extdg1id - Mathbox for Thierry Arnoux

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Theorem extdg1id 33030

Description: If the degree of the extension 𝐸/_FldExt𝐹 is 1, then 𝐸 and 𝐹 are identical. (Contributed by Thierry Arnoux, 6-Aug-2023.)

**Assertion**
Ref	Expression
extdg1id	⊢ ((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) → 𝐸 = 𝐹)

Proof of Theorem extdg1id Dummy variables 𝑎 𝑥 𝑏 𝑖 𝑣 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fldextress 33019	. . 3 ⊢ (𝐸/_FldExt𝐹 → 𝐹 = (𝐸 ↾_s (Base‘𝐹)))
2	1	adantr 479	. 2 ⊢ ((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) → 𝐹 = (𝐸 ↾_s (Base‘𝐹)))
3		fldextsralvec 33022	. . . . . . 7 ⊢ (𝐸/_FldExt𝐹 → ((subringAlg ‘𝐸)‘(Base‘𝐹)) ∈ LVec)
4	3	adantr 479	. . . . . 6 ⊢ ((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) → ((subringAlg ‘𝐸)‘(Base‘𝐹)) ∈ LVec)
5		eqid 2730	. . . . . . 7 ⊢ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹))) = (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))
6	5	lbsex 20923	. . . . . 6 ⊢ (((subringAlg ‘𝐸)‘(Base‘𝐹)) ∈ LVec → (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹))) ≠ ∅)
7	4, 6	syl 17	. . . . 5 ⊢ ((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) → (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹))) ≠ ∅)
8		n0 4345	. . . . 5 ⊢ ((LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹))) ≠ ∅ ↔ ∃𝑏 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
9	7, 8	sylib 217	. . . 4 ⊢ ((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) → ∃𝑏 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
10		simpr 483	. . . . . 6 ⊢ (((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
11	5	dimval 32973	. . . . . . . 8 ⊢ ((((subringAlg ‘𝐸)‘(Base‘𝐹)) ∈ LVec ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → (dim‘((subringAlg ‘𝐸)‘(Base‘𝐹))) = (♯‘𝑏))
12	4, 11	sylan 578	. . . . . . 7 ⊢ (((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → (dim‘((subringAlg ‘𝐸)‘(Base‘𝐹))) = (♯‘𝑏))
13		extdgval 33021	. . . . . . . . . 10 ⊢ (𝐸/_FldExt𝐹 → (𝐸[:]𝐹) = (dim‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
14	13	adantr 479	. . . . . . . . 9 ⊢ ((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) → (𝐸[:]𝐹) = (dim‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
15		simpr 483	. . . . . . . . 9 ⊢ ((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) → (𝐸[:]𝐹) = 1)
16	14, 15	eqtr3d 2772	. . . . . . . 8 ⊢ ((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) → (dim‘((subringAlg ‘𝐸)‘(Base‘𝐹))) = 1)
17	16	adantr 479	. . . . . . 7 ⊢ (((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → (dim‘((subringAlg ‘𝐸)‘(Base‘𝐹))) = 1)
18	12, 17	eqtr3d 2772	. . . . . 6 ⊢ (((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → (♯‘𝑏) = 1)
19		hash1snb 14383	. . . . . . 7 ⊢ (𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹))) → ((♯‘𝑏) = 1 ↔ ∃𝑥 𝑏 = {𝑥}))
20	19	biimpa 475	. . . . . 6 ⊢ ((𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹))) ∧ (♯‘𝑏) = 1) → ∃𝑥 𝑏 = {𝑥})
21	10, 18, 20	syl2anc 582	. . . . 5 ⊢ (((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → ∃𝑥 𝑏 = {𝑥})
22		simpr 483	. . . . . . . . . 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 (0_g‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹))))) ∧ 𝑢 = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σ_g (𝑖 ∈ 𝑏 ↦ ((𝑣‘𝑖)( ·_𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖)))) → 𝑢 = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σ_g (𝑖 ∈ 𝑏 ↦ ((𝑣‘𝑖)( ·_𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖))))
23		simplr 765	. . . . . . . . . . . . . . . 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 2731	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐸/_FldExt𝐹 → ((subringAlg ‘𝐸)‘(Base‘𝐹)) = ((subringAlg ‘𝐸)‘(Base‘𝐹)))
25		eqid 2730	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (Base‘𝐹) = (Base‘𝐹)
26	25	fldextsubrg 33018	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐸/_FldExt𝐹 → (Base‘𝐹) ∈ (SubRing‘𝐸))
27		eqid 2730	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (Base‘𝐸) = (Base‘𝐸)
28	27	subrgss 20462	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((Base‘𝐹) ∈ (SubRing‘𝐸) → (Base‘𝐹) ⊆ (Base‘𝐸))
29	26, 28	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐸/_FldExt𝐹 → (Base‘𝐹) ⊆ (Base‘𝐸))
30	24, 29	sravsca 20945	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐸/_FldExt𝐹 → (._r‘𝐸) = ( ·_𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
31	30	eqcomd 2736	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐸/_FldExt𝐹 → ( ·_𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹))) = (._r‘𝐸))
