Theorem extdgfialglem1 34105

Description: Lemma for extdgfialg 34107. (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 20870	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐸 ∈ DivRing)
4		extdgfialg.f	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐸))
5		eqid 2765	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐸 ↾s 𝐹) = (𝐸 ↾s 𝐹)
6	5	sdrgdrng 20922	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐹 ∈ (SubDRing‘𝐸) → (𝐸 ↾s 𝐹) ∈ DivRing)
7	4, 6	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝐸 ↾s 𝐹) ∈ DivRing)
8		sdrgsubrg 20923	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐹 ∈ (SubDRing‘𝐸) → 𝐹 ∈ (SubRing‘𝐸))
9	4, 8	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐹 ∈ (SubRing‘𝐸))
10		eqid 2765	. . . . . . . . . . . . . . . . . . 19 ⊢ ((subringAlg ‘𝐸)‘𝐹) = ((subringAlg ‘𝐸)‘𝐹)
11	10, 5	sralvec 33998	. . . . . . . . . . . . . . . . . 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 2765	. . . . . . . . . . . . . . . . 17 ⊢ (LBasis‘((subringAlg ‘𝐸)‘𝐹)) = (LBasis‘((subringAlg ‘𝐸)‘𝐹))
17	16	dimval 34014	. . . . . . . . . . . . . . . 16 ⊢ ((((subringAlg ‘𝐸)‘𝐹) ∈ LVec ∧ 𝑏 ∈ (LBasis‘((subringAlg ‘𝐸)‘𝐹))) → (dim‘((subringAlg ‘𝐸)‘𝐹)) = (♯‘𝑏))
18	15, 17	eqtrid 2812	. . . . . . . . . . . . . . 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 2866	. . . . . . . . . . . . 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 14407	. . . . . . . . . . . . . 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 14458	. . . . . . . . . . . 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 5896	. . . . . . . . . . . . . 14 ⊢ dom 𝐺 = dom (𝑛 ∈ (0...𝐷) ↦ (𝑛(.g‘(mulGrp‘((subringAlg ‘𝐸)‘𝐹)))𝑋))
30		eqid 2765	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ (0...𝐷) ↦ (𝑛(.g‘(mulGrp‘((subringAlg ‘𝐸)‘𝐹)))𝑋)) = (𝑛 ∈ (0...𝐷) ↦ (𝑛(.g‘(mulGrp‘((subringAlg ‘𝐸)‘𝐹)))𝑋))
31		ovexd 7451	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) ∧ 𝑛 ∈ (0...𝐷)) → (𝑛(.g‘(mulGrp‘((subringAlg ‘𝐸)‘𝐹)))𝑋) ∈ V)
32	30, 31	dmmptd 6684	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) → dom (𝑛 ∈ (0...𝐷) ↦ (𝑛(.g‘(mulGrp‘((subringAlg ‘𝐸)‘𝐹)))𝑋)) = (0...𝐷))
33		ovexd 7451	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) → (0...𝐷) ∈ V)
34	32, 33	eqeltrd 2865	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) → dom (𝑛 ∈ (0...𝐷) ↦ (𝑛(.g‘(mulGrp‘((subringAlg ‘𝐸)‘𝐹)))𝑋)) ∈ V)
35	29, 34	eqeltrid 2869	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) → dom 𝐺 ∈ V)
36		hashf1rn 14401	. . . . . . . . . . . . 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 5145	. . . . . . . . . 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 22014	. . . . . . . . . . . 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 3175	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) ∧ ran 𝐺 ∈ (LIndS‘((subringAlg ‘𝐸)‘𝐹))) → (♯‘𝐺) ≤ 𝐷)
