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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 20064 | . . 3 ⊢ (𝑋 ∈ 𝐸 ↔ (𝑋 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ((𝑋 · 𝑦) = 0 → 𝑦 = 0 ))) |
6 | 5 | simprbi 499 | . 2 ⊢ (𝑋 ∈ 𝐸 → ∀𝑦 ∈ 𝐵 ((𝑋 · 𝑦) = 0 → 𝑦 = 0 )) |
7 | oveq2 7167 | . . . . 5 ⊢ (𝑦 = 𝑌 → (𝑋 · 𝑦) = (𝑋 · 𝑌)) | |
8 | 7 | eqeq1d 2826 | . . . 4 ⊢ (𝑦 = 𝑌 → ((𝑋 · 𝑦) = 0 ↔ (𝑋 · 𝑌) = 0 )) |
9 | eqeq1 2828 | . . . 4 ⊢ (𝑦 = 𝑌 → (𝑦 = 0 ↔ 𝑌 = 0 )) | |
10 | 8, 9 | imbi12d 347 | . . 3 ⊢ (𝑦 = 𝑌 → (((𝑋 · 𝑦) = 0 → 𝑦 = 0 ) ↔ ((𝑋 · 𝑌) = 0 → 𝑌 = 0 ))) |
11 | 10 | rspcv 3621 | . 2 ⊢ (𝑌 ∈ 𝐵 → (∀𝑦 ∈ 𝐵 ((𝑋 · 𝑦) = 0 → 𝑦 = 0 ) → ((𝑋 · 𝑌) = 0 → 𝑌 = 0 ))) |
12 | 6, 11 | mpan9 509 | 1 ⊢ ((𝑋 ∈ 𝐸 ∧ 𝑌 ∈ 𝐵) → ((𝑋 · 𝑌) = 0 → 𝑌 = 0 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ∀wral 3141 ‘cfv 6358 (class class class)co 7159 Basecbs 16486 .rcmulr 16569 0gc0g 16716 RLRegcrlreg 20055 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-iota 6317 df-fun 6360 df-fv 6366 df-ov 7162 df-rlreg 20059 |
This theorem is referenced by: rrgeq0 20066 znrrg 20715 deg1mul2 24711 |
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