extdg1id - Mathbox for Thierry Arnoux

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Theorem extdg1id 33717

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

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

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

Step	Hyp	Ref	Expression
1		fldextress 33704	. . 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 33707	. . . . . . 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 2736	. . . . . . 7 ⊢ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹))) = (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))
6	5	lbsex 21168	. . . . . 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 4352	. . . . 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 33652	. . . . . . . 8 ⊢ ((((subringAlg ‘𝐸)‘(Base‘𝐹)) ∈ LVec ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → (dim‘((subringAlg ‘𝐸)‘(Base‘𝐹))) = (♯‘𝑏))
12	4, 11	sylan 580	. . . . . . 7 ⊢ (((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → (dim‘((subringAlg ‘𝐸)‘(Base‘𝐹))) = (♯‘𝑏))
13		extdgval 33706	. . . . . . . . . 10 ⊢ (𝐸/_FldExt𝐹 → (𝐸[:]𝐹) = (dim‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
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 2778	. . . . . . . 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 2778	. . . . . 6 ⊢ (((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → (♯‘𝑏) = 1)
19		hash1snb 14459	. . . . . . 7 ⊢ (𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹))) → ((♯‘𝑏) = 1 ↔ ∃𝑥 𝑏 = {𝑥}))
20	19	biimpa 476	. . . . . 6 ⊢ ((𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹))) ∧ (♯‘𝑏) = 1) → ∃𝑥 𝑏 = {𝑥})
21	10, 18, 20	syl2anc 584	. . . . 5 ⊢ (((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → ∃𝑥 𝑏 = {𝑥})
22		simpr 484	. . . . . . . . . 10 ⊢ ((((((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑢 ∈ (Base‘𝐸)) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑_m 𝑏)) ∧ 𝑣 finSupp (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 768	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑_m 𝑏)) → 𝑏 = {𝑥})
24		eqidd 2737	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐸/_FldExt𝐹 → ((subringAlg ‘𝐸)‘(Base‘𝐹)) = ((subringAlg ‘𝐸)‘(Base‘𝐹)))
25		eqid 2736	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (Base‘𝐹) = (Base‘𝐹)
26	25	fldextsubrg 33703	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐸/_FldExt𝐹 → (Base‘𝐹) ∈ (SubRing‘𝐸))
27		eqid 2736	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (Base‘𝐸) = (Base‘𝐸)
28	27	subrgss 20573	. . . . . . . . . . . . . . . . . . . . 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 21186	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐸/_FldExt𝐹 → (._r‘𝐸) = ( ·_𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
31	30	eqcomd 2742	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐸/_FldExt𝐹 → ( ·_𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹))) = (._r‘𝐸))
32	31	ad5antr 734	. . . . . . . . . . . . . . . . 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 7449	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑_m 𝑏)) ∧ 𝑖 ∈ 𝑏) → ((𝑣‘𝑖)( ·_𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖) = ((𝑣‘𝑖)(._r‘𝐸)𝑖))
34	23, 33	mpteq12dva 5230	. . . . . . . . . . . . . . 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 7448	. . . . . . . . . . . . . 14 ⊢ (((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑_m 𝑏)) → (𝐸 Σ_g (𝑖 ∈ 𝑏 ↦ ((𝑣‘𝑖)( ·_𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖))) = (𝐸 Σ_g (𝑖 ∈ {𝑥} ↦ ((𝑣‘𝑖)(._r‘𝐸)𝑖))))
36		eqid 2736	. . . . . . . . . . . . . . . . 17 ⊢ ((subringAlg ‘𝐸)‘(Base‘𝐹)) = ((subringAlg ‘𝐸)‘(Base‘𝐹))
37		fldextfld1 33701	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐸/_FldExt𝐹 → 𝐸 ∈ Field)
38		isfld 20741	. . . . . . . . . . . . . . . . . . . 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 2736	. . . . . . . . . . . . . . . . 17 ⊢ (𝐸 ↾_s (Base‘𝐹)) = (𝐸 ↾_s (Base‘𝐹))
44		fldextfld2 33702	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐸/_FldExt𝐹 → 𝐹 ∈ Field)
45		isfld 20741	. . . . . . . . . . . . . . . . . . . . 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 2841	. . . . . . . . . . . . . . . . . 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 33646	. . . . . . . . . . . . . . . 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 715	. . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . 14 ⊢ (((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑_m 𝑏)) → (𝐸 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 20737	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐸 ∈ DivRing → 𝐸 ∈ Ring)
