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Theorem extdg1id 32352
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) → 𝐸 = 𝐹)

Proof of Theorem extdg1id
Dummy variables 𝑎 𝑥 𝑏 𝑖 𝑣 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fldextress 32341 . . 3 (𝐸/FldExt𝐹𝐹 = (𝐸s (Base‘𝐹)))
21adantr 481 . 2 ((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) → 𝐹 = (𝐸s (Base‘𝐹)))
3 fldextsralvec 32344 . . . . . . 7 (𝐸/FldExt𝐹 → ((subringAlg ‘𝐸)‘(Base‘𝐹)) ∈ LVec)
43adantr 481 . . . . . 6 ((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) → ((subringAlg ‘𝐸)‘(Base‘𝐹)) ∈ LVec)
5 eqid 2736 . . . . . . 7 (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹))) = (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))
65lbsex 20626 . . . . . 6 (((subringAlg ‘𝐸)‘(Base‘𝐹)) ∈ LVec → (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹))) ≠ ∅)
74, 6syl 17 . . . . 5 ((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) → (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹))) ≠ ∅)
8 n0 4306 . . . . 5 ((LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹))) ≠ ∅ ↔ ∃𝑏 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
97, 8sylib 217 . . . 4 ((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) → ∃𝑏 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
10 simpr 485 . . . . . 6 (((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
115dimval 32300 . . . . . . . 8 ((((subringAlg ‘𝐸)‘(Base‘𝐹)) ∈ LVec ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → (dim‘((subringAlg ‘𝐸)‘(Base‘𝐹))) = (♯‘𝑏))
124, 11sylan 580 . . . . . . 7 (((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → (dim‘((subringAlg ‘𝐸)‘(Base‘𝐹))) = (♯‘𝑏))
13 extdgval 32343 . . . . . . . . . 10 (𝐸/FldExt𝐹 → (𝐸[:]𝐹) = (dim‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
1413adantr 481 . . . . . . . . 9 ((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) → (𝐸[:]𝐹) = (dim‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
15 simpr 485 . . . . . . . . 9 ((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) → (𝐸[:]𝐹) = 1)
1614, 15eqtr3d 2778 . . . . . . . 8 ((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) → (dim‘((subringAlg ‘𝐸)‘(Base‘𝐹))) = 1)
1716adantr 481 . . . . . . 7 (((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → (dim‘((subringAlg ‘𝐸)‘(Base‘𝐹))) = 1)
1812, 17eqtr3d 2778 . . . . . 6 (((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → (♯‘𝑏) = 1)
19 hash1snb 14319 . . . . . . 7 (𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹))) → ((♯‘𝑏) = 1 ↔ ∃𝑥 𝑏 = {𝑥}))
2019biimpa 477 . . . . . 6 ((𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹))) ∧ (♯‘𝑏) = 1) → ∃𝑥 𝑏 = {𝑥})
2110, 18, 20syl2anc 584 . . . . 5 (((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → ∃𝑥 𝑏 = {𝑥})
22 simpr 485 . . . . . . . . . 10 ((((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑢 ∈ (Base‘𝐸)) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑m 𝑏)) ∧ 𝑣 finSupp (0g‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹))))) ∧ 𝑢 = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σg (𝑖𝑏 ↦ ((𝑣𝑖)( ·𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖)))) → 𝑢 = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σg (𝑖𝑏 ↦ ((𝑣𝑖)( ·𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖))))
23 simplr 767 . . . . . . . . . . . . . . . 16 (((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑m 𝑏)) → 𝑏 = {𝑥})
24 eqidd 2737 . . . . . . . . . . . . . . . . . . . 20 (𝐸/FldExt𝐹 → ((subringAlg ‘𝐸)‘(Base‘𝐹)) = ((subringAlg ‘𝐸)‘(Base‘𝐹)))
25 eqid 2736 . . . . . . . . . . . . . . . . . . . . . 22 (Base‘𝐹) = (Base‘𝐹)
2625fldextsubrg 32340 . . . . . . . . . . . . . . . . . . . . 21 (𝐸/FldExt𝐹 → (Base‘𝐹) ∈ (SubRing‘𝐸))
27 eqid 2736 . . . . . . . . . . . . . . . . . . . . . 22 (Base‘𝐸) = (Base‘𝐸)
2827subrgss 20223 . . . . . . . . . . . . . . . . . . . . 21 ((Base‘𝐹) ∈ (SubRing‘𝐸) → (Base‘𝐹) ⊆ (Base‘𝐸))
2926, 28syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝐸/FldExt𝐹 → (Base‘𝐹) ⊆ (Base‘𝐸))
3024, 29sravsca 20648 . . . . . . . . . . . . . . . . . . 19 (𝐸/FldExt𝐹 → (.r𝐸) = ( ·𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
3130eqcomd 2742 . . . . . . . . . . . . . . . . . 18 (𝐸/FldExt𝐹 → ( ·𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹))) = (.r𝐸))
3231ad5antr 732 . . . . . . . . . . . . . . . . 17 ((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑m 𝑏)) ∧ 𝑖𝑏) → ( ·𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹))) = (.r𝐸))
3332oveqd 7374 . . . . . . . . . . . . . . . 16 ((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑m 𝑏)) ∧ 𝑖𝑏) → ((𝑣𝑖)( ·𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖) = ((𝑣𝑖)(.r𝐸)𝑖))
