extdg1id - Mathbox for Thierry Arnoux

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Theorem extdg1id 34079

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 34064	. . 3 ⊢ (𝐸/_FldExt𝐹 → 𝐹 = (𝐸 ↾_s (Base‘𝐹)))
2	1	adantr 486	. 2 ⊢ ((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) → 𝐹 = (𝐸 ↾_s (Base‘𝐹)))
3		fldextsralvec 34068	. . . . . . 7 ⊢ (𝐸/_FldExt𝐹 → ((subringAlg ‘𝐸)‘(Base‘𝐹)) ∈ LVec)
4	3	adantr 486	. . . . . 6 ⊢ ((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) → ((subringAlg ‘𝐸)‘(Base‘𝐹)) ∈ LVec)
5		eqid 2765	. . . . . . 7 ⊢ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹))) = (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))
6	5	lbsex 21318	. . . . . 6 ⊢ (((subringAlg ‘𝐸)‘(Base‘𝐹)) ∈ LVec → (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹))) ≠ ∅)
7	4, 6	syl 18	. . . . 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 221	. . . 4 ⊢ ((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) → ∃𝑏 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
10		simpr 490	. . . . . 6 ⊢ (((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
11	5	dimval 34014	. . . . . . . 8 ⊢ ((((subringAlg ‘𝐸)‘(Base‘𝐹)) ∈ LVec ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → (dim‘((subringAlg ‘𝐸)‘(Base‘𝐹))) = (♯‘𝑏))
12	4, 11	sylan 592	. . . . . . 7 ⊢ (((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → (dim‘((subringAlg ‘𝐸)‘(Base‘𝐹))) = (♯‘𝑏))
13		extdgval 34066	. . . . . . . . . 10 ⊢ (𝐸/_FldExt𝐹 → (𝐸[:]𝐹) = (dim‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
14	13	adantr 486	. . . . . . . . 9 ⊢ ((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) → (𝐸[:]𝐹) = (dim‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
15		simpr 490	. . . . . . . . 9 ⊢ ((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) → (𝐸[:]𝐹) = 1)
16	14, 15	eqtr3d 2802	. . . . . . . 8 ⊢ ((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) → (dim‘((subringAlg ‘𝐸)‘(Base‘𝐹))) = 1)
17	16	adantr 486	. . . . . . 7 ⊢ (((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → (dim‘((subringAlg ‘𝐸)‘(Base‘𝐹))) = 1)
18	12, 17	eqtr3d 2802	. . . . . 6 ⊢ (((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → (♯‘𝑏) = 1)
19		hash1snb 14469	. . . . . . 7 ⊢ (𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹))) → ((♯‘𝑏) = 1 ↔ ∃𝑥 𝑏 = {𝑥}))
20	19	biimpa 482	. . . . . 6 ⊢ ((𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹))) ∧ (♯‘𝑏) = 1) → ∃𝑥 𝑏 = {𝑥})
21	10, 18, 20	syl2anc 596	. . . . 5 ⊢ (((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → ∃𝑥 𝑏 = {𝑥})
22		simpr 490	. . . . . . . . . 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 781	. . . . . . . . . . . . . . . 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 2766	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐸/_FldExt𝐹 → ((subringAlg ‘𝐸)‘(Base‘𝐹)) = ((subringAlg ‘𝐸)‘(Base‘𝐹)))
25		eqid 2765	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (Base‘𝐹) = (Base‘𝐹)
26	25	fldextsubrg 34062	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐸/_FldExt𝐹 → (Base‘𝐹) ∈ (SubRing‘𝐸))
27		eqid 2765	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (Base‘𝐸) = (Base‘𝐸)
28	27	subrgss 20700	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((Base‘𝐹) ∈ (SubRing‘𝐸) → (Base‘𝐹) ⊆ (Base‘𝐸))
29	26, 28	syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐸/_FldExt𝐹 → (Base‘𝐹) ⊆ (Base‘𝐸))
30	24, 29	sravsca 21331	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐸/_FldExt𝐹 → (._r‘𝐸) = ( ·_𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
31	30	eqcomd 2771	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐸/_FldExt𝐹 → ( ·_𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹))) = (._r‘𝐸))
32	31	ad5antr 747	. . . . . . . . . . . . . . . . 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 7433	. . . . . . . . . . . . . . . 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 5199	. . . . . . . . . . . . . . 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 7432	. . . . . . . . . . . . . 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 2765	. . . . . . . . . . . . . . . . 17 ⊢ ((subringAlg ‘𝐸)‘(Base‘𝐹)) = ((subringAlg ‘𝐸)‘(Base‘𝐹))
