Theorem extdgfialglem1 34202

Description: Lemma for extdgfialg 34204. (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 781	. . . . . . . . . . . . 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 20904	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐸 ∈ DivRing)
4		extdgfialg.f	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))
5		eqid 2760	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐸 ↾s 𝐹) = (𝐸 ↾s 𝐹)
6	5	sdrgdrng 20956	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐹 ∈ (SubDRing‘𝐸) → (𝐸 ↾s 𝐹) ∈ DivRing)
7	4, 6	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐸 ↾s 𝐹) ∈ DivRing)
8		sdrgsubrg 20957	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐹 ∈ (SubDRing‘𝐸) → 𝐹 ∈ (SubRing‘𝐸))
9	4, 8	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐹 ∈ (SubRing‘𝐸))
10		eqid 2760	. . . . . . . . . . . . . . . . . . 19 ⊢ ((subringAlg ‘𝐸)‘𝐹) = ((subringAlg ‘𝐸)‘𝐹)
11	10, 5	sralvec 34095	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐸 ∈ DivRing ∧ (𝐸 ↾s 𝐹) ∈ DivRing ∧ 𝐹 ∈ (SubRing‘𝐸)) → ((subringAlg ‘𝐸)‘𝐹) ∈ LVec)
12	3, 7, 9, 11	syl3anc 1398	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((subringAlg ‘𝐸)‘𝐹) ∈ LVec)
13	12	ad2antrr 739	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) ∧ ran 𝐺 ∈ (LIndS‘((subringAlg ‘𝐸)‘𝐹))) → ((subringAlg ‘𝐸)‘𝐹) ∈ LVec)
14	13	ad2antrr 739	. . . . . . . . . . . . . . 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 2760	. . . . . . . . . . . . . . . . 17 ⊢ (LBasis‘((subringAlg ‘𝐸)‘𝐹)) = (LBasis‘((subringAlg ‘𝐸)‘𝐹))
17	16	dimval 34111	. . . . . . . . . . . . . . . 16 ⊢ ((((subringAlg ‘𝐸)‘𝐹) ∈ LVec ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘𝐹))) → (dim‘((subringAlg ‘𝐸)‘𝐹)) = (♯‘𝑏))
18	15, 17	eqtrid 2807	. . . . . . . . . . . . . . 15 ⊢ ((((subringAlg ‘𝐸)‘𝐹) ∈ LVec ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘𝐹))) → 𝐷 = (♯‘𝑏))
19	14, 1, 18	syl2anc 596	. . . . . . . . . . . . . 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 745	. . . . . . . . . . . . . 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 2861	. . . . . . . . . . . . 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 14422	. . . . . . . . . . . . . 14 ⊢ (𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘𝐹)) → (𝑏 ∈ Fin ↔ (♯‘𝑏) ∈ ℕ0))
24	23	biimpar 483	. . . . . . . . . . . . 13 ⊢ ((𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘𝐹)) ∧ (♯‘𝑏) ∈ ℕ0) → 𝑏 ∈ Fin)
25	1, 22, 24	syl2anc 596	. . . . . . . . . . . 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 14473	. . . . . . . . . . . 12 ⊢ ((𝑏 ∈ Fin ∧ ran 𝐺 ⊆ 𝑏) → (♯‘ran 𝐺) ≤ (♯‘𝑏))
27	25, 26	sylancom 600	. . . . . . . . . . 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 5888	. . . . . . . . . . . . . 14 ⊢ dom 𝐺 = dom (𝑛 ∈ (0...𝐷) ↦ (𝑛(.g‘(mulGrp‘((subringAlg ‘𝐸)‘𝐹)))𝑋))