32	31	ad5antr 730	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑_m 𝑏)) ∧ 𝑖 ∈ 𝑏) → ( ·_𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹))) = (._r‘𝐸))
33	32	oveqd 7428	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑_m 𝑏)) ∧ 𝑖 ∈ 𝑏) → ((𝑣‘𝑖)( ·_𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖) = ((𝑣‘𝑖)(._r‘𝐸)𝑖))
34	23, 33	mpteq12dva 5236	. . . . . . . . . . . . . . 15 ⊢ (((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑_m 𝑏)) → (𝑖 ∈ 𝑏 ↦ ((𝑣‘𝑖)( ·_𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖)) = (𝑖 ∈ {𝑥} ↦ ((𝑣‘𝑖)(._r‘𝐸)𝑖)))
35	34	oveq2d 7427	. . . . . . . . . . . . . 14 ⊢ (((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑_m 𝑏)) → (𝐸 Σ_g (𝑖 ∈ 𝑏 ↦ ((𝑣‘𝑖)( ·_𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖))) = (𝐸 Σ_g (𝑖 ∈ {𝑥} ↦ ((𝑣‘𝑖)(._r‘𝐸)𝑖))))
36		eqid 2730	. . . . . . . . . . . . . . . . 17 ⊢ ((subringAlg ‘𝐸)‘(Base‘𝐹)) = ((subringAlg ‘𝐸)‘(Base‘𝐹))
37		fldextfld1 33016	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐸/_FldExt𝐹 → 𝐸 ∈ Field)
38		isfld 20511	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐸 ∈ Field ↔ (𝐸 ∈ DivRing ∧ 𝐸 ∈ CRing))
39	38	simplbi 496	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐸 ∈ Field → 𝐸 ∈ DivRing)
40	37, 39	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐸/_FldExt𝐹 → 𝐸 ∈ DivRing)
41	40	adantr 479	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐸/_FldExt𝐹 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → 𝐸 ∈ DivRing)
42	26	adantr 479	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐸/_FldExt𝐹 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → (Base‘𝐹) ∈ (SubRing‘𝐸))
43		eqid 2730	. . . . . . . . . . . . . . . . 17 ⊢ (𝐸 ↾_s (Base‘𝐹)) = (𝐸 ↾_s (Base‘𝐹))
44		fldextfld2 33017	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐸/_FldExt𝐹 → 𝐹 ∈ Field)
45		isfld 20511	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐹 ∈ Field ↔ (𝐹 ∈ DivRing ∧ 𝐹 ∈ CRing))
46	45	simplbi 496	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐹 ∈ Field → 𝐹 ∈ DivRing)
47	44, 46	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐸/_FldExt𝐹 → 𝐹 ∈ DivRing)
48	1, 47	eqeltrrd 2832	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐸/_FldExt𝐹 → (𝐸 ↾_s (Base‘𝐹)) ∈ DivRing)
49	48	adantr 479	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐸/_FldExt𝐹 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → (𝐸 ↾_s (Base‘𝐹)) ∈ DivRing)
50		simpr 483	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐸/_FldExt𝐹 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
51	36, 41, 42, 43, 49, 50	drgextgsum 32969	. . . . . . . . . . . . . . . 16 ⊢ ((𝐸/_FldExt𝐹 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → (𝐸 Σ_g (𝑖 ∈ 𝑏 ↦ ((𝑣‘𝑖)( ·_𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖))) = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σ_g (𝑖 ∈ 𝑏 ↦ ((𝑣‘𝑖)( ·_𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖))))
52	51	adantlr 711	. . . . . . . . . . . . . . 15 ⊢ (((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → (𝐸 Σ_g (𝑖 ∈ 𝑏 ↦ ((𝑣‘𝑖)( ·_𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖))) = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σ_g (𝑖 ∈ 𝑏 ↦ ((𝑣‘𝑖)( ·_𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖))))
53	52	ad2antrr 722	. . . . . . . . . . . . . 14 ⊢ (((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑_m 𝑏)) → (𝐸 Σ_g (𝑖 ∈ 𝑏 ↦ ((𝑣‘𝑖)( ·_𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖))) = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σ_g (𝑖 ∈ 𝑏 ↦ ((𝑣‘𝑖)( ·_𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖))))
54		drngring 20507	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐸 ∈ DivRing → 𝐸 ∈ Ring)
55	37, 39, 54	3syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐸/_FldExt𝐹 → 𝐸 ∈ Ring)
56		ringmnd 20137	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐸 ∈ Ring → 𝐸 ∈ Mnd)
57	55, 56	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝐸/_FldExt𝐹 → 𝐸 ∈ Mnd)
58	57	ad4antr 728	. . . . . . . . . . . . . . . 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 3476	. . . . . . . . . . . . . . . . 17 ⊢ 𝑥 ∈ V
60	59	a1i 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)
61	55	ad3antrrr 726	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → 𝐸 ∈ Ring)
62	61	adantr 479	. . . . . . . . . . . . . . . . 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)
63	29	ad3antrrr 726	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → (Base‘𝐹) ⊆ (Base‘𝐸))
