Step | Hyp | Ref
| Expression |
1 | | zsupcllemex.bnd |
. 2
⊢ (𝜑 → ∃𝑗 ∈ (ℤ≥‘𝑀)∀𝑛 ∈ (ℤ≥‘𝑗) ¬ 𝜓) |
2 | | simpl 108 |
. . 3
⊢ ((𝜑 ∧ (𝑗 ∈ (ℤ≥‘𝑀) ∧ ∀𝑛 ∈
(ℤ≥‘𝑗) ¬ 𝜓)) → 𝜑) |
3 | | simprr 527 |
. . 3
⊢ ((𝜑 ∧ (𝑗 ∈ (ℤ≥‘𝑀) ∧ ∀𝑛 ∈
(ℤ≥‘𝑗) ¬ 𝜓)) → ∀𝑛 ∈ (ℤ≥‘𝑗) ¬ 𝜓) |
4 | | fveq2 5496 |
. . . . . . . 8
⊢ (𝑤 = 𝑀 → (ℤ≥‘𝑤) =
(ℤ≥‘𝑀)) |
5 | 4 | raleqdv 2671 |
. . . . . . 7
⊢ (𝑤 = 𝑀 → (∀𝑛 ∈ (ℤ≥‘𝑤) ¬ 𝜓 ↔ ∀𝑛 ∈ (ℤ≥‘𝑀) ¬ 𝜓)) |
6 | 5 | anbi2d 461 |
. . . . . 6
⊢ (𝑤 = 𝑀 → ((𝜑 ∧ ∀𝑛 ∈ (ℤ≥‘𝑤) ¬ 𝜓) ↔ (𝜑 ∧ ∀𝑛 ∈ (ℤ≥‘𝑀) ¬ 𝜓))) |
7 | 6 | imbi1d 230 |
. . . . 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 5496 |
. . . . . . . 8
⊢ (𝑤 = 𝑘 → (ℤ≥‘𝑤) =
(ℤ≥‘𝑘)) |
9 | 8 | raleqdv 2671 |
. . . . . . 7
⊢ (𝑤 = 𝑘 → (∀𝑛 ∈ (ℤ≥‘𝑤) ¬ 𝜓 ↔ ∀𝑛 ∈ (ℤ≥‘𝑘) ¬ 𝜓)) |
10 | 9 | anbi2d 461 |
. . . . . 6
⊢ (𝑤 = 𝑘 → ((𝜑 ∧ ∀𝑛 ∈ (ℤ≥‘𝑤) ¬ 𝜓) ↔ (𝜑 ∧ ∀𝑛 ∈ (ℤ≥‘𝑘) ¬ 𝜓))) |
11 | 10 | imbi1d 230 |
. . . . 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 5496 |
. . . . . . . 8
⊢ (𝑤 = (𝑘 + 1) →
(ℤ≥‘𝑤) = (ℤ≥‘(𝑘 + 1))) |
13 | 12 | raleqdv 2671 |
. . . . . . 7
⊢ (𝑤 = (𝑘 + 1) → (∀𝑛 ∈ (ℤ≥‘𝑤) ¬ 𝜓 ↔ ∀𝑛 ∈ (ℤ≥‘(𝑘 + 1)) ¬ 𝜓)) |
14 | 13 | anbi2d 461 |
. . . . . 6
⊢ (𝑤 = (𝑘 + 1) → ((𝜑 ∧ ∀𝑛 ∈ (ℤ≥‘𝑤) ¬ 𝜓) ↔ (𝜑 ∧ ∀𝑛 ∈ (ℤ≥‘(𝑘 + 1)) ¬ 𝜓))) |
15 | 14 | imbi1d 230 |
. . . . 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 5496 |
. . . . . . . 8
⊢ (𝑤 = 𝑗 → (ℤ≥‘𝑤) =
(ℤ≥‘𝑗)) |
17 | 16 | raleqdv 2671 |
. . . . . . 7
⊢ (𝑤 = 𝑗 → (∀𝑛 ∈ (ℤ≥‘𝑤) ¬ 𝜓 ↔ ∀𝑛 ∈ (ℤ≥‘𝑗) ¬ 𝜓)) |
18 | 17 | anbi2d 461 |
. . . . . 6
⊢ (𝑤 = 𝑗 → ((𝜑 ∧ ∀𝑛 ∈ (ℤ≥‘𝑤) ¬ 𝜓) ↔ (𝜑 ∧ ∀𝑛 ∈ (ℤ≥‘𝑗) ¬ 𝜓))) |
19 | 18 | imbi1d 230 |
. . . . 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 274 |
. . . . . . 7
⊢ ((𝜑 ∧ ∀𝑛 ∈ (ℤ≥‘𝑀) ¬ 𝜓) → 𝜒) |
22 | | zsupcllemex.m |
. . . . . . . . 9
⊢ (𝜑 → 𝑀 ∈ ℤ) |
23 | | uzid 9501 |
. . . . . . . . 9
⊢ (𝑀 ∈ ℤ → 𝑀 ∈
(ℤ≥‘𝑀)) |
24 | | zsupcllemex.sbm |
. . . . . . . . . . 11
⊢ (𝑛 = 𝑀 → (𝜓 ↔ 𝜒)) |
25 | 24 | notbid 662 |
. . . . . . . . . 10
⊢ (𝑛 = 𝑀 → (¬ 𝜓 ↔ ¬ 𝜒)) |
26 | 25 | rspcv 2830 |
. . . . . . . . 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 123 |
. . . . . . 7
⊢ ((𝜑 ∧ ∀𝑛 ∈ (ℤ≥‘𝑀) ¬ 𝜓) → ¬ 𝜒) |
29 | 21, 28 | pm2.21dd 615 |
. . . . . 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 11900 |
. . . . 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 9547 |
. . . 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 487 |
. . 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 431 |
. 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 2592 |
1
⊢ (𝜑 → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧))) |