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Definition | df-qpa 32901* | Define the completion of the 𝑝-adic rationals. Here we simply define it as the splitting field of a dense sequence of polynomials (using as the 𝑛-th set the collection of polynomials with degree less than 𝑛 and with coefficients < (𝑝↑𝑛)). Krasner's lemma will then show that all monic polynomials have splitting fields isomorphic to a sufficiently close Eisenstein polynomial from the list, and unramified extensions are generated by the polynomial 𝑥↑(𝑝↑𝑛) − 𝑥, which is in the list. Thus, every finite extension of Qp is a subfield of this field extension, so it is algebraically closed. (Contributed by Mario Carneiro, 2-Dec-2014.) |
⊢ _Qp = (𝑝 ∈ ℙ ↦ ⦋(Qp‘𝑝) / 𝑟⦌(𝑟 polySplitLim (𝑛 ∈ ℕ ↦ {𝑓 ∈ (Poly1‘𝑟) ∣ ((𝑟 deg1 𝑓) ≤ 𝑛 ∧ ∀𝑑 ∈ ran (coe1‘𝑓)(◡𝑑 “ (ℤ ∖ {0})) ⊆ (0...𝑛))}))) | ||
Definition | df-cp 32902 | Define the metric completion of the algebraic completion of the 𝑝 -adic rationals. (Contributed by Mario Carneiro, 2-Dec-2014.) |
⊢ Cp = ( cplMetSp ∘ _Qp) | ||
I hope someone will enjoy solving (proving) the simple equations, inequalities, and calculations from this mathbox. I have proved these problems (theorems) using the Milpgame proof assistant. (It can be downloaded from https://us.metamath.org/other/milpgame/milpgame.html.) | ||
Theorem | problem1 32903 | Practice problem 1. Clues: 5p4e9 11789 3p2e5 11782 eqtri 2844 oveq1i 7160. (Contributed by Filip Cernatescu, 16-Mar-2019.) (Proof modification is discouraged.) |
⊢ ((3 + 2) + 4) = 9 | ||
Theorem | problem2 32904 | Practice problem 2. Clues: oveq12i 7162 adddiri 10648 add4i 10858 mulcli 10642 recni 10649 2re 11705 3eqtri 2848 10re 12111 5re 11718 1re 10635 4re 11715 eqcomi 2830 5p4e9 11789 oveq1i 7160 df-3 11695. (Contributed by Filip Cernatescu, 16-Mar-2019.) (Revised by AV, 9-Sep-2021.) (Proof modification is discouraged.) |
⊢ (((2 · ;10) + 5) + ((1 · ;10) + 4)) = ((3 · ;10) + 9) | ||
Theorem | problem3 32905 | Practice problem 3. Clues: eqcomi 2830 eqtri 2844 subaddrii 10969 recni 10649 4re 11715 3re 11711 1re 10635 df-4 11696 addcomi 10825. (Contributed by Filip Cernatescu, 16-Mar-2019.) (Proof modification is discouraged.) |
⊢ 𝐴 ∈ ℂ & ⊢ (𝐴 + 3) = 4 ⇒ ⊢ 𝐴 = 1 | ||
Theorem | problem4 32906 | Practice problem 4. Clues: pm3.2i 473 eqcomi 2830 eqtri 2844 subaddrii 10969 recni 10649 7re 11724 6re 11721 ax-1cn 10589 df-7 11699 ax-mp 5 oveq1i 7160 3cn 11712 2cn 11706 df-3 11695 mulid2i 10640 subdiri 11084 mp3an 1457 mulcli 10642 subadd23 10892 oveq2i 7161 oveq12i 7162 3t2e6 11797 mulcomi 10643 subcli 10956 biimpri 230 subadd2i 10968. (Contributed by Filip Cernatescu, 16-Mar-2019.) (Proof modification is discouraged.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ & ⊢ (𝐴 + 𝐵) = 3 & ⊢ ((3 · 𝐴) + (2 · 𝐵)) = 7 ⇒ ⊢ (𝐴 = 1 ∧ 𝐵 = 2) | ||
Theorem | problem5 32907 | Practice problem 5. Clues: 3brtr3i 5087 mpbi 232 breqtri 5083 ltaddsubi 11195 remulcli 10651 2re 11705 3re 11711 9re 11730 eqcomi 2830 mvlladdi 10898 3cn 6cn 11722 eqtr3i 2846 6p3e9 11791 addcomi 10825 ltdiv1ii 11563 6re 11721 nngt0i 11670 2nn 11704 divcan3i 11380 recni 10649 2cn 11706 2ne0 11735 mpbir 233 eqtri 2844 mulcomi 10643 3t2e6 11797 divmuli 11388. (Contributed by Filip Cernatescu, 16-Mar-2019.) (Proof modification is discouraged.) |
