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Theorem zsupcllemstep 11878

Description: Lemma for zsupcl 11880. 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 9475	. . . . 5 ⊢ (𝐾 ∈ (ℤ_≥‘𝑀) → 𝐾 ∈ ℤ)
2	1	ad3antrrr 484	. . . 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 1516	. . . . . . . 8 ⊢ Ⅎ𝑦 𝐾 ∈ (ℤ_≥‘𝑀)
4		nfv 1516	. . . . . . . . 9 ⊢ Ⅎ𝑦(𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘𝐾) ¬ 𝜓)
5		nfcv 2308	. . . . . . . . . 10 ⊢ Ⅎ𝑦ℤ
6		nfra1 2497	. . . . . . . . . . 11 ⊢ Ⅎ𝑦∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦
7		nfra1 2497	. . . . . . . . . . 11 ⊢ Ⅎ𝑦∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)
8	6, 7	nfan 1553	. . . . . . . . . 10 ⊢ Ⅎ𝑦(∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧))
9	5, 8	nfrexya 2507	. . . . . . . . 9 ⊢ Ⅎ𝑦∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧))
10	4, 9	nfim 1560	. . . . . . . 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 1553	. . . . . . 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 1516	. . . . . . 7 ⊢ Ⅎ𝑦(𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘(𝐾 + 1)) ¬ 𝜓)
13	11, 12	nfan 1553	. . . . . 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 1516	. . . . . 6 ⊢ Ⅎ𝑦[𝐾 / 𝑛]𝜓
15	13, 14	nfan 1553	. . . . 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 2308	. . . . . . . . . . 11 ⊢ Ⅎ𝑛ℤ
17	16	elrabsf 2989	. . . . . . . . . 10 ⊢ (𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ↔ (𝑦 ∈ ℤ ∧ [𝑦 / 𝑛]𝜓))
18	17	simprbi 273	. . . . . . . . 9 ⊢ (𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} → [𝑦 / 𝑛]𝜓)
19		sbsbc 2955	. . . . . . . . 9 ⊢ ([𝑦 / 𝑛]𝜓 ↔ [𝑦 / 𝑛]𝜓)
20	18, 19	sylibr 133	. . . . . . . 8 ⊢ (𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} → [𝑦 / 𝑛]𝜓)
21	20	ad2antlr 481	. . . . . . 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 2879	. . . . . . . . . . 11 ⊢ (𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} → 𝑦 ∈ ℤ)
23		zltp1le 9245	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (𝐾 < 𝑦 ↔ (𝐾 + 1) ≤ 𝑦))
24	2, 22, 23	syl2an 287	. . . . . . . . . 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 294	. . . . . . . . 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 9316	. . . . . . . . . . 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 9479	. . . . . . . . . . 11 ⊢ (((𝐾 + 1) ∈ ℤ ∧ 𝑦 ∈ ℤ) → (𝑦 ∈ (ℤ_≥‘(𝐾 + 1)) ↔ (𝐾 + 1) ≤ 𝑦))
28	26, 22, 27	syl2an 287	. . . . . . . . . 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 274	. . . . . . . . 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 166	. . . . . . . 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 522	. . . . . . . . 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 484	. . . . . . . 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 1927	. . . . . . . . . 10 ⊢ Ⅎ𝑛[𝑦 / 𝑛]𝜓
34	33	nfn 1646	. . . . . . . . 9 ⊢ Ⅎ𝑛 ¬ [𝑦 / 𝑛]𝜓
35		sbequ12 1759	. . . . . . . . . 10 ⊢ (𝑛 = 𝑦 → (𝜓 ↔ [𝑦 / 𝑛]𝜓))
36	35	notbid 657	. . . . . . . . 9 ⊢ (𝑛 = 𝑦 → (¬ 𝜓 ↔ ¬ [𝑦 / 𝑛]𝜓))
37	34, 36	rspc 2824	. . . . . . . 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 651	. . . . . 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 114	. . . . 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 2537	. . . 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 480	. . . . . . . 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 524	. . . . . . . 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 2989	. . . . . . . 8 ⊢ (𝐾 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ↔ (𝐾 ∈ ℤ ∧ [𝐾 / 𝑛]𝜓))
45	42, 43, 44	sylanbrc 414	. . . . . . 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 3986	. . . . . . . 8 ⊢ (𝑧 = 𝐾 → (𝑦 < 𝑧 ↔ 𝑦 < 𝐾))
47	46	rspcev 2830	. . . . . . 7 ⊢ ((𝐾 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ∧ 𝑦 < 𝐾) → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)
