extdg1id - Mathbox for Thierry Arnoux

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Theorem extdg1id 31640

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 31629	. . 3 ⊢ (𝐸/_FldExt𝐹 → 𝐹 = (𝐸 ↾_s (Base‘𝐹)))
2	1	adantr 480	. 2 ⊢ ((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) → 𝐹 = (𝐸 ↾_s (Base‘𝐹)))
3		fldextsralvec 31632	. . . . . . 7 ⊢ (𝐸/_FldExt𝐹 → ((subringAlg ‘𝐸)‘(Base‘𝐹)) ∈ LVec)
4	3	adantr 480	. . . . . 6 ⊢ ((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) → ((subringAlg ‘𝐸)‘(Base‘𝐹)) ∈ LVec)
5		eqid 2738	. . . . . . 7 ⊢ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹))) = (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))
6	5	lbsex 20342	. . . . . 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 4277	. . . . 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 484	. . . . . 6 ⊢ (((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
11	5	dimval 31588	. . . . . . . 8 ⊢ ((((subringAlg ‘𝐸)‘(Base‘𝐹)) ∈ LVec ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → (dim‘((subringAlg ‘𝐸)‘(Base‘𝐹))) = (♯‘𝑏))
12	4, 11	sylan 579	. . . . . . 7 ⊢ (((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → (dim‘((subringAlg ‘𝐸)‘(Base‘𝐹))) = (♯‘𝑏))
13		extdgval 31631	. . . . . . . . . 10 ⊢ (𝐸/_FldExt𝐹 → (𝐸[:]𝐹) = (dim‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
14	13	adantr 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)
16	14, 15	eqtr3d 2780	. . . . . . . 8 ⊢ ((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) → (dim‘((subringAlg ‘𝐸)‘(Base‘𝐹))) = 1)
17	16	adantr 480	. . . . . . 7 ⊢ (((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → (dim‘((subringAlg ‘𝐸)‘(Base‘𝐹))) = 1)
18	12, 17	eqtr3d 2780	. . . . . 6 ⊢ (((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → (♯‘𝑏) = 1)
19		hash1snb 14062	. . . . . . 7 ⊢ (𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹))) → ((♯‘𝑏) = 1 ↔ ∃𝑥 𝑏 = {𝑥}))
20	19	biimpa 476	. . . . . 6 ⊢ ((𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹))) ∧ (♯‘𝑏) = 1) → ∃𝑥 𝑏 = {𝑥})
21	10, 18, 20	syl2anc 583	. . . . 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 (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 2739	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐸/_FldExt𝐹 → ((subringAlg ‘𝐸)‘(Base‘𝐹)) = ((subringAlg ‘𝐸)‘(Base‘𝐹)))
25		eqid 2738	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (Base‘𝐹) = (Base‘𝐹)
26	25	fldextsubrg 31628	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐸/_FldExt𝐹 → (Base‘𝐹) ∈ (SubRing‘𝐸))
27		eqid 2738	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (Base‘𝐸) = (Base‘𝐸)
28	27	subrgss 19940	. . . . . . . . . . . . . . . . . . . . 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 20363	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐸/_FldExt𝐹 → (._r‘𝐸) = ( ·_𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
31	30	eqcomd 2744	. . . . . . . . . . . . . . . . . 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 7272	. . . . . . . . . . . . . . . 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 5159	. . . . . . . . . . . . . . 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 7271	. . . . . . . . . . . . . 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 2738	. . . . . . . . . . . . . . . . 17 ⊢ ((subringAlg ‘𝐸)‘(Base‘𝐹)) = ((subringAlg ‘𝐸)‘(Base‘𝐹))
37		fldextfld1 31626	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐸/_FldExt𝐹 → 𝐸 ∈ Field)
38		isfld 19915	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐸 ∈ Field ↔ (𝐸 ∈ DivRing ∧ 𝐸 ∈ CRing))
39	38	simplbi 497	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐸 ∈ Field → 𝐸 ∈ DivRing)
40	37, 39	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐸/_FldExt𝐹 → 𝐸 ∈ DivRing)
41	40	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐸/_FldExt𝐹 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → 𝐸 ∈ DivRing)
42	26	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐸/_FldExt𝐹 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → (Base‘𝐹) ∈ (SubRing‘𝐸))
43		eqid 2738	. . . . . . . . . . . . . . . . 17 ⊢ (𝐸 ↾_s (Base‘𝐹)) = (𝐸 ↾_s (Base‘𝐹))
44		fldextfld2 31627	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐸/_FldExt𝐹 → 𝐹 ∈ Field)
45		isfld 19915	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐹 ∈ Field ↔ (𝐹 ∈ DivRing ∧ 𝐹 ∈ CRing))