⊢ 𝐴 ∈ ℝ & ⊢ ((2 · 𝐴) + 3) < 9 ⇒ ⊢ 𝐴 < 3 | ||
Theorem | quad3 32908 | Variant of quadratic equation with discriminant expanded. (Contributed by Filip Cernatescu, 19-Oct-2019.) |
⊢ 𝑋 ∈ ℂ & ⊢ 𝐴 ∈ ℂ & ⊢ 𝐴 ≠ 0 & ⊢ 𝐵 ∈ ℂ & ⊢ 𝐶 ∈ ℂ & ⊢ ((𝐴 · (𝑋↑2)) + ((𝐵 · 𝑋) + 𝐶)) = 0 ⇒ ⊢ (𝑋 = ((-𝐵 + (√‘((𝐵↑2) − (4 · (𝐴 · 𝐶))))) / (2 · 𝐴)) ∨ 𝑋 = ((-𝐵 − (√‘((𝐵↑2) − (4 · (𝐴 · 𝐶))))) / (2 · 𝐴))) | ||
Theorem | climuzcnv 32909* | Utility lemma to convert between 𝑚 ≤ 𝑘 and 𝑘 ∈ (ℤ≥‘𝑚) in limit theorems. (Contributed by Paul Chapman, 10-Nov-2012.) |
⊢ (𝑚 ∈ ℕ → ((𝑘 ∈ (ℤ≥‘𝑚) → 𝜑) ↔ (𝑘 ∈ ℕ → (𝑚 ≤ 𝑘 → 𝜑)))) | ||
Theorem | sinccvglem 32910* | ((sin‘𝑥) / 𝑥) ⇝ 1 as (real) 𝑥 ⇝ 0. (Contributed by Paul Chapman, 10-Nov-2012.) (Revised by Mario Carneiro, 21-May-2014.) |
⊢ (𝜑 → 𝐹:ℕ⟶(ℝ ∖ {0})) & ⊢ (𝜑 → 𝐹 ⇝ 0) & ⊢ 𝐺 = (𝑥 ∈ (ℝ ∖ {0}) ↦ ((sin‘𝑥) / 𝑥)) & ⊢ 𝐻 = (𝑥 ∈ ℂ ↦ (1 − ((𝑥↑2) / 3))) & ⊢ (𝜑 → 𝑀 ∈ ℕ) & ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (abs‘(𝐹‘𝑘)) < 1) ⇒ ⊢ (𝜑 → (𝐺 ∘ 𝐹) ⇝ 1) | ||
Theorem | sinccvg 32911* | ((sin‘𝑥) / 𝑥) ⇝ 1 as (real) 𝑥 ⇝ 0. (Contributed by Paul Chapman, 10-Nov-2012.) (Proof shortened by Mario Carneiro, 21-May-2014.) |
⊢ ((𝐹:ℕ⟶(ℝ ∖ {0}) ∧ 𝐹 ⇝ 0) → ((𝑥 ∈ (ℝ ∖ {0}) ↦ ((sin‘𝑥) / 𝑥)) ∘ 𝐹) ⇝ 1) | ||
Theorem | circum 32912* | The circumference of a circle of radius 𝑅, defined as the limit as 𝑛 ⇝ +∞ of the perimeter of an inscribed n-sided isogons, is ((2 · π) · 𝑅). (Contributed by Paul Chapman, 10-Nov-2012.) (Proof shortened by Mario Carneiro, 21-May-2014.) |
⊢ 𝐴 = ((2 · π) / 𝑛) & ⊢ 𝑃 = (𝑛 ∈ ℕ ↦ ((2 · 𝑛) · (𝑅 · (sin‘(𝐴 / 2))))) & ⊢ 𝑅 ∈ ℝ ⇒ ⊢ 𝑃 ⇝ ((2 · π) · 𝑅) | ||
Theorem | elfzm12 32913 | Membership in a curtailed finite sequence of integers. (Contributed by Paul Chapman, 17-Nov-2012.) |
⊢ (𝑁 ∈ ℕ → (𝑀 ∈ (1...(𝑁 − 1)) → 𝑀 ∈ (1...𝑁))) | ||
Theorem | nn0seqcvg 32914* | A strictly-decreasing nonnegative integer sequence with initial term 𝑁 reaches zero by the 𝑁 th term. Inference version. (Contributed by Paul Chapman, 31-Mar-2011.) |
⊢ 𝐹:ℕ0⟶ℕ0 & ⊢ 𝑁 = (𝐹‘0) & ⊢ (𝑘 ∈ ℕ0 → ((𝐹‘(𝑘 + 1)) ≠ 0 → (𝐹‘(𝑘 + 1)) < (𝐹‘𝑘))) ⇒ ⊢ (𝐹‘𝑁) = 0 | ||
Theorem | lediv2aALT 32915 | Division of both sides of 'less than or equal to' by a nonnegative number. (Contributed by Paul Chapman, 7-Sep-2007.) (New usage is discouraged.) (Proof modification is discouraged.) |
⊢ (((𝐴 ∈ ℝ ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶)) → (𝐴 ≤ 𝐵 → (𝐶 / 𝐵) ≤ (𝐶 / 𝐴))) | ||
Theorem | abs2sqlei 32916 | The absolute values of two numbers compare as their squares. (Contributed by Paul Chapman, 7-Sep-2007.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ ⇒ ⊢ ((abs‘𝐴) ≤ (abs‘𝐵) ↔ ((abs‘𝐴)↑2) ≤ ((abs‘𝐵)↑2)) | ||
Theorem | abs2sqlti 32917 | The absolute values of two numbers compare as their squares. (Contributed by Paul Chapman, 7-Sep-2007.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ ⇒ ⊢ ((abs‘𝐴) < (abs‘𝐵) ↔ ((abs‘𝐴)↑2) < ((abs‘𝐵)↑2)) | ||
Theorem | abs2sqle 32918 | The absolute values of two numbers compare as their squares. (Contributed by Paul Chapman, 7-Sep-2007.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((abs‘𝐴) ≤ (abs‘𝐵) ↔ ((abs‘𝐴)↑2) ≤ ((abs‘𝐵)↑2))) | ||
