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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | omeu 8201* | The division algorithm for ordinal multiplication. (Contributed by Mario Carneiro, 28-Feb-2013.) |
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ≠ ∅) → ∃!𝑧∃𝑥 ∈ On ∃𝑦 ∈ 𝐴 (𝑧 = 〈𝑥, 𝑦〉 ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝐵)) | ||
Theorem | oen0 8202 | Ordinal exponentiation with a nonzero mantissa is nonzero. Proposition 8.32 of [TakeutiZaring] p. 67. (Contributed by NM, 4-Jan-2005.) |
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐴) → ∅ ∈ (𝐴 ↑o 𝐵)) | ||
Theorem | oeordi 8203 | Ordering law for ordinal exponentiation. Proposition 8.33 of [TakeutiZaring] p. 67. (Contributed by NM, 5-Jan-2005.) (Revised by Mario Carneiro, 24-May-2015.) |
⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐴 ∈ 𝐵 → (𝐶 ↑o 𝐴) ∈ (𝐶 ↑o 𝐵))) | ||
Theorem | oeord 8204 | Ordering property of ordinal exponentiation. Corollary 8.34 of [TakeutiZaring] p. 68 and its converse. (Contributed by NM, 6-Jan-2005.) (Revised by Mario Carneiro, 24-May-2015.) |
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐴 ∈ 𝐵 ↔ (𝐶 ↑o 𝐴) ∈ (𝐶 ↑o 𝐵))) | ||
Theorem | oecan 8205 | Left cancellation law for ordinal exponentiation. (Contributed by NM, 6-Jan-2005.) (Revised by Mario Carneiro, 24-May-2015.) |
⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ↑o 𝐵) = (𝐴 ↑o 𝐶) ↔ 𝐵 = 𝐶)) | ||
Theorem | oeword 8206 | Weak ordering property of ordinal exponentiation. (Contributed by NM, 6-Jan-2005.) (Revised by Mario Carneiro, 24-May-2015.) |
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ (On ∖ 2o)) → (𝐴 ⊆ 𝐵 ↔ (𝐶 ↑o 𝐴) ⊆ (𝐶 ↑o 𝐵))) | ||
Theorem | oewordi 8207 | Weak ordering property of ordinal exponentiation. (Contributed by NM, 6-Jan-2005.) |
⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴 ⊆ 𝐵 → (𝐶 ↑o 𝐴) ⊆ (𝐶 ↑o 𝐵))) | ||
Theorem | oewordri 8208 | Weak ordering property of ordinal exponentiation. Proposition 8.35 of [TakeutiZaring] p. 68. (Contributed by NM, 6-Jan-2005.) |
⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐵 → (𝐴 ↑o 𝐶) ⊆ (𝐵 ↑o 𝐶))) | ||
Theorem | oeworde 8209 | Ordinal exponentiation compared to its exponent. Proposition 8.37 of [TakeutiZaring] p. 68. (Contributed by NM, 7-Jan-2005.) (Revised by Mario Carneiro, 24-May-2015.) |
⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ On) → 𝐵 ⊆ (𝐴 ↑o 𝐵)) | ||
Theorem | oeordsuc 8210 | Ordering property of ordinal exponentiation with a successor exponent. Corollary 8.36 of [TakeutiZaring] p. 68. (Contributed by NM, 7-Jan-2005.) |
⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐵 → (𝐴 ↑o suc 𝐶) ∈ (𝐵 ↑o suc 𝐶))) | ||
Theorem | oelim2 8211* | Ordinal exponentiation with a limit exponent. Part of Exercise 4.36 of [Mendelson] p. 250. (Contributed by NM, 6-Jan-2005.) |
⊢ ((𝐴 ∈ On ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → (𝐴 ↑o 𝐵) = ∪ 𝑥 ∈ (𝐵 ∖ 1o)(𝐴 ↑o 𝑥)) | ||
