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Theorem zsupcllemex 11650
Description: Lemma for zsupcl 11651. 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 521 . . 3 ((𝜑 ∧ (𝑗 ∈ (ℤ𝑀) ∧ ∀𝑛 ∈ (ℤ𝑗) ¬ 𝜓)) → ∀𝑛 ∈ (ℤ𝑗) ¬ 𝜓)
4 fveq2 5421 . . . . . . . 8 (𝑤 = 𝑀 → (ℤ𝑤) = (ℤ𝑀))
54raleqdv 2632 . . . . . . 7 (𝑤 = 𝑀 → (∀𝑛 ∈ (ℤ𝑤) ¬ 𝜓 ↔ ∀𝑛 ∈ (ℤ𝑀) ¬ 𝜓))
65anbi2d 459 . . . . . 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 5421 . . . . . . . 8 (𝑤 = 𝑘 → (ℤ𝑤) = (ℤ𝑘))
98raleqdv 2632 . . . . . . 7 (𝑤 = 𝑘 → (∀𝑛 ∈ (ℤ𝑤) ¬ 𝜓 ↔ ∀𝑛 ∈ (ℤ𝑘) ¬ 𝜓))
109anbi2d 459 . . . . . 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 5421 . . . . . . . 8 (𝑤 = (𝑘 + 1) → (ℤ𝑤) = (ℤ‘(𝑘 + 1)))
1312raleqdv 2632 . . . . . . 7 (𝑤 = (𝑘 + 1) → (∀𝑛 ∈ (ℤ𝑤) ¬ 𝜓 ↔ ∀𝑛 ∈ (ℤ‘(𝑘 + 1)) ¬ 𝜓))
1413anbi2d 459 . . . . . 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 5421 . . . . . . . 8 (𝑤 = 𝑗 → (ℤ𝑤) = (ℤ𝑗))
1716raleqdv 2632 . . . . . . 7 (𝑤 = 𝑗 → (∀𝑛 ∈ (ℤ𝑤) ¬ 𝜓 ↔ ∀𝑛 ∈ (ℤ𝑗) ¬ 𝜓))
1817anbi2d 459 . . . . . 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 9352 . . . . . . . . 9 (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ𝑀))
24 zsupcllemex.sbm . . . . . . . . . . 11 (𝑛 = 𝑀 → (𝜓𝜒))
2524notbid 656 . . . . . . . . . 10 (𝑛 = 𝑀 → (¬ 𝜓 ↔ ¬ 𝜒))
2625rspcv 2785 . . . . . . . . 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 609 . . . . . 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 11649 . . . . 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 9395 . . . 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 481 . . 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 429 . 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 2554 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 819   = wceq 1331  wcel 1480  wral 2416  wrex 2417  {crab 2420   class class class wbr 3929  cfv 5123  (class class class)co 5774  cr 7631  1c1 7633   + caddc 7635   < clt 7812  cz 9066  cuz 9338
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7723  ax-resscn 7724  ax-1cn 7725  ax-1re 7726  ax-icn 7727  ax-addcl 7728  ax-addrcl 7729  ax-mulcl 7730  ax-addcom 7732  ax-addass 7734  ax-distr 7736  ax-i2m1 7737  ax-0lt1 7738  ax-0id 7740  ax-rnegex 7741  ax-cnre 7743  ax-pre-ltirr 7744  ax-pre-ltwlin 7745  ax-pre-lttrn 7746  ax-pre-apti 7747  ax-pre-ltadd 7748
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-pnf 7814  df-mnf 7815  df-xr 7816  df-ltxr 7817  df-le 7818  df-sub 7947  df-neg 7948  df-inn 8733  df-n0 8990  df-z 9067  df-uz 9339  df-fz 9803  df-fzo 9932
This theorem is referenced by:  zsupcl  11651  infssuzex  11653  gcdsupex  11657
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