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Mirrors > Home > MPE Home > Th. List > Mathboxes > isdomn5 | Structured version Visualization version GIF version |
Description: The right conjunct in the right hand side of the equivalence of isdomn 20286 is logically equivalent to a less symmetric version where one of the variables is restricted to be nonzero. (Contributed by SN, 16-Sep-2024.) |
Ref | Expression |
---|---|
isdomn5 | ⊢ (∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝑎 · 𝑏) = 0 → (𝑎 = 0 ∨ 𝑏 = 0 )) ↔ ∀𝑎 ∈ (𝐵 ∖ { 0 })∀𝑏 ∈ 𝐵 ((𝑎 · 𝑏) = 0 → 𝑏 = 0 )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bi2.04 392 | . . . 4 ⊢ ((¬ 𝑎 = 0 → ((𝑎 · 𝑏) = 0 → 𝑏 = 0 )) ↔ ((𝑎 · 𝑏) = 0 → (¬ 𝑎 = 0 → 𝑏 = 0 ))) | |
2 | df-ne 2933 | . . . . 5 ⊢ (𝑎 ≠ 0 ↔ ¬ 𝑎 = 0 ) | |
3 | 2 | imbi1i 353 | . . . 4 ⊢ ((𝑎 ≠ 0 → ((𝑎 · 𝑏) = 0 → 𝑏 = 0 )) ↔ (¬ 𝑎 = 0 → ((𝑎 · 𝑏) = 0 → 𝑏 = 0 ))) |
4 | df-or 848 | . . . . 5 ⊢ ((𝑎 = 0 ∨ 𝑏 = 0 ) ↔ (¬ 𝑎 = 0 → 𝑏 = 0 )) | |
5 | 4 | imbi2i 339 | . . . 4 ⊢ (((𝑎 · 𝑏) = 0 → (𝑎 = 0 ∨ 𝑏 = 0 )) ↔ ((𝑎 · 𝑏) = 0 → (¬ 𝑎 = 0 → 𝑏 = 0 ))) |
6 | 1, 3, 5 | 3bitr4ri 307 | . . 3 ⊢ (((𝑎 · 𝑏) = 0 → (𝑎 = 0 ∨ 𝑏 = 0 )) ↔ (𝑎 ≠ 0 → ((𝑎 · 𝑏) = 0 → 𝑏 = 0 ))) |
7 | 6 | 2ralbii 3079 | . 2 ⊢ (∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝑎 · 𝑏) = 0 → (𝑎 = 0 ∨ 𝑏 = 0 )) ↔ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝑎 ≠ 0 → ((𝑎 · 𝑏) = 0 → 𝑏 = 0 ))) |
8 | r19.21v 3088 | . . 3 ⊢ (∀𝑏 ∈ 𝐵 (𝑎 ≠ 0 → ((𝑎 · 𝑏) = 0 → 𝑏 = 0 )) ↔ (𝑎 ≠ 0 → ∀𝑏 ∈ 𝐵 ((𝑎 · 𝑏) = 0 → 𝑏 = 0 ))) | |
9 | 8 | ralbii 3078 | . 2 ⊢ (∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝑎 ≠ 0 → ((𝑎 · 𝑏) = 0 → 𝑏 = 0 )) ↔ ∀𝑎 ∈ 𝐵 (𝑎 ≠ 0 → ∀𝑏 ∈ 𝐵 ((𝑎 · 𝑏) = 0 → 𝑏 = 0 ))) |
10 | raldifsnb 4695 | . 2 ⊢ (∀𝑎 ∈ 𝐵 (𝑎 ≠ 0 → ∀𝑏 ∈ 𝐵 ((𝑎 · 𝑏) = 0 → 𝑏 = 0 )) ↔ ∀𝑎 ∈ (𝐵 ∖ { 0 })∀𝑏 ∈ 𝐵 ((𝑎 · 𝑏) = 0 → 𝑏 = 0 )) | |
11 | 7, 9, 10 | 3bitri 300 | 1 ⊢ (∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝑎 · 𝑏) = 0 → (𝑎 = 0 ∨ 𝑏 = 0 )) ↔ ∀𝑎 ∈ (𝐵 ∖ { 0 })∀𝑏 ∈ 𝐵 ((𝑎 · 𝑏) = 0 → 𝑏 = 0 )) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∨ wo 847 = wceq 1543 ≠ wne 2932 ∀wral 3051 ∖ cdif 3850 {csn 4527 (class class class)co 7191 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-tru 1546 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-ne 2933 df-nel 3037 df-ral 3056 df-v 3400 df-dif 3856 df-sn 4528 |
This theorem is referenced by: isdomn4 39835 |
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