55	37, 39, 54	3syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐸/_FldExt𝐹 → 𝐸 ∈ Ring)
56		ringmnd 20241	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐸 ∈ Ring → 𝐸 ∈ Mnd)
57	55, 56	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝐸/_FldExt𝐹 → 𝐸 ∈ Mnd)
58	57	ad4antr 732	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑_m 𝑏)) → 𝐸 ∈ Mnd)
59		vex 3483	. . . . . . . . . . . . . . . . 17 ⊢ 𝑥 ∈ V
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 730	. . . . . . . . . . . . . . . . . 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 730	. . . . . . . . . . . . . . . . . . 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 8890	. . . . . . . . . . . . . . . . . . . . 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 4662	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑥 ∈ {𝑥}
68	67, 23	eleqtrrid 2847	. . . . . . . . . . . . . . . . . . . 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 7104	. . . . . . . . . . . . . . . . . . 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 21184	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐸/_FldExt𝐹 → (𝐸 ↾_s (Base‘𝐹)) = (Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
71	1, 70	eqtrd 2776	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐸/_FldExt𝐹 → 𝐹 = (Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
72	71	fveq2d 6909	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐸/_FldExt𝐹 → (Base‘𝐹) = (Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))))
73	72	ad4antr 732	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑_m 𝑏)) → (Base‘𝐹) = (Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))))
74	69, 73	eleqtrrd 2843	. . . . . . . . . . . . . . . . . 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 3983	. . . . . . . . . . . . . . . . 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 768	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
78		eqid 2736	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (Base‘((subringAlg ‘𝐸)‘(Base‘𝐹))) = (Base‘((subringAlg ‘𝐸)‘(Base‘𝐹)))
79	78, 5	lbsss 21077	. . . . . . . . . . . . . . . . . . . . . 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 4018	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → {𝑥} ⊆ (Base‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
82	59	snss 4784	. . . . . . . . . . . . . . . . . . . 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 2737	. . . . . . . . . . . . . . . . . . . 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 21178	. . . . . . . . . . . . . . . . . . 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 2843	. . . . . . . . . . . . . . . . . 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 2736	. . . . . . . . . . . . . . . . . 18 ⊢ (._r‘𝐸) = (._r‘𝐸)
89	27, 88	ringcl 20248	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐸 ∈ Ring ∧ (𝑣‘𝑥) ∈ (Base‘𝐸) ∧ 𝑥 ∈ (Base‘𝐸)) → ((𝑣‘𝑥)(._r‘𝐸)𝑥) ∈ (Base‘𝐸))
90	62, 75, 87, 89	syl3anc 1372	. . . . . . . . . . . . . . . 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 6909	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑_m 𝑏)) ∧ 𝑖 = 𝑥) → (𝑣‘𝑖) = (𝑣‘𝑥))
93	92, 91	oveq12d 7450	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑_m 𝑏)) ∧ 𝑖 = 𝑥) → ((𝑣‘𝑖)(._r‘𝐸)𝑖) = ((𝑣‘𝑥)(._r‘𝐸)𝑥))
94	27, 58, 60, 90, 93	gsumsnd 19971	. . . . . . . . . . . . . . 15 ⊢ (((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑_m 𝑏)) → (𝐸 Σ_g (𝑖 ∈ {𝑥} ↦ ((𝑣‘𝑖)(._r‘𝐸)𝑖))) = ((𝑣‘𝑥)(._r‘𝐸)𝑥))
95	1	fveq2d 6909	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐸/_FldExt𝐹 → (._r‘𝐹) = (._r‘(𝐸 ↾_s (Base‘𝐹))))
96	43, 88	ressmulr 17352	. . . . . . . . . . . . . . . . . . 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 2779	. . . . . . . . . . . . . . . . 17 ⊢ (𝐸/_FldExt𝐹 → (._r‘𝐹) = (._r‘𝐸))
99	98	ad4antr 732	. . . . . . . . . . . . . . . 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 7449	. . . . . . . . . . . . . . 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 2779	. . . . . . . . . . . . . 14 ⊢ (((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑_m 𝑏)) → (𝐸 Σ_g (𝑖 ∈ {𝑥} ↦ ((𝑣‘𝑖)(._r‘𝐸)𝑖))) = ((𝑣‘𝑥)(._r‘𝐹)𝑥))
