Theorem List for Metamath Proof Explorer - 42501-42600 *Has distinct variable
group(s)
| Type | Label | Description |
|---|
| Statement |
|---|
| |
| Theorem | 2uasbanhVD 42501* |
The following User's Proof is a Virtual Deduction proof (see wvd1 42159)
completed automatically by a Metamath tools program invoking mmj2 and
the Metamath Proof Assistant. 2uasbanh 42151 is 2uasbanhVD 42501 without
virtual deductions and was automatically derived from 2uasbanhVD 42501.
(Contributed by Alan Sare, 31-May-2014.)
(Proof modification is discouraged.) (New usage is discouraged.)
| h1:: | ⊢ (𝜒 ↔ (∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 =
𝑣) ∧ 𝜑) ∧ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜓)))
| | 100:1: | ⊢ (𝜒 → (∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 =
𝑣) ∧ 𝜑) ∧ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜓)))
| | 2:100: | ⊢ ( 𝜒 ▶ (∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦
= 𝑣) ∧ 𝜑) ∧ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜓)) )
| | 3:2: | ⊢ ( 𝜒 ▶ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 =
𝑣) ∧ 𝜑) )
| | 4:3: | ⊢ ( 𝜒 ▶ ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣
) )
| | 5:4: | ⊢ ( 𝜒 ▶ (¬ ∀𝑥𝑥 = 𝑦 ∨ 𝑢 = 𝑣)
)
| | 6:5: | ⊢ ( 𝜒 ▶ ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑
↔ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑)) )
| | 7:3,6: | ⊢ ( 𝜒 ▶ [𝑢 / 𝑥][𝑣 / 𝑦]𝜑 )
| | 8:2: | ⊢ ( 𝜒 ▶ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 =
𝑣) ∧ 𝜓) )
| | 9:5: | ⊢ ( 𝜒 ▶ ([𝑢 / 𝑥][𝑣 / 𝑦]𝜓
↔ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜓)) )
| | 10:8,9: | ⊢ ( 𝜒 ▶ [𝑢 / 𝑥][𝑣 / 𝑦]𝜓 )
| | 101:: | ⊢ ([𝑣 / 𝑦](𝜑 ∧ 𝜓) ↔ ([𝑣 /
𝑦]𝜑 ∧ [𝑣 / 𝑦]𝜓))
| | 102:101: | ⊢ ([𝑢 / 𝑥][𝑣 / 𝑦](𝜑 ∧ 𝜓)
↔ [𝑢 / 𝑥]([𝑣 / 𝑦]𝜑 ∧ [𝑣 / 𝑦]𝜓))
| | 103:: | ⊢ ([𝑢 / 𝑥]([𝑣 / 𝑦]𝜑 ∧ [𝑣 / 𝑦
]𝜓) ↔ ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜓))
| | 104:102,103: | ⊢ ([𝑢 / 𝑥][𝑣 / 𝑦](𝜑 ∧ 𝜓)
↔ ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ∧ [𝑢 / 𝑥][𝑣 / 𝑦]𝜓))
| | 11:7,10,104: | ⊢ ( 𝜒 ▶ [𝑢 / 𝑥][𝑣 / 𝑦](𝜑 ∧
𝜓) )
| | 110:5: | ⊢ ( 𝜒 ▶ ([𝑢 / 𝑥][𝑣 / 𝑦](𝜑
∧ 𝜓) ↔ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (𝜑 ∧ 𝜓))) )
| | 12:11,110: | ⊢ ( 𝜒 ▶ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 =
𝑣) ∧ (𝜑 ∧ 𝜓)) )
| | 120:12: | ⊢ (𝜒 → ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣
) ∧ (𝜑 ∧ 𝜓)))
| | 13:1,120: | ⊢ ((∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧
𝜑) ∧ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜓)) →
∃𝑥∃𝑦((𝑥 = 𝑢
∧ 𝑦 = 𝑣) ∧ (𝜑 ∧ 𝜓)))
| | 14:: | ⊢ ( ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (𝜑 ∧ 𝜓
)) ▶ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (𝜑 ∧ 𝜓)) )
| | 15:14: | ⊢ ( ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (𝜑 ∧ 𝜓
)) ▶ (𝑥 = 𝑢 ∧ 𝑦 = 𝑣) )
| | 16:14: | ⊢ ( ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (𝜑 ∧ 𝜓
)) ▶ (𝜑 ∧ 𝜓) )
| | 17:16: | ⊢ ( ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (𝜑 ∧ 𝜓
)) ▶ 𝜑 )
| | 18:15,17: | ⊢ ( ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (𝜑 ∧ 𝜓
)) ▶ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) )
| | 19:18: | ⊢ (((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (𝜑 ∧ 𝜓
)) → ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑))
| | 20:19: | ⊢ (∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (𝜑
∧ 𝜓)) → ∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑))
| | 21:20: | ⊢ (∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (
𝜑 ∧ 𝜓)) → ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑))
| | 22:16: | ⊢ ( ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (𝜑 ∧ 𝜓
)) ▶ 𝜓 )
| | 23:15,22: | ⊢ ( ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (𝜑 ∧ 𝜓
)) ▶ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜓) )
| | 24:23: | ⊢ (((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (𝜑 ∧ 𝜓
)) → ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜓))
| | 25:24: | ⊢ (∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (𝜑
∧ 𝜓)) → ∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜓))
| | 26:25: | ⊢ (∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (
𝜑 ∧ 𝜓)) → ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜓))
| | 27:21,26: | ⊢ (∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (
𝜑 ∧ 𝜓)) → (∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) ∧
∃𝑥∃𝑦(
(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜓)))
| | qed:13,27: | ⊢ (∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (
𝜑 ∧ 𝜓)) ↔ (∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) ∧
∃𝑥∃𝑦(
(𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜓)))
|
|
| ⊢ (𝜒 ↔ (∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) ∧ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜓))) ⇒ ⊢ (∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ (𝜑 ∧ 𝜓)) ↔ (∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑) ∧ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜓))) |
| |
| Theorem | e2ebindVD 42502 |
The following User's Proof is a Virtual Deduction proof (see wvd1 42159)
completed automatically by a Metamath tools program invoking mmj2 and the
Metamath Proof Assistant. e2ebind 42153 is e2ebindVD 42502 without virtual
deductions and was automatically derived from e2ebindVD 42502.
| 1:: | ⊢ (𝜑 ↔ 𝜑)
| | 2:1: | ⊢ (∀𝑦𝑦 = 𝑥 → (𝜑 ↔ 𝜑))
| | 3:2: | ⊢ (∀𝑦𝑦 = 𝑥 → (∃𝑦𝜑 ↔ ∃𝑥𝜑
))
| | 4:: | ⊢ ( ∀𝑦𝑦 = 𝑥 ▶ ∀𝑦𝑦 = 𝑥 )
| | 5:3,4: | ⊢ ( ∀𝑦𝑦 = 𝑥 ▶ (∃𝑦𝜑 ↔ ∃𝑥
𝜑) )
| | 6:: | ⊢ (∀𝑦𝑦 = 𝑥 → ∀𝑦∀𝑦𝑦 = 𝑥)
| | 7:5,6: | ⊢ ( ∀𝑦𝑦 = 𝑥 ▶ ∀𝑦(∃𝑦𝜑 ↔
∃𝑥𝜑) )
| | 8:7: | ⊢ ( ∀𝑦𝑦 = 𝑥 ▶ (∃𝑦∃𝑦𝜑 ↔
∃𝑦∃𝑥𝜑) )
| | 9:: | ⊢ (∃𝑦∃𝑥𝜑 ↔ ∃𝑥∃𝑦𝜑)
| | 10:8,9: | ⊢ ( ∀𝑦𝑦 = 𝑥 ▶ (∃𝑦∃𝑦𝜑 ↔
∃𝑥∃𝑦𝜑) )
| | 11:: | ⊢ (∃𝑦𝜑 → ∀𝑦∃𝑦𝜑)
| | 12:11: | ⊢ (∃𝑦∃𝑦𝜑 ↔ ∃𝑦𝜑)
| | 13:10,12: | ⊢ ( ∀𝑦𝑦 = 𝑥 ▶ (∃𝑥∃𝑦𝜑 ↔
∃𝑦𝜑) )
| | 14:13: | ⊢ (∀𝑦𝑦 = 𝑥 → (∃𝑥∃𝑦𝜑 ↔ ∃
𝑦𝜑))
| | 15:: | ⊢ (∀𝑦𝑦 = 𝑥 ↔ ∀𝑥𝑥 = 𝑦)
| | qed:14,15: | ⊢ (∀𝑥𝑥 = 𝑦 → (∃𝑥∃𝑦𝜑 ↔ ∃
𝑦𝜑))
|
(Contributed by Alan Sare, 27-Nov-2014.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
| ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥∃𝑦𝜑 ↔ ∃𝑦𝜑)) |
| |
| 20.36.8 Virtual Deduction transcriptions of
textbook proofs
|
| |
| Theorem | sb5ALTVD 42503* |
The following User's Proof is a Natural Deduction Sequent Calculus
transcription of the Fitch-style Natural Deduction proof of Unit 20
Excercise 3.a., which is sb5 2272, found in the "Answers to Starred
Exercises" on page 457 of "Understanding Symbolic Logic", Fifth
Edition (2008), by Virginia Klenk. The same proof may also be
interpreted as a Virtual Deduction Hilbert-style axiomatic proof. It
was completed automatically by the tools program completeusersproof.cmd,
which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof
Assistant. sb5ALT 42115 is sb5ALTVD 42503 without virtual deductions and
was automatically derived from sb5ALTVD 42503.
