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| Description: Property of a left-regular element. (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 | 
|---|---|
| rrgeq0i | ⊢ ((𝑋 ∈ 𝐸 ∧ 𝑌 ∈ 𝐵) → ((𝑋 · 𝑌) = 0 → 𝑌 = 0 )) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | rrgval.e | . . . 4 ⊢ 𝐸 = (RLReg‘𝑅) | |
| 2 | rrgval.b | . . . 4 ⊢ 𝐵 = (Base‘𝑅) | |
| 3 | rrgval.t | . . . 4 ⊢ · = (.r‘𝑅) | |
| 4 | rrgval.z | . . . 4 ⊢ 0 = (0g‘𝑅) | |
| 5 | 1, 2, 3, 4 | isrrg 20699 | . . 3 ⊢ (𝑋 ∈ 𝐸 ↔ (𝑋 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ((𝑋 · 𝑦) = 0 → 𝑦 = 0 ))) | 
| 6 | 5 | simprbi 496 | . 2 ⊢ (𝑋 ∈ 𝐸 → ∀𝑦 ∈ 𝐵 ((𝑋 · 𝑦) = 0 → 𝑦 = 0 )) | 
| 7 | oveq2 7440 | . . . . 5 ⊢ (𝑦 = 𝑌 → (𝑋 · 𝑦) = (𝑋 · 𝑌)) | |
| 8 | 7 | eqeq1d 2738 | . . . 4 ⊢ (𝑦 = 𝑌 → ((𝑋 · 𝑦) = 0 ↔ (𝑋 · 𝑌) = 0 )) | 
| 9 | eqeq1 2740 | . . . 4 ⊢ (𝑦 = 𝑌 → (𝑦 = 0 ↔ 𝑌 = 0 )) | |
| 10 | 8, 9 | imbi12d 344 | . . 3 ⊢ (𝑦 = 𝑌 → (((𝑋 · 𝑦) = 0 → 𝑦 = 0 ) ↔ ((𝑋 · 𝑌) = 0 → 𝑌 = 0 ))) | 
| 11 | 10 | rspcv 3617 | . 2 ⊢ (𝑌 ∈ 𝐵 → (∀𝑦 ∈ 𝐵 ((𝑋 · 𝑦) = 0 → 𝑦 = 0 ) → ((𝑋 · 𝑌) = 0 → 𝑌 = 0 ))) | 
| 12 | 6, 11 | mpan9 506 | 1 ⊢ ((𝑋 ∈ 𝐸 ∧ 𝑌 ∈ 𝐵) → ((𝑋 · 𝑌) = 0 → 𝑌 = 0 )) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ∀wral 3060 ‘cfv 6560 (class class class)co 7432 Basecbs 17248 .rcmulr 17299 0gc0g 17485 RLRegcrlreg 20692 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-iota 6513 df-fun 6562 df-fv 6568 df-ov 7435 df-rlreg 20695 | 
| This theorem is referenced by: rrgeq0 20701 znrrg 21585 deg1mul2 26154 rlocf1 33278 rrgsubm 33288 fracerl 33309 assalactf1o 33687 | 
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