Theorem | oeoalem 8212 | Lemma for oeoa 8213. (Contributed by Eric Schmidt, 26-May-2009.) |
⊢ 𝐴 ∈ On & ⊢ ∅ ∈ 𝐴 & ⊢ 𝐵 ∈ On ⇒ ⊢ (𝐶 ∈ On → (𝐴 ↑o (𝐵 +o 𝐶)) = ((𝐴 ↑o 𝐵) ·o (𝐴 ↑o 𝐶))) | ||
Theorem | oeoa 8213 | Sum of exponents law for ordinal exponentiation. Theorem 8R of [Enderton] p. 238. Also Proposition 8.41 of [TakeutiZaring] p. 69. (Contributed by Eric Schmidt, 26-May-2009.) |
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ↑o (𝐵 +o 𝐶)) = ((𝐴 ↑o 𝐵) ·o (𝐴 ↑o 𝐶))) | ||
Theorem | oeoelem 8214 | Lemma for oeoe 8215. (Contributed by Eric Schmidt, 26-May-2009.) |
⊢ 𝐴 ∈ On & ⊢ ∅ ∈ 𝐴 ⇒ ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ↑o 𝐵) ↑o 𝐶) = (𝐴 ↑o (𝐵 ·o 𝐶))) | ||
Theorem | oeoe 8215 | Product of exponents law for ordinal exponentiation. Theorem 8S of [Enderton] p. 238. Also Proposition 8.42 of [TakeutiZaring] p. 70. (Contributed by Eric Schmidt, 26-May-2009.) |
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ↑o 𝐵) ↑o 𝐶) = (𝐴 ↑o (𝐵 ·o 𝐶))) | ||
Theorem | oelimcl 8216 | The ordinal exponential with a limit ordinal is a limit ordinal. (Contributed by Mario Carneiro, 29-May-2015.) |
⊢ ((𝐴 ∈ (On ∖ 2o) ∧ (𝐵 ∈ 𝐶 ∧ Lim 𝐵)) → Lim (𝐴 ↑o 𝐵)) | ||
Theorem | oeeulem 8217* | Lemma for oeeu 8219. (Contributed by Mario Carneiro, 28-Feb-2013.) |
⊢ 𝑋 = ∪ ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)} ⇒ ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → (𝑋 ∈ On ∧ (𝐴 ↑o 𝑋) ⊆ 𝐵 ∧ 𝐵 ∈ (𝐴 ↑o suc 𝑋))) | ||
Theorem | oeeui 8218* | The division algorithm for ordinal exponentiation. (This version of oeeu 8219 gives an explicit expression for the unique solution of the equation, in terms of the solution 𝑃 to omeu 8201.) (Contributed by Mario Carneiro, 25-May-2015.) |
⊢ 𝑋 = ∪ ∩ {𝑥 ∈ On ∣ 𝐵 ∈ (𝐴 ↑o 𝑥)} & ⊢ 𝑃 = (℩𝑤∃𝑦 ∈ On ∃𝑧 ∈ (𝐴 ↑o 𝑋)(𝑤 = 〈𝑦, 𝑧〉 ∧ (((𝐴 ↑o 𝑋) ·o 𝑦) +o 𝑧) = 𝐵)) & ⊢ 𝑌 = (1st ‘𝑃) & ⊢ 𝑍 = (2nd ‘𝑃) ⇒ ⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → (((𝐶 ∈ On ∧ 𝐷 ∈ (𝐴 ∖ 1o) ∧ 𝐸 ∈ (𝐴 ↑o 𝐶)) ∧ (((𝐴 ↑o 𝐶) ·o 𝐷) +o 𝐸) = 𝐵) ↔ (𝐶 = 𝑋 ∧ 𝐷 = 𝑌 ∧ 𝐸 = 𝑍))) | ||
Theorem | oeeu 8219* | The division algorithm for ordinal exponentiation. (Contributed by Mario Carneiro, 25-May-2015.) |
⊢ ((𝐴 ∈ (On ∖ 2o) ∧ 𝐵 ∈ (On ∖ 1o)) → ∃!𝑤∃𝑥 ∈ On ∃𝑦 ∈ (𝐴 ∖ 1o)∃𝑧 ∈ (𝐴 ↑o 𝑥)(𝑤 = 〈𝑥, 𝑦, 𝑧〉 ∧ (((𝐴 ↑o 𝑥) ·o 𝑦) +o 𝑧) = 𝐵)) | ||
Theorem | nna0 8220 | Addition with zero. Theorem 4I(A1) of [Enderton] p. 79. (Contributed by NM, 20-Sep-1995.) |
⊢ (𝐴 ∈ ω → (𝐴 +o ∅) = 𝐴) | ||
Theorem | nnm0 8221 | Multiplication with zero. Theorem 4J(A1) of [Enderton] p. 80. (Contributed by NM, 20-Sep-1995.) |
⊢ (𝐴 ∈ ω → (𝐴 ·o ∅) = ∅) | ||
Theorem | nnasuc 8222 | Addition with successor. Theorem 4I(A2) of [Enderton] p. 79. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 14-Nov-2014.) |
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 +o suc 𝐵) = suc (𝐴 +o 𝐵)) | ||