3423, 33mpteq12dva 5194 . . . . . . . . . . . . . . 15 (((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑m 𝑏)) → (𝑖𝑏 ↦ ((𝑣𝑖)( ·𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖)) = (𝑖 ∈ {𝑥} ↦ ((𝑣𝑖)(.r𝐸)𝑖)))
3534oveq2d 7373 . . . . . . . . . . . . . 14 (((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑m 𝑏)) → (𝐸 Σg (𝑖𝑏 ↦ ((𝑣𝑖)( ·𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖))) = (𝐸 Σg (𝑖 ∈ {𝑥} ↦ ((𝑣𝑖)(.r𝐸)𝑖))))
36 eqid 2736 . . . . . . . . . . . . . . . . 17 ((subringAlg ‘𝐸)‘(Base‘𝐹)) = ((subringAlg ‘𝐸)‘(Base‘𝐹))
37 fldextfld1 32338 . . . . . . . . . . . . . . . . . . 19 (𝐸/FldExt𝐹𝐸 ∈ Field)
38 isfld 20196 . . . . . . . . . . . . . . . . . . . 20 (𝐸 ∈ Field ↔ (𝐸 ∈ DivRing ∧ 𝐸 ∈ CRing))
3938simplbi 498 . . . . . . . . . . . . . . . . . . 19 (𝐸 ∈ Field → 𝐸 ∈ DivRing)
4037, 39syl 17 . . . . . . . . . . . . . . . . . 18 (𝐸/FldExt𝐹𝐸 ∈ DivRing)
4140adantr 481 . . . . . . . . . . . . . . . . 17 ((𝐸/FldExt𝐹𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → 𝐸 ∈ DivRing)
4226adantr 481 . . . . . . . . . . . . . . . . 17 ((𝐸/FldExt𝐹𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → (Base‘𝐹) ∈ (SubRing‘𝐸))
43 eqid 2736 . . . . . . . . . . . . . . . . 17 (𝐸s (Base‘𝐹)) = (𝐸s (Base‘𝐹))
44 fldextfld2 32339 . . . . . . . . . . . . . . . . . . . 20 (𝐸/FldExt𝐹𝐹 ∈ Field)
45 isfld 20196 . . . . . . . . . . . . . . . . . . . . 21 (𝐹 ∈ Field ↔ (𝐹 ∈ DivRing ∧ 𝐹 ∈ CRing))
4645simplbi 498 . . . . . . . . . . . . . . . . . . . 20 (𝐹 ∈ Field → 𝐹 ∈ DivRing)
4744, 46syl 17 . . . . . . . . . . . . . . . . . . 19 (𝐸/FldExt𝐹𝐹 ∈ DivRing)
481, 47eqeltrrd 2839 . . . . . . . . . . . . . . . . . 18 (𝐸/FldExt𝐹 → (𝐸s (Base‘𝐹)) ∈ DivRing)
4948adantr 481 . . . . . . . . . . . . . . . . 17 ((𝐸/FldExt𝐹𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → (𝐸s (Base‘𝐹)) ∈ DivRing)
50 simpr 485 . . . . . . . . . . . . . . . . 17 ((𝐸/FldExt𝐹𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
5136, 41, 42, 43, 49, 50drgextgsum 32296 . . . . . . . . . . . . . . . 16 ((𝐸/FldExt𝐹𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → (𝐸 Σg (𝑖𝑏 ↦ ((𝑣𝑖)( ·𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖))) = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σg (𝑖𝑏 ↦ ((𝑣𝑖)( ·𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖))))
5251adantlr 713 . . . . . . . . . . . . . . 15 (((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → (𝐸 Σg (𝑖𝑏 ↦ ((𝑣𝑖)( ·𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖))) = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σg (𝑖𝑏 ↦ ((𝑣𝑖)( ·𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖))))
5352ad2antrr 724 . . . . . . . . . . . . . 14 (((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑m 𝑏)) → (𝐸 Σg (𝑖𝑏 ↦ ((𝑣𝑖)( ·𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖))) = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σg (𝑖𝑏 ↦ ((𝑣𝑖)( ·𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖))))
54 drngring 20192 . . . . . . . . . . . . . . . . . . 19 (𝐸 ∈ DivRing → 𝐸 ∈ Ring)
5537, 39, 543syl 18 . . . . . . . . . . . . . . . . . 18 (𝐸/FldExt𝐹𝐸 ∈ Ring)
56 ringmnd 19974 . . . . . . . . . . . . . . . . . 18 (𝐸 ∈ Ring → 𝐸 ∈ Mnd)
5755, 56syl 17 . . . . . . . . . . . . . . . . 17 (𝐸/FldExt𝐹𝐸 ∈ Mnd)
5857ad4antr 730 . . . . . . . . . . . . . . . 16 (((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑m 𝑏)) → 𝐸 ∈ Mnd)
59 vex 3449 . . . . . . . . . . . . . . . . 17 𝑥 ∈ V
6059a1i 11 . . . . . . . . . . . . . . . 16 (((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑m 𝑏)) → 𝑥 ∈ V)
6155ad3antrrr 728 . . . . . . . . . . . . . . . . . 18 ((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → 𝐸 ∈ Ring)
6261adantr 481 . . . . . . . . . . . . . . . . 17 (((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑m 𝑏)) → 𝐸 ∈ Ring)
6329ad3antrrr 728 . . . . . . . . . . . . . . . . . . 19 ((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → (Base‘𝐹) ⊆ (Base‘𝐸))
6463adantr 481 . . . . . . . . . . . . . . . . . 18 (((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑m 𝑏)) → (Base‘𝐹) ⊆ (Base‘𝐸))
65 elmapi 8787 . . . . . . . . . . . . . . . . . . . . 21 (𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑m 𝑏) → 𝑣:𝑏⟶(Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))))
6665adantl 482 . . . . . . . . . . . . . . . . . . . 20 (((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑m 𝑏)) → 𝑣:𝑏⟶(Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))))
67 vsnid 4623 . . . . . . . . . . . . . . . . . . . . 21 𝑥 ∈ {𝑥}
6867, 23eleqtrrid 2845 . . . . . . . . . . . . . . . . . . . 20 (((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑m 𝑏)) → 𝑥𝑏)
6966, 68ffvelcdmd 7036 . . . . . . . . . . . . . . . . . . 19 (((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑m 𝑏)) → (𝑣𝑥) ∈ (Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))))
