Theorem mplbas2 21955

Description: An alternative expression for the set of polynomials, as the smallest subalgebra of the set of power series that contains all the variable generators. (Contributed by Mario Carneiro, 10-Jan-2015.)

Hypotheses

Ref	Expression
mplbas2.p	⊢ 𝑃 = (𝐼 mPoly 𝑅)
mplbas2.s	⊢ 𝑆 = (𝐼 mPwSer 𝑅)
mplbas2.v	⊢ 𝑉 = (𝐼 mVar 𝑅)
mplbas2.a	⊢ 𝐴 = (AlgSpan‘𝑆)
mplbas2.i	⊢ (𝜑 → 𝐼 ∈ 𝑊)
mplbas2.r	⊢ (𝜑 → 𝑅 ∈ CRing)

Assertion

Ref	Expression
mplbas2	⊢ (𝜑 → (𝐴‘ran 𝑉) = (Base‘𝑃))

Proof of Theorem mplbas2

Dummy variables 𝑢 𝑘 𝑣 𝑥 𝑧 𝑦 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mplbas2.s	. . . . 5 ⊢ 𝑆 = (𝐼 mPwSer 𝑅)
2		mplbas2.i	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ 𝑊)
3		mplbas2.r	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ CRing)
4	1, 2, 3	psrassa 21888	. . . 4 ⊢ (𝜑 → 𝑆 ∈ AssAlg)
5		mplbas2.p	. . . . . 6 ⊢ 𝑃 = (𝐼 mPoly 𝑅)
6		eqid 2730	. . . . . 6 ⊢ (Base‘𝑃) = (Base‘𝑃)
7		eqid 2730	. . . . . 6 ⊢ (Base‘𝑆) = (Base‘𝑆)
8	5, 1, 6, 7	mplbasss 21912	. . . . 5 ⊢ (Base‘𝑃) ⊆ (Base‘𝑆)
9	8	a1i 11	. . . 4 ⊢ (𝜑 → (Base‘𝑃) ⊆ (Base‘𝑆))
10		mplbas2.v	. . . . . . . 8 ⊢ 𝑉 = (𝐼 mVar 𝑅)
11		crngring 20160	. . . . . . . . 9 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
12	3, 11	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ Ring)
13	1, 10, 7, 2, 12	mvrf 21900	. . . . . . 7 ⊢ (𝜑 → 𝑉:𝐼⟶(Base‘𝑆))
14	13	ffnd 6691	. . . . . 6 ⊢ (𝜑 → 𝑉 Fn 𝐼)
15	2	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐼 ∈ 𝑊)
16	12	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑅 ∈ Ring)
17		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑥 ∈ 𝐼)
18	5, 10, 6, 15, 16, 17	mvrcl 21907	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑉‘𝑥) ∈ (Base‘𝑃))
19	18	ralrimiva 3126	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 (𝑉‘𝑥) ∈ (Base‘𝑃))
20		ffnfv 7093	. . . . . 6 ⊢ (𝑉:𝐼⟶(Base‘𝑃) ↔ (𝑉 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝑉‘𝑥) ∈ (Base‘𝑃)))
21	14, 19, 20	sylanbrc 583	. . . . 5 ⊢ (𝜑 → 𝑉:𝐼⟶(Base‘𝑃))
22	21	frnd 6698	. . . 4 ⊢ (𝜑 → ran 𝑉 ⊆ (Base‘𝑃))
23		mplbas2.a	. . . . 5 ⊢ 𝐴 = (AlgSpan‘𝑆)
24	23, 7	aspss 21792	. . . 4 ⊢ ((𝑆 ∈ AssAlg ∧ (Base‘𝑃) ⊆ (Base‘𝑆) ∧ ran 𝑉 ⊆ (Base‘𝑃)) → (𝐴‘ran 𝑉) ⊆ (𝐴‘(Base‘𝑃)))
25	4, 9, 22, 24	syl3anc 1373	. . 3 ⊢ (𝜑 → (𝐴‘ran 𝑉) ⊆ (𝐴‘(Base‘𝑃)))
26	1, 5, 6, 2, 12	mplsubrg 21920	. . . 4 ⊢ (𝜑 → (Base‘𝑃) ∈ (SubRing‘𝑆))
27	1, 5, 6, 2, 12	mpllss 21918	. . . 4 ⊢ (𝜑 → (Base‘𝑃) ∈ (LSubSp‘𝑆))
28		eqid 2730	. . . . 5 ⊢ (LSubSp‘𝑆) = (LSubSp‘𝑆)