46	45	simplbi 497	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐹 ∈ Field → 𝐹 ∈ DivRing)
47	44, 46	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐸/_FldExt𝐹 → 𝐹 ∈ DivRing)
48	1, 47	eqeltrrd 2840	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐸/_FldExt𝐹 → (𝐸 ↾_s (Base‘𝐹)) ∈ DivRing)
49	48	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐸/_FldExt𝐹 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → (𝐸 ↾_s (Base‘𝐹)) ∈ DivRing)
50		simpr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐸/_FldExt𝐹 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
51	36, 41, 42, 43, 49, 50	drgextgsum 31584	. . . . . . . . . . . . . . . 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 19913	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐸 ∈ DivRing → 𝐸 ∈ Ring)
55	37, 39, 54	3syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐸/_FldExt𝐹 → 𝐸 ∈ Ring)
56		ringmnd 19708	. . . . . . . . . . . . . . . . . 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 3426	. . . . . . . . . . . . . . . . 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 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)
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 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 8595	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑_m 𝑏) → 𝑣:𝑏⟶(Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))))
66	65	adantl 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 4595	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑥 ∈ {𝑥}
68	67, 23	eleqtrrid 2846	. . . . . . . . . . . . . . . . . . . 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	ffvelrnd 6944	. . . . . . . . . . . . . . . . . . 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 20362	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐸/_FldExt𝐹 → (𝐸 ↾_s (Base‘𝐹)) = (Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
71	1, 70	eqtrd 2778	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐸/_FldExt𝐹 → 𝐹 = (Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
72	71	fveq2d 6760	. . . . . . . . . . . . . . . . . . . 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 2842	. . . . . . . . . . . . . . . . . 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 3918	. . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . . . . 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 2738	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (Base‘((subringAlg ‘𝐸)‘(Base‘𝐹))) = (Base‘((subringAlg ‘𝐸)‘(Base‘𝐹)))
79	78, 5	lbsss 20254	. . . . . . . . . . . . . . . . . . . . . 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 3956	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → {𝑥} ⊆ (Base‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
82	59	snss 4716	. . . . . . . . . . . . . . . . . . . 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 2739	. . . . . . . . . . . . . . . . . . . 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 20356	. . . . . . . . . . . . . . . . . . 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 2842	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → 𝑥 ∈ (Base‘𝐸))
87	86	adantr 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 2738	. . . . . . . . . . . . . . . . . 18 ⊢ (._r‘𝐸) = (._r‘𝐸)
89	27, 88	ringcl 19715	. . . . . . . . . . . . . . . . 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 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)
92	91	fveq2d 6760	. . . . . . . . . . . . . . . . 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 7273	. . . . . . . . . . . . . . . 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 19468	. . . . . . . . . . . . . . 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 6760	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐸/_FldExt𝐹 → (._r‘𝐹) = (._r‘(𝐸 ↾_s (Base‘𝐹))))
96	43, 88	ressmulr 16943	. . . . . . . . . . . . . . . . . . 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 2781	. . . . . . . . . . . . . . . . 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 7272	. . . . . . . . . . . . . . 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 2781	. . . . . . . . . . . . . 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 2786	. . . . . . . . . . . . 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 19913	. . . . . . . . . . . . . . 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 2738	. . . . . . . . . . . . . . . . . . . 20 ⊢ (1_r‘𝐸) = (1_r‘𝐸)
