Theorem zsupcllemstep 12082

Description: Lemma for zsupcl 12084. Induction step. (Contributed by Jim Kingdon, 7-Dec-2021.)

Hypothesis

Ref	Expression
zsupcllemstep.dc	⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → DECID 𝜓)

Assertion

Ref	Expression
zsupcllemstep	⊢ (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)))))

Distinct variable groups:   𝑛,𝐾,𝑥,𝑦,𝑧   𝑛,𝑀,𝑦   𝜑,𝑛,𝑦   𝜓,𝑥,𝑦,𝑧

Allowed substitution hints:   𝜑(𝑥,𝑧)   𝜓(𝑛)   𝑀(𝑥,𝑧)

Proof of Theorem zsupcllemstep

Step	Hyp	Ref	Expression
1		eluzelz 9601	. . . . 5 ⊢ (𝐾 ∈ (ℤ≥‘𝑀) → 𝐾 ∈ ℤ)
2	1	ad3antrrr 492	. . . 4 ⊢ ((((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)) ∧ [𝐾 / 𝑛]𝜓) → 𝐾 ∈ ℤ)
3		nfv 1539	. . . . . . . 8 ⊢ Ⅎ𝑦 𝐾 ∈ (ℤ≥‘𝑀)
4		nfv 1539	. . . . . . . . 9 ⊢ Ⅎ𝑦(𝜑 ∧ ∀𝑛 ∈ (ℤ≥‘𝐾) ¬ 𝜓)
5		nfcv 2336	. . . . . . . . . 10 ⊢ Ⅎ𝑦ℤ
6		nfra1 2525	. . . . . . . . . . 11 ⊢ Ⅎ𝑦∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦
7		nfra1 2525	. . . . . . . . . . 11 ⊢ Ⅎ𝑦∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)
8	6, 7	nfan 1576	. . . . . . . . . 10 ⊢ Ⅎ𝑦(∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧))
9	5, 8	nfrexya 2535	. . . . . . . . 9 ⊢ Ⅎ𝑦∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧))
10	4, 9	nfim 1583	. . . . . . . 8 ⊢ Ⅎ𝑦((𝜑 ∧ ∀𝑛 ∈ (ℤ≥‘𝐾) ¬ 𝜓) → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)))
11	3, 10	nfan 1576	. . . . . . 7 ⊢ Ⅎ𝑦(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))))
12		nfv 1539	. . . . . . 7 ⊢ Ⅎ𝑦(𝜑 ∧ ∀𝑛 ∈ (ℤ≥‘(𝐾 + 1)) ¬ 𝜓)
13	11, 12	nfan 1576	. . . . . 6 ⊢ Ⅎ𝑦((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))
14		nfv 1539	. . . . . 6 ⊢ Ⅎ𝑦[𝐾 / 𝑛]𝜓
15	13, 14	nfan 1576	. . . . 5 ⊢ Ⅎ𝑦(((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)) ∧ [𝐾 / 𝑛]𝜓)
16		nfcv 2336	. . . . . . . . . . 11 ⊢ Ⅎ𝑛ℤ
17	16	elrabsf 3024	. . . . . . . . . 10 ⊢ (𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ↔ (𝑦 ∈ ℤ ∧ [𝑦 / 𝑛]𝜓))
18	17	simprbi 275	. . . . . . . . 9 ⊢ (𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} → [𝑦 / 𝑛]𝜓)
19		sbsbc 2989	. . . . . . . . 9 ⊢ ([𝑦 / 𝑛]𝜓 ↔ [𝑦 / 𝑛]𝜓)
20	18, 19	sylibr 134	. . . . . . . 8 ⊢ (𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} → [𝑦 / 𝑛]𝜓)
21	20	ad2antlr 489	. . . . . . 7 ⊢ ((((((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)) ∧ [𝐾 / 𝑛]𝜓) ∧ 𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓}) ∧ 𝐾 < 𝑦) → [𝑦 / 𝑛]𝜓)
22		elrabi 2913	. . . . . . . . . . 11 ⊢ (𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} → 𝑦 ∈ ℤ)
23		zltp1le 9371	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (𝐾 < 𝑦 ↔ (𝐾 + 1) ≤ 𝑦))
