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Theorem extdg1id 33030
Description: If the degree of the extension 𝐸/FldExt𝐹 is 1, then 𝐸 and 𝐹 are identical. (Contributed by Thierry Arnoux, 6-Aug-2023.)
Assertion
Ref Expression
extdg1id ((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) β†’ 𝐸 = 𝐹)

Proof of Theorem extdg1id
Dummy variables π‘Ž π‘₯ 𝑏 𝑖 𝑣 𝑒 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fldextress 33019 . . 3 (𝐸/FldExt𝐹 β†’ 𝐹 = (𝐸 β†Ύs (Baseβ€˜πΉ)))
21adantr 479 . 2 ((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) β†’ 𝐹 = (𝐸 β†Ύs (Baseβ€˜πΉ)))
3 fldextsralvec 33022 . . . . . . 7 (𝐸/FldExt𝐹 β†’ ((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)) ∈ LVec)
43adantr 479 . . . . . 6 ((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) β†’ ((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)) ∈ LVec)
5 eqid 2730 . . . . . . 7 (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ))) = (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))
65lbsex 20923 . . . . . 6 (((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)) ∈ LVec β†’ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ))) β‰  βˆ…)
74, 6syl 17 . . . . 5 ((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) β†’ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ))) β‰  βˆ…)
8 n0 4345 . . . . 5 ((LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ))) β‰  βˆ… ↔ βˆƒπ‘ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ))))
97, 8sylib 217 . . . 4 ((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) β†’ βˆƒπ‘ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ))))
10 simpr 483 . . . . . 6 (((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) β†’ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ))))
115dimval 32973 . . . . . . . 8 ((((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)) ∈ LVec ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) β†’ (dimβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ))) = (β™―β€˜π‘))
124, 11sylan 578 . . . . . . 7 (((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) β†’ (dimβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ))) = (β™―β€˜π‘))
13 extdgval 33021 . . . . . . . . . 10 (𝐸/FldExt𝐹 β†’ (𝐸[:]𝐹) = (dimβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ))))
1413adantr 479 . . . . . . . . 9 ((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) β†’ (𝐸[:]𝐹) = (dimβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ))))
15 simpr 483 . . . . . . . . 9 ((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) β†’ (𝐸[:]𝐹) = 1)
1614, 15eqtr3d 2772 . . . . . . . 8 ((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) β†’ (dimβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ))) = 1)
1716adantr 479 . . . . . . 7 (((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) β†’ (dimβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ))) = 1)
1812, 17eqtr3d 2772 . . . . . 6 (((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) β†’ (β™―β€˜π‘) = 1)
19 hash1snb 14383 . . . . . . 7 (𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ))) β†’ ((β™―β€˜π‘) = 1 ↔ βˆƒπ‘₯ 𝑏 = {π‘₯}))
2019biimpa 475 . . . . . 6 ((𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ))) ∧ (β™―β€˜π‘) = 1) β†’ βˆƒπ‘₯ 𝑏 = {π‘₯})
2110, 18, 20syl2anc 582 . . . . 5 (((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) β†’ βˆƒπ‘₯ 𝑏 = {π‘₯})
22 simpr 483 . . . . . . . . . 10 ((((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) ∧ 𝑒 ∈ (Baseβ€˜πΈ)) ∧ 𝑣 ∈ ((Baseβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ↑m 𝑏)) ∧ 𝑣 finSupp (0gβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ))))) ∧ 𝑒 = (((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)) Ξ£g (𝑖 ∈ 𝑏 ↦ ((π‘£β€˜π‘–)( ·𝑠 β€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))𝑖)))) β†’ 𝑒 = (((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)) Ξ£g (𝑖 ∈ 𝑏 ↦ ((π‘£β€˜π‘–)( ·𝑠 β€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))𝑖))))
23 simplr 765 . . . . . . . . . . . . . . . 16 (((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) ∧ 𝑣 ∈ ((Baseβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ↑m 𝑏)) β†’ 𝑏 = {π‘₯})
24 eqidd 2731 . . . . . . . . . . . . . . . . . . . 20 (𝐸/FldExt𝐹 β†’ ((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)) = ((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))
25 eqid 2730 . . . . . . . . . . . . . . . . . . . . . 22 (Baseβ€˜πΉ) = (Baseβ€˜πΉ)
2625fldextsubrg 33018 . . . . . . . . . . . . . . . . . . . . 21 (𝐸/FldExt𝐹 β†’ (Baseβ€˜πΉ) ∈ (SubRingβ€˜πΈ))
27 eqid 2730 . . . . . . . . . . . . . . . . . . . . . 22 (Baseβ€˜πΈ) = (Baseβ€˜πΈ)
2827subrgss 20462 . . . . . . . . . . . . . . . . . . . . 21 ((Baseβ€˜πΉ) ∈ (SubRingβ€˜πΈ) β†’ (Baseβ€˜πΉ) βŠ† (Baseβ€˜πΈ))
2926, 28syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝐸/FldExt𝐹 β†’ (Baseβ€˜πΉ) βŠ† (Baseβ€˜πΈ))
3024, 29sravsca 20945 . . . . . . . . . . . . . . . . . . 19 (𝐸/FldExt𝐹 β†’ (.rβ€˜πΈ) = ( ·𝑠 β€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ))))
3130eqcomd 2736 . . . . . . . . . . . . . . . . . 18 (𝐸/FldExt𝐹 β†’ ( ·𝑠 β€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ))) = (.rβ€˜πΈ))
3231ad5antr 730 . . . . . . . . . . . . . . . . 17 ((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) ∧ 𝑣 ∈ ((Baseβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ↑m 𝑏)) ∧ 𝑖 ∈ 𝑏) β†’ ( ·𝑠 β€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ))) = (.rβ€˜πΈ))
3332oveqd 7428 . . . . . . . . . . . . . . . 16 ((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) ∧ 𝑣 ∈ ((Baseβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ↑m 𝑏)) ∧ 𝑖 ∈ 𝑏) β†’ ((π‘£β€˜π‘–)( ·𝑠 β€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))𝑖) = ((π‘£β€˜π‘–)(.rβ€˜πΈ)𝑖))
3423, 33mpteq12dva 5236 . . . . . . . . . . . . . . 15 (((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) ∧ 𝑣 ∈ ((Baseβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ↑m 𝑏)) β†’ (𝑖 ∈ 𝑏 ↦ ((π‘£β€˜π‘–)( ·𝑠 β€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))𝑖)) = (𝑖 ∈ {π‘₯} ↦ ((π‘£β€˜π‘–)(.rβ€˜πΈ)𝑖)))
