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Type | Label | Description |
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Statement | ||
Theorem | ltexprlem7 10201* | Lemma for Proposition 9-3.5(iv) of [Gleason] p. 123. (Contributed by NM, 8-Apr-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.) |
⊢ 𝐶 = {𝑥 ∣ ∃𝑦(¬ 𝑦 ∈ 𝐴 ∧ (𝑦 +Q 𝑥) ∈ 𝐵)} ⇒ ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ 𝐴 ⊊ 𝐵) → 𝐵 ⊆ (𝐴 +P 𝐶)) | ||
Theorem | ltexpri 10202* | Proposition 9-3.5(iv) of [Gleason] p. 123. (Contributed by NM, 13-May-1996.) (Revised by Mario Carneiro, 14-Jun-2013.) (New usage is discouraged.) |
⊢ (𝐴<P 𝐵 → ∃𝑥 ∈ P (𝐴 +P 𝑥) = 𝐵) | ||
Theorem | ltaprlem 10203 | Lemma for Proposition 9-3.5(v) of [Gleason] p. 123. (Contributed by NM, 8-Apr-1996.) (New usage is discouraged.) |
⊢ (𝐶 ∈ P → (𝐴<P 𝐵 → (𝐶 +P 𝐴)<P (𝐶 +P 𝐵))) | ||
Theorem | ltapr 10204 | Ordering property of addition. Proposition 9-3.5(v) of [Gleason] p. 123. (Contributed by NM, 8-Apr-1996.) (New usage is discouraged.) |
⊢ (𝐶 ∈ P → (𝐴<P 𝐵 ↔ (𝐶 +P 𝐴)<P (𝐶 +P 𝐵))) | ||
Theorem | addcanpr 10205 | Addition cancellation law for positive reals. Proposition 9-3.5(vi) of [Gleason] p. 123. (Contributed by NM, 9-Apr-1996.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((𝐴 +P 𝐵) = (𝐴 +P 𝐶) → 𝐵 = 𝐶)) | ||
Theorem | prlem936 10206* | Lemma 9-3.6 of [Gleason] p. 124. (Contributed by NM, 26-Apr-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ P ∧ 1Q <Q 𝐵) → ∃𝑥 ∈ 𝐴 ¬ (𝑥 ·Q 𝐵) ∈ 𝐴) | ||
Theorem | reclem2pr 10207* | Lemma for Proposition 9-3.7 of [Gleason] p. 124. (Contributed by NM, 30-Apr-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.) |
⊢ 𝐵 = {𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ ¬ (*Q‘𝑦) ∈ 𝐴)} ⇒ ⊢ (𝐴 ∈ P → 𝐵 ∈ P) | ||
Theorem | reclem3pr 10208* | Lemma for Proposition 9-3.7(v) of [Gleason] p. 124. (Contributed by NM, 30-Apr-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.) |
⊢ 𝐵 = {𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ ¬ (*Q‘𝑦) ∈ 𝐴)} ⇒ ⊢ (𝐴 ∈ P → 1P ⊆ (𝐴 ·P 𝐵)) | ||
Theorem | reclem4pr 10209* | Lemma for Proposition 9-3.7(v) of [Gleason] p. 124. (Contributed by NM, 30-Apr-1996.) (New usage is discouraged.) |
⊢ 𝐵 = {𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ ¬ (*Q‘𝑦) ∈ 𝐴)} ⇒ ⊢ (𝐴 ∈ P → (𝐴 ·P 𝐵) = 1P) | ||
Theorem | recexpr 10210* | The reciprocal of a positive real exists. Part of Proposition 9-3.7(v) of [Gleason] p. 124. (Contributed by NM, 15-May-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.) |
⊢ (𝐴 ∈ P → ∃𝑥 ∈ P (𝐴 ·P 𝑥) = 1P) | ||
Theorem | suplem1pr 10211* | The union of a nonempty, bounded set of positive reals is a positive real. Part of Proposition 9-3.3 of [Gleason] p. 122. (Contributed by NM, 19-May-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.) |
⊢ ((𝐴 ≠ ∅ ∧ ∃𝑥 ∈ P ∀𝑦 ∈ 𝐴 𝑦<P 𝑥) → ∪ 𝐴 ∈ P) | ||
