Theorem extdgfialglem1 33726

Description: Lemma for extdgfialg 33728. (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 768	. . . . . . . . . . . . 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 20658	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐸 ∈ DivRing)
4		extdgfialg.f	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))
5		eqid 2733	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐸 ↾s 𝐹) = (𝐸 ↾s 𝐹)
6	5	sdrgdrng 20707	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐹 ∈ (SubDRing‘𝐸) → (𝐸 ↾s 𝐹) ∈ DivRing)
7	4, 6	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐸 ↾s 𝐹) ∈ DivRing)
8		sdrgsubrg 20708	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐹 ∈ (SubDRing‘𝐸) → 𝐹 ∈ (SubRing‘𝐸))
9	4, 8	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐹 ∈ (SubRing‘𝐸))
10		eqid 2733	. . . . . . . . . . . . . . . . . . 19 ⊢ ((subringAlg ‘𝐸)‘𝐹) = ((subringAlg ‘𝐸)‘𝐹)
11	10, 5	sralvec 33618	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐸 ∈ DivRing ∧ (𝐸 ↾s 𝐹) ∈ DivRing ∧ 𝐹 ∈ (SubRing‘𝐸)) → ((subringAlg ‘𝐸)‘𝐹) ∈ LVec)
12	3, 7, 9, 11	syl3anc 1373	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((subringAlg ‘𝐸)‘𝐹) ∈ LVec)
13	12	ad2antrr 726	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) ∧ ran 𝐺 ∈ (LIndS‘((subringAlg ‘𝐸)‘𝐹))) → ((subringAlg ‘𝐸)‘𝐹) ∈ LVec)
14	13	ad2antrr 726	. . . . . . . . . . . . . . 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 2733	. . . . . . . . . . . . . . . . 17 ⊢ (LBasis‘((subringAlg ‘𝐸)‘𝐹)) = (LBasis‘((subringAlg ‘𝐸)‘𝐹))
17	16	dimval 33634	. . . . . . . . . . . . . . . 16 ⊢ ((((subringAlg ‘𝐸)‘𝐹) ∈ LVec ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘𝐹))) → (dim‘((subringAlg ‘𝐸)‘𝐹)) = (♯‘𝑏))
18	15, 17	eqtrid 2780	. . . . . . . . . . . . . . 15 ⊢ ((((subringAlg ‘𝐸)‘𝐹) ∈ LVec ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘𝐹))) → 𝐷 = (♯‘𝑏))
19	14, 1, 18	syl2anc 584	. . . . . . . . . . . . . 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 732	. . . . . . . . . . . . . 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 2834	. . . . . . . . . . . . 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 14267	. . . . . . . . . . . . . 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 584	. . . . . . . . . . . 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 14318	. . . . . . . . . . . 12 ⊢ ((𝑏 ∈ Fin ∧ ran 𝐺 ⊆ 𝑏) → (♯‘ran 𝐺) ≤ (♯‘𝑏))
27	25, 26	sylancom 588	. . . . . . . . . . 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 5848	. . . . . . . . . . . . . 14 ⊢ dom 𝐺 = dom (𝑛 ∈ (0...𝐷) ↦ (𝑛(.g‘(mulGrp‘((subringAlg ‘𝐸)‘𝐹)))𝑋))
30		eqid 2733	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ (0...𝐷) ↦ (𝑛(.g‘(mulGrp‘((subringAlg ‘𝐸)‘𝐹)))𝑋)) = (𝑛 ∈ (0...𝐷) ↦ (𝑛(.g‘(mulGrp‘((subringAlg ‘𝐸)‘𝐹)))𝑋))