3534oveq2d 7427 . . . . . . . . . . . . . 14 (((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) ∧ 𝑣 ∈ ((Baseβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ↑m 𝑏)) β†’ (𝐸 Ξ£g (𝑖 ∈ 𝑏 ↦ ((π‘£β€˜π‘–)( ·𝑠 β€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))𝑖))) = (𝐸 Ξ£g (𝑖 ∈ {π‘₯} ↦ ((π‘£β€˜π‘–)(.rβ€˜πΈ)𝑖))))
36 eqid 2730 . . . . . . . . . . . . . . . . 17 ((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)) = ((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ))
37 fldextfld1 33016 . . . . . . . . . . . . . . . . . . 19 (𝐸/FldExt𝐹 β†’ 𝐸 ∈ Field)
38 isfld 20511 . . . . . . . . . . . . . . . . . . . 20 (𝐸 ∈ Field ↔ (𝐸 ∈ DivRing ∧ 𝐸 ∈ CRing))
3938simplbi 496 . . . . . . . . . . . . . . . . . . 19 (𝐸 ∈ Field β†’ 𝐸 ∈ DivRing)
4037, 39syl 17 . . . . . . . . . . . . . . . . . 18 (𝐸/FldExt𝐹 β†’ 𝐸 ∈ DivRing)
4140adantr 479 . . . . . . . . . . . . . . . . 17 ((𝐸/FldExt𝐹 ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) β†’ 𝐸 ∈ DivRing)
4226adantr 479 . . . . . . . . . . . . . . . . 17 ((𝐸/FldExt𝐹 ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) β†’ (Baseβ€˜πΉ) ∈ (SubRingβ€˜πΈ))
43 eqid 2730 . . . . . . . . . . . . . . . . 17 (𝐸 β†Ύs (Baseβ€˜πΉ)) = (𝐸 β†Ύs (Baseβ€˜πΉ))
44 fldextfld2 33017 . . . . . . . . . . . . . . . . . . . 20 (𝐸/FldExt𝐹 β†’ 𝐹 ∈ Field)
45 isfld 20511 . . . . . . . . . . . . . . . . . . . . 21 (𝐹 ∈ Field ↔ (𝐹 ∈ DivRing ∧ 𝐹 ∈ CRing))
4645simplbi 496 . . . . . . . . . . . . . . . . . . . 20 (𝐹 ∈ Field β†’ 𝐹 ∈ DivRing)
4744, 46syl 17 . . . . . . . . . . . . . . . . . . 19 (𝐸/FldExt𝐹 β†’ 𝐹 ∈ DivRing)
481, 47eqeltrrd 2832 . . . . . . . . . . . . . . . . . 18 (𝐸/FldExt𝐹 β†’ (𝐸 β†Ύs (Baseβ€˜πΉ)) ∈ DivRing)
4948adantr 479 . . . . . . . . . . . . . . . . 17 ((𝐸/FldExt𝐹 ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) β†’ (𝐸 β†Ύs (Baseβ€˜πΉ)) ∈ DivRing)
50 simpr 483 . . . . . . . . . . . . . . . . 17 ((𝐸/FldExt𝐹 ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) β†’ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ))))
5136, 41, 42, 43, 49, 50drgextgsum 32969 . . . . . . . . . . . . . . . 16 ((𝐸/FldExt𝐹 ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) β†’ (𝐸 Ξ£g (𝑖 ∈ 𝑏 ↦ ((π‘£β€˜π‘–)( ·𝑠 β€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))𝑖))) = (((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)) Ξ£g (𝑖 ∈ 𝑏 ↦ ((π‘£β€˜π‘–)( ·𝑠 β€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))𝑖))))
5251adantlr 711 . . . . . . . . . . . . . . 15 (((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) β†’ (𝐸 Ξ£g (𝑖 ∈ 𝑏 ↦ ((π‘£β€˜π‘–)( ·𝑠 β€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))𝑖))) = (((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)) Ξ£g (𝑖 ∈ 𝑏 ↦ ((π‘£β€˜π‘–)( ·𝑠 β€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))𝑖))))
5352ad2antrr 722 . . . . . . . . . . . . . 14 (((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) ∧ 𝑣 ∈ ((Baseβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ↑m 𝑏)) β†’ (𝐸 Ξ£g (𝑖 ∈ 𝑏 ↦ ((π‘£β€˜π‘–)( ·𝑠 β€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))𝑖))) = (((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)) Ξ£g (𝑖 ∈ 𝑏 ↦ ((π‘£β€˜π‘–)( ·𝑠 β€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))𝑖))))
54 drngring 20507 . . . . . . . . . . . . . . . . . . 19 (𝐸 ∈ DivRing β†’ 𝐸 ∈ Ring)
5537, 39, 543syl 18 . . . . . . . . . . . . . . . . . 18 (𝐸/FldExt𝐹 β†’ 𝐸 ∈ Ring)
56 ringmnd 20137 . . . . . . . . . . . . . . . . . 18 (𝐸 ∈ Ring β†’ 𝐸 ∈ Mnd)
5755, 56syl 17 . . . . . . . . . . . . . . . . 17 (𝐸/FldExt𝐹 β†’ 𝐸 ∈ Mnd)
5857ad4antr 728 . . . . . . . . . . . . . . . 16 (((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) ∧ 𝑣 ∈ ((Baseβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ↑m 𝑏)) β†’ 𝐸 ∈ Mnd)
59 vex 3476 . . . . . . . . . . . . . . . . 17 π‘₯ ∈ V
6059a1i 11 . . . . . . . . . . . . . . . 16 (((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) ∧ 𝑣 ∈ ((Baseβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ↑m 𝑏)) β†’ π‘₯ ∈ V)
6155ad3antrrr 726 . . . . . . . . . . . . . . . . . 18 ((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) β†’ 𝐸 ∈ Ring)
6261adantr 479 . . . . . . . . . . . . . . . . 17 (((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) ∧ 𝑣 ∈ ((Baseβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ↑m 𝑏)) β†’ 𝐸 ∈ Ring)
6329ad3antrrr 726 . . . . . . . . . . . . . . . . . . 19 ((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) β†’ (Baseβ€˜πΉ) βŠ† (Baseβ€˜πΈ))
6463adantr 479 . . . . . . . . . . . . . . . . . 18 (((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) ∧ 𝑣 ∈ ((Baseβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ↑m 𝑏)) β†’ (Baseβ€˜πΉ) βŠ† (Baseβ€˜πΈ))
65 elmapi 8845 . . . . . . . . . . . . . . . . . . . . 21 (𝑣 ∈ ((Baseβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ↑m 𝑏) β†’ 𝑣:π‘βŸΆ(Baseβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))))
6665adantl 480 . . . . . . . . . . . . . . . . . . . 20 (((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) ∧ 𝑣 ∈ ((Baseβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ↑m 𝑏)) β†’ 𝑣:π‘βŸΆ(Baseβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))))
67 vsnid 4664 . . . . . . . . . . . . . . . . . . . . 21 π‘₯ ∈ {π‘₯}
6867, 23eleqtrrid 2838 . . . . . . . . . . . . . . . . . . . 20 (((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) ∧ 𝑣 ∈ ((Baseβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ↑m 𝑏)) β†’ π‘₯ ∈ 𝑏)
6966, 68ffvelcdmd 7086 . . . . . . . . . . . . . . . . . . 19 (((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) ∧ 𝑣 ∈ ((Baseβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ↑m 𝑏)) β†’ (π‘£β€˜π‘₯) ∈ (Baseβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))))
7024, 29srasca 20943 . . . . . . . . . . . . . . . . . . . . . 22 (𝐸/FldExt𝐹 β†’ (𝐸 β†Ύs (Baseβ€˜πΉ)) = (Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ))))
711, 70eqtrd 2770 . . . . . . . . . . . . . . . . . . . . 21 (𝐸/FldExt𝐹 β†’ 𝐹 = (Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ))))