24	2, 22, 23	syl2an 289	. . . . . . . . . 10 ⊢ (((((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)) ∧ [𝐾 / 𝑛]𝜓) ∧ 𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓}) → (𝐾 < 𝑦 ↔ (𝐾 + 1) ≤ 𝑦))
25	24	biimpa 296	. . . . . . . . 9 ⊢ ((((((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)) ∧ [𝐾 / 𝑛]𝜓) ∧ 𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓}) ∧ 𝐾 < 𝑦) → (𝐾 + 1) ≤ 𝑦)
26	2	peano2zd 9442	. . . . . . . . . . 11 ⊢ ((((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)) ∧ [𝐾 / 𝑛]𝜓) → (𝐾 + 1) ∈ ℤ)
27		eluz 9605	. . . . . . . . . . 11 ⊢ (((𝐾 + 1) ∈ ℤ ∧ 𝑦 ∈ ℤ) → (𝑦 ∈ (ℤ≥‘(𝐾 + 1)) ↔ (𝐾 + 1) ≤ 𝑦))
28	26, 22, 27	syl2an 289	. . . . . . . . . 10 ⊢ (((((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)) ∧ [𝐾 / 𝑛]𝜓) ∧ 𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓}) → (𝑦 ∈ (ℤ≥‘(𝐾 + 1)) ↔ (𝐾 + 1) ≤ 𝑦))
29	28	adantr 276	. . . . . . . . 9 ⊢ ((((((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)) ∧ [𝐾 / 𝑛]𝜓) ∧ 𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓}) ∧ 𝐾 < 𝑦) → (𝑦 ∈ (ℤ≥‘(𝐾 + 1)) ↔ (𝐾 + 1) ≤ 𝑦))
30	25, 29	mpbird 167	. . . . . . . 8 ⊢ ((((((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)) ∧ [𝐾 / 𝑛]𝜓) ∧ 𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓}) ∧ 𝐾 < 𝑦) → 𝑦 ∈ (ℤ≥‘(𝐾 + 1)))
31		simprr 531	. . . . . . . . 9 ⊢ (((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)) → ∀𝑛 ∈ (ℤ≥‘(𝐾 + 1)) ¬ 𝜓)
32	31	ad3antrrr 492	. . . . . . . 8 ⊢ ((((((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)) ∧ [𝐾 / 𝑛]𝜓) ∧ 𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓}) ∧ 𝐾 < 𝑦) → ∀𝑛 ∈ (ℤ≥‘(𝐾 + 1)) ¬ 𝜓)
33		nfs1v 1955	. . . . . . . . . 10 ⊢ Ⅎ𝑛[𝑦 / 𝑛]𝜓
34	33	nfn 1669	. . . . . . . . 9 ⊢ Ⅎ𝑛 ¬ [𝑦 / 𝑛]𝜓
35		sbequ12 1782	. . . . . . . . . 10 ⊢ (𝑛 = 𝑦 → (𝜓 ↔ [𝑦 / 𝑛]𝜓))
36	35	notbid 668	. . . . . . . . 9 ⊢ (𝑛 = 𝑦 → (¬ 𝜓 ↔ ¬ [𝑦 / 𝑛]𝜓))
37	34, 36	rspc 2858	. . . . . . . 8 ⊢ (𝑦 ∈ (ℤ≥‘(𝐾 + 1)) → (∀𝑛 ∈ (ℤ≥‘(𝐾 + 1)) ¬ 𝜓 → ¬ [𝑦 / 𝑛]𝜓))
38	30, 32, 37	sylc 62	. . . . . . 7 ⊢ ((((((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)) ∧ [𝐾 / 𝑛]𝜓) ∧ 𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓}) ∧ 𝐾 < 𝑦) → ¬ [𝑦 / 𝑛]𝜓)
39	21, 38	pm2.65da 662	. . . . . 6 ⊢ (((((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)) ∧ [𝐾 / 𝑛]𝜓) ∧ 𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓}) → ¬ 𝐾 < 𝑦)
40	39	ex 115	. . . . 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)) ∧ [𝐾 / 𝑛]𝜓) → (𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} → ¬ 𝐾 < 𝑦))
