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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 31063 | . . 3 ⊢ (𝐸/FldExt𝐹 → 𝐸 ∈ Field) | |
2 | fldextfld2 31064 | . . 3 ⊢ (𝐸/FldExt𝐹 → 𝐹 ∈ Field) | |
3 | breq12 5064 | . . . . 5 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (𝑒/FldExt𝑓 ↔ 𝐸/FldExt𝐹)) | |
4 | oveq12 7158 | . . . . . 6 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (𝑒[:]𝑓) = (𝐸[:]𝐹)) | |
5 | 4 | eleq1d 2896 | . . . . 5 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → ((𝑒[:]𝑓) ∈ ℕ0 ↔ (𝐸[:]𝐹) ∈ ℕ0)) |
6 | 3, 5 | anbi12d 632 | . . . 4 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → ((𝑒/FldExt𝑓 ∧ (𝑒[:]𝑓) ∈ ℕ0) ↔ (𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) ∈ ℕ0))) |
7 | df-finext 31058 | . . . 4 ⊢ /FinExt = {〈𝑒, 𝑓〉 ∣ (𝑒/FldExt𝑓 ∧ (𝑒[:]𝑓) ∈ ℕ0)} | |
8 | 6, 7 | brabga 5414 | . . 3 ⊢ ((𝐸 ∈ Field ∧ 𝐹 ∈ Field) → (𝐸/FinExt𝐹 ↔ (𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) ∈ ℕ0))) |
9 | 1, 2, 8 | syl2anc 586 | . 2 ⊢ (𝐸/FldExt𝐹 → (𝐸/FinExt𝐹 ↔ (𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) ∈ ℕ0))) |
10 | 9 | bianabs 544 | 1 ⊢ (𝐸/FldExt𝐹 → (𝐸/FinExt𝐹 ↔ (𝐸[:]𝐹) ∈ ℕ0)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1536 ∈ wcel 2113 class class class wbr 5059 (class class class)co 7149 ℕ0cn0 11891 Fieldcfield 19498 /FldExtcfldext 31052 /FinExtcfinext 31053 [:]cextdg 31055 |
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 2792 ax-sep 5196 ax-nul 5203 ax-pr 5323 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5060 df-opab 5122 df-xp 5554 df-iota 6307 df-fv 6356 df-ov 7152 df-fldext 31056 df-finext 31058 |
This theorem is referenced by: finexttrb 31076 |
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