7271fveq2d 6894 . . . . . . . . . . . . . . . . . . . 20 (𝐸/FldExt𝐹 β†’ (Baseβ€˜πΉ) = (Baseβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))))
7372ad4antr 728 . . . . . . . . . . . . . . . . . . 19 (((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) ∧ 𝑣 ∈ ((Baseβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ↑m 𝑏)) β†’ (Baseβ€˜πΉ) = (Baseβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))))
7469, 73eleqtrrd 2834 . . . . . . . . . . . . . . . . . 18 (((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) ∧ 𝑣 ∈ ((Baseβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ↑m 𝑏)) β†’ (π‘£β€˜π‘₯) ∈ (Baseβ€˜πΉ))
7564, 74sseldd 3982 . . . . . . . . . . . . . . . . 17 (((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) ∧ 𝑣 ∈ ((Baseβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ↑m 𝑏)) β†’ (π‘£β€˜π‘₯) ∈ (Baseβ€˜πΈ))
76 simpr 483 . . . . . . . . . . . . . . . . . . . . 21 ((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) β†’ 𝑏 = {π‘₯})
77 simplr 765 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) β†’ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ))))
78 eqid 2730 . . . . . . . . . . . . . . . . . . . . . . 23 (Baseβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ))) = (Baseβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))
7978, 5lbsss 20832 . . . . . . . . . . . . . . . . . . . . . 22 (𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ))) β†’ 𝑏 βŠ† (Baseβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ))))
8077, 79syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) β†’ 𝑏 βŠ† (Baseβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ))))
8176, 80eqsstrrd 4020 . . . . . . . . . . . . . . . . . . . 20 ((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) β†’ {π‘₯} βŠ† (Baseβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ))))
8259snss 4788 . . . . . . . . . . . . . . . . . . . 20 (π‘₯ ∈ (Baseβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ))) ↔ {π‘₯} βŠ† (Baseβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ))))
8381, 82sylibr 233 . . . . . . . . . . . . . . . . . . 19 ((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) β†’ π‘₯ ∈ (Baseβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ))))
84 eqidd 2731 . . . . . . . . . . . . . . . . . . . 20 ((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) β†’ ((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)) = ((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))
8584, 63srabase 20937 . . . . . . . . . . . . . . . . . . 19 ((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) β†’ (Baseβ€˜πΈ) = (Baseβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ))))
8683, 85eleqtrrd 2834 . . . . . . . . . . . . . . . . . 18 ((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) β†’ π‘₯ ∈ (Baseβ€˜πΈ))
8786adantr 479 . . . . . . . . . . . . . . . . 17 (((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) ∧ 𝑣 ∈ ((Baseβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ↑m 𝑏)) β†’ π‘₯ ∈ (Baseβ€˜πΈ))
88 eqid 2730 . . . . . . . . . . . . . . . . . 18 (.rβ€˜πΈ) = (.rβ€˜πΈ)
8927, 88ringcl 20144 . . . . . . . . . . . . . . . . 17 ((𝐸 ∈ Ring ∧ (π‘£β€˜π‘₯) ∈ (Baseβ€˜πΈ) ∧ π‘₯ ∈ (Baseβ€˜πΈ)) β†’ ((π‘£β€˜π‘₯)(.rβ€˜πΈ)π‘₯) ∈ (Baseβ€˜πΈ))
9062, 75, 87, 89syl3anc 1369 . . . . . . . . . . . . . . . 16 (((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) ∧ 𝑣 ∈ ((Baseβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ↑m 𝑏)) β†’ ((π‘£β€˜π‘₯)(.rβ€˜πΈ)π‘₯) ∈ (Baseβ€˜πΈ))
91 simpr 483 . . . . . . . . . . . . . . . . . 18 ((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) ∧ 𝑣 ∈ ((Baseβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ↑m 𝑏)) ∧ 𝑖 = π‘₯) β†’ 𝑖 = π‘₯)
9291fveq2d 6894 . . . . . . . . . . . . . . . . 17 ((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) ∧ 𝑣 ∈ ((Baseβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ↑m 𝑏)) ∧ 𝑖 = π‘₯) β†’ (π‘£β€˜π‘–) = (π‘£β€˜π‘₯))
9392, 91oveq12d 7429 . . . . . . . . . . . . . . . 16 ((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) ∧ 𝑣 ∈ ((Baseβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ↑m 𝑏)) ∧ 𝑖 = π‘₯) β†’ ((π‘£β€˜π‘–)(.rβ€˜πΈ)𝑖) = ((π‘£β€˜π‘₯)(.rβ€˜πΈ)π‘₯))
9427, 58, 60, 90, 93gsumsnd 19861 . . . . . . . . . . . . . . 15 (((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) ∧ 𝑣 ∈ ((Baseβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ↑m 𝑏)) β†’ (𝐸 Ξ£g (𝑖 ∈ {π‘₯} ↦ ((π‘£β€˜π‘–)(.rβ€˜πΈ)𝑖))) = ((π‘£β€˜π‘₯)(.rβ€˜πΈ)π‘₯))
951fveq2d 6894 . . . . . . . . . . . . . . . . . 18 (𝐸/FldExt𝐹 β†’ (.rβ€˜πΉ) = (.rβ€˜(𝐸 β†Ύs (Baseβ€˜πΉ))))
9643, 88ressmulr 17256 . . . . . . . . . . . . . . . . . . 19 ((Baseβ€˜πΉ) ∈ (SubRingβ€˜πΈ) β†’ (.rβ€˜πΈ) = (.rβ€˜(𝐸 β†Ύs (Baseβ€˜πΉ))))
9726, 96syl 17 . . . . . . . . . . . . . . . . . 18 (𝐸/FldExt𝐹 β†’ (.rβ€˜πΈ) = (.rβ€˜(𝐸 β†Ύs (Baseβ€˜πΉ))))
9895, 97eqtr4d 2773 . . . . . . . . . . . . . . . . 17 (𝐸/FldExt𝐹 β†’ (.rβ€˜πΉ) = (.rβ€˜πΈ))
9998ad4antr 728 . . . . . . . . . . . . . . . 16 (((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) ∧ 𝑣 ∈ ((Baseβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ↑m 𝑏)) β†’ (.rβ€˜πΉ) = (.rβ€˜πΈ))
10099oveqd 7428 . . . . . . . . . . . . . . 15 (((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) ∧ 𝑣 ∈ ((Baseβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ↑m 𝑏)) β†’ ((π‘£β€˜π‘₯)(.rβ€˜πΉ)π‘₯) = ((π‘£β€˜π‘₯)(.rβ€˜πΈ)π‘₯))
10194, 100eqtr4d 2773 . . . . . . . . . . . . . 14 (((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) ∧ 𝑣 ∈ ((Baseβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ↑m 𝑏)) β†’ (𝐸 Ξ£g (𝑖 ∈ {π‘₯} ↦ ((π‘£β€˜π‘–)(.rβ€˜πΈ)𝑖))) = ((π‘£β€˜π‘₯)(.rβ€˜πΉ)π‘₯))
10235, 53, 1013eqtr3d 2778 . . . . . . . . . . . . 13 (((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) ∧ 𝑣 ∈ ((Baseβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ↑m 𝑏)) β†’ (((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)) Ξ£g (𝑖 ∈ 𝑏 ↦ ((π‘£β€˜π‘–)( ·𝑠 β€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))𝑖))) = ((π‘£β€˜π‘₯)(.rβ€˜πΉ)π‘₯))