64	63	adantr 479	. . . . . . . . . . . . . . . . . 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 8845	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑_m 𝑏) → 𝑣:𝑏⟶(Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))))
66	65	adantl 480	. . . . . . . . . . . . . . . . . . . 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 4664	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑥 ∈ {𝑥}
68	67, 23	eleqtrrid 2838	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑_m 𝑏)) → 𝑥 ∈ 𝑏)
69	66, 68	ffvelcdmd 7086	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑_m 𝑏)) → (𝑣‘𝑥) ∈ (Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))))
70	24, 29	srasca 20943	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐸/_FldExt𝐹 → (𝐸 ↾_s (Base‘𝐹)) = (Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
71	1, 70	eqtrd 2770	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐸/_FldExt𝐹 → 𝐹 = (Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
72	71	fveq2d 6894	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐸/_FldExt𝐹 → (Base‘𝐹) = (Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))))
73	72	ad4antr 728	. . . . . . . . . . . . . . . . . . 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)))))
74	69, 73	eleqtrrd 2834	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑_m 𝑏)) → (𝑣‘𝑥) ∈ (Base‘𝐹))
75	64, 74	sseldd 3982	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑_m 𝑏)) → (𝑣‘𝑥) ∈ (Base‘𝐸))
76		simpr 483	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → 𝑏 = {𝑥})
77		simplr 765	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
78		eqid 2730	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (Base‘((subringAlg ‘𝐸)‘(Base‘𝐹))) = (Base‘((subringAlg ‘𝐸)‘(Base‘𝐹)))
79	78, 5	lbsss 20832	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹))) → 𝑏 ⊆ (Base‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
80	77, 79	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → 𝑏 ⊆ (Base‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
81	76, 80	eqsstrrd 4020	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → {𝑥} ⊆ (Base‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
82	59	snss 4788	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ (Base‘((subringAlg ‘𝐸)‘(Base‘𝐹))) ↔ {𝑥} ⊆ (Base‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
83	81, 82	sylibr 233	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → 𝑥 ∈ (Base‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
84		eqidd 2731	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → ((subringAlg ‘𝐸)‘(Base‘𝐹)) = ((subringAlg ‘𝐸)‘(Base‘𝐹)))
85	84, 63	srabase 20937	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → (Base‘𝐸) = (Base‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
86	83, 85	eleqtrrd 2834	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → 𝑥 ∈ (Base‘𝐸))
87	86	adantr 479	. . . . . . . . . . . . . . . . 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 2730	. . . . . . . . . . . . . . . . . 18 ⊢ (._r‘𝐸) = (._r‘𝐸)
89	27, 88	ringcl 20144	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐸 ∈ Ring ∧ (𝑣‘𝑥) ∈ (Base‘𝐸) ∧ 𝑥 ∈ (Base‘𝐸)) → ((𝑣‘𝑥)(._r‘𝐸)𝑥) ∈ (Base‘𝐸))
90	62, 75, 87, 89	syl3anc 1369	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑_m 𝑏)) → ((𝑣‘𝑥)(._r‘𝐸)𝑥) ∈ (Base‘𝐸))
91		simpr 483	. . . . . . . . . . . . . . . . . 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)
92	91	fveq2d 6894	. . . . . . . . . . . . . . . . 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) → (𝑣‘𝑖) = (𝑣‘𝑥))
93	92, 91	oveq12d 7429	. . . . . . . . . . . . . . . 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) → ((𝑣‘𝑖)(._r‘𝐸)𝑖) = ((𝑣‘𝑥)(._r‘𝐸)𝑥))
94	27, 58, 60, 90, 93	gsumsnd 19861	. . . . . . . . . . . . . . 15 ⊢ (((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑_m 𝑏)) → (𝐸 Σ_g (𝑖 ∈ {𝑥} ↦ ((𝑣‘𝑖)(._r‘𝐸)𝑖))) = ((𝑣‘𝑥)(._r‘𝐸)𝑥))
95	1	fveq2d 6894	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐸/_FldExt𝐹 → (._r‘𝐹) = (._r‘(𝐸 ↾_s (Base‘𝐹))))
96	43, 88	ressmulr 17256	. . . . . . . . . . . . . . . . . . 19 ⊢ ((Base‘𝐹) ∈ (SubRing‘𝐸) → (._r‘𝐸) = (._r‘(𝐸 ↾_s (Base‘𝐹))))
97	26, 96	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐸/_FldExt𝐹 → (._r‘𝐸) = (._r‘(𝐸 ↾_s (Base‘𝐹))))
98	95, 97	eqtr4d 2773	. . . . . . . . . . . . . . . . 17 ⊢ (𝐸/_FldExt𝐹 → (._r‘𝐹) = (._r‘𝐸))
99	98	ad4antr 728	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑_m 𝑏)) → (._r‘𝐹) = (._r‘𝐸))
100	99	oveqd 7428	. . . . . . . . . . . . . . 15 ⊢ (((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑_m 𝑏)) → ((𝑣‘𝑥)(._r‘𝐹)𝑥) = ((𝑣‘𝑥)(._r‘𝐸)𝑥))