37		fldextfld1 34060	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐸/_FldExt𝐹 → 𝐸 ∈ Field)
38		isfld 20869	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐸 ∈ Field ↔ (𝐸 ∈ DivRing ∧ 𝐸 ∈ CRing))
39	38	simplbi 502	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐸 ∈ Field → 𝐸 ∈ DivRing)
40	37, 39	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐸/_FldExt𝐹 → 𝐸 ∈ DivRing)
41	40	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐸/_FldExt𝐹 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → 𝐸 ∈ DivRing)
42	26	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐸/_FldExt𝐹 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → (Base‘𝐹) ∈ (SubRing‘𝐸))
43		eqid 2765	. . . . . . . . . . . . . . . . 17 ⊢ (𝐸 ↾_s (Base‘𝐹)) = (𝐸 ↾_s (Base‘𝐹))
44		fldextfld2 34061	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐸/_FldExt𝐹 → 𝐹 ∈ Field)
45		isfld 20869	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐹 ∈ Field ↔ (𝐹 ∈ DivRing ∧ 𝐹 ∈ CRing))
46	45	simplbi 502	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐹 ∈ Field → 𝐹 ∈ DivRing)
47	44, 46	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐸/_FldExt𝐹 → 𝐹 ∈ DivRing)
48	1, 47	eqeltrrd 2866	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐸/_FldExt𝐹 → (𝐸 ↾_s (Base‘𝐹)) ∈ DivRing)
49	48	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐸/_FldExt𝐹 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → (𝐸 ↾_s (Base‘𝐹)) ∈ DivRing)
50		simpr 490	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐸/_FldExt𝐹 ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
51	36, 41, 42, 43, 49, 50	drgextgsum 34008	. . . . . . . . . . . . . . . 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 728	. . . . . . . . . . . . . . 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 739	. . . . . . . . . . . . . 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 20863	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐸 ∈ DivRing → 𝐸 ∈ Ring)
55	37, 39, 54	3syl 19	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐸/_FldExt𝐹 → 𝐸 ∈ Ring)
56		ringmnd 20348	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐸 ∈ Ring → 𝐸 ∈ Mnd)
57	55, 56	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝐸/_FldExt𝐹 → 𝐸 ∈ Mnd)
58	57	ad4antr 745	. . . . . . . . . . . . . . . 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 3461	. . . . . . . . . . . . . . . . 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 743	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → 𝐸 ∈ Ring)
62	61	adantr 486	. . . . . . . . . . . . . . . . 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 743	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → (Base‘𝐹) ⊆ (Base‘𝐸))
64	63	adantr 486	. . . . . . . . . . . . . . . . . 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 8848	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑣 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ↑_m 𝑏) → 𝑣:𝑏⟶(Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))))
66	65	adantl 487	. . . . . . . . . . . . . . . . . . . 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 4631	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑥 ∈ {𝑥}
68	67, 23	eleqtrrid 2872	. . . . . . . . . . . . . . . . . . . 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 7084	. . . . . . . . . . . . . . . . . . 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 21330	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐸/_FldExt𝐹 → (𝐸 ↾_s (Base‘𝐹)) = (Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
71	1, 70	eqtrd 2800	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐸/_FldExt𝐹 → 𝐹 = (Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
72	71	fveq2d 6889	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐸/_FldExt𝐹 → (Base‘𝐹) = (Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))))
73	72	ad4antr 745	. . . . . . . . . . . . . . . . . . 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 2868	. . . . . . . . . . . . . . . . . 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 3939	. . . . . . . . . . . . . . . . 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 490	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → 𝑏 = {𝑥})
