Theorem extdgfialglem1 33852

Description: Lemma for extdgfialg 33854. (Contributed by Thierry Arnoux, 10-Jan-2026.)

Hypotheses

Ref	Expression
extdgfialg.b	⊢ 𝐵 = (Base‘𝐸)
extdgfialg.d	⊢ 𝐷 = (dim‘((subringAlg ‘𝐸)‘𝐹))
extdgfialg.e	⊢ (𝜑 → 𝐸 ∈ Field)
extdgfialg.f	⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))
extdgfialg.1	⊢ (𝜑 → 𝐷 ∈ ℕ0)
extdgfialglem1.2	⊢ 𝑍 = (0g‘𝐸)
extdgfialglem1.3	⊢ · = (.r‘𝐸)
extdgfialglem1.r	⊢ 𝐺 = (𝑛 ∈ (0...𝐷) ↦ (𝑛(.g‘(mulGrp‘((subringAlg ‘𝐸)‘𝐹)))𝑋))
extdgfialglem1.4	⊢ (𝜑 → 𝑋 ∈ 𝐵)

Assertion

Ref	Expression
extdgfialglem1	⊢ (𝜑 → ∃𝑎 ∈ (𝐹 ↑m (0...𝐷))(𝑎 finSupp 𝑍 ∧ ((𝐸 Σg (𝑎 ∘f · 𝐺)) = 𝑍 ∧ 𝑎 ≠ ((0...𝐷) × {𝑍}))))

Distinct variable groups:   · ,𝑛   𝐵,𝑛   𝐷,𝑛   𝑛,𝐸   𝑛,𝐹   𝑛,𝐺   𝑛,𝑋   𝑛,𝑍   𝜑,𝑛   𝐵,𝑎,𝑛   𝐷,𝑎   𝐸,𝑎   𝐹,𝑎   𝜑,𝑎   𝐺,𝑎   𝑋,𝑎

Allowed substitution hints:   · (𝑎)   𝑍(𝑎)

Proof of Theorem extdgfialglem1

Dummy variable 𝑏 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simplr 769	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) ∧ ran 𝐺 ∈ (LIndS‘((subringAlg ‘𝐸)‘𝐹))) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘𝐹))) ∧ ran 𝐺 ⊆ 𝑏) → 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘𝐹)))
2		extdgfialg.e	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐸 ∈ Field)
3	2	flddrngd 20709	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐸 ∈ DivRing)
4		extdgfialg.f	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))
5		eqid 2737	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐸 ↾s 𝐹) = (𝐸 ↾s 𝐹)
6	5	sdrgdrng 20758	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐹 ∈ (SubDRing‘𝐸) → (𝐸 ↾s 𝐹) ∈ DivRing)
7	4, 6	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐸 ↾s 𝐹) ∈ DivRing)
8		sdrgsubrg 20759	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐹 ∈ (SubDRing‘𝐸) → 𝐹 ∈ (SubRing‘𝐸))
9	4, 8	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐹 ∈ (SubRing‘𝐸))
10		eqid 2737	. . . . . . . . . . . . . . . . . . 19 ⊢ ((subringAlg ‘𝐸)‘𝐹) = ((subringAlg ‘𝐸)‘𝐹)
11	10, 5	sralvec 33744	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐸 ∈ DivRing ∧ (𝐸 ↾s 𝐹) ∈ DivRing ∧ 𝐹 ∈ (SubRing‘𝐸)) → ((subringAlg ‘𝐸)‘𝐹) ∈ LVec)
12	3, 7, 9, 11	syl3anc 1374	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((subringAlg ‘𝐸)‘𝐹) ∈ LVec)
13	12	ad2antrr 727	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) ∧ ran 𝐺 ∈ (LIndS‘((subringAlg ‘𝐸)‘𝐹))) → ((subringAlg ‘𝐸)‘𝐹) ∈ LVec)
14	13	ad2antrr 727	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) ∧ ran 𝐺 ∈ (LIndS‘((subringAlg ‘𝐸)‘𝐹))) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘𝐹))) ∧ ran 𝐺 ⊆ 𝑏) → ((subringAlg ‘𝐸)‘𝐹) ∈ LVec)
15		extdgfialg.d	. . . . . . . . . . . . . . . 16 ⊢ 𝐷 = (dim‘((subringAlg ‘𝐸)‘𝐹))
16		eqid 2737	. . . . . . . . . . . . . . . . 17 ⊢ (LBasis‘((subringAlg ‘𝐸)‘𝐹)) = (LBasis‘((subringAlg ‘𝐸)‘𝐹))
17	16	dimval 33760	. . . . . . . . . . . . . . . 16 ⊢ ((((subringAlg ‘𝐸)‘𝐹) ∈ LVec ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘𝐹))) → (dim‘((subringAlg ‘𝐸)‘𝐹)) = (♯‘𝑏))
18	15, 17	eqtrid 2784	. . . . . . . . . . . . . . 15 ⊢ ((((subringAlg ‘𝐸)‘𝐹) ∈ LVec ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘𝐹))) → 𝐷 = (♯‘𝑏))
19	14, 1, 18	syl2anc 585	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) ∧ ran 𝐺 ∈ (LIndS‘((subringAlg ‘𝐸)‘𝐹))) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘𝐹))) ∧ ran 𝐺 ⊆ 𝑏) → 𝐷 = (♯‘𝑏))
