Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  extdg1id Structured version   Visualization version   GIF version

Theorem extdg1id 33691
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 33680 . . 3 (𝐸/FldExt𝐹𝐹 = (𝐸s (Base‘𝐹)))
21adantr 480 . 2 ((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) → 𝐹 = (𝐸s (Base‘𝐹)))
3 fldextsralvec 33683 . . . . . . 7 (𝐸/FldExt𝐹 → ((subringAlg ‘𝐸)‘(Base‘𝐹)) ∈ LVec)
43adantr 480 . . . . . 6 ((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) → ((subringAlg ‘𝐸)‘(Base‘𝐹)) ∈ LVec)
5 eqid 2735 . . . . . . 7 (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹))) = (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))
65lbsex 21185 . . . . . 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 4359 . . . . 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 33628 . . . . . . . 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 33682 . . . . . . . . . 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 2777 . . . . . . . 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 2777 . . . . . 6 (((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → (♯‘𝑏) = 1)
19 hash1snb 14455 . . . . . . 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 769 . . . . . . . . . . . . . . . 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 2736 . . . . . . . . . . . . . . . . . . . 20 (𝐸/FldExt𝐹 → ((subringAlg ‘𝐸)‘(Base‘𝐹)) = ((subringAlg ‘𝐸)‘(Base‘𝐹)))
25 eqid 2735 . . . . . . . . . . . . . . . . . . . . . 22 (Base‘𝐹) = (Base‘𝐹)
2625fldextsubrg 33679 . . . . . . . . . . . . . . . . . . . . 21 (𝐸/FldExt𝐹 → (Base‘𝐹) ∈ (SubRing‘𝐸))
27 eqid 2735 . . . . . . . . . . . . . . . . . . . . . 22 (Base‘𝐸) = (Base‘𝐸)
2827subrgss 20589 . . . . . . . . . . . . . . . . . . . . 21 ((Base‘𝐹) ∈ (SubRing‘𝐸) → (Base‘𝐹) ⊆ (Base‘𝐸))
2926, 28syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝐸/FldExt𝐹 → (Base‘𝐹) ⊆ (Base‘𝐸))
3024, 29sravsca 21203 . . . . . . . . . . . . . . . . . . 19 (𝐸/FldExt𝐹 → (.r𝐸) = ( ·𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
3130eqcomd 2741 . . . . . . . . . . . . . . . . . 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 7448 . . . . . . . . . . . . . . . 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 5237 . . . . . . . . . . . . . . 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 7447 . . . . . . . . . . . . . 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 2735 . . . . . . . . . . . . . . . . 17 ((subringAlg ‘𝐸)‘(Base‘𝐹)) = ((subringAlg ‘𝐸)‘(Base‘𝐹))
37 fldextfld1 33677 . . . . . . . . . . . . . . . . . . 19 (𝐸/FldExt𝐹𝐸 ∈ Field)
38 isfld 20757 . . . . . . . . . . . . . . . . . . . 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 2735 . . . . . . . . . . . . . . . . 17 (𝐸s (Base‘𝐹)) = (𝐸s (Base‘𝐹))
44 fldextfld2 33678 . . . . . . . . . . . . . . . . . . . 20 (𝐸/FldExt𝐹𝐹 ∈ Field)
45 isfld 20757 . . . . . . . . . . . . . . . . . . . . 21 (𝐹 ∈ Field ↔ (𝐹 ∈ DivRing ∧ 𝐹 ∈ CRing))
4645simplbi 497 . . . . . . . . . . . . . . . . . . . 20 (𝐹 ∈ Field → 𝐹 ∈ DivRing)
4744, 46syl 17 . . . . . . . . . . . . . . . . . . 19 (𝐸/FldExt𝐹𝐹 ∈ DivRing)
481, 47eqeltrrd 2840 . . . . . . . . . . . . . . . . . 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 33624 . . . . . . . . . . . . . . . 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 20753 . . . . . . . . . . . . . . . . . . 19 (𝐸 ∈ DivRing → 𝐸 ∈ Ring)
5537, 39, 543syl 18 . . . . . . . . . . . . . . . . . 18 (𝐸/FldExt𝐹𝐸 ∈ Ring)
56 ringmnd 20261 . . . . . . . . . . . . . . . . . 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 3482 . . . . . . . . . . . . . . . . 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 8888 . . . . . . . . . . . . . . . . . . . . 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 4668 . . . . . . . . . . . . . . . . . . . . 21 𝑥 ∈ {𝑥}