30		eqid 2760	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ (0...𝐷) ↦ (𝑛(.g‘(mulGrp‘((subringAlg ‘𝐸)‘𝐹)))𝑋)) = (𝑛 ∈ (0...𝐷) ↦ (𝑛(.g‘(mulGrp‘((subringAlg ‘𝐸)‘𝐹)))𝑋))
31		ovexd 7448	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) ∧ 𝑛 ∈ (0...𝐷)) → (𝑛(.g‘(mulGrp‘((subringAlg ‘𝐸)‘𝐹)))𝑋) ∈ V)
32	30, 31	dmmptd 6677	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) → dom (𝑛 ∈ (0...𝐷) ↦ (𝑛(.g‘(mulGrp‘((subringAlg ‘𝐸)‘𝐹)))𝑋)) = (0...𝐷))
33		ovexd 7448	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) → (0...𝐷) ∈ V)
34	32, 33	eqeltrd 2860	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) → dom (𝑛 ∈ (0...𝐷) ↦ (𝑛(.g‘(mulGrp‘((subringAlg ‘𝐸)‘𝐹)))𝑋)) ∈ V)
35	29, 34	eqeltrid 2864	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) → dom 𝐺 ∈ V)
36		hashf1rn 14416	. . . . . . . . . . . . 13 ⊢ ((dom 𝐺 ∈ V ∧ 𝐺:dom 𝐺–1-1→V) → (♯‘𝐺) = (♯‘ran 𝐺))
37	35, 36	sylancom 600	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) → (♯‘𝐺) = (♯‘ran 𝐺))
38	37	ad3antrrr 743	. . . . . . . . . . 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 5137	. . . . . . . . . 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 22048	. . . . . . . . . . . 12 ⊢ (((subringAlg ‘𝐸)‘𝐹) ∈ LVec → (ran 𝐺 ∈ (LIndS‘((subringAlg ‘𝐸)‘𝐹)) ↔ ∃𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘𝐹))ran 𝐺 ⊆ 𝑏))
41	40	biimpa 482	. . . . . . . . . . 11 ⊢ ((((subringAlg ‘𝐸)‘𝐹) ∈ LVec ∧ ran 𝐺 ∈ (LIndS‘((subringAlg ‘𝐸)‘𝐹))) → ∃𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘𝐹))ran 𝐺 ⊆ 𝑏)
42	13, 41	sylancom 600	. . . . . . . . . 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 3170	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) ∧ ran 𝐺 ∈ (LIndS‘((subringAlg ‘𝐸)‘𝐹))) → (♯‘𝐺) ≤ 𝐷)
44	20	nn0red 12590	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐷 ∈ ℝ)
45	44	ad2antrr 739	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) ∧ ran 𝐺 ∈ (LIndS‘((subringAlg ‘𝐸)‘𝐹))) → 𝐷 ∈ ℝ)
46	45	ltp1d 12169	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) ∧ ran 𝐺 ∈ (LIndS‘((subringAlg ‘𝐸)‘𝐹))) → 𝐷 < (𝐷 + 1))
47		fzfid 14037	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (0...𝐷) ∈ Fin)
48	47	mptexd 7223	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑛 ∈ (0...𝐷) ↦ (𝑛(.g‘(mulGrp‘((subringAlg ‘𝐸)‘𝐹)))𝑋)) ∈ V)
49	28, 48	eqeltrid 2864	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐺 ∈ V)
50	49	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) → 𝐺 ∈ V)
51		f1f 6771	. . . . . . . . . . . . . . . 16 ⊢ (𝐺:dom 𝐺–1-1→V → 𝐺:dom 𝐺⟶V)
52	51	adantl 487	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) → 𝐺:dom 𝐺⟶V)
53	52	ffund 6707	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) → Fun 𝐺)