Theorem | nnmsuc 8223 | Multiplication with successor. Theorem 4J(A2) of [Enderton] p. 80. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 14-Nov-2014.) |
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·o suc 𝐵) = ((𝐴 ·o 𝐵) +o 𝐴)) | ||
Theorem | nnesuc 8224 | Exponentiation with a successor exponent. Definition 8.30 of [TakeutiZaring] p. 67. (Contributed by Mario Carneiro, 14-Nov-2014.) |
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ↑o suc 𝐵) = ((𝐴 ↑o 𝐵) ·o 𝐴)) | ||
Theorem | nna0r 8225 | Addition to zero. Remark in proof of Theorem 4K(2) of [Enderton] p. 81. Note: In this and later theorems, we deliberately avoid the more general ordinal versions of these theorems (in this case oa0r 8154) so that we can avoid ax-rep 5182, which is not needed for finite recursive definitions. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 14-Nov-2014.) |
⊢ (𝐴 ∈ ω → (∅ +o 𝐴) = 𝐴) | ||
Theorem | nnm0r 8226 | Multiplication with zero. Exercise 16 of [Enderton] p. 82. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 15-Nov-2014.) |
⊢ (𝐴 ∈ ω → (∅ ·o 𝐴) = ∅) | ||
Theorem | nnacl 8227 | Closure of addition of natural numbers. Proposition 8.9 of [TakeutiZaring] p. 59. (Contributed by NM, 20-Sep-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 +o 𝐵) ∈ ω) | ||
Theorem | nnmcl 8228 | Closure of multiplication of natural numbers. Proposition 8.17 of [TakeutiZaring] p. 63. (Contributed by NM, 20-Sep-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·o 𝐵) ∈ ω) | ||
Theorem | nnecl 8229 | Closure of exponentiation of natural numbers. Proposition 8.17 of [TakeutiZaring] p. 63. (Contributed by NM, 24-Mar-2007.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ↑o 𝐵) ∈ ω) | ||
Theorem | nnacli 8230 | ω is closed under addition. Inference form of nnacl 8227. (Contributed by Scott Fenton, 20-Apr-2012.) |
⊢ 𝐴 ∈ ω & ⊢ 𝐵 ∈ ω ⇒ ⊢ (𝐴 +o 𝐵) ∈ ω | ||
Theorem | nnmcli 8231 | ω is closed under multiplication. Inference form of nnmcl 8228. (Contributed by Scott Fenton, 20-Apr-2012.) |
⊢ 𝐴 ∈ ω & ⊢ 𝐵 ∈ ω ⇒ ⊢ (𝐴 ·o 𝐵) ∈ ω | ||
Theorem | nnarcl 8232 | Reverse closure law for addition of natural numbers. Exercise 1 of [TakeutiZaring] p. 62 and its converse. (Contributed by NM, 12-Dec-2004.) |
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 +o 𝐵) ∈ ω ↔ (𝐴 ∈ ω ∧ 𝐵 ∈ ω))) | ||
Theorem | nnacom 8233 | Addition of natural numbers is commutative. Theorem 4K(2) of [Enderton] p. 81. (Contributed by NM, 6-May-1995.) (Revised by Mario Carneiro, 15-Nov-2014.) |
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 +o 𝐵) = (𝐵 +o 𝐴)) | ||
Theorem | nnaordi 8234 | Ordering property of addition. Proposition 8.4 of [TakeutiZaring] p. 58, limited to natural numbers. (Contributed by NM, 3-Feb-1996.) (Revised by Mario Carneiro, 15-Nov-2014.) |
⊢ ((𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴 ∈ 𝐵 → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵))) | ||