20		extdgfialg.1	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐷 ∈ ℕ0)
21	20	ad4antr 733	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) ∧ ran 𝐺 ∈ (LIndS‘((subringAlg ‘𝐸)‘𝐹))) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘𝐹))) ∧ ran 𝐺 ⊆ 𝑏) → 𝐷 ∈ ℕ0)
22	19, 21	eqeltrrd 2838	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) ∧ ran 𝐺 ∈ (LIndS‘((subringAlg ‘𝐸)‘𝐹))) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘𝐹))) ∧ ran 𝐺 ⊆ 𝑏) → (♯‘𝑏) ∈ ℕ0)
23		hashclb 14311	. . . . . . . . . . . . . 14 ⊢ (𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘𝐹)) → (𝑏 ∈ Fin ↔ (♯‘𝑏) ∈ ℕ0))
24	23	biimpar 477	. . . . . . . . . . . . 13 ⊢ ((𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘𝐹)) ∧ (♯‘𝑏) ∈ ℕ0) → 𝑏 ∈ Fin)
25	1, 22, 24	syl2anc 585	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) ∧ ran 𝐺 ∈ (LIndS‘((subringAlg ‘𝐸)‘𝐹))) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘𝐹))) ∧ ran 𝐺 ⊆ 𝑏) → 𝑏 ∈ Fin)
26		hashss 14362	. . . . . . . . . . . 12 ⊢ ((𝑏 ∈ Fin ∧ ran 𝐺 ⊆ 𝑏) → (♯‘ran 𝐺) ≤ (♯‘𝑏))
27	25, 26	sylancom 589	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) ∧ ran 𝐺 ∈ (LIndS‘((subringAlg ‘𝐸)‘𝐹))) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘𝐹))) ∧ ran 𝐺 ⊆ 𝑏) → (♯‘ran 𝐺) ≤ (♯‘𝑏))
28		extdgfialglem1.r	. . . . . . . . . . . . . . 15 ⊢ 𝐺 = (𝑛 ∈ (0...𝐷) ↦ (𝑛(.g‘(mulGrp‘((subringAlg ‘𝐸)‘𝐹)))𝑋))
29	28	dmeqi 5853	. . . . . . . . . . . . . 14 ⊢ dom 𝐺 = dom (𝑛 ∈ (0...𝐷) ↦ (𝑛(.g‘(mulGrp‘((subringAlg ‘𝐸)‘𝐹)))𝑋))
30		eqid 2737	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ (0...𝐷) ↦ (𝑛(.g‘(mulGrp‘((subringAlg ‘𝐸)‘𝐹)))𝑋)) = (𝑛 ∈ (0...𝐷) ↦ (𝑛(.g‘(mulGrp‘((subringAlg ‘𝐸)‘𝐹)))𝑋))
31		ovexd 7395	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) ∧ 𝑛 ∈ (0...𝐷)) → (𝑛(.g‘(mulGrp‘((subringAlg ‘𝐸)‘𝐹)))𝑋) ∈ V)
32	30, 31	dmmptd 6637	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) → dom (𝑛 ∈ (0...𝐷) ↦ (𝑛(.g‘(mulGrp‘((subringAlg ‘𝐸)‘𝐹)))𝑋)) = (0...𝐷))
33		ovexd 7395	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) → (0...𝐷) ∈ V)
34	32, 33	eqeltrd 2837	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) → dom (𝑛 ∈ (0...𝐷) ↦ (𝑛(.g‘(mulGrp‘((subringAlg ‘𝐸)‘𝐹)))𝑋)) ∈ V)
35	29, 34	eqeltrid 2841	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) → dom 𝐺 ∈ V)
36		hashf1rn 14305	. . . . . . . . . . . . 13 ⊢ ((dom 𝐺 ∈ V ∧ 𝐺:dom 𝐺–1-1→V) → (♯‘𝐺) = (♯‘ran 𝐺))
37	35, 36	sylancom 589	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) → (♯‘𝐺) = (♯‘ran 𝐺))
38	37	ad3antrrr 731	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) ∧ ran 𝐺 ∈ (LIndS‘((subringAlg ‘𝐸)‘𝐹))) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘𝐹))) ∧ ran 𝐺 ⊆ 𝑏) → (♯‘𝐺) = (♯‘ran 𝐺))
39	27, 38, 19	3brtr4d 5118	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) ∧ ran 𝐺 ∈ (LIndS‘((subringAlg ‘𝐸)‘𝐹))) ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘𝐹))) ∧ ran 𝐺 ⊆ 𝑏) → (♯‘𝐺) ≤ 𝐷)
40	16	islinds4 21825	. . . . . . . . . . . 12 ⊢ (((subringAlg ‘𝐸)‘𝐹) ∈ LVec → (ran 𝐺 ∈ (LIndS‘((subringAlg ‘𝐸)‘𝐹)) ↔ ∃𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘𝐹))ran 𝐺 ⊆ 𝑏))
41	40	biimpa 476	. . . . . . . . . . 11 ⊢ ((((subringAlg ‘𝐸)‘𝐹) ∈ LVec ∧ ran 𝐺 ∈ (LIndS‘((subringAlg ‘𝐸)‘𝐹))) → ∃𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘𝐹))ran 𝐺 ⊆ 𝑏)
42	13, 41	sylancom 589	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) ∧ ran 𝐺 ∈ (LIndS‘((subringAlg ‘𝐸)‘𝐹))) → ∃𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘𝐹))ran 𝐺 ⊆ 𝑏)