109		eqid 2738	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Unit‘𝐸) = (Unit‘𝐸)
110		eqid 2738	. . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . . 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 496	. . . . . . . . . . . . . . . . . . . . . . 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 19716	. . . . . . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . . . . . . 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 7271	. . . . . . . . . . . . . . . . . . . . . . 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 2784	. . . . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . . . . 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 2778	. . . . . . . . . . . . . . . . . . . . 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 2781	. . . . . . . . . . . . . . . . . . . 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 2744	. . . . . . . . . . . . . . . . . . . 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 21733	. . . . . . . . . . . . . . . . . . 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 494	. . . . . . . . . . . . . . . . . 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 19832	. . . . . . . . . . . . . . . . . 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 2840	. . . . . . . . . . . . . . . . . 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 495	. . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . . 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 2839	. . . . . . . . . . . . . . . . . . 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 2739	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐸/_FldExt𝐹 → (0_g‘𝐸) = (0_g‘𝐸))
141	24, 140, 29	sralmod0 20371	. . . . . . . . . . . . . . . . . . . . . . . 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 20950	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹))) ⊆ (LIndS‘((subringAlg ‘𝐸)‘(Base‘𝐹)))
144	143, 10	sselid 3915	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → 𝑏 ∈ (LIndS‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
145		eqid 2738	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (0_g‘((subringAlg ‘𝐸)‘(Base‘𝐹))) = (0_g‘((subringAlg ‘𝐸)‘(Base‘𝐹)))
146	145	0nellinds 31468	. . . . . . . . . . . . . . . . . . . . . . . 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 2858	. . . . . . . . . . . . . . . . . . . . . 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 3041	. . . . . . . . . . . . . . . . . . . . 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 2738	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (0_g‘𝐸) = (0_g‘𝐸)
153	27, 152, 110	drnginvrn0 19924	. . . . . . . . . . . . . . . . . . . 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 4717	. . . . . . . . . . . . . . . . . . 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 582	. . . . . . . . . . . . . . . . . 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 6756	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = ((inv_r‘𝐸)‘𝑥) → ((inv_r‘𝐸)‘𝑎) = ((inv_r‘𝐸)‘((inv_r‘𝐸)‘𝑥)))
158	157	eleq1d 2823	. . . . . . . . . . . . . . . . . . 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 19964	. . . . . . . . . . . . . . . . . . . . 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 476	. . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . . 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 3554	. . . . . . . . . . . . . . . . . 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 2840	. . . . . . . . . . . . . . . 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 19722	. . . . . . . . . . . . . . . . . 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 2841	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → (1_r‘𝐸) ∈ (Base‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
170		eqid 2738	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) = (Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
171		eqid 2738	. . . . . . . . . . . . . . . . 17 ⊢ (Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹))) = (Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))
172		eqid 2738	. . . . . . . . . . . . . . . . 17 ⊢ (0_g‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) = (0_g‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