44	20	nn0red 12577	. . . . . . . . . . . . 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 12156	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) ∧ ran 𝐺 ∈ (LIndS‘((subringAlg ‘𝐸)‘𝐹))) → 𝐷 < (𝐷 + 1))
47		fzfid 14022	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (0...𝐷) ∈ Fin)
48	47	mptexd 7226	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑛 ∈ (0...𝐷) ↦ (𝑛(.g‘(mulGrp‘((subringAlg ‘𝐸)‘𝐹)))𝑋)) ∈ V)
49	28, 48	eqeltrid 2869	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐺 ∈ V)
50	49	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) → 𝐺 ∈ V)
51		f1f 6778	. . . . . . . . . . . . . . . 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 6714	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) → Fun 𝐺)
54		hashfundm 14492	. . . . . . . . . . . . . 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 6684	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) → dom 𝐺 = (0...𝐷))
57	56	fveq2d 6889	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) → (♯‘dom 𝐺) = (♯‘(0...𝐷)))
58		hashfz0 14482	. . . . . . . . . . . . . . 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 2804	. . . . . . . . . . . 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 5141	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐺:dom 𝐺–1-1→V) ∧ ran 𝐺 ∈ (LIndS‘((subringAlg ‘𝐸)‘𝐹))) → 𝐷 < (♯‘𝐺))
64	45	rexrd 11270	. . . . . . . . . . 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 14406	. . . . . . . . . . . 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 11288	. . . . . . . . . 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 21257	. . . . . . 7 ⊢ (𝜑 → ((subringAlg ‘𝐸)‘𝐹) ∈ LMod)
75		eqidd 2766	. . . . . . . . 9 ⊢ (𝜑 → ((subringAlg ‘𝐸)‘𝐹) = ((subringAlg ‘𝐸)‘𝐹))
76		extdgfialg.b	. . . . . . . . . . . 12 ⊢ 𝐵 = (Base‘𝐸)
77	76	sdrgss 20925	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (SubDRing‘𝐸) → 𝐹 ⊆ 𝐵)
78	4, 77	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ⊆ 𝐵)
79	78, 76	sseqtrdi 3978	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ⊆ (Base‘𝐸))
80	75, 79	srasca 21330	. . . . . . . 8 ⊢ (𝜑 → (𝐸 ↾s 𝐹) = (Scalar‘((subringAlg ‘𝐸)‘𝐹)))
81		drngnzr 20877	. . . . . . . . 9 ⊢ ((𝐸 ↾s 𝐹) ∈ DivRing → (𝐸 ↾s 𝐹) ∈ NzRing)
82	7, 81	syl 18	. . . . . . . 8 ⊢ (𝜑 → (𝐸 ↾s 𝐹) ∈ NzRing)
83	80, 82	eqeltrrd 2866	. . . . . . 7 ⊢ (𝜑 → (Scalar‘((subringAlg ‘𝐸)‘𝐹)) ∈ NzRing)
84		eqid 2765	. . . . . . . 8 ⊢ (Scalar‘((subringAlg ‘𝐸)‘𝐹)) = (Scalar‘((subringAlg ‘𝐸)‘𝐹))
85	84	islindf3 22005	. . . . . . 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 7451	. . . . . 6 ⊢ (𝜑 → (0...𝐷) ∈ V)
89		eqid 2765	. . . . . . . . 9 ⊢ (mulGrp‘((subringAlg ‘𝐸)‘𝐹)) = (mulGrp‘((subringAlg ‘𝐸)‘𝐹))