101	94, 100	eqtr4d 2773	. . . . . . . . . . . . . 14 ⊢ (((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑_m 𝑏)) → (𝐸 Σ_g (𝑖 ∈ {𝑥} ↦ ((𝑣‘𝑖)(._r‘𝐸)𝑖))) = ((𝑣‘𝑥)(._r‘𝐹)𝑥))
102	35, 53, 101	3eqtr3d 2778	. . . . . . . . . . . . 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)) Σ_g (𝑖 ∈ 𝑏 ↦ ((𝑣‘𝑖)( ·_𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖))) = ((𝑣‘𝑥)(._r‘𝐹)𝑥))
103	102	adantlr 711	. . . . . . . . . . . 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)) Σ_g (𝑖 ∈ 𝑏 ↦ ((𝑣‘𝑖)( ·_𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖))) = ((𝑣‘𝑥)(._r‘𝐹)𝑥))
104		drngring 20507	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∈ DivRing → 𝐹 ∈ Ring)
105	44, 46, 104	3syl 18	. . . . . . . . . . . . . 14 ⊢ (𝐸/_FldExt𝐹 → 𝐹 ∈ Ring)
106	105	ad5antr 730	. . . . . . . . . . . . 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)
107	74	adantlr 711	. . . . . . . . . . . . 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)) → (𝑣‘𝑥) ∈ (Base‘𝐹))
108		eqid 2730	. . . . . . . . . . . . . . . . . . . 20 ⊢ (1_r‘𝐸) = (1_r‘𝐸)
109		eqid 2730	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Unit‘𝐸) = (Unit‘𝐸)
110		eqid 2730	. . . . . . . . . . . . . . . . . . . 20 ⊢ (inv_r‘𝐸) = (inv_r‘𝐸)
111		simp-5l 781	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑_m 𝑏)) ∧ (1_r‘𝐸) = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σ_g (𝑖 ∈ 𝑏 ↦ ((𝑣‘𝑖)( ·_𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖)))) → 𝐸/_FldExt𝐹)
112	111, 55	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑_m 𝑏)) ∧ (1_r‘𝐸) = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σ_g (𝑖 ∈ 𝑏 ↦ ((𝑣‘𝑖)( ·_𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖)))) → 𝐸 ∈ Ring)
113	87	adantr 479	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑_m 𝑏)) ∧ (1_r‘𝐸) = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σ_g (𝑖 ∈ 𝑏 ↦ ((𝑣‘𝑖)( ·_𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖)))) → 𝑥 ∈ (Base‘𝐸))
114	75	adantr 479	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑_m 𝑏)) ∧ (1_r‘𝐸) = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σ_g (𝑖 ∈ 𝑏 ↦ ((𝑣‘𝑖)( ·_𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖)))) → (𝑣‘𝑥) ∈ (Base‘𝐸))
115	38	simprbi 495	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐸 ∈ Field → 𝐸 ∈ CRing)
116	111, 37, 115	3syl 18	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑_m 𝑏)) ∧ (1_r‘𝐸) = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σ_g (𝑖 ∈ 𝑏 ↦ ((𝑣‘𝑖)( ·_𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖)))) → 𝐸 ∈ CRing)
117	27, 88	crngcom 20145	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐸 ∈ CRing ∧ 𝑥 ∈ (Base‘𝐸) ∧ (𝑣‘𝑥) ∈ (Base‘𝐸)) → (𝑥(._r‘𝐸)(𝑣‘𝑥)) = ((𝑣‘𝑥)(._r‘𝐸)𝑥))
118	116, 113, 114, 117	syl3anc 1369	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑_m 𝑏)) ∧ (1_r‘𝐸) = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σ_g (𝑖 ∈ 𝑏 ↦ ((𝑣‘𝑖)( ·_𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖)))) → (𝑥(._r‘𝐸)(𝑣‘𝑥)) = ((𝑣‘𝑥)(._r‘𝐸)𝑥))
119		simpr 483	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑_m 𝑏)) ∧ (1_r‘𝐸) = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σ_g (𝑖 ∈ 𝑏 ↦ ((𝑣‘𝑖)( ·_𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖)))) → (1_r‘𝐸) = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σ_g (𝑖 ∈ 𝑏 ↦ ((𝑣‘𝑖)( ·_𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖))))
120	52	ad3antrrr 726	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑_m 𝑏)) ∧ (1_r‘𝐸) = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σ_g (𝑖 ∈ 𝑏 ↦ ((𝑣‘𝑖)( ·_𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖)))) → (𝐸 Σ_g (𝑖 ∈ 𝑏 ↦ ((𝑣‘𝑖)( ·_𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖))) = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σ_g (𝑖 ∈ 𝑏 ↦ ((𝑣‘𝑖)( ·_𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖))))
121	34	adantr 479	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑_m 𝑏)) ∧ (1_r‘𝐸) = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σ_g (𝑖 ∈ 𝑏 ↦ ((𝑣‘𝑖)( ·_𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖)))) → (𝑖 ∈ 𝑏 ↦ ((𝑣‘𝑖)( ·_𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖)) = (𝑖 ∈ {𝑥} ↦ ((𝑣‘𝑖)(._r‘𝐸)𝑖)))
122	121	oveq2d 7427	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑_m 𝑏)) ∧ (1_r‘𝐸) = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σ_g (𝑖 ∈ 𝑏 ↦ ((𝑣‘𝑖)( ·_𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖)))) → (𝐸 Σ_g (𝑖 ∈ 𝑏 ↦ ((𝑣‘𝑖)( ·_𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖))) = (𝐸 Σ_g (𝑖 ∈ {𝑥} ↦ ((𝑣‘𝑖)(._r‘𝐸)𝑖))))