103102adantlr 711 . . . . . . . . . . . 12 ((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) ∧ 𝑒 ∈ (Baseβ€˜πΈ)) ∧ 𝑣 ∈ ((Baseβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ↑m 𝑏)) β†’ (((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)) Ξ£g (𝑖 ∈ 𝑏 ↦ ((π‘£β€˜π‘–)( ·𝑠 β€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))𝑖))) = ((π‘£β€˜π‘₯)(.rβ€˜πΉ)π‘₯))
104 drngring 20507 . . . . . . . . . . . . . . 15 (𝐹 ∈ DivRing β†’ 𝐹 ∈ Ring)
10544, 46, 1043syl 18 . . . . . . . . . . . . . 14 (𝐸/FldExt𝐹 β†’ 𝐹 ∈ Ring)
106105ad5antr 730 . . . . . . . . . . . . 13 ((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) ∧ 𝑒 ∈ (Baseβ€˜πΈ)) ∧ 𝑣 ∈ ((Baseβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ↑m 𝑏)) β†’ 𝐹 ∈ Ring)
10774adantlr 711 . . . . . . . . . . . . 13 ((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) ∧ 𝑒 ∈ (Baseβ€˜πΈ)) ∧ 𝑣 ∈ ((Baseβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ↑m 𝑏)) β†’ (π‘£β€˜π‘₯) ∈ (Baseβ€˜πΉ))
108 eqid 2730 . . . . . . . . . . . . . . . . . . . 20 (1rβ€˜πΈ) = (1rβ€˜πΈ)
109 eqid 2730 . . . . . . . . . . . . . . . . . . . 20 (Unitβ€˜πΈ) = (Unitβ€˜πΈ)
110 eqid 2730 . . . . . . . . . . . . . . . . . . . 20 (invrβ€˜πΈ) = (invrβ€˜πΈ)
111 simp-5l 781 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) ∧ 𝑣 ∈ ((Baseβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ↑m 𝑏)) ∧ (1rβ€˜πΈ) = (((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)) Ξ£g (𝑖 ∈ 𝑏 ↦ ((π‘£β€˜π‘–)( ·𝑠 β€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))𝑖)))) β†’ 𝐸/FldExt𝐹)
112111, 55syl 17 . . . . . . . . . . . . . . . . . . . 20 ((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) ∧ 𝑣 ∈ ((Baseβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ↑m 𝑏)) ∧ (1rβ€˜πΈ) = (((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)) Ξ£g (𝑖 ∈ 𝑏 ↦ ((π‘£β€˜π‘–)( ·𝑠 β€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))𝑖)))) β†’ 𝐸 ∈ Ring)
11387adantr 479 . . . . . . . . . . . . . . . . . . . 20 ((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) ∧ 𝑣 ∈ ((Baseβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ↑m 𝑏)) ∧ (1rβ€˜πΈ) = (((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)) Ξ£g (𝑖 ∈ 𝑏 ↦ ((π‘£β€˜π‘–)( ·𝑠 β€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))𝑖)))) β†’ π‘₯ ∈ (Baseβ€˜πΈ))
11475adantr 479 . . . . . . . . . . . . . . . . . . . 20 ((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) ∧ 𝑣 ∈ ((Baseβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ↑m 𝑏)) ∧ (1rβ€˜πΈ) = (((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)) Ξ£g (𝑖 ∈ 𝑏 ↦ ((π‘£β€˜π‘–)( ·𝑠 β€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))𝑖)))) β†’ (π‘£β€˜π‘₯) ∈ (Baseβ€˜πΈ))
11538simprbi 495 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐸 ∈ Field β†’ 𝐸 ∈ CRing)
116111, 37, 1153syl 18 . . . . . . . . . . . . . . . . . . . . . 22 ((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) ∧ 𝑣 ∈ ((Baseβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ↑m 𝑏)) ∧ (1rβ€˜πΈ) = (((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)) Ξ£g (𝑖 ∈ 𝑏 ↦ ((π‘£β€˜π‘–)( ·𝑠 β€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))𝑖)))) β†’ 𝐸 ∈ CRing)
11727, 88crngcom 20145 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐸 ∈ CRing ∧ π‘₯ ∈ (Baseβ€˜πΈ) ∧ (π‘£β€˜π‘₯) ∈ (Baseβ€˜πΈ)) β†’ (π‘₯(.rβ€˜πΈ)(π‘£β€˜π‘₯)) = ((π‘£β€˜π‘₯)(.rβ€˜πΈ)π‘₯))
118116, 113, 114, 117syl3anc 1369 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) ∧ 𝑣 ∈ ((Baseβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ↑m 𝑏)) ∧ (1rβ€˜πΈ) = (((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)) Ξ£g (𝑖 ∈ 𝑏 ↦ ((π‘£β€˜π‘–)( ·𝑠 β€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))𝑖)))) β†’ (π‘₯(.rβ€˜πΈ)(π‘£β€˜π‘₯)) = ((π‘£β€˜π‘₯)(.rβ€˜πΈ)π‘₯))
119 simpr 483 . . . . . . . . . . . . . . . . . . . . . . 23 ((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) ∧ 𝑣 ∈ ((Baseβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ↑m 𝑏)) ∧ (1rβ€˜πΈ) = (((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)) Ξ£g (𝑖 ∈ 𝑏 ↦ ((π‘£β€˜π‘–)( ·𝑠 β€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))𝑖)))) β†’ (1rβ€˜πΈ) = (((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)) Ξ£g (𝑖 ∈ 𝑏 ↦ ((π‘£β€˜π‘–)( ·𝑠 β€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))𝑖))))
12052ad3antrrr 726 . . . . . . . . . . . . . . . . . . . . . . 23 ((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) ∧ 𝑣 ∈ ((Baseβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ↑m 𝑏)) ∧ (1rβ€˜πΈ) = (((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)) Ξ£g (𝑖 ∈ 𝑏 ↦ ((π‘£β€˜π‘–)( ·𝑠 β€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))𝑖)))) β†’ (𝐸 Ξ£g (𝑖 ∈ 𝑏 ↦ ((π‘£β€˜π‘–)( ·𝑠 β€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))𝑖))) = (((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)) Ξ£g (𝑖 ∈ 𝑏 ↦ ((π‘£β€˜π‘–)( ·𝑠 β€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))𝑖))))
12134adantr 479 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) ∧ 𝑣 ∈ ((Baseβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ↑m 𝑏)) ∧ (1rβ€˜πΈ) = (((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)) Ξ£g (𝑖 ∈ 𝑏 ↦ ((π‘£β€˜π‘–)( ·𝑠 β€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))𝑖)))) β†’ (𝑖 ∈ 𝑏 ↦ ((π‘£β€˜π‘–)( ·𝑠 β€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))𝑖)) = (𝑖 ∈ {π‘₯} ↦ ((π‘£β€˜π‘–)(.rβ€˜πΈ)𝑖)))
122121oveq2d 7427 . . . . . . . . . . . . . . . . . . . . . . 23 ((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) ∧ 𝑣 ∈ ((Baseβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ↑m 𝑏)) ∧ (1rβ€˜πΈ) = (((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)) Ξ£g (𝑖 ∈ 𝑏 ↦ ((π‘£β€˜π‘–)( ·𝑠 β€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))𝑖)))) β†’ (𝐸 Ξ£g (𝑖 ∈ 𝑏 ↦ ((π‘£β€˜π‘–)( ·𝑠 β€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))𝑖))) = (𝐸 Ξ£g (𝑖 ∈ {π‘₯} ↦ ((π‘£β€˜π‘–)(.rβ€˜πΈ)𝑖))))
