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Mirrors > Home > MPE Home > Th. List > gaorb | Structured version Visualization version GIF version |
Description: The orbit equivalence relation puts two points in the group action in the same equivalence class iff there is a group element that takes one element to the other. (Contributed by Mario Carneiro, 14-Jan-2015.) |
Ref | Expression |
---|---|
gaorb.1 | ⊢ ∼ = {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝑌 ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)} |
Ref | Expression |
---|---|
gaorb | ⊢ (𝐴 ∼ 𝐵 ↔ (𝐴 ∈ 𝑌 ∧ 𝐵 ∈ 𝑌 ∧ ∃ℎ ∈ 𝑋 (ℎ ⊕ 𝐴) = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 7276 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑔 ⊕ 𝑥) = (𝑔 ⊕ 𝐴)) | |
2 | eqeq12 2755 | . . . . . 6 ⊢ (((𝑔 ⊕ 𝑥) = (𝑔 ⊕ 𝐴) ∧ 𝑦 = 𝐵) → ((𝑔 ⊕ 𝑥) = 𝑦 ↔ (𝑔 ⊕ 𝐴) = 𝐵)) | |
3 | 1, 2 | sylan 580 | . . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ((𝑔 ⊕ 𝑥) = 𝑦 ↔ (𝑔 ⊕ 𝐴) = 𝐵)) |
4 | 3 | rexbidv 3224 | . . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦 ↔ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝐴) = 𝐵)) |
5 | oveq1 7275 | . . . . . 6 ⊢ (𝑔 = ℎ → (𝑔 ⊕ 𝐴) = (ℎ ⊕ 𝐴)) | |
6 | 5 | eqeq1d 2740 | . . . . 5 ⊢ (𝑔 = ℎ → ((𝑔 ⊕ 𝐴) = 𝐵 ↔ (ℎ ⊕ 𝐴) = 𝐵)) |
7 | 6 | cbvrexvw 3382 | . . . 4 ⊢ (∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝐴) = 𝐵 ↔ ∃ℎ ∈ 𝑋 (ℎ ⊕ 𝐴) = 𝐵) |
8 | 4, 7 | bitrdi 287 | . . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦 ↔ ∃ℎ ∈ 𝑋 (ℎ ⊕ 𝐴) = 𝐵)) |
9 | gaorb.1 | . . . 4 ⊢ ∼ = {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝑌 ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)} | |
10 | vex 3434 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
11 | vex 3434 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
12 | 10, 11 | prss 4754 | . . . . . 6 ⊢ ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌) ↔ {𝑥, 𝑦} ⊆ 𝑌) |
13 | 12 | anbi1i 624 | . . . . 5 ⊢ (((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌) ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦) ↔ ({𝑥, 𝑦} ⊆ 𝑌 ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)) |
14 | 13 | opabbii 5141 | . . . 4 ⊢ {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌) ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)} = {〈𝑥, 𝑦〉 ∣ ({𝑥, 𝑦} ⊆ 𝑌 ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)} |
15 | 9, 14 | eqtr4i 2769 | . . 3 ⊢ ∼ = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝑌 ∧ 𝑦 ∈ 𝑌) ∧ ∃𝑔 ∈ 𝑋 (𝑔 ⊕ 𝑥) = 𝑦)} |
16 | 8, 15 | brab2a 5675 | . 2 ⊢ (𝐴 ∼ 𝐵 ↔ ((𝐴 ∈ 𝑌 ∧ 𝐵 ∈ 𝑌) ∧ ∃ℎ ∈ 𝑋 (ℎ ⊕ 𝐴) = 𝐵)) |
17 | df-3an 1088 | . 2 ⊢ ((𝐴 ∈ 𝑌 ∧ 𝐵 ∈ 𝑌 ∧ ∃ℎ ∈ 𝑋 (ℎ ⊕ 𝐴) = 𝐵) ↔ ((𝐴 ∈ 𝑌 ∧ 𝐵 ∈ 𝑌) ∧ ∃ℎ ∈ 𝑋 (ℎ ⊕ 𝐴) = 𝐵)) | |
18 | 16, 17 | bitr4i 277 | 1 ⊢ (𝐴 ∼ 𝐵 ↔ (𝐴 ∈ 𝑌 ∧ 𝐵 ∈ 𝑌 ∧ ∃ℎ ∈ 𝑋 (ℎ ⊕ 𝐴) = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ∃wrex 3065 ⊆ wss 3887 {cpr 4564 class class class wbr 5074 {copab 5136 (class class class)co 7268 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 ax-sep 5222 ax-nul 5229 ax-pr 5351 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3432 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4258 df-if 4461 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4841 df-br 5075 df-opab 5137 df-xp 5591 df-iota 6385 df-fv 6435 df-ov 7271 |
This theorem is referenced by: gaorber 18902 orbsta 18907 sylow2alem1 19210 sylow2alem2 19211 sylow3lem3 19222 lsmsnorb 31565 |
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