54		hashfundm 14507	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ V ∧ Fun 𝐺) → (♯‘𝐺) = (♯‘dom 𝐺))
55	50, 53, 54	syl2anc 596	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) → (♯‘𝐺) = (♯‘dom 𝐺))
56	28, 31	dmmptd 6677	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) → dom 𝐺 = (0...𝐷))
57	56	fveq2d 6882	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) → (♯‘dom 𝐺) = (♯‘(0...𝐷)))
58		hashfz0 14497	. . . . . . . . . . . . . . 15 ⊢ (𝐷 ∈ ℕ0 → (♯‘(0...𝐷)) = (𝐷 + 1))
59	20, 58	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (♯‘(0...𝐷)) = (𝐷 + 1))
60	59	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) → (♯‘(0...𝐷)) = (𝐷 + 1))
61	55, 57, 60	3eqtrd 2799	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) → (♯‘𝐺) = (𝐷 + 1))
62	61	adantr 486	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) ∧ ran 𝐺 ∈ (LIndS‘((subringAlg ‘𝐸)‘𝐹))) → (♯‘𝐺) = (𝐷 + 1))
63	46, 62	breqtrrd 5133	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) ∧ ran 𝐺 ∈ (LIndS‘((subringAlg ‘𝐸)‘𝐹))) → 𝐷 < (♯‘𝐺))
64	45	rexrd 11283	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) ∧ ran 𝐺 ∈ (LIndS‘((subringAlg ‘𝐸)‘𝐹))) → 𝐷 ∈ ℝ*)
65	50	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) ∧ ran 𝐺 ∈ (LIndS‘((subringAlg ‘𝐸)‘𝐹))) → 𝐺 ∈ V)
66		hashxrcl 14421	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ V → (♯‘𝐺) ∈ ℝ*)
67	65, 66	syl 18	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) ∧ ran 𝐺 ∈ (LIndS‘((subringAlg ‘𝐸)‘𝐹))) → (♯‘𝐺) ∈ ℝ*)
68	64, 67	xrltnled 11301	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) ∧ ran 𝐺 ∈ (LIndS‘((subringAlg ‘𝐸)‘𝐹))) → (𝐷 < (♯‘𝐺) ↔ ¬ (♯‘𝐺) ≤ 𝐷))
69	63, 68	mpbid 235	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) ∧ ran 𝐺 ∈ (LIndS‘((subringAlg ‘𝐸)‘𝐹))) → ¬ (♯‘𝐺) ≤ 𝐷)
70	43, 69	pm2.65da 829	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) → ¬ ran 𝐺 ∈ (LIndS‘((subringAlg ‘𝐸)‘𝐹)))
71	70	ex 418	. . . . . . 7 ⊢ (𝜑 → (𝐺:dom 𝐺–1-1→V → ¬ ran 𝐺 ∈ (LIndS‘((subringAlg ‘𝐸)‘𝐹))))
72		imnan 405	. . . . . . 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 221	. . . . . 6 ⊢ (𝜑 → ¬ (𝐺:dom 𝐺–1-1→V ∧ ran 𝐺 ∈ (LIndS‘((subringAlg ‘𝐸)‘𝐹))))
74	12	lveclmodd 21291	. . . . . . 7 ⊢ (𝜑 → ((subringAlg ‘𝐸)‘𝐹) ∈ LMod)
75		eqidd 2761	. . . . . . . . 9 ⊢ (𝜑 → ((subringAlg ‘𝐸)‘𝐹) = ((subringAlg ‘𝐸)‘𝐹))
76		extdgfialg.b	. . . . . . . . . . . 12 ⊢ 𝐵 = (Base‘𝐸)
77	76	sdrgss 20959	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (SubDRing‘𝐸) → 𝐹 ⊆ 𝐵)
78	4, 77	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ⊆ 𝐵)
79	78, 76	sseqtrdi 3971	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ⊆ (Base‘𝐸))
80	75, 79	srasca 21364	. . . . . . . 8 ⊢ (𝜑 → (𝐸 ↾s 𝐹) = (Scalar‘((subringAlg ‘𝐸)‘𝐹)))