102	35, 53, 101	3eqtr3d 2784	. . . . . . . . . . . . 13 ⊢ (((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑_m 𝑏)) → (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σ_g (𝑖 ∈ 𝑏 ↦ ((𝑣‘𝑖)( ·_𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖))) = ((𝑣‘𝑥)(._r‘𝐹)𝑥))
103	102	adantlr 715	. . . . . . . . . . . 12 ⊢ ((((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑢 ∈ (Base‘𝐸)) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑_m 𝑏)) → (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σ_g (𝑖 ∈ 𝑏 ↦ ((𝑣‘𝑖)( ·_𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖))) = ((𝑣‘𝑥)(._r‘𝐹)𝑥))
104		drngring 20737	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∈ DivRing → 𝐹 ∈ Ring)
105	44, 46, 104	3syl 18	. . . . . . . . . . . . . 14 ⊢ (𝐸/_FldExt𝐹 → 𝐹 ∈ Ring)
106	105	ad5antr 734	. . . . . . . . . . . . 13 ⊢ ((((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑢 ∈ (Base‘𝐸)) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑_m 𝑏)) → 𝐹 ∈ Ring)
107	74	adantlr 715	. . . . . . . . . . . . 13 ⊢ ((((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑢 ∈ (Base‘𝐸)) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑_m 𝑏)) → (𝑣‘𝑥) ∈ (Base‘𝐹))
108		eqid 2736	. . . . . . . . . . . . . . . . . . . 20 ⊢ (1_r‘𝐸) = (1_r‘𝐸)
109		eqid 2736	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Unit‘𝐸) = (Unit‘𝐸)
110		eqid 2736	. . . . . . . . . . . . . . . . . . . 20 ⊢ (inv_r‘𝐸) = (inv_r‘𝐸)
111		simp-5l 784	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑_m 𝑏)) ∧ (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 20249	. . . . . . . . . . . . . . . . . . . . . 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 1372	. . . . . . . . . . . . . . . . . . . . 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 730	. . . . . . . . . . . . . . . . . . . . . . 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 7448	. . . . . . . . . . . . . . . . . . . . . . 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 2782	. . . . . . . . . . . . . . . . . . . . . 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 2776	. . . . . . . . . . . . . . . . . . . . 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 2779	. . . . . . . . . . . . . . . . . . . 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 2742	. . . . . . . . . . . . . . . . . . . 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 22683	. . . . . . . . . . . . . . . . . . 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 20392	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐸 ∈ Ring ∧ 𝑥 ∈ (Unit‘𝐸)) → ((inv_r‘𝐸)‘((inv_r‘𝐸)‘𝑥)) = 𝑥)
131	62, 129, 130	syl2an2r 685	. . . . . . . . . . . . . . . . 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 2841	. . . . . . . . . . . . . . . . . 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 2840	. . . . . . . . . . . . . . . . . . 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 2737	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐸/_FldExt𝐹 → (0_g‘𝐸) = (0_g‘𝐸))
141	24, 140, 29	sralmod0 21196	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐸/_FldExt𝐹 → (0_g‘𝐸) = (0_g‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
142	141	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → (0_g‘𝐸) = (0_g‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
143	5	lbslinds 21854	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹))) ⊆ (LIndS‘((subringAlg ‘𝐸)‘(Base‘𝐹)))
144	143, 10	sselid 3980	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → 𝑏 ∈ (LIndS‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
145		eqid 2736	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (0_g‘((subringAlg ‘𝐸)‘(Base‘𝐹))) = (0_g‘((subringAlg ‘𝐸)‘(Base‘𝐹)))
146	145	0nellinds 33399	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((subringAlg ‘𝐸)‘(Base‘𝐹)) ∈ LVec ∧ 𝑏 ∈ (LIndS‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → ¬ (0_g‘((subringAlg ‘𝐸)‘(Base‘𝐹))) ∈ 𝑏)
147	4, 144, 146	syl2an2r 685	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → ¬ (0_g‘((subringAlg ‘𝐸)‘(Base‘𝐹))) ∈ 𝑏)
148	142, 147	eqneltrd 2860	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → ¬ (0_g‘𝐸) ∈ 𝑏)
