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Mirrors > Home > MPE Home > Th. List > ecopoveq | Structured version Visualization version GIF version |
Description: This is the first of several theorems about equivalence relations of the kind used in construction of fractions and signed reals, involving operations on equivalent classes of ordered pairs. This theorem expresses the relation ∼ (specified by the hypothesis) in terms of its operation 𝐹. (Contributed by NM, 16-Aug-1995.) |
Ref | Expression |
---|---|
ecopopr.1 | ⊢ ∼ = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ (𝑆 × 𝑆) ∧ 𝑦 ∈ (𝑆 × 𝑆)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = 〈𝑧, 𝑤〉 ∧ 𝑦 = 〈𝑣, 𝑢〉) ∧ (𝑧 + 𝑢) = (𝑤 + 𝑣)))} |
Ref | Expression |
---|---|
ecopoveq | ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → (〈𝐴, 𝐵〉 ∼ 〈𝐶, 𝐷〉 ↔ (𝐴 + 𝐷) = (𝐵 + 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq12 6659 | . . . 4 ⊢ ((𝑧 = 𝐴 ∧ 𝑢 = 𝐷) → (𝑧 + 𝑢) = (𝐴 + 𝐷)) | |
2 | oveq12 6659 | . . . 4 ⊢ ((𝑤 = 𝐵 ∧ 𝑣 = 𝐶) → (𝑤 + 𝑣) = (𝐵 + 𝐶)) | |
3 | 1, 2 | eqeqan12d 2638 | . . 3 ⊢ (((𝑧 = 𝐴 ∧ 𝑢 = 𝐷) ∧ (𝑤 = 𝐵 ∧ 𝑣 = 𝐶)) → ((𝑧 + 𝑢) = (𝑤 + 𝑣) ↔ (𝐴 + 𝐷) = (𝐵 + 𝐶))) |
4 | 3 | an42s 870 | . 2 ⊢ (((𝑧 = 𝐴 ∧ 𝑤 = 𝐵) ∧ (𝑣 = 𝐶 ∧ 𝑢 = 𝐷)) → ((𝑧 + 𝑢) = (𝑤 + 𝑣) ↔ (𝐴 + 𝐷) = (𝐵 + 𝐶))) |
5 | ecopopr.1 | . 2 ⊢ ∼ = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ (𝑆 × 𝑆) ∧ 𝑦 ∈ (𝑆 × 𝑆)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = 〈𝑧, 𝑤〉 ∧ 𝑦 = 〈𝑣, 𝑢〉) ∧ (𝑧 + 𝑢) = (𝑤 + 𝑣)))} | |
6 | 4, 5 | opbrop 5198 | 1 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ (𝐶 ∈ 𝑆 ∧ 𝐷 ∈ 𝑆)) → (〈𝐴, 𝐵〉 ∼ 〈𝐶, 𝐷〉 ↔ (𝐴 + 𝐷) = (𝐵 + 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∃wex 1704 ∈ wcel 1990 〈cop 4183 class class class wbr 4653 {copab 4712 × cxp 5112 (class class class)co 6650 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-xp 5120 df-iota 5851 df-fv 5896 df-ov 6653 |
This theorem is referenced by: ecopovsym 7849 ecopovtrn 7850 ecopover 7851 ecopoverOLD 7852 enqbreq 9741 enrbreq 9885 prsrlem1 9893 |
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