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| 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 5830 | . 2 ⊢ Rel ∼ | 
| 3 | ecopopr.com | . . 3 ⊢ (𝑥 + 𝑦) = (𝑦 + 𝑥) | |
| 4 | 1, 3 | ecopovsym 8859 | . 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 8860 | . 2 ⊢ ((𝑓 ∼ 𝑔 ∧ 𝑔 ∼ ℎ) → 𝑓 ∼ ℎ) | 
| 9 | vex 3484 | . . . . . . . . 9 ⊢ 𝑔 ∈ V | |
| 10 | vex 3484 | . . . . . . . . 9 ⊢ ℎ ∈ V | |
| 11 | 9, 10, 3 | caovcom 7630 | . . . . . . . 8 ⊢ (𝑔 + ℎ) = (ℎ + 𝑔) | 
| 12 | 1 | ecopoveq 8858 | . . . . . . . 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 3199 | . . . . 5 ⊢ ∀𝑔 ∈ 𝑆 ∀ℎ ∈ 𝑆 〈𝑔, ℎ〉 ∼ 〈𝑔, ℎ〉 | 
| 16 | breq12 5148 | . . . . . . 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 5852 | . . . . 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 3250 | . . 3 ⊢ (𝑓 ∈ (𝑆 × 𝑆) → 𝑓 ∼ 𝑓) | 
| 21 | opabssxp 5778 | . . . . . 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 5188 | . . . 4 ⊢ (𝑓 ∼ 𝑓 → 𝑓((𝑆 × 𝑆) × (𝑆 × 𝑆))𝑓) | 
| 24 | brxp 5734 | . . . . 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 8772 | 1 ⊢ ∼ Er (𝑆 × 𝑆) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∃wex 1779 ∈ wcel 2108 ∀wral 3061 〈cop 4632 class class class wbr 5143 {copab 5205 × cxp 5683 (class class class)co 7431 Er wer 8742 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fv 6569 df-ov 7434 df-er 8745 | 
| This theorem is referenced by: enqer 10961 enrer 11103 | 
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