29	23, 7, 28	aspid 21790	. . . 4 ⊢ ((𝑆 ∈ AssAlg ∧ (Base‘𝑃) ∈ (SubRing‘𝑆) ∧ (Base‘𝑃) ∈ (LSubSp‘𝑆)) → (𝐴‘(Base‘𝑃)) = (Base‘𝑃))
30	4, 26, 27, 29	syl3anc 1373	. . 3 ⊢ (𝜑 → (𝐴‘(Base‘𝑃)) = (Base‘𝑃))
31	25, 30	sseqtrd 3985	. 2 ⊢ (𝜑 → (𝐴‘ran 𝑉) ⊆ (Base‘𝑃))
32		eqid 2730	. . . 4 ⊢ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}
33		eqid 2730	. . . 4 ⊢ (0g‘𝑅) = (0g‘𝑅)
34		eqid 2730	. . . 4 ⊢ (1r‘𝑅) = (1r‘𝑅)
35	2	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) → 𝐼 ∈ 𝑊)
36		eqid 2730	. . . 4 ⊢ ( ·𝑠 ‘𝑃) = ( ·𝑠 ‘𝑃)
37	12	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) → 𝑅 ∈ Ring)
38		simpr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) → 𝑥 ∈ (Base‘𝑃))
39	5, 32, 33, 34, 35, 6, 36, 37, 38	mplcoe1 21950	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) → 𝑥 = (𝑃 Σg (𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ ((𝑥‘𝑘)( ·𝑠 ‘𝑃)(𝑦 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = 𝑘, (1r‘𝑅), (0g‘𝑅)))))))
40		eqid 2730	. . . 4 ⊢ (0g‘𝑃) = (0g‘𝑃)
41	5, 2, 12	mplringd 21938	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ Ring)
42		ringabl 20196	. . . . . 6 ⊢ (𝑃 ∈ Ring → 𝑃 ∈ Abel)
43	41, 42	syl 17	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ Abel)
44	43	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) → 𝑃 ∈ Abel)
45		ovex 7422	. . . . . 6 ⊢ (ℕ0 ↑m 𝐼) ∈ V
46	45	rabex 5296	. . . . 5 ⊢ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∈ V
47	46	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) → {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∈ V)
48	13	frnd 6698	. . . . . . . 8 ⊢ (𝜑 → ran 𝑉 ⊆ (Base‘𝑆))
49	23, 7	aspsubrg 21791	. . . . . . . 8 ⊢ ((𝑆 ∈ AssAlg ∧ ran 𝑉 ⊆ (Base‘𝑆)) → (𝐴‘ran 𝑉) ∈ (SubRing‘𝑆))
50	4, 48, 49	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → (𝐴‘ran 𝑉) ∈ (SubRing‘𝑆))
51	5, 1, 6	mplval2 21911	. . . . . . . . 9 ⊢ 𝑃 = (𝑆 ↾s (Base‘𝑃))
52	51	subsubrg 20513	. . . . . . . 8 ⊢ ((Base‘𝑃) ∈ (SubRing‘𝑆) → ((𝐴‘ran 𝑉) ∈ (SubRing‘𝑃) ↔ ((𝐴‘ran 𝑉) ∈ (SubRing‘𝑆) ∧ (𝐴‘ran 𝑉) ⊆ (Base‘𝑃))))
53	26, 52	syl 17	. . . . . . 7 ⊢ (𝜑 → ((𝐴‘ran 𝑉) ∈ (SubRing‘𝑃) ↔ ((𝐴‘ran 𝑉) ∈ (SubRing‘𝑆) ∧ (𝐴‘ran 𝑉) ⊆ (Base‘𝑃))))
54	50, 31, 53	mpbir2and 713	. . . . . 6 ⊢ (𝜑 → (𝐴‘ran 𝑉) ∈ (SubRing‘𝑃))
55		subrgsubg 20492	. . . . . 6 ⊢ ((𝐴‘ran 𝑉) ∈ (SubRing‘𝑃) → (𝐴‘ran 𝑉) ∈ (SubGrp‘𝑃))
56	54, 55	syl 17	. . . . 5 ⊢ (𝜑 → (𝐴‘ran 𝑉) ∈ (SubGrp‘𝑃))
57	56	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) → (𝐴‘ran 𝑉) ∈ (SubGrp‘𝑃))