Theorem | abs2sqlt 32919 | The absolute values of two numbers compare as their squares. (Contributed by Paul Chapman, 7-Sep-2007.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((abs‘𝐴) < (abs‘𝐵) ↔ ((abs‘𝐴)↑2) < ((abs‘𝐵)↑2))) | ||
Theorem | abs2difi 32920 | Difference of absolute values. (Contributed by Paul Chapman, 7-Sep-2007.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ ⇒ ⊢ ((abs‘𝐴) − (abs‘𝐵)) ≤ (abs‘(𝐴 − 𝐵)) | ||
Theorem | abs2difabsi 32921 | Absolute value of difference of absolute values. (Contributed by Paul Chapman, 7-Sep-2007.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ ⇒ ⊢ (abs‘((abs‘𝐴) − (abs‘𝐵))) ≤ (abs‘(𝐴 − 𝐵)) | ||
Theorem | axextprim 32922 | ax-ext 2793 without distinct variable conditions or defined symbols. (Contributed by Scott Fenton, 13-Oct-2010.) |
⊢ ¬ ∀𝑥 ¬ ((𝑥 ∈ 𝑦 → 𝑥 ∈ 𝑧) → ((𝑥 ∈ 𝑧 → 𝑥 ∈ 𝑦) → 𝑦 = 𝑧)) | ||
Theorem | axrepprim 32923 | ax-rep 5182 without distinct variable conditions or defined symbols. (Contributed by Scott Fenton, 13-Oct-2010.) |
⊢ ¬ ∀𝑥 ¬ (¬ ∀𝑦 ¬ ∀𝑧(𝜑 → 𝑧 = 𝑦) → ∀𝑧 ¬ ((∀𝑦 𝑧 ∈ 𝑥 → ¬ ∀𝑥(∀𝑧 𝑥 ∈ 𝑦 → ¬ ∀𝑦𝜑)) → ¬ (¬ ∀𝑥(∀𝑧 𝑥 ∈ 𝑦 → ¬ ∀𝑦𝜑) → ∀𝑦 𝑧 ∈ 𝑥))) | ||
Theorem | axunprim 32924 | ax-un 7455 without distinct variable conditions or defined symbols. (Contributed by Scott Fenton, 13-Oct-2010.) |
⊢ ¬ ∀𝑥 ¬ ∀𝑦(¬ ∀𝑥(𝑦 ∈ 𝑥 → ¬ 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥) | ||
Theorem | axpowprim 32925 | ax-pow 5258 without distinct variable conditions or defined symbols. (Contributed by Scott Fenton, 13-Oct-2010.) |
⊢ (∀𝑥 ¬ ∀𝑦(∀𝑥(¬ ∀𝑧 ¬ 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥) → 𝑥 = 𝑦) | ||
Theorem | axregprim 32926 | ax-reg 9050 without distinct variable conditions or defined symbols. (Contributed by Scott Fenton, 13-Oct-2010.) |
⊢ (𝑥 ∈ 𝑦 → ¬ ∀𝑥(𝑥 ∈ 𝑦 → ¬ ∀𝑧(𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦))) | ||
Theorem | axinfprim 32927 | ax-inf 9095 without distinct variable conditions or defined symbols. (New usage is discouraged.) (Contributed by Scott Fenton, 13-Oct-2010.) |
⊢ ¬ ∀𝑥 ¬ (𝑦 ∈ 𝑧 → ¬ (𝑦 ∈ 𝑥 → ¬ ∀𝑦(𝑦 ∈ 𝑥 → ¬ ∀𝑧(𝑦 ∈ 𝑧 → ¬ 𝑧 ∈ 𝑥)))) | ||
Theorem | axacprim 32928 | ax-ac 9875 without distinct variable conditions or defined symbols. (New usage is discouraged.) (Contributed by Scott Fenton, 26-Oct-2010.) |
⊢ ¬ ∀𝑥 ¬ ∀𝑦∀𝑧(∀𝑥 ¬ (𝑦 ∈ 𝑧 → ¬ 𝑧 ∈ 𝑤) → ¬ ∀𝑤 ¬ ∀𝑦 ¬ ((¬ ∀𝑤(𝑦 ∈ 𝑧 → (𝑧 ∈ 𝑤 → (𝑦 ∈ 𝑤 → ¬ 𝑤 ∈ 𝑥))) → 𝑦 = 𝑤) → ¬ (𝑦 = 𝑤 → ¬ ∀𝑤(𝑦 ∈ 𝑧 → (𝑧 ∈ 𝑤 → (𝑦 ∈ 𝑤 → ¬ 𝑤 ∈ 𝑥)))))) | ||
Theorem | untelirr 32929* | We call a class "untanged" if all its members are not members of themselves. The term originates from Isbell (see citation in dfon2 33032). Using this concept, we can avoid a lot of the uses of the Axiom of Regularity. Here, we prove a series of properties of untanged classes. First, we prove that an untangled class is not a member of itself. (Contributed by Scott Fenton, 28-Feb-2011.) |
⊢ (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥 → ¬ 𝐴 ∈ 𝐴) | ||
Theorem | untuni 32930* | The union of a class is untangled iff all its members are untangled. (Contributed by Scott Fenton, 28-Feb-2011.) |
⊢ (∀𝑥 ∈ ∪ 𝐴 ¬ 𝑥 ∈ 𝑥 ↔ ∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝑦 ¬ 𝑥 ∈ 𝑥) | ||
Theorem | untsucf 32931* | If a class is untangled, then so is its successor. (Contributed by Scott Fenton, 28-Feb-2011.) (Revised by Mario Carneiro, 11-Dec-2016.) |