149	148	ad3antrrr 730	. . . . . . . . . . . . . . . . . . . . 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 3039	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ 𝑏 ∧ ¬ (0_g‘𝐸) ∈ 𝑏) → 𝑥 ≠ (0_g‘𝐸))
151	68, 149, 150	syl2an2r 685	. . . . . . . . . . . . . . . . . . . 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 2736	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (0_g‘𝐸) = (0_g‘𝐸)
153	27, 152, 110	drnginvrn0 20755	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐸 ∈ DivRing ∧ 𝑥 ∈ (Base‘𝐸) ∧ 𝑥 ≠ (0_g‘𝐸)) → ((inv_r‘𝐸)‘𝑥) ≠ (0_g‘𝐸))
154	132, 113, 151, 153	syl3anc 1372	. . . . . . . . . . . . . . . . . . 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 4785	. . . . . . . . . . . . . . . . . . 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 583	. . . . . . . . . . . . . . . . . 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 6905	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = ((inv_r‘𝐸)‘𝑥) → ((inv_r‘𝐸)‘𝑎) = ((inv_r‘𝐸)‘((inv_r‘𝐸)‘𝑥)))
158	157	eleq1d 2825	. . . . . . . . . . . . . . . . . . 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 20782	. . . . . . . . . . . . . . . . . . . . 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 3622	. . . . . . . . . . . . . . . . . 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 840	. . . . . . . . . . . . . . . . 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 2841	. . . . . . . . . . . . . . . 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 716	. . . . . . . . . . . . . . 15 ⊢ ((((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑_m 𝑏)) ∧ (𝑣 finSupp (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 20263	. . . . . . . . . . . . . . . . . 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 2842	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → (1_r‘𝐸) ∈ (Base‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
170		eqid 2736	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) = (Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
171		eqid 2736	. . . . . . . . . . . . . . . . 17 ⊢ (Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹))) = (Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))
172		eqid 2736	. . . . . . . . . . . . . . . . 17 ⊢ (0_g‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) = (0_g‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
173		eqid 2736	. . . . . . . . . . . . . . . . 17 ⊢ ( ·_𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹))) = ( ·_𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))
174	4	ad2antrr 726	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → ((subringAlg ‘𝐸)‘(Base‘𝐹)) ∈ LVec)
175		lveclmod 21106	. . . . . . . . . . . . . . . . . 18 ⊢ (((subringAlg ‘𝐸)‘(Base‘𝐹)) ∈ LVec → ((subringAlg ‘𝐸)‘(Base‘𝐹)) ∈ LMod)
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 33406	. . . . . . . . . . . . . . . 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 3161	. . . . . . . . . . . . . 14 ⊢ ((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → 𝑥 ∈ (Base‘𝐹))
180	179	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ ((((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑢 ∈ (Base‘𝐸)) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑_m 𝑏)) → 𝑥 ∈ (Base‘𝐹))
181		eqid 2736	. . . . . . . . . . . . . 14 ⊢ (._r‘𝐹) = (._r‘𝐹)
182	25, 181	ringcl 20248	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ Ring ∧ (𝑣‘𝑥) ∈ (Base‘𝐹) ∧ 𝑥 ∈ (Base‘𝐹)) → ((𝑣‘𝑥)(._r‘𝐹)𝑥) ∈ (Base‘𝐹))
183	106, 107, 180, 182	syl3anc 1372	. . . . . . . . . . . 12 ⊢ ((((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑢 ∈ (Base‘𝐸)) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑_m 𝑏)) → ((𝑣‘𝑥)(._r‘𝐹)𝑥) ∈ (Base‘𝐹))
184	103, 183	eqeltrd 2840	. . . . . . . . . . 11 ⊢ ((((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑢 ∈ (Base‘𝐸)) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑_m 𝑏)) → (((subringAlg ‘𝐸)‘(Base‘𝐹)) Σ_g (𝑖 ∈ 𝑏 ↦ ((𝑣‘𝑖)( ·_𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))𝑖))) ∈ (Base‘𝐹))