123119, 120, 1223eqtr2d 2776 . . . . . . . . . . . . . . . . . . . . . 22 ((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) ∧ 𝑣 ∈ ((Baseβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ↑m 𝑏)) ∧ (1rβ€˜πΈ) = (((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)) Ξ£g (𝑖 ∈ 𝑏 ↦ ((π‘£β€˜π‘–)( ·𝑠 β€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))𝑖)))) β†’ (1rβ€˜πΈ) = (𝐸 Ξ£g (𝑖 ∈ {π‘₯} ↦ ((π‘£β€˜π‘–)(.rβ€˜πΈ)𝑖))))
12494adantr 479 . . . . . . . . . . . . . . . . . . . . . 22 ((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) ∧ 𝑣 ∈ ((Baseβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ↑m 𝑏)) ∧ (1rβ€˜πΈ) = (((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)) Ξ£g (𝑖 ∈ 𝑏 ↦ ((π‘£β€˜π‘–)( ·𝑠 β€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))𝑖)))) β†’ (𝐸 Ξ£g (𝑖 ∈ {π‘₯} ↦ ((π‘£β€˜π‘–)(.rβ€˜πΈ)𝑖))) = ((π‘£β€˜π‘₯)(.rβ€˜πΈ)π‘₯))
125123, 124eqtrd 2770 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) ∧ 𝑣 ∈ ((Baseβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ↑m 𝑏)) ∧ (1rβ€˜πΈ) = (((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)) Ξ£g (𝑖 ∈ 𝑏 ↦ ((π‘£β€˜π‘–)( ·𝑠 β€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))𝑖)))) β†’ (1rβ€˜πΈ) = ((π‘£β€˜π‘₯)(.rβ€˜πΈ)π‘₯))
126118, 125eqtr4d 2773 . . . . . . . . . . . . . . . . . . . 20 ((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) ∧ 𝑣 ∈ ((Baseβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ↑m 𝑏)) ∧ (1rβ€˜πΈ) = (((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)) Ξ£g (𝑖 ∈ 𝑏 ↦ ((π‘£β€˜π‘–)( ·𝑠 β€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))𝑖)))) β†’ (π‘₯(.rβ€˜πΈ)(π‘£β€˜π‘₯)) = (1rβ€˜πΈ))
127125eqcomd 2736 . . . . . . . . . . . . . . . . . . . 20 ((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) ∧ 𝑣 ∈ ((Baseβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ↑m 𝑏)) ∧ (1rβ€˜πΈ) = (((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)) Ξ£g (𝑖 ∈ 𝑏 ↦ ((π‘£β€˜π‘–)( ·𝑠 β€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))𝑖)))) β†’ ((π‘£β€˜π‘₯)(.rβ€˜πΈ)π‘₯) = (1rβ€˜πΈ))
12827, 88, 108, 109, 110, 112, 113, 114, 126, 127invrvald 22398 . . . . . . . . . . . . . . . . . . 19 ((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) ∧ 𝑣 ∈ ((Baseβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ↑m 𝑏)) ∧ (1rβ€˜πΈ) = (((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)) Ξ£g (𝑖 ∈ 𝑏 ↦ ((π‘£β€˜π‘–)( ·𝑠 β€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))𝑖)))) β†’ (π‘₯ ∈ (Unitβ€˜πΈ) ∧ ((invrβ€˜πΈ)β€˜π‘₯) = (π‘£β€˜π‘₯)))
129128simpld 493 . . . . . . . . . . . . . . . . . 18 ((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) ∧ 𝑣 ∈ ((Baseβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ↑m 𝑏)) ∧ (1rβ€˜πΈ) = (((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)) Ξ£g (𝑖 ∈ 𝑏 ↦ ((π‘£β€˜π‘–)( ·𝑠 β€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))𝑖)))) β†’ π‘₯ ∈ (Unitβ€˜πΈ))
130109, 110unitinvinv 20282 . . . . . . . . . . . . . . . . . 18 ((𝐸 ∈ Ring ∧ π‘₯ ∈ (Unitβ€˜πΈ)) β†’ ((invrβ€˜πΈ)β€˜((invrβ€˜πΈ)β€˜π‘₯)) = π‘₯)
13162, 129, 130syl2an2r 681 . . . . . . . . . . . . . . . . 17 ((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) ∧ 𝑣 ∈ ((Baseβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ↑m 𝑏)) ∧ (1rβ€˜πΈ) = (((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)) Ξ£g (𝑖 ∈ 𝑏 ↦ ((π‘£β€˜π‘–)( ·𝑠 β€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))𝑖)))) β†’ ((invrβ€˜πΈ)β€˜((invrβ€˜πΈ)β€˜π‘₯)) = π‘₯)
132111, 37, 393syl 18 . . . . . . . . . . . . . . . . . 18 ((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) ∧ 𝑣 ∈ ((Baseβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ↑m 𝑏)) ∧ (1rβ€˜πΈ) = (((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)) Ξ£g (𝑖 ∈ 𝑏 ↦ ((π‘£β€˜π‘–)( ·𝑠 β€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))𝑖)))) β†’ 𝐸 ∈ DivRing)
133111, 26syl 17 . . . . . . . . . . . . . . . . . 18 ((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) ∧ 𝑣 ∈ ((Baseβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ↑m 𝑏)) ∧ (1rβ€˜πΈ) = (((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)) Ξ£g (𝑖 ∈ 𝑏 ↦ ((π‘£β€˜π‘–)( ·𝑠 β€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))𝑖)))) β†’ (Baseβ€˜πΉ) ∈ (SubRingβ€˜πΈ))
134111, 1syl 17 . . . . . . . . . . . . . . . . . . 19 ((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) ∧ 𝑣 ∈ ((Baseβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ↑m 𝑏)) ∧ (1rβ€˜πΈ) = (((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)) Ξ£g (𝑖 ∈ 𝑏 ↦ ((π‘£β€˜π‘–)( ·𝑠 β€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))𝑖)))) β†’ 𝐹 = (𝐸 β†Ύs (Baseβ€˜πΉ)))
135111, 44, 463syl 18 . . . . . . . . . . . . . . . . . . 19 ((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) ∧ 𝑣 ∈ ((Baseβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ↑m 𝑏)) ∧ (1rβ€˜πΈ) = (((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)) Ξ£g (𝑖 ∈ 𝑏 ↦ ((π‘£β€˜π‘–)( ·𝑠 β€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))𝑖)))) β†’ 𝐹 ∈ DivRing)
136134, 135eqeltrrd 2832 . . . . . . . . . . . . . . . . . 18 ((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) ∧ 𝑣 ∈ ((Baseβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ↑m 𝑏)) ∧ (1rβ€˜πΈ) = (((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)) Ξ£g (𝑖 ∈ 𝑏 ↦ ((π‘£β€˜π‘–)( ·𝑠 β€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))𝑖)))) β†’ (𝐸 β†Ύs (Baseβ€˜πΉ)) ∈ DivRing)
137128simprd 494 . . . . . . . . . . . . . . . . . . . 20 ((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) ∧ 𝑣 ∈ ((Baseβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ↑m 𝑏)) ∧ (1rβ€˜πΈ) = (((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)) Ξ£g (𝑖 ∈ 𝑏 ↦ ((π‘£β€˜π‘–)( ·𝑠 β€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))𝑖)))) β†’ ((invrβ€˜πΈ)β€˜π‘₯) = (π‘£β€˜π‘₯))
13874adantr 479 . . . . . . . . . . . . . . . . . . . 20 ((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) ∧ 𝑣 ∈ ((Baseβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ↑m 𝑏)) ∧ (1rβ€˜πΈ) = (((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)) Ξ£g (𝑖 ∈ 𝑏 ↦ ((π‘£β€˜π‘–)( ·𝑠 β€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))𝑖)))) β†’ (π‘£β€˜π‘₯) ∈ (Baseβ€˜πΉ))