90		eqid 2765	. . . . . . . . 9 ⊢ (Base‘((subringAlg ‘𝐸)‘𝐹)) = (Base‘((subringAlg ‘𝐸)‘𝐹))
91	89, 90	mgpbas 20244	. . . . . . . 8 ⊢ (Base‘((subringAlg ‘𝐸)‘𝐹)) = (Base‘(mulGrp‘((subringAlg ‘𝐸)‘𝐹)))
92		eqid 2765	. . . . . . . 8 ⊢ (.g‘(mulGrp‘((subringAlg ‘𝐸)‘𝐹))) = (.g‘(mulGrp‘((subringAlg ‘𝐸)‘𝐹)))
93	2	fldcrngd 20871	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐸 ∈ CRing)
94	93	crngringd 20351	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ∈ Ring)
95	10, 76	sraring 21336	. . . . . . . . . . 11 ⊢ ((𝐸 ∈ Ring ∧ 𝐹 ⊆ 𝐵) → ((subringAlg ‘𝐸)‘𝐹) ∈ Ring)
96	94, 78, 95	syl2anc 596	. . . . . . . . . 10 ⊢ (𝜑 → ((subringAlg ‘𝐸)‘𝐹) ∈ Ring)
97	89	ringmgp 20344	. . . . . . . . . 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 13662	. . . . . . . . . 10 ⊢ (0...𝐷) ⊆ ℕ0
101	100	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → (0...𝐷) ⊆ ℕ0)
102	101	sselda 3938	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...𝐷)) → 𝑛 ∈ ℕ0)
103		extdgfialglem1.4	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
104	75, 79	srabase 21327	. . . . . . . . . . 11 ⊢ (𝜑 → (Base‘𝐸) = (Base‘((subringAlg ‘𝐸)‘𝐹)))
105	76, 104	eqtr2id 2813	. . . . . . . . . 10 ⊢ (𝜑 → (Base‘((subringAlg ‘𝐸)‘𝐹)) = 𝐵)
106	103, 105	eleqtrrd 2868	. . . . . . . . 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 19184	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (0...𝐷)) → (𝑛(.g‘(mulGrp‘((subringAlg ‘𝐸)‘𝐹)))𝑋) ∈ (Base‘((subringAlg ‘𝐸)‘𝐹)))
109	108, 28	fmptd 7113	. . . . . 6 ⊢ (𝜑 → 𝐺:(0...𝐷)⟶(Base‘((subringAlg ‘𝐸)‘𝐹)))
110		eqid 2765	. . . . . . 7 ⊢ ( ·𝑠 ‘((subringAlg ‘𝐸)‘𝐹)) = ( ·𝑠 ‘((subringAlg ‘𝐸)‘𝐹))
111		eqid 2765	. . . . . . 7 ⊢ (0g‘((subringAlg ‘𝐸)‘𝐹)) = (0g‘((subringAlg ‘𝐸)‘𝐹))
112		eqid 2765	. . . . . . 7 ⊢ (0g‘(Scalar‘((subringAlg ‘𝐸)‘𝐹))) = (0g‘(Scalar‘((subringAlg ‘𝐸)‘𝐹)))
113		eqid 2765	. . . . . . 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 22017	. . . . . 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 3121	. . . 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 6898	. . . . . . 7 ⊢ (Scalar‘((subringAlg ‘𝐸)‘𝐹)) ∈ V
120		ovex 7449	. . . . . . 7 ⊢ (0...𝐷) ∈ V
121		eqid 2765	. . . . . . . 8 ⊢ ((Scalar‘((subringAlg ‘𝐸)‘𝐹)) freeLMod (0...𝐷)) = ((Scalar‘((subringAlg ‘𝐸)‘𝐹)) freeLMod (0...𝐷))
122		eqid 2765	. . . . . . . 8 ⊢ (Base‘(Scalar‘((subringAlg ‘𝐸)‘𝐹))) = (Base‘(Scalar‘((subringAlg ‘𝐸)‘𝐹)))
123	121, 122, 112, 113	frlmelbas 21935	. . . . . . 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 2961	. . . . . . 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 3110	. . 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 17315	. . . . . 6 ⊢ (𝐹 ⊆ 𝐵 → 𝐹 = (Base‘(𝐸 ↾s 𝐹)))
134	78, 133	syl 18	. . . . 5 ⊢ (𝜑 → 𝐹 = (Base‘(𝐸 ↾s 𝐹)))
135	80	fveq2d 6889	. . . . 5 ⊢ (𝜑 → (Base‘(𝐸 ↾s 𝐹)) = (Base‘(Scalar‘((subringAlg ‘𝐸)‘𝐹))))