185	184	ad2antrr 726	. . . . . . . . . 10 ⊢ ((((((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑢 ∈ (Base‘𝐸)) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑_m 𝑏)) ∧ 𝑣 finSupp (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 2840	. . . . . . . . 9 ⊢ ((((((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑢 ∈ (Base‘𝐸)) ∧ 𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑_m 𝑏)) ∧ 𝑣 finSupp (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 2826	. . . . . . . . . 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 33406	. . . . . . . . . 10 ⊢ ((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → (𝑢 ∈ (Base‘((subringAlg ‘𝐸)‘(Base‘𝐹))) ↔ ∃𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑_m 𝑏)(𝑣 finSupp (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 3161	. . . . . . 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 3988	. . . . 5 ⊢ ((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → (Base‘𝐸) ⊆ (Base‘𝐹))
195	21, 194	exlimddv 1934	. . . 4 ⊢ (((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → (Base‘𝐸) ⊆ (Base‘𝐹))
196	9, 195	exlimddv 1934	. . 3 ⊢ ((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) → (Base‘𝐸) ⊆ (Base‘𝐹))
197		simpr 484	. . . 4 ⊢ (((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ (Base‘𝐸) ⊆ (Base‘𝐹)) → (Base‘𝐸) ⊆ (Base‘𝐹))
198	37	ad2antrr 726	. . . 4 ⊢ (((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ (Base‘𝐸) ⊆ (Base‘𝐹)) → 𝐸 ∈ Field)
199		fvexd 6920	. . . 4 ⊢ (((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ (Base‘𝐸) ⊆ (Base‘𝐹)) → (Base‘𝐹) ∈ V)
200	43, 27	ressid2 17279	. . . 4 ⊢ (((Base‘𝐸) ⊆ (Base‘𝐹) ∧ 𝐸 ∈ Field ∧ (Base‘𝐹) ∈ V) → (𝐸 ↾_s (Base‘𝐹)) = 𝐸)
201	197, 198, 199, 200	syl3anc 1372	. . 3 ⊢ (((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ (Base‘𝐸) ⊆ (Base‘𝐹)) → (𝐸 ↾_s (Base‘𝐹)) = 𝐸)
202	196, 201	mpdan 687	. 2 ⊢ ((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) → (𝐸 ↾_s (Base‘𝐹)) = 𝐸)
203	2, 202	eqtr2d 2777	1 ⊢ ((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) → 𝐸 = 𝐹)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1539 ∃wex 1778 ∈ wcel 2107 ≠ wne 2939 ∀wral 3060 ∃wrex 3069 Vcvv 3479 ∖ cdif 3947 ⊆ wss 3950 ∅c0 4332 {csn 4625 class class class wbr 5142 ↦ cmpt 5224 ⟶wf 6556 ‘cfv 6560 (class class class)co 7432 ↑_m cmap 8867 finSupp cfsupp 9402 1c1 11157 ♯chash 14370 Basecbs 17248 ↾_s cress 17275 ._rcmulr 17299 Scalarcsca 17301 ·_𝑠 cvsca 17302 0_gc0g 17485 Σ_g cgsu 17486 Mndcmnd 18748 1_rcur 20179 Ringcrg 20231 CRingccrg 20232 Unitcui 20356 inv_rcinvr 20388 SubRingcsubrg 20570 DivRingcdr 20730 Fieldcfield 20731 LModclmod 20859 LBasisclbs 21074 LVecclvec 21102 subringAlg csra 21171 LIndSclinds 21826 dimcldim 33650 /_FldExtcfldext 33690 [:]cextdg 33693

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-reg 9633 ax-inf2 9682 ax-ac2 10504 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-iin 4993 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-se 5637 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-isom 6569 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-of 7698 df-rpss 7744 df-om 7889 df-1st 8015 df-2nd 8016 df-supp 8187 df-tpos 8252 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-2o 8508 df-oadd 8511 df-er 8746 df-map 8869 df-ixp 8939 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-fsupp 9403 df-sup 9483 df-oi 9551 df-r1 9805 df-rank 9806 df-dju 9942 df-card 9980 df-acn 9983 df-ac 10157 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12529 df-xnn0 12602 df-z 12616 df-dec 12736 df-uz 12880 df-fz 13549 df-fzo 13696 df-seq 14044 df-hash 14371 df-struct 17185 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-ress 17276 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ocomp 17319 df-ds 17320 df-hom 17322 df-cco 17323 df-0g 17487 df-gsum 17488 df-prds 17493 df-pws 17495 df-mre 17630 df-mrc 17631 df-mri 17632 df-acs 17633 df-proset 18341 df-drs 18342 df-poset 18360 df-ipo 18574 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-mhm 18797 df-submnd 18798 df-grp 18955 df-minusg 18956 df-sbg 18957 df-mulg 19087 df-subg 19142 df-ghm 19232 df-cntz 19336 df-cmn 19801 df-abl 19802 df-mgp 20139 df-rng 20151 df-ur 20180 df-ring 20233 df-cring 20234 df-oppr 20335 df-dvdsr 20358 df-unit 20359 df-invr 20389 df-nzr 20514 df-subrg 20571 df-drng 20732 df-field 20733 df-lmod 20861 df-lss 20931 df-lsp 20971 df-lmhm 21022 df-lbs 21075 df-lvec 21103 df-sra 21173 df-rgmod 21174 df-dsmm 21753 df-frlm 21768 df-uvc 21804 df-lindf 21827 df-linds 21828 df-dim 33651 df-fldext 33694 df-extdg 33695

This theorem is referenced by: extdg1b 33718 rtelextdg2 33769

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