123	119, 120, 122	3eqtr2d 2776	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑_m 𝑏)) ∧ (1_r‘𝐸) = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σ_g (𝑖 ∈ 𝑏 ↦ ((𝑣‘𝑖)( ·_𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖)))) → (1_r‘𝐸) = (𝐸 Σ_g (𝑖 ∈ {𝑥} ↦ ((𝑣‘𝑖)(._r‘𝐸)𝑖))))
124	94	adantr 479	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑_m 𝑏)) ∧ (1_r‘𝐸) = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σ_g (𝑖 ∈ 𝑏 ↦ ((𝑣‘𝑖)( ·_𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖)))) → (𝐸 Σ_g (𝑖 ∈ {𝑥} ↦ ((𝑣‘𝑖)(._r‘𝐸)𝑖))) = ((𝑣‘𝑥)(._r‘𝐸)𝑥))
125	123, 124	eqtrd 2770	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑_m 𝑏)) ∧ (1_r‘𝐸) = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σ_g (𝑖 ∈ 𝑏 ↦ ((𝑣‘𝑖)( ·_𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖)))) → (1_r‘𝐸) = ((𝑣‘𝑥)(._r‘𝐸)𝑥))
126	118, 125	eqtr4d 2773	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑_m 𝑏)) ∧ (1_r‘𝐸) = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σ_g (𝑖 ∈ 𝑏 ↦ ((𝑣‘𝑖)( ·_𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖)))) → (𝑥(._r‘𝐸)(𝑣‘𝑥)) = (1_r‘𝐸))
127	125	eqcomd 2736	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑_m 𝑏)) ∧ (1_r‘𝐸) = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σ_g (𝑖 ∈ 𝑏 ↦ ((𝑣‘𝑖)( ·_𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖)))) → ((𝑣‘𝑥)(._r‘𝐸)𝑥) = (1_r‘𝐸))
128	27, 88, 108, 109, 110, 112, 113, 114, 126, 127	invrvald 22398	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑_m 𝑏)) ∧ (1_r‘𝐸) = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σ_g (𝑖 ∈ 𝑏 ↦ ((𝑣‘𝑖)( ·_𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖)))) → (𝑥 ∈ (Unit‘𝐸) ∧ ((inv_r‘𝐸)‘𝑥) = (𝑣‘𝑥)))
129	128	simpld 493	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑_m 𝑏)) ∧ (1_r‘𝐸) = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σ_g (𝑖 ∈ 𝑏 ↦ ((𝑣‘𝑖)( ·_𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖)))) → 𝑥 ∈ (Unit‘𝐸))
130	109, 110	unitinvinv 20282	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐸 ∈ Ring ∧ 𝑥 ∈ (Unit‘𝐸)) → ((inv_r‘𝐸)‘((inv_r‘𝐸)‘𝑥)) = 𝑥)
131	62, 129, 130	syl2an2r 681	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑_m 𝑏)) ∧ (1_r‘𝐸) = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σ_g (𝑖 ∈ 𝑏 ↦ ((𝑣‘𝑖)( ·_𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖)))) → ((inv_r‘𝐸)‘((inv_r‘𝐸)‘𝑥)) = 𝑥)
132	111, 37, 39	3syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑_m 𝑏)) ∧ (1_r‘𝐸) = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σ_g (𝑖 ∈ 𝑏 ↦ ((𝑣‘𝑖)( ·_𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖)))) → 𝐸 ∈ DivRing)
133	111, 26	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑_m 𝑏)) ∧ (1_r‘𝐸) = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σ_g (𝑖 ∈ 𝑏 ↦ ((𝑣‘𝑖)( ·_𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖)))) → (Base‘𝐹) ∈ (SubRing‘𝐸))
134	111, 1	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑_m 𝑏)) ∧ (1_r‘𝐸) = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σ_g (𝑖 ∈ 𝑏 ↦ ((𝑣‘𝑖)( ·_𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖)))) → 𝐹 = (𝐸 ↾_s (Base‘𝐹)))
135	111, 44, 46	3syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑_m 𝑏)) ∧ (1_r‘𝐸) = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σ_g (𝑖 ∈ 𝑏 ↦ ((𝑣‘𝑖)( ·_𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖)))) → 𝐹 ∈ DivRing)
136	134, 135	eqeltrrd 2832	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑_m 𝑏)) ∧ (1_r‘𝐸) = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σ_g (𝑖 ∈ 𝑏 ↦ ((𝑣‘𝑖)( ·_𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖)))) → (𝐸 ↾_s (Base‘𝐹)) ∈ DivRing)
137	128	simprd 494	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑_m 𝑏)) ∧ (1_r‘𝐸) = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σ_g (𝑖 ∈ 𝑏 ↦ ((𝑣‘𝑖)( ·_𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖)))) → ((inv_r‘𝐸)‘𝑥) = (𝑣‘𝑥))
138	74	adantr 479	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑_m 𝑏)) ∧ (1_r‘𝐸) = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σ_g (𝑖 ∈ 𝑏 ↦ ((𝑣‘𝑖)( ·_𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖)))) → (𝑣‘𝑥) ∈ (Base‘𝐹))
139	137, 138	eqeltrd 2831	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑_m 𝑏)) ∧ (1_r‘𝐸) = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σ_g (𝑖 ∈ 𝑏 ↦ ((𝑣‘𝑖)( ·_𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖)))) → ((inv_r‘𝐸)‘𝑥) ∈ (Base‘𝐹))
140		eqidd 2731	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐸/_FldExt𝐹 → (0_g‘𝐸) = (0_g‘𝐸))
141	24, 140, 29	sralmod0 20955	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐸/_FldExt𝐹 → (0_g‘𝐸) = (0_g‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