⊢ Ⅎ𝑦𝐴 ⇒ ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥 → ∀𝑦 ∈ suc 𝐴 ¬ 𝑦 ∈ 𝑦) | ||
Theorem | unt0 32932 | The null set is untangled. (Contributed by Scott Fenton, 10-Mar-2011.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
⊢ ∀𝑥 ∈ ∅ ¬ 𝑥 ∈ 𝑥 | ||
Theorem | untint 32933* | If there is an untangled element of a class, then the intersection of the class is untangled. (Contributed by Scott Fenton, 1-Mar-2011.) |
⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝑦 → ∀𝑦 ∈ ∩ 𝐴 ¬ 𝑦 ∈ 𝑦) | ||
Theorem | efrunt 32934* | If 𝐴 is well-founded by E, then it is untangled. (Contributed by Scott Fenton, 1-Mar-2011.) |
⊢ ( E Fr 𝐴 → ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥) | ||
Theorem | untangtr 32935* | A transitive class is untangled iff its elements are. (Contributed by Scott Fenton, 7-Mar-2011.) |
⊢ (Tr 𝐴 → (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝑦)) | ||
Theorem | 3orel2 32936 | Partial elimination of a triple disjunction by denial of a disjunct. (Contributed by Scott Fenton, 26-Mar-2011.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
⊢ (¬ 𝜓 → ((𝜑 ∨ 𝜓 ∨ 𝜒) → (𝜑 ∨ 𝜒))) | ||
Theorem | 3orel3 32937 | Partial elimination of a triple disjunction by denial of a disjunct. (Contributed by Scott Fenton, 26-Mar-2011.) |
⊢ (¬ 𝜒 → ((𝜑 ∨ 𝜓 ∨ 𝜒) → (𝜑 ∨ 𝜓))) | ||
Theorem | 3pm3.2ni 32938 | Triple negated disjunction introduction. (Contributed by Scott Fenton, 20-Apr-2011.) |
⊢ ¬ 𝜑 & ⊢ ¬ 𝜓 & ⊢ ¬ 𝜒 ⇒ ⊢ ¬ (𝜑 ∨ 𝜓 ∨ 𝜒) | ||
Theorem | 3jaodd 32939 | Double deduction form of 3jaoi 1423. (Contributed by Scott Fenton, 20-Apr-2011.) |
⊢ (𝜑 → (𝜓 → (𝜒 → 𝜂))) & ⊢ (𝜑 → (𝜓 → (𝜃 → 𝜂))) & ⊢ (𝜑 → (𝜓 → (𝜏 → 𝜂))) ⇒ ⊢ (𝜑 → (𝜓 → ((𝜒 ∨ 𝜃 ∨ 𝜏) → 𝜂))) | ||
Theorem | 3orit 32940 | Closed form of 3ori 1420. (Contributed by Scott Fenton, 20-Apr-2011.) |
⊢ ((𝜑 ∨ 𝜓 ∨ 𝜒) ↔ ((¬ 𝜑 ∧ ¬ 𝜓) → 𝜒)) | ||
Theorem | biimpexp 32941 | A biconditional in the antecedent is the same as two implications. (Contributed by Scott Fenton, 12-Dec-2010.) |
⊢ (((𝜑 ↔ 𝜓) → 𝜒) ↔ ((𝜑 → 𝜓) → ((𝜓 → 𝜑) → 𝜒))) | ||
Theorem | 3orel13 32942 | Elimination of two disjuncts in a triple disjunction. (Contributed by Scott Fenton, 9-Jun-2011.) |
⊢ ((¬ 𝜑 ∧ ¬ 𝜒) → ((𝜑 ∨ 𝜓 ∨ 𝜒) → 𝜓)) | ||
Theorem | nepss 32943 | Two classes are unequal iff their intersection is a proper subset of one of them. (Contributed by Scott Fenton, 23-Feb-2011.) |
⊢ (𝐴 ≠ 𝐵 ↔ ((𝐴 ∩ 𝐵) ⊊ 𝐴 ∨ (𝐴 ∩ 𝐵) ⊊ 𝐵)) | ||
Theorem | 3ccased 32944 | Triple disjunction form of ccased 1033. (Contributed by Scott Fenton, 27-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ (𝜑 → ((𝜒 ∧ 𝜂) → 𝜓)) & ⊢ (𝜑 → ((𝜒 ∧ 𝜁) → 𝜓)) & ⊢ (𝜑 → ((𝜒 ∧ 𝜎) → 𝜓)) & ⊢ (𝜑 → ((𝜃 ∧ 𝜂) → 𝜓)) & ⊢ (𝜑 → ((𝜃 ∧ 𝜁) → 𝜓)) & ⊢ (𝜑 → ((𝜃 ∧ 𝜎) → 𝜓)) & ⊢ (𝜑 → ((𝜏 ∧ 𝜂) → 𝜓)) & ⊢ (𝜑 → ((𝜏 ∧ 𝜁) → 𝜓)) & ⊢ (𝜑 → ((𝜏 ∧ 𝜎) → 𝜓)) ⇒ ⊢ (𝜑 → (((𝜒 ∨ 𝜃 ∨ 𝜏) ∧ (𝜂 ∨ 𝜁 ∨ 𝜎)) → 𝜓)) | ||
Theorem | dfso3 32945* | Expansion of the definition of a strict order. (Contributed by Scott Fenton, 6-Jun-2016.) |
⊢ (𝑅 Or 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧) ∧ (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥))) | ||
Theorem | brtpid1 32946 | A binary relation involving unordered triplets. (Contributed by Scott Fenton, 7-Jun-2016.) |
⊢ 𝐴{〈𝐴, 𝐵〉, 𝐶, 𝐷}𝐵 | ||
Theorem | brtpid2 32947 | A binary relation involving unordered triplets. (Contributed by Scott Fenton, 7-Jun-2016.) |