81		drngnzr 20911	. . . . . . . . 9 ⊢ ((𝐸 ↾s 𝐹) ∈ DivRing → (𝐸 ↾s 𝐹) ∈ NzRing)
82	7, 81	syl 18	. . . . . . . 8 ⊢ (𝜑 → (𝐸 ↾s 𝐹) ∈ NzRing)
83	80, 82	eqeltrrd 2861	. . . . . . 7 ⊢ (𝜑 → (Scalar‘((subringAlg ‘𝐸)‘𝐹)) ∈ NzRing)
84		eqid 2760	. . . . . . . 8 ⊢ (Scalar‘((subringAlg ‘𝐸)‘𝐹)) = (Scalar‘((subringAlg ‘𝐸)‘𝐹))
85	84	islindf3 22039	. . . . . . 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 596	. . . . . 6 ⊢ (𝜑 → (𝐺 LIndF ((subringAlg ‘𝐸)‘𝐹) ↔ (𝐺:dom 𝐺–1-1→V ∧ ran 𝐺 ∈ (LIndS‘((subringAlg ‘𝐸)‘𝐹)))))
87	73, 86	mtbird 328	. . . . 5 ⊢ (𝜑 → ¬ 𝐺 LIndF ((subringAlg ‘𝐸)‘𝐹))
88		ovexd 7448	. . . . . 6 ⊢ (𝜑 → (0...𝐷) ∈ V)
89		eqid 2760	. . . . . . . . 9 ⊢ (mulGrp‘((subringAlg ‘𝐸)‘𝐹)) = (mulGrp‘((subringAlg ‘𝐸)‘𝐹))
90		eqid 2760	. . . . . . . . 9 ⊢ (Base‘((subringAlg ‘𝐸)‘𝐹)) = (Base‘((subringAlg ‘𝐸)‘𝐹))
91	89, 90	mgpbas 20278	. . . . . . . 8 ⊢ (Base‘((subringAlg ‘𝐸)‘𝐹)) = (Base‘(mulGrp‘((subringAlg ‘𝐸)‘𝐹)))
92		eqid 2760	. . . . . . . 8 ⊢ (.g‘(mulGrp‘((subringAlg ‘𝐸)‘𝐹))) = (.g‘(mulGrp‘((subringAlg ‘𝐸)‘𝐹)))
93	2	fldcrngd 20905	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐸 ∈ CRing)
94	93	crngringd 20385	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ∈ Ring)
95	10, 76	sraring 21370	. . . . . . . . . . 11 ⊢ ((𝐸 ∈ Ring ∧ 𝐹 ⊆ 𝐵) → ((subringAlg ‘𝐸)‘𝐹) ∈ Ring)
96	94, 78, 95	syl2anc 596	. . . . . . . . . 10 ⊢ (𝜑 → ((subringAlg ‘𝐸)‘𝐹) ∈ Ring)
97	89	ringmgp 20378	. . . . . . . . . 10 ⊢ (((subringAlg ‘𝐸)‘𝐹) ∈ Ring → (mulGrp‘((subringAlg ‘𝐸)‘𝐹)) ∈ Mnd)
98	96, 97	syl 18	. . . . . . . . 9 ⊢ (𝜑 → (mulGrp‘((subringAlg ‘𝐸)‘𝐹)) ∈ Mnd)
99	98	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...𝐷)) → (mulGrp‘((subringAlg ‘𝐸)‘𝐹)) ∈ Mnd)
100		fz0ssnn0 13677	. . . . . . . . . 10 ⊢ (0...𝐷) ⊆ ℕ0
101	100	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → (0...𝐷) ⊆ ℕ0)
102	101	sselda 3931	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...𝐷)) → 𝑛 ∈ ℕ0)
103		extdgfialglem1.4	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
104	75, 79	srabase 21361	. . . . . . . . . . 11 ⊢ (𝜑 → (Base‘𝐸) = (Base‘((subringAlg ‘𝐸)‘𝐹)))
105	76, 104	eqtr2id 2808	. . . . . . . . . 10 ⊢ (𝜑 → (Base‘((subringAlg ‘𝐸)‘𝐹)) = 𝐵)
106	103, 105	eleqtrrd 2863	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ (Base‘((subringAlg ‘𝐸)‘𝐹)))
107	106	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...𝐷)) → 𝑋 ∈ (Base‘((subringAlg ‘𝐸)‘𝐹)))
108	91, 92, 99, 102, 107	mulgnn0cld 19218	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...𝐷)) → (𝑛(.g‘(mulGrp‘((subringAlg ‘𝐸)‘𝐹)))𝑋) ∈ (Base‘((subringAlg ‘𝐸)‘𝐹)))
109	108, 28	fmptd 7107	. . . . . 6 ⊢ (𝜑 → 𝐺:(0...𝐷)⟶(Base‘((subringAlg ‘𝐸)‘𝐹)))
110		eqid 2760	. . . . . . 7 ⊢ ( ·𝑠 ‘((subringAlg ‘𝐸)‘𝐹)) = ( ·𝑠 ‘((subringAlg ‘𝐸)‘𝐹))
