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Mirrors > Home > MPE Home > Th. List > releqg | Structured version Visualization version GIF version |
Description: The left coset equivalence relation is a relation. (Contributed by Mario Carneiro, 14-Jun-2015.) |
Ref | Expression |
---|---|
releqg.r | ⊢ 𝑅 = (𝐺 ~QG 𝑆) |
Ref | Expression |
---|---|
releqg | ⊢ Rel 𝑅 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-eqg 18216 | . . 3 ⊢ ~QG = (𝑔 ∈ V, 𝑠 ∈ V ↦ {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ (Base‘𝑔) ∧ (((invg‘𝑔)‘𝑥)(+g‘𝑔)𝑦) ∈ 𝑠)}) | |
2 | 1 | relmpoopab 7778 | . 2 ⊢ Rel (𝐺 ~QG 𝑆) |
3 | releqg.r | . . 3 ⊢ 𝑅 = (𝐺 ~QG 𝑆) | |
4 | 3 | releqi 5645 | . 2 ⊢ (Rel 𝑅 ↔ Rel (𝐺 ~QG 𝑆)) |
5 | 2, 4 | mpbir 232 | 1 ⊢ Rel 𝑅 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1528 ∈ wcel 2105 Vcvv 3492 ⊆ wss 3933 {cpr 4559 Rel wrel 5553 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 +gcplusg 16553 invgcminusg 18042 ~QG cqg 18213 |
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-pow 5257 ax-pr 5320 ax-un 7450 |
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-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-1st 7678 df-2nd 7679 df-eqg 18216 |
This theorem is referenced by: eqger 18268 eqgid 18270 tgptsmscls 22685 |
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