142	141	ad2antrr 722	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → (0_g‘𝐸) = (0_g‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
143	5	lbslinds 21607	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹))) ⊆ (LIndS‘((subringAlg ‘𝐸)‘(Base‘𝐹)))
144	143, 10	sselid 3979	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → 𝑏 ∈ (LIndS‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
145		eqid 2730	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (0_g‘((subringAlg ‘𝐸)‘(Base‘𝐹))) = (0_g‘((subringAlg ‘𝐸)‘(Base‘𝐹)))
146	145	0nellinds 32757	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((subringAlg ‘𝐸)‘(Base‘𝐹)) ∈ LVec ∧ 𝑏 ∈ (LIndS‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → ¬ (0_g‘((subringAlg ‘𝐸)‘(Base‘𝐹))) ∈ 𝑏)
147	4, 144, 146	syl2an2r 681	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → ¬ (0_g‘((subringAlg ‘𝐸)‘(Base‘𝐹))) ∈ 𝑏)
148	142, 147	eqneltrd 2851	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → ¬ (0_g‘𝐸) ∈ 𝑏)
149	148	ad3antrrr 726	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑_m 𝑏)) ∧ (1_r‘𝐸) = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σ_g (𝑖 ∈ 𝑏 ↦ ((𝑣‘𝑖)( ·_𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖)))) → ¬ (0_g‘𝐸) ∈ 𝑏)
150		nelne2 3038	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ 𝑏 ∧ ¬ (0_g‘𝐸) ∈ 𝑏) → 𝑥 ≠ (0_g‘𝐸))
151	68, 149, 150	syl2an2r 681	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑_m 𝑏)) ∧ (1_r‘𝐸) = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σ_g (𝑖 ∈ 𝑏 ↦ ((𝑣‘𝑖)( ·_𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖)))) → 𝑥 ≠ (0_g‘𝐸))
152		eqid 2730	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (0_g‘𝐸) = (0_g‘𝐸)
153	27, 152, 110	drnginvrn0 20523	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐸 ∈ DivRing ∧ 𝑥 ∈ (Base‘𝐸) ∧ 𝑥 ≠ (0_g‘𝐸)) → ((inv_r‘𝐸)‘𝑥) ≠ (0_g‘𝐸))
154	132, 113, 151, 153	syl3anc 1369	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑_m 𝑏)) ∧ (1_r‘𝐸) = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σ_g (𝑖 ∈ 𝑏 ↦ ((𝑣‘𝑖)( ·_𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖)))) → ((inv_r‘𝐸)‘𝑥) ≠ (0_g‘𝐸))
155		eldifsn 4789	. . . . . . . . . . . . . . . . . . 19 ⊢ (((inv_r‘𝐸)‘𝑥) ∈ ((Base‘𝐹) ∖ {(0_g‘𝐸)}) ↔ (((inv_r‘𝐸)‘𝑥) ∈ (Base‘𝐹) ∧ ((inv_r‘𝐸)‘𝑥) ≠ (0_g‘𝐸)))
156	139, 154, 155	sylanbrc 581	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑_m 𝑏)) ∧ (1_r‘𝐸) = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σ_g (𝑖 ∈ 𝑏 ↦ ((𝑣‘𝑖)( ·_𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖)))) → ((inv_r‘𝐸)‘𝑥) ∈ ((Base‘𝐹) ∖ {(0_g‘𝐸)}))
157		fveq2 6890	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = ((inv_r‘𝐸)‘𝑥) → ((inv_r‘𝐸)‘𝑎) = ((inv_r‘𝐸)‘((inv_r‘𝐸)‘𝑥)))
158	157	eleq1d 2816	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = ((inv_r‘𝐸)‘𝑥) → (((inv_r‘𝐸)‘𝑎) ∈ (Base‘𝐹) ↔ ((inv_r‘𝐸)‘((inv_r‘𝐸)‘𝑥)) ∈ (Base‘𝐹)))
159	43, 152, 110	issubdrg 20544	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐸 ∈ DivRing ∧ (Base‘𝐹) ∈ (SubRing‘𝐸)) → ((𝐸 ↾_s (Base‘𝐹)) ∈ DivRing ↔ ∀𝑎 ∈ ((Base‘𝐹) ∖ {(0_g‘𝐸)})((inv_r‘𝐸)‘𝑎) ∈ (Base‘𝐹)))
160	159	biimpa 475	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐸 ∈ DivRing ∧ (Base‘𝐹) ∈ (SubRing‘𝐸)) ∧ (𝐸 ↾_s (Base‘𝐹)) ∈ DivRing) → ∀𝑎 ∈ ((Base‘𝐹) ∖ {(0_g‘𝐸)})((inv_r‘𝐸)‘𝑎) ∈ (Base‘𝐹))
161	160	adantr 479	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐸 ∈ DivRing ∧ (Base‘𝐹) ∈ (SubRing‘𝐸)) ∧ (𝐸 ↾_s (Base‘𝐹)) ∈ DivRing) ∧ ((inv_r‘𝐸)‘𝑥) ∈ ((Base‘𝐹) ∖ {(0_g‘𝐸)})) → ∀𝑎 ∈ ((Base‘𝐹) ∖ {(0_g‘𝐸)})((inv_r‘𝐸)‘𝑎) ∈ (Base‘𝐹))
162		simpr 483	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐸 ∈ DivRing ∧ (Base‘𝐹) ∈ (SubRing‘𝐸)) ∧ (𝐸 ↾_s (Base‘𝐹)) ∈ DivRing) ∧ ((inv_r‘𝐸)‘𝑥) ∈ ((Base‘𝐹) ∖ {(0_g‘𝐸)})) → ((inv_r‘𝐸)‘𝑥) ∈ ((Base‘𝐹) ∖ {(0_g‘𝐸)}))
163	158, 161, 162	rspcdva 3612	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐸 ∈ DivRing ∧ (Base‘𝐹) ∈ (SubRing‘𝐸)) ∧ (𝐸 ↾_s (Base‘𝐹)) ∈ DivRing) ∧ ((inv_r‘𝐸)‘𝑥) ∈ ((Base‘𝐹) ∖ {(0_g‘𝐸)})) → ((inv_r‘𝐸)‘((inv_r‘𝐸)‘𝑥)) ∈ (Base‘𝐹))
164	132, 133, 136, 156, 163	syl1111anc 836	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑_m 𝑏)) ∧ (1_r‘𝐸) = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σ_g (𝑖 ∈ 𝑏 ↦ ((𝑣‘𝑖)( ·_𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖)))) → ((inv_r‘𝐸)‘((inv_r‘𝐸)‘𝑥)) ∈ (Base‘𝐹))
165	131, 164	eqeltrrd 2832	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑_m 𝑏)) ∧ (1_r‘𝐸) = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σ_g (𝑖 ∈ 𝑏 ↦ ((𝑣‘𝑖)( ·_𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖)))) → 𝑥 ∈ (Base‘𝐹))