| 1:: | ⊢ ( [𝑦 / 𝑥]𝜑 ▶ [𝑦 / 𝑥]𝜑 )
| | 2:: | ⊢ [𝑦 / 𝑥]𝑥 = 𝑦
| | 3:1,2: | ⊢ ( [𝑦 / 𝑥]𝜑 ▶ [𝑦 / 𝑥](𝑥 = 𝑦
∧ 𝜑) )
| | 4:3: | ⊢ ( [𝑦 / 𝑥]𝜑 ▶ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑
) )
| | 5:4: | ⊢ ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)
)
| | 6:: | ⊢ ( ∃𝑥(𝑥 = 𝑦 ∧ 𝜑) ▶ ∃𝑥(𝑥 =
𝑦 ∧ 𝜑) )
| | 7:: | ⊢ ( ∃𝑥(𝑥 = 𝑦 ∧ 𝜑) , (𝑥 = 𝑦 ∧ 𝜑
) ▶ (𝑥 = 𝑦 ∧ 𝜑) )
| | 8:7: | ⊢ ( ∃𝑥(𝑥 = 𝑦 ∧ 𝜑) , (𝑥 = 𝑦 ∧ 𝜑
) ▶ 𝜑 )
| | 9:7: | ⊢ ( ∃𝑥(𝑥 = 𝑦 ∧ 𝜑) , (𝑥 = 𝑦 ∧ 𝜑
) ▶ 𝑥 = 𝑦 )
| | 10:8,9: | ⊢ ( ∃𝑥(𝑥 = 𝑦 ∧ 𝜑) , (𝑥 = 𝑦 ∧ 𝜑
) ▶ [𝑦 / 𝑥]𝜑 )
| | 101:: | ⊢ ([𝑦 / 𝑥]𝜑 → ∀𝑥[𝑦 / 𝑥]𝜑)
| | 11:101,10: | ⊢ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → [𝑦 / 𝑥]𝜑
)
| | 12:5,11: | ⊢ (([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑
)) ∧ (∃𝑥(𝑥 = 𝑦 ∧ 𝜑) → [𝑦 / 𝑥]𝜑))
| | qed:12: | ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)
)
|
(Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
| ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) |
| |
| Theorem | vk15.4jVD 42504 |
The following User's Proof is a Natural Deduction Sequent Calculus
transcription of the Fitch-style Natural Deduction proof of Unit 15
Excercise 4.f. found in the "Answers to Starred Exercises" on page 442
of "Understanding Symbolic Logic", Fifth Edition (2008), by Virginia
Klenk. The same proof may also be interpreted to be a Virtual Deduction
Hilbert-style axiomatic proof. It was completed automatically by the
tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2
and Norm Megill's Metamath Proof Assistant. vk15.4j 42118 is vk15.4jVD 42504
without virtual deductions and was automatically derived
from vk15.4jVD 42504. Step numbers greater than 25 are additional steps
necessary for the sequent calculus proof not contained in the
Fitch-style proof. Otherwise, step i of the User's Proof corresponds to
step i of the Fitch-style proof.
| h1:: | ⊢ ¬ (∃𝑥¬ 𝜑 ∧ ∃𝑥(𝜓 ∧
¬ 𝜒))
| | h2:: | ⊢ (∀𝑥𝜒 → ¬ ∃𝑥(𝜃 ∧ 𝜏
))
| | h3:: | ⊢ ¬ ∀𝑥(𝜏 → 𝜑)
| | 4:: | ⊢ ( ¬ ∃𝑥¬ 𝜃 ▶ ¬ ∃𝑥¬
𝜃 )
| | 5:4: | ⊢ ( ¬ ∃𝑥¬ 𝜃 ▶ ∀𝑥𝜃 )
| | 6:3: | ⊢ ∃𝑥(𝜏 ∧ ¬ 𝜑)
| | 7:: | ⊢ ( ¬ ∃𝑥¬ 𝜃 , (𝜏 ∧ ¬
𝜑) ▶ (𝜏 ∧ ¬ 𝜑) )
| | 8:7: | ⊢ ( ¬ ∃𝑥¬ 𝜃 , (𝜏 ∧ ¬
𝜑) ▶ 𝜏 )
| | 9:7: | ⊢ ( ¬ ∃𝑥¬ 𝜃 , (𝜏 ∧ ¬
𝜑) ▶ ¬ 𝜑 )
| | 10:5: | ⊢ ( ¬ ∃𝑥¬ 𝜃 ▶ 𝜃 )
| | 11:10,8: | ⊢ ( ¬ ∃𝑥¬ 𝜃 , (𝜏 ∧ ¬
𝜑) ▶ (𝜃 ∧ 𝜏) )
| | 12:11: | ⊢ ( ¬ ∃𝑥¬ 𝜃 , (𝜏 ∧ ¬
𝜑) ▶ ∃𝑥(𝜃 ∧ 𝜏) )
| | 13:12: | ⊢ ( ¬ ∃𝑥¬ 𝜃 , (𝜏 ∧ ¬
𝜑) ▶ ¬ ¬ ∃𝑥(𝜃 ∧ 𝜏) )
| | 14:2,13: | ⊢ ( ¬ ∃𝑥¬ 𝜃 , (𝜏 ∧ ¬
𝜑) ▶ ¬ ∀𝑥𝜒 )
| | 140:: | ⊢ (∃𝑥¬ 𝜃 → ∀𝑥∃𝑥¬ 𝜃
)
| | 141:140: | ⊢ (¬ ∃𝑥¬ 𝜃 → ∀𝑥¬ ∃𝑥
¬ 𝜃)
| | 142:: | ⊢ (∀𝑥𝜒 → ∀𝑥∀𝑥𝜒)
| | 143:142: | ⊢ (¬ ∀𝑥𝜒 → ∀𝑥¬ ∀𝑥𝜒
)
| | 144:6,14,141,143: | ⊢ ( ¬ ∃𝑥¬ 𝜃 ▶ ¬ ∀𝑥𝜒
)
| | 15:1: | ⊢ (¬ ∃𝑥¬ 𝜑 ∨ ¬ ∃𝑥(𝜓
∧ ¬ 𝜒))
| | 16:9: | ⊢ ( ¬ ∃𝑥¬ 𝜃 , (𝜏 ∧ ¬
𝜑) ▶ ∃𝑥¬ 𝜑 )
| | 161:: | ⊢ (∃𝑥¬ 𝜑 → ∀𝑥∃𝑥¬ 𝜑
)
| | 162:6,16,141,161: | ⊢ ( ¬ ∃𝑥¬ 𝜃 ▶ ∃𝑥¬ 𝜑
)
| | 17:162: | ⊢ ( ¬ ∃𝑥¬ 𝜃 ▶ ¬ ¬ ∃𝑥
¬ 𝜑 )
| | 18:15,17: | ⊢ ( ¬ ∃𝑥¬ 𝜃 ▶ ¬ ∃𝑥(
𝜓 ∧ ¬ 𝜒) )
| | 19:18: | ⊢ ( ¬ ∃𝑥¬ 𝜃 ▶ ∀𝑥(𝜓
→ 𝜒) )
| | 20:144: | ⊢ ( ¬ ∃𝑥¬ 𝜃 ▶ ∃𝑥¬ 𝜒
)
| | 21:: | ⊢ ( ¬ ∃𝑥¬ 𝜃 , ¬ 𝜒 ▶ ¬
𝜒 )
| | 22:19: | ⊢ ( ¬ ∃𝑥¬ 𝜃 ▶ (𝜓 → 𝜒
) )
| | 23:21,22: | ⊢ ( ¬ ∃𝑥¬ 𝜃 , ¬ 𝜒 ▶ ¬
𝜓 )
| | 24:23: | ⊢ ( ¬ ∃𝑥¬ 𝜃 , ¬ 𝜒 ▶ ∃
𝑥¬ 𝜓 )
| | 240:: | ⊢ (∃𝑥¬ 𝜓 → ∀𝑥∃𝑥¬ 𝜓
)
| | 241:20,24,141,240: | ⊢ ( ¬ ∃𝑥¬ 𝜃 ▶ ∃𝑥¬ 𝜓
)
| | 25:241: | ⊢ ( ¬ ∃𝑥¬ 𝜃 ▶ ¬ ∀𝑥𝜓
)
| | qed:25: | ⊢ (¬ ∃𝑥¬ 𝜃 → ¬ ∀𝑥𝜓)
|
(Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
| ⊢ ¬
(∃𝑥 ¬ 𝜑 ∧ ∃𝑥(𝜓 ∧ ¬ 𝜒)) & ⊢ (∀𝑥𝜒 → ¬ ∃𝑥(𝜃 ∧ 𝜏)) & ⊢ ¬
∀𝑥(𝜏 → 𝜑) ⇒ ⊢ (¬ ∃𝑥 ¬ 𝜃 → ¬ ∀𝑥𝜓) |
| |
| Theorem | notnotrALTVD 42505 |
The following User's Proof is a Natural Deduction Sequent Calculus
transcription of the Fitch-style Natural Deduction proof of Theorem 5 of
Section 14 of [Margaris] p. 59 (which is notnotr 130). The same proof
may also be interpreted as a Virtual Deduction Hilbert-style
axiomatic proof. It was completed automatically by the tools program
completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm
Megill's Metamath Proof Assistant. notnotrALT 42119 is notnotrALTVD 42505
without virtual deductions and was automatically derived
from notnotrALTVD 42505. Step i of the User's Proof corresponds to
step i of the Fitch-style proof.
| 1:: | ⊢ ( ¬ ¬ 𝜑 ▶ ¬ ¬ 𝜑 )
| | 2:: | ⊢ (¬ ¬ 𝜑 → (¬ 𝜑 → ¬ ¬ ¬ 𝜑))
| | 3:1: | ⊢ ( ¬ ¬ 𝜑 ▶ (¬ 𝜑 → ¬ ¬ ¬ 𝜑) )
| | 4:: | ⊢ ((¬ 𝜑 → ¬ ¬ ¬ 𝜑) → (¬ ¬ 𝜑 →
𝜑))
| | 5:3: | ⊢ ( ¬ ¬ 𝜑 ▶ (¬ ¬ 𝜑 → 𝜑) )
| | 6:5,1: | ⊢ ( ¬ ¬ 𝜑 ▶ 𝜑 )
| | qed:6: | ⊢ (¬ ¬ 𝜑 → 𝜑)
|
(Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
| ⊢ (¬ ¬
𝜑 → 𝜑) |
| |
| Theorem | con3ALTVD 42506 |
The following User's Proof is a Natural Deduction Sequent Calculus
transcription of the Fitch-style Natural Deduction proof of Theorem 7 of
Section 14 of [Margaris] p. 60 (which is con3 153). The same proof may
also be interpreted to be a Virtual Deduction Hilbert-style axiomatic
proof. It was completed automatically by the tools program
completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm
Megill's Metamath Proof Assistant. con3ALT2 42120 is con3ALTVD 42506 without
virtual deductions and was automatically derived from con3ALTVD 42506.
