extdg1id - Mathbox for Thierry Arnoux

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Theorem extdg1id 33843

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 33828	. . 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 33832	. . . . . . 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 2737	. . . . . . 7 ⊢ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹))) = (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))
6	5	lbsex 21132	. . . . . 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 4307	. . . . 5 ⊢ ((LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹))) ≠ ∅ ↔ ∃𝑏 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
9	7, 8	sylib 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))))
11	5	dimval 33777	. . . . . . . 8 ⊢ ((((subringAlg ‘𝐸)‘(Base‘𝐹)) ∈ LVec ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → (dim‘((subringAlg ‘𝐸)‘(Base‘𝐹))) = (♯‘𝑏))
12	4, 11	sylan 581	. . . . . . 7 ⊢ (((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → (dim‘((subringAlg ‘𝐸)‘(Base‘𝐹))) = (♯‘𝑏))
13		extdgval 33830	. . . . . . . . . 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 2774	. . . . . . . 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 2774	. . . . . 6 ⊢ (((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → (♯‘𝑏) = 1)
19		hash1snb 14354	. . . . . . 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 585	. . . . 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 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 2738	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐸/_FldExt𝐹 → ((subringAlg ‘𝐸)‘(Base‘𝐹)) = ((subringAlg ‘𝐸)‘(Base‘𝐹)))
25		eqid 2737	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (Base‘𝐹) = (Base‘𝐹)
26	25	fldextsubrg 33826	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐸/_FldExt𝐹 → (Base‘𝐹) ∈ (SubRing‘𝐸))
27		eqid 2737	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (Base‘𝐸) = (Base‘𝐸)
28	27	subrgss 20517	. . . . . . . . . . . . . . . . . . . . 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 21145	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐸/_FldExt𝐹 → (._r‘𝐸) = ( ·_𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
31	30	eqcomd 2743	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐸/_FldExt𝐹 → ( ·_𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹))) = (._r‘𝐸))
32	31	ad5antr 735	. . . . . . . . . . . . . . . . 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 7385	. . . . . . . . . . . . . . . 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 5186	. . . . . . . . . . . . . . 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 7384	. . . . . . . . . . . . . 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 2737	. . . . . . . . . . . . . . . . 17 ⊢ ((subringAlg ‘𝐸)‘(Base‘𝐹)) = ((subringAlg ‘𝐸)‘(Base‘𝐹))
37		fldextfld1 33824	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐸/_FldExt𝐹 → 𝐸 ∈ Field)
38		isfld 20685	. . . . . . . . . . . . . . . . . . . 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 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 2737	. . . . . . . . . . . . . . . . 17 ⊢ (𝐸 ↾_s (Base‘𝐹)) = (𝐸 ↾_s (Base‘𝐹))
44		fldextfld2 33825	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐸/_FldExt𝐹 → 𝐹 ∈ Field)
45		isfld 20685	. . . . . . . . . . . . . . . . . . . . 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 2838	. . . . . . . . . . . . . . . . . 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 33771	. . . . . . . . . . . . . . . 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 716	. . . . . . . . . . . . . . 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 727	. . . . . . . . . . . . . 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 20681	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐸 ∈ DivRing → 𝐸 ∈ Ring)
55	37, 39, 54	3syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐸/_FldExt𝐹 → 𝐸 ∈ Ring)
56		ringmnd 20190	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐸 ∈ Ring → 𝐸 ∈ Mnd)
57	55, 56	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝐸/_FldExt𝐹 → 𝐸 ∈ Mnd)
58	57	ad4antr 733	. . . . . . . . . . . . . . . 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 3446	. . . . . . . . . . . . . . . . 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 731	. . . . . . . . . . . . . . . . . 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 731	. . . . . . . . . . . . . . . . . . 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 8798	. . . . . . . . . . . . . . . . . . . . 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 4622	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑥 ∈ {𝑥}