77		simplr 781	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
78		eqid 2765	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (Base‘((subringAlg ‘𝐸)‘(Base‘𝐹))) = (Base‘((subringAlg ‘𝐸)‘(Base‘𝐹)))
79	78, 5	lbsss 21227	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹))) → 𝑏 ⊆ (Base‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
80	77, 79	syl 18	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → 𝑏 ⊆ (Base‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
81	76, 80	eqsstrrd 3973	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → {𝑥} ⊆ (Base‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
82	59	snss 4752	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ (Base‘((subringAlg ‘𝐸)‘(Base‘𝐹))) ↔ {𝑥} ⊆ (Base‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
83	81, 82	sylibr 237	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → 𝑥 ∈ (Base‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
84		eqidd 2766	. . . . . . . . . . . . . . . . . . . 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 21327	. . . . . . . . . . . . . . . . . . 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 2868	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → 𝑥 ∈ (Base‘𝐸))
87	86	adantr 486	. . . . . . . . . . . . . . . . 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 2765	. . . . . . . . . . . . . . . . . 18 ⊢ (._r‘𝐸) = (._r‘𝐸)
89	27, 88	ringcl 20355	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐸 ∈ Ring ∧ (𝑣‘𝑥) ∈ (Base‘𝐸) ∧ 𝑥 ∈ (Base‘𝐸)) → ((𝑣‘𝑥)(._r‘𝐸)𝑥) ∈ (Base‘𝐸))
90	62, 75, 87, 89	syl3anc 1398	. . . . . . . . . . . . . . . 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 490	. . . . . . . . . . . . . . . . . 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 6889	. . . . . . . . . . . . . . . . 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 7434	. . . . . . . . . . . . . . . 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 20045	. . . . . . . . . . . . . . 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 6889	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐸/_FldExt𝐹 → (._r‘𝐹) = (._r‘(𝐸 ↾_s (Base‘𝐹))))
96	43, 88	ressmulr 17377	. . . . . . . . . . . . . . . . . . 19 ⊢ ((Base‘𝐹) ∈ (SubRing‘𝐸) → (._r‘𝐸) = (._r‘(𝐸 ↾_s (Base‘𝐹))))
97	26, 96	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐸/_FldExt𝐹 → (._r‘𝐸) = (._r‘(𝐸 ↾_s (Base‘𝐹))))
98	95, 97	eqtr4d 2803	. . . . . . . . . . . . . . . . 17 ⊢ (𝐸/_FldExt𝐹 → (._r‘𝐹) = (._r‘𝐸))
99	98	ad4antr 745	. . . . . . . . . . . . . . . 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 7433	. . . . . . . . . . . . . . 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 2803	. . . . . . . . . . . . . 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 2808	. . . . . . . . . . . . 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 728	. . . . . . . . . . . 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 20863	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∈ DivRing → 𝐹 ∈ Ring)
105	44, 46, 104	3syl 19	. . . . . . . . . . . . . 14 ⊢ (𝐸/_FldExt𝐹 → 𝐹 ∈ Ring)
106	105	ad5antr 747	. . . . . . . . . . . . 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 728	. . . . . . . . . . . . 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 2765	. . . . . . . . . . . . . . . . . . . 20 ⊢ (1_r‘𝐸) = (1_r‘𝐸)
109		eqid 2765	. . . . . . . . . . . . . . . . . . . 20 ⊢ (Unit‘𝐸) = (Unit‘𝐸)
110		eqid 2765	. . . . . . . . . . . . . . . . . . . 20 ⊢ (inv_r‘𝐸) = (inv_r‘𝐸)
111		simp-5l 797	. . . . . . . . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . . . . . 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 503	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐸 ∈ Field → 𝐸 ∈ CRing)