139137, 138eqeltrd 2831 . . . . . . . . . . . . . . . . . . 19 ((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) ∧ 𝑣 ∈ ((Baseβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ↑m 𝑏)) ∧ (1rβ€˜πΈ) = (((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)) Ξ£g (𝑖 ∈ 𝑏 ↦ ((π‘£β€˜π‘–)( ·𝑠 β€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))𝑖)))) β†’ ((invrβ€˜πΈ)β€˜π‘₯) ∈ (Baseβ€˜πΉ))
140 eqidd 2731 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐸/FldExt𝐹 β†’ (0gβ€˜πΈ) = (0gβ€˜πΈ))
14124, 140, 29sralmod0 20955 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐸/FldExt𝐹 β†’ (0gβ€˜πΈ) = (0gβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ))))
142141ad2antrr 722 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) β†’ (0gβ€˜πΈ) = (0gβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ))))
1435lbslinds 21607 . . . . . . . . . . . . . . . . . . . . . . . . 25 (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ))) βŠ† (LIndSβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))
144143, 10sselid 3979 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) β†’ 𝑏 ∈ (LIndSβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ))))
145 eqid 2730 . . . . . . . . . . . . . . . . . . . . . . . . 25 (0gβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ))) = (0gβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))
1461450nellinds 32757 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)) ∈ LVec ∧ 𝑏 ∈ (LIndSβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) β†’ Β¬ (0gβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ))) ∈ 𝑏)
1474, 144, 146syl2an2r 681 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) β†’ Β¬ (0gβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ))) ∈ 𝑏)
148142, 147eqneltrd 2851 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) β†’ Β¬ (0gβ€˜πΈ) ∈ 𝑏)
149148ad3antrrr 726 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) ∧ 𝑣 ∈ ((Baseβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ↑m 𝑏)) ∧ (1rβ€˜πΈ) = (((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)) Ξ£g (𝑖 ∈ 𝑏 ↦ ((π‘£β€˜π‘–)( ·𝑠 β€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))𝑖)))) β†’ Β¬ (0gβ€˜πΈ) ∈ 𝑏)
150 nelne2 3038 . . . . . . . . . . . . . . . . . . . . 21 ((π‘₯ ∈ 𝑏 ∧ Β¬ (0gβ€˜πΈ) ∈ 𝑏) β†’ π‘₯ β‰  (0gβ€˜πΈ))
15168, 149, 150syl2an2r 681 . . . . . . . . . . . . . . . . . . . 20 ((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) ∧ 𝑣 ∈ ((Baseβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ↑m 𝑏)) ∧ (1rβ€˜πΈ) = (((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)) Ξ£g (𝑖 ∈ 𝑏 ↦ ((π‘£β€˜π‘–)( ·𝑠 β€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))𝑖)))) β†’ π‘₯ β‰  (0gβ€˜πΈ))
152 eqid 2730 . . . . . . . . . . . . . . . . . . . . 21 (0gβ€˜πΈ) = (0gβ€˜πΈ)
15327, 152, 110drnginvrn0 20523 . . . . . . . . . . . . . . . . . . . 20 ((𝐸 ∈ DivRing ∧ π‘₯ ∈ (Baseβ€˜πΈ) ∧ π‘₯ β‰  (0gβ€˜πΈ)) β†’ ((invrβ€˜πΈ)β€˜π‘₯) β‰  (0gβ€˜πΈ))
154132, 113, 151, 153syl3anc 1369 . . . . . . . . . . . . . . . . . . 19 ((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) ∧ 𝑣 ∈ ((Baseβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ↑m 𝑏)) ∧ (1rβ€˜πΈ) = (((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)) Ξ£g (𝑖 ∈ 𝑏 ↦ ((π‘£β€˜π‘–)( ·𝑠 β€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))𝑖)))) β†’ ((invrβ€˜πΈ)β€˜π‘₯) β‰  (0gβ€˜πΈ))
155 eldifsn 4789 . . . . . . . . . . . . . . . . . . 19 (((invrβ€˜πΈ)β€˜π‘₯) ∈ ((Baseβ€˜πΉ) βˆ– {(0gβ€˜πΈ)}) ↔ (((invrβ€˜πΈ)β€˜π‘₯) ∈ (Baseβ€˜πΉ) ∧ ((invrβ€˜πΈ)β€˜π‘₯) β‰  (0gβ€˜πΈ)))
156139, 154, 155sylanbrc 581 . . . . . . . . . . . . . . . . . 18 ((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) ∧ 𝑣 ∈ ((Baseβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ↑m 𝑏)) ∧ (1rβ€˜πΈ) = (((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)) Ξ£g (𝑖 ∈ 𝑏 ↦ ((π‘£β€˜π‘–)( ·𝑠 β€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))𝑖)))) β†’ ((invrβ€˜πΈ)β€˜π‘₯) ∈ ((Baseβ€˜πΉ) βˆ– {(0gβ€˜πΈ)}))
157 fveq2 6890 . . . . . . . . . . . . . . . . . . . 20 (π‘Ž = ((invrβ€˜πΈ)β€˜π‘₯) β†’ ((invrβ€˜πΈ)β€˜π‘Ž) = ((invrβ€˜πΈ)β€˜((invrβ€˜πΈ)β€˜π‘₯)))
158157eleq1d 2816 . . . . . . . . . . . . . . . . . . 19 (π‘Ž = ((invrβ€˜πΈ)β€˜π‘₯) β†’ (((invrβ€˜πΈ)β€˜π‘Ž) ∈ (Baseβ€˜πΉ) ↔ ((invrβ€˜πΈ)β€˜((invrβ€˜πΈ)β€˜π‘₯)) ∈ (Baseβ€˜πΉ)))
15943, 152, 110issubdrg 20544 . . . . . . . . . . . . . . . . . . . . 21 ((𝐸 ∈ DivRing ∧ (Baseβ€˜πΉ) ∈ (SubRingβ€˜πΈ)) β†’ ((𝐸 β†Ύs (Baseβ€˜πΉ)) ∈ DivRing ↔ βˆ€π‘Ž ∈ ((Baseβ€˜πΉ) βˆ– {(0gβ€˜πΈ)})((invrβ€˜πΈ)β€˜π‘Ž) ∈ (Baseβ€˜πΉ)))
160159biimpa 475 . . . . . . . . . . . . . . . . . . . 20 (((𝐸 ∈ DivRing ∧ (Baseβ€˜πΉ) ∈ (SubRingβ€˜πΈ)) ∧ (𝐸 β†Ύs (Baseβ€˜πΉ)) ∈ DivRing) β†’ βˆ€π‘Ž ∈ ((Baseβ€˜πΉ) βˆ– {(0gβ€˜πΈ)})((invrβ€˜πΈ)β€˜π‘Ž) ∈ (Baseβ€˜πΉ))
161160adantr 479 . . . . . . . . . . . . . . . . . . 19 ((((𝐸 ∈ DivRing ∧ (Baseβ€˜πΉ) ∈ (SubRingβ€˜πΈ)) ∧ (𝐸 β†Ύs (Baseβ€˜πΉ)) ∈ DivRing) ∧ ((invrβ€˜πΈ)β€˜π‘₯) ∈ ((Baseβ€˜πΉ) βˆ– {(0gβ€˜πΈ)})) β†’ βˆ€π‘Ž ∈ ((Baseβ€˜πΉ) βˆ– {(0gβ€˜πΈ)})((invrβ€˜πΈ)β€˜π‘Ž) ∈ (Baseβ€˜πΉ))
162 simpr 483 . . . . . . . . . . . . . . . . . . 19 ((((𝐸 ∈ DivRing ∧ (Baseβ€˜πΉ) ∈ (SubRingβ€˜πΈ)) ∧ (𝐸 β†Ύs (Baseβ€˜πΉ)) ∈ DivRing) ∧ ((invrβ€˜πΈ)β€˜π‘₯) ∈ ((Baseβ€˜πΉ) βˆ– {(0gβ€˜πΈ)})) β†’ ((invrβ€˜πΈ)β€˜π‘₯) ∈ ((Baseβ€˜πΉ) βˆ– {(0gβ€˜πΈ)}))
163158, 161, 162rspcdva 3612 . . . . . . . . . . . . . . . . . 18 ((((𝐸 ∈ DivRing ∧ (Baseβ€˜πΉ) ∈ (SubRingβ€˜πΈ)) ∧ (𝐸 β†Ύs (Baseβ€˜πΉ)) ∈ DivRing) ∧ ((invrβ€˜πΈ)β€˜π‘₯) ∈ ((Baseβ€˜πΉ) βˆ– {(0gβ€˜πΈ)})) β†’ ((invrβ€˜πΈ)β€˜((invrβ€˜πΈ)β€˜π‘₯)) ∈ (Baseβ€˜πΉ))
164132, 133, 136, 156, 163syl1111anc 836 . . . . . . . . . . . . . . . . 17 ((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) ∧ 𝑣 ∈ ((Baseβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ↑m 𝑏)) ∧ (1rβ€˜πΈ) = (((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)) Ξ£g (𝑖 ∈ 𝑏 ↦ ((π‘£β€˜π‘–)( ·𝑠 β€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))𝑖)))) β†’ ((invrβ€˜πΈ)β€˜((invrβ€˜πΈ)β€˜π‘₯)) ∈ (Baseβ€˜πΉ))
