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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > extdgval | Structured version Visualization version GIF version |
Description: Value of the field extension degree operation. (Contributed by Thierry Arnoux, 29-Jul-2023.) |
Ref | Expression |
---|---|
extdgval | ⊢ (𝐸/FldExt𝐹 → (𝐸[:]𝐹) = (dim‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relfldext 32392 | . . 3 ⊢ Rel /FldExt | |
2 | 1 | brrelex1i 5689 | . 2 ⊢ (𝐸/FldExt𝐹 → 𝐸 ∈ V) |
3 | elrelimasn 6038 | . . . 4 ⊢ (Rel /FldExt → (𝐹 ∈ (/FldExt “ {𝐸}) ↔ 𝐸/FldExt𝐹)) | |
4 | 1, 3 | ax-mp 5 | . . 3 ⊢ (𝐹 ∈ (/FldExt “ {𝐸}) ↔ 𝐸/FldExt𝐹) |
5 | 4 | biimpri 227 | . 2 ⊢ (𝐸/FldExt𝐹 → 𝐹 ∈ (/FldExt “ {𝐸})) |
6 | fvexd 6858 | . 2 ⊢ (𝐸/FldExt𝐹 → (dim‘((subringAlg ‘𝐸)‘(Base‘𝐹))) ∈ V) | |
7 | simpl 484 | . . . . . 6 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → 𝑒 = 𝐸) | |
8 | 7 | fveq2d 6847 | . . . . 5 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (subringAlg ‘𝑒) = (subringAlg ‘𝐸)) |
9 | simpr 486 | . . . . . 6 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → 𝑓 = 𝐹) | |
10 | 9 | fveq2d 6847 | . . . . 5 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (Base‘𝑓) = (Base‘𝐹)) |
11 | 8, 10 | fveq12d 6850 | . . . 4 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → ((subringAlg ‘𝑒)‘(Base‘𝑓)) = ((subringAlg ‘𝐸)‘(Base‘𝐹))) |
12 | 11 | fveq2d 6847 | . . 3 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (dim‘((subringAlg ‘𝑒)‘(Base‘𝑓))) = (dim‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) |
13 | sneq 4597 | . . . 4 ⊢ (𝑒 = 𝐸 → {𝑒} = {𝐸}) | |
14 | 13 | imaeq2d 6014 | . . 3 ⊢ (𝑒 = 𝐸 → (/FldExt “ {𝑒}) = (/FldExt “ {𝐸})) |
15 | df-extdg 32389 | . . 3 ⊢ [:] = (𝑒 ∈ V, 𝑓 ∈ (/FldExt “ {𝑒}) ↦ (dim‘((subringAlg ‘𝑒)‘(Base‘𝑓)))) | |
16 | 12, 14, 15 | ovmpox 7509 | . 2 ⊢ ((𝐸 ∈ V ∧ 𝐹 ∈ (/FldExt “ {𝐸}) ∧ (dim‘((subringAlg ‘𝐸)‘(Base‘𝐹))) ∈ V) → (𝐸[:]𝐹) = (dim‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) |
17 | 2, 5, 6, 16 | syl3anc 1372 | 1 ⊢ (𝐸/FldExt𝐹 → (𝐸[:]𝐹) = (dim‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 Vcvv 3444 {csn 4587 class class class wbr 5106 “ cima 5637 Rel wrel 5639 ‘cfv 6497 (class class class)co 7358 Basecbs 17088 subringAlg csra 20645 dimcldim 32353 /FldExtcfldext 32384 [:]cextdg 32387 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-sbc 3741 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fv 6505 df-ov 7361 df-oprab 7362 df-mpo 7363 df-fldext 32388 df-extdg 32389 |
This theorem is referenced by: extdgcl 32402 extdggt0 32403 extdgid 32406 extdgmul 32407 extdg1id 32409 ccfldextdgrr 32413 |
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