![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > isrrg | Structured version Visualization version GIF version |
Description: Membership in the set of left-regular elements. (Contributed by Stefan O'Rear, 22-Mar-2015.) |
Ref | Expression |
---|---|
rrgval.e | ⊢ 𝐸 = (RLReg‘𝑅) |
rrgval.b | ⊢ 𝐵 = (Base‘𝑅) |
rrgval.t | ⊢ · = (.r‘𝑅) |
rrgval.z | ⊢ 0 = (0g‘𝑅) |
Ref | Expression |
---|---|
isrrg | ⊢ (𝑋 ∈ 𝐸 ↔ (𝑋 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ((𝑋 · 𝑦) = 0 → 𝑦 = 0 ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq1 7438 | . . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥 · 𝑦) = (𝑋 · 𝑦)) | |
2 | 1 | eqeq1d 2737 | . . . 4 ⊢ (𝑥 = 𝑋 → ((𝑥 · 𝑦) = 0 ↔ (𝑋 · 𝑦) = 0 )) |
3 | 2 | imbi1d 341 | . . 3 ⊢ (𝑥 = 𝑋 → (((𝑥 · 𝑦) = 0 → 𝑦 = 0 ) ↔ ((𝑋 · 𝑦) = 0 → 𝑦 = 0 ))) |
4 | 3 | ralbidv 3176 | . 2 ⊢ (𝑥 = 𝑋 → (∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → 𝑦 = 0 ) ↔ ∀𝑦 ∈ 𝐵 ((𝑋 · 𝑦) = 0 → 𝑦 = 0 ))) |
5 | rrgval.e | . . 3 ⊢ 𝐸 = (RLReg‘𝑅) | |
6 | rrgval.b | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
7 | rrgval.t | . . 3 ⊢ · = (.r‘𝑅) | |
8 | rrgval.z | . . 3 ⊢ 0 = (0g‘𝑅) | |
9 | 5, 6, 7, 8 | rrgval 20714 | . 2 ⊢ 𝐸 = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → 𝑦 = 0 )} |
10 | 4, 9 | elrab2 3698 | 1 ⊢ (𝑋 ∈ 𝐸 ↔ (𝑋 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ((𝑋 · 𝑦) = 0 → 𝑦 = 0 ))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∀wral 3059 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 .rcmulr 17299 0gc0g 17486 RLRegcrlreg 20708 |
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-mo 2538 df-eu 2567 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-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fv 6571 df-ov 7434 df-rlreg 20711 |
This theorem is referenced by: rrgeq0i 20716 unitrrg 20720 isdomn2 20728 isdomn2OLD 20729 rrgsubm 33268 zringidom 33559 |
Copyright terms: Public domain | W3C validator |