111		eqid 2760	. . . . . . 7 ⊢ (0g‘((subringAlg ‘𝐸)‘𝐹)) = (0g‘((subringAlg ‘𝐸)‘𝐹))
112		eqid 2760	. . . . . . 7 ⊢ (0g‘(Scalar‘((subringAlg ‘𝐸)‘𝐹))) = (0g‘(Scalar‘((subringAlg ‘𝐸)‘𝐹)))
113		eqid 2760	. . . . . . 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 22051	. . . . . 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 1398	. . . . 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 327	. . . 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 3116	. . . 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 237	. . 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 6891	. . . . . . 7 ⊢ (Scalar‘((subringAlg ‘𝐸)‘𝐹)) ∈ V
120		ovex 7446	. . . . . . 7 ⊢ (0...𝐷) ∈ V
121		eqid 2760	. . . . . . . 8 ⊢ ((Scalar‘((subringAlg ‘𝐸)‘𝐹)) freeLMod (0...𝐷)) = ((Scalar‘((subringAlg ‘𝐸)‘𝐹)) freeLMod (0...𝐷))
122		eqid 2760	. . . . . . . 8 ⊢ (Base‘(Scalar‘((subringAlg ‘𝐸)‘𝐹))) = (Base‘(Scalar‘((subringAlg ‘𝐸)‘𝐹)))
123	121, 122, 112, 113	frlmelbas 21969	. . . . . . 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 705	. . . . . 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 636	. . . . 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 2956	. . . . . . 7 ⊢ (𝑎 ≠ ((0...𝐷) × {(0g‘(Scalar‘((subringAlg ‘𝐸)‘𝐹)))}) ↔ ¬ 𝑎 = ((0...𝐷) × {(0g‘(Scalar‘((subringAlg ‘𝐸)‘𝐹)))}))
127	126	anbi2i 635	. . . . . 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 635	. . . . 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 474	. . . . 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 304	. . . 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 3105	. . 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 221	. 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 17330	. . . . . 6 ⊢ (𝐹 ⊆ 𝐵 → 𝐹 = (Base‘(𝐸 ↾s 𝐹)))
134	78, 133	syl 18	. . . . 5 ⊢ (𝜑 → 𝐹 = (Base‘(𝐸 ↾s 𝐹)))
135	80	fveq2d 6882	. . . . 5 ⊢ (𝜑 → (Base‘(𝐸 ↾s 𝐹)) = (Base‘(Scalar‘((subringAlg ‘𝐸)‘𝐹))))
136	134, 135	eqtr2d 2796	. . . 4 ⊢ (𝜑 → (Base‘(Scalar‘((subringAlg ‘𝐸)‘𝐹))) = 𝐹)
137	136	oveq1d 7428	. . 3 ⊢ (𝜑 → ((Base‘(Scalar‘((subringAlg ‘𝐸)‘𝐹))) ↑m (0...𝐷)) = (𝐹 ↑m (0...𝐷)))
138	93	crnggrpd 20386	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ Grp)
139	138	grpmndd 19070	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ Mnd)
140		subrgsubg 20739	. . . . . . . . . 10 ⊢ (𝐹 ∈ (SubRing‘𝐸) → 𝐹 ∈ (SubGrp‘𝐸))
141	9, 140	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ (SubGrp‘𝐸))
142		eqid 2760	. . . . . . . . . 10 ⊢ (0g‘𝐸) = (0g‘𝐸)
143	142	subg0cl 19257	. . . . . . . . 9 ⊢ (𝐹 ∈ (SubGrp‘𝐸) → (0g‘𝐸) ∈ 𝐹)
144	141, 143	syl 18	. . . . . . . 8 ⊢ (𝜑 → (0g‘𝐸) ∈ 𝐹)
145	5, 76, 142	ress0g 18867	. . . . . . . 8 ⊢ ((𝐸 ∈ Mnd ∧ (0g‘𝐸) ∈ 𝐹 ∧ 𝐹 ⊆ 𝐵) → (0g‘𝐸) = (0g‘(𝐸 ↾s 𝐹)))
