Theorem extdgfialglem1 33849

Description: Lemma for extdgfialg 33851. (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 20674	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐸 ∈ DivRing)
4		extdgfialg.f	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))
5		eqid 2736	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐸 ↾s 𝐹) = (𝐸 ↾s 𝐹)
6	5	sdrgdrng 20723	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐹 ∈ (SubDRing‘𝐸) → (𝐸 ↾s 𝐹) ∈ DivRing)
7	4, 6	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐸 ↾s 𝐹) ∈ DivRing)
8		sdrgsubrg 20724	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐹 ∈ (SubDRing‘𝐸) → 𝐹 ∈ (SubRing‘𝐸))
9	4, 8	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐹 ∈ (SubRing‘𝐸))
10		eqid 2736	. . . . . . . . . . . . . . . . . . 19 ⊢ ((subringAlg ‘𝐸)‘𝐹) = ((subringAlg ‘𝐸)‘𝐹)
11	10, 5	sralvec 33741	. . . . . . . . . . . . . . . . . 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 2736	. . . . . . . . . . . . . . . . 17 ⊢ (LBasis‘((subringAlg ‘𝐸)‘𝐹)) = (LBasis‘((subringAlg ‘𝐸)‘𝐹))
17	16	dimval 33757	. . . . . . . . . . . . . . . 16 ⊢ ((((subringAlg ‘𝐸)‘𝐹) ∈ LVec ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘𝐹))) → (dim‘((subringAlg ‘𝐸)‘𝐹)) = (♯‘𝑏))
18	15, 17	eqtrid 2783	. . . . . . . . . . . . . . 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 2837	. . . . . . . . . . . . 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 14281	. . . . . . . . . . . . . 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 14332	. . . . . . . . . . . 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 5853	. . . . . . . . . . . . . 14 ⊢ dom 𝐺 = dom (𝑛 ∈ (0...𝐷) ↦ (𝑛(.g‘(mulGrp‘((subringAlg ‘𝐸)‘𝐹)))𝑋))
30		eqid 2736	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ (0...𝐷) ↦ (𝑛(.g‘(mulGrp‘((subringAlg ‘𝐸)‘𝐹)))𝑋)) = (𝑛 ∈ (0...𝐷) ↦ (𝑛(.g‘(mulGrp‘((subringAlg ‘𝐸)‘𝐹)))𝑋))
31		ovexd 7393	. . . . . . . . . . . . . . . 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 7393	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) → (0...𝐷) ∈ V)
34	32, 33	eqeltrd 2836	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) → dom (𝑛 ∈ (0...𝐷) ↦ (𝑛(.g‘(mulGrp‘((subringAlg ‘𝐸)‘𝐹)))𝑋)) ∈ V)
35	29, 34	eqeltrid 2840	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) → dom 𝐺 ∈ V)
36		hashf1rn 14275	. . . . . . . . . . . . 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 5130	. . . . . . . . . 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 21790	. . . . . . . . . . . 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 3144	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) ∧ ran 𝐺 ∈ (LIndS‘((subringAlg ‘𝐸)‘𝐹))) → (♯‘𝐺) ≤ 𝐷)
44	20	nn0red 12463	. . . . . . . . . . . . 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 12072	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) ∧ ran 𝐺 ∈ (LIndS‘((subringAlg ‘𝐸)‘𝐹))) → 𝐷 < (𝐷 + 1))
47		fzfid 13896	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (0...𝐷) ∈ Fin)
48	47	mptexd 7170	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑛 ∈ (0...𝐷) ↦ (𝑛(.g‘(mulGrp‘((subringAlg ‘𝐸)‘𝐹)))𝑋)) ∈ V)