⊢ 𝐴{𝐶, 〈𝐴, 𝐵〉, 𝐷}𝐵 | ||
Theorem | brtpid3 32948 | A binary relation involving unordered triplets. (Contributed by Scott Fenton, 7-Jun-2016.) |
⊢ 𝐴{𝐶, 𝐷, 〈𝐴, 𝐵〉}𝐵 | ||
Theorem | ceqsrexv2 32949* | Alternate elimitation of a restricted existential quantifier, using implicit substitution. (Contributed by Scott Fenton, 5-Sep-2017.) |
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∃𝑥 ∈ 𝐵 (𝑥 = 𝐴 ∧ 𝜑) ↔ (𝐴 ∈ 𝐵 ∧ 𝜓)) | ||
Theorem | iota5f 32950* | A method for computing iota. (Contributed by Scott Fenton, 13-Dec-2017.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝐴 & ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (𝜓 ↔ 𝑥 = 𝐴)) ⇒ ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (℩𝑥𝜓) = 𝐴) | ||
Theorem | ceqsralv2 32951* | Alternate elimination of a restricted universal quantifier, using implicit substitution. (Contributed by Scott Fenton, 7-Dec-2020.) |
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) ⇒ ⊢ (∀𝑥 ∈ 𝐵 (𝑥 = 𝐴 → 𝜑) ↔ (𝐴 ∈ 𝐵 → 𝜓)) | ||
Theorem | dford5 32952 | A class is ordinal iff it is a subclass of On and transitive. (Contributed by Scott Fenton, 21-Nov-2021.) |
⊢ (Ord 𝐴 ↔ (𝐴 ⊆ On ∧ Tr 𝐴)) | ||
Theorem | jath 32953 | Closed form of ja 188. Proved using the completeness script. (Proof modification is discouraged.) (Contributed by Scott Fenton, 13-Dec-2021.) |
⊢ ((¬ 𝜑 → 𝜒) → ((𝜓 → 𝜒) → ((𝜑 → 𝜓) → 𝜒))) | ||
Theorem | sqdivzi 32954 | Distribution of square over division. (Contributed by Scott Fenton, 7-Jun-2013.) |
⊢ 𝐴 ∈ ℂ & ⊢ 𝐵 ∈ ℂ ⇒ ⊢ (𝐵 ≠ 0 → ((𝐴 / 𝐵)↑2) = ((𝐴↑2) / (𝐵↑2))) | ||
Theorem | supfz 32955 | The supremum of a finite sequence of integers. (Contributed by Scott Fenton, 8-Aug-2013.) |
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → sup((𝑀...𝑁), ℤ, < ) = 𝑁) | ||
Theorem | inffz 32956 | The infimum of a finite sequence of integers. (Contributed by Scott Fenton, 8-Aug-2013.) (Revised by AV, 10-Oct-2021.) |
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → inf((𝑀...𝑁), ℤ, < ) = 𝑀) | ||
Theorem | fz0n 32957 | The sequence (0...(𝑁 − 1)) is empty iff 𝑁 is zero. (Contributed by Scott Fenton, 16-May-2014.) |
⊢ (𝑁 ∈ ℕ0 → ((0...(𝑁 − 1)) = ∅ ↔ 𝑁 = 0)) | ||
Theorem | shftvalg 32958 | Value of a sequence shifted by 𝐴. (Contributed by Scott Fenton, 16-Dec-2017.) |
⊢ ((𝐹 ∈ 𝑉 ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐹 shift 𝐴)‘𝐵) = (𝐹‘(𝐵 − 𝐴))) | ||
Theorem | divcnvlin 32959* | Limit of the ratio of two linear functions. (Contributed by Scott Fenton, 17-Dec-2017.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝐵 ∈ ℤ) & ⊢ (𝜑 → 𝐹 ∈ 𝑉) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = ((𝑘 + 𝐴) / (𝑘 + 𝐵))) ⇒ ⊢ (𝜑 → 𝐹 ⇝ 1) | ||
Theorem | climlec3 32960* | Comparison of a constant to the limit of a sequence. (Contributed by Scott Fenton, 5-Jan-2018.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐵 ∈ ℝ) & ⊢ (𝜑 → 𝐹 ⇝ 𝐴) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ≤ 𝐵) ⇒ ⊢ (𝜑 → 𝐴 ≤ 𝐵) | ||
Theorem | logi 32961 | Calculate the logarithm of i. (Contributed by Scott Fenton, 13-Apr-2020.) |
⊢ (log‘i) = (i · (π / 2)) | ||
Theorem | iexpire 32962 | i raised to itself is real. (Contributed by Scott Fenton, 13-Apr-2020.) |
⊢ (i↑𝑐i) ∈ ℝ | ||