146	139, 144, 78, 145	syl3anc 1398	. . . . . . 7 ⊢ (𝜑 → (0g‘𝐸) = (0g‘(𝐸 ↾s 𝐹)))
147	80	fveq2d 6882	. . . . . . 7 ⊢ (𝜑 → (0g‘(𝐸 ↾s 𝐹)) = (0g‘(Scalar‘((subringAlg ‘𝐸)‘𝐹))))
148	146, 147	eqtr2d 2796	. . . . . 6 ⊢ (𝜑 → (0g‘(Scalar‘((subringAlg ‘𝐸)‘𝐹))) = (0g‘𝐸))
149		extdgfialglem1.2	. . . . . 6 ⊢ 𝑍 = (0g‘𝐸)
150	148, 149	eqtr4di 2813	. . . . 5 ⊢ (𝜑 → (0g‘(Scalar‘((subringAlg ‘𝐸)‘𝐹))) = 𝑍)
151	150	breq2d 5115	. . . 4 ⊢ (𝜑 → (𝑎 finSupp (0g‘(Scalar‘((subringAlg ‘𝐸)‘𝐹))) ↔ 𝑎 finSupp 𝑍))
152		extdgfialglem1.3	. . . . . . . . . . 11 ⊢ · = (.r‘𝐸)
153	75, 79	sravsca 21365	. . . . . . . . . . 11 ⊢ (𝜑 → (.r‘𝐸) = ( ·𝑠 ‘((subringAlg ‘𝐸)‘𝐹)))
154	152, 153	eqtr2id 2808	. . . . . . . . . 10 ⊢ (𝜑 → ( ·𝑠 ‘((subringAlg ‘𝐸)‘𝐹)) = · )
155	154	ofeqd 7680	. . . . . . . . 9 ⊢ (𝜑 → ∘f ( ·𝑠 ‘((subringAlg ‘𝐸)‘𝐹)) = ∘f · )
156	155	oveqd 7430	. . . . . . . 8 ⊢ (𝜑 → (𝑎 ∘f ( ·𝑠 ‘((subringAlg ‘𝐸)‘𝐹))𝐺) = (𝑎 ∘f · 𝐺))
157	156	oveq2d 7429	. . . . . . 7 ⊢ (𝜑 → (((subringAlg ‘𝐸)‘𝐹) Σg (𝑎 ∘f ( ·𝑠 ‘((subringAlg ‘𝐸)‘𝐹))𝐺)) = (((subringAlg ‘𝐸)‘𝐹) Σg (𝑎 ∘f · 𝐺)))
158		ovexd 7448	. . . . . . . 8 ⊢ (𝜑 → (𝑎 ∘f · 𝐺) ∈ V)
159	10, 158, 2, 12, 79	gsumsra 33487	. . . . . . 7 ⊢ (𝜑 → (𝐸 Σg (𝑎 ∘f · 𝐺)) = (((subringAlg ‘𝐸)‘𝐹) Σg (𝑎 ∘f · 𝐺)))
160	157, 159	eqtr4d 2798	. . . . . 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 21372	. . . . . . 7 ⊢ (𝜑 → 𝑍 = (0g‘((subringAlg ‘𝐸)‘𝐹)))
163	162	eqcomd 2766	. . . . . 6 ⊢ (𝜑 → (0g‘((subringAlg ‘𝐸)‘𝐹)) = 𝑍)
164	160, 163	eqeq12d 2776	. . . . 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 4596	. . . . . . 7 ⊢ (𝜑 → {(0g‘(Scalar‘((subringAlg ‘𝐸)‘𝐹)))} = {𝑍})
166	165	xpeq2d 5685	. . . . . 6 ⊢ (𝜑 → ((0...𝐷) × {(0g‘(Scalar‘((subringAlg ‘𝐸)‘𝐹)))}) = ((0...𝐷) × {𝑍}))
167	166	neeq2d 3015	. . . . 5 ⊢ (𝜑 → (𝑎 ≠ ((0...𝐷) × {(0g‘(Scalar‘((subringAlg ‘𝐸)‘𝐹)))}) ↔ 𝑎 ≠ ((0...𝐷) × {𝑍})))
168	164, 167	anbi12d 644	. . . 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 644	. . 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 3335	. 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 235	1 ⊢ (𝜑 → ∃𝑎 ∈ (𝐹 ↑m (0...𝐷))(𝑎 finSupp 𝑍 ∧ ((𝐸 Σg (𝑎 ∘f · 𝐺)) = 𝑍 ∧ 𝑎 ≠ ((0...𝐷) × {𝑍}))))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ≠ wne 2955  ∀wral 3076  ∃wrex 3086  Vcvv 3450   ⊆ wss 3899  {csn 4584   class class class wbr 5103   ↦ cmpt 5186   × cxp 5653  dom cdm 5655  ran crn 5656  Fun wfun 6527  ⟶wf 6529  –1-1→wf1 6530  ‘cfv 6533  (class class class)co 7413   ∘_f cof 7676   ↑_m cmap 8826  Fincfn 8952   finSupp cfsupp 9331  ℝcr 11123  0cc0 11124  1c1 11125   + caddc 11127  ℝ^*cxr 11266   < clt 11267   ≤ cle 11268  ℕ₀cn0 12528  ...cfz 13561  ♯chash 14394  Basecbs 17301   ↾_s cress 17322  ._rcmulr 17343  Scalarcsca 17345   ·_𝑠 cvsca 17346  0_gc0g 17524   Σ_g cgsu 17525  Mndcmnd 18836  ._gcmg 19190  SubGrpcsubg 19243  mulGrpcmgp 20273  Ringcrg 20372  NzRingcnzr 20672  SubRingcsubrg 20731  DivRingcdr 20890  Fieldcfield 20891  SubDRingcsdrg 20952  LModclmod 21044  LBasisclbs 21258  LVecclvec 21286  subringAlg csra 21355   freeLMod cfrlm 21959   LIndF clindf 22017  LIndSclinds 22018  dimcldim 34109