7024, 29srasca 20646 . . . . . . . . . . . . . . . . . . . . . 22 (𝐸/FldExt𝐹 → (𝐸s (Base‘𝐹)) = (Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
711, 70eqtrd 2776 . . . . . . . . . . . . . . . . . . . . 21 (𝐸/FldExt𝐹𝐹 = (Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
7271fveq2d 6846 . . . . . . . . . . . . . . . . . . . 20 (𝐸/FldExt𝐹 → (Base‘𝐹) = (Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))))
7372ad4antr 730 . . . . . . . . . . . . . . . . . . 19 (((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑m 𝑏)) → (Base‘𝐹) = (Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))))
7469, 73eleqtrrd 2841 . . . . . . . . . . . . . . . . . 18 (((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑m 𝑏)) → (𝑣𝑥) ∈ (Base‘𝐹))
7564, 74sseldd 3945 . . . . . . . . . . . . . . . . 17 (((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑m 𝑏)) → (𝑣𝑥) ∈ (Base‘𝐸))
76 simpr 485 . . . . . . . . . . . . . . . . . . . . 21 ((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → 𝑏 = {𝑥})
77 simplr 767 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
78 eqid 2736 . . . . . . . . . . . . . . . . . . . . . . 23 (Base‘((subringAlg ‘𝐸)‘(Base‘𝐹))) = (Base‘((subringAlg ‘𝐸)‘(Base‘𝐹)))
7978, 5lbsss 20538 . . . . . . . . . . . . . . . . . . . . . 22 (𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹))) → 𝑏 ⊆ (Base‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
8077, 79syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → 𝑏 ⊆ (Base‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
8176, 80eqsstrrd 3983 . . . . . . . . . . . . . . . . . . . 20 ((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → {𝑥} ⊆ (Base‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
8259snss 4746 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ (Base‘((subringAlg ‘𝐸)‘(Base‘𝐹))) ↔ {𝑥} ⊆ (Base‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
8381, 82sylibr 233 . . . . . . . . . . . . . . . . . . 19 ((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → 𝑥 ∈ (Base‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
84 eqidd 2737 . . . . . . . . . . . . . . . . . . . 20 ((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → ((subringAlg ‘𝐸)‘(Base‘𝐹)) = ((subringAlg ‘𝐸)‘(Base‘𝐹)))
8584, 63srabase 20640 . . . . . . . . . . . . . . . . . . 19 ((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → (Base‘𝐸) = (Base‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
8683, 85eleqtrrd 2841 . . . . . . . . . . . . . . . . . 18 ((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → 𝑥 ∈ (Base‘𝐸))
8786adantr 481 . . . . . . . . . . . . . . . . 17 (((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑m 𝑏)) → 𝑥 ∈ (Base‘𝐸))
88 eqid 2736 . . . . . . . . . . . . . . . . . 18 (.r𝐸) = (.r𝐸)
8927, 88ringcl 19981 . . . . . . . . . . . . . . . . 17 ((𝐸 ∈ Ring ∧ (𝑣𝑥) ∈ (Base‘𝐸) ∧ 𝑥 ∈ (Base‘𝐸)) → ((𝑣𝑥)(.r𝐸)𝑥) ∈ (Base‘𝐸))
9062, 75, 87, 89syl3anc 1371 . . . . . . . . . . . . . . . 16 (((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑m 𝑏)) → ((𝑣𝑥)(.r𝐸)𝑥) ∈ (Base‘𝐸))
91 simpr 485 . . . . . . . . . . . . . . . . . 18 ((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑m 𝑏)) ∧ 𝑖 = 𝑥) → 𝑖 = 𝑥)
9291fveq2d 6846 . . . . . . . . . . . . . . . . 17 ((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑m 𝑏)) ∧ 𝑖 = 𝑥) → (𝑣𝑖) = (𝑣𝑥))
9392, 91oveq12d 7375 . . . . . . . . . . . . . . . 16 ((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑m 𝑏)) ∧ 𝑖 = 𝑥) → ((𝑣𝑖)(.r𝐸)𝑖) = ((𝑣𝑥)(.r𝐸)𝑥))
9427, 58, 60, 90, 93gsumsnd 19729 . . . . . . . . . . . . . . 15 (((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑m 𝑏)) → (𝐸 Σg (𝑖 ∈ {𝑥} ↦ ((𝑣𝑖)(.r𝐸)𝑖))) = ((𝑣𝑥)(.r𝐸)𝑥))
951fveq2d 6846 . . . . . . . . . . . . . . . . . 18 (𝐸/FldExt𝐹 → (.r𝐹) = (.r‘(𝐸s (Base‘𝐹))))
9643, 88ressmulr 17188 . . . . . . . . . . . . . . . . . . 19 ((Base‘𝐹) ∈ (SubRing‘𝐸) → (.r𝐸) = (.r‘(𝐸s (Base‘𝐹))))
9726, 96syl 17 . . . . . . . . . . . . . . . . . 18 (𝐸/FldExt𝐹 → (.r𝐸) = (.r‘(𝐸s (Base‘𝐹))))
9895, 97eqtr4d 2779 . . . . . . . . . . . . . . . . 17 (𝐸/FldExt𝐹 → (.r𝐹) = (.r𝐸))
9998ad4antr 730 . . . . . . . . . . . . . . . 16 (((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑m 𝑏)) → (.r𝐹) = (.r𝐸))
10099oveqd 7374 . . . . . . . . . . . . . . 15 (((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑m 𝑏)) → ((𝑣𝑥)(.r𝐹)𝑥) = ((𝑣𝑥)(.r𝐸)𝑥))
10194, 100eqtr4d 2779 . . . . . . . . . . . . . 14 (((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑m 𝑏)) → (𝐸 Σg (𝑖 ∈ {𝑥} ↦ ((𝑣𝑖)(.r𝐸)𝑖))) = ((𝑣𝑥)(.r𝐹)𝑥))
10235, 53, 1013eqtr3d 2784 . . . . . . . . . . . . 13 (((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑m 𝑏)) → (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σg (𝑖𝑏 ↦ ((𝑣𝑖)( ·𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖))) = ((𝑣𝑥)(.r𝐹)𝑥))
