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Mirrors > Home > MPE Home > Th. List > rrgeq0i | Structured version Visualization version GIF version |
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 19288 | . . 3 ⊢ (𝑋 ∈ 𝐸 ↔ (𝑋 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ((𝑋 · 𝑦) = 0 → 𝑦 = 0 ))) |
6 | 5 | simprbi 480 | . 2 ⊢ (𝑋 ∈ 𝐸 → ∀𝑦 ∈ 𝐵 ((𝑋 · 𝑦) = 0 → 𝑦 = 0 )) |
7 | oveq2 6658 | . . . . 5 ⊢ (𝑦 = 𝑌 → (𝑋 · 𝑦) = (𝑋 · 𝑌)) | |
8 | 7 | eqeq1d 2624 | . . . 4 ⊢ (𝑦 = 𝑌 → ((𝑋 · 𝑦) = 0 ↔ (𝑋 · 𝑌) = 0 )) |
9 | eqeq1 2626 | . . . 4 ⊢ (𝑦 = 𝑌 → (𝑦 = 0 ↔ 𝑌 = 0 )) | |
10 | 8, 9 | imbi12d 334 | . . 3 ⊢ (𝑦 = 𝑌 → (((𝑋 · 𝑦) = 0 → 𝑦 = 0 ) ↔ ((𝑋 · 𝑌) = 0 → 𝑌 = 0 ))) |
11 | 10 | rspcv 3305 | . 2 ⊢ (𝑌 ∈ 𝐵 → (∀𝑦 ∈ 𝐵 ((𝑋 · 𝑦) = 0 → 𝑦 = 0 ) → ((𝑋 · 𝑌) = 0 → 𝑌 = 0 ))) |
12 | 6, 11 | mpan9 486 | 1 ⊢ ((𝑋 ∈ 𝐸 ∧ 𝑌 ∈ 𝐵) → ((𝑋 · 𝑌) = 0 → 𝑌 = 0 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 .rcmulr 15942 0gc0g 16100 RLRegcrlreg 19279 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-ov 6653 df-rlreg 19283 |
This theorem is referenced by: rrgeq0 19290 znrrg 19914 deg1mul2 23874 |
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