43	39, 42	r19.29a 3146	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) ∧ ran 𝐺 ∈ (LIndS‘((subringAlg ‘𝐸)‘𝐹))) → (♯‘𝐺) ≤ 𝐷)
44	20	nn0red 12490	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐷 ∈ ℝ)
45	44	ad2antrr 727	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) ∧ ran 𝐺 ∈ (LIndS‘((subringAlg ‘𝐸)‘𝐹))) → 𝐷 ∈ ℝ)
46	45	ltp1d 12077	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) ∧ ran 𝐺 ∈ (LIndS‘((subringAlg ‘𝐸)‘𝐹))) → 𝐷 < (𝐷 + 1))
47		fzfid 13926	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (0...𝐷) ∈ Fin)
48	47	mptexd 7172	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑛 ∈ (0...𝐷) ↦ (𝑛(.g‘(mulGrp‘((subringAlg ‘𝐸)‘𝐹)))𝑋)) ∈ V)
49	28, 48	eqeltrid 2841	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐺 ∈ V)
50	49	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) → 𝐺 ∈ V)
51		f1f 6730	. . . . . . . . . . . . . . . 16 ⊢ (𝐺:dom 𝐺–1-1→V → 𝐺:dom 𝐺⟶V)
52	51	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) → 𝐺:dom 𝐺⟶V)
53	52	ffund 6666	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) → Fun 𝐺)
54		hashfundm 14395	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ V ∧ Fun 𝐺) → (♯‘𝐺) = (♯‘dom 𝐺))
55	50, 53, 54	syl2anc 585	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) → (♯‘𝐺) = (♯‘dom 𝐺))
56	28, 31	dmmptd 6637	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) → dom 𝐺 = (0...𝐷))
57	56	fveq2d 6838	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) → (♯‘dom 𝐺) = (♯‘(0...𝐷)))
58		hashfz0 14385	. . . . . . . . . . . . . . 15 ⊢ (𝐷 ∈ ℕ0 → (♯‘(0...𝐷)) = (𝐷 + 1))
59	20, 58	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (♯‘(0...𝐷)) = (𝐷 + 1))
60	59	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) → (♯‘(0...𝐷)) = (𝐷 + 1))
61	55, 57, 60	3eqtrd 2776	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) → (♯‘𝐺) = (𝐷 + 1))
62	61	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) ∧ ran 𝐺 ∈ (LIndS‘((subringAlg ‘𝐸)‘𝐹))) → (♯‘𝐺) = (𝐷 + 1))
63	46, 62	breqtrrd 5114	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) ∧ ran 𝐺 ∈ (LIndS‘((subringAlg ‘𝐸)‘𝐹))) → 𝐷 < (♯‘𝐺))
64	45	rexrd 11186	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) ∧ ran 𝐺 ∈ (LIndS‘((subringAlg ‘𝐸)‘𝐹))) → 𝐷 ∈ ℝ*)
65	50	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) ∧ ran 𝐺 ∈ (LIndS‘((subringAlg ‘𝐸)‘𝐹))) → 𝐺 ∈ V)
66		hashxrcl 14310	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ V → (♯‘𝐺) ∈ ℝ*)
67	65, 66	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) ∧ ran 𝐺 ∈ (LIndS‘((subringAlg ‘𝐸)‘𝐹))) → (♯‘𝐺) ∈ ℝ*)
68	64, 67	xrltnled 11204	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) ∧ ran 𝐺 ∈ (LIndS‘((subringAlg ‘𝐸)‘𝐹))) → (𝐷 < (♯‘𝐺) ↔ ¬ (♯‘𝐺) ≤ 𝐷))
69	63, 68	mpbid 232	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) ∧ ran 𝐺 ∈ (LIndS‘((subringAlg ‘𝐸)‘𝐹))) → ¬ (♯‘𝐺) ≤ 𝐷)
70	43, 69	pm2.65da 817	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) → ¬ ran 𝐺 ∈ (LIndS‘((subringAlg ‘𝐸)‘𝐹)))
71	70	ex 412	. . . . . . 7 ⊢ (𝜑 → (𝐺:dom 𝐺–1-1→V → ¬ ran 𝐺 ∈ (LIndS‘((subringAlg ‘𝐸)‘𝐹))))
72		imnan 399	. . . . . . 7 ⊢ ((𝐺:dom 𝐺–1-1→V → ¬ ran 𝐺 ∈ (LIndS‘((subringAlg ‘𝐸)‘𝐹))) ↔ ¬ (𝐺:dom 𝐺–1-1→V ∧ ran 𝐺 ∈ (LIndS‘((subringAlg ‘𝐸)‘𝐹))))
73	71, 72	sylib 218	. . . . . 6 ⊢ (𝜑 → ¬ (𝐺:dom 𝐺–1-1→V ∧ ran 𝐺 ∈ (LIndS‘((subringAlg ‘𝐸)‘𝐹))))