58	5, 2, 12	mpllmodd 21939	. . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ LMod)
59	58	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → 𝑃 ∈ LMod)
60	23, 7, 28	asplss 21789	. . . . . . . . 9 ⊢ ((𝑆 ∈ AssAlg ∧ ran 𝑉 ⊆ (Base‘𝑆)) → (𝐴‘ran 𝑉) ∈ (LSubSp‘𝑆))
61	4, 48, 60	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → (𝐴‘ran 𝑉) ∈ (LSubSp‘𝑆))
62	1, 2, 12	psrlmod 21875	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ LMod)
63		eqid 2730	. . . . . . . . . 10 ⊢ (LSubSp‘𝑃) = (LSubSp‘𝑃)
64	51, 28, 63	lsslss 20873	. . . . . . . . 9 ⊢ ((𝑆 ∈ LMod ∧ (Base‘𝑃) ∈ (LSubSp‘𝑆)) → ((𝐴‘ran 𝑉) ∈ (LSubSp‘𝑃) ↔ ((𝐴‘ran 𝑉) ∈ (LSubSp‘𝑆) ∧ (𝐴‘ran 𝑉) ⊆ (Base‘𝑃))))
65	62, 27, 64	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → ((𝐴‘ran 𝑉) ∈ (LSubSp‘𝑃) ↔ ((𝐴‘ran 𝑉) ∈ (LSubSp‘𝑆) ∧ (𝐴‘ran 𝑉) ⊆ (Base‘𝑃))))
66	61, 31, 65	mpbir2and 713	. . . . . . 7 ⊢ (𝜑 → (𝐴‘ran 𝑉) ∈ (LSubSp‘𝑃))
67	66	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝐴‘ran 𝑉) ∈ (LSubSp‘𝑃))
68		eqid 2730	. . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅)
69	5, 68, 6, 32, 38	mplelf 21913	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) → 𝑥:{𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
70	69	ffvelcdmda 7058	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝑥‘𝑘) ∈ (Base‘𝑅))
71	5, 35, 37	mplsca 21928	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) → 𝑅 = (Scalar‘𝑃))
72	71	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → 𝑅 = (Scalar‘𝑃))
73	72	fveq2d 6864	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (Base‘𝑅) = (Base‘(Scalar‘𝑃)))
74	70, 73	eleqtrd 2831	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝑥‘𝑘) ∈ (Base‘(Scalar‘𝑃)))
75	2	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → 𝐼 ∈ 𝑊)
76		eqid 2730	. . . . . . . 8 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃)
77		eqid 2730	. . . . . . . 8 ⊢ (.g‘(mulGrp‘𝑃)) = (.g‘(mulGrp‘𝑃))
78	3	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → 𝑅 ∈ CRing)
79		simpr 484	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin})
80	5, 32, 33, 34, 75, 76, 77, 10, 78, 79	mplcoe2 21954	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝑦 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = 𝑘, (1r‘𝑅), (0g‘𝑅))) = ((mulGrp‘𝑃) Σg (𝑧 ∈ 𝐼 ↦ ((𝑘‘𝑧)(.g‘(mulGrp‘𝑃))(𝑉‘𝑧)))))
81		eqid 2730	. . . . . . . . 9 ⊢ (1r‘𝑃) = (1r‘𝑃)
82	76, 81	ringidval 20098	. . . . . . . 8 ⊢ (1r‘𝑃) = (0g‘(mulGrp‘𝑃))
83	5	mplcrng 21936	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ CRing) → 𝑃 ∈ CRing)