49	28, 48	eqeltrid 2840	. . . . . . . . . . . . . . 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 14365	. . . . . . . . . . . . . 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 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 14355	. . . . . . . . . . . . . . 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 2775	. . . . . . . . . . . 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 5126	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) ∧ ran 𝐺 ∈ (LIndS‘((subringAlg ‘𝐸)‘𝐹))) → 𝐷 < (♯‘𝐺))
64	45	rexrd 11182	. . . . . . . . . . 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 14280	. . . . . . . . . . . 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 11200	. . . . . . . . . 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 21059	. . . . . . 7 ⊢ (𝜑 → ((subringAlg ‘𝐸)‘𝐹) ∈ LMod)
75		eqidd 2737	. . . . . . . . 9 ⊢ (𝜑 → ((subringAlg ‘𝐸)‘𝐹) = ((subringAlg ‘𝐸)‘𝐹))
76		extdgfialg.b	. . . . . . . . . . . 12 ⊢ 𝐵 = (Base‘𝐸)
77	76	sdrgss 20726	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (SubDRing‘𝐸) → 𝐹 ⊆ 𝐵)
78	4, 77	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ⊆ 𝐵)
79	78, 76	sseqtrdi 3974	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ⊆ (Base‘𝐸))
80	75, 79	srasca 21132	. . . . . . . 8 ⊢ (𝜑 → (𝐸 ↾s 𝐹) = (Scalar‘((subringAlg ‘𝐸)‘𝐹)))
81		drngnzr 20681	. . . . . . . . 9 ⊢ ((𝐸 ↾s 𝐹) ∈ DivRing → (𝐸 ↾s 𝐹) ∈ NzRing)
82	7, 81	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝐸 ↾s 𝐹) ∈ NzRing)
83	80, 82	eqeltrrd 2837	. . . . . . 7 ⊢ (𝜑 → (Scalar‘((subringAlg ‘𝐸)‘𝐹)) ∈ NzRing)
84		eqid 2736	. . . . . . . 8 ⊢ (Scalar‘((subringAlg ‘𝐸)‘𝐹)) = (Scalar‘((subringAlg ‘𝐸)‘𝐹))
85	84	islindf3 21781	. . . . . . 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 7393	. . . . . 6 ⊢ (𝜑 → (0...𝐷) ∈ V)
89		eqid 2736	. . . . . . . . 9 ⊢ (mulGrp‘((subringAlg ‘𝐸)‘𝐹)) = (mulGrp‘((subringAlg ‘𝐸)‘𝐹))
90		eqid 2736	. . . . . . . . 9 ⊢ (Base‘((subringAlg ‘𝐸)‘𝐹)) = (Base‘((subringAlg ‘𝐸)‘𝐹))
91	89, 90	mgpbas 20080	. . . . . . . 8 ⊢ (Base‘((subringAlg ‘𝐸)‘𝐹)) = (Base‘(mulGrp‘((subringAlg ‘𝐸)‘𝐹)))
92		eqid 2736	. . . . . . . 8 ⊢ (.g‘(mulGrp‘((subringAlg ‘𝐸)‘𝐹))) = (.g‘(mulGrp‘((subringAlg ‘𝐸)‘𝐹)))
93	2	fldcrngd 20675	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐸 ∈ CRing)
94	93	crngringd 20181	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ∈ Ring)
95	10, 76	sraring 21138	. . . . . . . . . . 11 ⊢ ((𝐸 ∈ Ring ∧ 𝐹 ⊆ 𝐵) → ((subringAlg ‘𝐸)‘𝐹) ∈ Ring)
96	94, 78, 95	syl2anc 584	. . . . . . . . . 10 ⊢ (𝜑 → ((subringAlg ‘𝐸)‘𝐹) ∈ Ring)
97	89	ringmgp 20174	. . . . . . . . . 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 13538	. . . . . . . . . 10 ⊢ (0...𝐷) ⊆ ℕ0
101	100	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → (0...𝐷) ⊆ ℕ0)
102	101	sselda 3933	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...𝐷)) → 𝑛 ∈ ℕ0)
103		extdgfialglem1.4	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
104	75, 79	srabase 21129	. . . . . . . . . . 11 ⊢ (𝜑 → (Base‘𝐸) = (Base‘((subringAlg ‘𝐸)‘𝐹)))
105	76, 104	eqtr2id 2784	. . . . . . . . . 10 ⊢ (𝜑 → (Base‘((subringAlg ‘𝐸)‘𝐹)) = 𝐵)
106	103, 105	eleqtrrd 2839	. . . . . . . . 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 19025	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...𝐷)) → (𝑛(.g‘(mulGrp‘((subringAlg ‘𝐸)‘𝐹)))𝑋) ∈ (Base‘((subringAlg ‘𝐸)‘𝐹)))
109	108, 28	fmptd 7059	. . . . . 6 ⊢ (𝜑 → 𝐺:(0...𝐷)⟶(Base‘((subringAlg ‘𝐸)‘𝐹)))
110		eqid 2736	. . . . . . 7 ⊢ ( ·𝑠 ‘((subringAlg ‘𝐸)‘𝐹)) = ( ·𝑠 ‘((subringAlg ‘𝐸)‘𝐹))