Step i of the User's Proof corresponds to step i of the Fitch-style proof.
| 1:: | ⊢ ( (𝜑 → 𝜓) ▶ (𝜑 → 𝜓) )
| | 2:: | ⊢ ( (𝜑 → 𝜓) , ¬ ¬ 𝜑 ▶ ¬ ¬ 𝜑 )
| | 3:: | ⊢ (¬ ¬ 𝜑 → 𝜑)
| | 4:2: | ⊢ ( (𝜑 → 𝜓) , ¬ ¬ 𝜑 ▶ 𝜑 )
| | 5:1,4: | ⊢ ( (𝜑 → 𝜓) , ¬ ¬ 𝜑 ▶ 𝜓 )
| | 6:: | ⊢ (𝜓 → ¬ ¬ 𝜓)
| | 7:6,5: | ⊢ ( (𝜑 → 𝜓) , ¬ ¬ 𝜑 ▶ ¬ ¬ 𝜓 )
| | 8:7: | ⊢ ( (𝜑 → 𝜓) ▶ (¬ ¬ 𝜑 → ¬ ¬ 𝜓
) )
| | 9:: | ⊢ ((¬ ¬ 𝜑 → ¬ ¬ 𝜓) → (¬ 𝜓 →
¬ 𝜑))
| | 10:8: | ⊢ ( (𝜑 → 𝜓) ▶ (¬ 𝜓 → ¬ 𝜑) )
| | qed:10: | ⊢ ((𝜑 → 𝜓) → (¬ 𝜓 → ¬ 𝜑))
|
(Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is
discouraged.) (New usage is discouraged.)
|
| ⊢ ((𝜑 → 𝜓) → (¬ 𝜓 → ¬ 𝜑)) |
| |
| 20.36.9 Theorems proved using conjunction-form
Virtual Deduction
|
| |
| Theorem | elpwgdedVD 42507 |
Membership in a power class. Theorem 86 of [Suppes] p. 47. Derived
from elpwg 4542. In form of VD deduction with 𝜑 and 𝜓 as
variable virtual hypothesis collections based on Mario Carneiro's
metavariable concept. elpwgded 42154 is elpwgdedVD 42507 using conventional
notation. (Contributed by Alan Sare, 23-Apr-2015.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ ( 𝜑 ▶ 𝐴 ∈ V ) & ⊢ ( 𝜓 ▶ 𝐴 ⊆ 𝐵 )
⇒ ⊢ ( ( 𝜑 , 𝜓 ) ▶ 𝐴 ∈ 𝒫 𝐵 ) |
| |
| Theorem | sspwimp 42508 |
If a class is a subclass of another class, then its power class is a
subclass of that other class's power class. Left-to-right implication
of Exercise 18 of [TakeutiZaring]
p. 18. For the biconditional, see
sspwb 5369. The proof sspwimp 42508, using conventional notation, was
translated from virtual deduction form, sspwimpVD 42509, using a
translation program. (Contributed by Alan Sare, 23-Apr-2015.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ (𝐴 ⊆ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵) |
| |
| Theorem | sspwimpVD 42509 |
The following User's Proof is a Virtual Deduction proof (see wvd1 42159)
using conjunction-form virtual hypothesis collections. It was completed
manually, but has the potential to be completed automatically by a tools
program which would invoke Mel L. O'Cat's mmj2 and Norm Megill's
Metamath Proof Assistant.
sspwimp 42508 is sspwimpVD 42509 without virtual deductions and was derived
from sspwimpVD 42509. (Contributed by Alan Sare, 23-Apr-2015.)
(Proof modification is discouraged.) (New usage is discouraged.)
| 1:: | ⊢ ( 𝐴 ⊆ 𝐵 ▶ 𝐴 ⊆ 𝐵 )
| | 2:: | ⊢ ( .............. 𝑥 ∈ 𝒫 𝐴
▶ 𝑥 ∈ 𝒫 𝐴 )
| | 3:2: | ⊢ ( .............. 𝑥 ∈ 𝒫 𝐴
▶ 𝑥 ⊆ 𝐴 )
| | 4:3,1: | ⊢ ( ( 𝐴 ⊆ 𝐵 , 𝑥 ∈ 𝒫 𝐴 ) ▶ 𝑥 ⊆ 𝐵 )
| | 5:: | ⊢ 𝑥 ∈ V
| | 6:4,5: | ⊢ ( ( 𝐴 ⊆ 𝐵 , 𝑥 ∈ 𝒫 𝐴 ) ▶ 𝑥 ∈ 𝒫 𝐵
)
| | 7:6: | ⊢ ( 𝐴 ⊆ 𝐵 ▶ (𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝒫 𝐵)
)
| | 8:7: | ⊢ ( 𝐴 ⊆ 𝐵 ▶ ∀𝑥(𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈
𝒫 𝐵) )
| | 9:8: | ⊢ ( 𝐴 ⊆ 𝐵 ▶ 𝒫 𝐴 ⊆ 𝒫 𝐵 )
| | qed:9: | ⊢ (𝐴 ⊆ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)
|
|
| ⊢ (𝐴 ⊆ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵) |
| |
| Theorem | sspwimpcf 42510 |
If a class is a subclass of another class, then its power class is a
subclass of that other class's power class. Left-to-right implication
of Exercise 18 of [TakeutiZaring]
p. 18. sspwimpcf 42510, using
conventional notation, was translated from its virtual deduction form,
sspwimpcfVD 42511, using a translation program. (Contributed
by Alan Sare,
13-Jun-2015.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
| ⊢ (𝐴 ⊆ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵) |
| |
| Theorem | sspwimpcfVD 42511 |
The following User's Proof is a Virtual Deduction proof (see wvd1 42159)
using conjunction-form virtual hypothesis collections. It was completed
automatically by a tools program which would invokes Mel L. O'Cat's mmj2
and Norm Megill's Metamath Proof Assistant.
sspwimpcf 42510 is sspwimpcfVD 42511 without virtual deductions and was derived
from sspwimpcfVD 42511.
The version of completeusersproof.cmd used is capable of only generating
conjunction-form unification theorems, not unification deductions.
(Contributed by Alan Sare, 13-Jun-2015.)
(Proof modification is discouraged.) (New usage is discouraged.)
| 1:: | ⊢ ( 𝐴 ⊆ 𝐵 ▶ 𝐴 ⊆ 𝐵 )
| | 2:: | ⊢ ( ........... 𝑥 ∈ 𝒫 𝐴
▶ 𝑥 ∈ 𝒫 𝐴 )
| | 3:2: | ⊢ ( ........... 𝑥 ∈ 𝒫 𝐴
▶ 𝑥 ⊆ 𝐴 )
| | 4:3,1: | ⊢ ( ( 𝐴 ⊆ 𝐵 , 𝑥 ∈ 𝒫 𝐴 ) ▶ 𝑥 ⊆ 𝐵 )
| | 5:: | ⊢ 𝑥 ∈ V
| | 6:4,5: | ⊢ ( ( 𝐴 ⊆ 𝐵 , 𝑥 ∈ 𝒫 𝐴 ) ▶ 𝑥 ∈ 𝒫 𝐵
)
| | 7:6: | ⊢ ( 𝐴 ⊆ 𝐵 ▶ (𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝒫 𝐵)
)
| | 8:7: | ⊢ ( 𝐴 ⊆ 𝐵 ▶ ∀𝑥(𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈
𝒫 𝐵) )
| | 9:8: | ⊢ ( 𝐴 ⊆ 𝐵 ▶ 𝒫 𝐴 ⊆ 𝒫 𝐵 )
| | qed:9: | ⊢ (𝐴 ⊆ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)
|
|
| ⊢ (𝐴 ⊆ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵) |
| |
| Theorem | suctrALTcf 42512 |
The sucessor of a transitive class is transitive. suctrALTcf 42512, using
conventional notation, was translated from virtual deduction form,
suctrALTcfVD 42513, using a translation program. (Contributed
by Alan
Sare, 13-Jun-2015.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
| ⊢ (Tr 𝐴 → Tr suc 𝐴) |
| |
| Theorem | suctrALTcfVD 42513 |
The following User's Proof is a Virtual Deduction proof (see wvd1 42159)
using conjunction-form virtual hypothesis collections. The
conjunction-form version of completeusersproof.cmd. It allows the User
to avoid superflous virtual hypotheses. This proof was completed
automatically by a tools program which invokes Mel L. O'Cat's
mmj2 and Norm Megill's Metamath Proof Assistant. suctrALTcf 42512
is suctrALTcfVD 42513 without virtual deductions and was derived
automatically from suctrALTcfVD 42513. The version of
completeusersproof.cmd used is capable of only generating
conjunction-form unification theorems, not unification deductions.