68	67, 23	eleqtrrid 2844	. . . . . . . . . . . . . . . . . . . 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 7039	. . . . . . . . . . . . . . . . . . 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 21144	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐸/_FldExt𝐹 → (𝐸 ↾_s (Base‘𝐹)) = (Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
71	1, 70	eqtrd 2772	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐸/_FldExt𝐹 → 𝐹 = (Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
72	71	fveq2d 6846	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐸/_FldExt𝐹 → (Base‘𝐹) = (Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))))
73	72	ad4antr 733	. . . . . . . . . . . . . . . . . . 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 2840	. . . . . . . . . . . . . . . . . 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 3936	. . . . . . . . . . . . . . . . 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 769	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
78		eqid 2737	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (Base‘((subringAlg ‘𝐸)‘(Base‘𝐹))) = (Base‘((subringAlg ‘𝐸)‘(Base‘𝐹)))
79	78, 5	lbsss 21041	. . . . . . . . . . . . . . . . . . . . . 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 3971	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → {𝑥} ⊆ (Base‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
82	59	snss 4743	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ (Base‘((subringAlg ‘𝐸)‘(Base‘𝐹))) ↔ {𝑥} ⊆ (Base‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
83	81, 82	sylibr 234	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → 𝑥 ∈ (Base‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
84		eqidd 2738	. . . . . . . . . . . . . . . . . . . 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 21141	. . . . . . . . . . . . . . . . . . 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 2840	. . . . . . . . . . . . . . . . . 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 2737	. . . . . . . . . . . . . . . . . 18 ⊢ (._r‘𝐸) = (._r‘𝐸)
89	27, 88	ringcl 20197	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐸 ∈ Ring ∧ (𝑣‘𝑥) ∈ (Base‘𝐸) ∧ 𝑥 ∈ (Base‘𝐸)) → ((𝑣‘𝑥)(._r‘𝐸)𝑥) ∈ (Base‘𝐸))
90	62, 75, 87, 89	syl3anc 1374	. . . . . . . . . . . . . . . 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 6846	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑_m 𝑏)) ∧ 𝑖 = 𝑥) → (𝑣‘𝑖) = (𝑣‘𝑥))
93	92, 91	oveq12d 7386	. . . . . . . . . . . . . . . 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 19893	. . . . . . . . . . . . . . 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 6846	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐸/_FldExt𝐹 → (._r‘𝐹) = (._r‘(𝐸 ↾_s (Base‘𝐹))))
96	43, 88	ressmulr 17239	. . . . . . . . . . . . . . . . . . 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 2775	. . . . . . . . . . . . . . . . 17 ⊢ (𝐸/_FldExt𝐹 → (._r‘𝐹) = (._r‘𝐸))
99	98	ad4antr 733	. . . . . . . . . . . . . . . 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 7385	. . . . . . . . . . . . . . 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 2775	. . . . . . . . . . . . . 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 2780	. . . . . . . . . . . . 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 716	. . . . . . . . . . . 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 20681	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∈ DivRing → 𝐹 ∈ Ring)
105	44, 46, 104	3syl 18	. . . . . . . . . . . . . 14 ⊢ (𝐸/_FldExt𝐹 → 𝐹 ∈ Ring)
106	105	ad5antr 735	. . . . . . . . . . . . 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 716	. . . . . . . . . . . . 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 2737	. . . . . . . . . . . . . . . . . . . 20 ⊢ (1_r‘𝐸) = (1_r‘𝐸)
109		eqid 2737	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Unit‘𝐸) = (Unit‘𝐸)
110		eqid 2737	. . . . . . . . . . . . . . . . . . . 20 ⊢ (inv_r‘𝐸) = (inv_r‘𝐸)
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)) ∧ (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 497	. . . . . . . . . . . . . . . . . . . . . . 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 20198	. . . . . . . . . . . . . . . . . . . . . 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 1374	. . . . . . . . . . . . . . . . . . . . 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 731	. . . . . . . . . . . . . . . . . . . . . . 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 7384	. . . . . . . . . . . . . . . . . . . . . . 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 2778	. . . . . . . . . . . . . . . . . . . . . 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 2772	. . . . . . . . . . . . . . . . . . . . 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 2775	. . . . . . . . . . . . . . . . . . . 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 2743	. . . . . . . . . . . . . . . . . . . 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 22632	. . . . . . . . . . . . . . . . . . 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 20339	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐸 ∈ Ring ∧ 𝑥 ∈ (Unit‘𝐸)) → ((inv_r‘𝐸)‘((inv_r‘𝐸)‘𝑥)) = 𝑥)