116	111, 37, 115	3syl 19	. . . . . . . . . . . . . . . . . . . . . 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 20356	. . . . . . . . . . . . . . . . . . . . . 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 1398	. . . . . . . . . . . . . . . . . . . . 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 490	. . . . . . . . . . . . . . . . . . . . . . 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 743	. . . . . . . . . . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . . . . . . . . . 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 7432	. . . . . . . . . . . . . . . . . . . . . . 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 2806	. . . . . . . . . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . . . . . . . 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 2800	. . . . . . . . . . . . . . . . . . . . 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 2803	. . . . . . . . . . . . . . . . . . . 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 2771	. . . . . . . . . . . . . . . . . . . 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 22862	. . . . . . . . . . . . . . . . . . 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 500	. . . . . . . . . . . . . . . . . 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 20498	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐸 ∈ Ring ∧ 𝑥 ∈ (Unit‘𝐸)) → ((inv_r‘𝐸)‘((inv_r‘𝐸)‘𝑥)) = 𝑥)
131	62, 129, 130	syl2an2r 698	. . . . . . . . . . . . . . . . 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 19	. . . . . . . . . . . . . . . . . 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 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)))) → (Base‘𝐹) ∈ (SubRing‘𝐸))
134	111, 1	syl 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 = (𝐸 ↾_s (Base‘𝐹)))
135	111, 44, 46	3syl 19	. . . . . . . . . . . . . . . . . . 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 2866	. . . . . . . . . . . . . . . . . 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 501	. . . . . . . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . . . . . 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 2865	. . . . . . . . . . . . . . . . . . 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 2766	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐸/_FldExt𝐹 → (0_g‘𝐸) = (0_g‘𝐸))
141	24, 140, 29	sralmod0 21338	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝐸/_FldExt𝐹 → (0_g‘𝐸) = (0_g‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
142	141	ad2antrr 739	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → (0_g‘𝐸) = (0_g‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
143	5	lbslinds 22012	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹))) ⊆ (LIndS‘((subringAlg ‘𝐸)‘(Base‘𝐹)))
144	143, 10	sselid 3936	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → 𝑏 ∈ (LIndS‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
145		eqid 2765	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (0_g‘((subringAlg ‘𝐸)‘(Base‘𝐹))) = (0_g‘((subringAlg ‘𝐸)‘(Base‘𝐹)))
146	145	0nellinds 33708	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((subringAlg ‘𝐸)‘(Base‘𝐹)) ∈ LVec ∧ 𝑏 ∈ (LIndS‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → ¬ (0_g‘((subringAlg ‘𝐸)‘(Base‘𝐹))) ∈ 𝑏)
147	4, 144, 146	syl2an2r 698	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → ¬ (0_g‘((subringAlg ‘𝐸)‘(Base‘𝐹))) ∈ 𝑏)
148	142, 147	eqneltrd 2885	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → ¬ (0_g‘𝐸) ∈ 𝑏)
149	148	ad3antrrr 743	. . . . . . . . . . . . . . . . . . . . 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 3058	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ 𝑏 ∧ ¬ (0_g‘𝐸) ∈ 𝑏) → 𝑥 ≠ (0_g‘𝐸))
151	68, 149, 150	syl2an2r 698	. . . . . . . . . . . . . . . . . . . 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 2765	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (0_g‘𝐸) = (0_g‘𝐸)
153	27, 152, 110	drnginvrn0 20887	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐸 ∈ DivRing ∧ 𝑥 ∈ (Base‘𝐸) ∧ 𝑥 ≠ (0_g‘𝐸)) → ((inv_r‘𝐸)‘𝑥) ≠ (0_g‘𝐸))
154	132, 113, 151, 153	syl3anc 1398	. . . . . . . . . . . . . . . . . . 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 4755	. . . . . . . . . . . . . . . . . . 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 595	. . . . . . . . . . . . . . . . . 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 6885	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = ((inv_r‘𝐸)‘𝑥) → ((inv_r‘𝐸)‘𝑎) = ((inv_r‘𝐸)‘((inv_r‘𝐸)‘𝑥)))
