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Theorem extdg1id 33717
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 33704 . . 3 (𝐸/FldExt𝐹𝐹 = (𝐸s (Base‘𝐹)))
21adantr 480 . 2 ((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) → 𝐹 = (𝐸s (Base‘𝐹)))
3 fldextsralvec 33707 . . . . . . 7 (𝐸/FldExt𝐹 → ((subringAlg ‘𝐸)‘(Base‘𝐹)) ∈ LVec)
43adantr 480 . . . . . 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 21168 . . . . . 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 4352 . . . . 5 ((LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹))) ≠ ∅ ↔ ∃𝑏 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
97, 8sylib 218 . . . 4 ((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) → ∃𝑏 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
10 simpr 484 . . . . . 6 (((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
115dimval 33652 . . . . . . . 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 33706 . . . . . . . . . 10 (𝐸/FldExt𝐹 → (𝐸[:]𝐹) = (dim‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
1413adantr 480 . . . . . . . . 9 ((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) → (𝐸[:]𝐹) = (dim‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
15 simpr 484 . . . . . . . . 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 480 . . . . . . 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 14459 . . . . . . 7 (𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹))) → ((♯‘𝑏) = 1 ↔ ∃𝑥 𝑏 = {𝑥}))
2019biimpa 476 . . . . . 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 484 . . . . . . . . . 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 768 . . . . . . . . . . . . . . . 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 33703 . . . . . . . . . . . . . . . . . . . . 21 (𝐸/FldExt𝐹 → (Base‘𝐹) ∈ (SubRing‘𝐸))
27 eqid 2736 . . . . . . . . . . . . . . . . . . . . . 22 (Base‘𝐸) = (Base‘𝐸)
2827subrgss 20573 . . . . . . . . . . . . . . . . . . . . 21 ((Base‘𝐹) ∈ (SubRing‘𝐸) → (Base‘𝐹) ⊆ (Base‘𝐸))
2926, 28syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝐸/FldExt𝐹 → (Base‘𝐹) ⊆ (Base‘𝐸))
3024, 29sravsca 21186 . . . . . . . . . . . . . . . . . . 19 (𝐸/FldExt𝐹 → (.r𝐸) = ( ·𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
3130eqcomd 2742 . . . . . . . . . . . . . . . . . 18 (𝐸/FldExt𝐹 → ( ·𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹))) = (.r𝐸))
3231ad5antr 734 . . . . . . . . . . . . . . . . 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 7449 . . . . . . . . . . . . . . . 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 5230 . . . . . . . . . . . . . . 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 7448 . . . . . . . . . . . . . 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 33701 . . . . . . . . . . . . . . . . . . 19 (𝐸/FldExt𝐹𝐸 ∈ Field)
38 isfld 20741 . . . . . . . . . . . . . . . . . . . 20 (𝐸 ∈ Field ↔ (𝐸 ∈ DivRing ∧ 𝐸 ∈ CRing))
3938simplbi 497 . . . . . . . . . . . . . . . . . . 19 (𝐸 ∈ Field → 𝐸 ∈ DivRing)
4037, 39syl 17 . . . . . . . . . . . . . . . . . 18 (𝐸/FldExt𝐹𝐸 ∈ DivRing)
4140adantr 480 . . . . . . . . . . . . . . . . 17 ((𝐸/FldExt𝐹𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → 𝐸 ∈ DivRing)
4226adantr 480 . . . . . . . . . . . . . . . . 17 ((𝐸/FldExt𝐹𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → (Base‘𝐹) ∈ (SubRing‘𝐸))
43 eqid 2736 . . . . . . . . . . . . . . . . 17 (𝐸s (Base‘𝐹)) = (𝐸s (Base‘𝐹))
44 fldextfld2 33702 . . . . . . . . . . . . . . . . . . . 20 (𝐸/FldExt𝐹𝐹 ∈ Field)
45 isfld 20741 . . . . . . . . . . . . . . . . . . . . 21 (𝐹 ∈ Field ↔ (𝐹 ∈ DivRing ∧ 𝐹 ∈ CRing))
4645simplbi 497 . . . . . . . . . . . . . . . . . . . 20 (𝐹 ∈ Field → 𝐹 ∈ DivRing)
4744, 46syl 17 . . . . . . . . . . . . . . . . . . 19 (𝐸/FldExt𝐹𝐹 ∈ DivRing)
481, 47eqeltrrd 2841 . . . . . . . . . . . . . . . . . 18 (𝐸/FldExt𝐹 → (𝐸s (Base‘𝐹)) ∈ DivRing)