31		ovexd 7387	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) ∧ 𝑛 ∈ (0...𝐷)) → (𝑛(.g‘(mulGrp‘((subringAlg ‘𝐸)‘𝐹)))𝑋) ∈ V)
32	30, 31	dmmptd 6631	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) → dom (𝑛 ∈ (0...𝐷) ↦ (𝑛(.g‘(mulGrp‘((subringAlg ‘𝐸)‘𝐹)))𝑋)) = (0...𝐷))
33		ovexd 7387	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) → (0...𝐷) ∈ V)
34	32, 33	eqeltrd 2833	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) → dom (𝑛 ∈ (0...𝐷) ↦ (𝑛(.g‘(mulGrp‘((subringAlg ‘𝐸)‘𝐹)))𝑋)) ∈ V)
35	29, 34	eqeltrid 2837	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) → dom 𝐺 ∈ V)
36		hashf1rn 14261	. . . . . . . . . . . . 13 ⊢ ((dom 𝐺 ∈ V ∧ 𝐺:dom 𝐺–1-1→V) → (♯‘𝐺) = (♯‘ran 𝐺))
37	35, 36	sylancom 588	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) → (♯‘𝐺) = (♯‘ran 𝐺))
38	37	ad3antrrr 730	. . . . . . . . . . 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 5125	. . . . . . . . . 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 21774	. . . . . . . . . . . 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 588	. . . . . . . . . 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 3141	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) ∧ ran 𝐺 ∈ (LIndS‘((subringAlg ‘𝐸)‘𝐹))) → (♯‘𝐺) ≤ 𝐷)
44	20	nn0red 12450	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐷 ∈ ℝ)
45	44	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) ∧ ran 𝐺 ∈ (LIndS‘((subringAlg ‘𝐸)‘𝐹))) → 𝐷 ∈ ℝ)
46	45	ltp1d 12059	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) ∧ ran 𝐺 ∈ (LIndS‘((subringAlg ‘𝐸)‘𝐹))) → 𝐷 < (𝐷 + 1))
47		fzfid 13882	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (0...𝐷) ∈ Fin)
48	47	mptexd 7164	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑛 ∈ (0...𝐷) ↦ (𝑛(.g‘(mulGrp‘((subringAlg ‘𝐸)‘𝐹)))𝑋)) ∈ V)
49	28, 48	eqeltrid 2837	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐺 ∈ V)
50	49	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) → 𝐺 ∈ V)
51		f1f 6724	. . . . . . . . . . . . . . . 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 6660	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) → Fun 𝐺)
54		hashfundm 14351	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ V ∧ Fun 𝐺) → (♯‘𝐺) = (♯‘dom 𝐺))
55	50, 53, 54	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) → (♯‘𝐺) = (♯‘dom 𝐺))
56	28, 31	dmmptd 6631	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) → dom 𝐺 = (0...𝐷))
57	56	fveq2d 6832	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) → (♯‘dom 𝐺) = (♯‘(0...𝐷)))
58		hashfz0 14341	. . . . . . . . . . . . . . 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 2772	. . . . . . . . . . . 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 5121	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) ∧ ran 𝐺 ∈ (LIndS‘((subringAlg ‘𝐸)‘𝐹))) → 𝐷 < (♯‘𝐺))
