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Theorem brfldext 32714
Description: The field extension relation explicited. (Contributed by Thierry Arnoux, 29-Jul-2023.)
Assertion
Ref Expression
brfldext ((𝐸 ∈ Field ∧ 𝐹 ∈ Field) β†’ (𝐸/FldExt𝐹 ↔ (𝐹 = (𝐸 β†Ύs (Baseβ€˜πΉ)) ∧ (Baseβ€˜πΉ) ∈ (SubRingβ€˜πΈ))))

Proof of Theorem brfldext
Dummy variables 𝑒 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 483 . . . . . 6 ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) β†’ 𝑒 = 𝐸)
21eleq1d 2818 . . . . 5 ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) β†’ (𝑒 ∈ Field ↔ 𝐸 ∈ Field))
3 simpr 485 . . . . . 6 ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) β†’ 𝑓 = 𝐹)
43eleq1d 2818 . . . . 5 ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) β†’ (𝑓 ∈ Field ↔ 𝐹 ∈ Field))
52, 4anbi12d 631 . . . 4 ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) β†’ ((𝑒 ∈ Field ∧ 𝑓 ∈ Field) ↔ (𝐸 ∈ Field ∧ 𝐹 ∈ Field)))
63fveq2d 6892 . . . . . . 7 ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) β†’ (Baseβ€˜π‘“) = (Baseβ€˜πΉ))
71, 6oveq12d 7423 . . . . . 6 ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) β†’ (𝑒 β†Ύs (Baseβ€˜π‘“)) = (𝐸 β†Ύs (Baseβ€˜πΉ)))
83, 7eqeq12d 2748 . . . . 5 ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) β†’ (𝑓 = (𝑒 β†Ύs (Baseβ€˜π‘“)) ↔ 𝐹 = (𝐸 β†Ύs (Baseβ€˜πΉ))))
91fveq2d 6892 . . . . . 6 ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) β†’ (SubRingβ€˜π‘’) = (SubRingβ€˜πΈ))
106, 9eleq12d 2827 . . . . 5 ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) β†’ ((Baseβ€˜π‘“) ∈ (SubRingβ€˜π‘’) ↔ (Baseβ€˜πΉ) ∈ (SubRingβ€˜πΈ)))
118, 10anbi12d 631 . . . 4 ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) β†’ ((𝑓 = (𝑒 β†Ύs (Baseβ€˜π‘“)) ∧ (Baseβ€˜π‘“) ∈ (SubRingβ€˜π‘’)) ↔ (𝐹 = (𝐸 β†Ύs (Baseβ€˜πΉ)) ∧ (Baseβ€˜πΉ) ∈ (SubRingβ€˜πΈ))))
125, 11anbi12d 631 . . 3 ((𝑒 = 𝐸 ∧ 𝑓 = 𝐹) β†’ (((𝑒 ∈ Field ∧ 𝑓 ∈ Field) ∧ (𝑓 = (𝑒 β†Ύs (Baseβ€˜π‘“)) ∧ (Baseβ€˜π‘“) ∈ (SubRingβ€˜π‘’))) ↔ ((𝐸 ∈ Field ∧ 𝐹 ∈ Field) ∧ (𝐹 = (𝐸 β†Ύs (Baseβ€˜πΉ)) ∧ (Baseβ€˜πΉ) ∈ (SubRingβ€˜πΈ)))))
13 df-fldext 32709 . . 3 /FldExt = {βŸ¨π‘’, π‘“βŸ© ∣ ((𝑒 ∈ Field ∧ 𝑓 ∈ Field) ∧ (𝑓 = (𝑒 β†Ύs (Baseβ€˜π‘“)) ∧ (Baseβ€˜π‘“) ∈ (SubRingβ€˜π‘’)))}
1412, 13brabga 5533 . 2 ((𝐸 ∈ Field ∧ 𝐹 ∈ Field) β†’ (𝐸/FldExt𝐹 ↔ ((𝐸 ∈ Field ∧ 𝐹 ∈ Field) ∧ (𝐹 = (𝐸 β†Ύs (Baseβ€˜πΉ)) ∧ (Baseβ€˜πΉ) ∈ (SubRingβ€˜πΈ)))))
1514bianabs 542 1 ((𝐸 ∈ Field ∧ 𝐹 ∈ Field) β†’ (𝐸/FldExt𝐹 ↔ (𝐹 = (𝐸 β†Ύs (Baseβ€˜πΉ)) ∧ (Baseβ€˜πΉ) ∈ (SubRingβ€˜πΈ))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106   class class class wbr 5147  β€˜cfv 6540  (class class class)co 7405  Basecbs 17140   β†Ύs cress 17169  Fieldcfield 20308  SubRingcsubrg 20351  /FldExtcfldext 32705
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-iota 6492  df-fv 6548  df-ov 7408  df-fldext 32709
This theorem is referenced by:  ccfldextrr  32715  fldextsubrg  32718  fldextress  32719  fldexttr  32725  fldextid  32726  extdgmul  32728  algextdeglem1  32760
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