(Contributed by Alan Sare, 13-Jun-2015.)
(Proof modification is discouraged.) (New usage is discouraged.)
| 1:: | ⊢ ( Tr 𝐴 ▶ Tr 𝐴 )
| | 2:: | ⊢ ( ......... (𝑧 ∈ 𝑦 ∧ 𝑦 ∈
suc 𝐴) ▶ (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴) )
| | 3:2: | ⊢ ( ......... (𝑧 ∈ 𝑦 ∧ 𝑦 ∈
suc 𝐴) ▶ 𝑧 ∈ 𝑦 )
| | 4:: | ⊢ ( ...................................
....... 𝑦 ∈ 𝐴 ▶ 𝑦 ∈ 𝐴 )
| | 5:1,3,4: | ⊢ ( ( Tr 𝐴 , (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴)
, 𝑦 ∈ 𝐴 ) ▶ 𝑧 ∈ 𝐴 )
| | 6:: | ⊢ 𝐴 ⊆ suc 𝐴
| | 7:5,6: | ⊢ ( ( Tr 𝐴 , (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴)
, 𝑦 ∈ 𝐴 ) ▶ 𝑧 ∈ suc 𝐴 )
| | 8:7: | ⊢ ( ( Tr 𝐴 , (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴)
) ▶ (𝑦 ∈ 𝐴 → 𝑧 ∈ suc 𝐴) )
| | 9:: | ⊢ ( ...................................
...... 𝑦 = 𝐴 ▶ 𝑦 = 𝐴 )
| | 10:3,9: | ⊢ ( ........ ( (𝑧 ∈ 𝑦 ∧ 𝑦 ∈
suc 𝐴), 𝑦 = 𝐴 ) ▶ 𝑧 ∈ 𝐴 )
| | 11:10,6: | ⊢ ( ........ ( (𝑧 ∈ 𝑦 ∧ 𝑦 ∈
suc 𝐴), 𝑦 = 𝐴 ) ▶ 𝑧 ∈ suc 𝐴 )
| | 12:11: | ⊢ ( .......... (𝑧 ∈ 𝑦 ∧ 𝑦 ∈
suc 𝐴) ▶ (𝑦 = 𝐴 → 𝑧 ∈ suc 𝐴) )
| | 13:2: | ⊢ ( .......... (𝑧 ∈ 𝑦 ∧ 𝑦 ∈
suc 𝐴) ▶ 𝑦 ∈ suc 𝐴 )
| | 14:13: | ⊢ ( .......... (𝑧 ∈ 𝑦 ∧ 𝑦 ∈
suc 𝐴) ▶ (𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴) )
| | 15:8,12,14: | ⊢ ( ( Tr 𝐴 , (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ suc 𝐴)
) ▶ 𝑧 ∈ suc 𝐴 )
| | 16:15: | ⊢ ( Tr 𝐴 ▶ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈
suc 𝐴) → 𝑧 ∈ suc 𝐴) )
| | 17:16: | ⊢ ( Tr 𝐴 ▶ ∀𝑧∀𝑦((𝑧 ∈
𝑦 ∧ 𝑦 ∈ suc 𝐴) → 𝑧 ∈ suc 𝐴) )
| | 18:17: | ⊢ ( Tr 𝐴 ▶ Tr suc 𝐴 )
| | qed:18: | ⊢ (Tr 𝐴 → Tr suc 𝐴)
|
|
| ⊢ (Tr 𝐴 → Tr suc 𝐴) |
| |
| 20.36.10 Theorems with a VD proof in
conventional notation derived from a VD proof
|
| |
| Theorem | suctrALT3 42514 |
The successor of a transitive class is transitive. suctrALT3 42514 is the
completed proof in conventional notation of the Virtual Deduction proof
https://us.metamath.org/other/completeusersproof/suctralt3vd.html 42514.
It was completed manually. The potential for automated derivation from
the VD proof exists. See wvd1 42159 for a description of Virtual
Deduction.
Some sub-theorems of the proof were completed using a unification
deduction (e.g., the sub-theorem whose assertion is step 19 used
jaoded 42156). Unification deductions employ Mario
Carneiro's metavariable
concept. Some sub-theorems were completed using a unification theorem
(e.g., the sub-theorem whose assertion is step 24 used dftr2 5198) .
(Contributed by Alan Sare, 3-Dec-2015.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ (Tr 𝐴 → Tr suc 𝐴) |
| |
| Theorem | sspwimpALT 42515 |
If a class is a subclass of another class, then its power class is a
subclass of that other class's power class. Left-to-right implication
of Exercise 18 of [TakeutiZaring]
p. 18. sspwimpALT 42515 is the completed
proof in conventional notation of the Virtual Deduction proof
https://us.metamath.org/other/completeusersproof/sspwimpaltvd.html 42515.
It was completed manually. The potential for automated derivation from
the VD proof exists. See wvd1 42159 for a description of Virtual
Deduction.
Some sub-theorems of the proof were completed using a unification
deduction (e.g., the sub-theorem whose assertion is step 9 used
elpwgded 42154). Unification deductions employ Mario
Carneiro's
metavariable concept. Some sub-theorems were completed using a
unification theorem (e.g., the sub-theorem whose assertion is step 5
used elpwi 4548). (Contributed by Alan Sare, 3-Dec-2015.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ (𝐴 ⊆ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵) |
| |
| Theorem | unisnALT 42516 |
A set equals the union of its singleton. Theorem 8.2 of [Quine] p. 53.
The User manually input on a mmj2 Proof Worksheet, without labels, all
steps of unisnALT 42516 except 1, 11, 15, 21, and 30. With
execution of the
mmj2 unification command, mmj2 could find labels for all steps except
for 2, 12, 16, 22, and 31 (and the then non-existing steps 1, 11, 15,
21, and 30). mmj2 could not find reference theorems for those five steps
because the hypothesis field of each of these steps was empty and none
of those steps unifies with a theorem in set.mm. Each of these five
steps is a semantic variation of a theorem in set.mm and is 2-step
provable. mmj2 does not have the ability to automatically generate the
semantic variation in set.mm of a theorem in a mmj2 Proof Worksheet
unless the theorem in the Proof Worksheet is labeled with a 1-hypothesis
deduction whose hypothesis is a theorem in set.mm which unifies with the
theorem in the Proof Worksheet. The stepprover.c program, which invokes
mmj2, has this capability. stepprover.c automatically generated steps 1,
11, 15, 21, and 30, labeled all steps, and generated the RPN proof of
unisnALT 42516. Roughly speaking, stepprover.c added to
the Proof
Worksheet a labeled duplicate step of each non-unifying theorem for each
label in a text file, labels.txt, containing a list of labels provided
by the User. Upon mmj2 unification, stepprover.c identified a label for
each of the five theorems which 2-step proves it. For unisnALT 42516, the
label list is a list of all 1-hypothesis propositional calculus
deductions in set.mm. stepproverp.c is the same as stepprover.c except
that it intermittently pauses during execution, allowing the User to
observe the changes to a text file caused by the execution of particular
statements of the program. (Contributed by Alan Sare, 19-Aug-2016.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ 𝐴 ∈
V ⇒ ⊢ ∪
{𝐴} = 𝐴 |
| |
| 20.36.11 Theorems with a proof in conventional
notation derived from a VD proof
Theorems with a proof in conventional notation automatically derived by
completeusersproof.c from a Virtual Deduction User's Proof.
|
| |
| Theorem | notnotrALT2 42517 |
Converse of double negation. Theorem *2.14 of [WhiteheadRussell] p. 102.
Proof derived by completeusersproof.c from User's Proof in
VirtualDeductionProofs.txt. (Contributed by Alan Sare, 11-Sep-2016.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ (¬ ¬
𝜑 → 𝜑) |
| |
| Theorem | sspwimpALT2 42518 |
If a class is a subclass of another class, then its power class is a
subclass of that other class's power class. Left-to-right implication
of Exercise 18 of [TakeutiZaring]
p. 18. Proof derived by
completeusersproof.c from User's Proof in VirtualDeductionProofs.txt.
The User's Proof in html format is displayed in
https://us.metamath.org/other/completeusersproof/sspwimpaltvd.html.
(Contributed by Alan Sare, 11-Sep-2016.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ (𝐴 ⊆ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵) |
| |
| Theorem | e2ebindALT 42519 |
Absorption of an existential quantifier of a double existential quantifier
of non-distinct variables. The proof is derived by completeusersproof.c
from User's Proof in VirtualDeductionProofs.txt. The User's Proof in html
format is displayed in e2ebindVD 42502. (Contributed by Alan Sare,
11-Sep-2016.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
| ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥∃𝑦𝜑 ↔ ∃𝑦𝜑)) |
| |
| Theorem | ax6e2ndALT 42520* |
If at least two sets exist (dtru 5363) , then the same is true expressed
in an alternate form similar to the form of ax6e 2385.
The proof is
derived by completeusersproof.c from User's Proof in
VirtualDeductionProofs.txt. The User's Proof in html format is
displayed in ax6e2ndVD 42498. (Contributed by Alan Sare, 11-Sep-2016.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ (¬
∀𝑥 𝑥 = 𝑦 → ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)) |
| |
| Theorem | ax6e2ndeqALT 42521* |
"At least two sets exist" expressed in the form of dtru 5363
is logically
equivalent to the same expressed in a form similar to ax6e 2385
if dtru 5363
is false implies 𝑢 = 𝑣. Proof derived by
completeusersproof.c from
User's Proof in VirtualDeductionProofs.txt. The User's Proof in html
format is displayed in ax6e2ndeqVD 42499. (Contributed by Alan Sare,
11-Sep-2016.) (Proof modification is discouraged.)