64	45	rexrd 11169	. . . . . . . . . . 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 14266	. . . . . . . . . . . 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 11187	. . . . . . . . . 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 816	. . . . . . . 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 21043	. . . . . . 7 ⊢ (𝜑 → ((subringAlg ‘𝐸)‘𝐹) ∈ LMod)
75		eqidd 2734	. . . . . . . . 9 ⊢ (𝜑 → ((subringAlg ‘𝐸)‘𝐹) = ((subringAlg ‘𝐸)‘𝐹))
76		extdgfialg.b	. . . . . . . . . . . 12 ⊢ 𝐵 = (Base‘𝐸)
77	76	sdrgss 20710	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (SubDRing‘𝐸) → 𝐹 ⊆ 𝐵)
78	4, 77	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ⊆ 𝐵)
79	78, 76	sseqtrdi 3971	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ⊆ (Base‘𝐸))
80	75, 79	srasca 21116	. . . . . . . 8 ⊢ (𝜑 → (𝐸 ↾s 𝐹) = (Scalar‘((subringAlg ‘𝐸)‘𝐹)))
81		drngnzr 20665	. . . . . . . . 9 ⊢ ((𝐸 ↾s 𝐹) ∈ DivRing → (𝐸 ↾s 𝐹) ∈ NzRing)
82	7, 81	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝐸 ↾s 𝐹) ∈ NzRing)
83	80, 82	eqeltrrd 2834	. . . . . . 7 ⊢ (𝜑 → (Scalar‘((subringAlg ‘𝐸)‘𝐹)) ∈ NzRing)
84		eqid 2733	. . . . . . . 8 ⊢ (Scalar‘((subringAlg ‘𝐸)‘𝐹)) = (Scalar‘((subringAlg ‘𝐸)‘𝐹))
85	84	islindf3 21765	. . . . . . 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 584	. . . . . 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 7387	. . . . . 6 ⊢ (𝜑 → (0...𝐷) ∈ V)
89		eqid 2733	. . . . . . . . 9 ⊢ (mulGrp‘((subringAlg ‘𝐸)‘𝐹)) = (mulGrp‘((subringAlg ‘𝐸)‘𝐹))
90		eqid 2733	. . . . . . . . 9 ⊢ (Base‘((subringAlg ‘𝐸)‘𝐹)) = (Base‘((subringAlg ‘𝐸)‘𝐹))
91	89, 90	mgpbas 20065	. . . . . . . 8 ⊢ (Base‘((subringAlg ‘𝐸)‘𝐹)) = (Base‘(mulGrp‘((subringAlg ‘𝐸)‘𝐹)))
92		eqid 2733	. . . . . . . 8 ⊢ (.g‘(mulGrp‘((subringAlg ‘𝐸)‘𝐹))) = (.g‘(mulGrp‘((subringAlg ‘𝐸)‘𝐹)))
93	2	fldcrngd 20659	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐸 ∈ CRing)
94	93	crngringd 20166	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ∈ Ring)
95	10, 76	sraring 21122	. . . . . . . . . . 11 ⊢ ((𝐸 ∈ Ring ∧ 𝐹 ⊆ 𝐵) → ((subringAlg ‘𝐸)‘𝐹) ∈ Ring)
96	94, 78, 95	syl2anc 584	. . . . . . . . . 10 ⊢ (𝜑 → ((subringAlg ‘𝐸)‘𝐹) ∈ Ring)
97	89	ringmgp 20159	. . . . . . . . . 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 13524	. . . . . . . . . 10 ⊢ (0...𝐷) ⊆ ℕ0
101	100	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → (0...𝐷) ⊆ ℕ0)
102	101	sselda 3930	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...𝐷)) → 𝑛 ∈ ℕ0)
103		extdgfialglem1.4	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
104	75, 79	srabase 21113	. . . . . . . . . . 11 ⊢ (𝜑 → (Base‘𝐸) = (Base‘((subringAlg ‘𝐸)‘𝐹)))
105	76, 104	eqtr2id 2781	. . . . . . . . . 10 ⊢ (𝜑 → (Base‘((subringAlg ‘𝐸)‘𝐹)) = 𝐵)