103102adantlr 713 . . . . . . . . . . . 12 ((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑢 ∈ (Base‘𝐸)) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑m 𝑏)) → (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σg (𝑖𝑏 ↦ ((𝑣𝑖)( ·𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖))) = ((𝑣𝑥)(.r𝐹)𝑥))
104 drngring 20192 . . . . . . . . . . . . . . 15 (𝐹 ∈ DivRing → 𝐹 ∈ Ring)
10544, 46, 1043syl 18 . . . . . . . . . . . . . 14 (𝐸/FldExt𝐹𝐹 ∈ Ring)
106105ad5antr 732 . . . . . . . . . . . . 13 ((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑢 ∈ (Base‘𝐸)) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑m 𝑏)) → 𝐹 ∈ Ring)
10774adantlr 713 . . . . . . . . . . . . 13 ((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑢 ∈ (Base‘𝐸)) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑m 𝑏)) → (𝑣𝑥) ∈ (Base‘𝐹))
108 eqid 2736 . . . . . . . . . . . . . . . . . . . 20 (1r𝐸) = (1r𝐸)
109 eqid 2736 . . . . . . . . . . . . . . . . . . . 20 (Unit‘𝐸) = (Unit‘𝐸)
110 eqid 2736 . . . . . . . . . . . . . . . . . . . 20 (invr𝐸) = (invr𝐸)
111 simp-5l 783 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑m 𝑏)) ∧ (1r𝐸) = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σg (𝑖𝑏 ↦ ((𝑣𝑖)( ·𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖)))) → 𝐸/FldExt𝐹)
112111, 55syl 17 . . . . . . . . . . . . . . . . . . . 20 ((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑m 𝑏)) ∧ (1r𝐸) = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σg (𝑖𝑏 ↦ ((𝑣𝑖)( ·𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖)))) → 𝐸 ∈ Ring)
11387adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑m 𝑏)) ∧ (1r𝐸) = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σg (𝑖𝑏 ↦ ((𝑣𝑖)( ·𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖)))) → 𝑥 ∈ (Base‘𝐸))
11475adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑m 𝑏)) ∧ (1r𝐸) = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σg (𝑖𝑏 ↦ ((𝑣𝑖)( ·𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖)))) → (𝑣𝑥) ∈ (Base‘𝐸))
11538simprbi 497 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐸 ∈ Field → 𝐸 ∈ CRing)
116111, 37, 1153syl 18 . . . . . . . . . . . . . . . . . . . . . 22 ((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑m 𝑏)) ∧ (1r𝐸) = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σg (𝑖𝑏 ↦ ((𝑣𝑖)( ·𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖)))) → 𝐸 ∈ CRing)
11727, 88crngcom 19982 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐸 ∈ CRing ∧ 𝑥 ∈ (Base‘𝐸) ∧ (𝑣𝑥) ∈ (Base‘𝐸)) → (𝑥(.r𝐸)(𝑣𝑥)) = ((𝑣𝑥)(.r𝐸)𝑥))
118116, 113, 114, 117syl3anc 1371 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑m 𝑏)) ∧ (1r𝐸) = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σg (𝑖𝑏 ↦ ((𝑣𝑖)( ·𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖)))) → (𝑥(.r𝐸)(𝑣𝑥)) = ((𝑣𝑥)(.r𝐸)𝑥))
119 simpr 485 . . . . . . . . . . . . . . . . . . . . . . 23 ((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑m 𝑏)) ∧ (1r𝐸) = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σg (𝑖𝑏 ↦ ((𝑣𝑖)( ·𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖)))) → (1r𝐸) = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σg (𝑖𝑏 ↦ ((𝑣𝑖)( ·𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖))))
12052ad3antrrr 728 . . . . . . . . . . . . . . . . . . . . . . 23 ((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑m 𝑏)) ∧ (1r𝐸) = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σg (𝑖𝑏 ↦ ((𝑣𝑖)( ·𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖)))) → (𝐸 Σg (𝑖𝑏 ↦ ((𝑣𝑖)( ·𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖))) = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σg (𝑖𝑏 ↦ ((𝑣𝑖)( ·𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖))))
12134adantr 481 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑m 𝑏)) ∧ (1r𝐸) = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σg (𝑖𝑏 ↦ ((𝑣𝑖)( ·𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖)))) → (𝑖𝑏 ↦ ((𝑣𝑖)( ·𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖)) = (𝑖 ∈ {𝑥} ↦ ((𝑣𝑖)(.r𝐸)𝑖)))
122121oveq2d 7373 . . . . . . . . . . . . . . . . . . . . . . 23 ((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑m 𝑏)) ∧ (1r𝐸) = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σg (𝑖𝑏 ↦ ((𝑣𝑖)( ·𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖)))) → (𝐸 Σg (𝑖𝑏 ↦ ((𝑣𝑖)( ·𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖))) = (𝐸 Σg (𝑖 ∈ {𝑥} ↦ ((𝑣𝑖)(.r𝐸)𝑖))))
123119, 120, 1223eqtr2d 2782 . . . . . . . . . . . . . . . . . . . . . 22 ((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑m 𝑏)) ∧ (1r𝐸) = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σg (𝑖𝑏 ↦ ((𝑣𝑖)( ·𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖)))) → (1r𝐸) = (𝐸 Σg (𝑖 ∈ {𝑥} ↦ ((𝑣𝑖)(.r𝐸)𝑖))))
12494adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 ((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑m 𝑏)) ∧ (1r𝐸) = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σg (𝑖𝑏 ↦ ((𝑣𝑖)( ·𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖)))) → (𝐸 Σg (𝑖 ∈ {𝑥} ↦ ((𝑣𝑖)(.r𝐸)𝑖))) = ((𝑣𝑥)(.r𝐸)𝑥))