41	15, 40	ralrimi 2565	. . . 4 ⊢ ((((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)) ∧ [𝐾 / 𝑛]𝜓) → ∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝐾 < 𝑦)
42	2	ad2antrr 488	. . . . . . . 8 ⊢ ((((((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)) ∧ [𝐾 / 𝑛]𝜓) ∧ 𝑦 ∈ ℝ) ∧ 𝑦 < 𝐾) → 𝐾 ∈ ℤ)
43		simpllr 534	. . . . . . . 8 ⊢ ((((((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)) ∧ [𝐾 / 𝑛]𝜓) ∧ 𝑦 ∈ ℝ) ∧ 𝑦 < 𝐾) → [𝐾 / 𝑛]𝜓)
44	16	elrabsf 3024	. . . . . . . 8 ⊢ (𝐾 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ↔ (𝐾 ∈ ℤ ∧ [𝐾 / 𝑛]𝜓))
45	42, 43, 44	sylanbrc 417	. . . . . . 7 ⊢ ((((((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)) ∧ [𝐾 / 𝑛]𝜓) ∧ 𝑦 ∈ ℝ) ∧ 𝑦 < 𝐾) → 𝐾 ∈ {𝑛 ∈ ℤ ∣ 𝜓})
46		breq2 4033	. . . . . . . 8 ⊢ (𝑧 = 𝐾 → (𝑦 < 𝑧 ↔ 𝑦 < 𝐾))
47	46	rspcev 2864	. . . . . . 7 ⊢ ((𝐾 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ∧ 𝑦 < 𝐾) → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)
48	45, 47	sylancom 420	. . . . . 6 ⊢ ((((((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)) ∧ [𝐾 / 𝑛]𝜓) ∧ 𝑦 ∈ ℝ) ∧ 𝑦 < 𝐾) → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)
49	48	exp31 364	. . . . 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)) ∧ [𝐾 / 𝑛]𝜓) → (𝑦 ∈ ℝ → (𝑦 < 𝐾 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)))
50	15, 49	ralrimi 2565	. . . 4 ⊢ ((((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)) ∧ [𝐾 / 𝑛]𝜓) → ∀𝑦 ∈ ℝ (𝑦 < 𝐾 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧))
51		breq1 4032	. . . . . . . 8 ⊢ (𝑥 = 𝐾 → (𝑥 < 𝑦 ↔ 𝐾 < 𝑦))
52	51	notbid 668	. . . . . . 7 ⊢ (𝑥 = 𝐾 → (¬ 𝑥 < 𝑦 ↔ ¬ 𝐾 < 𝑦))
53	52	ralbidv 2494	. . . . . 6 ⊢ (𝑥 = 𝐾 → (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ↔ ∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝐾 < 𝑦))
54		breq2 4033	. . . . . . . 8 ⊢ (𝑥 = 𝐾 → (𝑦 < 𝑥 ↔ 𝑦 < 𝐾))
55	54	imbi1d 231	. . . . . . 7 ⊢ (𝑥 = 𝐾 → ((𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧) ↔ (𝑦 < 𝐾 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)))
56	55	ralbidv 2494	. . . . . 6 ⊢ (𝑥 = 𝐾 → (∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧) ↔ ∀𝑦 ∈ ℝ (𝑦 < 𝐾 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)))
57	53, 56	anbi12d 473	. . . . 5 ⊢ (𝑥 = 𝐾 → ((∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)) ↔ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝐾 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝐾 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧))))
58	57	rspcev 2864	. . . 4 ⊢ ((𝐾 ∈ ℤ ∧ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝐾 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝐾 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧))) → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)))