106	103, 105	eleqtrrd 2836	. . . . . . . . 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 19010	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...𝐷)) → (𝑛(.g‘(mulGrp‘((subringAlg ‘𝐸)‘𝐹)))𝑋) ∈ (Base‘((subringAlg ‘𝐸)‘𝐹)))
109	108, 28	fmptd 7053	. . . . . 6 ⊢ (𝜑 → 𝐺:(0...𝐷)⟶(Base‘((subringAlg ‘𝐸)‘𝐹)))
110		eqid 2733	. . . . . . 7 ⊢ ( ·𝑠 ‘((subringAlg ‘𝐸)‘𝐹)) = ( ·𝑠 ‘((subringAlg ‘𝐸)‘𝐹))
111		eqid 2733	. . . . . . 7 ⊢ (0g‘((subringAlg ‘𝐸)‘𝐹)) = (0g‘((subringAlg ‘𝐸)‘𝐹))
112		eqid 2733	. . . . . . 7 ⊢ (0g‘(Scalar‘((subringAlg ‘𝐸)‘𝐹))) = (0g‘(Scalar‘((subringAlg ‘𝐸)‘𝐹)))
113		eqid 2733	. . . . . . 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 21777	. . . . . 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 1373	. . . . 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 3087	. . . 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 6841	. . . . . . 7 ⊢ (Scalar‘((subringAlg ‘𝐸)‘𝐹)) ∈ V
120		ovex 7385	. . . . . . 7 ⊢ (0...𝐷) ∈ V
121		eqid 2733	. . . . . . . 8 ⊢ ((Scalar‘((subringAlg ‘𝐸)‘𝐹)) freeLMod (0...𝐷)) = ((Scalar‘((subringAlg ‘𝐸)‘𝐹)) freeLMod (0...𝐷))
122		eqid 2733	. . . . . . . 8 ⊢ (Base‘(Scalar‘((subringAlg ‘𝐸)‘𝐹))) = (Base‘(Scalar‘((subringAlg ‘𝐸)‘𝐹)))
123	121, 122, 112, 113	frlmelbas 21695	. . . . . . 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 692	. . . . . 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 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))) ↑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 2930	. . . . . . 7 ⊢ (𝑎 ≠ ((0...𝐷) × {(0g‘(Scalar‘((subringAlg ‘𝐸)‘𝐹)))}) ↔ ¬ 𝑎 = ((0...𝐷) × {(0g‘(Scalar‘((subringAlg ‘𝐸)‘𝐹)))}))
127	126	anbi2i 623	. . . . . 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 623	. . . . 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 3076	. . 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 17151	. . . . . 6 ⊢ (𝐹 ⊆ 𝐵 → 𝐹 = (Base‘(𝐸 ↾s 𝐹)))
134	78, 133	syl 17	. . . . 5 ⊢ (𝜑 → 𝐹 = (Base‘(𝐸 ↾s 𝐹)))
135	80	fveq2d 6832	. . . . 5 ⊢ (𝜑 → (Base‘(𝐸 ↾s 𝐹)) = (Base‘(Scalar‘((subringAlg ‘𝐸)‘𝐹))))
136	134, 135	eqtr2d 2769	. . . 4 ⊢ (𝜑 → (Base‘(Scalar‘((subringAlg ‘𝐸)‘𝐹))) = 𝐹)
137	136	oveq1d 7367	. . 3 ⊢ (𝜑 → ((Base‘(Scalar‘((subringAlg ‘𝐸)‘𝐹))) ↑m (0...𝐷)) = (𝐹 ↑m (0...𝐷)))
138	93	crnggrpd 20167	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ Grp)
139	138	grpmndd 18861	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ Mnd)
140		subrgsubg 20494	. . . . . . . . . 10 ⊢ (𝐹 ∈ (SubRing‘𝐸) → 𝐹 ∈ (SubGrp‘𝐸))
141	9, 140	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ (SubGrp‘𝐸))
142		eqid 2733	. . . . . . . . . 10 ⊢ (0g‘𝐸) = (0g‘𝐸)
143	142	subg0cl 19049	. . . . . . . . 9 ⊢ (𝐹 ∈ (SubGrp‘𝐸) → (0g‘𝐸) ∈ 𝐹)
144	141, 143	syl 17	. . . . . . . 8 ⊢ (𝜑 → (0g‘𝐸) ∈ 𝐹)
145	5, 76, 142	ress0g 18672	. . . . . . . 8 ⊢ ((𝐸 ∈ Mnd ∧ (0g‘𝐸) ∈ 𝐹 ∧ 𝐹 ⊆ 𝐵) → (0g‘𝐸) = (0g‘(𝐸 ↾s 𝐹)))