Theorem | suplem2pr 10212* | The union of a set of positive reals (if a positive real) is its supremum (the least upper bound). Part of Proposition 9-3.3 of [Gleason] p. 122. (Contributed by NM, 19-May-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.) |
⊢ (𝐴 ⊆ P → ((𝑦 ∈ 𝐴 → ¬ ∪ 𝐴<P 𝑦) ∧ (𝑦<P ∪ 𝐴 → ∃𝑧 ∈ 𝐴 𝑦<P 𝑧))) | ||
Theorem | supexpr 10213* | The union of a nonempty, bounded set of positive reals has a supremum. Part of Proposition 9-3.3 of [Gleason] p. 122. (Contributed by NM, 19-May-1996.) (New usage is discouraged.) |
⊢ ((𝐴 ≠ ∅ ∧ ∃𝑥 ∈ P ∀𝑦 ∈ 𝐴 𝑦<P 𝑥) → ∃𝑥 ∈ P (∀𝑦 ∈ 𝐴 ¬ 𝑥<P 𝑦 ∧ ∀𝑦 ∈ P (𝑦<P 𝑥 → ∃𝑧 ∈ 𝐴 𝑦<P 𝑧))) | ||
Definition | df-enr 10214* | Define equivalence relation for signed reals. This is a "temporary" set used in the construction of complex numbers df-c 10280, and is intended to be used only by the construction. From Proposition 9-4.1 of [Gleason] p. 126. (Contributed by NM, 25-Jul-1995.) (New usage is discouraged.) |
⊢ ~R = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ (P × P) ∧ 𝑦 ∈ (P × P)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = 〈𝑧, 𝑤〉 ∧ 𝑦 = 〈𝑣, 𝑢〉) ∧ (𝑧 +P 𝑢) = (𝑤 +P 𝑣)))} | ||
Definition | df-nr 10215 | Define class of signed reals. This is a "temporary" set used in the construction of complex numbers df-c 10280, and is intended to be used only by the construction. From Proposition 9-4.2 of [Gleason] p. 126. (Contributed by NM, 25-Jul-1995.) (New usage is discouraged.) |
⊢ R = ((P × P) / ~R ) | ||
Definition | df-plr 10216* | Define addition on signed reals. This is a "temporary" set used in the construction of complex numbers df-c 10280, and is intended to be used only by the construction. From Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 25-Aug-1995.) (New usage is discouraged.) |
⊢ +R = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = [〈𝑤, 𝑣〉] ~R ∧ 𝑦 = [〈𝑢, 𝑓〉] ~R ) ∧ 𝑧 = [〈(𝑤 +P 𝑢), (𝑣 +P 𝑓)〉] ~R ))} | ||
Definition | df-mr 10217* | Define multiplication on signed reals. This is a "temporary" set used in the construction of complex numbers df-c 10280, and is intended to be used only by the construction. From Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 25-Aug-1995.) (New usage is discouraged.) |
⊢ ·R = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = [〈𝑤, 𝑣〉] ~R ∧ 𝑦 = [〈𝑢, 𝑓〉] ~R ) ∧ 𝑧 = [〈((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑓)), ((𝑤 ·P 𝑓) +P (𝑣 ·P 𝑢))〉] ~R ))} | ||
Definition | df-ltr 10218* | Define ordering relation on signed reals. This is a "temporary" set used in the construction of complex numbers df-c 10280, and is intended to be used only by the construction. From Proposition 9-4.4 of [Gleason] p. 127. (Contributed by NM, 14-Feb-1996.) (New usage is discouraged.) |
⊢ <R = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ R ∧ 𝑦 ∈ R) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = [〈𝑧, 𝑤〉] ~R ∧ 𝑦 = [〈𝑣, 𝑢〉] ~R ) ∧ (𝑧 +P 𝑢)<P (𝑤 +P 𝑣)))} | ||