74	12	lveclmodd 21094	. . . . . . 7 ⊢ (𝜑 → ((subringAlg ‘𝐸)‘𝐹) ∈ LMod)
75		eqidd 2738	. . . . . . . . 9 ⊢ (𝜑 → ((subringAlg ‘𝐸)‘𝐹) = ((subringAlg ‘𝐸)‘𝐹))
76		extdgfialg.b	. . . . . . . . . . . 12 ⊢ 𝐵 = (Base‘𝐸)
77	76	sdrgss 20761	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (SubDRing‘𝐸) → 𝐹 ⊆ 𝐵)
78	4, 77	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ⊆ 𝐵)
79	78, 76	sseqtrdi 3963	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ⊆ (Base‘𝐸))
80	75, 79	srasca 21167	. . . . . . . 8 ⊢ (𝜑 → (𝐸 ↾s 𝐹) = (Scalar‘((subringAlg ‘𝐸)‘𝐹)))
81		drngnzr 20716	. . . . . . . . 9 ⊢ ((𝐸 ↾s 𝐹) ∈ DivRing → (𝐸 ↾s 𝐹) ∈ NzRing)
82	7, 81	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝐸 ↾s 𝐹) ∈ NzRing)
83	80, 82	eqeltrrd 2838	. . . . . . 7 ⊢ (𝜑 → (Scalar‘((subringAlg ‘𝐸)‘𝐹)) ∈ NzRing)
84		eqid 2737	. . . . . . . 8 ⊢ (Scalar‘((subringAlg ‘𝐸)‘𝐹)) = (Scalar‘((subringAlg ‘𝐸)‘𝐹))
85	84	islindf3 21816	. . . . . . 7 ⊢ ((((subringAlg ‘𝐸)‘𝐹) ∈ LMod ∧ (Scalar‘((subringAlg ‘𝐸)‘𝐹)) ∈ NzRing) → (𝐺 LIndF ((subringAlg ‘𝐸)‘𝐹) ↔ (𝐺:dom 𝐺–1-1→V ∧ ran 𝐺 ∈ (LIndS‘((subringAlg ‘𝐸)‘𝐹)))))
86	74, 83, 85	syl2anc 585	. . . . . 6 ⊢ (𝜑 → (𝐺 LIndF ((subringAlg ‘𝐸)‘𝐹) ↔ (𝐺:dom 𝐺–1-1→V ∧ ran 𝐺 ∈ (LIndS‘((subringAlg ‘𝐸)‘𝐹)))))
87	73, 86	mtbird 325	. . . . 5 ⊢ (𝜑 → ¬ 𝐺 LIndF ((subringAlg ‘𝐸)‘𝐹))
88		ovexd 7395	. . . . . 6 ⊢ (𝜑 → (0...𝐷) ∈ V)
89		eqid 2737	. . . . . . . . 9 ⊢ (mulGrp‘((subringAlg ‘𝐸)‘𝐹)) = (mulGrp‘((subringAlg ‘𝐸)‘𝐹))
90		eqid 2737	. . . . . . . . 9 ⊢ (Base‘((subringAlg ‘𝐸)‘𝐹)) = (Base‘((subringAlg ‘𝐸)‘𝐹))
91	89, 90	mgpbas 20117	. . . . . . . 8 ⊢ (Base‘((subringAlg ‘𝐸)‘𝐹)) = (Base‘(mulGrp‘((subringAlg ‘𝐸)‘𝐹)))
92		eqid 2737	. . . . . . . 8 ⊢ (.g‘(mulGrp‘((subringAlg ‘𝐸)‘𝐹))) = (.g‘(mulGrp‘((subringAlg ‘𝐸)‘𝐹)))
93	2	fldcrngd 20710	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐸 ∈ CRing)
94	93	crngringd 20218	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ∈ Ring)
95	10, 76	sraring 21173	. . . . . . . . . . 11 ⊢ ((𝐸 ∈ Ring ∧ 𝐹 ⊆ 𝐵) → ((subringAlg ‘𝐸)‘𝐹) ∈ Ring)
96	94, 78, 95	syl2anc 585	. . . . . . . . . 10 ⊢ (𝜑 → ((subringAlg ‘𝐸)‘𝐹) ∈ Ring)
97	89	ringmgp 20211	. . . . . . . . . 10 ⊢ (((subringAlg ‘𝐸)‘𝐹) ∈ Ring → (mulGrp‘((subringAlg ‘𝐸)‘𝐹)) ∈ Mnd)
98	96, 97	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (mulGrp‘((subringAlg ‘𝐸)‘𝐹)) ∈ Mnd)
99	98	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...𝐷)) → (mulGrp‘((subringAlg ‘𝐸)‘𝐹)) ∈ Mnd)
100		fz0ssnn0 13567	. . . . . . . . . 10 ⊢ (0...𝐷) ⊆ ℕ0
101	100	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → (0...𝐷) ⊆ ℕ0)
102	101	sselda 3922	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...𝐷)) → 𝑛 ∈ ℕ0)
103		extdgfialglem1.4	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
104	75, 79	srabase 21164	. . . . . . . . . . 11 ⊢ (𝜑 → (Base‘𝐸) = (Base‘((subringAlg ‘𝐸)‘𝐹)))
105	76, 104	eqtr2id 2785	. . . . . . . . . 10 ⊢ (𝜑 → (Base‘((subringAlg ‘𝐸)‘𝐹)) = 𝐵)
106	103, 105	eleqtrrd 2840	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ (Base‘((subringAlg ‘𝐸)‘𝐹)))
107	106	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...𝐷)) → 𝑋 ∈ (Base‘((subringAlg ‘𝐸)‘𝐹)))
108	91, 92, 99, 102, 107	mulgnn0cld 19062	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...𝐷)) → (𝑛(.g‘(mulGrp‘((subringAlg ‘𝐸)‘𝐹)))𝑋) ∈ (Base‘((subringAlg ‘𝐸)‘𝐹)))
109	108, 28	fmptd 7060	. . . . . 6 ⊢ (𝜑 → 𝐺:(0...𝐷)⟶(Base‘((subringAlg ‘𝐸)‘𝐹)))