Theorem | bcneg1 32963 | The binomial coefficent over negative one is zero. (Contributed by Scott Fenton, 29-May-2020.) |
⊢ (𝑁 ∈ ℕ0 → (𝑁C-1) = 0) | ||
Theorem | bcm1nt 32964 | The proportion of one bionmial coefficient to another with 𝑁 decreased by 1. (Contributed by Scott Fenton, 23-Jun-2020.) |
⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...(𝑁 − 1))) → (𝑁C𝐾) = (((𝑁 − 1)C𝐾) · (𝑁 / (𝑁 − 𝐾)))) | ||
Theorem | bcprod 32965* | A product identity for binomial coefficents. (Contributed by Scott Fenton, 23-Jun-2020.) |
⊢ (𝑁 ∈ ℕ → ∏𝑘 ∈ (1...(𝑁 − 1))((𝑁 − 1)C𝑘) = ∏𝑘 ∈ (1...(𝑁 − 1))(𝑘↑((2 · 𝑘) − 𝑁))) | ||
Theorem | bccolsum 32966* | A column-sum rule for binomial coefficents. (Contributed by Scott Fenton, 24-Jun-2020.) |
⊢ ((𝑁 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → Σ𝑘 ∈ (0...𝑁)(𝑘C𝐶) = ((𝑁 + 1)C(𝐶 + 1))) | ||
Theorem | iprodefisumlem 32967 | Lemma for iprodefisum 32968. (Contributed by Scott Fenton, 11-Feb-2018.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐹:𝑍⟶ℂ) ⇒ ⊢ (𝜑 → seq𝑀( · , (exp ∘ 𝐹)) = (exp ∘ seq𝑀( + , 𝐹))) | ||
Theorem | iprodefisum 32968* | Applying the exponential function to an infinite sum yields an infinite product. (Contributed by Scott Fenton, 11-Feb-2018.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐵) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℂ) & ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ ) ⇒ ⊢ (𝜑 → ∏𝑘 ∈ 𝑍 (exp‘𝐵) = (exp‘Σ𝑘 ∈ 𝑍 𝐵)) | ||
Theorem | iprodgam 32969* | An infinite product version of Euler's gamma function. (Contributed by Scott Fenton, 12-Feb-2018.) |
⊢ (𝜑 → 𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ))) ⇒ ⊢ (𝜑 → (Γ‘𝐴) = (∏𝑘 ∈ ℕ (((1 + (1 / 𝑘))↑𝑐𝐴) / (1 + (𝐴 / 𝑘))) / 𝐴)) | ||
Theorem | faclimlem1 32970* | Lemma for faclim 32973. Closed form for a particular sequence. (Contributed by Scott Fenton, 15-Dec-2017.) |
⊢ (𝑀 ∈ ℕ0 → seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (𝑀 / 𝑛)) · (1 + (1 / 𝑛))) / (1 + ((𝑀 + 1) / 𝑛))))) = (𝑥 ∈ ℕ ↦ ((𝑀 + 1) · ((𝑥 + 1) / (𝑥 + (𝑀 + 1)))))) | ||
Theorem | faclimlem2 32971* | Lemma for faclim 32973. Show a limit for the inductive step. (Contributed by Scott Fenton, 15-Dec-2017.) |
⊢ (𝑀 ∈ ℕ0 → seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (𝑀 / 𝑛)) · (1 + (1 / 𝑛))) / (1 + ((𝑀 + 1) / 𝑛))))) ⇝ (𝑀 + 1)) | ||
Theorem | faclimlem3 32972 | Lemma for faclim 32973. Algebraic manipulation for the final induction. (Contributed by Scott Fenton, 15-Dec-2017.) |
⊢ ((𝑀 ∈ ℕ0 ∧ 𝐵 ∈ ℕ) → (((1 + (1 / 𝐵))↑(𝑀 + 1)) / (1 + ((𝑀 + 1) / 𝐵))) = ((((1 + (1 / 𝐵))↑𝑀) / (1 + (𝑀 / 𝐵))) · (((1 + (𝑀 / 𝐵)) · (1 + (1 / 𝐵))) / (1 + ((𝑀 + 1) / 𝐵))))) | ||
Theorem | faclim 32973* | An infinite product expression relating to factorials. Originally due to Euler. (Contributed by Scott Fenton, 22-Nov-2017.) |
⊢ 𝐹 = (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝐴) / (1 + (𝐴 / 𝑛)))) ⇒ ⊢ (𝐴 ∈ ℕ0 → seq1( · , 𝐹) ⇝ (!‘𝐴)) | ||
Theorem | iprodfac 32974* | An infinite product expression for factorial. (Contributed by Scott Fenton, 15-Dec-2017.) |
⊢ (𝐴 ∈ ℕ0 → (!‘𝐴) = ∏𝑘 ∈ ℕ (((1 + (1 / 𝑘))↑𝐴) / (1 + (𝐴 / 𝑘)))) | ||
Theorem | faclim2 32975* | Another factorial limit due to Euler. (Contributed by Scott Fenton, 17-Dec-2017.) |
⊢ 𝐹 = (𝑛 ∈ ℕ ↦ (((!‘𝑛) · ((𝑛 + 1)↑𝑀)) / (!‘(𝑛 + 𝑀)))) ⇒ ⊢ (𝑀 ∈ ℕ0 → 𝐹 ⇝ 1) | ||