Definition | df-0r 10219 | Define signed real constant 0. This is a "temporary" set used in the construction of complex numbers df-c 10280, and is intended to be used only by the construction. From Proposition 9-4.2 of [Gleason] p. 126. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.) |
⊢ 0R = [〈1P, 1P〉] ~R | ||
Definition | df-1r 10220 | Define signed real constant 1. This is a "temporary" set used in the construction of complex numbers df-c 10280, and is intended to be used only by the construction. From Proposition 9-4.2 of [Gleason] p. 126. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.) |
⊢ 1R = [〈(1P +P 1P), 1P〉] ~R | ||
Definition | df-m1r 10221 | Define signed real constant -1. This is a "temporary" set used in the construction of complex numbers df-c 10280, and is intended to be used only by the construction. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.) |
⊢ -1R = [〈1P, (1P +P 1P)〉] ~R | ||
Theorem | enrer 10222 | The equivalence relation for signed reals is an equivalence relation. Proposition 9-4.1 of [Gleason] p. 126. (Contributed by NM, 3-Sep-1995.) (Revised by Mario Carneiro, 6-Jul-2015.) (New usage is discouraged.) |
⊢ ~R Er (P × P) | ||
Theorem | nrex1 10223 | The class of signed reals is a set. Note that a shorter proof is possible using qsex 8091 (and not requiring enrer 10222), but it would add a dependency on ax-rep 5008. (Contributed by Mario Carneiro, 17-Nov-2014.) Extract proof from that of axcnex 10306. (Revised by BJ, 4-Feb-2023.) (New usage is discouraged.) |
⊢ R ∈ V | ||
Theorem | enrbreq 10224 | Equivalence relation for signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.) (New usage is discouraged.) |
⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ P)) → (〈𝐴, 𝐵〉 ~R 〈𝐶, 𝐷〉 ↔ (𝐴 +P 𝐷) = (𝐵 +P 𝐶))) | ||
Theorem | enreceq 10225 | Equivalence class equality of positive fractions in terms of positive integers. (Contributed by NM, 29-Nov-1995.) (New usage is discouraged.) |
⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ P)) → ([〈𝐴, 𝐵〉] ~R = [〈𝐶, 𝐷〉] ~R ↔ (𝐴 +P 𝐷) = (𝐵 +P 𝐶))) | ||
Theorem | enrex 10226 | The equivalence relation for signed reals exists. (Contributed by NM, 25-Jul-1995.) (New usage is discouraged.) |
⊢ ~R ∈ V | ||
Theorem | ltrelsr 10227 | Signed real 'less than' is a relation on signed reals. (Contributed by NM, 14-Feb-1996.) (New usage is discouraged.) |
⊢ <R ⊆ (R × R) | ||
Theorem | addcmpblnr 10228 | Lemma showing compatibility of addition. (Contributed by NM, 3-Sep-1995.) (New usage is discouraged.) |
⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ P)) ∧ ((𝐹 ∈ P ∧ 𝐺 ∈ P) ∧ (𝑅 ∈ P ∧ 𝑆 ∈ P))) → (((𝐴 +P 𝐷) = (𝐵 +P 𝐶) ∧ (𝐹 +P 𝑆) = (𝐺 +P 𝑅)) → 〈(𝐴 +P 𝐹), (𝐵 +P 𝐺)〉 ~R 〈(𝐶 +P 𝑅), (𝐷 +P 𝑆)〉)) | ||
Theorem | mulcmpblnrlem 10229 | Lemma used in lemma showing compatibility of multiplication. (Contributed by NM, 4-Sep-1995.) (New usage is discouraged.) |