166	165	adantrl 712	. . . . . . . . . . . . . . 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 (0_g‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ (1_r‘𝐸) = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σ_g (𝑖 ∈ 𝑏 ↦ ((𝑣‘𝑖)( ·_𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖))))) → 𝑥 ∈ (Base‘𝐹))
167	27, 108	ringidcl 20154	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐸 ∈ Ring → (1_r‘𝐸) ∈ (Base‘𝐸))
168	61, 167	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → (1_r‘𝐸) ∈ (Base‘𝐸))
169	168, 85	eleqtrd 2833	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → (1_r‘𝐸) ∈ (Base‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
170		eqid 2730	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) = (Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
171		eqid 2730	. . . . . . . . . . . . . . . . 17 ⊢ (Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹))) = (Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))
172		eqid 2730	. . . . . . . . . . . . . . . . 17 ⊢ (0_g‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) = (0_g‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
173		eqid 2730	. . . . . . . . . . . . . . . . 17 ⊢ ( ·_𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹))) = ( ·_𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))
174	4	ad2antrr 722	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → ((subringAlg ‘𝐸)‘(Base‘𝐹)) ∈ LVec)
175		lveclmod 20861	. . . . . . . . . . . . . . . . . 18 ⊢ (((subringAlg ‘𝐸)‘(Base‘𝐹)) ∈ LVec → ((subringAlg ‘𝐸)‘(Base‘𝐹)) ∈ LMod)
176	174, 175	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → ((subringAlg ‘𝐸)‘(Base‘𝐹)) ∈ LMod)
177	78, 170, 171, 172, 173, 176, 77	lbslsp 32767	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → ((1_r‘𝐸) ∈ (Base‘((subringAlg ‘𝐸)‘(Base‘𝐹))) ↔ ∃𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑_m 𝑏)(𝑣 finSupp (0_g‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ (1_r‘𝐸) = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σ_g (𝑖 ∈ 𝑏 ↦ ((𝑣‘𝑖)( ·_𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖))))))
178	169, 177	mpbid 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 (0_g‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ (1_r‘𝐸) = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σ_g (𝑖 ∈ 𝑏 ↦ ((𝑣‘𝑖)( ·_𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖)))))
179	166, 178	r19.29a 3160	. . . . . . . . . . . . . 14 ⊢ ((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → 𝑥 ∈ (Base‘𝐹))
180	179	ad2antrr 722	. . . . . . . . . . . . 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 2730	. . . . . . . . . . . . . 14 ⊢ (._r‘𝐹) = (._r‘𝐹)
182	25, 181	ringcl 20144	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ Ring ∧ (𝑣‘𝑥) ∈ (Base‘𝐹) ∧ 𝑥 ∈ (Base‘𝐹)) → ((𝑣‘𝑥)(._r‘𝐹)𝑥) ∈ (Base‘𝐹))
183	106, 107, 180, 182	syl3anc 1369	. . . . . . . . . . . 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)) → ((𝑣‘𝑥)(._r‘𝐹)𝑥) ∈ (Base‘𝐹))
184	103, 183	eqeltrd 2831	. . . . . . . . . . 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)) Σ_g (𝑖 ∈ 𝑏 ↦ ((𝑣‘𝑖)( ·_𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖))) ∈ (Base‘𝐹))
185	184	ad2antrr 722	. . . . . . . . . 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 (0_g‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹))))) ∧ 𝑢 = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σ_g (𝑖 ∈ 𝑏 ↦ ((𝑣‘𝑖)( ·_𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖)))) → (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σ_g (𝑖 ∈ 𝑏 ↦ ((𝑣‘𝑖)( ·_𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖))) ∈ (Base‘𝐹))
186	22, 185	eqeltrd 2831	. . . . . . . . 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 (0_g‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹))))) ∧ 𝑢 = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σ_g (𝑖 ∈ 𝑏 ↦ ((𝑣‘𝑖)( ·_𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖)))) → 𝑢 ∈ (Base‘𝐹))
187	186	anasss 465	. . . . . . . 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 (0_g‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑢 = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σ_g (𝑖 ∈ 𝑏 ↦ ((𝑣‘𝑖)( ·_𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖))))) → 𝑢 ∈ (Base‘𝐹))
188	85	eleq2d 2817	. . . . . . . . . 10 ⊢ ((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → (𝑢 ∈ (Base‘𝐸) ↔ 𝑢 ∈ (Base‘((subringAlg ‘𝐸)‘(Base‘𝐹)))))
