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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 | relopabi 5687 | . 2 ⊢ Rel ∼ |
3 | ecopopr.com | . . 3 ⊢ (𝑥 + 𝑦) = (𝑦 + 𝑥) | |
4 | 1, 3 | ecopovsym 8388 | . 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 8389 | . 2 ⊢ ((𝑓 ∼ 𝑔 ∧ 𝑔 ∼ ℎ) → 𝑓 ∼ ℎ) |
9 | vex 3495 | . . . . . . . . 9 ⊢ 𝑔 ∈ V | |
10 | vex 3495 | . . . . . . . . 9 ⊢ ℎ ∈ V | |
11 | 9, 10, 3 | caovcom 7334 | . . . . . . . 8 ⊢ (𝑔 + ℎ) = (ℎ + 𝑔) |
12 | 1 | ecopoveq 8387 | . . . . . . . 8 ⊢ (((𝑔 ∈ 𝑆 ∧ ℎ ∈ 𝑆) ∧ (𝑔 ∈ 𝑆 ∧ ℎ ∈ 𝑆)) → (〈𝑔, ℎ〉 ∼ 〈𝑔, ℎ〉 ↔ (𝑔 + ℎ) = (ℎ + 𝑔))) |
13 | 11, 12 | mpbiri 259 | . . . . . . 7 ⊢ (((𝑔 ∈ 𝑆 ∧ ℎ ∈ 𝑆) ∧ (𝑔 ∈ 𝑆 ∧ ℎ ∈ 𝑆)) → 〈𝑔, ℎ〉 ∼ 〈𝑔, ℎ〉) |
14 | 13 | anidms 567 | . . . . . 6 ⊢ ((𝑔 ∈ 𝑆 ∧ ℎ ∈ 𝑆) → 〈𝑔, ℎ〉 ∼ 〈𝑔, ℎ〉) |
15 | 14 | rgen2 3200 | . . . . 5 ⊢ ∀𝑔 ∈ 𝑆 ∀ℎ ∈ 𝑆 〈𝑔, ℎ〉 ∼ 〈𝑔, ℎ〉 |
16 | breq12 5062 | . . . . . . 7 ⊢ ((𝑓 = 〈𝑔, ℎ〉 ∧ 𝑓 = 〈𝑔, ℎ〉) → (𝑓 ∼ 𝑓 ↔ 〈𝑔, ℎ〉 ∼ 〈𝑔, ℎ〉)) | |
17 | 16 | anidms 567 | . . . . . 6 ⊢ (𝑓 = 〈𝑔, ℎ〉 → (𝑓 ∼ 𝑓 ↔ 〈𝑔, ℎ〉 ∼ 〈𝑔, ℎ〉)) |
18 | 17 | ralxp 5705 | . . . . 5 ⊢ (∀𝑓 ∈ (𝑆 × 𝑆)𝑓 ∼ 𝑓 ↔ ∀𝑔 ∈ 𝑆 ∀ℎ ∈ 𝑆 〈𝑔, ℎ〉 ∼ 〈𝑔, ℎ〉) |
19 | 15, 18 | mpbir 232 | . . . 4 ⊢ ∀𝑓 ∈ (𝑆 × 𝑆)𝑓 ∼ 𝑓 |
20 | 19 | rspec 3204 | . . 3 ⊢ (𝑓 ∈ (𝑆 × 𝑆) → 𝑓 ∼ 𝑓) |
21 | opabssxp 5636 | . . . . . 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 3998 | . . . . 5 ⊢ ∼ ⊆ ((𝑆 × 𝑆) × (𝑆 × 𝑆)) |
23 | 22 | ssbri 5102 | . . . 4 ⊢ (𝑓 ∼ 𝑓 → 𝑓((𝑆 × 𝑆) × (𝑆 × 𝑆))𝑓) |
24 | brxp 5594 | . . . . 5 ⊢ (𝑓((𝑆 × 𝑆) × (𝑆 × 𝑆))𝑓 ↔ (𝑓 ∈ (𝑆 × 𝑆) ∧ 𝑓 ∈ (𝑆 × 𝑆))) | |
25 | 24 | simplbi 498 | . . . 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 210 | . 2 ⊢ (𝑓 ∈ (𝑆 × 𝑆) ↔ 𝑓 ∼ 𝑓) |
28 | 2, 4, 8, 27 | iseri 8305 | 1 ⊢ ∼ Er (𝑆 × 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∃wex 1771 ∈ wcel 2105 ∀wral 3135 〈cop 4563 class class class wbr 5057 {copab 5119 × cxp 5546 (class class class)co 7145 Er wer 8275 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6307 df-fv 6356 df-ov 7148 df-er 8278 |
This theorem is referenced by: enqer 10331 enrer 10473 |
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