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Theorem zsupcllemex 11434
Description: Lemma for zsupcl 11435. 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 502 . . 3 ((𝜑 ∧ (𝑗 ∈ (ℤ𝑀) ∧ ∀𝑛 ∈ (ℤ𝑗) ¬ 𝜓)) → ∀𝑛 ∈ (ℤ𝑗) ¬ 𝜓)
4 fveq2 5353 . . . . . . . 8 (𝑤 = 𝑀 → (ℤ𝑤) = (ℤ𝑀))
54raleqdv 2590 . . . . . . 7 (𝑤 = 𝑀 → (∀𝑛 ∈ (ℤ𝑤) ¬ 𝜓 ↔ ∀𝑛 ∈ (ℤ𝑀) ¬ 𝜓))
65anbi2d 455 . . . . . 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 5353 . . . . . . . 8 (𝑤 = 𝑘 → (ℤ𝑤) = (ℤ𝑘))
98raleqdv 2590 . . . . . . 7 (𝑤 = 𝑘 → (∀𝑛 ∈ (ℤ𝑤) ¬ 𝜓 ↔ ∀𝑛 ∈ (ℤ𝑘) ¬ 𝜓))
109anbi2d 455 . . . . . 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 5353 . . . . . . . 8 (𝑤 = (𝑘 + 1) → (ℤ𝑤) = (ℤ‘(𝑘 + 1)))
1312raleqdv 2590 . . . . . . 7 (𝑤 = (𝑘 + 1) → (∀𝑛 ∈ (ℤ𝑤) ¬ 𝜓 ↔ ∀𝑛 ∈ (ℤ‘(𝑘 + 1)) ¬ 𝜓))
1413anbi2d 455 . . . . . 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 5353 . . . . . . . 8 (𝑤 = 𝑗 → (ℤ𝑤) = (ℤ𝑗))
1716raleqdv 2590 . . . . . . 7 (𝑤 = 𝑗 → (∀𝑛 ∈ (ℤ𝑤) ¬ 𝜓 ↔ ∀𝑛 ∈ (ℤ𝑗) ¬ 𝜓))
1817anbi2d 455 . . . . . 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 272 . . . . . . 7 ((𝜑 ∧ ∀𝑛 ∈ (ℤ𝑀) ¬ 𝜓) → 𝜒)
22 zsupcllemex.m . . . . . . . . 9 (𝜑𝑀 ∈ ℤ)
23 uzid 9190 . . . . . . . . 9 (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ𝑀))
24 zsupcllemex.sbm . . . . . . . . . . 11 (𝑛 = 𝑀 → (𝜓𝜒))
2524notbid 633 . . . . . . . . . 10 (𝑛 = 𝑀 → (¬ 𝜓 ↔ ¬ 𝜒))
2625rspcv 2740 . . . . . . . . 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 590 . . . . . 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 11433 . . . . 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 9233 . . . 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 477 . . 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 427 . 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 2513 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 786   = wceq 1299  wcel 1448  wral 2375  wrex 2376  {crab 2379   class class class wbr 3875  cfv 5059  (class class class)co 5706  cr 7499  1c1 7501   + caddc 7503   < clt 7672  cz 8906  cuz 9176
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-cnex 7586  ax-resscn 7587  ax-1cn 7588  ax-1re 7589  ax-icn 7590  ax-addcl 7591  ax-addrcl 7592  ax-mulcl 7593  ax-addcom 7595  ax-addass 7597  ax-distr 7599  ax-i2m1 7600  ax-0lt1 7601  ax-0id 7603  ax-rnegex 7604  ax-cnre 7606  ax-pre-ltirr 7607  ax-pre-ltwlin 7608  ax-pre-lttrn 7609  ax-pre-apti 7610  ax-pre-ltadd 7611
This theorem depends on definitions:  df-bi 116  df-dc 787  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-nel 2363  df-ral 2380  df-rex 2381  df-reu 2382  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-id 4153  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-fv 5067  df-riota 5662  df-ov 5709  df-oprab 5710  df-mpo 5711  df-1st 5969  df-2nd 5970  df-pnf 7674  df-mnf 7675  df-xr 7676  df-ltxr 7677  df-le 7678  df-sub 7806  df-neg 7807  df-inn 8579  df-n0 8830  df-z 8907  df-uz 9177  df-fz 9632  df-fzo 9761
This theorem is referenced by:  zsupcl  11435  infssuzex  11437  gcdsupex  11441
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