158	157	eleq1d 2850	. . . . . . . . . . . . . . . . . . 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 20912	. . . . . . . . . . . . . . . . . . . . 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 482	. . . . . . . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . . . . 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 490	. . . . . . . . . . . . . . . . . . 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 3584	. . . . . . . . . . . . . . . . . 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 854	. . . . . . . . . . . . . . . . 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 2866	. . . . . . . . . . . . . . . 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 729	. . . . . . . . . . . . . . 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 20372	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐸 ∈ Ring → (1_r‘𝐸) ∈ (Base‘𝐸))
168	61, 167	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → (1_r‘𝐸) ∈ (Base‘𝐸))
169	168, 85	eleqtrd 2867	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → (1_r‘𝐸) ∈ (Base‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
170		eqid 2765	. . . . . . . . . . . . . . . . 17 ⊢ (Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) = (Base‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
171		eqid 2765	. . . . . . . . . . . . . . . . 17 ⊢ (Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹))) = (Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))
172		eqid 2765	. . . . . . . . . . . . . . . . 17 ⊢ (0_g‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) = (0_g‘(Scalar‘((subringAlg ‘𝐸)‘(Base‘𝐹))))
173		eqid 2765	. . . . . . . . . . . . . . . . 17 ⊢ ( ·_𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹))) = ( ·_𝑠 ‘((subringAlg ‘𝐸)‘(Base‘𝐹)))
174	4	ad2antrr 739	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → ((subringAlg ‘𝐸)‘(Base‘𝐹)) ∈ LVec)
175		lveclmod 21256	. . . . . . . . . . . . . . . . . 18 ⊢ (((subringAlg ‘𝐸)‘(Base‘𝐹)) ∈ LVec → ((subringAlg ‘𝐸)‘(Base‘𝐹)) ∈ LMod)
176	174, 175	syl 18	. . . . . . . . . . . . . . . . 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 33713	. . . . . . . . . . . . . . . 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 235	. . . . . . . . . . . . . . 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 3175	. . . . . . . . . . . . . 14 ⊢ ((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → 𝑥 ∈ (Base‘𝐹))
180	179	ad2antrr 739	. . . . . . . . . . . . 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 2765	. . . . . . . . . . . . . 14 ⊢ (._r‘𝐹) = (._r‘𝐹)
182	25, 181	ringcl 20355	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ Ring ∧ (𝑣‘𝑥) ∈ (Base‘𝐹) ∧ 𝑥 ∈ (Base‘𝐹)) → ((𝑣‘𝑥)(._r‘𝐹)𝑥) ∈ (Base‘𝐹))
183	106, 107, 180, 182	syl3anc 1398	. . . . . . . . . . . 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 2865	. . . . . . . . . . 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 739	. . . . . . . . . 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 2865	. . . . . . . . 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 472	. . . . . . . 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 2851	. . . . . . . . . 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 33713	. . . . . . . . . 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 282	. . . . . . . . 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 482	. . . . . . . 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 3175	. . . . . . 7 ⊢ (((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) ∧ 𝑢 ∈ (Base‘𝐸)) → 𝑢 ∈ (Base‘𝐹))
193	192	ex 418	. . . . . 6 ⊢ ((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → (𝑢 ∈ (Base‘𝐸) → 𝑢 ∈ (Base‘𝐹)))
194	193	ssrdv 3944	. . . . 5 ⊢ ((((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) ∧ 𝑏 = {𝑥}) → (Base‘𝐸) ⊆ (Base‘𝐹))
195	21, 194	exlimddv 1968	. . . 4 ⊢ (((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘(Base‘𝐹)))) → (Base‘𝐸) ⊆ (Base‘𝐹))