(New usage is discouraged.)
|
| ⊢ ((¬
∀𝑥 𝑥 = 𝑦 ∨ 𝑢 = 𝑣) ↔ ∃𝑥∃𝑦(𝑥 = 𝑢 ∧ 𝑦 = 𝑣)) |
| |
| Theorem | 2sb5ndALT 42522* |
Equivalence for double substitution 2sb5 2276 without distinct 𝑥,
𝑦 requirement. 2sb5nd 42150 is derived from 2sb5ndVD 42500. The proof is
derived by completeusersproof.c from User's Proof in
VirtualDeductionProofs.txt. The User's Proof in html format is
displayed in 2sb5ndVD 42500. (Contributed by Alan Sare, 19-Sep-2016.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ ((¬
∀𝑥 𝑥 = 𝑦 ∨ 𝑢 = 𝑣) → ([𝑢 / 𝑥][𝑣 / 𝑦]𝜑 ↔ ∃𝑥∃𝑦((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) ∧ 𝜑))) |
| |
| Theorem | chordthmALT 42523* |
The intersecting chords theorem. If points A, B, C, and D lie on a
circle (with center Q, say), and the point P is on the interior of the
segments AB and CD, then the two products of lengths PA · PB and
PC · PD are equal. The Euclidean plane is identified with the
complex plane, and the fact that P is on AB and on CD is expressed by
the hypothesis that the angles APB and CPD are equal to π. The
result is proven by using chordthmlem5 25984 twice to show that PA
· PB and PC · PD both equal BQ
2
−
PQ
2
. This is similar to the proof of the
theorem given in Euclid's Elements, where it is Proposition
III.35.
Proven by David Moews on 28-Feb-2017 as chordthm 25985.
https://us.metamath.org/other/completeusersproof/chordthmaltvd.html 25985 is
a Virtual
Deduction User's Proof transcription of chordthm 25985. That VD User's
Proof was input into completeusersproof, automatically generating this
chordthmALT 42523 Metamath proof. (Contributed by Alan Sare,
19-Sep-2017.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ 𝐹 = (𝑥 ∈ (ℂ ∖ {0}), 𝑦 ∈ (ℂ ∖ {0})
↦ (ℑ‘(log‘(𝑦 / 𝑥)))) & ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℂ) & ⊢ (𝜑 → 𝐶 ∈ ℂ) & ⊢ (𝜑 → 𝐷 ∈ ℂ) & ⊢ (𝜑 → 𝑃 ∈ ℂ) & ⊢ (𝜑 → 𝐴 ≠ 𝑃)
& ⊢ (𝜑 → 𝐵 ≠ 𝑃)
& ⊢ (𝜑 → 𝐶 ≠ 𝑃)
& ⊢ (𝜑 → 𝐷 ≠ 𝑃)
& ⊢ (𝜑 → ((𝐴 − 𝑃)𝐹(𝐵 − 𝑃)) = π) & ⊢ (𝜑 → ((𝐶 − 𝑃)𝐹(𝐷 − 𝑃)) = π) & ⊢ (𝜑 → 𝑄 ∈ ℂ) & ⊢ (𝜑 → (abs‘(𝐴 − 𝑄)) = (abs‘(𝐵 − 𝑄))) & ⊢ (𝜑 → (abs‘(𝐴 − 𝑄)) = (abs‘(𝐶 − 𝑄))) & ⊢ (𝜑 → (abs‘(𝐴 − 𝑄)) = (abs‘(𝐷 − 𝑄))) ⇒ ⊢ (𝜑 → ((abs‘(𝑃 − 𝐴)) · (abs‘(𝑃 − 𝐵))) = ((abs‘(𝑃 − 𝐶)) · (abs‘(𝑃 − 𝐷)))) |
| |
| Theorem | isosctrlem1ALT 42524 |
Lemma for isosctr 25969. This proof was automatically derived by
completeusersproof from its Virtual Deduction proof counterpart
https://us.metamath.org/other/completeusersproof/isosctrlem1altvd.html 25969.
As it is verified by the Metamath program, isosctrlem1ALT 42524 verifies
https://us.metamath.org/other/completeusersproof/isosctrlem1altvd.html 42524.
(Contributed by Alan Sare, 22-Apr-2018.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ ((𝐴 ∈ ℂ ∧
(abs‘𝐴) = 1 ∧
¬ 1 = 𝐴) →
(ℑ‘(log‘(1 − 𝐴))) ≠ π) |
| |
| Theorem | iunconnlem2 42525* |
The indexed union of connected overlapping subspaces sharing a common
point is connected. This proof was automatically derived by
completeusersproof from its Virtual Deduction proof counterpart
https://us.metamath.org/other/completeusersproof/iunconlem2vd.html.
As it is verified by the Metamath program, iunconnlem2 42525 verifies
https://us.metamath.org/other/completeusersproof/iunconlem2vd.html 42525.
(Contributed by Alan Sare, 22-Apr-2018.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ (𝜓 ↔ ((((((𝜑 ∧ 𝑢 ∈ 𝐽) ∧ 𝑣 ∈ 𝐽) ∧ (𝑢 ∩ ∪
𝑘 ∈ 𝐴 𝐵) ≠ ∅) ∧ (𝑣 ∩ ∪
𝑘 ∈ 𝐴 𝐵) ≠ ∅) ∧ (𝑢 ∩ 𝑣) ⊆ (𝑋 ∖ ∪ 𝑘 ∈ 𝐴 𝐵)) ∧ ∪ 𝑘 ∈ 𝐴 𝐵 ⊆ (𝑢 ∪ 𝑣))) & ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ⊆ 𝑋)
& ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑃 ∈ 𝐵)
& ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐽 ↾t 𝐵) ∈ Conn)
⇒ ⊢ (𝜑 → (𝐽 ↾t ∪ 𝑘 ∈ 𝐴 𝐵) ∈ Conn) |
| |
| Theorem | iunconnALT 42526* |
The indexed union of connected overlapping subspaces sharing a common
point is connected. This proof was automatically derived by
completeusersproof from its Virtual Deduction proof counterpart
https://us.metamath.org/other/completeusersproof/iunconaltvd.html.
As it is verified by the Metamath program, iunconnALT 42526 verifies
https://us.metamath.org/other/completeusersproof/iunconaltvd.html 42526.
(Contributed by Alan Sare, 22-Apr-2018.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ⊆ 𝑋)
& ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑃 ∈ 𝐵)
& ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐽 ↾t 𝐵) ∈ Conn)
⇒ ⊢ (𝜑 → (𝐽 ↾t ∪ 𝑘 ∈ 𝐴 𝐵) ∈ Conn) |
| |
| Theorem | sineq0ALT 42527 |
A complex number whose sine is zero is an integer multiple of π.
The Virtual Deduction form of the proof is
https://us.metamath.org/other/completeusersproof/sineq0altvd.html.
The
Metamath form of the proof is sineq0ALT 42527. The Virtual Deduction proof
is based on Mario Carneiro's revision of Norm Megill's proof of sineq0 25678.
The Virtual Deduction proof is verified by automatically transforming it
into the Metamath form of the proof using completeusersproof, which is
verified by the Metamath program. The proof of
https://us.metamath.org/other/completeusersproof/sineq0altro.html 25678 is a
form of the completed proof which preserves the Virtual Deduction proof's
step numbers and their ordering. (Contributed by Alan Sare, 13-Jun-2018.)
(Proof modification is discouraged.) (New usage is discouraged.)
|
| ⊢ (𝐴 ∈ ℂ →
((sin‘𝐴) = 0 ↔
(𝐴 / π) ∈
ℤ)) |
| |
| 20.37 Mathbox for Glauco
Siliprandi
|
| |
| 20.37.1 Miscellanea
|
| |
| Theorem | evth2f 42528* |
A version of evth2 24121 using bound-variable hypotheses instead of
distinct
variable conditions. (Contributed by Glauco Siliprandi,
20-Apr-2017.)
|
| ⊢
Ⅎ𝑥𝐹
& ⊢ Ⅎ𝑦𝐹
& ⊢ Ⅎ𝑥𝑋
& ⊢ Ⅎ𝑦𝑋
& ⊢ 𝑋 = ∪ 𝐽 & ⊢ 𝐾 = (topGen‘ran
(,))
& ⊢ (𝜑 → 𝐽 ∈ Comp) & ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) & ⊢ (𝜑 → 𝑋 ≠ ∅)
⇒ ⊢ (𝜑 → ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐹‘𝑥) ≤ (𝐹‘𝑦)) |
| |
| Theorem | elunif 42529* |
A version of eluni 4848 using bound-variable hypotheses instead of
distinct
variable conditions. (Contributed by Glauco Siliprandi,
20-Apr-2017.)
|
| ⊢
Ⅎ𝑥𝐴
& ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ (𝐴 ∈ ∪ 𝐵 ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵)) |
| |
| Theorem | rzalf 42530 |
A version of rzal 4445 using bound-variable hypotheses instead of
distinct
variable conditions. (Contributed by Glauco Siliprandi,
20-Apr-2017.)
|
| ⊢ Ⅎ𝑥 𝐴 = ∅ ⇒ ⊢ (𝐴 = ∅ → ∀𝑥 ∈ 𝐴 𝜑) |
| |
| Theorem | fvelrnbf 42531 |
A version of fvelrnb 6827 using bound-variable hypotheses instead of
distinct variable conditions. (Contributed by Glauco Siliprandi,
20-Apr-2017.)
|
| ⊢
Ⅎ𝑥𝐴
& ⊢ Ⅎ𝑥𝐵
& ⊢ Ⅎ𝑥𝐹 ⇒ ⊢ (𝐹 Fn 𝐴 → (𝐵 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵)) |
| |
| Theorem | rfcnpre1 42532 |
If F is a continuous function with respect to the standard topology,
then the preimage A of the values greater than a given extended real B
is an open set. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
|
| ⊢
Ⅎ𝑥𝐵
& ⊢ Ⅎ𝑥𝐹
& ⊢ Ⅎ𝑥𝜑
& ⊢ 𝐾 = (topGen‘ran (,)) & ⊢ 𝑋 = ∪
𝐽 & ⊢ 𝐴 = {𝑥 ∈ 𝑋 ∣ 𝐵 < (𝐹‘𝑥)}
& ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) ⇒ ⊢ (𝜑 → 𝐴 ∈ 𝐽) |
| |
| Theorem | ubelsupr 42533* |
If U belongs to A and U is an upper bound, then U is the sup of A.