131	62, 129, 130	syl2an2r 686	. . . . . . . . . . . . . . . . 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 2838	. . . . . . . . . . . . . . . . . 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 2837	. . . . . . . . . . . . . . . . . . 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 2738	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐸/_FldExt𝐹 → (0_g‘𝐸) = (0_g‘𝐸))
141	24, 140, 29	sralmod0 21152	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐸/_FldExt𝐹 → (0_g‘𝐸) = (0_g‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
142	141	ad2antrr 727	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → (0_g‘𝐸) = (0_g‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
143	5	lbslinds 21800	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹))) ⊆ (LIndS‘((subringAlg ‘𝐸)‘(Base‘𝐹)))
144	143, 10	sselid 3933	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → 𝑏 ∈ (LIndS‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
145		eqid 2737	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (0_g‘((subringAlg ‘𝐸)‘(Base‘𝐹))) = (0_g‘((subringAlg ‘𝐸)‘(Base‘𝐹)))
146	145	0nellinds 33462	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((subringAlg ‘𝐸)‘(Base‘𝐹)) ∈ LVec ∧ 𝑏 ∈ (LIndS‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → ¬ (0_g‘((subringAlg ‘𝐸)‘(Base‘𝐹))) ∈ 𝑏)
147	4, 144, 146	syl2an2r 686	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → ¬ (0_g‘((subringAlg ‘𝐸)‘(Base‘𝐹))) ∈ 𝑏)
148	142, 147	eqneltrd 2857	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → ¬ (0_g‘𝐸) ∈ 𝑏)
149	148	ad3antrrr 731	. . . . . . . . . . . . . . . . . . . . 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 3031	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ 𝑏 ∧ ¬ (0_g‘𝐸) ∈ 𝑏) → 𝑥 ≠ (0_g‘𝐸))
151	68, 149, 150	syl2an2r 686	. . . . . . . . . . . . . . . . . . . 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 2737	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (0_g‘𝐸) = (0_g‘𝐸)
153	27, 152, 110	drnginvrn0 20699	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐸 ∈ DivRing ∧ 𝑥 ∈ (Base‘𝐸) ∧ 𝑥 ≠ (0_g‘𝐸)) → ((inv_r‘𝐸)‘𝑥) ≠ (0_g‘𝐸))
154	132, 113, 151, 153	syl3anc 1374	. . . . . . . . . . . . . . . . . . 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 4744	. . . . . . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . . . . . 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 6842	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = ((inv_r‘𝐸)‘𝑥) → ((inv_r‘𝐸)‘𝑎) = ((inv_r‘𝐸)‘((inv_r‘𝐸)‘𝑥)))
158	157	eleq1d 2822	. . . . . . . . . . . . . . . . . . 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 20725	. . . . . . . . . . . . . . . . . . . . 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 3579	. . . . . . . . . . . . . . . . . 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 841	. . . . . . . . . . . . . . . . 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 2838	. . . . . . . . . . . . . . . 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 717	. . . . . . . . . . . . . . 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 20212	. . . . . . . . . . . . . . . . . 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 2839	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → (1_r‘𝐸) ∈ (Base‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
170		eqid 2737	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) = (Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
171		eqid 2737	. . . . . . . . . . . . . . . . 17 ⊢ (Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹))) = (Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))
172		eqid 2737	. . . . . . . . . . . . . . . . 17 ⊢ (0_g‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) = (0_g‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
173		eqid 2737	. . . . . . . . . . . . . . . . 17 ⊢ ( ·_𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹))) = ( ·_𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))
174	4	ad2antrr 727	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → ((subringAlg ‘𝐸)‘(Base‘𝐹)) ∈ LVec)
175		lveclmod 21070	. . . . . . . . . . . . . . . . . 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 33469	. . . . . . . . . . . . . . . 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 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 (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 3146	. . . . . . . . . . . . . 14 ⊢ ((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → 𝑥 ∈ (Base‘𝐹))