Theorem | nnaord 8235 | Ordering property of addition. Proposition 8.4 of [TakeutiZaring] p. 58, limited to natural numbers, and its converse. (Contributed by NM, 7-Mar-1996.) (Revised by Mario Carneiro, 15-Nov-2014.) |
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴 ∈ 𝐵 ↔ (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵))) | ||
Theorem | nnaordr 8236 | Ordering property of addition of natural numbers. (Contributed by NM, 9-Nov-2002.) |
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴 ∈ 𝐵 ↔ (𝐴 +o 𝐶) ∈ (𝐵 +o 𝐶))) | ||
Theorem | nnawordi 8237 | Adding to both sides of an inequality in ω. (Contributed by Scott Fenton, 16-Apr-2012.) (Revised by Mario Carneiro, 12-May-2012.) |
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴 ⊆ 𝐵 → (𝐴 +o 𝐶) ⊆ (𝐵 +o 𝐶))) | ||
Theorem | nnaass 8238 | Addition of natural numbers is associative. Theorem 4K(1) of [Enderton] p. 81. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 15-Nov-2014.) |
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐴 +o 𝐵) +o 𝐶) = (𝐴 +o (𝐵 +o 𝐶))) | ||
Theorem | nndi 8239 | Distributive law for natural numbers (left-distributivity). Theorem 4K(3) of [Enderton] p. 81. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 15-Nov-2014.) |
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴 ·o (𝐵 +o 𝐶)) = ((𝐴 ·o 𝐵) +o (𝐴 ·o 𝐶))) | ||
Theorem | nnmass 8240 | Multiplication of natural numbers is associative. Theorem 4K(4) of [Enderton] p. 81. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 15-Nov-2014.) |
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐴 ·o 𝐵) ·o 𝐶) = (𝐴 ·o (𝐵 ·o 𝐶))) | ||
Theorem | nnmsucr 8241 | Multiplication with successor. Exercise 16 of [Enderton] p. 82. (Contributed by NM, 21-Sep-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc 𝐴 ·o 𝐵) = ((𝐴 ·o 𝐵) +o 𝐵)) | ||
Theorem | nnmcom 8242 | Multiplication of natural numbers is commutative. Theorem 4K(5) of [Enderton] p. 81. (Contributed by NM, 21-Sep-1995.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·o 𝐵) = (𝐵 ·o 𝐴)) | ||
Theorem | nnaword 8243 | Weak ordering property of addition. (Contributed by NM, 17-Sep-1995.) (Revised by Mario Carneiro, 15-Nov-2014.) |
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴 ⊆ 𝐵 ↔ (𝐶 +o 𝐴) ⊆ (𝐶 +o 𝐵))) | ||
Theorem | nnacan 8244 | Cancellation law for addition of natural numbers. (Contributed by NM, 27-Oct-1995.) (Revised by Mario Carneiro, 15-Nov-2014.) |
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐴 +o 𝐵) = (𝐴 +o 𝐶) ↔ 𝐵 = 𝐶)) | ||
Theorem | nnaword1 8245 | Weak ordering property of addition. (Contributed by NM, 9-Nov-2002.) (Revised by Mario Carneiro, 15-Nov-2014.) |
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → 𝐴 ⊆ (𝐴 +o 𝐵)) | ||
Theorem | nnaword2 8246 | Weak ordering property of addition. (Contributed by NM, 9-Nov-2002.) |
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → 𝐴 ⊆ (𝐵 +o 𝐴)) | ||