4948adantr 480 . . . . . . . . . . . . . . . . 17 ((𝐸/FldExt𝐹𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → (𝐸s (Base‘𝐹)) ∈ DivRing)
50 simpr 484 . . . . . . . . . . . . . . . . 17 ((𝐸/FldExt𝐹𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
5136, 41, 42, 43, 49, 50drgextgsum 33646 . . . . . . . . . . . . . . . 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 715 . . . . . . . . . . . . . . 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 726 . . . . . . . . . . . . . 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 20737 . . . . . . . . . . . . . . . . . . 19 (𝐸 ∈ DivRing → 𝐸 ∈ Ring)
5537, 39, 543syl 18 . . . . . . . . . . . . . . . . . 18 (𝐸/FldExt𝐹𝐸 ∈ Ring)
56 ringmnd 20241 . . . . . . . . . . . . . . . . . 18 (𝐸 ∈ Ring → 𝐸 ∈ Mnd)
5755, 56syl 17 . . . . . . . . . . . . . . . . 17 (𝐸/FldExt𝐹𝐸 ∈ Mnd)
5857ad4antr 732 . . . . . . . . . . . . . . . 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 3483 . . . . . . . . . . . . . . . . 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 730 . . . . . . . . . . . . . . . . . 18 ((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → 𝐸 ∈ Ring)
6261adantr 480 . . . . . . . . . . . . . . . . 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 730 . . . . . . . . . . . . . . . . . . 19 ((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → (Base‘𝐹) ⊆ (Base‘𝐸))
6463adantr 480 . . . . . . . . . . . . . . . . . 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 8890 . . . . . . . . . . . . . . . . . . . . 21 (𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑m 𝑏) → 𝑣:𝑏⟶(Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))))
6665adantl 481 . . . . . . . . . . . . . . . . . . . 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 4662 . . . . . . . . . . . . . . . . . . . . 21 𝑥 ∈ {𝑥}
6867, 23eleqtrrid 2847 . . . . . . . . . . . . . . . . . . . 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 7104 . . . . . . . . . . . . . . . . . . 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 21184 . . . . . . . . . . . . . . . . . . . . . 22 (𝐸/FldExt𝐹 → (𝐸s (Base‘𝐹)) = (Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
711, 70eqtrd 2776 . . . . . . . . . . . . . . . . . . . . 21 (𝐸/FldExt𝐹𝐹 = (Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
7271fveq2d 6909 . . . . . . . . . . . . . . . . . . . 20 (𝐸/FldExt𝐹 → (Base‘𝐹) = (Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))))
7372ad4antr 732 . . . . . . . . . . . . . . . . . . 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 2843 . . . . . . . . . . . . . . . . . 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 3983 . . . . . . . . . . . . . . . . 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 484 . . . . . . . . . . . . . . . . . . . . 21 ((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → 𝑏 = {𝑥})
77 simplr 768 . . . . . . . . . . . . . . . . . . . . . 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 21077 . . . . . . . . . . . . . . . . . . . . . 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 4018 . . . . . . . . . . . . . . . . . . . 20 ((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → {𝑥} ⊆ (Base‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
8259snss 4784 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ (Base‘((subringAlg ‘𝐸)‘(Base‘𝐹))) ↔ {𝑥} ⊆ (Base‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
8381, 82sylibr 234 . . . . . . . . . . . . . . . . . . 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 21178 . . . . . . . . . . . . . . . . . . 19 ((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → (Base‘𝐸) = (Base‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
8683, 85eleqtrrd 2843 . . . . . . . . . . . . . . . . . 18 ((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → 𝑥 ∈ (Base‘𝐸))
