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Mirrors > Home > MPE Home > Th. List > ecopover | Structured version Visualization version GIF version |
Description: Assuming that operation 𝐹 is commutative (second hypothesis), closed (third hypothesis), associative (fourth hypothesis), and has the cancellation property (fifth hypothesis), show that the relation ∼, specified by the first hypothesis, is an equivalence relation. (Contributed by NM, 16-Feb-1996.) (Revised by Mario Carneiro, 12-Aug-2015.) (Proof shortened by AV, 1-May-2021.) |
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)))} |
ecopopr.com | ⊢ (𝑥 + 𝑦) = (𝑦 + 𝑥) |
ecopopr.cl | ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥 + 𝑦) ∈ 𝑆) |
ecopopr.ass | ⊢ ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)) |
ecopopr.can | ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → ((𝑥 + 𝑦) = (𝑥 + 𝑧) → 𝑦 = 𝑧)) |
Ref | Expression |
---|---|
ecopover | ⊢ ∼ Er (𝑆 × 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ecopopr.1 | . . 3 ⊢ ∼ = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ (𝑆 × 𝑆) ∧ 𝑦 ∈ (𝑆 × 𝑆)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = 〈𝑧, 𝑤〉 ∧ 𝑦 = 〈𝑣, 𝑢〉) ∧ (𝑧 + 𝑢) = (𝑤 + 𝑣)))} | |
2 | 1 | relopabiv 5833 | . 2 ⊢ Rel ∼ |
3 | ecopopr.com | . . 3 ⊢ (𝑥 + 𝑦) = (𝑦 + 𝑥) | |
4 | 1, 3 | ecopovsym 8858 | . 2 ⊢ (𝑓 ∼ 𝑔 → 𝑔 ∼ 𝑓) |
5 | ecopopr.cl | . . 3 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥 + 𝑦) ∈ 𝑆) | |
6 | ecopopr.ass | . . 3 ⊢ ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)) | |
7 | ecopopr.can | . . 3 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → ((𝑥 + 𝑦) = (𝑥 + 𝑧) → 𝑦 = 𝑧)) | |
8 | 1, 3, 5, 6, 7 | ecopovtrn 8859 | . 2 ⊢ ((𝑓 ∼ 𝑔 ∧ 𝑔 ∼ ℎ) → 𝑓 ∼ ℎ) |
9 | vex 3482 | . . . . . . . . 9 ⊢ 𝑔 ∈ V | |
10 | vex 3482 | . . . . . . . . 9 ⊢ ℎ ∈ V | |
11 | 9, 10, 3 | caovcom 7630 | . . . . . . . 8 ⊢ (𝑔 + ℎ) = (ℎ + 𝑔) |
12 | 1 | ecopoveq 8857 | . . . . . . . 8 ⊢ (((𝑔 ∈ 𝑆 ∧ ℎ ∈ 𝑆) ∧ (𝑔 ∈ 𝑆 ∧ ℎ ∈ 𝑆)) → (〈𝑔, ℎ〉 ∼ 〈𝑔, ℎ〉 ↔ (𝑔 + ℎ) = (ℎ + 𝑔))) |
13 | 11, 12 | mpbiri 258 | . . . . . . 7 ⊢ (((𝑔 ∈ 𝑆 ∧ ℎ ∈ 𝑆) ∧ (𝑔 ∈ 𝑆 ∧ ℎ ∈ 𝑆)) → 〈𝑔, ℎ〉 ∼ 〈𝑔, ℎ〉) |
14 | 13 | anidms 566 | . . . . . 6 ⊢ ((𝑔 ∈ 𝑆 ∧ ℎ ∈ 𝑆) → 〈𝑔, ℎ〉 ∼ 〈𝑔, ℎ〉) |
15 | 14 | rgen2 3197 | . . . . 5 ⊢ ∀𝑔 ∈ 𝑆 ∀ℎ ∈ 𝑆 〈𝑔, ℎ〉 ∼ 〈𝑔, ℎ〉 |
16 | breq12 5153 | . . . . . . 7 ⊢ ((𝑓 = 〈𝑔, ℎ〉 ∧ 𝑓 = 〈𝑔, ℎ〉) → (𝑓 ∼ 𝑓 ↔ 〈𝑔, ℎ〉 ∼ 〈𝑔, ℎ〉)) | |
17 | 16 | anidms 566 | . . . . . 6 ⊢ (𝑓 = 〈𝑔, ℎ〉 → (𝑓 ∼ 𝑓 ↔ 〈𝑔, ℎ〉 ∼ 〈𝑔, ℎ〉)) |
18 | 17 | ralxp 5855 | . . . . 5 ⊢ (∀𝑓 ∈ (𝑆 × 𝑆)𝑓 ∼ 𝑓 ↔ ∀𝑔 ∈ 𝑆 ∀ℎ ∈ 𝑆 〈𝑔, ℎ〉 ∼ 〈𝑔, ℎ〉) |
19 | 15, 18 | mpbir 231 | . . . 4 ⊢ ∀𝑓 ∈ (𝑆 × 𝑆)𝑓 ∼ 𝑓 |
20 | 19 | rspec 3248 | . . 3 ⊢ (𝑓 ∈ (𝑆 × 𝑆) → 𝑓 ∼ 𝑓) |
21 | opabssxp 5781 | . . . . . 6 ⊢ {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ (𝑆 × 𝑆) ∧ 𝑦 ∈ (𝑆 × 𝑆)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = 〈𝑧, 𝑤〉 ∧ 𝑦 = 〈𝑣, 𝑢〉) ∧ (𝑧 + 𝑢) = (𝑤 + 𝑣)))} ⊆ ((𝑆 × 𝑆) × (𝑆 × 𝑆)) | |
22 | 1, 21 | eqsstri 4030 | . . . . 5 ⊢ ∼ ⊆ ((𝑆 × 𝑆) × (𝑆 × 𝑆)) |
23 | 22 | ssbri 5193 | . . . 4 ⊢ (𝑓 ∼ 𝑓 → 𝑓((𝑆 × 𝑆) × (𝑆 × 𝑆))𝑓) |
24 | brxp 5738 | . . . . 5 ⊢ (𝑓((𝑆 × 𝑆) × (𝑆 × 𝑆))𝑓 ↔ (𝑓 ∈ (𝑆 × 𝑆) ∧ 𝑓 ∈ (𝑆 × 𝑆))) | |
25 | 24 | simplbi 497 | . . . 4 ⊢ (𝑓((𝑆 × 𝑆) × (𝑆 × 𝑆))𝑓 → 𝑓 ∈ (𝑆 × 𝑆)) |
26 | 23, 25 | syl 17 | . . 3 ⊢ (𝑓 ∼ 𝑓 → 𝑓 ∈ (𝑆 × 𝑆)) |
27 | 20, 26 | impbii 209 | . 2 ⊢ (𝑓 ∈ (𝑆 × 𝑆) ↔ 𝑓 ∼ 𝑓) |
28 | 2, 4, 8, 27 | iseri 8771 | 1 ⊢ ∼ Er (𝑆 × 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∃wex 1776 ∈ wcel 2106 ∀wral 3059 〈cop 4637 class class class wbr 5148 {copab 5210 × cxp 5687 (class class class)co 7431 Er wer 8741 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fv 6571 df-ov 7434 df-er 8744 |
This theorem is referenced by: enqer 10959 enrer 11101 |
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