59	2, 41, 50, 58	syl12anc 1247	. . 3 ⊢ ((((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)) ∧ [𝐾 / 𝑛]𝜓) → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)))
60		sbcng 3026	. . . . . . . 8 ⊢ (𝐾 ∈ (ℤ≥‘𝑀) → ([𝐾 / 𝑛] ¬ 𝜓 ↔ ¬ [𝐾 / 𝑛]𝜓))
61	60	ad2antrr 488	. . . . . . 7 ⊢ (((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)) → ([𝐾 / 𝑛] ¬ 𝜓 ↔ ¬ [𝐾 / 𝑛]𝜓))
62	61	biimpar 297	. . . . . 6 ⊢ ((((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)) ∧ ¬ [𝐾 / 𝑛]𝜓) → [𝐾 / 𝑛] ¬ 𝜓)
63		sbcsng 3677	. . . . . . 7 ⊢ (𝐾 ∈ (ℤ≥‘𝑀) → ([𝐾 / 𝑛] ¬ 𝜓 ↔ ∀𝑛 ∈ {𝐾} ¬ 𝜓))
64	63	ad3antrrr 492	. . . . . 6 ⊢ ((((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)) ∧ ¬ [𝐾 / 𝑛]𝜓) → ([𝐾 / 𝑛] ¬ 𝜓 ↔ ∀𝑛 ∈ {𝐾} ¬ 𝜓))
65	62, 64	mpbid 147	. . . . 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)) ∧ ¬ [𝐾 / 𝑛]𝜓) → ∀𝑛 ∈ {𝐾} ¬ 𝜓)
66		simplrr 536	. . . . 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)) ∧ ¬ [𝐾 / 𝑛]𝜓) → ∀𝑛 ∈ (ℤ≥‘(𝐾 + 1)) ¬ 𝜓)
67		uzid 9606	. . . . . . . . . . 11 ⊢ (𝐾 ∈ ℤ → 𝐾 ∈ (ℤ≥‘𝐾))
68		peano2uz 9648	. . . . . . . . . . 11 ⊢ (𝐾 ∈ (ℤ≥‘𝐾) → (𝐾 + 1) ∈ (ℤ≥‘𝐾))
69	67, 68	syl 14	. . . . . . . . . 10 ⊢ (𝐾 ∈ ℤ → (𝐾 + 1) ∈ (ℤ≥‘𝐾))
70		fzouzsplit 10246	. . . . . . . . . 10 ⊢ ((𝐾 + 1) ∈ (ℤ≥‘𝐾) → (ℤ≥‘𝐾) = ((𝐾..^(𝐾 + 1)) ∪ (ℤ≥‘(𝐾 + 1))))
71	1, 69, 70	3syl 17	. . . . . . . . 9 ⊢ (𝐾 ∈ (ℤ≥‘𝑀) → (ℤ≥‘𝐾) = ((𝐾..^(𝐾 + 1)) ∪ (ℤ≥‘(𝐾 + 1))))
72		fzosn 10272	. . . . . . . . . . 11 ⊢ (𝐾 ∈ ℤ → (𝐾..^(𝐾 + 1)) = {𝐾})
73	1, 72	syl 14	. . . . . . . . . 10 ⊢ (𝐾 ∈ (ℤ≥‘𝑀) → (𝐾..^(𝐾 + 1)) = {𝐾})
74	73	uneq1d 3312	. . . . . . . . 9 ⊢ (𝐾 ∈ (ℤ≥‘𝑀) → ((𝐾..^(𝐾 + 1)) ∪ (ℤ≥‘(𝐾 + 1))) = ({𝐾} ∪ (ℤ≥‘(𝐾 + 1))))
75	71, 74	eqtrd 2226	. . . . . . . 8 ⊢ (𝐾 ∈ (ℤ≥‘𝑀) → (ℤ≥‘𝐾) = ({𝐾} ∪ (ℤ≥‘(𝐾 + 1))))
76	75	raleqdv 2696	. . . . . . 7 ⊢ (𝐾 ∈ (ℤ≥‘𝑀) → (∀𝑛 ∈ (ℤ≥‘𝐾) ¬ 𝜓 ↔ ∀𝑛 ∈ ({𝐾} ∪ (ℤ≥‘(𝐾 + 1))) ¬ 𝜓))
77		ralunb 3340	. . . . . . 7 ⊢ (∀𝑛 ∈ ({𝐾} ∪ (ℤ≥‘(𝐾 + 1))) ¬ 𝜓 ↔ (∀𝑛 ∈ {𝐾} ¬ 𝜓 ∧ ∀𝑛 ∈ (ℤ≥‘(𝐾 + 1)) ¬ 𝜓))
78	76, 77	bitrdi 196	. . . . . 6 ⊢ (𝐾 ∈ (ℤ≥‘𝑀) → (∀𝑛 ∈ (ℤ≥‘𝐾) ¬ 𝜓 ↔ (∀𝑛 ∈ {𝐾} ¬ 𝜓 ∧ ∀𝑛 ∈ (ℤ≥‘(𝐾 + 1)) ¬ 𝜓)))
79	78	ad3antrrr 492	. . . . 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)) ∧ ¬ [𝐾 / 𝑛]𝜓) → (∀𝑛 ∈ (ℤ≥‘𝐾) ¬ 𝜓 ↔ (∀𝑛 ∈ {𝐾} ¬ 𝜓 ∧ ∀𝑛 ∈ (ℤ≥‘(𝐾 + 1)) ¬ 𝜓)))
80	65, 66, 79	mpbir2and 946	. . . 4 ⊢ ((((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)) ∧ ¬ [𝐾 / 𝑛]𝜓) → ∀𝑛 ∈ (ℤ≥‘𝐾) ¬ 𝜓)
81		simprl 529	. . . . . 6 ⊢ (((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)) → 𝜑)
