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

Theorem zsupcllemex 11890
Description: Lemma for zsupcl 11891. 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 108 . . 3 ((𝜑 ∧ (𝑗 ∈ (ℤ𝑀) ∧ ∀𝑛 ∈ (ℤ𝑗) ¬ 𝜓)) → 𝜑)
3 simprr 527 . . 3 ((𝜑 ∧ (𝑗 ∈ (ℤ𝑀) ∧ ∀𝑛 ∈ (ℤ𝑗) ¬ 𝜓)) → ∀𝑛 ∈ (ℤ𝑗) ¬ 𝜓)
4 fveq2 5494 . . . . . . . 8 (𝑤 = 𝑀 → (ℤ𝑤) = (ℤ𝑀))
54raleqdv 2671 . . . . . . 7 (𝑤 = 𝑀 → (∀𝑛 ∈ (ℤ𝑤) ¬ 𝜓 ↔ ∀𝑛 ∈ (ℤ𝑀) ¬ 𝜓))
65anbi2d 461 . . . . . 6 (𝑤 = 𝑀 → ((𝜑 ∧ ∀𝑛 ∈ (ℤ𝑤) ¬ 𝜓) ↔ (𝜑 ∧ ∀𝑛 ∈ (ℤ𝑀) ¬ 𝜓)))
76imbi1d 230 . . . . 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 5494 . . . . . . . 8 (𝑤 = 𝑘 → (ℤ𝑤) = (ℤ𝑘))
98raleqdv 2671 . . . . . . 7 (𝑤 = 𝑘 → (∀𝑛 ∈ (ℤ𝑤) ¬ 𝜓 ↔ ∀𝑛 ∈ (ℤ𝑘) ¬ 𝜓))
109anbi2d 461 . . . . . 6 (𝑤 = 𝑘 → ((𝜑 ∧ ∀𝑛 ∈ (ℤ𝑤) ¬ 𝜓) ↔ (𝜑 ∧ ∀𝑛 ∈ (ℤ𝑘) ¬ 𝜓)))
1110imbi1d 230 . . . . 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 5494 . . . . . . . 8 (𝑤 = (𝑘 + 1) → (ℤ𝑤) = (ℤ‘(𝑘 + 1)))
1312raleqdv 2671 . . . . . . 7 (𝑤 = (𝑘 + 1) → (∀𝑛 ∈ (ℤ𝑤) ¬ 𝜓 ↔ ∀𝑛 ∈ (ℤ‘(𝑘 + 1)) ¬ 𝜓))
1413anbi2d 461 . . . . . 6 (𝑤 = (𝑘 + 1) → ((𝜑 ∧ ∀𝑛 ∈ (ℤ𝑤) ¬ 𝜓) ↔ (𝜑 ∧ ∀𝑛 ∈ (ℤ‘(𝑘 + 1)) ¬ 𝜓)))
1514imbi1d 230 . . . . 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 5494 . . . . . . . 8 (𝑤 = 𝑗 → (ℤ𝑤) = (ℤ𝑗))
1716raleqdv 2671 . . . . . . 7 (𝑤 = 𝑗 → (∀𝑛 ∈ (ℤ𝑤) ¬ 𝜓 ↔ ∀𝑛 ∈ (ℤ𝑗) ¬ 𝜓))
1817anbi2d 461 . . . . . 6 (𝑤 = 𝑗 → ((𝜑 ∧ ∀𝑛 ∈ (ℤ𝑤) ¬ 𝜓) ↔ (𝜑 ∧ ∀𝑛 ∈ (ℤ𝑗) ¬ 𝜓)))
1918imbi1d 230 . . . . 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 274 . . . . . . 7 ((𝜑 ∧ ∀𝑛 ∈ (ℤ𝑀) ¬ 𝜓) → 𝜒)
22 zsupcllemex.m . . . . . . . . 9 (𝜑𝑀 ∈ ℤ)
23 uzid 9490 . . . . . . . . 9 (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ𝑀))
24 zsupcllemex.sbm . . . . . . . . . . 11 (𝑛 = 𝑀 → (𝜓𝜒))
2524notbid 662 . . . . . . . . . 10 (𝑛 = 𝑀 → (¬ 𝜓 ↔ ¬ 𝜒))
2625rspcv 2830 . . . . . . . . 9 (𝑀 ∈ (ℤ𝑀) → (∀𝑛 ∈ (ℤ𝑀) ¬ 𝜓 → ¬ 𝜒))
2722, 23, 263syl 17 . . . . . . . 8 (𝜑 → (∀𝑛 ∈ (ℤ𝑀) ¬ 𝜓 → ¬ 𝜒))
2827imp 123 . . . . . . 7 ((𝜑 ∧ ∀𝑛 ∈ (ℤ𝑀) ¬ 𝜓) → ¬ 𝜒)
2921, 28pm2.21dd 615 . . . . . 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 11889 . . . . 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 9536 . . . 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 487 . . 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 431 . 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 2592 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 103  wb 104  DECID wdc 829   = wceq 1348  wcel 2141  wral 2448  wrex 2449  {crab 2452   class class class wbr 3987  cfv 5196  (class class class)co 5851  cr 7762  1c1 7764   + caddc 7766   < clt 7943  cz 9201  cuz 9476
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-cnex 7854  ax-resscn 7855  ax-1cn 7856  ax-1re 7857  ax-icn 7858  ax-addcl 7859  ax-addrcl 7860  ax-mulcl 7861  ax-addcom 7863  ax-addass 7865  ax-distr 7867  ax-i2m1 7868  ax-0lt1 7869  ax-0id 7871  ax-rnegex 7872  ax-cnre 7874  ax-pre-ltirr 7875  ax-pre-ltwlin 7876  ax-pre-lttrn 7877  ax-pre-apti 7878  ax-pre-ltadd 7879
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-fv 5204  df-riota 5807  df-ov 5854  df-oprab 5855  df-mpo 5856  df-1st 6117  df-2nd 6118  df-pnf 7945  df-mnf 7946  df-xr 7947  df-ltxr 7948  df-le 7949  df-sub 8081  df-neg 8082  df-inn 8868  df-n0 9125  df-z 9202  df-uz 9477  df-fz 9955  df-fzo 10088
This theorem is referenced by:  zsupcl  11891  infssuzex  11893  gcdsupex  11901
  Copyright terms: Public domain W3C validator