⊢ (((𝐴 +P 𝐷) = (𝐵 +P 𝐶) ∧ (𝐹 +P 𝑆) = (𝐺 +P 𝑅)) → ((𝐷 ·P 𝐹) +P (((𝐴 ·P 𝐹) +P (𝐵 ·P 𝐺)) +P ((𝐶 ·P 𝑆) +P (𝐷 ·P 𝑅)))) = ((𝐷 ·P 𝐹) +P (((𝐴 ·P 𝐺) +P (𝐵 ·P 𝐹)) +P ((𝐶 ·P 𝑅) +P (𝐷 ·P 𝑆))))) | ||
Theorem | mulcmpblnr 10230 | Lemma showing compatibility of multiplication. (Contributed by NM, 5-Sep-1995.) (New usage is discouraged.) |
⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ P)) ∧ ((𝐹 ∈ P ∧ 𝐺 ∈ P) ∧ (𝑅 ∈ P ∧ 𝑆 ∈ P))) → (((𝐴 +P 𝐷) = (𝐵 +P 𝐶) ∧ (𝐹 +P 𝑆) = (𝐺 +P 𝑅)) → 〈((𝐴 ·P 𝐹) +P (𝐵 ·P 𝐺)), ((𝐴 ·P 𝐺) +P (𝐵 ·P 𝐹))〉 ~R 〈((𝐶 ·P 𝑅) +P (𝐷 ·P 𝑆)), ((𝐶 ·P 𝑆) +P (𝐷 ·P 𝑅))〉)) | ||
Theorem | prsrlem1 10231* | Decomposing signed reals into positive reals. Lemma for addsrpr 10234 and mulsrpr 10235. (Contributed by Jim Kingdon, 30-Dec-2019.) |
⊢ (((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) ∧ ((𝐴 = [〈𝑤, 𝑣〉] ~R ∧ 𝐵 = [〈𝑢, 𝑡〉] ~R ) ∧ (𝐴 = [〈𝑠, 𝑓〉] ~R ∧ 𝐵 = [〈𝑔, ℎ〉] ~R ))) → ((((𝑤 ∈ P ∧ 𝑣 ∈ P) ∧ (𝑠 ∈ P ∧ 𝑓 ∈ P)) ∧ ((𝑢 ∈ P ∧ 𝑡 ∈ P) ∧ (𝑔 ∈ P ∧ ℎ ∈ P))) ∧ ((𝑤 +P 𝑓) = (𝑣 +P 𝑠) ∧ (𝑢 +P ℎ) = (𝑡 +P 𝑔)))) | ||
Theorem | addsrmo 10232* | There is at most one result from adding signed reals. (Contributed by Jim Kingdon, 30-Dec-2019.) |
⊢ ((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) → ∃*𝑧∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [〈𝑤, 𝑣〉] ~R ∧ 𝐵 = [〈𝑢, 𝑡〉] ~R ) ∧ 𝑧 = [〈(𝑤 +P 𝑢), (𝑣 +P 𝑡)〉] ~R )) | ||
Theorem | mulsrmo 10233* | There is at most one result from multiplying signed reals. (Contributed by Jim Kingdon, 30-Dec-2019.) |
⊢ ((𝐴 ∈ ((P × P) / ~R ) ∧ 𝐵 ∈ ((P × P) / ~R )) → ∃*𝑧∃𝑤∃𝑣∃𝑢∃𝑡((𝐴 = [〈𝑤, 𝑣〉] ~R ∧ 𝐵 = [〈𝑢, 𝑡〉] ~R ) ∧ 𝑧 = [〈((𝑤 ·P 𝑢) +P (𝑣 ·P 𝑡)), ((𝑤 ·P 𝑡) +P (𝑣 ·P 𝑢))〉] ~R )) | ||
Theorem | addsrpr 10234 | Addition of signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.) (Revised by Mario Carneiro, 12-Aug-2015.) (New usage is discouraged.) |
⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ P)) → ([〈𝐴, 𝐵〉] ~R +R [〈𝐶, 𝐷〉] ~R ) = [〈(𝐴 +P 𝐶), (𝐵 +P 𝐷)〉] ~R ) | ||
Theorem | mulsrpr 10235 | Multiplication of signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.) (Revised by Mario Carneiro, 12-Aug-2015.) (New usage is discouraged.) |
⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝐶 ∈ P ∧ 𝐷 ∈ P)) → ([〈𝐴, 𝐵〉] ~R ·R [〈𝐶, 𝐷〉] ~R ) = [〈((𝐴 ·P 𝐶) +P (𝐵 ·P 𝐷)), ((𝐴 ·P 𝐷) +P (𝐵 ·P 𝐶))〉] ~R ) | ||
Theorem | ltsrpr 10236 | Ordering of signed reals in terms of positive reals. (Contributed by NM, 20-Feb-1996.) (Revised by Mario Carneiro, 12-Aug-2015.) (New usage is discouraged.) |
⊢ ([〈𝐴, 𝐵〉] ~R <R [〈𝐶, 𝐷〉] ~R ↔ (𝐴 +P 𝐷)<P (𝐵 +P 𝐶)) | ||
Theorem | gt0srpr 10237 | Greater than zero in terms of positive reals. (Contributed by NM, 13-May-1996.) (New usage is discouraged.) |
⊢ (0R <R [〈𝐴, 𝐵〉] ~R ↔ 𝐵<P 𝐴) | ||
Theorem | 0nsr 10238 | The empty set is not a signed real. (Contributed by NM, 25-Aug-1995.) (Revised by Mario Carneiro, 10-Jul-2014.) (New usage is discouraged.) |
