Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
||
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 31060 | . . 3 ⊢ Rel /FldExt | |
2 | 1 | brrelex1i 5601 | . 2 ⊢ (𝐸/FldExt𝐹 → 𝐸 ∈ V) |
3 | elrelimasn 5946 | . . . 4 ⊢ (Rel /FldExt → (𝐹 ∈ (/FldExt “ {𝐸}) ↔ 𝐸/FldExt𝐹)) | |
4 | 1, 3 | ax-mp 5 | . . 3 ⊢ (𝐹 ∈ (/FldExt “ {𝐸}) ↔ 𝐸/FldExt𝐹) |
5 | 4 | biimpri 230 | . 2 ⊢ (𝐸/FldExt𝐹 → 𝐹 ∈ (/FldExt “ {𝐸})) |
6 | fvexd 6678 | . 2 ⊢ (𝐸/FldExt𝐹 → (dim‘((subringAlg ‘𝐸)‘(Base‘𝐹))) ∈ V) | |
7 | simpl 485 | . . . . . 6 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → 𝑒 = 𝐸) | |
8 | 7 | fveq2d 6667 | . . . . 5 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (subringAlg ‘𝑒) = (subringAlg ‘𝐸)) |
9 | simpr 487 | . . . . . 6 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → 𝑓 = 𝐹) | |
10 | 9 | fveq2d 6667 | . . . . 5 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (Base‘𝑓) = (Base‘𝐹)) |
11 | 8, 10 | fveq12d 6670 | . . . 4 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → ((subringAlg ‘𝑒)‘(Base‘𝑓)) = ((subringAlg ‘𝐸)‘(Base‘𝐹))) |
12 | 11 | fveq2d 6667 | . . 3 ⊢ ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) → (dim‘((subringAlg ‘𝑒)‘(Base‘𝑓))) = (dim‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) |
13 | sneq 4570 | . . . 4 ⊢ (𝑒 = 𝐸 → {𝑒} = {𝐸}) | |
14 | 13 | imaeq2d 5922 | . . 3 ⊢ (𝑒 = 𝐸 → (/FldExt “ {𝑒}) = (/FldExt “ {𝐸})) |
15 | df-extdg 31057 | . . 3 ⊢ [:] = (𝑒 ∈ V, 𝑓 ∈ (/FldExt “ {𝑒}) ↦ (dim‘((subringAlg ‘𝑒)‘(Base‘𝑓)))) | |
16 | 12, 14, 15 | ovmpox 7296 | . 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 1366 | 1 ⊢ (𝐸/FldExt𝐹 → (𝐸[:]𝐹) = (dim‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1536 ∈ wcel 2113 Vcvv 3491 {csn 4560 class class class wbr 5059 “ cima 5551 Rel wrel 5553 ‘cfv 6348 (class class class)co 7149 Basecbs 16478 subringAlg csra 19935 dimcldim 31023 /FldExtcfldext 31052 [:]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-ne 3016 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-sbc 3769 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-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fv 6356 df-ov 7152 df-oprab 7153 df-mpo 7154 df-fldext 31056 df-extdg 31057 |
This theorem is referenced by: extdgcl 31070 extdggt0 31071 extdgid 31074 extdgmul 31075 extdg1id 31077 ccfldextdgrr 31081 |
Copyright terms: Public domain | W3C validator |