125123, 124eqtrd 2776 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑m 𝑏)) ∧ (1r𝐸) = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σg (𝑖𝑏 ↦ ((𝑣𝑖)( ·𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖)))) → (1r𝐸) = ((𝑣𝑥)(.r𝐸)𝑥))
126118, 125eqtr4d 2779 . . . . . . . . . . . . . . . . . . . 20 ((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑m 𝑏)) ∧ (1r𝐸) = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σg (𝑖𝑏 ↦ ((𝑣𝑖)( ·𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖)))) → (𝑥(.r𝐸)(𝑣𝑥)) = (1r𝐸))
127125eqcomd 2742 . . . . . . . . . . . . . . . . . . . 20 ((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑m 𝑏)) ∧ (1r𝐸) = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σg (𝑖𝑏 ↦ ((𝑣𝑖)( ·𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖)))) → ((𝑣𝑥)(.r𝐸)𝑥) = (1r𝐸))
12827, 88, 108, 109, 110, 112, 113, 114, 126, 127invrvald 22025 . . . . . . . . . . . . . . . . . . 19 ((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑m 𝑏)) ∧ (1r𝐸) = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σg (𝑖𝑏 ↦ ((𝑣𝑖)( ·𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖)))) → (𝑥 ∈ (Unit‘𝐸) ∧ ((invr𝐸)‘𝑥) = (𝑣𝑥)))
129128simpld 495 . . . . . . . . . . . . . . . . . 18 ((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑m 𝑏)) ∧ (1r𝐸) = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σg (𝑖𝑏 ↦ ((𝑣𝑖)( ·𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖)))) → 𝑥 ∈ (Unit‘𝐸))
130109, 110unitinvinv 20104 . . . . . . . . . . . . . . . . . 18 ((𝐸 ∈ Ring ∧ 𝑥 ∈ (Unit‘𝐸)) → ((invr𝐸)‘((invr𝐸)‘𝑥)) = 𝑥)
13162, 129, 130syl2an2r 683 . . . . . . . . . . . . . . . . 17 ((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑m 𝑏)) ∧ (1r𝐸) = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σg (𝑖𝑏 ↦ ((𝑣𝑖)( ·𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖)))) → ((invr𝐸)‘((invr𝐸)‘𝑥)) = 𝑥)
132111, 37, 393syl 18 . . . . . . . . . . . . . . . . . 18 ((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑m 𝑏)) ∧ (1r𝐸) = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σg (𝑖𝑏 ↦ ((𝑣𝑖)( ·𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖)))) → 𝐸 ∈ DivRing)
133111, 26syl 17 . . . . . . . . . . . . . . . . . 18 ((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑m 𝑏)) ∧ (1r𝐸) = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σg (𝑖𝑏 ↦ ((𝑣𝑖)( ·𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖)))) → (Base‘𝐹) ∈ (SubRing‘𝐸))
134111, 1syl 17 . . . . . . . . . . . . . . . . . . 19 ((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑m 𝑏)) ∧ (1r𝐸) = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σg (𝑖𝑏 ↦ ((𝑣𝑖)( ·𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖)))) → 𝐹 = (𝐸s (Base‘𝐹)))
135111, 44, 463syl 18 . . . . . . . . . . . . . . . . . . 19 ((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑m 𝑏)) ∧ (1r𝐸) = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σg (𝑖𝑏 ↦ ((𝑣𝑖)( ·𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖)))) → 𝐹 ∈ DivRing)
136134, 135eqeltrrd 2839 . . . . . . . . . . . . . . . . . 18 ((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑m 𝑏)) ∧ (1r𝐸) = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σg (𝑖𝑏 ↦ ((𝑣𝑖)( ·𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖)))) → (𝐸s (Base‘𝐹)) ∈ DivRing)
137128simprd 496 . . . . . . . . . . . . . . . . . . . 20 ((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑m 𝑏)) ∧ (1r𝐸) = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σg (𝑖𝑏 ↦ ((𝑣𝑖)( ·𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖)))) → ((invr𝐸)‘𝑥) = (𝑣𝑥))
13874adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑m 𝑏)) ∧ (1r𝐸) = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σg (𝑖𝑏 ↦ ((𝑣𝑖)( ·𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖)))) → (𝑣𝑥) ∈ (Base‘𝐹))
139137, 138eqeltrd 2838 . . . . . . . . . . . . . . . . . . 19 ((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑m 𝑏)) ∧ (1r𝐸) = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σg (𝑖𝑏 ↦ ((𝑣𝑖)( ·𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖)))) → ((invr𝐸)‘𝑥) ∈ (Base‘𝐹))
140 eqidd 2737 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐸/FldExt𝐹 → (0g𝐸) = (0g𝐸))
14124, 140, 29sralmod0 20657 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐸/FldExt𝐹 → (0g𝐸) = (0g‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
142141ad2antrr 724 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → (0g𝐸) = (0g‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
1435lbslinds 21239 . . . . . . . . . . . . . . . . . . . . . . . . 25 (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹))) ⊆ (LIndS‘((subringAlg ‘𝐸)‘(Base‘𝐹)))
