Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > brfinext | Structured version Visualization version GIF version |
Description: The finite field extension relation explicited. (Contributed by Thierry Arnoux, 29-Jul-2023.) |
Ref | Expression |
---|---|
brfinext | ⊢ (𝐸/FldExt𝐹 → (𝐸/FinExt𝐹 ↔ (𝐸[:]𝐹) ∈ ℕ0)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fldextfld1 31462 | . . 3 ⊢ (𝐸/FldExt𝐹 → 𝐸 ∈ Field) | |
2 | fldextfld2 31463 | . . 3 ⊢ (𝐸/FldExt𝐹 → 𝐹 ∈ Field) | |
3 | breq12 5072 | . . . . 5 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (𝑒/FldExt𝑓 ↔ 𝐸/FldExt𝐹)) | |
4 | oveq12 7240 | . . . . . 6 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (𝑒[:]𝑓) = (𝐸[:]𝐹)) | |
5 | 4 | eleq1d 2823 | . . . . 5 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → ((𝑒[:]𝑓) ∈ ℕ0 ↔ (𝐸[:]𝐹) ∈ ℕ0)) |
6 | 3, 5 | anbi12d 634 | . . . 4 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → ((𝑒/FldExt𝑓 ∧ (𝑒[:]𝑓) ∈ ℕ0) ↔ (𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) ∈ ℕ0))) |
7 | df-finext 31457 | . . . 4 ⊢ /FinExt = {〈𝑒, 𝑓〉 ∣ (𝑒/FldExt𝑓 ∧ (𝑒[:]𝑓) ∈ ℕ0)} | |
8 | 6, 7 | brabga 5429 | . . 3 ⊢ ((𝐸 ∈ Field ∧ 𝐹 ∈ Field) → (𝐸/FinExt𝐹 ↔ (𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) ∈ ℕ0))) |
9 | 1, 2, 8 | syl2anc 587 | . 2 ⊢ (𝐸/FldExt𝐹 → (𝐸/FinExt𝐹 ↔ (𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) ∈ ℕ0))) |
10 | 9 | bianabs 545 | 1 ⊢ (𝐸/FldExt𝐹 → (𝐸/FinExt𝐹 ↔ (𝐸[:]𝐹) ∈ ℕ0)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2111 class class class wbr 5067 (class class class)co 7231 ℕ0cn0 12114 Fieldcfield 19792 /FldExtcfldext 31451 /FinExtcfinext 31452 [:]cextdg 31454 |
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 2016 ax-8 2113 ax-9 2121 ax-ext 2709 ax-sep 5206 ax-nul 5213 ax-pr 5336 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2072 df-clab 2716 df-cleq 2730 df-clel 2817 df-ral 3067 df-rex 3068 df-rab 3071 df-v 3422 df-dif 3883 df-un 3885 df-in 3887 df-ss 3897 df-nul 4252 df-if 4454 df-sn 4556 df-pr 4558 df-op 4562 df-uni 4834 df-br 5068 df-opab 5130 df-xp 5571 df-iota 6355 df-fv 6405 df-ov 7234 df-fldext 31455 df-finext 31457 |
This theorem is referenced by: finexttrb 31475 |
Copyright terms: Public domain | W3C validator |