6867, 23eleqtrrid 2846 . . . . . . . . . . . . . . . . . . . 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 7105 . . . . . . . . . . . . . . . . . . 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 21201 . . . . . . . . . . . . . . . . . . . . . 22 (𝐸/FldExt𝐹 → (𝐸s (Base‘𝐹)) = (Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
711, 70eqtrd 2775 . . . . . . . . . . . . . . . . . . . . 21 (𝐸/FldExt𝐹𝐹 = (Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
7271fveq2d 6911 . . . . . . . . . . . . . . . . . . . 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 2842 . . . . . . . . . . . . . . . . . 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 3996 . . . . . . . . . . . . . . . . 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 769 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
78 eqid 2735 . . . . . . . . . . . . . . . . . . . . . . 23 (Base‘((subringAlg ‘𝐸)‘(Base‘𝐹))) = (Base‘((subringAlg ‘𝐸)‘(Base‘𝐹)))
7978, 5lbsss 21094 . . . . . . . . . . . . . . . . . . . . . 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 4035 . . . . . . . . . . . . . . . . . . . 20 ((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → {𝑥} ⊆ (Base‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
8259snss 4790 . . . . . . . . . . . . . . . . . . . 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 2736 . . . . . . . . . . . . . . . . . . . 20 ((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → ((subringAlg ‘𝐸)‘(Base‘𝐹)) = ((subringAlg ‘𝐸)‘(Base‘𝐹)))
8584, 63srabase 21195 . . . . . . . . . . . . . . . . . . 19 ((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → (Base‘𝐸) = (Base‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
8683, 85eleqtrrd 2842 . . . . . . . . . . . . . . . . . 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 2735 . . . . . . . . . . . . . . . . . 18 (.r𝐸) = (.r𝐸)
8927, 88ringcl 20268 . . . . . . . . . . . . . . . . 17 ((𝐸 ∈ Ring ∧ (𝑣𝑥) ∈ (Base‘𝐸) ∧ 𝑥 ∈ (Base‘𝐸)) → ((𝑣𝑥)(.r𝐸)𝑥) ∈ (Base‘𝐸))
9062, 75, 87, 89syl3anc 1370 . . . . . . . . . . . . . . . 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 6911 . . . . . . . . . . . . . . . . 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 7449 . . . . . . . . . . . . . . . 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 19985 . . . . . . . . . . . . . . 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 6911 . . . . . . . . . . . . . . . . . 18 (𝐸/FldExt𝐹 → (.r𝐹) = (.r‘(𝐸s (Base‘𝐹))))
9643, 88ressmulr 17353 . . . . . . . . . . . . . . . . . . 19 ((Base‘𝐹) ∈ (SubRing‘𝐸) → (.r𝐸) = (.r‘(𝐸s (Base‘𝐹))))
9726, 96syl 17 . . . . . . . . . . . . . . . . . 18 (𝐸/FldExt𝐹 → (.r𝐸) = (.r‘(𝐸s (Base‘𝐹))))
9895, 97eqtr4d 2778 . . . . . . . . . . . . . . . . 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 7448 . . . . . . . . . . . . . . 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 2778 . . . . . . . . . . . . . 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 2783 . . . . . . . . . . . . 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 20753 . . . . . . . . . . . . . . 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 2735 . . . . . . . . . . . . . . . . . . . 20 (1r𝐸) = (1r𝐸)
109 eqid 2735 . . . . . . . . . . . . . . . . . . . 20 (Unit‘𝐸) = (Unit‘𝐸)
110 eqid 2735 . . . . . . . . . . . . . . . . . . . 20 (invr𝐸) = (invr𝐸)
111 simp-5l 785 . . . . . . . . . . . . . . . . . . . . 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 20269 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐸 ∈ CRing ∧ 𝑥 ∈ (Base‘𝐸) ∧ (𝑣𝑥) ∈ (Base‘𝐸)) → (𝑥(.r𝐸)(𝑣𝑥)) = ((𝑣𝑥)(.r𝐸)𝑥))
