Theorem zsupcllemex 11949

Description: Lemma for zsupcl 11950. 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 5517	. . . . . . . 8 ⊢ (𝑤 = 𝑀 → (ℤ≥‘𝑤) = (ℤ≥‘𝑀))
5	4	raleqdv 2679	. . . . . . 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 5517	. . . . . . . 8 ⊢ (𝑤 = 𝑘 → (ℤ≥‘𝑤) = (ℤ≥‘𝑘))
9	8	raleqdv 2679	. . . . . . 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 5517	. . . . . . . 8 ⊢ (𝑤 = (𝑘 + 1) → (ℤ≥‘𝑤) = (ℤ≥‘(𝑘 + 1)))
13	12	raleqdv 2679	. . . . . . 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 5517	. . . . . . . 8 ⊢ (𝑤 = 𝑗 → (ℤ≥‘𝑤) = (ℤ≥‘𝑗))
17	16	raleqdv 2679	. . . . . . 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 9544	. . . . . . . . 9 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀))
24		zsupcllemex.sbm	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑀 → (𝜓 ↔ 𝜒))
25	24	notbid 667	. . . . . . . . . 10 ⊢ (𝑛 = 𝑀 → (¬ 𝜓 ↔ ¬ 𝜒))
26	25	rspcv 2839	. . . . . . . . 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 620	. . . . . 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 11948	. . . . 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 9590	. . . 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 2599	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 834   = wceq 1353   ∈ wcel 2148  ∀wral 2455  ∃wrex 2456  {crab 2459   class class class wbr 4005  ‘cfv 5218  (class class class)co 5877  ℝcr 7812  1c1 7814   + caddc 7816   < clt 7994  ℤcz 9255  ℤ_≥cuz 9530

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-addcom 7913  ax-addass 7915  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-0id 7921  ax-rnegex 7922  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-apti 7928  ax-pre-ltadd 7929

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-inn 8922  df-n0 9179  df-z 9256  df-uz 9531  df-fz 10011  df-fzo 10145

This theorem is referenced by:  zsupcl  11950  infssuzex  11952  gcdsupex  11960