⊢ ¬ ∅ ∈ R | ||
Theorem | 0r 10239 | The constant 0R is a signed real. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.) |
⊢ 0R ∈ R | ||
Theorem | 1sr 10240 | The constant 1R is a signed real. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.) |
⊢ 1R ∈ R | ||
Theorem | m1r 10241 | The constant -1R is a signed real. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.) |
⊢ -1R ∈ R | ||
Theorem | addclsr 10242 | Closure of addition on signed reals. (Contributed by NM, 25-Jul-1995.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → (𝐴 +R 𝐵) ∈ R) | ||
Theorem | mulclsr 10243 | Closure of multiplication on signed reals. (Contributed by NM, 10-Aug-1995.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → (𝐴 ·R 𝐵) ∈ R) | ||
Theorem | dmaddsr 10244 | Domain of addition on signed reals. (Contributed by NM, 25-Aug-1995.) (New usage is discouraged.) |
⊢ dom +R = (R × R) | ||
Theorem | dmmulsr 10245 | Domain of multiplication on signed reals. (Contributed by NM, 25-Aug-1995.) (New usage is discouraged.) |
⊢ dom ·R = (R × R) | ||
Theorem | addcomsr 10246 | Addition of signed reals is commutative. (Contributed by NM, 31-Aug-1995.) (Revised by Mario Carneiro, 28-Apr-2015.) (New usage is discouraged.) |
⊢ (𝐴 +R 𝐵) = (𝐵 +R 𝐴) | ||
Theorem | addasssr 10247 | Addition of signed reals is associative. (Contributed by NM, 2-Sep-1995.) (Revised by Mario Carneiro, 28-Apr-2015.) (New usage is discouraged.) |
⊢ ((𝐴 +R 𝐵) +R 𝐶) = (𝐴 +R (𝐵 +R 𝐶)) | ||
Theorem | mulcomsr 10248 | Multiplication of signed reals is commutative. (Contributed by NM, 31-Aug-1995.) (Revised by Mario Carneiro, 28-Apr-2015.) (New usage is discouraged.) |
⊢ (𝐴 ·R 𝐵) = (𝐵 ·R 𝐴) | ||
Theorem | mulasssr 10249 | Multiplication of signed reals is associative. (Contributed by NM, 2-Sep-1995.) (Revised by Mario Carneiro, 28-Apr-2015.) (New usage is discouraged.) |
⊢ ((𝐴 ·R 𝐵) ·R 𝐶) = (𝐴 ·R (𝐵 ·R 𝐶)) | ||
Theorem | distrsr 10250 | Multiplication of signed reals is distributive. (Contributed by NM, 2-Sep-1995.) (Revised by Mario Carneiro, 28-Apr-2015.) (New usage is discouraged.) |
⊢ (𝐴 ·R (𝐵 +R 𝐶)) = ((𝐴 ·R 𝐵) +R (𝐴 ·R 𝐶)) | ||
Theorem | m1p1sr 10251 | Minus one plus one is zero for signed reals. (Contributed by NM, 5-May-1996.) (New usage is discouraged.) |
⊢ (-1R +R 1R) = 0R | ||
Theorem | m1m1sr 10252 | Minus one times minus one is plus one for signed reals. (Contributed by NM, 14-May-1996.) (New usage is discouraged.) |
⊢ (-1R ·R -1R) = 1R | ||
Theorem | ltsosr 10253 | Signed real 'less than' is a strict ordering. (Contributed by NM, 19-Feb-1996.) (New usage is discouraged.) |
⊢ <R Or R | ||
Theorem | 0lt1sr 10254 | 0 is less than 1 for signed reals. (Contributed by NM, 26-Mar-1996.) (New usage is discouraged.) |
⊢ 0R <R 1R | ||
Theorem | 1ne0sr 10255 | 1 and 0 are distinct for signed reals. (Contributed by NM, 26-Mar-1996.) (New usage is discouraged.) |