146	139, 144, 78, 145	syl3anc 1373	. . . . . . 7 ⊢ (𝜑 → (0g‘𝐸) = (0g‘(𝐸 ↾s 𝐹)))
147	80	fveq2d 6832	. . . . . . 7 ⊢ (𝜑 → (0g‘(𝐸 ↾s 𝐹)) = (0g‘(Scalar‘((subringAlg ‘𝐸)‘𝐹))))
148	146, 147	eqtr2d 2769	. . . . . 6 ⊢ (𝜑 → (0g‘(Scalar‘((subringAlg ‘𝐸)‘𝐹))) = (0g‘𝐸))
149		extdgfialglem1.2	. . . . . 6 ⊢ 𝑍 = (0g‘𝐸)
150	148, 149	eqtr4di 2786	. . . . 5 ⊢ (𝜑 → (0g‘(Scalar‘((subringAlg ‘𝐸)‘𝐹))) = 𝑍)
151	150	breq2d 5105	. . . 4 ⊢ (𝜑 → (𝑎 finSupp (0g‘(Scalar‘((subringAlg ‘𝐸)‘𝐹))) ↔ 𝑎 finSupp 𝑍))
152		extdgfialglem1.3	. . . . . . . . . . 11 ⊢ · = (.r‘𝐸)
153	75, 79	sravsca 21117	. . . . . . . . . . 11 ⊢ (𝜑 → (.r‘𝐸) = ( ·𝑠 ‘((subringAlg ‘𝐸)‘𝐹)))
154	152, 153	eqtr2id 2781	. . . . . . . . . 10 ⊢ (𝜑 → ( ·𝑠 ‘((subringAlg ‘𝐸)‘𝐹)) = · )
155	154	ofeqd 7618	. . . . . . . . 9 ⊢ (𝜑 → ∘f ( ·𝑠 ‘((subringAlg ‘𝐸)‘𝐹)) = ∘f · )
156	155	oveqd 7369	. . . . . . . 8 ⊢ (𝜑 → (𝑎 ∘f ( ·𝑠 ‘((subringAlg ‘𝐸)‘𝐹))𝐺) = (𝑎 ∘f · 𝐺))
157	156	oveq2d 7368	. . . . . . 7 ⊢ (𝜑 → (((subringAlg ‘𝐸)‘𝐹) Σg (𝑎 ∘f ( ·𝑠 ‘((subringAlg ‘𝐸)‘𝐹))𝐺)) = (((subringAlg ‘𝐸)‘𝐹) Σg (𝑎 ∘f · 𝐺)))
158		ovexd 7387	. . . . . . . 8 ⊢ (𝜑 → (𝑎 ∘f · 𝐺) ∈ V)
159	10, 158, 2, 12, 79	gsumsra 33034	. . . . . . 7 ⊢ (𝜑 → (𝐸 Σg (𝑎 ∘f · 𝐺)) = (((subringAlg ‘𝐸)‘𝐹) Σg (𝑎 ∘f · 𝐺)))
160	157, 159	eqtr4d 2771	. . . . . 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 21124	. . . . . . 7 ⊢ (𝜑 → 𝑍 = (0g‘((subringAlg ‘𝐸)‘𝐹)))
163	162	eqcomd 2739	. . . . . 6 ⊢ (𝜑 → (0g‘((subringAlg ‘𝐸)‘𝐹)) = 𝑍)
164	160, 163	eqeq12d 2749	. . . . 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 4587	. . . . . . 7 ⊢ (𝜑 → {(0g‘(Scalar‘((subringAlg ‘𝐸)‘𝐹)))} = {𝑍})
166	165	xpeq2d 5649	. . . . . 6 ⊢ (𝜑 → ((0...𝐷) × {(0g‘(Scalar‘((subringAlg ‘𝐸)‘𝐹)))}) = ((0...𝐷) × {𝑍}))
167	166	neeq2d 2989	. . . . 5 ⊢ (𝜑 → (𝑎 ≠ ((0...𝐷) × {(0g‘(Scalar‘((subringAlg ‘𝐸)‘𝐹)))}) ↔ 𝑎 ≠ ((0...𝐷) × {𝑍})))
168	164, 167	anbi12d 632	. . . 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 632	. . 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 3314	. 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 1541   ∈ wcel 2113   ≠ wne 2929  ∀wral 3048  ∃wrex 3057  Vcvv 3437   ⊆ wss 3898  {csn 4575   class class class wbr 5093   ↦ cmpt 5174   × cxp 5617  dom cdm 5619  ran crn 5620  Fun wfun 6480  ⟶wf 6482  –1-1→wf1 6483  ‘cfv 6486  (class class class)co 7352   ∘_f cof 7614   ↑_m cmap 8756  Fincfn 8875   finSupp cfsupp 9252  ℝcr 11012  0cc0 11013  1c1 11014   + caddc 11016  ℝ^*cxr 11152   < clt 11153   ≤ cle 11154  ℕ₀cn0 12388  ...cfz 13409  ♯chash 14239  Basecbs 17122   ↾_s cress 17143  ._rcmulr 17164  Scalarcsca 17166   ·_𝑠 cvsca 17167  0_gc0g 17345   Σ_g cgsu 17346  Mndcmnd 18644  ._gcmg 18982  SubGrpcsubg 19035  mulGrpcmgp 20060  Ringcrg 20153  NzRingcnzr 20429  SubRingcsubrg 20486  DivRingcdr 20646  Fieldcfield 20647  SubDRingcsdrg 20703  LModclmod 20795  LBasisclbs 21010  LVecclvec 21038  subringAlg csra 21107   freeLMod cfrlm 21685   LIndF clindf 21743  LIndSclinds 21744  dimcldim 33632

