Theorem zsupcllemex 10445

Description: Lemma for zsupcl 10446. 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 5626	. . . . . . . 8 ⊢ (𝑤 = 𝑀 → (ℤ≥‘𝑤) = (ℤ≥‘𝑀))
5	4	raleqdv 2734	. . . . . . 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 5626	. . . . . . . 8 ⊢ (𝑤 = 𝑘 → (ℤ≥‘𝑤) = (ℤ≥‘𝑘))
9	8	raleqdv 2734	. . . . . . 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 5626	. . . . . . . 8 ⊢ (𝑤 = (𝑘 + 1) → (ℤ≥‘𝑤) = (ℤ≥‘(𝑘 + 1)))
13	12	raleqdv 2734	. . . . . . 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 5626	. . . . . . . 8 ⊢ (𝑤 = 𝑗 → (ℤ≥‘𝑤) = (ℤ≥‘𝑗))
17	16	raleqdv 2734	. . . . . . 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 9732	. . . . . . . . 9 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀))
24		zsupcllemex.sbm	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑀 → (𝜓 ↔ 𝜒))
25	24	notbid 671	. . . . . . . . . 10 ⊢ (𝑛 = 𝑀 → (¬ 𝜓 ↔ ¬ 𝜒))
26	25	rspcv 2903	. . . . . . . . 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 623	. . . . . 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 10444	. . . . 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 9779	. . . 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 2653	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 839   = wceq 1395   ∈ wcel 2200  ∀wral 2508  ∃wrex 2509  {crab 2512   class class class wbr 4082  ‘cfv 5317  (class class class)co 6000  ℝcr 7994  1c1 7996   + caddc 7998   < clt 8177  ℤcz 9442  ℤ_≥cuz 9718

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-addcom 8095  ax-addass 8097  ax-distr 8099  ax-i2m1 8100  ax-0lt1 8101  ax-0id 8103  ax-rnegex 8104  ax-cnre 8106  ax-pre-ltirr 8107  ax-pre-ltwlin 8108  ax-pre-lttrn 8109  ax-pre-apti 8110  ax-pre-ltadd 8111

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-fv 5325  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-pnf 8179  df-mnf 8180  df-xr 8181  df-ltxr 8182  df-le 8183  df-sub 8315  df-neg 8316  df-inn 9107  df-n0 9366  df-z 9443  df-uz 9719  df-fz 10201  df-fzo 10335

This theorem is referenced by:  zsupcl  10446  infssuzex  10448  gcdsupex  12473