110		eqid 2737	. . . . . . 7 ⊢ ( ·𝑠 ‘((subringAlg ‘𝐸)‘𝐹)) = ( ·𝑠 ‘((subringAlg ‘𝐸)‘𝐹))
111		eqid 2737	. . . . . . 7 ⊢ (0g‘((subringAlg ‘𝐸)‘𝐹)) = (0g‘((subringAlg ‘𝐸)‘𝐹))
112		eqid 2737	. . . . . . 7 ⊢ (0g‘(Scalar‘((subringAlg ‘𝐸)‘𝐹))) = (0g‘(Scalar‘((subringAlg ‘𝐸)‘𝐹)))
113		eqid 2737	. . . . . . 7 ⊢ (Base‘((Scalar‘((subringAlg ‘𝐸)‘𝐹)) freeLMod (0...𝐷))) = (Base‘((Scalar‘((subringAlg ‘𝐸)‘𝐹)) freeLMod (0...𝐷)))
114	90, 84, 110, 111, 112, 113	islindf4 21828	. . . . . 6 ⊢ ((((subringAlg ‘𝐸)‘𝐹) ∈ LMod ∧ (0...𝐷) ∈ V ∧ 𝐺:(0...𝐷)⟶(Base‘((subringAlg ‘𝐸)‘𝐹))) → (𝐺 LIndF ((subringAlg ‘𝐸)‘𝐹) ↔ ∀𝑎 ∈ (Base‘((Scalar‘((subringAlg ‘𝐸)‘𝐹)) freeLMod (0...𝐷)))((((subringAlg ‘𝐸)‘𝐹) Σg (𝑎 ∘f ( ·𝑠 ‘((subringAlg ‘𝐸)‘𝐹))𝐺)) = (0g‘((subringAlg ‘𝐸)‘𝐹)) → 𝑎 = ((0...𝐷) × {(0g‘(Scalar‘((subringAlg ‘𝐸)‘𝐹)))}))))
115	74, 88, 109, 114	syl3anc 1374	. . . . 5 ⊢ (𝜑 → (𝐺 LIndF ((subringAlg ‘𝐸)‘𝐹) ↔ ∀𝑎 ∈ (Base‘((Scalar‘((subringAlg ‘𝐸)‘𝐹)) freeLMod (0...𝐷)))((((subringAlg ‘𝐸)‘𝐹) Σg (𝑎 ∘f ( ·𝑠 ‘((subringAlg ‘𝐸)‘𝐹))𝐺)) = (0g‘((subringAlg ‘𝐸)‘𝐹)) → 𝑎 = ((0...𝐷) × {(0g‘(Scalar‘((subringAlg ‘𝐸)‘𝐹)))}))))
116	87, 115	mtbid 324	. . . 4 ⊢ (𝜑 → ¬ ∀𝑎 ∈ (Base‘((Scalar‘((subringAlg ‘𝐸)‘𝐹)) freeLMod (0...𝐷)))((((subringAlg ‘𝐸)‘𝐹) Σg (𝑎 ∘f ( ·𝑠 ‘((subringAlg ‘𝐸)‘𝐹))𝐺)) = (0g‘((subringAlg ‘𝐸)‘𝐹)) → 𝑎 = ((0...𝐷) × {(0g‘(Scalar‘((subringAlg ‘𝐸)‘𝐹)))})))
117		rexanali 3092	. . . 4 ⊢ (∃𝑎 ∈ (Base‘((Scalar‘((subringAlg ‘𝐸)‘𝐹)) freeLMod (0...𝐷)))((((subringAlg ‘𝐸)‘𝐹) Σg (𝑎 ∘f ( ·𝑠 ‘((subringAlg ‘𝐸)‘𝐹))𝐺)) = (0g‘((subringAlg ‘𝐸)‘𝐹)) ∧ ¬ 𝑎 = ((0...𝐷) × {(0g‘(Scalar‘((subringAlg ‘𝐸)‘𝐹)))})) ↔ ¬ ∀𝑎 ∈ (Base‘((Scalar‘((subringAlg ‘𝐸)‘𝐹)) freeLMod (0...𝐷)))((((subringAlg ‘𝐸)‘𝐹) Σg (𝑎 ∘f ( ·𝑠 ‘((subringAlg ‘𝐸)‘𝐹))𝐺)) = (0g‘((subringAlg ‘𝐸)‘𝐹)) → 𝑎 = ((0...𝐷) × {(0g‘(Scalar‘((subringAlg ‘𝐸)‘𝐹)))})))
118	116, 117	sylibr 234	. . 3 ⊢ (𝜑 → ∃𝑎 ∈ (Base‘((Scalar‘((subringAlg ‘𝐸)‘𝐹)) freeLMod (0...𝐷)))((((subringAlg ‘𝐸)‘𝐹) Σg (𝑎 ∘f ( ·𝑠 ‘((subringAlg ‘𝐸)‘𝐹))𝐺)) = (0g‘((subringAlg ‘𝐸)‘𝐹)) ∧ ¬ 𝑎 = ((0...𝐷) × {(0g‘(Scalar‘((subringAlg ‘𝐸)‘𝐹)))})))
119		fvex 6847	. . . . . . 7 ⊢ (Scalar‘((subringAlg ‘𝐸)‘𝐹)) ∈ V
120		ovex 7393	. . . . . . 7 ⊢ (0...𝐷) ∈ V
121		eqid 2737	. . . . . . . 8 ⊢ ((Scalar‘((subringAlg ‘𝐸)‘𝐹)) freeLMod (0...𝐷)) = ((Scalar‘((subringAlg ‘𝐸)‘𝐹)) freeLMod (0...𝐷))
122		eqid 2737	. . . . . . . 8 ⊢ (Base‘(Scalar‘((subringAlg ‘𝐸)‘𝐹))) = (Base‘(Scalar‘((subringAlg ‘𝐸)‘𝐹)))
123	121, 122, 112, 113	frlmelbas 21746	. . . . . . 7 ⊢ (((Scalar‘((subringAlg ‘𝐸)‘𝐹)) ∈ V ∧ (0...𝐷) ∈ V) → (𝑎 ∈ (Base‘((Scalar‘((subringAlg ‘𝐸)‘𝐹)) freeLMod (0...𝐷))) ↔ (𝑎 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘𝐹))) ↑m (0...𝐷)) ∧ 𝑎 finSupp (0g‘(Scalar‘((subringAlg ‘𝐸)‘𝐹))))))
124	119, 120, 123	mp2an 693	. . . . . 6 ⊢ (𝑎 ∈ (Base‘((Scalar‘((subringAlg ‘𝐸)‘𝐹)) freeLMod (0...𝐷))) ↔ (𝑎 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘𝐹))) ↑m (0...𝐷)) ∧ 𝑎 finSupp (0g‘(Scalar‘((subringAlg ‘𝐸)‘𝐹)))))
125	124	anbi1i 625	. . . . 5 ⊢ ((𝑎 ∈ (Base‘((Scalar‘((subringAlg ‘𝐸)‘𝐹)) freeLMod (0...𝐷))) ∧ ((((subringAlg ‘𝐸)‘𝐹) Σg (𝑎 ∘f ( ·𝑠 ‘((subringAlg ‘𝐸)‘𝐹))𝐺)) = (0g‘((subringAlg ‘𝐸)‘𝐹)) ∧ 𝑎 ≠ ((0...𝐷) × {(0g‘(Scalar‘((subringAlg ‘𝐸)‘𝐹)))}))) ↔ ((𝑎 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘𝐹))) ↑m (0...𝐷)) ∧ 𝑎 finSupp (0g‘(Scalar‘((subringAlg ‘𝐸)‘𝐹)))) ∧ ((((subringAlg ‘𝐸)‘𝐹) Σg (𝑎 ∘f ( ·𝑠 ‘((subringAlg ‘𝐸)‘𝐹))𝐺)) = (0g‘((subringAlg ‘𝐸)‘𝐹)) ∧ 𝑎 ≠ ((0...𝐷) × {(0g‘(Scalar‘((subringAlg ‘𝐸)‘𝐹)))}))))
