Theorem zsupcllemex 10371

Description: Lemma for zsupcl 10372. Existence of the supremum. (Contributed by Jim Kingdon, 7-Dec-2021.)

Hypotheses

Ref	Expression
zsupcllemex.m	⊢ (𝜑 → 𝑀 ∈ ℤ)
zsupcllemex.sbm	⊢ (𝑛 = 𝑀 → (𝜓 ↔ 𝜒))
zsupcllemex.mtru	⊢ (𝜑 → 𝜒)
zsupcllemex.dc	⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → DECID 𝜓)
zsupcllemex.bnd	⊢ (𝜑 → ∃𝑗 ∈ (ℤ≥‘𝑀)∀𝑛 ∈ (ℤ≥‘𝑗) ¬ 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.

Step	Hyp	Ref	Expression
1		zsupcllemex.bnd	. 2 ⊢ (𝜑 → ∃𝑗 ∈ (ℤ≥‘𝑀)∀𝑛 ∈ (ℤ≥‘𝑗) ¬ 𝜓)
2		simpl 109	. . 3 ⊢ ((𝜑 ∧ (𝑗 ∈ (ℤ≥‘𝑀) ∧ ∀𝑛 ∈ (ℤ≥‘𝑗) ¬ 𝜓)) → 𝜑)
3		simprr 531	. . 3 ⊢ ((𝜑 ∧ (𝑗 ∈ (ℤ≥‘𝑀) ∧ ∀𝑛 ∈ (ℤ≥‘𝑗) ¬ 𝜓)) → ∀𝑛 ∈ (ℤ≥‘𝑗) ¬ 𝜓)
4		fveq2 5575	. . . . . . . 8 ⊢ (𝑤 = 𝑀 → (ℤ≥‘𝑤) = (ℤ≥‘𝑀))
5	4	raleqdv 2707	. . . . . . 7 ⊢ (𝑤 = 𝑀 → (∀𝑛 ∈ (ℤ≥‘𝑤) ¬ 𝜓 ↔ ∀𝑛 ∈ (ℤ≥‘𝑀) ¬ 𝜓))
6	5	anbi2d 464	. . . . . 6 ⊢ (𝑤 = 𝑀 → ((𝜑 ∧ ∀𝑛 ∈ (ℤ≥‘𝑤) ¬ 𝜓) ↔ (𝜑 ∧ ∀𝑛 ∈ (ℤ≥‘𝑀) ¬ 𝜓)))
7	6	imbi1d 231	. . . . 5 ⊢ (𝑤 = 𝑀 → (((𝜑 ∧ ∀𝑛 ∈ (ℤ≥‘𝑤) ¬ 𝜓) → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧))) ↔ ((𝜑 ∧ ∀𝑛 ∈ (ℤ≥‘𝑀) ¬ 𝜓) → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)))))
8		fveq2 5575	. . . . . . . 8 ⊢ (𝑤 = 𝑘 → (ℤ≥‘𝑤) = (ℤ≥‘𝑘))
9	8	raleqdv 2707	. . . . . . 7 ⊢ (𝑤 = 𝑘 → (∀𝑛 ∈ (ℤ≥‘𝑤) ¬ 𝜓 ↔ ∀𝑛 ∈ (ℤ≥‘𝑘) ¬ 𝜓))
10	9	anbi2d 464	. . . . . 6 ⊢ (𝑤 = 𝑘 → ((𝜑 ∧ ∀𝑛 ∈ (ℤ≥‘𝑤) ¬ 𝜓) ↔ (𝜑 ∧ ∀𝑛 ∈ (ℤ≥‘𝑘) ¬ 𝜓)))
11	10	imbi1d 231	. . . . 5 ⊢ (𝑤 = 𝑘 → (((𝜑 ∧ ∀𝑛 ∈ (ℤ≥‘𝑤) ¬ 𝜓) → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧))) ↔ ((𝜑 ∧ ∀𝑛 ∈ (ℤ≥‘𝑘) ¬ 𝜓) → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)))))
12		fveq2 5575	. . . . . . . 8 ⊢ (𝑤 = (𝑘 + 1) → (ℤ≥‘𝑤) = (ℤ≥‘(𝑘 + 1)))
13	12	raleqdv 2707	. . . . . . 7 ⊢ (𝑤 = (𝑘 + 1) → (∀𝑛 ∈ (ℤ≥‘𝑤) ¬ 𝜓 ↔ ∀𝑛 ∈ (ℤ≥‘(𝑘 + 1)) ¬ 𝜓))
14	13	anbi2d 464	. . . . . 6 ⊢ (𝑤 = (𝑘 + 1) → ((𝜑 ∧ ∀𝑛 ∈ (ℤ≥‘𝑤) ¬ 𝜓) ↔ (𝜑 ∧ ∀𝑛 ∈ (ℤ≥‘(𝑘 + 1)) ¬ 𝜓)))
15	14	imbi1d 231	. . . . 5 ⊢ (𝑤 = (𝑘 + 1) → (((𝜑 ∧ ∀𝑛 ∈ (ℤ≥‘𝑤) ¬ 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 5575	. . . . . . . 8 ⊢ (𝑤 = 𝑗 → (ℤ≥‘𝑤) = (ℤ≥‘𝑗))
17	16	raleqdv 2707	. . . . . . 7 ⊢ (𝑤 = 𝑗 → (∀𝑛 ∈ (ℤ≥‘𝑤) ¬ 𝜓 ↔ ∀𝑛 ∈ (ℤ≥‘𝑗) ¬ 𝜓))
18	17	anbi2d 464	. . . . . 6 ⊢ (𝑤 = 𝑗 → ((𝜑 ∧ ∀𝑛 ∈ (ℤ≥‘𝑤) ¬ 𝜓) ↔ (𝜑 ∧ ∀𝑛 ∈ (ℤ≥‘𝑗) ¬ 𝜓)))
19	18	imbi1d 231	. . . . 5 ⊢ (𝑤 = 𝑗 → (((𝜑 ∧ ∀𝑛 ∈ (ℤ≥‘𝑤) ¬ 𝜓) → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧))) ↔ ((𝜑 ∧ ∀𝑛 ∈ (ℤ≥‘𝑗) ¬ 𝜓) → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)))))
20		zsupcllemex.mtru	. . . . . . . 8 ⊢ (𝜑 → 𝜒)
21	20	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ ∀𝑛 ∈ (ℤ≥‘𝑀) ¬ 𝜓) → 𝜒)
22		zsupcllemex.m	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℤ)