48	45, 47	sylancom 417	. . . . . 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 362	. . . . 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 2537	. . . 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 3985	. . . . . . . 8 ⊢ (𝑥 = 𝐾 → (𝑥 < 𝑦 ↔ 𝐾 < 𝑦))
52	51	notbid 657	. . . . . . 7 ⊢ (𝑥 = 𝐾 → (¬ 𝑥 < 𝑦 ↔ ¬ 𝐾 < 𝑦))
53	52	ralbidv 2466	. . . . . 6 ⊢ (𝑥 = 𝐾 → (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ↔ ∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝐾 < 𝑦))
54		breq2 3986	. . . . . . . 8 ⊢ (𝑥 = 𝐾 → (𝑦 < 𝑥 ↔ 𝑦 < 𝐾))
55	54	imbi1d 230	. . . . . . 7 ⊢ (𝑥 = 𝐾 → ((𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧) ↔ (𝑦 < 𝐾 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)))
56	55	ralbidv 2466	. . . . . 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 465	. . . . 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 2830	. . . 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 1226	. . 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 2991	. . . . . . . 8 ⊢ (𝐾 ∈ (ℤ_≥‘𝑀) → ([𝐾 / 𝑛] ¬ 𝜓 ↔ ¬ [𝐾 / 𝑛]𝜓))
61	60	ad2antrr 480	. . . . . . 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 295	. . . . . 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 3635	. . . . . . 7 ⊢ (𝐾 ∈ (ℤ_≥‘𝑀) → ([𝐾 / 𝑛] ¬ 𝜓 ↔ ∀𝑛 ∈ {𝐾} ¬ 𝜓))
64	63	ad3antrrr 484	. . . . . 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 146	. . . . 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 526	. . . . 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 9480	. . . . . . . . . . 11 ⊢ (𝐾 ∈ ℤ → 𝐾 ∈ (ℤ_≥‘𝐾))
68		peano2uz 9521	. . . . . . . . . . 11 ⊢ (𝐾 ∈ (ℤ_≥‘𝐾) → (𝐾 + 1) ∈ (ℤ_≥‘𝐾))
69	67, 68	syl 14	. . . . . . . . . 10 ⊢ (𝐾 ∈ ℤ → (𝐾 + 1) ∈ (ℤ_≥‘𝐾))
70		fzouzsplit 10114	. . . . . . . . . 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 10140	. . . . . . . . . . 11 ⊢ (𝐾 ∈ ℤ → (𝐾..^(𝐾 + 1)) = {𝐾})
73	1, 72	syl 14	. . . . . . . . . 10 ⊢ (𝐾 ∈ (ℤ_≥‘𝑀) → (𝐾..^(𝐾 + 1)) = {𝐾})
74	73	uneq1d 3275	. . . . . . . . 9 ⊢ (𝐾 ∈ (ℤ_≥‘𝑀) → ((𝐾..^(𝐾 + 1)) ∪ (ℤ_≥‘(𝐾 + 1))) = ({𝐾} ∪ (ℤ_≥‘(𝐾 + 1))))
75	71, 74	eqtrd 2198	. . . . . . . 8 ⊢ (𝐾 ∈ (ℤ_≥‘𝑀) → (ℤ_≥‘𝐾) = ({𝐾} ∪ (ℤ_≥‘(𝐾 + 1))))
76	75	raleqdv 2667	. . . . . . 7 ⊢ (𝐾 ∈ (ℤ_≥‘𝑀) → (∀𝑛 ∈ (ℤ_≥‘𝐾) ¬ 𝜓 ↔ ∀𝑛 ∈ ({𝐾} ∪ (ℤ_≥‘(𝐾 + 1))) ¬ 𝜓))
77		ralunb 3303	. . . . . . 7 ⊢ (∀𝑛 ∈ ({𝐾} ∪ (ℤ_≥‘(𝐾 + 1))) ¬ 𝜓 ↔ (∀𝑛 ∈ {𝐾} ¬ 𝜓 ∧ ∀𝑛 ∈ (ℤ_≥‘(𝐾 + 1)) ¬ 𝜓))
78	76, 77	bitrdi 195	. . . . . 6 ⊢ (𝐾 ∈ (ℤ_≥‘𝑀) → (∀𝑛 ∈ (ℤ_≥‘𝐾) ¬ 𝜓 ↔ (∀𝑛 ∈ {𝐾} ¬ 𝜓 ∧ ∀𝑛 ∈ (ℤ_≥‘(𝐾 + 1)) ¬ 𝜓)))
79	78	ad3antrrr 484	. . . . 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 934	. . . 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 521	. . . . . 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 520	. . . . . 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 426	. . . . 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 274	. . . 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 2539	. . . . . 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 2969	. . . . . . . 8 ⊢ Ⅎ𝑛[𝐾 / 𝑛]𝜓
90	89	nfdc 1647	. . . . . . 7 ⊢ Ⅎ𝑛DECID [𝐾 / 𝑛]𝜓
91		sbceq1a 2960	. . . . . . . 8 ⊢ (𝑛 = 𝐾 → (𝜓 ↔ [𝐾 / 𝑛]𝜓))
92	91	dcbid 828	. . . . . . 7 ⊢ (𝑛 = 𝐾 → (DECID 𝜓 ↔ DECID [𝐾 / 𝑛]𝜓))
93	90, 92	rspc 2824	. . . . . 6 ⊢ (𝐾 ∈ (ℤ_≥‘𝑀) → (∀𝑛 ∈ (ℤ_≥‘𝑀)DECID 𝜓 → DECID [𝐾 / 𝑛]𝜓))
94	93	ad2antrr 480	. . . . 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 826	. . . 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 788	. 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 362	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)))))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 ∨ wo 698 DECID wdc 824 = wceq 1343 [wsb 1750 ∈ wcel 2136 ∀wral 2444 ∃wrex 2445 {crab 2448 [wsbc 2951 ∪ cun 3114 {csn 3576 class class class wbr 3982 ‘cfv 5188 (class class class)co 5842 ℝcr 7752 1c1 7754 + caddc 7756 < clt 7933 ≤ cle 7934 ℤcz 9191 ℤ_≥cuz 9466 ..^cfzo 10077