8786adantr 480 . . . . . . . . . . . . . . . . 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 20248 . . . . . . . . . . . . . . . . 17 ((𝐸 ∈ Ring ∧ (𝑣𝑥) ∈ (Base‘𝐸) ∧ 𝑥 ∈ (Base‘𝐸)) → ((𝑣𝑥)(.r𝐸)𝑥) ∈ (Base‘𝐸))
9062, 75, 87, 89syl3anc 1372 . . . . . . . . . . . . . . . 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 484 . . . . . . . . . . . . . . . . . 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 6909 . . . . . . . . . . . . . . . . 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 7450 . . . . . . . . . . . . . . . 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 19971 . . . . . . . . . . . . . . 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 6909 . . . . . . . . . . . . . . . . . 18 (𝐸/FldExt𝐹 → (.r𝐹) = (.r‘(𝐸s (Base‘𝐹))))
9643, 88ressmulr 17352 . . . . . . . . . . . . . . . . . . 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 732 . . . . . . . . . . . . . . . 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 7449 . . . . . . . . . . . . . . 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 715 . . . . . . . . . . . 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 20737 . . . . . . . . . . . . . . 15 (𝐹 ∈ DivRing → 𝐹 ∈ Ring)
10544, 46, 1043syl 18 . . . . . . . . . . . . . 14 (𝐸/FldExt𝐹𝐹 ∈ Ring)
106105ad5antr 734 . . . . . . . . . . . . 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 715 . . . . . . . . . . . . 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 784 . . . . . . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . . . . . 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 496 . . . . . . . . . . . . . . . . . . . . . . 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 20249 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐸 ∈ CRing ∧ 𝑥 ∈ (Base‘𝐸) ∧ (𝑣𝑥) ∈ (Base‘𝐸)) → (𝑥(.r𝐸)(𝑣𝑥)) = ((𝑣𝑥)(.r𝐸)𝑥))
118116, 113, 114, 117syl3anc 1372 . . . . . . . . . . . . . . . . . . . . 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 484 . . . . . . . . . . . . . . . . . . . . . . 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 730 . . . . . . . . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . . . . . . . . . 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 7448 . . . . . . . . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . . . . . . . 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 22683 . . . . . . . . . . . . . . . . . . 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 494 . . . . . . . . . . . . . . . . . 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 20392 . . . . . . . . . . . . . . . . . 18 ((𝐸 ∈ Ring ∧ 𝑥 ∈ (Unit‘𝐸)) → ((invr𝐸)‘((invr𝐸)‘𝑥)) = 𝑥)
13162, 129, 130syl2an2r 685 . . . . . . . . . . . . . . . . 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 2841 . . . . . . . . . . . . . . . . . 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 495 . . . . . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . . . . . 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 2840 . . . . . . . . . . . . . . . . . . 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 21196 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐸/FldExt𝐹 → (0g𝐸) = (0g‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
142141ad2antrr 726 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → (0g𝐸) = (0g‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
1435lbslinds 21854 . . . . . . . . . . . . . . . . . . . . . . . . 25 (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹))) ⊆ (LIndS‘((subringAlg ‘𝐸)‘(Base‘𝐹)))
144143, 10sselid 3980 . . . . . . . . . . . . . . . . . . . . . . . 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 33399 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((subringAlg ‘𝐸)‘(Base‘𝐹)) ∈ LVec ∧ 𝑏 ∈ (LIndS‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → ¬ (0g‘((subringAlg ‘𝐸)‘(Base‘𝐹))) ∈ 𝑏)
1474, 144, 146syl2an2r 685 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → ¬ (0g‘((subringAlg ‘𝐸)‘(Base‘𝐹))) ∈ 𝑏)