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674  ax-reg 9485  ax-inf2 9538  ax-ac2 10361  ax-cnex 11069  ax-resscn 11070  ax-1cn 11071  ax-icn 11072  ax-addcl 11073  ax-addrcl 11074  ax-mulcl 11075  ax-mulrcl 11076  ax-mulcom 11077  ax-addass 11078  ax-mulass 11079  ax-distr 11080  ax-i2m1 11081  ax-1ne0 11082  ax-1rid 11083  ax-rnegex 11084  ax-rrecex 11085  ax-cnre 11086  ax-pre-lttri 11087  ax-pre-lttrn 11088  ax-pre-ltadd 11089  ax-pre-mulgt0 11090

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-nel 3034  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-tp 4580  df-op 4582  df-uni 4859  df-int 4898  df-iun 4943  df-iin 4944  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-isom 6495  df-riota 7309  df-ov 7355  df-oprab 7356  df-mpo 7357  df-of 7616  df-rpss 7662  df-om 7803  df-1st 7927  df-2nd 7928  df-supp 8097  df-tpos 8162  df-frecs 8217  df-wrecs 8248  df-recs 8297  df-rdg 8335  df-1o 8391  df-2o 8392  df-oadd 8395  df-er 8628  df-map 8758  df-ixp 8828  df-en 8876  df-dom 8877  df-sdom 8878  df-fin 8879  df-fsupp 9253  df-sup 9333  df-oi 9403  df-r1 9664  df-rank 9665  df-dju 9801  df-card 9839  df-acn 9842  df-ac 10014  df-pnf 11155  df-mnf 11156  df-xr 11157  df-ltxr 11158  df-le 11159  df-sub 11353  df-neg 11354  df-nn 12133  df-2 12195  df-3 12196  df-4 12197  df-5 12198  df-6 12199  df-7 12200  df-8 12201  df-9 12202  df-n0 12389  df-xnn0 12462  df-z 12476  df-dec 12595  df-uz 12739  df-fz 13410  df-fzo 13557  df-seq 13911  df-hash 14240  df-struct 17060  df-sets 17077  df-slot 17095  df-ndx 17107  df-base 17123  df-ress 17144  df-plusg 17176  df-mulr 17177  df-sca 17179  df-vsca 17180  df-ip 17181  df-tset 17182  df-ple 17183  df-ocomp 17184  df-ds 17185  df-hom 17187  df-cco 17188  df-0g 17347  df-gsum 17348  df-prds 17353  df-pws 17355  df-mre 17490  df-mrc 17491  df-mri 17492  df-acs 17493  df-proset 18202  df-drs 18203  df-poset 18221  df-ipo 18436  df-mgm 18550  df-sgrp 18629  df-mnd 18645  df-mhm 18693  df-submnd 18694  df-grp 18851  df-minusg 18852  df-sbg 18853  df-mulg 18983  df-subg 19038  df-ghm 19127  df-cntz 19231  df-cmn 19696  df-abl 19697  df-mgp 20061  df-rng 20073  df-ur 20102  df-ring 20155  df-cring 20156  df-oppr 20257  df-dvdsr 20277  df-unit 20278  df-invr 20308  df-nzr 20430  df-subrg 20487  df-drng 20648  df-field 20649  df-sdrg 20704  df-lmod 20797  df-lss 20867  df-lsp 20907  df-lmhm 20958  df-lbs 21011  df-lvec 21039  df-sra 21109  df-rgmod 21110  df-dsmm 21671  df-frlm 21686  df-uvc 21722  df-lindf 21745  df-linds 21746  df-dim 33633

This theorem is referenced by:  extdgfialg  33728