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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 33697 | . . 3 ⊢ Rel /FldExt | |
| 2 | 1 | brrelex1i 5741 | . 2 ⊢ (𝐸/FldExt𝐹 → 𝐸 ∈ V) |
| 3 | elrelimasn 6104 | . . . 4 ⊢ (Rel /FldExt → (𝐹 ∈ (/FldExt “ {𝐸}) ↔ 𝐸/FldExt𝐹)) | |
| 4 | 1, 3 | ax-mp 5 | . . 3 ⊢ (𝐹 ∈ (/FldExt “ {𝐸}) ↔ 𝐸/FldExt𝐹) |
| 5 | 4 | biimpri 228 | . 2 ⊢ (𝐸/FldExt𝐹 → 𝐹 ∈ (/FldExt “ {𝐸})) |
| 6 | fvexd 6921 | . 2 ⊢ (𝐸/FldExt𝐹 → (dim‘((subringAlg ‘𝐸)‘(Base‘𝐹))) ∈ V) | |
| 7 | simpl 482 | . . . . . 6 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → 𝑒 = 𝐸) | |
| 8 | 7 | fveq2d 6910 | . . . . 5 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (subringAlg ‘𝑒) = (subringAlg ‘𝐸)) |
| 9 | simpr 484 | . . . . . 6 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → 𝑓 = 𝐹) | |
| 10 | 9 | fveq2d 6910 | . . . . 5 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (Base‘𝑓) = (Base‘𝐹)) |
| 11 | 8, 10 | fveq12d 6913 | . . . 4 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → ((subringAlg ‘𝑒)‘(Base‘𝑓)) = ((subringAlg ‘𝐸)‘(Base‘𝐹))) |
| 12 | 11 | fveq2d 6910 | . . 3 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (dim‘((subringAlg ‘𝑒)‘(Base‘𝑓))) = (dim‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) |
| 13 | sneq 4636 | . . . 4 ⊢ (𝑒 = 𝐸 → {𝑒} = {𝐸}) | |
| 14 | 13 | imaeq2d 6078 | . . 3 ⊢ (𝑒 = 𝐸 → (/FldExt “ {𝑒}) = (/FldExt “ {𝐸})) |
| 15 | df-extdg 33694 | . . 3 ⊢ [:] = (𝑒 ∈ V, 𝑓 ∈ (/FldExt “ {𝑒}) ↦ (dim‘((subringAlg ‘𝑒)‘(Base‘𝑓)))) | |
| 16 | 12, 14, 15 | ovmpox 7586 | . 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 1373 | 1 ⊢ (𝐸/FldExt𝐹 → (𝐸[:]𝐹) = (dim‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 {csn 4626 class class class wbr 5143 “ cima 5688 Rel wrel 5690 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 subringAlg csra 21170 dimcldim 33649 /FldExtcfldext 33689 [:]cextdg 33692 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-fldext 33693 df-extdg 33694 |
| This theorem is referenced by: extdgcl 33707 extdggt0 33708 extdgid 33711 extdgmul 33714 extdg1id 33716 ccfldextdgrr 33722 fldextrspunlem1 33725 fldextrspundgle 33728 algextdeglem4 33761 |
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