136	134, 135	eqtr2d 2801	. . . 4 ⊢ (𝜑 → (Base‘(Scalar‘((subringAlg ‘𝐸)‘𝐹))) = 𝐹)
137	136	oveq1d 7431	. . 3 ⊢ (𝜑 → ((Base‘(Scalar‘((subringAlg ‘𝐸)‘𝐹))) ↑m (0...𝐷)) = (𝐹 ↑m (0...𝐷)))
138	93	crnggrpd 20352	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ Grp)
139	138	grpmndd 19036	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ Mnd)
140		subrgsubg 20705	. . . . . . . . . 10 ⊢ (𝐹 ∈ (SubRing‘𝐸) → 𝐹 ∈ (SubGrp‘𝐸))
141	9, 140	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ (SubGrp‘𝐸))
142		eqid 2765	. . . . . . . . . 10 ⊢ (0g‘𝐸) = (0g‘𝐸)
143	142	subg0cl 19223	. . . . . . . . 9 ⊢ (𝐹 ∈ (SubGrp‘𝐸) → (0g‘𝐸) ∈ 𝐹)
144	141, 143	syl 18	. . . . . . . 8 ⊢ (𝜑 → (0g‘𝐸) ∈ 𝐹)
145	5, 76, 142	ress0g 18841	. . . . . . . 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 6889	. . . . . . 7 ⊢ (𝜑 → (0g‘(𝐸 ↾s 𝐹)) = (0g‘(Scalar‘((subringAlg ‘𝐸)‘𝐹))))
148	146, 147	eqtr2d 2801	. . . . . 6 ⊢ (𝜑 → (0g‘(Scalar‘((subringAlg ‘𝐸)‘𝐹))) = (0g‘𝐸))
149		extdgfialglem1.2	. . . . . 6 ⊢ 𝑍 = (0g‘𝐸)
150	148, 149	eqtr4di 2818	. . . . 5 ⊢ (𝜑 → (0g‘(Scalar‘((subringAlg ‘𝐸)‘𝐹))) = 𝑍)
151	150	breq2d 5123	. . . 4 ⊢ (𝜑 → (𝑎 finSupp (0g‘(Scalar‘((subringAlg ‘𝐸)‘𝐹))) ↔ 𝑎 finSupp 𝑍))
152		extdgfialglem1.3	. . . . . . . . . . 11 ⊢ · = (.r‘𝐸)
153	75, 79	sravsca 21331	. . . . . . . . . . 11 ⊢ (𝜑 → (.r‘𝐸) = ( ·𝑠 ‘((subringAlg ‘𝐸)‘𝐹)))
154	152, 153	eqtr2id 2813	. . . . . . . . . 10 ⊢ (𝜑 → ( ·𝑠 ‘((subringAlg ‘𝐸)‘𝐹)) = · )
155	154	ofeqd 7682	. . . . . . . . 9 ⊢ (𝜑 → ∘f ( ·𝑠 ‘((subringAlg ‘𝐸)‘𝐹)) = ∘f · )
156	155	oveqd 7433	. . . . . . . 8 ⊢ (𝜑 → (𝑎 ∘f ( ·𝑠 ‘((subringAlg ‘𝐸)‘𝐹))𝐺) = (𝑎 ∘f · 𝐺))
157	156	oveq2d 7432	. . . . . . 7 ⊢ (𝜑 → (((subringAlg ‘𝐸)‘𝐹) Σg (𝑎 ∘f ( ·𝑠 ‘((subringAlg ‘𝐸)‘𝐹))𝐺)) = (((subringAlg ‘𝐸)‘𝐹) Σg (𝑎 ∘f · 𝐺)))
158		ovexd 7451	. . . . . . . 8 ⊢ (𝜑 → (𝑎 ∘f · 𝐺) ∈ V)
159	10, 158, 2, 12, 79	gsumsra 33390	. . . . . . 7 ⊢ (𝜑 → (𝐸 Σg (𝑎 ∘f · 𝐺)) = (((subringAlg ‘𝐸)‘𝐹) Σg (𝑎 ∘f · 𝐺)))
160	157, 159	eqtr4d 2803	. . . . . 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 21338	. . . . . . 7 ⊢ (𝜑 → 𝑍 = (0g‘((subringAlg ‘𝐸)‘𝐹)))
163	162	eqcomd 2771	. . . . . 6 ⊢ (𝜑 → (0g‘((subringAlg ‘𝐸)‘𝐹)) = 𝑍)
164	160, 163	eqeq12d 2781	. . . . 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 4603	. . . . . . 7 ⊢ (𝜑 → {(0g‘(Scalar‘((subringAlg ‘𝐸)‘𝐹)))} = {𝑍})
166	165	xpeq2d 5693	. . . . . 6 ⊢ (𝜑 → ((0...𝐷) × {(0g‘(Scalar‘((subringAlg ‘𝐸)‘𝐹)))}) = ((0...𝐷) × {𝑍}))