Theorem | nnmordi 8247 | Ordering property of multiplication. Half of Proposition 8.19 of [TakeutiZaring] p. 63, limited to natural numbers. (Contributed by NM, 18-Sep-1995.) (Revised by Mario Carneiro, 15-Nov-2014.) |
⊢ (((𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐴 ∈ 𝐵 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵))) | ||
Theorem | nnmord 8248 | Ordering property of multiplication. Proposition 8.19 of [TakeutiZaring] p. 63, limited to natural numbers. (Contributed by NM, 22-Jan-1996.) (Revised by Mario Carneiro, 15-Nov-2014.) |
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → ((𝐴 ∈ 𝐵 ∧ ∅ ∈ 𝐶) ↔ (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵))) | ||
Theorem | nnmword 8249 | Weak ordering property of ordinal multiplication. (Contributed by Mario Carneiro, 17-Nov-2014.) |
⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐶) → (𝐴 ⊆ 𝐵 ↔ (𝐶 ·o 𝐴) ⊆ (𝐶 ·o 𝐵))) | ||
Theorem | nnmcan 8250 | Cancellation law for multiplication of natural numbers. (Contributed by NM, 26-Oct-1995.) (Revised by Mario Carneiro, 15-Nov-2014.) |
⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) ∧ ∅ ∈ 𝐴) → ((𝐴 ·o 𝐵) = (𝐴 ·o 𝐶) ↔ 𝐵 = 𝐶)) | ||
Theorem | nnmwordi 8251 | Weak ordering property of multiplication. (Contributed by Mario Carneiro, 17-Nov-2014.) |
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴 ⊆ 𝐵 → (𝐶 ·o 𝐴) ⊆ (𝐶 ·o 𝐵))) | ||
Theorem | nnmwordri 8252 | Weak ordering property of ordinal multiplication. Proposition 8.21 of [TakeutiZaring] p. 63, limited to natural numbers. (Contributed by Mario Carneiro, 17-Nov-2014.) |
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 ∈ ω) → (𝐴 ⊆ 𝐵 → (𝐴 ·o 𝐶) ⊆ (𝐵 ·o 𝐶))) | ||
Theorem | nnawordex 8253* | Equivalence for weak ordering of natural numbers. (Contributed by NM, 8-Nov-2002.) (Revised by Mario Carneiro, 15-Nov-2014.) |
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 ↔ ∃𝑥 ∈ ω (𝐴 +o 𝑥) = 𝐵)) | ||
Theorem | nnaordex 8254* | Equivalence for ordering. Compare Exercise 23 of [Enderton] p. 88. (Contributed by NM, 5-Dec-1995.) (Revised by Mario Carneiro, 15-Nov-2014.) |
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∈ 𝐵 ↔ ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵))) | ||
Theorem | 1onn 8255 | One is a natural number. (Contributed by NM, 29-Oct-1995.) |
⊢ 1o ∈ ω | ||
Theorem | 2onn 8256 | The ordinal 2 is a natural number. (Contributed by NM, 28-Sep-2004.) |
⊢ 2o ∈ ω | ||
Theorem | 3onn 8257 | The ordinal 3 is a natural number. (Contributed by Mario Carneiro, 5-Jan-2016.) |
⊢ 3o ∈ ω | ||
Theorem | 4onn 8258 | The ordinal 4 is a natural number. (Contributed by Mario Carneiro, 5-Jan-2016.) |
⊢ 4o ∈ ω | ||
Theorem | 1one2o 8259 | Ordinal one is not ordinal two. Analogous to 1ne2 11834. (Contributed by AV, 17-Oct-2023.) |
⊢ 1o ≠ 2o | ||
Theorem | oaabslem 8260 | Lemma for oaabs 8261. (Contributed by NM, 9-Dec-2004.) |
⊢ ((ω ∈ On ∧ 𝐴 ∈ ω) → (𝐴 +o ω) = ω) | ||