⊢ ¬ 1R = 0R | ||
Theorem | 0idsr 10256 | The signed real number 0 is an identity element for addition of signed reals. (Contributed by NM, 10-Apr-1996.) (New usage is discouraged.) |
⊢ (𝐴 ∈ R → (𝐴 +R 0R) = 𝐴) | ||
Theorem | 1idsr 10257 | 1 is an identity element for multiplication. (Contributed by NM, 2-May-1996.) (New usage is discouraged.) |
⊢ (𝐴 ∈ R → (𝐴 ·R 1R) = 𝐴) | ||
Theorem | 00sr 10258 | A signed real times 0 is 0. (Contributed by NM, 10-Apr-1996.) (New usage is discouraged.) |
⊢ (𝐴 ∈ R → (𝐴 ·R 0R) = 0R) | ||
Theorem | ltasr 10259 | Ordering property of addition. (Contributed by NM, 10-May-1996.) (New usage is discouraged.) |
⊢ (𝐶 ∈ R → (𝐴 <R 𝐵 ↔ (𝐶 +R 𝐴) <R (𝐶 +R 𝐵))) | ||
Theorem | pn0sr 10260 | A signed real plus its negative is zero. (Contributed by NM, 14-May-1996.) (New usage is discouraged.) |
⊢ (𝐴 ∈ R → (𝐴 +R (𝐴 ·R -1R)) = 0R) | ||
Theorem | negexsr 10261* | Existence of negative signed real. Part of Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 2-May-1996.) (New usage is discouraged.) |
⊢ (𝐴 ∈ R → ∃𝑥 ∈ R (𝐴 +R 𝑥) = 0R) | ||
Theorem | recexsrlem 10262* | The reciprocal of a positive signed real exists. Part of Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 15-May-1996.) (New usage is discouraged.) |
⊢ (0R <R 𝐴 → ∃𝑥 ∈ R (𝐴 ·R 𝑥) = 1R) | ||
Theorem | addgt0sr 10263 | The sum of two positive signed reals is positive. (Contributed by NM, 14-May-1996.) (New usage is discouraged.) |
⊢ ((0R <R 𝐴 ∧ 0R <R 𝐵) → 0R <R (𝐴 +R 𝐵)) | ||
Theorem | mulgt0sr 10264 | The product of two positive signed reals is positive. (Contributed by NM, 13-May-1996.) (New usage is discouraged.) |
⊢ ((0R <R 𝐴 ∧ 0R <R 𝐵) → 0R <R (𝐴 ·R 𝐵)) | ||
Theorem | sqgt0sr 10265 | The square of a nonzero signed real is positive. (Contributed by NM, 14-May-1996.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ R ∧ 𝐴 ≠ 0R) → 0R <R (𝐴 ·R 𝐴)) | ||
Theorem | recexsr 10266* | The reciprocal of a nonzero signed real exists. Part of Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 15-May-1996.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ R ∧ 𝐴 ≠ 0R) → ∃𝑥 ∈ R (𝐴 ·R 𝑥) = 1R) | ||
Theorem | mappsrpr 10267 | Mapping from positive signed reals to positive reals. (Contributed by NM, 17-May-1996.) (Revised by Mario Carneiro, 15-Jun-2013.) (New usage is discouraged.) |
⊢ 𝐶 ∈ R ⇒ ⊢ ((𝐶 +R -1R) <R (𝐶 +R [〈𝐴, 1P〉] ~R ) ↔ 𝐴 ∈ P) | ||
Theorem | ltpsrpr 10268 | Mapping of order from positive signed reals to positive reals. (Contributed by NM, 17-May-1996.) (Revised by Mario Carneiro, 15-Jun-2013.) (New usage is discouraged.) |
⊢ 𝐶 ∈ R ⇒ ⊢ ((𝐶 +R [〈𝐴, 1P〉] ~R ) <R (𝐶 +R [〈𝐵, 1P〉] ~R ) ↔ 𝐴<P 𝐵) | ||
Theorem | map2psrpr 10269* | Equivalence for positive signed real. (Contributed by NM, 17-May-1996.) (Revised by Mario Carneiro, 15-Jun-2013.) (New usage is discouraged.) |
⊢ 𝐶 ∈ R ⇒ ⊢ ((𝐶 +R -1R) <R 𝐴 ↔ ∃𝑥 ∈ P (𝐶 +R [〈𝑥, 1P〉] ~R ) = 𝐴) | ||