189	78, 170, 171, 172, 173, 176, 77	lbslsp 32767	. . . . . . . . . 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 (0_g‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑢 = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σ_g (𝑖 ∈ 𝑏 ↦ ((𝑣‘𝑖)( ·_𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖))))))
190	188, 189	bitrd 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 (0_g‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑢 = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σ_g (𝑖 ∈ 𝑏 ↦ ((𝑣‘𝑖)( ·_𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖))))))
191	190	biimpa 475	. . . . . . . 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 (0_g‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑢 = (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σ_g (𝑖 ∈ 𝑏 ↦ ((𝑣‘𝑖)( ·_𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖)))))
192	187, 191	r19.29a 3160	. . . . . . 7 ⊢ (((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑢 ∈ (Base‘𝐸)) → 𝑢 ∈ (Base‘𝐹))
193	192	ex 411	. . . . . 6 ⊢ ((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → (𝑢 ∈ (Base‘𝐸) → 𝑢 ∈ (Base‘𝐹)))
194	193	ssrdv 3987	. . . . 5 ⊢ ((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → (Base‘𝐸) ⊆ (Base‘𝐹))
195	21, 194	exlimddv 1936	. . . 4 ⊢ (((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → (Base‘𝐸) ⊆ (Base‘𝐹))
196	9, 195	exlimddv 1936	. . 3 ⊢ ((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) → (Base‘𝐸) ⊆ (Base‘𝐹))
197		simpr 483	. . . 4 ⊢ (((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ (Base‘𝐸) ⊆ (Base‘𝐹)) → (Base‘𝐸) ⊆ (Base‘𝐹))
198	37	ad2antrr 722	. . . 4 ⊢ (((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ (Base‘𝐸) ⊆ (Base‘𝐹)) → 𝐸 ∈ Field)
199		fvexd 6905	. . . 4 ⊢ (((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ (Base‘𝐸) ⊆ (Base‘𝐹)) → (Base‘𝐹) ∈ V)
200	43, 27	ressid2 17181	. . . 4 ⊢ (((Base‘𝐸) ⊆ (Base‘𝐹) ∧ 𝐸 ∈ Field ∧ (Base‘𝐹) ∈ V) → (𝐸 ↾_s (Base‘𝐹)) = 𝐸)
201	197, 198, 199, 200	syl3anc 1369	. . 3 ⊢ (((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ (Base‘𝐸) ⊆ (Base‘𝐹)) → (𝐸 ↾_s (Base‘𝐹)) = 𝐸)
202	196, 201	mpdan 683	. 2 ⊢ ((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) → (𝐸 ↾_s (Base‘𝐹)) = 𝐸)
203	2, 202	eqtr2d 2771	1 ⊢ ((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) → 𝐸 = 𝐹)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 = wceq 1539 ∃wex 1779 ∈ wcel 2104 ≠ wne 2938 ∀wral 3059 ∃wrex 3068 Vcvv 3472 ∖ cdif 3944 ⊆ wss 3947 ∅c0 4321 {csn 4627 class class class wbr 5147 ↦ cmpt 5230 ⟶wf 6538 ‘cfv 6542 (class class class)co 7411 ↑_m cmap 8822 finSupp cfsupp 9363 1c1 11113 ♯chash 14294 Basecbs 17148 ↾_s cress 17177 ._rcmulr 17202 Scalarcsca 17204 ·_𝑠 cvsca 17205 0_gc0g 17389 Σ_g cgsu 17390 Mndcmnd 18659 1_rcur 20075 Ringcrg 20127 CRingccrg 20128 Unitcui 20246 inv_rcinvr 20278 SubRingcsubrg 20457 DivRingcdr 20500 Fieldcfield 20501 LModclmod 20614 LBasisclbs 20829 LVecclvec 20857 subringAlg csra 20926 LIndSclinds 21579 dimcldim 32971 /_FldExtcfldext 33005 [:]cextdg 33008

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-reg 9589 ax-inf2 9638 ax-ac2 10460 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7672 df-rpss 7715 df-om 7858 df-1st 7977 df-2nd 7978 df-supp 8149 df-tpos 8213 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-oadd 8472 df-er 8705 df-map 8824 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-sup 9439 df-oi 9507 df-r1 9761 df-rank 9762 df-dju 9898 df-card 9936 df-acn 9939 df-ac 10113 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-xnn0 12549 df-z 12563 df-dec 12682 df-uz 12827 df-fz 13489 df-fzo 13632 df-seq 13971 df-hash 14295 df-struct 17084 df-sets 17101 df-slot 17119 df-ndx 17131 df-base 17149 df-ress 17178 df-plusg 17214 df-mulr 17215 df-sca 17217 df-vsca 17218 df-ip 17219 df-tset 17220 df-ple 17221 df-ocomp 17222 df-ds 17223 df-hom 17225 df-cco 17226 df-0g 17391 df-gsum 17392 df-prds 17397 df-pws 17399 df-mre 17534 df-mrc 17535 df-mri 17536 df-acs 17537 df-proset 18252 df-drs 18253 df-poset 18270 df-ipo 18485 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-mhm 18705 df-submnd 18706 df-grp 18858 df-minusg 18859 df-sbg 18860 df-mulg 18987 df-subg 19039 df-ghm 19128 df-cntz 19222 df-cmn 19691 df-abl 19692 df-mgp 20029 df-rng 20047 df-ur 20076 df-ring 20129 df-cring 20130 df-oppr 20225 df-dvdsr 20248 df-unit 20249 df-invr 20279 df-nzr 20404 df-subrg 20459 df-drng 20502 df-field 20503 df-lmod 20616 df-lss 20687 df-lsp 20727 df-lmhm 20777 df-lbs 20830 df-lvec 20858 df-sra 20930 df-rgmod 20931 df-dsmm 21506 df-frlm 21521 df-uvc 21557 df-lindf 21580 df-linds 21581 df-dim 32972 df-fldext 33009 df-extdg 33010

This theorem is referenced by: extdg1b 33031

W3C validator