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Theorem zsupcllemex 10736
Description: Lemma for zsupcl 10737. 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 107 . . 3 ((𝜑 ∧ (𝑗 ∈ (ℤ𝑀) ∧ ∀𝑛 ∈ (ℤ𝑗) ¬ 𝜓)) → 𝜑)
3 simprr 499 . . 3 ((𝜑 ∧ (𝑗 ∈ (ℤ𝑀) ∧ ∀𝑛 ∈ (ℤ𝑗) ¬ 𝜓)) → ∀𝑛 ∈ (ℤ𝑗) ¬ 𝜓)
4 fveq2 5256 . . . . . . . 8 (𝑤 = 𝑀 → (ℤ𝑤) = (ℤ𝑀))
54raleqdv 2563 . . . . . . 7 (𝑤 = 𝑀 → (∀𝑛 ∈ (ℤ𝑤) ¬ 𝜓 ↔ ∀𝑛 ∈ (ℤ𝑀) ¬ 𝜓))
65anbi2d 452 . . . . . 6 (𝑤 = 𝑀 → ((𝜑 ∧ ∀𝑛 ∈ (ℤ𝑤) ¬ 𝜓) ↔ (𝜑 ∧ ∀𝑛 ∈ (ℤ𝑀) ¬ 𝜓)))
76imbi1d 229 . . . . 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 5256 . . . . . . . 8 (𝑤 = 𝑘 → (ℤ𝑤) = (ℤ𝑘))
98raleqdv 2563 . . . . . . 7 (𝑤 = 𝑘 → (∀𝑛 ∈ (ℤ𝑤) ¬ 𝜓 ↔ ∀𝑛 ∈ (ℤ𝑘) ¬ 𝜓))
109anbi2d 452 . . . . . 6 (𝑤 = 𝑘 → ((𝜑 ∧ ∀𝑛 ∈ (ℤ𝑤) ¬ 𝜓) ↔ (𝜑 ∧ ∀𝑛 ∈ (ℤ𝑘) ¬ 𝜓)))
1110imbi1d 229 . . . . 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 5256 . . . . . . . 8 (𝑤 = (𝑘 + 1) → (ℤ𝑤) = (ℤ‘(𝑘 + 1)))
1312raleqdv 2563 . . . . . . 7 (𝑤 = (𝑘 + 1) → (∀𝑛 ∈ (ℤ𝑤) ¬ 𝜓 ↔ ∀𝑛 ∈ (ℤ‘(𝑘 + 1)) ¬ 𝜓))
1413anbi2d 452 . . . . . 6 (𝑤 = (𝑘 + 1) → ((𝜑 ∧ ∀𝑛 ∈ (ℤ𝑤) ¬ 𝜓) ↔ (𝜑 ∧ ∀𝑛 ∈ (ℤ‘(𝑘 + 1)) ¬ 𝜓)))
1514imbi1d 229 . . . . 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 5256 . . . . . . . 8 (𝑤 = 𝑗 → (ℤ𝑤) = (ℤ𝑗))
1716raleqdv 2563 . . . . . . 7 (𝑤 = 𝑗 → (∀𝑛 ∈ (ℤ𝑤) ¬ 𝜓 ↔ ∀𝑛 ∈ (ℤ𝑗) ¬ 𝜓))
1817anbi2d 452 . . . . . 6 (𝑤 = 𝑗 → ((𝜑 ∧ ∀𝑛 ∈ (ℤ𝑤) ¬ 𝜓) ↔ (𝜑 ∧ ∀𝑛 ∈ (ℤ𝑗) ¬ 𝜓)))
1918imbi1d 229 . . . . 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 270 . . . . . . 7 ((𝜑 ∧ ∀𝑛 ∈ (ℤ𝑀) ¬ 𝜓) → 𝜒)
22 zsupcllemex.m . . . . . . . . 9 (𝜑𝑀 ∈ ℤ)
23 uzid 8942 . . . . . . . . 9 (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ𝑀))
24 zsupcllemex.sbm . . . . . . . . . . 11 (𝑛 = 𝑀 → (𝜓𝜒))
2524notbid 625 . . . . . . . . . 10 (𝑛 = 𝑀 → (¬ 𝜓 ↔ ¬ 𝜒))
2625rspcv 2710 . . . . . . . . 9 (𝑀 ∈ (ℤ𝑀) → (∀𝑛 ∈ (ℤ𝑀) ¬ 𝜓 → ¬ 𝜒))
2722, 23, 263syl 17 . . . . . . . 8 (𝜑 → (∀𝑛 ∈ (ℤ𝑀) ¬ 𝜓 → ¬ 𝜒))
2827imp 122 . . . . . . 7 ((𝜑 ∧ ∀𝑛 ∈ (ℤ𝑀) ¬ 𝜓) → ¬ 𝜒)
2921, 28pm2.21dd 583 . . . . . 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 10735 . . . . 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 8985 . . . 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 474 . . 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 424 . 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 2489 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 102  wb 103  DECID wdc 778   = wceq 1287  wcel 1436  wral 2355  wrex 2356  {crab 2359   class class class wbr 3814  cfv 4972  (class class class)co 5594  cr 7270  1c1 7272   + caddc 7274   < clt 7443  cz 8660  cuz 8928
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3925  ax-pow 3977  ax-pr 4003  ax-un 4227  ax-setind 4319  ax-cnex 7357  ax-resscn 7358  ax-1cn 7359  ax-1re 7360  ax-icn 7361  ax-addcl 7362  ax-addrcl 7363  ax-mulcl 7364  ax-addcom 7366  ax-addass 7368  ax-distr 7370  ax-i2m1 7371  ax-0lt1 7372  ax-0id 7374  ax-rnegex 7375  ax-cnre 7377  ax-pre-ltirr 7378  ax-pre-ltwlin 7379  ax-pre-lttrn 7380  ax-pre-apti 7381  ax-pre-ltadd 7382
This theorem depends on definitions:  df-bi 115  df-dc 779  df-3or 923  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-nel 2347  df-ral 2360  df-rex 2361  df-reu 2362  df-rab 2364  df-v 2616  df-sbc 2829  df-csb 2922  df-dif 2988  df-un 2990  df-in 2992  df-ss 2999  df-pw 3411  df-sn 3431  df-pr 3432  df-op 3434  df-uni 3631  df-int 3666  df-iun 3709  df-br 3815  df-opab 3869  df-mpt 3870  df-id 4087  df-xp 4410  df-rel 4411  df-cnv 4412  df-co 4413  df-dm 4414  df-rn 4415  df-res 4416  df-ima 4417  df-iota 4937  df-fun 4974  df-fn 4975  df-f 4976  df-fv 4980  df-riota 5550  df-ov 5597  df-oprab 5598  df-mpt2 5599  df-1st 5849  df-2nd 5850  df-pnf 7445  df-mnf 7446  df-xr 7447  df-ltxr 7448  df-le 7449  df-sub 7576  df-neg 7577  df-inn 8335  df-n0 8584  df-z 8661  df-uz 8929  df-fz 9334  df-fzo 9458
This theorem is referenced by:  zsupcl  10737  infssuzex  10739  gcdsupex  10743
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