Theorem | supsrlem 10270* | Lemma for supremum theorem. (Contributed by NM, 21-May-1996.) (Revised by Mario Carneiro, 15-Jun-2013.) (New usage is discouraged.) |
⊢ 𝐵 = {𝑤 ∣ (𝐶 +R [〈𝑤, 1P〉] ~R ) ∈ 𝐴} & ⊢ 𝐶 ∈ R ⇒ ⊢ ((𝐶 ∈ 𝐴 ∧ ∃𝑥 ∈ R ∀𝑦 ∈ 𝐴 𝑦 <R 𝑥) → ∃𝑥 ∈ R (∀𝑦 ∈ 𝐴 ¬ 𝑥 <R 𝑦 ∧ ∀𝑦 ∈ R (𝑦 <R 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 <R 𝑧))) | ||
Theorem | supsr 10271* | A nonempty, bounded set of signed reals has a supremum. (Contributed by NM, 21-May-1996.) (Revised by Mario Carneiro, 15-Jun-2013.) (New usage is discouraged.) |
⊢ ((𝐴 ≠ ∅ ∧ ∃𝑥 ∈ R ∀𝑦 ∈ 𝐴 𝑦 <R 𝑥) → ∃𝑥 ∈ R (∀𝑦 ∈ 𝐴 ¬ 𝑥 <R 𝑦 ∧ ∀𝑦 ∈ R (𝑦 <R 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 <R 𝑧))) | ||
Syntax | cc 10272 | Class of complex numbers. |
class ℂ | ||
Syntax | cr 10273 | Class of real numbers. |
class ℝ | ||
Syntax | cc0 10274 | Extend class notation to include the complex number 0. |
class 0 | ||
Syntax | c1 10275 | Extend class notation to include the complex number 1. |
class 1 | ||
Syntax | ci 10276 | Extend class notation to include the complex number i. |
class i | ||
Syntax | caddc 10277 | Addition on complex numbers. |
class + | ||
Syntax | cltrr 10278 | 'Less than' predicate (defined over real subset of complex numbers). |
class <ℝ | ||
Syntax | cmul 10279 | Multiplication on complex numbers. The token · is a center dot. |
class · | ||
Definition | df-c 10280 | Define the set of complex numbers. The 23 axioms for complex numbers start at axresscn 10307. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.) |
⊢ ℂ = (R × R) | ||
Definition | df-0 10281 | Define the complex number 0. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.) |
⊢ 0 = 〈0R, 0R〉 | ||
Definition | df-1 10282 | Define the complex number 1. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.) |
⊢ 1 = 〈1R, 0R〉 | ||
Definition | df-i 10283 | Define the complex number i (the imaginary unit). (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.) |
⊢ i = 〈0R, 1R〉 | ||
Definition | df-r 10284 | Define the set of real numbers. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.) |
⊢ ℝ = (R × {0R}) | ||
Definition | df-add 10285* | Define addition over complex numbers. (Contributed by NM, 28-May-1995.) (New usage is discouraged.) |
⊢ + = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = 〈𝑤, 𝑣〉 ∧ 𝑦 = 〈𝑢, 𝑓〉) ∧ 𝑧 = 〈(𝑤 +R 𝑢), (𝑣 +R 𝑓)〉))} | ||
Definition | df-mul 10286* | Define multiplication over complex numbers. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.) |
⊢ · = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = 〈𝑤, 𝑣〉 ∧ 𝑦 = 〈𝑢, 𝑓〉) ∧ 𝑧 = 〈((𝑤 ·R 𝑢) +R (-1R ·R (𝑣 ·R 𝑓))), ((𝑣 ·R 𝑢) +R (𝑤 ·R 𝑓))〉))} | ||
Definition | df-lt 10287* | Define 'less than' on the real subset of complex numbers. Proofs should typically use < instead; see df-ltxr 10418. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.) |