118116, 113, 114, 117syl3anc 1370 . . . . . . . . . . . . . . . . . . . . 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 7447 . . . . . . . . . . . . . . . . . . . . . . 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 2781 . . . . . . . . . . . . . . . . . . . . . 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 2775 . . . . . . . . . . . . . . . . . . . . 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 2778 . . . . . . . . . . . . . . . . . . . 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 2741 . . . . . . . . . . . . . . . . . . . 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 22698 . . . . . . . . . . . . . . . . . . 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 20408 . . . . . . . . . . . . . . . . . 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 2840 . . . . . . . . . . . . . . . . . 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 2839 . . . . . . . . . . . . . . . . . . 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 2736 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐸/FldExt𝐹 → (0g𝐸) = (0g𝐸))
14124, 140, 29sralmod0 21213 . . . . . . . . . . . . . . . . . . . . . . . 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 21871 . . . . . . . . . . . . . . . . . . . . . . . . 25 (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹))) ⊆ (LIndS‘((subringAlg ‘𝐸)‘(Base‘𝐹)))
144143, 10sselid 3993 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → 𝑏 ∈ (LIndS‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
145 eqid 2735 . . . . . . . . . . . . . . . . . . . . . . . . 25 (0g‘((subringAlg ‘𝐸)‘(Base‘𝐹))) = (0g‘((subringAlg ‘𝐸)‘(Base‘𝐹)))
1461450nellinds 33378 . . . . . . . . . . . . . . . . . . . . . . . 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 2859 . . . . . . . . . . . . . . . . . . . . . 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 3038 . . . . . . . . . . . . . . . . . . . . 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 2735 . . . . . . . . . . . . . . . . . . . . 21 (0g𝐸) = (0g𝐸)
15327, 152, 110drnginvrn0 20771 . . . . . . . . . . . . . . . . . . . 20 ((𝐸 ∈ DivRing ∧ 𝑥 ∈ (Base‘𝐸) ∧ 𝑥 ≠ (0g𝐸)) → ((invr𝐸)‘𝑥) ≠ (0g𝐸))
154132, 113, 151, 153syl3anc 1370 . . . . . . . . . . . . . . . . . . 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 4791 . . . . . . . . . . . . . . . . . . 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 6907 . . . . . . . . . . . . . . . . . . . 20 (𝑎 = ((invr𝐸)‘𝑥) → ((invr𝐸)‘𝑎) = ((invr𝐸)‘((invr𝐸)‘𝑥)))
158157eleq1d 2824 . . . . . . . . . . . . . . . . . . 19 (𝑎 = ((invr𝐸)‘𝑥) → (((invr𝐸)‘𝑎) ∈ (Base‘𝐹) ↔ ((invr𝐸)‘((invr𝐸)‘𝑥)) ∈ (Base‘𝐹)))
15943, 152, 110issubdrg 20798 . . . . . . . . . . . . . . . . . . . . 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 3623 . . . . . . . . . . . . . . . . . 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 2840 . . . . . . . . . . . . . . . 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 20280 . . . . . . . . . . . . . . . . . 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 2841 . . . . . . . . . . . . . . . 16 ((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → (1r𝐸) ∈ (Base‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
170 eqid 2735 . . . . . . . . . . . . . . . . 17 (Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) = (Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
171 eqid 2735 . . . . . . . . . . . . . . . . 17 (Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹))) = (Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))
172 eqid 2735 . . . . . . . . . . . . . . . . 17 (0g‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) = (0g‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
173 eqid 2735 . . . . . . . . . . . . . . . . 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 21123 . . . . . . . . . . . . . . . . . 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 33385 . . . . . . . . . . . . . . . 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 3160 . . . . . . . . . . . . . 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 2735 . . . . . . . . . . . . . 14 (.r𝐹) = (.r𝐹)