(Contributed by Glauco Siliprandi, 20-Apr-2017.)
|
| ⊢ ((𝐴 ⊆ ℝ ∧ 𝑈 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑥 ≤ 𝑈) → 𝑈 = sup(𝐴, ℝ, < )) |
| |
| Theorem | fsumcnf 42534* |
A finite sum of functions to complex numbers from a common topological
space is continuous, without disjoint var constraint x ph. The class
expression for B normally contains free variables k and x to index it.
(Contributed by Glauco Siliprandi, 20-Apr-2017.)
|
| ⊢ 𝐾 =
(TopOpen‘ℂfld) & ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐾)) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝐴 𝐵) ∈ (𝐽 Cn 𝐾)) |
| |
| Theorem | mulltgt0 42535 |
The product of a negative and a positive number is negative. (Contributed
by Glauco Siliprandi, 20-Apr-2017.)
|
| ⊢ (((𝐴 ∈ ℝ ∧ 𝐴 < 0) ∧ (𝐵 ∈ ℝ ∧ 0 <
𝐵)) → (𝐴 · 𝐵) < 0) |
| |
| Theorem | rspcegf 42536 |
A version of rspcev 3561 using bound-variable hypotheses instead of
distinct variable conditions. (Contributed by Glauco Siliprandi,
20-Apr-2017.)
|
| ⊢ Ⅎ𝑥𝜓
& ⊢ Ⅎ𝑥𝐴
& ⊢ Ⅎ𝑥𝐵
& ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜓) → ∃𝑥 ∈ 𝐵 𝜑) |
| |
| Theorem | rabexgf 42537 |
A version of rabexg 5259 using bound-variable hypotheses instead of
distinct variable conditions. (Contributed by Glauco Siliprandi,
20-Apr-2017.)
|
| ⊢
Ⅎ𝑥𝐴 ⇒ ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V) |
| |
| Theorem | fcnre 42538 |
A function continuous with respect to the standard topology, is a real
mapping. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
|
| ⊢ 𝐾 = (topGen‘ran
(,))
& ⊢ 𝑇 = ∪ 𝐽 & ⊢ 𝐶 = (𝐽 Cn 𝐾)
& ⊢ (𝜑 → 𝐹 ∈ 𝐶) ⇒ ⊢ (𝜑 → 𝐹:𝑇⟶ℝ) |
| |
| Theorem | sumsnd 42539* |
A sum of a singleton is the term. The deduction version of sumsn 15456.
(Contributed by Glauco Siliprandi, 20-Apr-2017.)
|
| ⊢ (𝜑 → Ⅎ𝑘𝐵)
& ⊢ Ⅎ𝑘𝜑
& ⊢ ((𝜑 ∧ 𝑘 = 𝑀) → 𝐴 = 𝐵)
& ⊢ (𝜑 → 𝑀 ∈ 𝑉)
& ⊢ (𝜑 → 𝐵 ∈ ℂ)
⇒ ⊢ (𝜑 → Σ𝑘 ∈ {𝑀}𝐴 = 𝐵) |
| |
| Theorem | evthf 42540* |
A version of evth 24120 using bound-variable hypotheses instead of
distinct
variable conditions. (Contributed by Glauco Siliprandi,
20-Apr-2017.)
|
| ⊢
Ⅎ𝑥𝐹
& ⊢ Ⅎ𝑦𝐹
& ⊢ Ⅎ𝑥𝑋
& ⊢ Ⅎ𝑦𝑋
& ⊢ Ⅎ𝑥𝜑
& ⊢ Ⅎ𝑦𝜑
& ⊢ 𝑋 = ∪ 𝐽 & ⊢ 𝐾 = (topGen‘ran
(,))
& ⊢ (𝜑 → 𝐽 ∈ Comp) & ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) & ⊢ (𝜑 → 𝑋 ≠ ∅)
⇒ ⊢ (𝜑 → ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐹‘𝑦) ≤ (𝐹‘𝑥)) |
| |
| Theorem | cnfex 42541 |
The class of continuous functions between two topologies is a set.
(Contributed by Glauco Siliprandi, 20-Apr-2017.)
|
| ⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 Cn 𝐾) ∈ V) |
| |
| Theorem | fnchoice 42542* |
For a finite set, a choice function exists, without using the axiom of
choice. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
|
| ⊢ (𝐴 ∈ Fin → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ≠ ∅ → (𝑓‘𝑥) ∈ 𝑥))) |
| |
| Theorem | refsumcn 42543* |
A finite sum of continuous real functions, from a common topological
space, is continuous. The class expression for B normally contains free
variables k and x to index it. See fsumcn 24031 for the analogous theorem
on continuous complex functions. (Contributed by Glauco Siliprandi,
20-Apr-2017.)
|
| ⊢ Ⅎ𝑥𝜑
& ⊢ 𝐾 = (topGen‘ran (,)) & ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐾)) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝐴 𝐵) ∈ (𝐽 Cn 𝐾)) |
| |
| Theorem | rfcnpre2 42544 |
If 𝐹 is a continuous function with
respect to the standard topology,
then the preimage A of the values smaller than a given extended real
𝐵, is an open set. (Contributed by
Glauco Siliprandi,
20-Apr-2017.)
|
| ⊢
Ⅎ𝑥𝐵
& ⊢ Ⅎ𝑥𝐹
& ⊢ Ⅎ𝑥𝜑
& ⊢ 𝐾 = (topGen‘ran (,)) & ⊢ 𝑋 = ∪
𝐽 & ⊢ 𝐴 = {𝑥 ∈ 𝑋 ∣ (𝐹‘𝑥) < 𝐵}
& ⊢ (𝜑 → 𝐵 ∈ ℝ*) & ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) ⇒ ⊢ (𝜑 → 𝐴 ∈ 𝐽) |
| |
| Theorem | cncmpmax 42545* |
When the hypothesis for the extreme value theorem hold, then the sup of
the range of the function belongs to the range, it is real and it an
upper bound of the range. (Contributed by Glauco Siliprandi,
20-Apr-2017.)
|
| ⊢ 𝑇 = ∪
𝐽 & ⊢ 𝐾 = (topGen‘ran
(,))
& ⊢ (𝜑 → 𝐽 ∈ Comp) & ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) & ⊢ (𝜑 → 𝑇 ≠ ∅)
⇒ ⊢ (𝜑 → (sup(ran 𝐹, ℝ, < ) ∈ ran 𝐹 ∧ sup(ran 𝐹, ℝ, < ) ∈
ℝ ∧ ∀𝑡
∈ 𝑇 (𝐹‘𝑡) ≤ sup(ran 𝐹, ℝ, < ))) |
| |
| Theorem | rfcnpre3 42546* |
If F is a continuous function with respect to the standard topology,
then the preimage A of the values greater than or equal to a given real
B is a closed set. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
|
| ⊢
Ⅎ𝑡𝐹
& ⊢ 𝐾 = (topGen‘ran (,)) & ⊢ 𝑇 = ∪
𝐽 & ⊢ 𝐴 = {𝑡 ∈ 𝑇 ∣ 𝐵 ≤ (𝐹‘𝑡)}
& ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) ⇒ ⊢ (𝜑 → 𝐴 ∈ (Clsd‘𝐽)) |
| |
| Theorem | rfcnpre4 42547* |
If F is a continuous function with respect to the standard topology,
then the preimage A of the values less than or equal to a given real B
is a closed set. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
|
| ⊢
Ⅎ𝑡𝐹
& ⊢ 𝐾 = (topGen‘ran (,)) & ⊢ 𝑇 = ∪
𝐽 & ⊢ 𝐴 = {𝑡 ∈ 𝑇 ∣ (𝐹‘𝑡) ≤ 𝐵}
& ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) ⇒ ⊢ (𝜑 → 𝐴 ∈ (Clsd‘𝐽)) |
| |
| Theorem | sumpair 42548* |
Sum of two distinct complex values. The class expression for 𝐴 and
𝐵 normally contain free variable 𝑘 to
index it. (Contributed by
Glauco Siliprandi, 20-Apr-2017.)
|
| ⊢ (𝜑 → Ⅎ𝑘𝐷)
& ⊢ (𝜑 → Ⅎ𝑘𝐸)
& ⊢ (𝜑 → 𝐴 ∈ 𝑉)
& ⊢ (𝜑 → 𝐵 ∈ 𝑊)
& ⊢ (𝜑 → 𝐷 ∈ ℂ) & ⊢ (𝜑 → 𝐸 ∈ ℂ) & ⊢ (𝜑 → 𝐴 ≠ 𝐵)
& ⊢ ((𝜑 ∧ 𝑘 = 𝐴) → 𝐶 = 𝐷)
& ⊢ ((𝜑 ∧ 𝑘 = 𝐵) → 𝐶 = 𝐸) ⇒ ⊢ (𝜑 → Σ𝑘 ∈ {𝐴, 𝐵}𝐶 = (𝐷 + 𝐸)) |
| |
| Theorem | rfcnnnub 42549* |
Given a real continuous function 𝐹 defined on a compact topological
space, there is always a positive integer that is a strict upper bound
of its range. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
|
| ⊢
Ⅎ𝑡𝐹
& ⊢ Ⅎ𝑡𝜑
& ⊢ 𝐾 = (topGen‘ran (,)) & ⊢ (𝜑 → 𝐽 ∈ Comp) & ⊢ 𝑇 = ∪
𝐽 & ⊢ (𝜑 → 𝑇 ≠ ∅) & ⊢ 𝐶 = (𝐽 Cn 𝐾)
& ⊢ (𝜑 → 𝐹 ∈ 𝐶) ⇒ ⊢ (𝜑 → ∃𝑛 ∈ ℕ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑛) |
| |
| Theorem | refsum2cnlem1 42550* |
This is the core Lemma for refsum2cn 42551: the sum of two continuous real
functions (from a common topological space) is continuous. (Contributed
by Glauco Siliprandi, 20-Apr-2017.)