180	179	ad2antrr 727	. . . . . . . . . . . . 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 2737	. . . . . . . . . . . . . 14 ⊢ (._r‘𝐹) = (._r‘𝐹)
182	25, 181	ringcl 20197	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ Ring ∧ (𝑣‘𝑥) ∈ (Base‘𝐹) ∧ 𝑥 ∈ (Base‘𝐹)) → ((𝑣‘𝑥)(._r‘𝐹)𝑥) ∈ (Base‘𝐹))
183	106, 107, 180, 182	syl3anc 1374	. . . . . . . . . . . 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 2837	. . . . . . . . . . 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 727	. . . . . . . . . 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 2837	. . . . . . . . 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 2823	. . . . . . . . . 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 33469	. . . . . . . . . 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 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 (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 3146	. . . . . . 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 3941	. . . . 5 ⊢ ((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → (Base‘𝐸) ⊆ (Base‘𝐹))
195	21, 194	exlimddv 1937	. . . 4 ⊢ (((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → (Base‘𝐸) ⊆ (Base‘𝐹))
196	9, 195	exlimddv 1937	. . 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 727	. . . 4 ⊢ (((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ (Base‘𝐸) ⊆ (Base‘𝐹)) → 𝐸 ∈ Field)
199		fvexd 6857	. . . 4 ⊢ (((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ (Base‘𝐸) ⊆ (Base‘𝐹)) → (Base‘𝐹) ∈ V)
200	43, 27	ressid2 17173	. . . 4 ⊢ (((Base‘𝐸) ⊆ (Base‘𝐹) ∧ 𝐸 ∈ Field ∧ (Base‘𝐹) ∈ V) → (𝐸 ↾_s (Base‘𝐹)) = 𝐸)
201	197, 198, 199, 200	syl3anc 1374	. . 3 ⊢ (((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ (Base‘𝐸) ⊆ (Base‘𝐹)) → (𝐸 ↾_s (Base‘𝐹)) = 𝐸)
202	196, 201	mpdan 688	. 2 ⊢ ((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) → (𝐸 ↾_s (Base‘𝐹)) = 𝐸)
203	2, 202	eqtr2d 2773	1 ⊢ ((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) → 𝐸 = 𝐹)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1542 ∃wex 1781 ∈ wcel 2114 ≠ wne 2933 ∀wral 3052 ∃wrex 3062 Vcvv 3442 ∖ cdif 3900 ⊆ wss 3903 ∅c0 4287 {csn 4582 class class class wbr 5100 ↦ cmpt 5181 ⟶wf 6496 ‘cfv 6500 (class class class)co 7368 ↑_m cmap 8775 finSupp cfsupp 9276 1c1 11039 ♯chash 14265 Basecbs 17148 ↾_s cress 17169 ._rcmulr 17190 Scalarcsca 17192 ·_𝑠 cvsca 17193 0_gc0g 17371 Σ_g cgsu 17372 Mndcmnd 18671 1_rcur 20128 Ringcrg 20180 CRingccrg 20181 Unitcui 20303 inv_rcinvr 20335 SubRingcsubrg 20514 DivRingcdr 20674 Fieldcfield 20675 LModclmod 20823 LBasisclbs 21038 LVecclvec 21066 subringAlg csra 21135 LIndSclinds 21772 dimcldim 33775 /_FldExtcfldext 33815 [:]cextdg 33817

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-reg 9509 ax-inf2 9562 ax-ac2 10385 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-iin 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-se 5586 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-of 7632 df-rpss 7678 df-om 7819 df-1st 7943 df-2nd 7944 df-supp 8113 df-tpos 8178 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-2o 8408 df-oadd 8411 df-er 8645 df-map 8777 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9277 df-sup 9357 df-oi 9427 df-r1 9688 df-rank 9689 df-dju 9825 df-card 9863 df-acn 9866 df-ac 10038 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-nn 12158 df-2 12220 df-3 12221 df-4 12222 df-5 12223 df-6 12224 df-7 12225 df-8 12226 df-9 12227 df-n0 12414 df-xnn0 12487 df-z 12501 df-dec 12620 df-uz 12764 df-fz 13436 df-fzo 13583 df-seq 13937 df-hash 14266 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17149 df-ress 17170 df-plusg 17202 df-mulr 17203 df-sca 17205 df-vsca 17206 df-ip 17207 df-tset 17208 df-ple 17209 df-ocomp 17210 df-ds 17211 df-hom 17213 df-cco 17214 df-0g 17373 df-gsum 17374 df-prds 17379 df-pws 17381 df-mre 17517 df-mrc 17518 df-mri 17519 df-acs 17520 df-proset 18229 df-drs 18230 df-poset 18248 df-ipo 18463 df-mgm 18577 df-sgrp 18656 df-mnd 18672 df-mhm 18720 df-submnd 18721 df-grp 18878 df-minusg 18879 df-sbg 18880 df-mulg 19010 df-subg 19065 df-ghm 19154 df-cntz 19258 df-cmn 19723 df-abl 19724 df-mgp 20088 df-rng 20100 df-ur 20129 df-ring 20182 df-cring 20183 df-oppr 20285 df-dvdsr 20305 df-unit 20306 df-invr 20336 df-nzr 20458 df-subrg 20515 df-drng 20676 df-field 20677 df-lmod 20825 df-lss 20895 df-lsp 20935 df-lmhm 20986 df-lbs 21039 df-lvec 21067 df-sra 21137 df-rgmod 21138 df-dsmm 21699 df-frlm 21714 df-uvc 21750 df-lindf 21773 df-linds 21774 df-dim 33776 df-fldext 33818 df-extdg 33819

This theorem is referenced by: extdg1b 33844 rtelextdg2 33904

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