Theorem | oaabs 8261 | Ordinal addition absorbs a natural number added to the left of a transfinite number. Proposition 8.10 of [TakeutiZaring] p. 59. (Contributed by NM, 9-Dec-2004.) (Proof shortened by Mario Carneiro, 29-May-2015.) |
⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ On) ∧ ω ⊆ 𝐵) → (𝐴 +o 𝐵) = 𝐵) | ||
Theorem | oaabs2 8262 | The absorption law oaabs 8261 is also a property of higher powers of ω. (Contributed by Mario Carneiro, 29-May-2015.) |
⊢ (((𝐴 ∈ (ω ↑o 𝐶) ∧ 𝐵 ∈ On) ∧ (ω ↑o 𝐶) ⊆ 𝐵) → (𝐴 +o 𝐵) = 𝐵) | ||
Theorem | omabslem 8263 | Lemma for omabs 8264. (Contributed by Mario Carneiro, 30-May-2015.) |
⊢ ((ω ∈ On ∧ 𝐴 ∈ ω ∧ ∅ ∈ 𝐴) → (𝐴 ·o ω) = ω) | ||
Theorem | omabs 8264 | Ordinal multiplication is also absorbed by powers of ω. (Contributed by Mario Carneiro, 30-May-2015.) |
⊢ (((𝐴 ∈ ω ∧ ∅ ∈ 𝐴) ∧ (𝐵 ∈ On ∧ ∅ ∈ 𝐵)) → (𝐴 ·o (ω ↑o 𝐵)) = (ω ↑o 𝐵)) | ||
Theorem | nnm1 8265 | Multiply an element of ω by 1o. (Contributed by Mario Carneiro, 17-Nov-2014.) |
⊢ (𝐴 ∈ ω → (𝐴 ·o 1o) = 𝐴) | ||
Theorem | nnm2 8266 | Multiply an element of ω by 2o. (Contributed by Scott Fenton, 18-Apr-2012.) (Revised by Mario Carneiro, 17-Nov-2014.) |
⊢ (𝐴 ∈ ω → (𝐴 ·o 2o) = (𝐴 +o 𝐴)) | ||
Theorem | nn2m 8267 | Multiply an element of ω by 2o. (Contributed by Scott Fenton, 16-Apr-2012.) (Revised by Mario Carneiro, 17-Nov-2014.) |
⊢ (𝐴 ∈ ω → (2o ·o 𝐴) = (𝐴 +o 𝐴)) | ||
Theorem | nnneo 8268 | If a natural number is even, its successor is odd. (Contributed by Mario Carneiro, 16-Nov-2014.) |
⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω ∧ 𝐶 = (2o ·o 𝐴)) → ¬ suc 𝐶 = (2o ·o 𝐵)) | ||
Theorem | nneob 8269* | A natural number is even iff its successor is odd. (Contributed by NM, 26-Jan-2006.) (Revised by Mario Carneiro, 15-Nov-2014.) |
⊢ (𝐴 ∈ ω → (∃𝑥 ∈ ω 𝐴 = (2o ·o 𝑥) ↔ ¬ ∃𝑥 ∈ ω suc 𝐴 = (2o ·o 𝑥))) | ||
Theorem | omsmolem 8270* | Lemma for omsmo 8271. (Contributed by NM, 30-Nov-2003.) (Revised by David Abernethy, 1-Jan-2014.) |
⊢ (𝑦 ∈ ω → (((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹‘𝑥) ∈ (𝐹‘suc 𝑥)) → (𝑧 ∈ 𝑦 → (𝐹‘𝑧) ∈ (𝐹‘𝑦)))) | ||
Theorem | omsmo 8271* | A strictly monotonic ordinal function on the set of natural numbers is one-to-one. (Contributed by NM, 30-Nov-2003.) (Revised by David Abernethy, 1-Jan-2014.) |
⊢ (((𝐴 ⊆ On ∧ 𝐹:ω⟶𝐴) ∧ ∀𝑥 ∈ ω (𝐹‘𝑥) ∈ (𝐹‘suc 𝑥)) → 𝐹:ω–1-1→𝐴) | ||
Theorem | omopthlem1 8272 | Lemma for omopthi 8274. (Contributed by Scott Fenton, 18-Apr-2012.) (Revised by Mario Carneiro, 17-Nov-2014.) |
⊢ 𝐴 ∈ ω & ⊢ 𝐶 ∈ ω ⇒ ⊢ (𝐴 ∈ 𝐶 → ((𝐴 ·o 𝐴) +o (𝐴 ·o 2o)) ∈ (𝐶 ·o 𝐶)) | ||
Theorem | omopthlem2 8273 | Lemma for omopthi 8274. (Contributed by Scott Fenton, 16-Apr-2012.) (Revised by Mario Carneiro, 17-Nov-2014.) |
⊢ 𝐴 ∈ ω & ⊢ 𝐵 ∈ ω & ⊢ 𝐶 ∈ ω & ⊢ 𝐷 ∈ ω ⇒ ⊢ ((𝐴 +o 𝐵) ∈ 𝐶 → ¬ ((𝐶 ·o 𝐶) +o 𝐷) = (((𝐴 +o 𝐵) ·o (𝐴 +o 𝐵)) +o 𝐵)) | ||
Theorem | omopthi 8274 | An ordered pair theorem for ω. Theorem 17.3 of [Quine] p. 124. This proof is adapted from nn0opthi 13620. (Contributed by Scott Fenton, 16-Apr-2012.) (Revised by Mario Carneiro, 17-Nov-2014.) |