|
| ⊢
Ⅎ𝑥𝐴
& ⊢ Ⅎ𝑥𝐹
& ⊢ Ⅎ𝑥𝐺
& ⊢ Ⅎ𝑥𝜑
& ⊢ 𝐴 = (𝑘 ∈ {1, 2} ↦ if(𝑘 = 1, 𝐹, 𝐺)) & ⊢ 𝐾 = (topGen‘ran
(,))
& ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) & ⊢ (𝜑 → 𝐺 ∈ (𝐽 Cn 𝐾)) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ((𝐹‘𝑥) + (𝐺‘𝑥))) ∈ (𝐽 Cn 𝐾)) |
| |
| Theorem | refsum2cn 42551* |
The sum of two continuus real functions (from a common topological
space) is continuous. (Contributed by Glauco Siliprandi,
20-Apr-2017.)
|
| ⊢
Ⅎ𝑥𝐹
& ⊢ Ⅎ𝑥𝐺
& ⊢ Ⅎ𝑥𝜑
& ⊢ 𝐾 = (topGen‘ran (,)) & ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) & ⊢ (𝜑 → 𝐺 ∈ (𝐽 Cn 𝐾)) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ((𝐹‘𝑥) + (𝐺‘𝑥))) ∈ (𝐽 Cn 𝐾)) |
| |
| Theorem | elunnel2 42552 |
A member of a union that is not a member of the second class, is a member
of the first class. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
|
| ⊢ ((𝐴 ∈ (𝐵 ∪ 𝐶) ∧ ¬ 𝐴 ∈ 𝐶) → 𝐴 ∈ 𝐵) |
| |
| Theorem | adantlllr 42553 |
Deduction adding a conjunct to antecedent. (Contributed by Glauco
Siliprandi, 11-Dec-2019.)
|
| ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏) ⇒ ⊢ (((((𝜑 ∧ 𝜂) ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏) |
| |
| Theorem | 3adantlr3 42554 |
Deduction adding a conjunct to antecedent. (Contributed by Glauco
Siliprandi, 11-Dec-2019.)
|
| ⊢ (((𝜑 ∧ (𝜓 ∧ 𝜒)) ∧ 𝜃) → 𝜏) ⇒ ⊢ (((𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜂)) ∧ 𝜃) → 𝜏) |
| |
| Theorem | nnxrd 42555 |
A natural number is an extended real. (Contributed by Glauco
Siliprandi, 11-Dec-2019.)
|
| ⊢ (𝜑 → 𝐴 ∈ ℕ)
⇒ ⊢ (𝜑 → 𝐴 ∈
ℝ*) |
| |
| Theorem | 3adantll2 42556 |
Deduction adding a conjunct to antecedent. (Contributed by Glauco
Siliprandi, 11-Dec-2019.)
|
| ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏) ⇒ ⊢ ((((𝜑 ∧ 𝜂 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏) |
| |
| Theorem | 3adantll3 42557 |
Deduction adding a conjunct to antecedent. (Contributed by Glauco
Siliprandi, 11-Dec-2019.)
|
| ⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏) ⇒ ⊢ ((((𝜑 ∧ 𝜓 ∧ 𝜂) ∧ 𝜒) ∧ 𝜃) → 𝜏) |
| |
| Theorem | ssnel 42558 |
If not element of a set, then not element of a subset. (Contributed by
Glauco Siliprandi, 11-Dec-2019.)
|
| ⊢ ((𝐴 ⊆ 𝐵 ∧ ¬ 𝐶 ∈ 𝐵) → ¬ 𝐶 ∈ 𝐴) |
| |
| Theorem | elabrexg 42559* |
Elementhood in an image set. (Contributed by Glauco Siliprandi,
11-Dec-2019.)
|
| ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → 𝐵 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵}) |
| |
| Theorem | sncldre 42560 |
A singleton is closed w.r.t. the standard topology on the reals.
(Contributed by Glauco Siliprandi, 11-Dec-2019.)
|
| ⊢ (𝐴 ∈ ℝ → {𝐴} ∈
(Clsd‘(topGen‘ran (,)))) |
| |
| Theorem | n0p 42561 |
A polynomial with a nonzero coefficient is not the zero polynomial.
(Contributed by Glauco Siliprandi, 5-Apr-2020.)
|
| ⊢ ((𝑃 ∈ (Poly‘ℤ)
∧ 𝑁 ∈
ℕ0 ∧ ((coeff‘𝑃)‘𝑁) ≠ 0) → 𝑃 ≠
0𝑝) |
| |
| Theorem | pm2.65ni 42562 |
Inference rule for proof by contradiction. (Contributed by Glauco
Siliprandi, 5-Apr-2020.)
|
| ⊢ (¬ 𝜑 → 𝜓)
& ⊢ (¬ 𝜑 → ¬ 𝜓) ⇒ ⊢ 𝜑 |
| |
| Theorem | pwssfi 42563 |
Every element of the power set of 𝐴 is finite if and only if 𝐴
is finite. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|
| ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ Fin ↔ 𝒫 𝐴 ⊆ Fin)) |
| |
| Theorem | iuneq2df 42564 |
Equality deduction for indexed union. (Contributed by Glauco
Siliprandi, 17-Aug-2020.)
|
| ⊢ Ⅎ𝑥𝜑
& ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → ∪
𝑥 ∈ 𝐴 𝐵 = ∪
𝑥 ∈ 𝐴 𝐶) |
| |
| Theorem | nnfoctb 42565* |
There exists a mapping from ℕ onto any (nonempty)
countable set.
(Contributed by Glauco Siliprandi, 17-Aug-2020.)
|
| ⊢ ((𝐴 ≼ ω ∧ 𝐴 ≠ ∅) →
∃𝑓 𝑓:ℕ–onto→𝐴) |
| |
| Theorem | ssinss1d 42566 |
Intersection preserves subclass relationship. (Contributed by Glauco
Siliprandi, 17-Aug-2020.)
|
| ⊢ (𝜑 → 𝐴 ⊆ 𝐶) ⇒ ⊢ (𝜑 → (𝐴 ∩ 𝐵) ⊆ 𝐶) |
| |
| Theorem | elpwinss 42567 |
An element of the powerset of 𝐵 intersected with anything, is a
subset
of 𝐵. (Contributed by Glauco Siliprandi,
17-Aug-2020.)
|
| ⊢ (𝐴 ∈ (𝒫 𝐵 ∩ 𝐶) → 𝐴 ⊆ 𝐵) |
| |
| Theorem | unidmex 42568 |
If 𝐹 is a set, then ∪ dom 𝐹 is a set. (Contributed by Glauco
Siliprandi, 17-Aug-2020.)
|
| ⊢ (𝜑 → 𝐹 ∈ 𝑉)
& ⊢ 𝑋 = ∪ dom 𝐹 ⇒ ⊢ (𝜑 → 𝑋 ∈ V) |
| |
| Theorem | ndisj2 42569* |
A non-disjointness condition. (Contributed by Glauco Siliprandi,
17-Aug-2020.)
|
| ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) ⇒ ⊢ (¬ Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 (𝑥 ≠ 𝑦 ∧ (𝐵 ∩ 𝐶) ≠ ∅)) |
| |
| Theorem | zenom 42570 |
The set of integer numbers is equinumerous to omega (the set of finite
ordinal numbers). (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|
| ⊢ ℤ
≈ ω |
| |
| Theorem | uzwo4 42571* |
Well-ordering principle: any nonempty subset of an upper set of integers
has the least element. (Contributed by Glauco Siliprandi,
17-Aug-2020.)
|
| ⊢ Ⅎ𝑗𝜓
& ⊢ (𝑗 = 𝑘 → (𝜑 ↔ 𝜓)) ⇒ ⊢ ((𝑆 ⊆
(ℤ≥‘𝑀) ∧ ∃𝑗 ∈ 𝑆 𝜑) → ∃𝑗 ∈ 𝑆 (𝜑 ∧ ∀𝑘 ∈ 𝑆 (𝑘 < 𝑗 → ¬ 𝜓))) |
| |
| Theorem | unisn0 42572 |
The union of the singleton of the empty set is the empty set.
(Contributed by Glauco Siliprandi, 17-Aug-2020.)
|
| ⊢ ∪ {∅} = ∅ |
| |
| Theorem | ssin0 42573 |
If two classes are disjoint, two respective subclasses are disjoint.