196	9, 195	exlimddv 1968	. . 3 ⊢ ((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) → (Base‘𝐸) ⊆ (Base‘𝐹))
197		simpr 490	. . . 4 ⊢ (((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ (Base‘𝐸) ⊆ (Base‘𝐹)) → (Base‘𝐸) ⊆ (Base‘𝐹))
198	37	ad2antrr 739	. . . 4 ⊢ (((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ (Base‘𝐸) ⊆ (Base‘𝐹)) → 𝐸 ∈ Field)
199		fvexd 6900	. . . 4 ⊢ (((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ (Base‘𝐸) ⊆ (Base‘𝐹)) → (Base‘𝐹) ∈ V)
200	43, 27	ressid2 17311	. . . 4 ⊢ (((Base‘𝐸) ⊆ (Base‘𝐹) ∧ 𝐸 ∈ Field ∧ (Base‘𝐹) ∈ V) → (𝐸 ↾_s (Base‘𝐹)) = 𝐸)
201	197, 198, 199, 200	syl3anc 1398	. . 3 ⊢ (((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) ∧ (Base‘𝐸) ⊆ (Base‘𝐹)) → (𝐸 ↾_s (Base‘𝐹)) = 𝐸)
202	196, 201	mpdan 700	. 2 ⊢ ((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) → (𝐸 ↾_s (Base‘𝐹)) = 𝐸)
203	2, 202	eqtr2d 2801	1 ⊢ ((𝐸/_FldExt𝐹 ∧ (𝐸[:]𝐹) = 1) → 𝐸 = 𝐹)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 = wceq 1570 ∃wex 1812 ∈ wcel 2146 ≠ wne 2960 ∀wral 3081 ∃wrex 3091 Vcvv 3457 ∖ cdif 3903 ⊆ wss 3906 ∅c0 4286 {csn 4591 class class class wbr 5111 ↦ cmpt 5194 ⟶wf 6536 ‘cfv 6540 (class class class)co 7416 ↑_m cmap 8826 finSupp cfsupp 9324 1c1 11112 ♯chash 14379 Basecbs 17286 ↾_s cress 17307 ._rcmulr 17328 Scalarcsca 17330 ·_𝑠 cvsca 17331 0_gc0g 17509 Σ_g cgsu 17510 Mndcmnd 18813 1_rcur 20286 Ringcrg 20338 CRingccrg 20339 Unitcui 20462 inv_rcinvr 20494 SubRingcsubrg 20697 DivRingcdr 20856 Fieldcfield 20857 LModclmod 21010 LBasisclbs 21224 LVecclvec 21252 subringAlg csra 21321 LIndSclinds 21984 dimcldim 34012 /_FldExtcfldext 34051 [:]cextdg 34053

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7738 ax-reg 9557 ax-inf2 9613 ax-ac2 10458 ax-cnex 11167 ax-resscn 11168 ax-1cn 11169 ax-icn 11170 ax-addcl 11171 ax-addrcl 11172 ax-mulcl 11173 ax-mulrcl 11174 ax-mulcom 11175 ax-addass 11176 ax-mulass 11177 ax-distr 11178 ax-i2m1 11179 ax-1ne0 11180 ax-1rid 11181 ax-rnegex 11182 ax-rrecex 11183 ax-cnre 11184 ax-pre-lttri 11185 ax-pre-lttrn 11186 ax-pre-ltadd 11187 ax-pre-mulgt0 11188

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4875 df-int 4915 df-iun 4960 df-iin 4961 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-se 5617 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7680 df-rpss 7726 df-om 7865 df-1st 7988 df-2nd 7989 df-supp 8159 df-tpos 8224 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-1o 8455 df-2o 8456 df-oadd 8459 df-er 8696 df-map 8828 df-ixp 8898 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-fsupp 9325 df-sup 9405 df-oi 9475 df-r1 9739 df-rank 9740 df-scott 9861 df-dju 9899 df-card 9937 df-acn 9940 df-ac 10112 df-pnf 11256 df-mnf 11257 df-xr 11258 df-ltxr 11259 df-le 11260 df-sub 11454 df-neg 11455 df-nn 12245 df-2 12314 df-3 12315 df-4 12316 df-5 12317 df-6 12318 df-7 12319 df-8 12320 df-9 12321 df-n0 12516 df-xnn0 12589 df-z 12603 df-dec 12723 df-uz 12874 df-fz 13547 df-fzo 13695 df-seq 14051 df-hash 14380 df-struct 17224 df-sets 17241 df-slot 17259 df-ndx 17271 df-base 17287 df-ress 17308 df-plusg 17340 df-mulr 17341 df-sca 17343 df-vsca 17344 df-ip 17345 df-tset 17346 df-ple 17347 df-ocomp 17348 df-ds 17349 df-hom 17351 df-cco 17352 df-0g 17511 df-gsum 17512 df-prds 17517 df-pws 17519 df-mre 17655 df-mrc 17656 df-mri 17657 df-acs 17658 df-proset 18367 df-drs 18368 df-poset 18386 df-ipo 18601 df-mgm 18715 df-sgrp 18798 df-mnd 18814 df-mhm 18864 df-submnd 18865 df-grp 19026 df-minusg 19027 df-sbg 19028 df-mulg 19157 df-subg 19212 df-ghm 19307 df-cntz 19410 df-cmn 19875 df-abl 19876 df-mgp 20240 df-rng 20254 df-ur 20287 df-ring 20340 df-cring 20341 df-oppr 20444 df-dvdsr 20464 df-unit 20465 df-invr 20495 df-nzr 20639 df-subrg 20698 df-drng 20858 df-field 20859 df-lmod 21012 df-lss 21082 df-lsp 21122 df-lmhm 21172 df-lbs 21225 df-lvec 21253 df-sra 21323 df-rgmod 21324 df-dsmm 21911 df-frlm 21926 df-uvc 21962 df-lindf 21985 df-linds 21986 df-dim 34013 df-fldext 34054 df-extdg 34055

This theorem is used by: extdg1b 34080 rtelextdg2 34140

W3C validator