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

Theorem zsupcllemex 11982
Description: Lemma for zsupcl 11983. Existence of the supremum. (Contributed by Jim Kingdon, 7-Dec-2021.)
Hypotheses
Ref Expression
zsupcllemex.m (𝜑𝑀 ∈ ℤ)
zsupcllemex.sbm (𝑛 = 𝑀 → (𝜓𝜒))
zsupcllemex.mtru (𝜑𝜒)
zsupcllemex.dc ((𝜑𝑛 ∈ (ℤ𝑀)) → DECID 𝜓)
zsupcllemex.bnd (𝜑 → ∃𝑗 ∈ (ℤ𝑀)∀𝑛 ∈ (ℤ𝑗) ¬ 𝜓)
Assertion
Ref Expression
zsupcllemex (𝜑 → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)))
Distinct variable groups:   𝑛,𝑀,𝑦   𝜒,𝑛   𝑗,𝑛,𝜑,𝑦   𝜓,𝑗,𝑥,𝑧,𝑦   𝑥,𝑛,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑧)   𝜓(𝑛)   𝜒(𝑥,𝑦,𝑧,𝑗)   𝑀(𝑥,𝑧,𝑗)

Proof of Theorem zsupcllemex
Dummy variables 𝑘 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 zsupcllemex.bnd . 2 (𝜑 → ∃𝑗 ∈ (ℤ𝑀)∀𝑛 ∈ (ℤ𝑗) ¬ 𝜓)
2 simpl 109 . . 3 ((𝜑 ∧ (𝑗 ∈ (ℤ𝑀) ∧ ∀𝑛 ∈ (ℤ𝑗) ¬ 𝜓)) → 𝜑)
3 simprr 531 . . 3 ((𝜑 ∧ (𝑗 ∈ (ℤ𝑀) ∧ ∀𝑛 ∈ (ℤ𝑗) ¬ 𝜓)) → ∀𝑛 ∈ (ℤ𝑗) ¬ 𝜓)
4 fveq2 5534 . . . . . . . 8 (𝑤 = 𝑀 → (ℤ𝑤) = (ℤ𝑀))
54raleqdv 2692 . . . . . . 7 (𝑤 = 𝑀 → (∀𝑛 ∈ (ℤ𝑤) ¬ 𝜓 ↔ ∀𝑛 ∈ (ℤ𝑀) ¬ 𝜓))
65anbi2d 464 . . . . . 6 (𝑤 = 𝑀 → ((𝜑 ∧ ∀𝑛 ∈ (ℤ𝑤) ¬ 𝜓) ↔ (𝜑 ∧ ∀𝑛 ∈ (ℤ𝑀) ¬ 𝜓)))
76imbi1d 231 . . . . 5 (𝑤 = 𝑀 → (((𝜑 ∧ ∀𝑛 ∈ (ℤ𝑤) ¬ 𝜓) → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧))) ↔ ((𝜑 ∧ ∀𝑛 ∈ (ℤ𝑀) ¬ 𝜓) → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)))))
8 fveq2 5534 . . . . . . . 8 (𝑤 = 𝑘 → (ℤ𝑤) = (ℤ𝑘))
98raleqdv 2692 . . . . . . 7 (𝑤 = 𝑘 → (∀𝑛 ∈ (ℤ𝑤) ¬ 𝜓 ↔ ∀𝑛 ∈ (ℤ𝑘) ¬ 𝜓))
109anbi2d 464 . . . . . 6 (𝑤 = 𝑘 → ((𝜑 ∧ ∀𝑛 ∈ (ℤ𝑤) ¬ 𝜓) ↔ (𝜑 ∧ ∀𝑛 ∈ (ℤ𝑘) ¬ 𝜓)))
1110imbi1d 231 . . . . 5 (𝑤 = 𝑘 → (((𝜑 ∧ ∀𝑛 ∈ (ℤ𝑤) ¬ 𝜓) → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧))) ↔ ((𝜑 ∧ ∀𝑛 ∈ (ℤ𝑘) ¬ 𝜓) → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)))))
12 fveq2 5534 . . . . . . . 8 (𝑤 = (𝑘 + 1) → (ℤ𝑤) = (ℤ‘(𝑘 + 1)))
1312raleqdv 2692 . . . . . . 7 (𝑤 = (𝑘 + 1) → (∀𝑛 ∈ (ℤ𝑤) ¬ 𝜓 ↔ ∀𝑛 ∈ (ℤ‘(𝑘 + 1)) ¬ 𝜓))
1413anbi2d 464 . . . . . 6 (𝑤 = (𝑘 + 1) → ((𝜑 ∧ ∀𝑛 ∈ (ℤ𝑤) ¬ 𝜓) ↔ (𝜑 ∧ ∀𝑛 ∈ (ℤ‘(𝑘 + 1)) ¬ 𝜓)))
1514imbi1d 231 . . . . 5 (𝑤 = (𝑘 + 1) → (((𝜑 ∧ ∀𝑛 ∈ (ℤ𝑤) ¬ 𝜓) → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧))) ↔ ((𝜑 ∧ ∀𝑛 ∈ (ℤ‘(𝑘 + 1)) ¬ 𝜓) → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)))))
16 fveq2 5534 . . . . . . . 8 (𝑤 = 𝑗 → (ℤ𝑤) = (ℤ𝑗))
1716raleqdv 2692 . . . . . . 7 (𝑤 = 𝑗 → (∀𝑛 ∈ (ℤ𝑤) ¬ 𝜓 ↔ ∀𝑛 ∈ (ℤ𝑗) ¬ 𝜓))
1817anbi2d 464 . . . . . 6 (𝑤 = 𝑗 → ((𝜑 ∧ ∀𝑛 ∈ (ℤ𝑤) ¬ 𝜓) ↔ (𝜑 ∧ ∀𝑛 ∈ (ℤ𝑗) ¬ 𝜓)))
1918imbi1d 231 . . . . 5 (𝑤 = 𝑗 → (((𝜑 ∧ ∀𝑛 ∈ (ℤ𝑤) ¬ 𝜓) → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧))) ↔ ((𝜑 ∧ ∀𝑛 ∈ (ℤ𝑗) ¬ 𝜓) → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)))))