111		eqid 2736	. . . . . . 7 ⊢ (0g‘((subringAlg ‘𝐸)‘𝐹)) = (0g‘((subringAlg ‘𝐸)‘𝐹))
112		eqid 2736	. . . . . . 7 ⊢ (0g‘(Scalar‘((subringAlg ‘𝐸)‘𝐹))) = (0g‘(Scalar‘((subringAlg ‘𝐸)‘𝐹)))
113		eqid 2736	. . . . . . 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 21793	. . . . . 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 3090	. . . 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 7391	. . . . . . 7 ⊢ (0...𝐷) ∈ V
121		eqid 2736	. . . . . . . 8 ⊢ ((Scalar‘((subringAlg ‘𝐸)‘𝐹)) freeLMod (0...𝐷)) = ((Scalar‘((subringAlg ‘𝐸)‘𝐹)) freeLMod (0...𝐷))
122		eqid 2736	. . . . . . . 8 ⊢ (Base‘(Scalar‘((subringAlg ‘𝐸)‘𝐹))) = (Base‘(Scalar‘((subringAlg ‘𝐸)‘𝐹)))
123	121, 122, 112, 113	frlmelbas 21711	. . . . . . 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 2933	. . . . . . 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 3079	. . 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 17165	. . . . . 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 2772	. . . 4 ⊢ (𝜑 → (Base‘(Scalar‘((subringAlg ‘𝐸)‘𝐹))) = 𝐹)
137	136	oveq1d 7373	. . 3 ⊢ (𝜑 → ((Base‘(Scalar‘((subringAlg ‘𝐸)‘𝐹))) ↑m (0...𝐷)) = (𝐹 ↑m (0...𝐷)))
138	93	crnggrpd 20182	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ Grp)
139	138	grpmndd 18876	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ Mnd)
140		subrgsubg 20510	. . . . . . . . . 10 ⊢ (𝐹 ∈ (SubRing‘𝐸) → 𝐹 ∈ (SubGrp‘𝐸))
141	9, 140	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ (SubGrp‘𝐸))
142		eqid 2736	. . . . . . . . . 10 ⊢ (0g‘𝐸) = (0g‘𝐸)
143	142	subg0cl 19064	. . . . . . . . 9 ⊢ (𝐹 ∈ (SubGrp‘𝐸) → (0g‘𝐸) ∈ 𝐹)
144	141, 143	syl 17	. . . . . . . 8 ⊢ (𝜑 → (0g‘𝐸) ∈ 𝐹)
145	5, 76, 142	ress0g 18687	. . . . . . . 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 6838	. . . . . . 7 ⊢ (𝜑 → (0g‘(𝐸 ↾s 𝐹)) = (0g‘(Scalar‘((subringAlg ‘𝐸)‘𝐹))))
148	146, 147	eqtr2d 2772	. . . . . 6 ⊢ (𝜑 → (0g‘(Scalar‘((subringAlg ‘𝐸)‘𝐹))) = (0g‘𝐸))
149		extdgfialglem1.2	. . . . . 6 ⊢ 𝑍 = (0g‘𝐸)
150	148, 149	eqtr4di 2789	. . . . 5 ⊢ (𝜑 → (0g‘(Scalar‘((subringAlg ‘𝐸)‘𝐹))) = 𝑍)
151	150	breq2d 5110	. . . 4 ⊢ (𝜑 → (𝑎 finSupp (0g‘(Scalar‘((subringAlg ‘𝐸)‘𝐹))) ↔ 𝑎 finSupp 𝑍))
152		extdgfialglem1.3	. . . . . . . . . . 11 ⊢ · = (.r‘𝐸)
153	75, 79	sravsca 21133	. . . . . . . . . . 11 ⊢ (𝜑 → (.r‘𝐸) = ( ·𝑠 ‘((subringAlg ‘𝐸)‘𝐹)))
154	152, 153	eqtr2id 2784	. . . . . . . . . 10 ⊢ (𝜑 → ( ·𝑠 ‘((subringAlg ‘𝐸)‘𝐹)) = · )
155	154	ofeqd 7624	. . . . . . . . 9 ⊢ (𝜑 → ∘f ( ·𝑠 ‘((subringAlg ‘𝐸)‘𝐹)) = ∘f · )
156	155	oveqd 7375	. . . . . . . 8 ⊢ (𝜑 → (𝑎 ∘f ( ·𝑠 ‘((subringAlg ‘𝐸)‘𝐹))𝐺) = (𝑎 ∘f · 𝐺))
157	156	oveq2d 7374	. . . . . . 7 ⊢ (𝜑 → (((subringAlg ‘𝐸)‘𝐹) Σg (𝑎 ∘f ( ·𝑠 ‘((subringAlg ‘𝐸)‘𝐹))𝐺)) = (((subringAlg ‘𝐸)‘𝐹) Σg (𝑎 ∘f · 𝐺)))
158		ovexd 7393	. . . . . . . 8 ⊢ (𝜑 → (𝑎 ∘f · 𝐺) ∈ V)
159	10, 158, 2, 12, 79	gsumsra 33130	. . . . . . 7 ⊢ (𝜑 → (𝐸 Σg (𝑎 ∘f · 𝐺)) = (((subringAlg ‘𝐸)‘𝐹) Σg (𝑎 ∘f · 𝐺)))
160	157, 159	eqtr4d 2774	. . . . . 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 21140	. . . . . . 7 ⊢ (𝜑 → 𝑍 = (0g‘((subringAlg ‘𝐸)‘𝐹)))
163	162	eqcomd 2742	. . . . . 6 ⊢ (𝜑 → (0g‘((subringAlg ‘𝐸)‘𝐹)) = 𝑍)