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-sep 4100 ax-pow 4153 ax-pr 4187 ax-un 4411 ax-setind 4514 ax-cnex 7844 ax-resscn 7845 ax-1cn 7846 ax-1re 7847 ax-icn 7848 ax-addcl 7849 ax-addrcl 7850 ax-mulcl 7851 ax-addcom 7853 ax-addass 7855 ax-distr 7857 ax-i2m1 7858 ax-0lt1 7859 ax-0id 7861 ax-rnegex 7862 ax-cnre 7864 ax-pre-ltirr 7865 ax-pre-ltwlin 7866 ax-pre-lttrn 7867 ax-pre-apti 7868 ax-pre-ltadd 7869

This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 969 df-3an 970 df-tru 1346 df-fal 1349 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ne 2337 df-nel 2432 df-ral 2449 df-rex 2450 df-reu 2451 df-rab 2453 df-v 2728 df-sbc 2952 df-csb 3046 df-dif 3118 df-un 3120 df-in 3122 df-ss 3129 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-int 3825 df-iun 3868 df-br 3983 df-opab 4044 df-mpt 4045 df-id 4271 df-xp 4610 df-rel 4611 df-cnv 4612 df-co 4613 df-dm 4614 df-rn 4615 df-res 4616 df-ima 4617 df-iota 5153 df-fun 5190 df-fn 5191 df-f 5192 df-fv 5196 df-riota 5798 df-ov 5845 df-oprab 5846 df-mpo 5847 df-1st 6108 df-2nd 6109 df-pnf 7935 df-mnf 7936 df-xr 7937 df-ltxr 7938 df-le 7939 df-sub 8071 df-neg 8072 df-inn 8858 df-n0 9115 df-z 9192 df-uz 9467 df-fz 9945 df-fzo 10078

This theorem is referenced by: zsupcllemex 11879

W3C validator