18225, 181ringcl 20268 . . . . . . . . . . . . 13 ((𝐹 ∈ Ring ∧ (𝑣𝑥) ∈ (Base‘𝐹) ∧ 𝑥 ∈ (Base‘𝐹)) → ((𝑣𝑥)(.r𝐹)𝑥) ∈ (Base‘𝐹))
183106, 107, 180, 182syl3anc 1370 . . . . . . . . . . . 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 2839 . . . . . . . . . . 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 2839 . . . . . . . . 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 2825 . . . . . . . . . 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 33385 . . . . . . . . . 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 3160 . . . . . . 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 4001 . . . . 5 ((((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → (Base‘𝐸) ⊆ (Base‘𝐹))
19521, 194exlimddv 1933 . . . 4 (((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → (Base‘𝐸) ⊆ (Base‘𝐹))
1969, 195exlimddv 1933 . . 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 6922 . . . 4 (((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ (Base‘𝐸) ⊆ (Base‘𝐹)) → (Base‘𝐹) ∈ V)
20043, 27ressid2 17278 . . . 4 (((Base‘𝐸) ⊆ (Base‘𝐹) ∧ 𝐸 ∈ Field ∧ (Base‘𝐹) ∈ V) → (𝐸s (Base‘𝐹)) = 𝐸)
201197, 198, 199, 200syl3anc 1370 . . 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 2776 1 ((𝐸/FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) → 𝐸 = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1537  wex 1776  wcel 2106  wne 2938  wral 3059  wrex 3068  Vcvv 3478  cdif 3960  wss 3963  c0 4339  {csn 4631   class class class wbr 5148  cmpt 5231  wf 6559  cfv 6563  (class class class)co 7431  m cmap 8865   finSupp cfsupp 9399  1c1 11154  chash 14366  Basecbs 17245  s cress 17274  .rcmulr 17299  Scalarcsca 17301   ·𝑠 cvsca 17302  0gc0g 17486   Σg cgsu 17487  Mndcmnd 18760  1rcur 20199  Ringcrg 20251  CRingccrg 20252  Unitcui 20372  invrcinvr 20404  SubRingcsubrg 20586  DivRingcdr 20746  Fieldcfield 20747  LModclmod 20875  LBasisclbs 21091  LVecclvec 21119  subringAlg csra 21188  LIndSclinds 21843  dimcldim 33626  /FldExtcfldext 33666  [:]cextdg 33669
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-reg 9630  ax-inf2 9679  ax-ac2 10501  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-rpss 7742  df-om 7888  df-1st 8013  df-2nd 8014  df-supp 8185  df-tpos 8250  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-oadd 8509  df-er 8744  df-map 8867  df-ixp 8937  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-fsupp 9400  df-sup 9480  df-oi 9548  df-r1 9802  df-rank 9803  df-dju 9939  df-card 9977  df-acn 9980  df-ac 10154  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-xnn0 12598  df-z 12612  df-dec 12732  df-uz 12877  df-fz 13545  df-fzo 13692  df-seq 14040  df-hash 14367  df-struct 17181  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  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 17488  df-gsum 17489  df-prds 17494  df-pws 17496  df-mre 17631  df-mrc 17632  df-mri 17633  df-acs 17634  df-proset 18352  df-drs 18353  df-poset 18371  df-ipo 18586  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-mhm 18809  df-submnd 18810  df-grp 18967  df-minusg 18968  df-sbg 18969  df-mulg 19099  df-subg 19154  df-ghm 19244  df-cntz 19348  df-cmn 19815  df-abl 19816  df-mgp 20153  df-rng 20171  df-ur 20200  df-ring 20253  df-cring 20254  df-oppr 20351  df-dvdsr 20374  df-unit 20375  df-invr 20405  df-nzr 20530  df-subrg 20587  df-drng 20748  df-field 20749  df-lmod 20877  df-lss 20948  df-lsp 20988  df-lmhm 21039  df-lbs 21092  df-lvec 21120  df-sra 21190  df-rgmod 21191  df-dsmm 21770  df-frlm 21785  df-uvc 21821  df-lindf 21844  df-linds 21845  df-dim 33627  df-fldext 33670  df-extdg 33671
This theorem is referenced by:  extdg1b  33692  rtelextdg2  33733
  Copyright terms: Public domain W3C validator