom 4584  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-recs 6296  df-irdg 6361  df-1o 6407  df-oadd 6411  df-omul 6412  df-er 6525  df-ec 6527  df-qs 6531  df-ni 7278  df-pli 7279  df-mi 7280  df-lti 7281  df-plpq 7318  df-mpq 7319  df-enq 7321  df-nqqs 7322  df-plqqs 7323  df-mqqs 7324  df-1nqqs 7325  df-rq 7326  df-ltnqqs 7327  df-inp 7440  df-iltp 7444
This theorem is referenced by:  cauappcvgprlemlim  7635
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