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736  ax-reg 9564  ax-inf2 9620  ax-ac2 10465  ax-cnex 11180  ax-resscn 11181  ax-1cn 11182  ax-icn 11183  ax-addcl 11184  ax-addrcl 11185  ax-mulcl 11186  ax-mulrcl 11187  ax-mulcom 11188  ax-addass 11189  ax-mulass 11190  ax-distr 11191  ax-i2m1 11192  ax-1ne0 11193  ax-1rid 11194  ax-rnegex 11195  ax-rrecex 11196  ax-cnre 11197  ax-pre-lttri 11198  ax-pre-lttrn 11199  ax-pre-ltadd 11200  ax-pre-mulgt0 11201

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 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-iin 4954  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-isom 6542  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-of 7678  df-rpss 7724  df-om 7863  df-1st 7986  df-2nd 7987  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 8905  df-en 8953  df-dom 8954  df-sdom 8955  df-fin 8956  df-fsupp 9332  df-sup 9412  df-oi 9482  df-r1 9746  df-rank 9747  df-scott 9868  df-dju 9906  df-card 9944  df-acn 9947  df-ac 10119  df-pnf 11269  df-mnf 11270  df-xr 11271  df-ltxr 11272  df-le 11273  df-sub 11467  df-neg 11468  df-nn 12258  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12529  df-xnn0 12602  df-z 12616  df-dec 12737  df-uz 12888  df-fz 13562  df-fzo 13710  df-seq 14066  df-hash 14395  df-struct 17239  df-sets 17256  df-slot 17274  df-ndx 17286  df-base 17302  df-ress 17323  df-plusg 17355  df-mulr 17356  df-sca 17358  df-vsca 17359  df-ip 17360  df-tset 17361  df-ple 17362  df-ocomp 17363  df-ds 17364  df-hom 17366  df-cco 17367  df-0g 17526  df-gsum 17527  df-prds 17532  df-pws 17534  df-mre 17670  df-mrc 17671  df-mri 17672  df-acs 17673  df-proset 18382  df-drs 18383  df-poset 18401  df-ipo 18616  df-mgm 18730  df-sgrp 18821  df-mnd 18837  df-mhm 18891  df-submnd 18892  df-grp 19060  df-minusg 19061  df-sbg 19062  df-mulg 19191  df-subg 19246  df-ghm 19341  df-cntz 19444  df-cmn 19909  df-abl 19910  df-mgp 20274  df-rng 20288  df-ur 20321  df-ring 20374  df-cring 20375  df-oppr 20478  df-dvdsr 20498  df-unit 20499  df-invr 20529  df-nzr 20673  df-subrg 20732  df-drng 20892  df-field 20893  df-sdrg 20953  df-lmod 21046  df-lss 21116  df-lsp 21156  df-lmhm 21206  df-lbs 21259  df-lvec 21287  df-sra 21357  df-rgmod 21358  df-dsmm 21945  df-frlm 21960  df-uvc 21996  df-lindf 22019  df-linds 22020  df-dim 34110

This theorem is used by:  extdgfialg  34204