126		df-ne 2934	. . . . . . 7 ⊢ (𝑎 ≠ ((0...𝐷) × {(0g‘(Scalar‘((subringAlg ‘𝐸)‘𝐹)))}) ↔ ¬ 𝑎 = ((0...𝐷) × {(0g‘(Scalar‘((subringAlg ‘𝐸)‘𝐹)))}))
127	126	anbi2i 624	. . . . . 6 ⊢ (((((subringAlg ‘𝐸)‘𝐹) Σg (𝑎 ∘f ( ·𝑠 ‘((subringAlg ‘𝐸)‘𝐹))𝐺)) = (0g‘((subringAlg ‘𝐸)‘𝐹)) ∧ 𝑎 ≠ ((0...𝐷) × {(0g‘(Scalar‘((subringAlg ‘𝐸)‘𝐹)))})) ↔ ((((subringAlg ‘𝐸)‘𝐹) Σg (𝑎 ∘f ( ·𝑠 ‘((subringAlg ‘𝐸)‘𝐹))𝐺)) = (0g‘((subringAlg ‘𝐸)‘𝐹)) ∧ ¬ 𝑎 = ((0...𝐷) × {(0g‘(Scalar‘((subringAlg ‘𝐸)‘𝐹)))})))
128	127	anbi2i 624	. . . . 5 ⊢ ((𝑎 ∈ (Base‘((Scalar‘((subringAlg ‘𝐸)‘𝐹)) freeLMod (0...𝐷))) ∧ ((((subringAlg ‘𝐸)‘𝐹) Σg (𝑎 ∘f ( ·𝑠 ‘((subringAlg ‘𝐸)‘𝐹))𝐺)) = (0g‘((subringAlg ‘𝐸)‘𝐹)) ∧ 𝑎 ≠ ((0...𝐷) × {(0g‘(Scalar‘((subringAlg ‘𝐸)‘𝐹)))}))) ↔ (𝑎 ∈ (Base‘((Scalar‘((subringAlg ‘𝐸)‘𝐹)) freeLMod (0...𝐷))) ∧ ((((subringAlg ‘𝐸)‘𝐹) Σg (𝑎 ∘f ( ·𝑠 ‘((subringAlg ‘𝐸)‘𝐹))𝐺)) = (0g‘((subringAlg ‘𝐸)‘𝐹)) ∧ ¬ 𝑎 = ((0...𝐷) × {(0g‘(Scalar‘((subringAlg ‘𝐸)‘𝐹)))}))))
129		anass 468	. . . . 5 ⊢ (((𝑎 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘𝐹))) ↑m (0...𝐷)) ∧ 𝑎 finSupp (0g‘(Scalar‘((subringAlg ‘𝐸)‘𝐹)))) ∧ ((((subringAlg ‘𝐸)‘𝐹) Σg (𝑎 ∘f ( ·𝑠 ‘((subringAlg ‘𝐸)‘𝐹))𝐺)) = (0g‘((subringAlg ‘𝐸)‘𝐹)) ∧ 𝑎 ≠ ((0...𝐷) × {(0g‘(Scalar‘((subringAlg ‘𝐸)‘𝐹)))}))) ↔ (𝑎 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘𝐹))) ↑m (0...𝐷)) ∧ (𝑎 finSupp (0g‘(Scalar‘((subringAlg ‘𝐸)‘𝐹))) ∧ ((((subringAlg ‘𝐸)‘𝐹) Σg (𝑎 ∘f ( ·𝑠 ‘((subringAlg ‘𝐸)‘𝐹))𝐺)) = (0g‘((subringAlg ‘𝐸)‘𝐹)) ∧ 𝑎 ≠ ((0...𝐷) × {(0g‘(Scalar‘((subringAlg ‘𝐸)‘𝐹)))})))))
130	125, 128, 129	3bitr3i 301	. . . 4 ⊢ ((𝑎 ∈ (Base‘((Scalar‘((subringAlg ‘𝐸)‘𝐹)) freeLMod (0...𝐷))) ∧ ((((subringAlg ‘𝐸)‘𝐹) Σg (𝑎 ∘f ( ·𝑠 ‘((subringAlg ‘𝐸)‘𝐹))𝐺)) = (0g‘((subringAlg ‘𝐸)‘𝐹)) ∧ ¬ 𝑎 = ((0...𝐷) × {(0g‘(Scalar‘((subringAlg ‘𝐸)‘𝐹)))}))) ↔ (𝑎 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘𝐹))) ↑m (0...𝐷)) ∧ (𝑎 finSupp (0g‘(Scalar‘((subringAlg ‘𝐸)‘𝐹))) ∧ ((((subringAlg ‘𝐸)‘𝐹) Σg (𝑎 ∘f ( ·𝑠 ‘((subringAlg ‘𝐸)‘𝐹))𝐺)) = (0g‘((subringAlg ‘𝐸)‘𝐹)) ∧ 𝑎 ≠ ((0...𝐷) × {(0g‘(Scalar‘((subringAlg ‘𝐸)‘𝐹)))})))))
131	130	rexbii2 3081	. . 3 ⊢ (∃𝑎 ∈ (Base‘((Scalar‘((subringAlg ‘𝐸)‘𝐹)) freeLMod (0...𝐷)))((((subringAlg ‘𝐸)‘𝐹) Σg (𝑎 ∘f ( ·𝑠 ‘((subringAlg ‘𝐸)‘𝐹))𝐺)) = (0g‘((subringAlg ‘𝐸)‘𝐹)) ∧ ¬ 𝑎 = ((0...𝐷) × {(0g‘(Scalar‘((subringAlg ‘𝐸)‘𝐹)))})) ↔ ∃𝑎 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘𝐹))) ↑m (0...𝐷))(𝑎 finSupp (0g‘(Scalar‘((subringAlg ‘𝐸)‘𝐹))) ∧ ((((subringAlg ‘𝐸)‘𝐹) Σg (𝑎 ∘f ( ·𝑠 ‘((subringAlg ‘𝐸)‘𝐹))𝐺)) = (0g‘((subringAlg ‘𝐸)‘𝐹)) ∧ 𝑎 ≠ ((0...𝐷) × {(0g‘(Scalar‘((subringAlg ‘𝐸)‘𝐹)))}))))
132	118, 131	sylib 218	. 2 ⊢ (𝜑 → ∃𝑎 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘𝐹))) ↑m (0...𝐷))(𝑎 finSupp (0g‘(Scalar‘((subringAlg ‘𝐸)‘𝐹))) ∧ ((((subringAlg ‘𝐸)‘𝐹) Σg (𝑎 ∘f ( ·𝑠 ‘((subringAlg ‘𝐸)‘𝐹))𝐺)) = (0g‘((subringAlg ‘𝐸)‘𝐹)) ∧ 𝑎 ≠ ((0...𝐷) × {(0g‘(Scalar‘((subringAlg ‘𝐸)‘𝐹)))}))))
133	5, 76	ressbas2 17199	. . . . . 6 ⊢ (𝐹 ⊆ 𝐵 → 𝐹 = (Base‘(𝐸 ↾s 𝐹)))
134	78, 133	syl 17	. . . . 5 ⊢ (𝜑 → 𝐹 = (Base‘(𝐸 ↾s 𝐹)))
135	80	fveq2d 6838	. . . . 5 ⊢ (𝜑 → (Base‘(𝐸 ↾s 𝐹)) = (Base‘(Scalar‘((subringAlg ‘𝐸)‘𝐹))))
136	134, 135	eqtr2d 2773	. . . 4 ⊢ (𝜑 → (Base‘(Scalar‘((subringAlg ‘𝐸)‘𝐹))) = 𝐹)
137	136	oveq1d 7375	. . 3 ⊢ (𝜑 → ((Base‘(Scalar‘((subringAlg ‘𝐸)‘𝐹))) ↑m (0...𝐷)) = (𝐹 ↑m (0...𝐷)))
138	93	crnggrpd 20219	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ Grp)