(Contributed by Glauco Siliprandi, 17-Aug-2020.)
|
| ⊢ (((𝐴 ∩ 𝐵) = ∅ ∧ 𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐵) → (𝐶 ∩ 𝐷) = ∅) |
| |
| Theorem | inabs3 42574 |
Absorption law for intersection. (Contributed by Glauco Siliprandi,
17-Aug-2020.)
|
| ⊢ (𝐶 ⊆ 𝐵 → ((𝐴 ∩ 𝐵) ∩ 𝐶) = (𝐴 ∩ 𝐶)) |
| |
| Theorem | pwpwuni 42575 |
Relationship between power class and union. (Contributed by Glauco
Siliprandi, 17-Aug-2020.)
|
| ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝒫 𝒫 𝐵 ↔ ∪ 𝐴
∈ 𝒫 𝐵)) |
| |
| Theorem | disjiun2 42576* |
In a disjoint collection, an indexed union is disjoint from an
additional term. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|
| ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 𝐵)
& ⊢ (𝜑 → 𝐶 ⊆ 𝐴)
& ⊢ (𝜑 → 𝐷 ∈ (𝐴 ∖ 𝐶)) & ⊢ (𝑥 = 𝐷 → 𝐵 = 𝐸) ⇒ ⊢ (𝜑 → (∪ 𝑥 ∈ 𝐶 𝐵 ∩ 𝐸) = ∅) |
| |
| Theorem | 0pwfi 42577 |
The empty set is in any power set, and it's finite. (Contributed by
Glauco Siliprandi, 17-Aug-2020.)
|
| ⊢ ∅ ∈
(𝒫 𝐴 ∩
Fin) |
| |
| Theorem | ssinss2d 42578 |
Intersection preserves subclass relationship. (Contributed by Glauco
Siliprandi, 17-Aug-2020.)
|
| ⊢ (𝜑 → 𝐵 ⊆ 𝐶) ⇒ ⊢ (𝜑 → (𝐴 ∩ 𝐵) ⊆ 𝐶) |
| |
| Theorem | zct 42579 |
The set of integer numbers is countable. (Contributed by Glauco
Siliprandi, 17-Aug-2020.)
|
| ⊢ ℤ
≼ ω |
| |
| Theorem | pwfin0 42580 |
A finite set always belongs to a power class. (Contributed by Glauco
Siliprandi, 17-Aug-2020.)
|
| ⊢ (𝒫
𝐴 ∩ Fin) ≠
∅ |
| |
| Theorem | uzct 42581 |
An upper integer set is countable. (Contributed by Glauco Siliprandi,
17-Aug-2020.)
|
| ⊢ 𝑍 =
(ℤ≥‘𝑁) ⇒ ⊢ 𝑍 ≼ ω |
| |
| Theorem | iunxsnf 42582* |
A singleton index picks out an instance of an indexed union's argument.
(Contributed by Glauco Siliprandi, 17-Aug-2020.)
|
| ⊢
Ⅎ𝑥𝐶
& ⊢ 𝐴 ∈ V & ⊢ (𝑥 = 𝐴 → 𝐵 = 𝐶) ⇒ ⊢ ∪ 𝑥 ∈ {𝐴}𝐵 = 𝐶 |
| |
| Theorem | fiiuncl 42583* |
If a set is closed under the union of two sets, then it is closed under
finite indexed union. (Contributed by Glauco Siliprandi,
17-Aug-2020.)
|
| ⊢ Ⅎ𝑥𝜑
& ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐷)
& ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷 ∧ 𝑧 ∈ 𝐷) → (𝑦 ∪ 𝑧) ∈ 𝐷)
& ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐴 ≠ ∅)
⇒ ⊢ (𝜑 → ∪
𝑥 ∈ 𝐴 𝐵 ∈ 𝐷) |
| |
| Theorem | iunp1 42584* |
The addition of the next set to a union indexed by a finite set of
sequential integers. (Contributed by Glauco Siliprandi,
17-Aug-2020.)
|
| ⊢
Ⅎ𝑘𝐵
& ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) & ⊢ (𝑘 = (𝑁 + 1) → 𝐴 = 𝐵) ⇒ ⊢ (𝜑 → ∪
𝑘 ∈ (𝑀...(𝑁 + 1))𝐴 = (∪
𝑘 ∈ (𝑀...𝑁)𝐴 ∪ 𝐵)) |
| |
| Theorem | fiunicl 42585* |
If a set is closed under the union of two sets, then it is closed under
finite union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
|
| ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 ∪ 𝑦) ∈ 𝐴)
& ⊢ (𝜑 → 𝐴 ∈ Fin) & ⊢ (𝜑 → 𝐴 ≠ ∅)
⇒ ⊢ (𝜑 → ∪ 𝐴 ∈ 𝐴) |
| |
| Theorem | ixpeq2d 42586 |
Equality theorem for infinite Cartesian product. (Contributed by Glauco
Siliprandi, 11-Oct-2020.)
|
| ⊢ Ⅎ𝑥𝜑
& ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → X𝑥 ∈ 𝐴 𝐵 = X𝑥 ∈ 𝐴 𝐶) |
| |
| Theorem | disjxp1 42587* |
The sets of a cartesian product are disjoint if the sets in the first
argument are disjoint. (Contributed by Glauco Siliprandi,
11-Oct-2020.)
|
| ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 𝐵) ⇒ ⊢ (𝜑 → Disj 𝑥 ∈ 𝐴 (𝐵 × 𝐶)) |
| |
| Theorem | disjsnxp 42588* |
The sets in the cartesian product of singletons with other sets, are
disjoint. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
|
| ⊢ Disj
𝑗 ∈ 𝐴 ({𝑗} × 𝐵) |
| |
| Theorem | eliind 42589* |
Membership in indexed intersection. (Contributed by Glauco Siliprandi,
24-Dec-2020.)
|
| ⊢ (𝜑 → 𝐴 ∈ ∩
𝑥 ∈ 𝐵 𝐶)
& ⊢ (𝜑 → 𝐾 ∈ 𝐵)
& ⊢ (𝑥 = 𝐾 → (𝐴 ∈ 𝐶 ↔ 𝐴 ∈ 𝐷)) ⇒ ⊢ (𝜑 → 𝐴 ∈ 𝐷) |
| |
| Theorem | rspcef 42590 |
Restricted existential specialization, using implicit substitution.
(Contributed by Glauco Siliprandi, 24-Dec-2020.)
|
| ⊢ Ⅎ𝑥𝜓
& ⊢ Ⅎ𝑥𝐴
& ⊢ Ⅎ𝑥𝐵
& ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜓) → ∃𝑥 ∈ 𝐵 𝜑) |
| |
| Theorem | inn0f 42591 |
A nonempty intersection. (Contributed by Glauco Siliprandi,
24-Dec-2020.)
|
| ⊢
Ⅎ𝑥𝐴
& ⊢ Ⅎ𝑥𝐵 ⇒ ⊢ ((𝐴 ∩ 𝐵) ≠ ∅ ↔ ∃𝑥 ∈ 𝐴 𝑥 ∈ 𝐵) |
| |
| Theorem | ixpssmapc 42592* |
An infinite Cartesian product is a subset of set exponentiation.
(Contributed by Glauco Siliprandi, 24-Dec-2020.)
|
| ⊢ Ⅎ𝑥𝜑
& ⊢ (𝜑 → 𝐶 ∈ 𝑉)
& ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ⊆ 𝐶) ⇒ ⊢ (𝜑 → X𝑥 ∈ 𝐴 𝐵 ⊆ (𝐶 ↑m 𝐴)) |
| |
| Theorem | inn0 42593* |
A nonempty intersection. (Contributed by Glauco Siliprandi,
24-Dec-2020.)
|
| ⊢ ((𝐴 ∩ 𝐵) ≠ ∅ ↔ ∃𝑥 ∈ 𝐴 𝑥 ∈ 𝐵) |
| |
| Theorem | elintd 42594* |
Membership in class intersection. (Contributed by Glauco Siliprandi,
3-Jan-2021.)
|
| ⊢ Ⅎ𝑥𝜑
& ⊢ (𝜑 → 𝐴 ∈ 𝑉)
& ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐴 ∈ 𝑥) ⇒ ⊢ (𝜑 → 𝐴 ∈ ∩ 𝐵) |
| |
| Theorem | ssdf 42595* |
A sufficient condition for a subclass relationship. (Contributed by
Glauco Siliprandi, 3-Jan-2021.)
|
| ⊢ Ⅎ𝑥𝜑
& ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐵) ⇒ ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
| |
| Theorem | brneqtrd 42596 |
Substitution of equal classes into the negation of a binary relation.
(Contributed by Glauco Siliprandi, 3-Jan-2021.)
|
| ⊢ (𝜑 → ¬ 𝐴𝑅𝐵)
& ⊢ (𝜑 → 𝐵 = 𝐶) ⇒ ⊢ (𝜑 → ¬ 𝐴𝑅𝐶) |
| |
| Theorem | ssnct 42597 |
A set containing an uncountable set is itself uncountable. (Contributed
by Glauco Siliprandi, 3-Jan-2021.)
|
| ⊢ (𝜑 → ¬ 𝐴 ≼ ω) & ⊢ (𝜑 → 𝐴 ⊆ 𝐵) ⇒ ⊢ (𝜑 → ¬ 𝐵 ≼ ω) |
| |
| Theorem | ssuniint 42598* |
Sufficient condition for being a subclass of the union of an
intersection. (Contributed by Glauco Siliprandi, 3-Jan-2021.)
|
| ⊢ Ⅎ𝑥𝜑
& ⊢ (𝜑 → 𝐴 ∈ 𝑉)
& ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐴 ∈ 𝑥) ⇒ ⊢ (𝜑 → 𝐴 ⊆ ∪ ∩ 𝐵) |
| |
| Theorem | elintdv 42599* |
Membership in class intersection. (Contributed by Glauco Siliprandi,
3-Jan-2021.)
|
| ⊢ (𝜑 → 𝐴 ∈ 𝑉)
& ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐴 ∈ 𝑥) ⇒ ⊢ (𝜑 → 𝐴 ∈ ∩ 𝐵) |
| |
| Theorem | ssd 42600* |
A sufficient condition for a subclass relationship. (Contributed by
Glauco Siliprandi, 3-Jan-2021.)
|
| ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐵) ⇒ ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
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