Theorem | pdivsq 32976 | Condition for a prime dividing a square. (Contributed by Scott Fenton, 8-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ ((𝑃 ∈ ℙ ∧ 𝑀 ∈ ℤ) → (𝑃 ∥ 𝑀 ↔ 𝑃 ∥ (𝑀↑2))) | ||
Theorem | dvdspw 32977 | Exponentiation law for divisibility. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝐾 ∥ 𝑀 → 𝐾 ∥ (𝑀↑𝑁))) | ||
Theorem | gcd32 32978 | Swap the second and third arguments of a gcd. (Contributed by Scott Fenton, 8-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → ((𝐴 gcd 𝐵) gcd 𝐶) = ((𝐴 gcd 𝐶) gcd 𝐵)) | ||
Theorem | gcdabsorb 32979 | Absorption law for gcd. (Contributed by Scott Fenton, 8-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴 gcd 𝐵) gcd 𝐵) = (𝐴 gcd 𝐵)) | ||
Theorem | brtp 32980 | A condition for a binary relation over an unordered triple. (Contributed by Scott Fenton, 8-Jun-2011.) |
⊢ 𝑋 ∈ V & ⊢ 𝑌 ∈ V ⇒ ⊢ (𝑋{〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉, 〈𝐸, 𝐹〉}𝑌 ↔ ((𝑋 = 𝐴 ∧ 𝑌 = 𝐵) ∨ (𝑋 = 𝐶 ∧ 𝑌 = 𝐷) ∨ (𝑋 = 𝐸 ∧ 𝑌 = 𝐹))) | ||
Theorem | dftr6 32981 | A potential definition of transitivity for sets. (Contributed by Scott Fenton, 18-Mar-2012.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (Tr 𝐴 ↔ 𝐴 ∈ (V ∖ ran (( E ∘ E ) ∖ E ))) | ||
Theorem | coep 32982* | Composition with the membership relation. (Contributed by Scott Fenton, 18-Feb-2013.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴( E ∘ 𝑅)𝐵 ↔ ∃𝑥 ∈ 𝐵 𝐴𝑅𝑥) | ||
Theorem | coepr 32983* | Composition with the converse membership relation. (Contributed by Scott Fenton, 18-Feb-2013.) |
⊢ 𝐴 ∈ V & ⊢ 𝐵 ∈ V ⇒ ⊢ (𝐴(𝑅 ∘ ◡ E )𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑥𝑅𝐵) | ||
Theorem | dffr5 32984 | A quantifier free definition of a well-founded relationship. (Contributed by Scott Fenton, 11-Apr-2011.) |
⊢ (𝑅 Fr 𝐴 ↔ (𝒫 𝐴 ∖ {∅}) ⊆ ran ( E ∖ ( E ∘ ◡𝑅))) | ||
Theorem | dfso2 32985 | Quantifier free definition of a strict order. (Contributed by Scott Fenton, 22-Feb-2013.) |
⊢ (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ (𝐴 × 𝐴) ⊆ (𝑅 ∪ ( I ∪ ◡𝑅)))) | ||
Theorem | dfpo2 32986 | Quantifier free definition of a partial ordering. (Contributed by Scott Fenton, 22-Feb-2013.) (Proof shortened by Peter Mazsa, 2-Oct-2022.) |
⊢ (𝑅 Po 𝐴 ↔ ((𝑅 ∩ ( I ↾ 𝐴)) = ∅ ∧ ((𝑅 ∩ (𝐴 × 𝐴)) ∘ (𝑅 ∩ (𝐴 × 𝐴))) ⊆ 𝑅)) | ||
Theorem | br8 32987* | Substitution for an eight-place predicate. (Contributed by Scott Fenton, 26-Sep-2013.) (Revised by Mario Carneiro, 3-May-2015.) |
⊢ (𝑎 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ (𝑏 = 𝐵 → (𝜓 ↔ 𝜒)) & ⊢ (𝑐 = 𝐶 → (𝜒 ↔ 𝜃)) & ⊢ (𝑑 = 𝐷 → (𝜃 ↔ 𝜏)) & ⊢ (𝑒 = 𝐸 → (𝜏 ↔ 𝜂)) & ⊢ (𝑓 = 𝐹 → (𝜂 ↔ 𝜁)) & ⊢ (𝑔 = 𝐺 → (𝜁 ↔ 𝜎)) & ⊢ (ℎ = 𝐻 → (𝜎 ↔ 𝜌)) & ⊢ (𝑥 = 𝑋 → 𝑃 = 𝑄) & ⊢ 𝑅 = {〈𝑝, 𝑞〉 ∣ ∃𝑥 ∈ 𝑆 ∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 ∃𝑐 ∈ 𝑃 ∃𝑑 ∈ 𝑃 ∃𝑒 ∈ 𝑃 ∃𝑓 ∈ 𝑃 ∃𝑔 ∈ 𝑃 ∃ℎ ∈ 𝑃 (𝑝 = 〈〈𝑎, 𝑏〉, 〈𝑐, 𝑑〉〉 ∧ 𝑞 = 〈〈𝑒, 𝑓〉, 〈𝑔, ℎ〉〉 ∧ 𝜑)} ⇒ ⊢ (((𝑋 ∈ 𝑆 ∧ 𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄) ∧ (𝐶 ∈ 𝑄 ∧ 𝐷 ∈ 𝑄 ∧ 𝐸 ∈ 𝑄) ∧ (𝐹 ∈ 𝑄 ∧ 𝐺 ∈ 𝑄 ∧ 𝐻 ∈ 𝑄)) → (〈〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉〉𝑅〈〈𝐸, 𝐹〉, 〈𝐺, 𝐻〉〉 ↔ 𝜌)) | ||
Theorem | br6 32988* | Substitution for a six-place predicate. (Contributed by Scott Fenton, 4-Oct-2013.) (Revised by Mario Carneiro, 3-May-2015.) |