144143, 10sselid 3942 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → 𝑏 ∈ (LIndS‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
145 eqid 2736 . . . . . . . . . . . . . . . . . . . . . . . . 25 (0g‘((subringAlg ‘𝐸)‘(Base‘𝐹))) = (0g‘((subringAlg ‘𝐸)‘(Base‘𝐹)))
1461450nellinds 32159 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((subringAlg ‘𝐸)‘(Base‘𝐹)) ∈ LVec ∧ 𝑏 ∈ (LIndS‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → ¬ (0g‘((subringAlg ‘𝐸)‘(Base‘𝐹))) ∈ 𝑏)
1474, 144, 146syl2an2r 683 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → ¬ (0g‘((subringAlg ‘𝐸)‘(Base‘𝐹))) ∈ 𝑏)
148142, 147eqneltrd 2857 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → ¬ (0g𝐸) ∈ 𝑏)
149148ad3antrrr 728 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑m 𝑏)) ∧ (1r𝐸) = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σg (𝑖𝑏 ↦ ((𝑣𝑖)( ·𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖)))) → ¬ (0g𝐸) ∈ 𝑏)
150 nelne2 3042 . . . . . . . . . . . . . . . . . . . . 21 ((𝑥𝑏 ∧ ¬ (0g𝐸) ∈ 𝑏) → 𝑥 ≠ (0g𝐸))
15168, 149, 150syl2an2r 683 . . . . . . . . . . . . . . . . . . . 20 ((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑m 𝑏)) ∧ (1r𝐸) = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σg (𝑖𝑏 ↦ ((𝑣𝑖)( ·𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖)))) → 𝑥 ≠ (0g𝐸))
152 eqid 2736 . . . . . . . . . . . . . . . . . . . . 21 (0g𝐸) = (0g𝐸)
15327, 152, 110drnginvrn0 20206 . . . . . . . . . . . . . . . . . . . 20 ((𝐸 ∈ DivRing ∧ 𝑥 ∈ (Base‘𝐸) ∧ 𝑥 ≠ (0g𝐸)) → ((invr𝐸)‘𝑥) ≠ (0g𝐸))
154132, 113, 151, 153syl3anc 1371 . . . . . . . . . . . . . . . . . . 19 ((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑m 𝑏)) ∧ (1r𝐸) = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σg (𝑖𝑏 ↦ ((𝑣𝑖)( ·𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖)))) → ((invr𝐸)‘𝑥) ≠ (0g𝐸))
155 eldifsn 4747 . . . . . . . . . . . . . . . . . . 19 (((invr𝐸)‘𝑥) ∈ ((Base‘𝐹) ∖ {(0g𝐸)}) ↔ (((invr𝐸)‘𝑥) ∈ (Base‘𝐹) ∧ ((invr𝐸)‘𝑥) ≠ (0g𝐸)))
156139, 154, 155sylanbrc 583 . . . . . . . . . . . . . . . . . 18 ((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑m 𝑏)) ∧ (1r𝐸) = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σg (𝑖𝑏 ↦ ((𝑣𝑖)( ·𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖)))) → ((invr𝐸)‘𝑥) ∈ ((Base‘𝐹) ∖ {(0g𝐸)}))
157 fveq2 6842 . . . . . . . . . . . . . . . . . . . 20 (𝑎 = ((invr𝐸)‘𝑥) → ((invr𝐸)‘𝑎) = ((invr𝐸)‘((invr𝐸)‘𝑥)))
158157eleq1d 2822 . . . . . . . . . . . . . . . . . . 19 (𝑎 = ((invr𝐸)‘𝑥) → (((invr𝐸)‘𝑎) ∈ (Base‘𝐹) ↔ ((invr𝐸)‘((invr𝐸)‘𝑥)) ∈ (Base‘𝐹)))
15943, 152, 110issubdrg 20247 . . . . . . . . . . . . . . . . . . . . 21 ((𝐸 ∈ DivRing ∧ (Base‘𝐹) ∈ (SubRing‘𝐸)) → ((𝐸s (Base‘𝐹)) ∈ DivRing ↔ ∀𝑎 ∈ ((Base‘𝐹) ∖ {(0g𝐸)})((invr𝐸)‘𝑎) ∈ (Base‘𝐹)))
160159biimpa 477 . . . . . . . . . . . . . . . . . . . 20 (((𝐸 ∈ DivRing ∧ (Base‘𝐹) ∈ (SubRing‘𝐸)) ∧ (𝐸s (Base‘𝐹)) ∈ DivRing) → ∀𝑎 ∈ ((Base‘𝐹) ∖ {(0g𝐸)})((invr𝐸)‘𝑎) ∈ (Base‘𝐹))
161160adantr 481 . . . . . . . . . . . . . . . . . . 19 ((((𝐸 ∈ DivRing ∧ (Base‘𝐹) ∈ (SubRing‘𝐸)) ∧ (𝐸s (Base‘𝐹)) ∈ DivRing) ∧ ((invr𝐸)‘𝑥) ∈ ((Base‘𝐹) ∖ {(0g𝐸)})) → ∀𝑎 ∈ ((Base‘𝐹) ∖ {(0g𝐸)})((invr𝐸)‘𝑎) ∈ (Base‘𝐹))
162 simpr 485 . . . . . . . . . . . . . . . . . . 19 ((((𝐸 ∈ DivRing ∧ (Base‘𝐹) ∈ (SubRing‘𝐸)) ∧ (𝐸s (Base‘𝐹)) ∈ DivRing) ∧ ((invr𝐸)‘𝑥) ∈ ((Base‘𝐹) ∖ {(0g𝐸)})) → ((invr𝐸)‘𝑥) ∈ ((Base‘𝐹) ∖ {(0g𝐸)}))
163158, 161, 162rspcdva 3582 . . . . . . . . . . . . . . . . . 18 ((((𝐸 ∈ DivRing ∧ (Base‘𝐹) ∈ (SubRing‘𝐸)) ∧ (𝐸s (Base‘𝐹)) ∈ DivRing) ∧ ((invr𝐸)‘𝑥) ∈ ((Base‘𝐹) ∖ {(0g𝐸)})) → ((invr𝐸)‘((invr𝐸)‘𝑥)) ∈ (Base‘𝐹))
164132, 133, 136, 156, 163syl1111anc 838 . . . . . . . . . . . . . . . . 17 ((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑m 𝑏)) ∧ (1r𝐸) = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σg (𝑖𝑏 ↦ ((𝑣𝑖)( ·𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖)))) → ((invr𝐸)‘((invr𝐸)‘𝑥)) ∈ (Base‘𝐹))
165131, 164eqeltrrd 2839 . . . . . . . . . . . . . . . 16 ((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑m 𝑏)) ∧ (1r𝐸) = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σg (𝑖𝑏 ↦ ((𝑣𝑖)( ·𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖)))) → 𝑥 ∈ (Base‘𝐹))
166165adantrl 714 . . . . . . . . . . . . . . 15 ((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑m 𝑏)) ∧ (𝑣 finSupp (0g‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ (1r𝐸) = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σg (𝑖𝑏 ↦ ((𝑣𝑖)( ·𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖))))) → 𝑥 ∈ (Base‘𝐹))
16727, 108ringidcl 19989 . . . . . . . . . . . . . . . . . 18 (𝐸 ∈ Ring → (1r𝐸) ∈ (Base‘𝐸))
16861, 167syl 17 . . . . . . . . . . . . . . . . 17 ((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → (1r𝐸) ∈ (Base‘𝐸))
169168, 85eleqtrd 2840 . . . . . . . . . . . . . . . 16 ((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → (1r𝐸) ∈ (Base‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