82		simplr 528	. . . . . 6 ⊢ (((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)) → ((𝜑 ∧ ∀𝑛 ∈ (ℤ≥‘𝐾) ¬ 𝜓) → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧))))
83	81, 82	mpand 429	. . . . 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)) → (∀𝑛 ∈ (ℤ≥‘𝐾) ¬ 𝜓 → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧))))
84	83	adantr 276	. . . 4 ⊢ ((((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)) ∧ ¬ [𝐾 / 𝑛]𝜓) → (∀𝑛 ∈ (ℤ≥‘𝐾) ¬ 𝜓 → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧))))
85	80, 84	mpd 13	. . 3 ⊢ ((((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)) ∧ ¬ [𝐾 / 𝑛]𝜓) → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)))
86		zsupcllemstep.dc	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝑀)) → DECID 𝜓)
87	86	ralrimiva 2567	. . . . . 6 ⊢ (𝜑 → ∀𝑛 ∈ (ℤ≥‘𝑀)DECID 𝜓)
88	81, 87	syl 14	. . . . 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)) → ∀𝑛 ∈ (ℤ≥‘𝑀)DECID 𝜓)
89		nfsbc1v 3004	. . . . . . . 8 ⊢ Ⅎ𝑛[𝐾 / 𝑛]𝜓
90	89	nfdc 1670	. . . . . . 7 ⊢ Ⅎ𝑛DECID [𝐾 / 𝑛]𝜓
91		sbceq1a 2995	. . . . . . . 8 ⊢ (𝑛 = 𝐾 → (𝜓 ↔ [𝐾 / 𝑛]𝜓))
92	91	dcbid 839	. . . . . . 7 ⊢ (𝑛 = 𝐾 → (DECID 𝜓 ↔ DECID [𝐾 / 𝑛]𝜓))
93	90, 92	rspc 2858	. . . . . 6 ⊢ (𝐾 ∈ (ℤ≥‘𝑀) → (∀𝑛 ∈ (ℤ≥‘𝑀)DECID 𝜓 → DECID [𝐾 / 𝑛]𝜓))
94	93	ad2antrr 488	. . . . 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)) → (∀𝑛 ∈ (ℤ≥‘𝑀)DECID 𝜓 → DECID [𝐾 / 𝑛]𝜓))
95	88, 94	mpd 13	. . . 4 ⊢ (((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)) → DECID [𝐾 / 𝑛]𝜓)
96		exmiddc 837	. . . 4 ⊢ (DECID [𝐾 / 𝑛]𝜓 → ([𝐾 / 𝑛]𝜓 ∨ ¬ [𝐾 / 𝑛]𝜓))
97	95, 96	syl 14	. . 3 ⊢ (((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)) → ([𝐾 / 𝑛]𝜓 ∨ ¬ [𝐾 / 𝑛]𝜓))
98	59, 85, 97	mpjaodan 799	. 2 ⊢ (((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)))
99	98	exp31 364	1 ⊢ (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)))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 709  DECID wdc 835   = wceq 1364  [wsb 1773   ∈ wcel 2164  ∀wral 2472  ∃wrex 2473  {crab 2476  [wsbc 2985   ∪ cun 3151  {csn 3618   class class class wbr 4029  ‘cfv 5254  (class class class)co 5918  ℝcr 7871  1c1 7873   + caddc 7875   < clt 8054   ≤ cle 8055  ℤcz 9317  ℤ_≥cuz 9592  ..^cfzo 10208

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-addass 7974  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-0id 7980  ax-rnegex 7981  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988

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 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-inn 8983  df-n0 9241  df-z 9318  df-uz 9593  df-fz 10075  df-fzo 10209

This theorem is referenced by:  zsupcllemex  12083