⊢ (𝑎 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ (𝑏 = 𝐵 → (𝜓 ↔ 𝜒)) & ⊢ (𝑐 = 𝐶 → (𝜒 ↔ 𝜃)) & ⊢ (𝑑 = 𝐷 → (𝜃 ↔ 𝜏)) & ⊢ (𝑒 = 𝐸 → (𝜏 ↔ 𝜂)) & ⊢ (𝑓 = 𝐹 → (𝜂 ↔ 𝜁)) & ⊢ (𝑥 = 𝑋 → 𝑃 = 𝑄) & ⊢ 𝑅 = {〈𝑝, 𝑞〉 ∣ ∃𝑥 ∈ 𝑆 ∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 ∃𝑐 ∈ 𝑃 ∃𝑑 ∈ 𝑃 ∃𝑒 ∈ 𝑃 ∃𝑓 ∈ 𝑃 (𝑝 = 〈𝑎, 〈𝑏, 𝑐〉〉 ∧ 𝑞 = 〈𝑑, 〈𝑒, 𝑓〉〉 ∧ 𝜑)} ⇒ ⊢ ((𝑋 ∈ 𝑆 ∧ (𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄 ∧ 𝐶 ∈ 𝑄) ∧ (𝐷 ∈ 𝑄 ∧ 𝐸 ∈ 𝑄 ∧ 𝐹 ∈ 𝑄)) → (〈𝐴, 〈𝐵, 𝐶〉〉𝑅〈𝐷, 〈𝐸, 𝐹〉〉 ↔ 𝜁)) | ||
Theorem | br4 32989* | Substitution for a four-place predicate. (Contributed by Scott Fenton, 9-Oct-2013.) (Revised by Mario Carneiro, 14-Oct-2013.) |
⊢ (𝑎 = 𝐴 → (𝜑 ↔ 𝜓)) & ⊢ (𝑏 = 𝐵 → (𝜓 ↔ 𝜒)) & ⊢ (𝑐 = 𝐶 → (𝜒 ↔ 𝜃)) & ⊢ (𝑑 = 𝐷 → (𝜃 ↔ 𝜏)) & ⊢ (𝑥 = 𝑋 → 𝑃 = 𝑄) & ⊢ 𝑅 = {〈𝑝, 𝑞〉 ∣ ∃𝑥 ∈ 𝑆 ∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 ∃𝑐 ∈ 𝑃 ∃𝑑 ∈ 𝑃 (𝑝 = 〈𝑎, 𝑏〉 ∧ 𝑞 = 〈𝑐, 𝑑〉 ∧ 𝜑)} ⇒ ⊢ ((𝑋 ∈ 𝑆 ∧ (𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄) ∧ (𝐶 ∈ 𝑄 ∧ 𝐷 ∈ 𝑄)) → (〈𝐴, 𝐵〉𝑅〈𝐶, 𝐷〉 ↔ 𝜏)) | ||
Theorem | cnvco1 32990 | Another distributive law of converse over class composition. (Contributed by Scott Fenton, 3-May-2014.) |
⊢ ◡(◡𝐴 ∘ 𝐵) = (◡𝐵 ∘ 𝐴) | ||
Theorem | cnvco2 32991 | Another distributive law of converse over class composition. (Contributed by Scott Fenton, 3-May-2014.) |
⊢ ◡(𝐴 ∘ ◡𝐵) = (𝐵 ∘ ◡𝐴) | ||
Theorem | eldm3 32992 | Quantifier-free definition of membership in a domain. (Contributed by Scott Fenton, 21-Jan-2017.) |
⊢ (𝐴 ∈ dom 𝐵 ↔ (𝐵 ↾ {𝐴}) ≠ ∅) | ||
Theorem | elrn3 32993 | Quantifier-free definition of membership in a range. (Contributed by Scott Fenton, 21-Jan-2017.) |
⊢ (𝐴 ∈ ran 𝐵 ↔ (𝐵 ∩ (V × {𝐴})) ≠ ∅) | ||
Theorem | pocnv 32994 | The converse of a partial ordering is still a partial ordering. (Contributed by Scott Fenton, 13-Jun-2018.) |
⊢ (𝑅 Po 𝐴 → ◡𝑅 Po 𝐴) | ||
Theorem | socnv 32995 | The converse of a strict ordering is still a strict ordering. (Contributed by Scott Fenton, 13-Jun-2018.) |
⊢ (𝑅 Or 𝐴 → ◡𝑅 Or 𝐴) | ||
Theorem | sotrd 32996 | Transitivity law for strict orderings, deduction form. (Contributed by Scott Fenton, 24-Nov-2021.) |
⊢ (𝜑 → 𝑅 Or 𝐴) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → 𝑌 ∈ 𝐴) & ⊢ (𝜑 → 𝑍 ∈ 𝐴) & ⊢ (𝜑 → 𝑋𝑅𝑌) & ⊢ (𝜑 → 𝑌𝑅𝑍) ⇒ ⊢ (𝜑 → 𝑋𝑅𝑍) | ||
Theorem | sotr3 32997 | Transitivity law for strict orderings. (Contributed by Scott Fenton, 24-Nov-2021.) |
⊢ ((𝑅 Or 𝐴 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑍 ∈ 𝐴)) → ((𝑋𝑅𝑌 ∧ ¬ 𝑍𝑅𝑌) → 𝑋𝑅𝑍)) | ||
Theorem | sotrine 32998 | Trichotomy law for strict orderings. (Contributed by Scott Fenton, 8-Dec-2021.) |
⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝐵 ≠ 𝐶 ↔ (𝐵𝑅𝐶 ∨ 𝐶𝑅𝐵))) | ||
Theorem | eqfunresadj 32999 | Law for adjoining an element to restrictions of functions. (Contributed by Scott Fenton, 6-Dec-2021.) |
⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐹 ↾ 𝑋) = (𝐺 ↾ 𝑋) ∧ (𝑌 ∈ dom 𝐹 ∧ 𝑌 ∈ dom 𝐺 ∧ (𝐹‘𝑌) = (𝐺‘𝑌))) → (𝐹 ↾ (𝑋 ∪ {𝑌})) = (𝐺 ↾ (𝑋 ∪ {𝑌}))) | ||
Theorem | eqfunressuc 33000 | Law for equality of restriction to successors. This is primarily useful when 𝑋 is an ordinal, but it does not require that. (Contributed by Scott Fenton, 6-Dec-2021.) |
⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐹 ↾ 𝑋) = (𝐺 ↾ 𝑋) ∧ (𝑋 ∈ dom 𝐹 ∧ 𝑋 ∈ dom 𝐺 ∧ (𝐹‘𝑋) = (𝐺‘𝑋))) → (𝐹 ↾ suc 𝑋) = (𝐺 ↾ suc 𝑋)) |
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