⊢ 𝐴 ∈ ω & ⊢ 𝐵 ∈ ω & ⊢ 𝐶 ∈ ω & ⊢ 𝐷 ∈ ω ⇒ ⊢ ((((𝐴 +o 𝐵) ·o (𝐴 +o 𝐵)) +o 𝐵) = (((𝐶 +o 𝐷) ·o (𝐶 +o 𝐷)) +o 𝐷) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) | ||
Theorem | omopth 8275 | An ordered pair theorem for finite integers. Analogous to nn0opthi 13620. (Contributed by Scott Fenton, 1-May-2012.) |
⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐶 ∈ ω ∧ 𝐷 ∈ ω)) → ((((𝐴 +o 𝐵) ·o (𝐴 +o 𝐵)) +o 𝐵) = (((𝐶 +o 𝐷) ·o (𝐶 +o 𝐷)) +o 𝐷) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))) | ||
Syntax | wer 8276 | Extend the definition of a wff to include the equivalence predicate. |
wff 𝑅 Er 𝐴 | ||
Syntax | cec 8277 | Extend the definition of a class to include equivalence class. |
class [𝐴]𝑅 | ||
Syntax | cqs 8278 | Extend the definition of a class to include quotient set. |
class (𝐴 / 𝑅) | ||
Definition | df-er 8279 | Define the equivalence relation predicate. Our notation is not standard. A formal notation doesn't seem to exist in the literature; instead only informal English tends to be used. The present definition, although somewhat cryptic, nicely avoids dummy variables. In dfer2 8280 we derive a more typical definition. We show that an equivalence relation is reflexive, symmetric, and transitive in erref 8299, ersymb 8293, and ertr 8294. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 2-Nov-2015.) |
⊢ (𝑅 Er 𝐴 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ (◡𝑅 ∪ (𝑅 ∘ 𝑅)) ⊆ 𝑅)) | ||
Theorem | dfer2 8280* | Alternate definition of equivalence predicate. (Contributed by NM, 3-Jan-1997.) (Revised by Mario Carneiro, 12-Aug-2015.) |
⊢ (𝑅 Er 𝐴 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ ∀𝑥∀𝑦∀𝑧((𝑥𝑅𝑦 → 𝑦𝑅𝑥) ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))) | ||
Definition | df-ec 8281 | Define the 𝑅-coset of 𝐴. Exercise 35 of [Enderton] p. 61. This is called the equivalence class of 𝐴 modulo 𝑅 when 𝑅 is an equivalence relation (i.e. when Er 𝑅; see dfer2 8280). In this case, 𝐴 is a representative (member) of the equivalence class [𝐴]𝑅, which contains all sets that are equivalent to 𝐴. Definition of [Enderton] p. 57 uses the notation [𝐴] (subscript) 𝑅, although we simply follow the brackets by 𝑅 since we don't have subscripted expressions. For an alternate definition, see dfec2 8282. (Contributed by NM, 23-Jul-1995.) |
⊢ [𝐴]𝑅 = (𝑅 “ {𝐴}) | ||
Theorem | dfec2 8282* | Alternate definition of 𝑅-coset of 𝐴. Definition 34 of [Suppes] p. 81. (Contributed by NM, 3-Jan-1997.) (Proof shortened by Mario Carneiro, 9-Jul-2014.) |
⊢ (𝐴 ∈ 𝑉 → [𝐴]𝑅 = {𝑦 ∣ 𝐴𝑅𝑦}) | ||
Theorem | ecexg 8283 | An equivalence class modulo a set is a set. (Contributed by NM, 24-Jul-1995.) |
⊢ (𝑅 ∈ 𝐵 → [𝐴]𝑅 ∈ V) | ||
Theorem | ecexr 8284 | A nonempty equivalence class implies the representative is a set. (Contributed by Mario Carneiro, 9-Jul-2014.) |