173		eqid 2738	. . . . . . . . . . . . . . . . 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 20283	. . . . . . . . . . . . . . . . . 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 31474	. . . . . . . . . . . . . . . 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 3217	. . . . . . . . . . . . . 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 2738	. . . . . . . . . . . . . 14 ⊢ (._r‘𝐹) = (._r‘𝐹)
182	25, 181	ringcl 19715	. . . . . . . . . . . . 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 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)) 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 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 (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 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 (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 2824	. . . . . . . . . 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 31474	. . . . . . . . . 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 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 (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 3217	. . . . . . 7 ⊢ (((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑢 ∈ (Base‘𝐸)) → 𝑢 ∈ (Base‘𝐹))
193	192	ex 412	. . . . . 6 ⊢ ((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → (𝑢 ∈ (Base‘𝐸) → 𝑢 ∈ (Base‘𝐹)))
194	193	ssrdv 3923	. . . . 5 ⊢ ((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → (Base‘𝐸) ⊆ (Base‘𝐹))
195	21, 194	exlimddv 1939	. . . 4 ⊢ (((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → (Base‘𝐸) ⊆ (Base‘𝐹))
196	9, 195	exlimddv 1939	. . 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))
198	37	ad2antrr 722	. . . 4 ⊢ (((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ (Base‘𝐸) ⊆ (Base‘𝐹)) → 𝐸 ∈ Field)
199		fvexd 6771	. . . 4 ⊢ (((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ (Base‘𝐸) ⊆ (Base‘𝐹)) → (Base‘𝐹) ∈ V)
200	43, 27	ressid2 16871	. . . 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 2779	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 1783 ∈ wcel 2108 ≠ wne 2942 ∀wral 3063 ∃wrex 3064 Vcvv 3422 ∖ cdif 3880 ⊆ wss 3883 ∅c0 4253 {csn 4558 class class class wbr 5070 ↦ cmpt 5153 ⟶wf 6414 ‘cfv 6418 (class class class)co 7255 ↑_m cmap 8573 finSupp cfsupp 9058 1c1 10803 ♯chash 13972 Basecbs 16840 ↾_s cress 16867 ._rcmulr 16889 Scalarcsca 16891 ·_𝑠 cvsca 16892 0_gc0g 17067 Σ_g cgsu 17068 Mndcmnd 18300 1_rcur 19652 Ringcrg 19698 CRingccrg 19699 Unitcui 19796 inv_rcinvr 19828 DivRingcdr 19906 Fieldcfield 19907 SubRingcsubrg 19935 LModclmod 20038 LBasisclbs 20251 LVecclvec 20279 subringAlg csra 20345 LIndSclinds 20922 dimcldim 31586 /_FldExtcfldext 31615 [:]cextdg 31618

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-reg 9281 ax-inf2 9329 ax-ac2 10150 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-rpss 7554 df-om 7688 df-1st 7804 df-2nd 7805 df-supp 7949 df-tpos 8013 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-oadd 8271 df-er 8456 df-map 8575 df-ixp 8644 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fsupp 9059 df-sup 9131 df-oi 9199 df-r1 9453 df-rank 9454 df-dju 9590 df-card 9628 df-acn 9631 df-ac 9803 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-xnn0 12236 df-z 12250 df-dec 12367 df-uz 12512 df-fz 13169 df-fzo 13312 df-seq 13650 df-hash 13973 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-sca 16904 df-vsca 16905 df-ip 16906 df-tset 16907 df-ple 16908 df-ocomp 16909 df-ds 16910 df-hom 16912 df-cco 16913 df-0g 17069 df-gsum 17070 df-prds 17075 df-pws 17077 df-mre 17212 df-mrc 17213 df-mri 17214 df-acs 17215 df-proset 17928 df-drs 17929 df-poset 17946 df-ipo 18161 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-mhm 18345 df-submnd 18346 df-grp 18495 df-minusg 18496 df-sbg 18497 df-mulg 18616 df-subg 18667 df-ghm 18747 df-cntz 18838 df-cmn 19303 df-abl 19304 df-mgp 19636 df-ur 19653 df-ring 19700 df-cring 19701 df-oppr 19777 df-dvdsr 19798 df-unit 19799 df-invr 19829 df-drng 19908 df-field 19909 df-subrg 19937 df-lmod 20040 df-lss 20109 df-lsp 20149 df-lmhm 20199 df-lbs 20252 df-lvec 20280 df-sra 20349 df-rgmod 20350 df-nzr 20442 df-dsmm 20849 df-frlm 20864 df-uvc 20900 df-lindf 20923 df-linds 20924 df-dim 31587 df-fldext 31619 df-extdg 31620

This theorem is referenced by: extdg1b 31641

W3C validator