148142, 147eqneltrd 2860 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → ¬ (0g𝐸) ∈ 𝑏)
149148ad3antrrr 730 . . . . . . . . . . . . . . . . . . . . 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 3039 . . . . . . . . . . . . . . . . . . . . 21 ((𝑥𝑏 ∧ ¬ (0g𝐸) ∈ 𝑏) → 𝑥 ≠ (0g𝐸))
15168, 149, 150syl2an2r 685 . . . . . . . . . . . . . . . . . . . 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 20755 . . . . . . . . . . . . . . . . . . . 20 ((𝐸 ∈ DivRing ∧ 𝑥 ∈ (Base‘𝐸) ∧ 𝑥 ≠ (0g𝐸)) → ((invr𝐸)‘𝑥) ≠ (0g𝐸))
154132, 113, 151, 153syl3anc 1372 . . . . . . . . . . . . . . . . . . 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 4785 . . . . . . . . . . . . . . . . . . 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 6905 . . . . . . . . . . . . . . . . . . . 20 (𝑎 = ((invr𝐸)‘𝑥) → ((invr𝐸)‘𝑎) = ((invr𝐸)‘((invr𝐸)‘𝑥)))
158157eleq1d 2825 . . . . . . . . . . . . . . . . . . 19 (𝑎 = ((invr𝐸)‘𝑥) → (((invr𝐸)‘𝑎) ∈ (Base‘𝐹) ↔ ((invr𝐸)‘((invr𝐸)‘𝑥)) ∈ (Base‘𝐹)))
15943, 152, 110issubdrg 20782 . . . . . . . . . . . . . . . . . . . . 21 ((𝐸 ∈ DivRing ∧ (Base‘𝐹) ∈ (SubRing‘𝐸)) → ((𝐸s (Base‘𝐹)) ∈ DivRing ↔ ∀𝑎 ∈ ((Base‘𝐹) ∖ {(0g𝐸)})((invr𝐸)‘𝑎) ∈ (Base‘𝐹)))
160159biimpa 476 . . . . . . . . . . . . . . . . . . . 20 (((𝐸 ∈ DivRing ∧ (Base‘𝐹) ∈ (SubRing‘𝐸)) ∧ (𝐸s (Base‘𝐹)) ∈ DivRing) → ∀𝑎 ∈ ((Base‘𝐹) ∖ {(0g𝐸)})((invr𝐸)‘𝑎) ∈ (Base‘𝐹))
161160adantr 480 . . . . . . . . . . . . . . . . . . 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 484 . . . . . . . . . . . . . . . . . . 19 ((((𝐸 ∈ DivRing ∧ (Base‘𝐹) ∈ (SubRing‘𝐸)) ∧ (𝐸s (Base‘𝐹)) ∈ DivRing) ∧ ((invr𝐸)‘𝑥) ∈ ((Base‘𝐹) ∖ {(0g𝐸)})) → ((invr𝐸)‘𝑥) ∈ ((Base‘𝐹) ∖ {(0g𝐸)}))
163158, 161, 162rspcdva 3622 . . . . . . . . . . . . . . . . . 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 840 . . . . . . . . . . . . . . . . 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 2841 . . . . . . . . . . . . . . . 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 716 . . . . . . . . . . . . . . 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 20263 . . . . . . . . . . . . . . . . . 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 2842 . . . . . . . . . . . . . . . 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 726 . . . . . . . . . . . . . . . . . 18 ((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → ((subringAlg ‘𝐸)‘(Base‘𝐹)) ∈ LVec)
175 lveclmod 21106 . . . . . . . . . . . . . . . . . 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 33406 . . . . . . . . . . . . . . . 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 232 . . . . . . . . . . . . . . 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 3161 . . . . . . . . . . . . . 14 ((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → 𝑥 ∈ (Base‘𝐹))
180179ad2antrr 726 . . . . . . . . . . . . 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 20248 . . . . . . . . . . . . 13 ((𝐹 ∈ Ring ∧ (𝑣𝑥) ∈ (Base‘𝐹) ∧ 𝑥 ∈ (Base‘𝐹)) → ((𝑣𝑥)(.r𝐹)𝑥) ∈ (Base‘𝐹))
183106, 107, 180, 182syl3anc 1372 . . . . . . . . . . . 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 2840 . . . . . . . . . . 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 726 . . . . . . . . . 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 2840 . . . . . . . . 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 466 . . . . . . . 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 2826 . . . . . . . . . 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 33406 . . . . . . . . . 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 279 . . . . . . . . 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 476 . . . . . . . 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 3161 . . . . . . 7 (((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑢 ∈ (Base‘𝐸)) → 𝑢 ∈ (Base‘𝐹))
193192ex 412 . . . . . 6 ((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → (𝑢 ∈ (Base‘𝐸) → 𝑢 ∈ (Base‘𝐹)))