20 zsupcllemex.mtru . . . . . . . 8 (𝜑𝜒)
2120adantr 276 . . . . . . 7 ((𝜑 ∧ ∀𝑛 ∈ (ℤ𝑀) ¬ 𝜓) → 𝜒)
22 zsupcllemex.m . . . . . . . . 9 (𝜑𝑀 ∈ ℤ)
23 uzid 9573 . . . . . . . . 9 (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ𝑀))
24 zsupcllemex.sbm . . . . . . . . . . 11 (𝑛 = 𝑀 → (𝜓𝜒))
2524notbid 668 . . . . . . . . . 10 (𝑛 = 𝑀 → (¬ 𝜓 ↔ ¬ 𝜒))
2625rspcv 2852 . . . . . . . . 9 (𝑀 ∈ (ℤ𝑀) → (∀𝑛 ∈ (ℤ𝑀) ¬ 𝜓 → ¬ 𝜒))
2722, 23, 263syl 17 . . . . . . . 8 (𝜑 → (∀𝑛 ∈ (ℤ𝑀) ¬ 𝜓 → ¬ 𝜒))
2827imp 124 . . . . . . 7 ((𝜑 ∧ ∀𝑛 ∈ (ℤ𝑀) ¬ 𝜓) → ¬ 𝜒)
2921, 28pm2.21dd 621 . . . . . 6 ((𝜑 ∧ ∀𝑛 ∈ (ℤ𝑀) ¬ 𝜓) → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)))
3029a1i 9 . . . . 5 (𝑀 ∈ ℤ → ((𝜑 ∧ ∀𝑛 ∈ (ℤ𝑀) ¬ 𝜓) → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧))))
31 zsupcllemex.dc . . . . . 6 ((𝜑𝑛 ∈ (ℤ𝑀)) → DECID 𝜓)
3231zsupcllemstep 11981 . . . . 5 (𝑘 ∈ (ℤ𝑀) → (((𝜑 ∧ ∀𝑛 ∈ (ℤ𝑘) ¬ 𝜓) → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧))) → ((𝜑 ∧ ∀𝑛 ∈ (ℤ‘(𝑘 + 1)) ¬ 𝜓) → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)))))
337, 11, 15, 19, 30, 32uzind4 9620 . . . 4 (𝑗 ∈ (ℤ𝑀) → ((𝜑 ∧ ∀𝑛 ∈ (ℤ𝑗) ¬ 𝜓) → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧))))
3433ad2antrl 490 . . 3 ((𝜑 ∧ (𝑗 ∈ (ℤ𝑀) ∧ ∀𝑛 ∈ (ℤ𝑗) ¬ 𝜓)) → ((𝜑 ∧ ∀𝑛 ∈ (ℤ𝑗) ¬ 𝜓) → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧))))
352, 3, 34mp2and 433 . 2 ((𝜑 ∧ (𝑗 ∈ (ℤ𝑀) ∧ ∀𝑛 ∈ (ℤ𝑗) ¬ 𝜓)) → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)))
361, 35rexlimddv 2612 1 (𝜑 → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wb 105  DECID wdc 835   = wceq 1364  wcel 2160  wral 2468  wrex 2469  {crab 2472   class class class wbr 4018  cfv 5235  (class class class)co 5897  cr 7841  1c1 7843   + caddc 7845   < clt 8023  cz 9284  cuz 9559
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7933  ax-resscn 7934  ax-1cn 7935  ax-1re 7936  ax-icn 7937  ax-addcl 7938  ax-addrcl 7939  ax-mulcl 7940  ax-addcom 7942  ax-addass 7944  ax-distr 7946  ax-i2m1 7947  ax-0lt1 7948  ax-0id 7950  ax-rnegex 7951  ax-cnre 7953  ax-pre-ltirr 7954  ax-pre-ltwlin 7955  ax-pre-lttrn 7956  ax-pre-apti 7957  ax-pre-ltadd 7958
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-fv 5243  df-riota 5852  df-ov 5900  df-oprab 5901  df-mpo 5902  df-1st 6166  df-2nd 6167  df-pnf 8025  df-mnf 8026  df-xr 8027  df-ltxr 8028  df-le 8029  df-sub 8161  df-neg 8162  df-inn 8951  df-n0 9208  df-z 9285  df-uz 9560  df-fz 10041  df-fzo 10175
This theorem is referenced by:  zsupcl  11983  infssuzex  11985  gcdsupex  11993
  Copyright terms: Public domain W3C validator