Theorem zsupcllemex 10337

Description: Lemma for zsupcl 10338. 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 5561	. . . . . . . 8 ⊢ (𝑤 = 𝑀 → (ℤ≥‘𝑤) = (ℤ≥‘𝑀))
5	4	raleqdv 2699	. . . . . . 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 5561	. . . . . . . 8 ⊢ (𝑤 = 𝑘 → (ℤ≥‘𝑤) = (ℤ≥‘𝑘))
9	8	raleqdv 2699	. . . . . . 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 5561	. . . . . . . 8 ⊢ (𝑤 = (𝑘 + 1) → (ℤ≥‘𝑤) = (ℤ≥‘(𝑘 + 1)))
13	12	raleqdv 2699	. . . . . . 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 5561	. . . . . . . 8 ⊢ (𝑤 = 𝑗 → (ℤ≥‘𝑤) = (ℤ≥‘𝑗))
17	16	raleqdv 2699	. . . . . . 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 9632	. . . . . . . . 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 2864	. . . . . . . . 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 10336	. . . . 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 9679	. . . 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 2619	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 1364   ∈ wcel 2167  ∀wral 2475  ∃wrex 2476  {crab 2479   class class class wbr 4034  ‘cfv 5259  (class class class)co 5925  ℝcr 7895  1c1 7897   + caddc 7899   < clt 8078  ℤcz 9343  ℤ_≥cuz 9618

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-addcom 7996  ax-addass 7998  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-0id 8004  ax-rnegex 8005  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-apti 8011  ax-pre-ltadd 8012

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-inn 9008  df-n0 9267  df-z 9344  df-uz 9619  df-fz 10101  df-fzo 10235

This theorem is referenced by:  zsupcl  10338  infssuzex  10340  gcdsupex  12149