165131, 164eqeltrrd 2832 . . . . . . . . . . . . . . . 16 ((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) ∧ 𝑣 ∈ ((Baseβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ↑m 𝑏)) ∧ (1rβ€˜πΈ) = (((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)) Ξ£g (𝑖 ∈ 𝑏 ↦ ((π‘£β€˜π‘–)( ·𝑠 β€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))𝑖)))) β†’ π‘₯ ∈ (Baseβ€˜πΉ))
166165adantrl 712 . . . . . . . . . . . . . . 15 ((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) ∧ 𝑣 ∈ ((Baseβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ↑m 𝑏)) ∧ (𝑣 finSupp (0gβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ (1rβ€˜πΈ) = (((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)) Ξ£g (𝑖 ∈ 𝑏 ↦ ((π‘£β€˜π‘–)( ·𝑠 β€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))𝑖))))) β†’ π‘₯ ∈ (Baseβ€˜πΉ))
16727, 108ringidcl 20154 . . . . . . . . . . . . . . . . . 18 (𝐸 ∈ Ring β†’ (1rβ€˜πΈ) ∈ (Baseβ€˜πΈ))
16861, 167syl 17 . . . . . . . . . . . . . . . . 17 ((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) β†’ (1rβ€˜πΈ) ∈ (Baseβ€˜πΈ))
169168, 85eleqtrd 2833 . . . . . . . . . . . . . . . 16 ((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) β†’ (1rβ€˜πΈ) ∈ (Baseβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ))))
170 eqid 2730 . . . . . . . . . . . . . . . . 17 (Baseβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) = (Baseβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ))))
171 eqid 2730 . . . . . . . . . . . . . . . . 17 (Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ))) = (Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))
172 eqid 2730 . . . . . . . . . . . . . . . . 17 (0gβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) = (0gβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ))))
173 eqid 2730 . . . . . . . . . . . . . . . . 17 ( ·𝑠 β€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ))) = ( ·𝑠 β€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))
1744ad2antrr 722 . . . . . . . . . . . . . . . . . 18 ((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) β†’ ((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)) ∈ LVec)
175 lveclmod 20861 . . . . . . . . . . . . . . . . . 18 (((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)) ∈ LVec β†’ ((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)) ∈ LMod)
176174, 175syl 17 . . . . . . . . . . . . . . . . 17 ((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) β†’ ((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)) ∈ LMod)
17778, 170, 171, 172, 173, 176, 77lbslsp 32767 . . . . . . . . . . . . . . . 16 ((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) β†’ ((1rβ€˜πΈ) ∈ (Baseβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ))) ↔ βˆƒπ‘£ ∈ ((Baseβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ↑m 𝑏)(𝑣 finSupp (0gβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ (1rβ€˜πΈ) = (((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)) Ξ£g (𝑖 ∈ 𝑏 ↦ ((π‘£β€˜π‘–)( ·𝑠 β€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))𝑖))))))
178169, 177mpbid 231 . . . . . . . . . . . . . . 15 ((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) β†’ βˆƒπ‘£ ∈ ((Baseβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ↑m 𝑏)(𝑣 finSupp (0gβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ (1rβ€˜πΈ) = (((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)) Ξ£g (𝑖 ∈ 𝑏 ↦ ((π‘£β€˜π‘–)( ·𝑠 β€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))𝑖)))))
179166, 178r19.29a 3160 . . . . . . . . . . . . . 14 ((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) β†’ π‘₯ ∈ (Baseβ€˜πΉ))
180179ad2antrr 722 . . . . . . . . . . . . 13 ((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) ∧ 𝑒 ∈ (Baseβ€˜πΈ)) ∧ 𝑣 ∈ ((Baseβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ↑m 𝑏)) β†’ π‘₯ ∈ (Baseβ€˜πΉ))
181 eqid 2730 . . . . . . . . . . . . . 14 (.rβ€˜πΉ) = (.rβ€˜πΉ)
18225, 181ringcl 20144 . . . . . . . . . . . . 13 ((𝐹 ∈ Ring ∧ (π‘£β€˜π‘₯) ∈ (Baseβ€˜πΉ) ∧ π‘₯ ∈ (Baseβ€˜πΉ)) β†’ ((π‘£β€˜π‘₯)(.rβ€˜πΉ)π‘₯) ∈ (Baseβ€˜πΉ))
183106, 107, 180, 182syl3anc 1369 . . . . . . . . . . . 12 ((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) ∧ 𝑒 ∈ (Baseβ€˜πΈ)) ∧ 𝑣 ∈ ((Baseβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ↑m 𝑏)) β†’ ((π‘£β€˜π‘₯)(.rβ€˜πΉ)π‘₯) ∈ (Baseβ€˜πΉ))
184103, 183eqeltrd 2831 . . . . . . . . . . 11 ((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) ∧ 𝑒 ∈ (Baseβ€˜πΈ)) ∧ 𝑣 ∈ ((Baseβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ↑m 𝑏)) β†’ (((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)) Ξ£g (𝑖 ∈ 𝑏 ↦ ((π‘£β€˜π‘–)( ·𝑠 β€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))𝑖))) ∈ (Baseβ€˜πΉ))
185184ad2antrr 722 . . . . . . . . . 10 ((((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) ∧ 𝑒 ∈ (Baseβ€˜πΈ)) ∧ 𝑣 ∈ ((Baseβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ↑m 𝑏)) ∧ 𝑣 finSupp (0gβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ))))) ∧ 𝑒 = (((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)) Ξ£g (𝑖 ∈ 𝑏 ↦ ((π‘£β€˜π‘–)( ·𝑠 β€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))𝑖)))) β†’ (((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)) Ξ£g (𝑖 ∈ 𝑏 ↦ ((π‘£β€˜π‘–)( ·𝑠 β€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))𝑖))) ∈ (Baseβ€˜πΉ))
18622, 185eqeltrd 2831 . . . . . . . . 9 ((((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) ∧ 𝑒 ∈ (Baseβ€˜πΈ)) ∧ 𝑣 ∈ ((Baseβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ↑m 𝑏)) ∧ 𝑣 finSupp (0gβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ))))) ∧ 𝑒 = (((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)) Ξ£g (𝑖 ∈ 𝑏 ↦ ((π‘£β€˜π‘–)( ·𝑠 β€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))𝑖)))) β†’ 𝑒 ∈ (Baseβ€˜πΉ))
187186anasss 465 . . . . . . . 8 (((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) ∧ 𝑒 ∈ (Baseβ€˜πΈ)) ∧ 𝑣 ∈ ((Baseβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ↑m 𝑏)) ∧ (𝑣 finSupp (0gβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑒 = (((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)) Ξ£g (𝑖 ∈ 𝑏 ↦ ((π‘£β€˜π‘–)( ·𝑠 β€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))𝑖))))) β†’ 𝑒 ∈ (Baseβ€˜πΉ))