167	166	neeq2d 3020	. . . . 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 3341	. 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 2146   ≠ wne 2960  ∀wral 3081  ∃wrex 3091  Vcvv 3457   ⊆ wss 3906  {csn 4591   class class class wbr 5111   ↦ cmpt 5194   × cxp 5661  dom cdm 5663  ran crn 5664  Fun wfun 6534  ⟶wf 6536  –1-1→wf1 6537  ‘cfv 6540  (class class class)co 7416   ∘_f cof 7678   ↑_m cmap 8826  Fincfn 8945   finSupp cfsupp 9324  ℝcr 11110  0cc0 11111  1c1 11112   + caddc 11114  ℝ^*cxr 11253   < clt 11254   ≤ cle 11255  ℕ₀cn0 12515  ...cfz 13546  ♯chash 14379  Basecbs 17286   ↾_s cress 17307  ._rcmulr 17328  Scalarcsca 17330   ·_𝑠 cvsca 17331  0_gc0g 17509   Σ_g cgsu 17510  Mndcmnd 18813  ._gcmg 19156  SubGrpcsubg 19209  mulGrpcmgp 20239  Ringcrg 20338  NzRingcnzr 20638  SubRingcsubrg 20697  DivRingcdr 20856  Fieldcfield 20857  SubDRingcsdrg 20918  LModclmod 21010  LBasisclbs 21224  LVecclvec 21252  subringAlg csra 21321   freeLMod cfrlm 21925   LIndF clindf 21983  LIndSclinds 21984  dimcldim 34012

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

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4875  df-int 4915  df-iun 4960  df-iin 4961  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-se 5617  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-of 7680  df-rpss 7726  df-om 7865  df-1st 7988  df-2nd 7989  df-supp 8159  df-tpos 8224  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-1o 8455  df-2o 8456  df-oadd 8459  df-er 8696  df-map 8828  df-ixp 8898  df-en 8946  df-dom 8947  df-sdom 8948  df-fin 8949  df-fsupp 9325  df-sup 9405  df-oi 9475  df-r1 9739  df-rank 9740  df-scott 9861  df-dju 9899  df-card 9937  df-acn 9940  df-ac 10112  df-pnf 11256  df-mnf 11257  df-xr 11258  df-ltxr 11259  df-le 11260  df-sub 11454  df-neg 11455  df-nn 12245  df-2 12314  df-3 12315  df-4 12316  df-5 12317  df-6 12318  df-7 12319  df-8 12320  df-9 12321  df-n0 12516  df-xnn0 12589  df-z 12603  df-dec 12723  df-uz 12874  df-fz 13547  df-fzo 13695  df-seq 14051  df-hash 14380  df-struct 17224  df-sets 17241  df-slot 17259  df-ndx 17271  df-base 17287  df-ress 17308  df-plusg 17340  df-mulr 17341  df-sca 17343  df-vsca 17344  df-ip 17345  df-tset 17346  df-ple 17347  df-ocomp 17348  df-ds 17349  df-hom 17351  df-cco 17352  df-0g 17511  df-gsum 17512  df-prds 17517  df-pws 17519  df-mre 17655  df-mrc 17656  df-mri 17657  df-acs 17658  df-proset 18367  df-drs 18368  df-poset 18386  df-ipo 18601  df-mgm 18715  df-sgrp 18798  df-mnd 18814  df-mhm 18864  df-submnd 18865  df-grp 19026  df-minusg 19027  df-sbg 19028  df-mulg 19157  df-subg 19212  df-ghm 19307  df-cntz 19410  df-cmn 19875  df-abl 19876  df-mgp 20240  df-rng 20254  df-ur 20287  df-ring 20340  df-cring 20341  df-oppr 20444  df-dvdsr 20464  df-unit 20465  df-invr 20495  df-nzr 20639  df-subrg 20698  df-drng 20858  df-field 20859  df-sdrg 20919  df-lmod 21012  df-lss 21082  df-lsp 21122  df-lmhm 21172  df-lbs 21225  df-lvec 21253  df-sra 21323  df-rgmod 21324  df-dsmm 21911  df-frlm 21926  df-uvc 21962  df-lindf 21985  df-linds 21986  df-dim 34013

This theorem is used by:  extdgfialg  34107