23		uzid 9661	. . . . . . . . 9 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀))
24		zsupcllemex.sbm	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑀 → (𝜓 ↔ 𝜒))
25	24	notbid 668	. . . . . . . . . 10 ⊢ (𝑛 = 𝑀 → (¬ 𝜓 ↔ ¬ 𝜒))
26	25	rspcv 2872	. . . . . . . . 9 ⊢ (𝑀 ∈ (ℤ≥‘𝑀) → (∀𝑛 ∈ (ℤ≥‘𝑀) ¬ 𝜓 → ¬ 𝜒))
27	22, 23, 26	3syl 17	. . . . . . . 8 ⊢ (𝜑 → (∀𝑛 ∈ (ℤ≥‘𝑀) ¬ 𝜓 → ¬ 𝜒))
28	27	imp 124	. . . . . . 7 ⊢ ((𝜑 ∧ ∀𝑛 ∈ (ℤ≥‘𝑀) ¬ 𝜓) → ¬ 𝜒)
29	21, 28	pm2.21dd 621	. . . . . 6 ⊢ ((𝜑 ∧ ∀𝑛 ∈ (ℤ≥‘𝑀) ¬ 𝜓) → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)))
30	29	a1i 9	. . . . 5 ⊢ (𝑀 ∈ ℤ → ((𝜑 ∧ ∀𝑛 ∈ (ℤ≥‘𝑀) ¬ 𝜓) → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧))))
31		zsupcllemex.dc	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → DECID 𝜓)
32	31	zsupcllemstep 10370	. . . . 5 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → (((𝜑 ∧ ∀𝑛 ∈ (ℤ≥‘𝑘) ¬ 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)))))
33	7, 11, 15, 19, 30, 32	uzind4 9708	. . . 4 ⊢ (𝑗 ∈ (ℤ≥‘𝑀) → ((𝜑 ∧ ∀𝑛 ∈ (ℤ≥‘𝑗) ¬ 𝜓) → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧))))
34	33	ad2antrl 490	. . 3 ⊢ ((𝜑 ∧ (𝑗 ∈ (ℤ≥‘𝑀) ∧ ∀𝑛 ∈ (ℤ≥‘𝑗) ¬ 𝜓)) → ((𝜑 ∧ ∀𝑛 ∈ (ℤ≥‘𝑗) ¬ 𝜓) → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧))))
35	2, 3, 34	mp2and 433	. 2 ⊢ ((𝜑 ∧ (𝑗 ∈ (ℤ≥‘𝑀) ∧ ∀𝑛 ∈ (ℤ≥‘𝑗) ¬ 𝜓)) → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)))
36	1, 35	rexlimddv 2627	1 ⊢ (𝜑 → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105  DECID wdc 835   = wceq 1372   ∈ wcel 2175  ∀wral 2483  ∃wrex 2484  {crab 2487   class class class wbr 4043  ‘cfv 5270  (class class class)co 5943  ℝcr 7923  1c1 7925   + caddc 7927   < clt 8106  ℤcz 9371  ℤ_≥cuz 9647

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-cnex 8015  ax-resscn 8016  ax-1cn 8017  ax-1re 8018  ax-icn 8019  ax-addcl 8020  ax-addrcl 8021  ax-mulcl 8022  ax-addcom 8024  ax-addass 8026  ax-distr 8028  ax-i2m1 8029  ax-0lt1 8030  ax-0id 8032  ax-rnegex 8033  ax-cnre 8035  ax-pre-ltirr 8036  ax-pre-ltwlin 8037  ax-pre-lttrn 8038  ax-pre-apti 8039  ax-pre-ltadd 8040

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-nel 2471  df-ral 2488  df-rex 2489  df-reu 2490  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-fv 5278  df-riota 5898  df-ov 5946  df-oprab 5947  df-mpo 5948  df-1st 6225  df-2nd 6226  df-pnf 8108  df-mnf 8109  df-xr 8110  df-ltxr 8111  df-le 8112  df-sub 8244  df-neg 8245  df-inn 9036  df-n0 9295  df-z 9372  df-uz 9648  df-fz 10130  df-fzo 10264

This theorem is referenced by:  zsupcl  10372  infssuzex  10374  gcdsupex  12220