139	138	grpmndd 18913	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ Mnd)
140		subrgsubg 20545	. . . . . . . . . 10 ⊢ (𝐹 ∈ (SubRing‘𝐸) → 𝐹 ∈ (SubGrp‘𝐸))
141	9, 140	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ (SubGrp‘𝐸))
142		eqid 2737	. . . . . . . . . 10 ⊢ (0g‘𝐸) = (0g‘𝐸)
143	142	subg0cl 19101	. . . . . . . . 9 ⊢ (𝐹 ∈ (SubGrp‘𝐸) → (0g‘𝐸) ∈ 𝐹)
144	141, 143	syl 17	. . . . . . . 8 ⊢ (𝜑 → (0g‘𝐸) ∈ 𝐹)
145	5, 76, 142	ress0g 18721	. . . . . . . 8 ⊢ ((𝐸 ∈ Mnd ∧ (0g‘𝐸) ∈ 𝐹 ∧ 𝐹 ⊆ 𝐵) → (0g‘𝐸) = (0g‘(𝐸 ↾s 𝐹)))
146	139, 144, 78, 145	syl3anc 1374	. . . . . . 7 ⊢ (𝜑 → (0g‘𝐸) = (0g‘(𝐸 ↾s 𝐹)))
147	80	fveq2d 6838	. . . . . . 7 ⊢ (𝜑 → (0g‘(𝐸 ↾s 𝐹)) = (0g‘(Scalar‘((subringAlg ‘𝐸)‘𝐹))))
148	146, 147	eqtr2d 2773	. . . . . 6 ⊢ (𝜑 → (0g‘(Scalar‘((subringAlg ‘𝐸)‘𝐹))) = (0g‘𝐸))
149		extdgfialglem1.2	. . . . . 6 ⊢ 𝑍 = (0g‘𝐸)
150	148, 149	eqtr4di 2790	. . . . 5 ⊢ (𝜑 → (0g‘(Scalar‘((subringAlg ‘𝐸)‘𝐹))) = 𝑍)
151	150	breq2d 5098	. . . 4 ⊢ (𝜑 → (𝑎 finSupp (0g‘(Scalar‘((subringAlg ‘𝐸)‘𝐹))) ↔ 𝑎 finSupp 𝑍))
152		extdgfialglem1.3	. . . . . . . . . . 11 ⊢ · = (.r‘𝐸)
153	75, 79	sravsca 21168	. . . . . . . . . . 11 ⊢ (𝜑 → (.r‘𝐸) = ( ·𝑠 ‘((subringAlg ‘𝐸)‘𝐹)))
154	152, 153	eqtr2id 2785	. . . . . . . . . 10 ⊢ (𝜑 → ( ·𝑠 ‘((subringAlg ‘𝐸)‘𝐹)) = · )
155	154	ofeqd 7626	. . . . . . . . 9 ⊢ (𝜑 → ∘f ( ·𝑠 ‘((subringAlg ‘𝐸)‘𝐹)) = ∘f · )
156	155	oveqd 7377	. . . . . . . 8 ⊢ (𝜑 → (𝑎 ∘f ( ·𝑠 ‘((subringAlg ‘𝐸)‘𝐹))𝐺) = (𝑎 ∘f · 𝐺))
157	156	oveq2d 7376	. . . . . . 7 ⊢ (𝜑 → (((subringAlg ‘𝐸)‘𝐹) Σg (𝑎 ∘f ( ·𝑠 ‘((subringAlg ‘𝐸)‘𝐹))𝐺)) = (((subringAlg ‘𝐸)‘𝐹) Σg (𝑎 ∘f · 𝐺)))
158		ovexd 7395	. . . . . . . 8 ⊢ (𝜑 → (𝑎 ∘f · 𝐺) ∈ V)
159	10, 158, 2, 12, 79	gsumsra 33123	. . . . . . 7 ⊢ (𝜑 → (𝐸 Σg (𝑎 ∘f · 𝐺)) = (((subringAlg ‘𝐸)‘𝐹) Σg (𝑎 ∘f · 𝐺)))
160	157, 159	eqtr4d 2775	. . . . . 6 ⊢ (𝜑 → (((subringAlg ‘𝐸)‘𝐹) Σg (𝑎 ∘f ( ·𝑠 ‘((subringAlg ‘𝐸)‘𝐹))𝐺)) = (𝐸 Σg (𝑎 ∘f · 𝐺)))
161	149	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 𝑍 = (0g‘𝐸))
162	75, 161, 79	sralmod0 21175	. . . . . . 7 ⊢ (𝜑 → 𝑍 = (0g‘((subringAlg ‘𝐸)‘𝐹)))
163	162	eqcomd 2743	. . . . . 6 ⊢ (𝜑 → (0g‘((subringAlg ‘𝐸)‘𝐹)) = 𝑍)
164	160, 163	eqeq12d 2753	. . . . 5 ⊢ (𝜑 → ((((subringAlg ‘𝐸)‘𝐹) Σg (𝑎 ∘f ( ·𝑠 ‘((subringAlg ‘𝐸)‘𝐹))𝐺)) = (0g‘((subringAlg ‘𝐸)‘𝐹)) ↔ (𝐸 Σg (𝑎 ∘f · 𝐺)) = 𝑍))
165	150	sneqd 4580	. . . . . . 7 ⊢ (𝜑 → {(0g‘(Scalar‘((subringAlg ‘𝐸)‘𝐹)))} = {𝑍})
166	165	xpeq2d 5654	. . . . . 6 ⊢ (𝜑 → ((0...𝐷) × {(0g‘(Scalar‘((subringAlg ‘𝐸)‘𝐹)))}) = ((0...𝐷) × {𝑍}))
167	166	neeq2d 2993	. . . . 5 ⊢ (𝜑 → (𝑎 ≠ ((0...𝐷) × {(0g‘(Scalar‘((subringAlg ‘𝐸)‘𝐹)))}) ↔ 𝑎 ≠ ((0...𝐷) × {𝑍})))
168	164, 167	anbi12d 633	. . . 4 ⊢ (𝜑 → (((((subringAlg ‘𝐸)‘𝐹) Σg (𝑎 ∘f ( ·𝑠 ‘((subringAlg ‘𝐸)‘𝐹))𝐺)) = (0g‘((subringAlg ‘𝐸)‘𝐹)) ∧ 𝑎 ≠ ((0...𝐷) × {(0g‘(Scalar‘((subringAlg ‘𝐸)‘𝐹)))})) ↔ ((𝐸 Σg (𝑎 ∘f · 𝐺)) = 𝑍 ∧ 𝑎 ≠ ((0...𝐷) × {𝑍}))))
169	151, 168	anbi12d 633	. . 3 ⊢ (𝜑 → ((𝑎 finSupp (0g‘(Scalar‘((subringAlg ‘𝐸)‘𝐹))) ∧ ((((subringAlg ‘𝐸)‘𝐹) Σg (𝑎 ∘f ( ·𝑠 ‘((subringAlg ‘𝐸)‘𝐹))𝐺)) = (0g‘((subringAlg ‘𝐸)‘𝐹)) ∧ 𝑎 ≠ ((0...𝐷) × {(0g‘(Scalar‘((subringAlg ‘𝐸)‘𝐹)))}))) ↔ (𝑎 finSupp 𝑍 ∧ ((𝐸 Σg (𝑎 ∘f · 𝐺)) = 𝑍 ∧ 𝑎 ≠ ((0...𝐷) × {𝑍})))))
170	137, 169	rexeqbidv 3313	. 2 ⊢ (𝜑 → (∃𝑎 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐸)‘𝐹))) ↑m (0...𝐷))(𝑎 finSupp (0g‘(Scalar‘((subringAlg ‘𝐸)‘𝐹))) ∧ ((((subringAlg ‘𝐸)‘𝐹) Σg (𝑎 ∘f ( ·𝑠 ‘((subringAlg ‘𝐸)‘𝐹))𝐺)) = (0g‘((subringAlg ‘𝐸)‘𝐹)) ∧ 𝑎 ≠ ((0...𝐷) × {(0g‘(Scalar‘((subringAlg ‘𝐸)‘𝐹)))}))) ↔ ∃𝑎 ∈ (𝐹 ↑m (0...𝐷))(𝑎 finSupp 𝑍 ∧ ((𝐸 Σg (𝑎 ∘f · 𝐺)) = 𝑍 ∧ 𝑎 ≠ ((0...𝐷) × {𝑍})))))