18885eleq2d 2817 . . . . . . . . . 10 ((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) β†’ (𝑒 ∈ (Baseβ€˜πΈ) ↔ 𝑒 ∈ (Baseβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))))
18978, 170, 171, 172, 173, 176, 77lbslsp 32767 . . . . . . . . . 10 ((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) β†’ (𝑒 ∈ (Baseβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ))) ↔ βˆƒπ‘£ ∈ ((Baseβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ↑m 𝑏)(𝑣 finSupp (0gβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑒 = (((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)) Ξ£g (𝑖 ∈ 𝑏 ↦ ((π‘£β€˜π‘–)( ·𝑠 β€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))𝑖))))))
190188, 189bitrd 278 . . . . . . . . 9 ((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) β†’ (𝑒 ∈ (Baseβ€˜πΈ) ↔ βˆƒπ‘£ ∈ ((Baseβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ↑m 𝑏)(𝑣 finSupp (0gβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑒 = (((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)) Ξ£g (𝑖 ∈ 𝑏 ↦ ((π‘£β€˜π‘–)( ·𝑠 β€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))𝑖))))))
191190biimpa 475 . . . . . . . 8 (((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) ∧ 𝑒 ∈ (Baseβ€˜πΈ)) β†’ βˆƒπ‘£ ∈ ((Baseβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ↑m 𝑏)(𝑣 finSupp (0gβ€˜(Scalarβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑒 = (((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)) Ξ£g (𝑖 ∈ 𝑏 ↦ ((π‘£β€˜π‘–)( ·𝑠 β€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))𝑖)))))
192187, 191r19.29a 3160 . . . . . . 7 (((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) ∧ 𝑒 ∈ (Baseβ€˜πΈ)) β†’ 𝑒 ∈ (Baseβ€˜πΉ))
193192ex 411 . . . . . 6 ((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) β†’ (𝑒 ∈ (Baseβ€˜πΈ) β†’ 𝑒 ∈ (Baseβ€˜πΉ)))
194193ssrdv 3987 . . . . 5 ((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) ∧ 𝑏 = {π‘₯}) β†’ (Baseβ€˜πΈ) βŠ† (Baseβ€˜πΉ))
19521, 194exlimddv 1936 . . . 4 (((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasisβ€˜((subringAlg β€˜πΈ)β€˜(Baseβ€˜πΉ)))) β†’ (Baseβ€˜πΈ) βŠ† (Baseβ€˜πΉ))
1969, 195exlimddv 1936 . . 3 ((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) β†’ (Baseβ€˜πΈ) βŠ† (Baseβ€˜πΉ))
197 simpr 483 . . . 4 (((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ (Baseβ€˜πΈ) βŠ† (Baseβ€˜πΉ)) β†’ (Baseβ€˜πΈ) βŠ† (Baseβ€˜πΉ))
19837ad2antrr 722 . . . 4 (((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ (Baseβ€˜πΈ) βŠ† (Baseβ€˜πΉ)) β†’ 𝐸 ∈ Field)
199 fvexd 6905 . . . 4 (((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ (Baseβ€˜πΈ) βŠ† (Baseβ€˜πΉ)) β†’ (Baseβ€˜πΉ) ∈ V)
20043, 27ressid2 17181 . . . 4 (((Baseβ€˜πΈ) βŠ† (Baseβ€˜πΉ) ∧ 𝐸 ∈ Field ∧ (Baseβ€˜πΉ) ∈ V) β†’ (𝐸 β†Ύs (Baseβ€˜πΉ)) = 𝐸)
201197, 198, 199, 200syl3anc 1369 . . 3 (((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ (Baseβ€˜πΈ) βŠ† (Baseβ€˜πΉ)) β†’ (𝐸 β†Ύs (Baseβ€˜πΉ)) = 𝐸)
202196, 201mpdan 683 . 2 ((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) β†’ (𝐸 β†Ύs (Baseβ€˜πΉ)) = 𝐸)
2032, 202eqtr2d 2771 1 ((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) β†’ 𝐸 = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 394   = wceq 1539  βˆƒwex 1779   ∈ wcel 2104   β‰  wne 2938  βˆ€wral 3059  βˆƒwrex 3068  Vcvv 3472   βˆ– cdif 3944   βŠ† wss 3947  βˆ…c0 4321  {csn 4627   class class class wbr 5147   ↦ cmpt 5230  βŸΆwf 6538  β€˜cfv 6542  (class class class)co 7411   ↑m cmap 8822   finSupp cfsupp 9363  1c1 11113  β™―chash 14294  Basecbs 17148   β†Ύs cress 17177  .rcmulr 17202  Scalarcsca 17204   ·𝑠 cvsca 17205  0gc0g 17389   Ξ£g cgsu 17390  Mndcmnd 18659  1rcur 20075  Ringcrg 20127  CRingccrg 20128  Unitcui 20246  invrcinvr 20278  SubRingcsubrg 20457  DivRingcdr 20500  Fieldcfield 20501  LModclmod 20614  LBasisclbs 20829  LVecclvec 20857  subringAlg csra 20926  LIndSclinds 21579  dimcldim 32971  /FldExtcfldext 33005  [:]cextdg 33008
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-reg 9589  ax-inf2 9638  ax-ac2 10460  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-tp 4632  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-iin 4999  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-of 7672  df-rpss 7715  df-om 7858  df-1st 7977  df-2nd 7978  df-supp 8149  df-tpos 8213  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-oadd 8472  df-er 8705  df-map 8824  df-ixp 8894  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-fsupp 9364  df-sup 9439  df-oi 9507  df-r1 9761  df-rank 9762  df-dju 9898  df-card 9936  df-acn 9939  df-ac 10113  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-2 12279  df-3 12280  df-4 12281  df-5 12282  df-6 12283  df-7 12284  df-8 12285  df-9 12286  df-n0 12477  df-xnn0 12549  df-z 12563  df-dec 12682  df-uz 12827  df-fz 13489  df-fzo 13632  df-seq 13971  df-hash 14295  df-struct 17084  df-sets 17101  df-slot 17119  df-ndx 17131  df-base 17149  df-ress 17178  df-plusg 17214  df-mulr 17215  df-sca 17217  df-vsca 17218  df-ip 17219  df-tset 17220  df-ple 17221  df-ocomp 17222  df-ds 17223  df-hom 17225  df-cco 17226  df-0g 17391  df-gsum 17392  df-prds 17397  df-pws 17399  df-mre 17534  df-mrc 17535  df-mri 17536  df-acs 17537  df-proset 18252  df-drs 18253  df-poset 18270  df-ipo 18485  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-mhm 18705  df-submnd 18706  df-grp 18858  df-minusg 18859  df-sbg 18860  df-mulg 18987  df-subg 19039  df-ghm 19128  df-cntz 19222  df-cmn 19691  df-abl 19692  df-mgp 20029  df-rng 20047  df-ur 20076  df-ring 20129  df-cring 20130  df-oppr 20225  df-dvdsr 20248  df-unit 20249  df-invr 20279  df-nzr 20404  df-subrg 20459  df-drng 20502  df-field 20503  df-lmod 20616  df-lss 20687  df-lsp 20727  df-lmhm 20777  df-lbs 20830  df-lvec 20858  df-sra 20930  df-rgmod 20931  df-dsmm 21506  df-frlm 21521  df-uvc 21557  df-lindf 21580  df-linds 21581  df-dim 32972  df-fldext 33009  df-extdg 33010
This theorem is referenced by:  extdg1b  33031
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