164	160, 163	eqeq12d 2752	. . . . 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 4592	. . . . . . 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 2992	. . . . 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 3317	. 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 2932  ∀wral 3051  ∃wrex 3060  Vcvv 3440   ⊆ wss 3901  {csn 4580   class class class wbr 5098   ↦ cmpt 5179   × cxp 5622  dom cdm 5624  ran crn 5625  Fun wfun 6486  ⟶wf 6488  –1-1→wf1 6489  ‘cfv 6492  (class class class)co 7358   ∘_f cof 7620   ↑_m cmap 8763  Fincfn 8883   finSupp cfsupp 9264  ℝcr 11025  0cc0 11026  1c1 11027   + caddc 11029  ℝ^*cxr 11165   < clt 11166   ≤ cle 11167  ℕ₀cn0 12401  ...cfz 13423  ♯chash 14253  Basecbs 17136   ↾_s cress 17157  ._rcmulr 17178  Scalarcsca 17180   ·_𝑠 cvsca 17181  0_gc0g 17359   Σ_g cgsu 17360  Mndcmnd 18659  ._gcmg 18997  SubGrpcsubg 19050  mulGrpcmgp 20075  Ringcrg 20168  NzRingcnzr 20445  SubRingcsubrg 20502  DivRingcdr 20662  Fieldcfield 20663  SubDRingcsdrg 20719  LModclmod 20811  LBasisclbs 21026  LVecclvec 21054  subringAlg csra 21123   freeLMod cfrlm 21701   LIndF clindf 21759  LIndSclinds 21760  dimcldim 33755

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 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-reg 9497  ax-inf2 9550  ax-ac2 10373  ax-cnex 11082  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-pre-mulgt0 11103

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-tp 4585  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-iin 4949  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  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 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-of 7622  df-rpss 7668  df-om 7809  df-1st 7933  df-2nd 7934  df-supp 8103  df-tpos 8168  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-2o 8398  df-oadd 8401  df-er 8635  df-map 8765  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9265  df-sup 9345  df-oi 9415  df-r1 9676  df-rank 9677  df-dju 9813  df-card 9851  df-acn 9854  df-ac 10026  df-pnf 11168  df-mnf 11169  df-xr 11170  df-ltxr 11171  df-le 11172  df-sub 11366  df-neg 11367  df-nn 12146  df-2 12208  df-3 12209  df-4 12210  df-5 12211  df-6 12212  df-7 12213  df-8 12214  df-9 12215  df-n0 12402  df-xnn0 12475  df-z 12489  df-dec 12608  df-uz 12752  df-fz 13424  df-fzo 13571  df-seq 13925  df-hash 14254  df-struct 17074  df-sets 17091  df-slot 17109  df-ndx 17121  df-base 17137  df-ress 17158  df-plusg 17190  df-mulr 17191  df-sca 17193  df-vsca 17194  df-ip 17195  df-tset 17196  df-ple 17197  df-ocomp 17198  df-ds 17199  df-hom 17201  df-cco 17202  df-0g 17361  df-gsum 17362  df-prds 17367  df-pws 17369  df-mre 17505  df-mrc 17506  df-mri 17507  df-acs 17508  df-proset 18217  df-drs 18218  df-poset 18236  df-ipo 18451  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-mhm 18708  df-submnd 18709  df-grp 18866  df-minusg 18867  df-sbg 18868  df-mulg 18998  df-subg 19053  df-ghm 19142  df-cntz 19246  df-cmn 19711  df-abl 19712  df-mgp 20076  df-rng 20088  df-ur 20117  df-ring 20170  df-cring 20171  df-oppr 20273  df-dvdsr 20293  df-unit 20294  df-invr 20324  df-nzr 20446  df-subrg 20503  df-drng 20664  df-field 20665  df-sdrg 20720  df-lmod 20813  df-lss 20883  df-lsp 20923  df-lmhm 20974  df-lbs 21027  df-lvec 21055  df-sra 21125  df-rgmod 21126  df-dsmm 21687  df-frlm 21702  df-uvc 21738  df-lindf 21761  df-linds 21762  df-dim 33756

This theorem is referenced by:  extdgfialg  33851