194193ssrdv 3988 . . . . 5 ((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → (Base‘𝐸) ⊆ (Base‘𝐹))
19521, 194exlimddv 1934 . . . 4 (((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → (Base‘𝐸) ⊆ (Base‘𝐹))
1969, 195exlimddv 1934 . . 3 ((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) → (Base‘𝐸) ⊆ (Base‘𝐹))
197 simpr 484 . . . 4 (((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ (Base‘𝐸) ⊆ (Base‘𝐹)) → (Base‘𝐸) ⊆ (Base‘𝐹))
19837ad2antrr 726 . . . 4 (((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ (Base‘𝐸) ⊆ (Base‘𝐹)) → 𝐸 ∈ Field)
199 fvexd 6920 . . . 4 (((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ (Base‘𝐸) ⊆ (Base‘𝐹)) → (Base‘𝐹) ∈ V)
20043, 27ressid2 17279 . . . 4 (((Base‘𝐸) ⊆ (Base‘𝐹) ∧ 𝐸 ∈ Field ∧ (Base‘𝐹) ∈ V) → (𝐸s (Base‘𝐹)) = 𝐸)
201197, 198, 199, 200syl3anc 1372 . . 3 (((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ (Base‘𝐸) ⊆ (Base‘𝐹)) → (𝐸s (Base‘𝐹)) = 𝐸)
202196, 201mpdan 687 . 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 395   = wceq 1539  wex 1778  wcel 2107  wne 2939  wral 3060  wrex 3069  Vcvv 3479  cdif 3947  wss 3950  c0 4332  {csn 4625   class class class wbr 5142  cmpt 5224  wf 6556  cfv 6560  (class class class)co 7432  m cmap 8867   finSupp cfsupp 9402  1c1 11157  chash 14370  Basecbs 17248  s cress 17275  .rcmulr 17299  Scalarcsca 17301   ·𝑠 cvsca 17302  0gc0g 17485   Σg cgsu 17486  Mndcmnd 18748  1rcur 20179  Ringcrg 20231  CRingccrg 20232  Unitcui 20356  invrcinvr 20388  SubRingcsubrg 20570  DivRingcdr 20730  Fieldcfield 20731  LModclmod 20859  LBasisclbs 21074  LVecclvec 21102  subringAlg csra 21171  LIndSclinds 21826  dimcldim 33650  /FldExtcfldext 33690  [:]cextdg 33693
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-reg 9633  ax-inf2 9682  ax-ac2 10504  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-iin 4993  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-se 5637  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-isom 6569  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-of 7698  df-rpss 7744  df-om 7889  df-1st 8015  df-2nd 8016  df-supp 8187  df-tpos 8252  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-2o 8508  df-oadd 8511  df-er 8746  df-map 8869  df-ixp 8939  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-fsupp 9403  df-sup 9483  df-oi 9551  df-r1 9805  df-rank 9806  df-dju 9942  df-card 9980  df-acn 9983  df-ac 10157  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-nn 12268  df-2 12330  df-3 12331  df-4 12332  df-5 12333  df-6 12334  df-7 12335  df-8 12336  df-9 12337  df-n0 12529  df-xnn0 12602  df-z 12616  df-dec 12736  df-uz 12880  df-fz 13549  df-fzo 13696  df-seq 14044  df-hash 14371  df-struct 17185  df-sets 17202  df-slot 17220  df-ndx 17232  df-base 17249  df-ress 17276  df-plusg 17311  df-mulr 17312  df-sca 17314  df-vsca 17315  df-ip 17316  df-tset 17317  df-ple 17318  df-ocomp 17319  df-ds 17320  df-hom 17322  df-cco 17323  df-0g 17487  df-gsum 17488  df-prds 17493  df-pws 17495  df-mre 17630  df-mrc 17631  df-mri 17632  df-acs 17633  df-proset 18341  df-drs 18342  df-poset 18360  df-ipo 18574  df-mgm 18654  df-sgrp 18733  df-mnd 18749  df-mhm 18797  df-submnd 18798  df-grp 18955  df-minusg 18956  df-sbg 18957  df-mulg 19087  df-subg 19142  df-ghm 19232  df-cntz 19336  df-cmn 19801  df-abl 19802  df-mgp 20139  df-rng 20151  df-ur 20180  df-ring 20233  df-cring 20234  df-oppr 20335  df-dvdsr 20358  df-unit 20359  df-invr 20389  df-nzr 20514  df-subrg 20571  df-drng 20732  df-field 20733  df-lmod 20861  df-lss 20931  df-lsp 20971  df-lmhm 21022  df-lbs 21075  df-lvec 21103  df-sra 21173  df-rgmod 21174  df-dsmm 21753  df-frlm 21768  df-uvc 21804  df-lindf 21827  df-linds 21828  df-dim 33651  df-fldext 33694  df-extdg 33695
This theorem is referenced by:  extdg1b  33718  rtelextdg2  33769
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