170 eqid 2736 . . . . . . . . . . . . . . . . 17 (Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) = (Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
171 eqid 2736 . . . . . . . . . . . . . . . . 17 (Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹))) = (Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))
172 eqid 2736 . . . . . . . . . . . . . . . . 17 (0g‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) = (0g‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
173 eqid 2736 . . . . . . . . . . . . . . . . 17 ( ·𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹))) = ( ·𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))
1744ad2antrr 724 . . . . . . . . . . . . . . . . . 18 ((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → ((subringAlg ‘𝐸)‘(Base‘𝐹)) ∈ LVec)
175 lveclmod 20567 . . . . . . . . . . . . . . . . . 18 (((subringAlg ‘𝐸)‘(Base‘𝐹)) ∈ LVec → ((subringAlg ‘𝐸)‘(Base‘𝐹)) ∈ LMod)
176174, 175syl 17 . . . . . . . . . . . . . . . . 17 ((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → ((subringAlg ‘𝐸)‘(Base‘𝐹)) ∈ LMod)
17778, 170, 171, 172, 173, 176, 77lbslsp 32165 . . . . . . . . . . . . . . . 16 ((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → ((1r𝐸) ∈ (Base‘((subringAlg ‘𝐸)‘(Base‘𝐹))) ↔ ∃𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑m 𝑏)(𝑣 finSupp (0g‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ (1r𝐸) = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σg (𝑖𝑏 ↦ ((𝑣𝑖)( ·𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖))))))
178169, 177mpbid 231 . . . . . . . . . . . . . . 15 ((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → ∃𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑m 𝑏)(𝑣 finSupp (0g‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ (1r𝐸) = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σg (𝑖𝑏 ↦ ((𝑣𝑖)( ·𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖)))))
179166, 178r19.29a 3159 . . . . . . . . . . . . . 14 ((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → 𝑥 ∈ (Base‘𝐹))
180179ad2antrr 724 . . . . . . . . . . . . 13 ((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑢 ∈ (Base‘𝐸)) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑m 𝑏)) → 𝑥 ∈ (Base‘𝐹))
181 eqid 2736 . . . . . . . . . . . . . 14 (.r𝐹) = (.r𝐹)
18225, 181ringcl 19981 . . . . . . . . . . . . 13 ((𝐹 ∈ Ring ∧ (𝑣𝑥) ∈ (Base‘𝐹) ∧ 𝑥 ∈ (Base‘𝐹)) → ((𝑣𝑥)(.r𝐹)𝑥) ∈ (Base‘𝐹))
183106, 107, 180, 182syl3anc 1371 . . . . . . . . . . . 12 ((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑢 ∈ (Base‘𝐸)) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑m 𝑏)) → ((𝑣𝑥)(.r𝐹)𝑥) ∈ (Base‘𝐹))
184103, 183eqeltrd 2838 . . . . . . . . . . 11 ((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑢 ∈ (Base‘𝐸)) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑m 𝑏)) → (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σg (𝑖𝑏 ↦ ((𝑣𝑖)( ·𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖))) ∈ (Base‘𝐹))
185184ad2antrr 724 . . . . . . . . . 10 ((((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑢 ∈ (Base‘𝐸)) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑m 𝑏)) ∧ 𝑣 finSupp (0g‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹))))) ∧ 𝑢 = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σg (𝑖𝑏 ↦ ((𝑣𝑖)( ·𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖)))) → (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σg (𝑖𝑏 ↦ ((𝑣𝑖)( ·𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖))) ∈ (Base‘𝐹))
18622, 185eqeltrd 2838 . . . . . . . . 9 ((((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑢 ∈ (Base‘𝐸)) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑m 𝑏)) ∧ 𝑣 finSupp (0g‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹))))) ∧ 𝑢 = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σg (𝑖𝑏 ↦ ((𝑣𝑖)( ·𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖)))) → 𝑢 ∈ (Base‘𝐹))
187186anasss 467 . . . . . . . 8 (((((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑢 ∈ (Base‘𝐸)) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑m 𝑏)) ∧ (𝑣 finSupp (0g‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑢 = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σg (𝑖𝑏 ↦ ((𝑣𝑖)( ·𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖))))) → 𝑢 ∈ (Base‘𝐹))
18885eleq2d 2823 . . . . . . . . . 10 ((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → (𝑢 ∈ (Base‘𝐸) ↔ 𝑢 ∈ (Base‘((subringAlg ‘𝐸)‘(Base‘𝐹)))))
18978, 170, 171, 172, 173, 176, 77lbslsp 32165 . . . . . . . . . 10 ((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → (𝑢 ∈ (Base‘((subringAlg ‘𝐸)‘(Base‘𝐹))) ↔ ∃𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑m 𝑏)(𝑣 finSupp (0g‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑢 = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σg (𝑖𝑏 ↦ ((𝑣𝑖)( ·𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖))))))