⊢ (𝐴 ∈ [𝐵]𝑅 → 𝐵 ∈ V) | ||
Definition | df-qs 8285* | Define quotient set. 𝑅 is usually an equivalence relation. Definition of [Enderton] p. 58. (Contributed by NM, 23-Jul-1995.) |
⊢ (𝐴 / 𝑅) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝑅} | ||
Theorem | ereq1 8286 | Equality theorem for equivalence predicate. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 12-Aug-2015.) |
⊢ (𝑅 = 𝑆 → (𝑅 Er 𝐴 ↔ 𝑆 Er 𝐴)) | ||
Theorem | ereq2 8287 | Equality theorem for equivalence predicate. (Contributed by Mario Carneiro, 12-Aug-2015.) |
⊢ (𝐴 = 𝐵 → (𝑅 Er 𝐴 ↔ 𝑅 Er 𝐵)) | ||
Theorem | errel 8288 | An equivalence relation is a relation. (Contributed by Mario Carneiro, 12-Aug-2015.) |
⊢ (𝑅 Er 𝐴 → Rel 𝑅) | ||
Theorem | erdm 8289 | The domain of an equivalence relation. (Contributed by Mario Carneiro, 12-Aug-2015.) |
⊢ (𝑅 Er 𝐴 → dom 𝑅 = 𝐴) | ||
Theorem | ercl 8290 | Elementhood in the field of an equivalence relation. (Contributed by Mario Carneiro, 12-Aug-2015.) |
⊢ (𝜑 → 𝑅 Er 𝑋) & ⊢ (𝜑 → 𝐴𝑅𝐵) ⇒ ⊢ (𝜑 → 𝐴 ∈ 𝑋) | ||
Theorem | ersym 8291 | An equivalence relation is symmetric. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 12-Aug-2015.) |
⊢ (𝜑 → 𝑅 Er 𝑋) & ⊢ (𝜑 → 𝐴𝑅𝐵) ⇒ ⊢ (𝜑 → 𝐵𝑅𝐴) | ||
Theorem | ercl2 8292 | Elementhood in the field of an equivalence relation. (Contributed by Mario Carneiro, 12-Aug-2015.) |
⊢ (𝜑 → 𝑅 Er 𝑋) & ⊢ (𝜑 → 𝐴𝑅𝐵) ⇒ ⊢ (𝜑 → 𝐵 ∈ 𝑋) | ||
Theorem | ersymb 8293 | An equivalence relation is symmetric. (Contributed by NM, 30-Jul-1995.) (Revised by Mario Carneiro, 12-Aug-2015.) |
⊢ (𝜑 → 𝑅 Er 𝑋) ⇒ ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ 𝐵𝑅𝐴)) | ||
Theorem | ertr 8294 | An equivalence relation is transitive. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 12-Aug-2015.) |
⊢ (𝜑 → 𝑅 Er 𝑋) ⇒ ⊢ (𝜑 → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶)) | ||
Theorem | ertrd 8295 | A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.) |
⊢ (𝜑 → 𝑅 Er 𝑋) & ⊢ (𝜑 → 𝐴𝑅𝐵) & ⊢ (𝜑 → 𝐵𝑅𝐶) ⇒ ⊢ (𝜑 → 𝐴𝑅𝐶) | ||
Theorem | ertr2d 8296 | A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.) |
⊢ (𝜑 → 𝑅 Er 𝑋) & ⊢ (𝜑 → 𝐴𝑅𝐵) & ⊢ (𝜑 → 𝐵𝑅𝐶) ⇒ ⊢ (𝜑 → 𝐶𝑅𝐴) | ||
Theorem | ertr3d 8297 | A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.) |
⊢ (𝜑 → 𝑅 Er 𝑋) & ⊢ (𝜑 → 𝐵𝑅𝐴) & ⊢ (𝜑 → 𝐵𝑅𝐶) ⇒ ⊢ (𝜑 → 𝐴𝑅𝐶) | ||
Theorem | ertr4d 8298 | A transitivity relation for equivalences. (Contributed by Mario Carneiro, 9-Jul-2014.) |
⊢ (𝜑 → 𝑅 Er 𝑋) & ⊢ (𝜑 → 𝐴𝑅𝐵) & ⊢ (𝜑 → 𝐶𝑅𝐵) ⇒ ⊢ (𝜑 → 𝐴𝑅𝐶) | ||
Theorem | erref 8299 | An equivalence relation is reflexive on its field. Compare Theorem 3M of [Enderton] p. 56. (Contributed by Mario Carneiro, 6-May-2013.) (Revised by Mario Carneiro, 12-Aug-2015.) |
⊢ (𝜑 → 𝑅 Er 𝑋) & ⊢ (𝜑 → 𝐴 ∈ 𝑋) ⇒ ⊢ (𝜑 → 𝐴𝑅𝐴) | ||
Theorem | ercnv 8300 | The converse of an equivalence relation is itself. (Contributed by Mario Carneiro, 12-Aug-2015.) |
⊢ (𝑅 Er 𝐴 → ◡𝑅 = 𝑅) |
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