⊢ <ℝ = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ∃𝑧∃𝑤((𝑥 = 〈𝑧, 0R〉 ∧ 𝑦 = 〈𝑤, 0R〉) ∧ 𝑧 <R 𝑤))} | ||
Theorem | opelcn 10288 | Ordered pair membership in the class of complex numbers. (Contributed by NM, 14-May-1996.) (New usage is discouraged.) |
⊢ (〈𝐴, 𝐵〉 ∈ ℂ ↔ (𝐴 ∈ R ∧ 𝐵 ∈ R)) | ||
Theorem | opelreal 10289 | Ordered pair membership in class of real subset of complex numbers. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.) |
⊢ (〈𝐴, 0R〉 ∈ ℝ ↔ 𝐴 ∈ R) | ||
Theorem | elreal 10290* | Membership in class of real numbers. (Contributed by NM, 31-Mar-1996.) (New usage is discouraged.) |
⊢ (𝐴 ∈ ℝ ↔ ∃𝑥 ∈ R 〈𝑥, 0R〉 = 𝐴) | ||
Theorem | elreal2 10291 | Ordered pair membership in the class of complex numbers. (Contributed by Mario Carneiro, 15-Jun-2013.) (New usage is discouraged.) |
⊢ (𝐴 ∈ ℝ ↔ ((1st ‘𝐴) ∈ R ∧ 𝐴 = 〈(1st ‘𝐴), 0R〉)) | ||
Theorem | 0ncn 10292 | The empty set is not a complex number. Note: do not use this after the real number axioms are developed, since it is a construction-dependent property. (Contributed by NM, 2-May-1996.) (New usage is discouraged.) |
⊢ ¬ ∅ ∈ ℂ | ||
Theorem | ltrelre 10293 | 'Less than' is a relation on real numbers. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.) |
⊢ <ℝ ⊆ (ℝ × ℝ) | ||
Theorem | addcnsr 10294 | Addition of complex numbers in terms of signed reals. (Contributed by NM, 28-May-1995.) (New usage is discouraged.) |
⊢ (((𝐴 ∈ R ∧ 𝐵 ∈ R) ∧ (𝐶 ∈ R ∧ 𝐷 ∈ R)) → (〈𝐴, 𝐵〉 + 〈𝐶, 𝐷〉) = 〈(𝐴 +R 𝐶), (𝐵 +R 𝐷)〉) | ||
Theorem | mulcnsr 10295 | Multiplication of complex numbers in terms of signed reals. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.) |
⊢ (((𝐴 ∈ R ∧ 𝐵 ∈ R) ∧ (𝐶 ∈ R ∧ 𝐷 ∈ R)) → (〈𝐴, 𝐵〉 · 〈𝐶, 𝐷〉) = 〈((𝐴 ·R 𝐶) +R (-1R ·R (𝐵 ·R 𝐷))), ((𝐵 ·R 𝐶) +R (𝐴 ·R 𝐷))〉) | ||
Theorem | eqresr 10296 | Equality of real numbers in terms of intermediate signed reals. (Contributed by NM, 10-May-1996.) (New usage is discouraged.) |
⊢ 𝐴 ∈ V ⇒ ⊢ (〈𝐴, 0R〉 = 〈𝐵, 0R〉 ↔ 𝐴 = 𝐵) | ||
Theorem | addresr 10297 | Addition of real numbers in terms of intermediate signed reals. (Contributed by NM, 10-May-1996.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → (〈𝐴, 0R〉 + 〈𝐵, 0R〉) = 〈(𝐴 +R 𝐵), 0R〉) | ||
Theorem | mulresr 10298 | Multiplication of real numbers in terms of intermediate signed reals. (Contributed by NM, 10-May-1996.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ R ∧ 𝐵 ∈ R) → (〈𝐴, 0R〉 · 〈𝐵, 0R〉) = 〈(𝐴 ·R 𝐵), 0R〉) | ||
Theorem | ltresr 10299 | Ordering of real subset of complex numbers in terms of signed reals. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.) |
⊢ (〈𝐴, 0R〉 <ℝ 〈𝐵, 0R〉 ↔ 𝐴 <R 𝐵) | ||
Theorem | ltresr2 10300 | Ordering of real subset of complex numbers in terms of signed reals. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.) |
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 <ℝ 𝐵 ↔ (1st ‘𝐴) <R (1st ‘𝐵))) |
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