190188, 189bitrd 278 . . . . . . . . 9 ((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → (𝑢 ∈ (Base‘𝐸) ↔ ∃𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑m 𝑏)(𝑣 finSupp (0g‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑢 = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σg (𝑖𝑏 ↦ ((𝑣𝑖)( ·𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖))))))
191190biimpa 477 . . . . . . . 8 (((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑢 ∈ (Base‘𝐸)) → ∃𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑m 𝑏)(𝑣 finSupp (0g‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑢 = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σg (𝑖𝑏 ↦ ((𝑣𝑖)( ·𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖)))))
192187, 191r19.29a 3159 . . . . . . 7 (((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑢 ∈ (Base‘𝐸)) → 𝑢 ∈ (Base‘𝐹))
193192ex 413 . . . . . 6 ((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → (𝑢 ∈ (Base‘𝐸) → 𝑢 ∈ (Base‘𝐹)))
194193ssrdv 3950 . . . . 5 ((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → (Base‘𝐸) ⊆ (Base‘𝐹))
19521, 194exlimddv 1938 . . . 4 (((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → (Base‘𝐸) ⊆ (Base‘𝐹))
1969, 195exlimddv 1938 . . 3 ((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) → (Base‘𝐸) ⊆ (Base‘𝐹))
197 simpr 485 . . . 4 (((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ (Base‘𝐸) ⊆ (Base‘𝐹)) → (Base‘𝐸) ⊆ (Base‘𝐹))
19837ad2antrr 724 . . . 4 (((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ (Base‘𝐸) ⊆ (Base‘𝐹)) → 𝐸 ∈ Field)
199 fvexd 6857 . . . 4 (((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ (Base‘𝐸) ⊆ (Base‘𝐹)) → (Base‘𝐹) ∈ V)
20043, 27ressid2 17116 . . . 4 (((Base‘𝐸) ⊆ (Base‘𝐹) ∧ 𝐸 ∈ Field ∧ (Base‘𝐹) ∈ V) → (𝐸s (Base‘𝐹)) = 𝐸)
201197, 198, 199, 200syl3anc 1371 . . 3 (((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ (Base‘𝐸) ⊆ (Base‘𝐹)) → (𝐸s (Base‘𝐹)) = 𝐸)
202196, 201mpdan 685 . 2 ((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) → (𝐸s (Base‘𝐹)) = 𝐸)
2032, 202eqtr2d 2777 1 ((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) → 𝐸 = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1541  wex 1781  wcel 2106  wne 2943  wral 3064  wrex 3073  Vcvv 3445  cdif 3907  wss 3910  c0 4282  {csn 4586   class class class wbr 5105  cmpt 5188  wf 6492  cfv 6496  (class class class)co 7357  m cmap 8765   finSupp cfsupp 9305  1c1 11052  chash 14230  Basecbs 17083  s cress 17112  .rcmulr 17134  Scalarcsca 17136   ·𝑠 cvsca 17137  0gc0g 17321   Σg cgsu 17322  Mndcmnd 18556  1rcur 19913  Ringcrg 19964  CRingccrg 19965  Unitcui 20068  invrcinvr 20100  DivRingcdr 20185  Fieldcfield 20186  SubRingcsubrg 20218  LModclmod 20322  LBasisclbs 20535  LVecclvec 20563  subringAlg csra 20629  LIndSclinds 21211  dimcldim 32298  /FldExtcfldext 32327  [:]cextdg 32330
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-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-reg 9528  ax-inf2 9577  ax-ac2 10399  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-rpss 7660  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-tpos 8157  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-oadd 8416  df-er 8648  df-map 8767  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9306  df-sup 9378  df-oi 9446  df-r1 9700  df-rank 9701  df-dju 9837  df-card 9875  df-acn 9878  df-ac 10052  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-xnn0 12486  df-z 12500  df-dec 12619  df-uz 12764  df-fz 13425  df-fzo 13568  df-seq 13907  df-hash 14231  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-sca 17149  df-vsca 17150  df-ip 17151  df-tset 17152  df-ple 17153  df-ocomp 17154  df-ds 17155  df-hom 17157  df-cco 17158  df-0g 17323  df-gsum 17324  df-prds 17329  df-pws 17331  df-mre 17466  df-mrc 17467  df-mri 17468  df-acs 17469  df-proset 18184  df-drs 18185  df-poset 18202  df-ipo 18417  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-mhm 18601  df-submnd 18602  df-grp 18751  df-minusg 18752  df-sbg 18753  df-mulg 18873  df-subg 18925  df-ghm 19006  df-cntz 19097  df-cmn 19564  df-abl 19565  df-mgp 19897  df-ur 19914  df-ring 19966  df-cring 19967  df-oppr 20049  df-dvdsr 20070  df-unit 20071  df-invr 20101  df-drng 20187  df-field 20188  df-subrg 20220  df-lmod 20324  df-lss 20393  df-lsp 20433  df-lmhm 20483  df-lbs 20536  df-lvec 20564  df-sra 20633  df-rgmod 20634  df-nzr 20728  df-dsmm 21138  df-frlm 21153  df-uvc 21189  df-lindf 21212  df-linds 21213  df-dim 32299  df-fldext 32331  df-extdg 32332
This theorem is referenced by:  extdg1b  32353
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