171	132, 170	mpbid 232	1 ⊢ (𝜑 → ∃𝑎 ∈ (𝐹 ↑m (0...𝐷))(𝑎 finSupp 𝑍 ∧ ((𝐸 Σg (𝑎 ∘f · 𝐺)) = 𝑍 ∧ 𝑎 ≠ ((0...𝐷) × {𝑍}))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  ∃wrex 3062  Vcvv 3430   ⊆ wss 3890  {csn 4568   class class class wbr 5086   ↦ cmpt 5167   × cxp 5622  dom cdm 5624  ran crn 5625  Fun wfun 6486  ⟶wf 6488  –1-1→wf1 6489  ‘cfv 6492  (class class class)co 7360   ∘_f cof 7622   ↑_m cmap 8766  Fincfn 8886   finSupp cfsupp 9267  ℝcr 11028  0cc0 11029  1c1 11030   + caddc 11032  ℝ^*cxr 11169   < clt 11170   ≤ cle 11171  ℕ₀cn0 12428  ...cfz 13452  ♯chash 14283  Basecbs 17170   ↾_s cress 17191  ._rcmulr 17212  Scalarcsca 17214   ·_𝑠 cvsca 17215  0_gc0g 17393   Σ_g cgsu 17394  Mndcmnd 18693  ._gcmg 19034  SubGrpcsubg 19087  mulGrpcmgp 20112  Ringcrg 20205  NzRingcnzr 20480  SubRingcsubrg 20537  DivRingcdr 20697  Fieldcfield 20698  SubDRingcsdrg 20754  LModclmod 20846  LBasisclbs 21061  LVecclvec 21089  subringAlg csra 21158   freeLMod cfrlm 21736   LIndF clindf 21794  LIndSclinds 21795  dimcldim 33758

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 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-reg 9500  ax-inf2 9553  ax-ac2 10376  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106

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 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-iin 4937  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-of 7624  df-rpss 7670  df-om 7811  df-1st 7935  df-2nd 7936  df-supp 8104  df-tpos 8169  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-1o 8398  df-2o 8399  df-oadd 8402  df-er 8636  df-map 8768  df-ixp 8839  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-fsupp 9268  df-sup 9348  df-oi 9418  df-r1 9679  df-rank 9680  df-dju 9816  df-card 9854  df-acn 9857  df-ac 10029  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12166  df-2 12235  df-3 12236  df-4 12237  df-5 12238  df-6 12239  df-7 12240  df-8 12241  df-9 12242  df-n0 12429  df-xnn0 12502  df-z 12516  df-dec 12636  df-uz 12780  df-fz 13453  df-fzo 13600  df-seq 13955  df-hash 14284  df-struct 17108  df-sets 17125  df-slot 17143  df-ndx 17155  df-base 17171  df-ress 17192  df-plusg 17224  df-mulr 17225  df-sca 17227  df-vsca 17228  df-ip 17229  df-tset 17230  df-ple 17231  df-ocomp 17232  df-ds 17233  df-hom 17235  df-cco 17236  df-0g 17395  df-gsum 17396  df-prds 17401  df-pws 17403  df-mre 17539  df-mrc 17540  df-mri 17541  df-acs 17542  df-proset 18251  df-drs 18252  df-poset 18270  df-ipo 18485  df-mgm 18599  df-sgrp 18678  df-mnd 18694  df-mhm 18742  df-submnd 18743  df-grp 18903  df-minusg 18904  df-sbg 18905  df-mulg 19035  df-subg 19090  df-ghm 19179  df-cntz 19283  df-cmn 19748  df-abl 19749  df-mgp 20113  df-rng 20125  df-ur 20154  df-ring 20207  df-cring 20208  df-oppr 20308  df-dvdsr 20328  df-unit 20329  df-invr 20359  df-nzr 20481  df-subrg 20538  df-drng 20699  df-field 20700  df-sdrg 20755  df-lmod 20848  df-lss 20918  df-lsp 20958  df-lmhm 21009  df-lbs 21062  df-lvec 21090  df-sra 21160  df-rgmod 21161  df-dsmm 21722  df-frlm 21737  df-uvc 21773  df-lindf 21796  df-linds 21797  df-dim 33759

This theorem is referenced by:  extdgfialg  33854