84	2, 3, 83	syl2anc 584	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ CRing)
85	76	crngmgp 20156	. . . . . . . . . 10 ⊢ (𝑃 ∈ CRing → (mulGrp‘𝑃) ∈ CMnd)
86	84, 85	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (mulGrp‘𝑃) ∈ CMnd)
87	86	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (mulGrp‘𝑃) ∈ CMnd)
88	54	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝐴‘ran 𝑉) ∈ (SubRing‘𝑃))
89	76	subrgsubm 20500	. . . . . . . . 9 ⊢ ((𝐴‘ran 𝑉) ∈ (SubRing‘𝑃) → (𝐴‘ran 𝑉) ∈ (SubMnd‘(mulGrp‘𝑃)))
90	88, 89	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝐴‘ran 𝑉) ∈ (SubMnd‘(mulGrp‘𝑃)))
91		simplll 774	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧 ∈ 𝐼) → 𝜑)
92	32	psrbag 21832	. . . . . . . . . . . . . 14 ⊢ (𝐼 ∈ 𝑊 → (𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↔ (𝑘:𝐼⟶ℕ0 ∧ (◡𝑘 “ ℕ) ∈ Fin)))
93	35, 92	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) → (𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↔ (𝑘:𝐼⟶ℕ0 ∧ (◡𝑘 “ ℕ) ∈ Fin)))
94	93	biimpa 476	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝑘:𝐼⟶ℕ0 ∧ (◡𝑘 “ ℕ) ∈ Fin))
95	94	simpld 494	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → 𝑘:𝐼⟶ℕ0)
96	95	ffvelcdmda 7058	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧 ∈ 𝐼) → (𝑘‘𝑧) ∈ ℕ0)
97	23, 7	aspssid 21793	. . . . . . . . . . . . 13 ⊢ ((𝑆 ∈ AssAlg ∧ ran 𝑉 ⊆ (Base‘𝑆)) → ran 𝑉 ⊆ (𝐴‘ran 𝑉))
98	4, 48, 97	syl2anc 584	. . . . . . . . . . . 12 ⊢ (𝜑 → ran 𝑉 ⊆ (𝐴‘ran 𝑉))
99	98	ad3antrrr 730	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧 ∈ 𝐼) → ran 𝑉 ⊆ (𝐴‘ran 𝑉))
100	14	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → 𝑉 Fn 𝐼)
101		fnfvelrn 7054	. . . . . . . . . . . 12 ⊢ ((𝑉 Fn 𝐼 ∧ 𝑧 ∈ 𝐼) → (𝑉‘𝑧) ∈ ran 𝑉)
102	100, 101	sylan 580	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧 ∈ 𝐼) → (𝑉‘𝑧) ∈ ran 𝑉)
103	99, 102	sseldd 3949	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧 ∈ 𝐼) → (𝑉‘𝑧) ∈ (𝐴‘ran 𝑉))
104	76, 6	mgpbas 20060	. . . . . . . . . . 11 ⊢ (Base‘𝑃) = (Base‘(mulGrp‘𝑃))
105		eqid 2730	. . . . . . . . . . . 12 ⊢ (.r‘𝑃) = (.r‘𝑃)
106	76, 105	mgpplusg 20059	. . . . . . . . . . 11 ⊢ (.r‘𝑃) = (+g‘(mulGrp‘𝑃))
107	105	subrgmcl 20499	. . . . . . . . . . . 12 ⊢ (((𝐴‘ran 𝑉) ∈ (SubRing‘𝑃) ∧ 𝑢 ∈ (𝐴‘ran 𝑉) ∧ 𝑣 ∈ (𝐴‘ran 𝑉)) → (𝑢(.r‘𝑃)𝑣) ∈ (𝐴‘ran 𝑉))
108	54, 107	syl3an1 1163	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐴‘ran 𝑉) ∧ 𝑣 ∈ (𝐴‘ran 𝑉)) → (𝑢(.r‘𝑃)𝑣) ∈ (𝐴‘ran 𝑉))
109	81	subrg1cl 20495	. . . . . . . . . . . 12 ⊢ ((𝐴‘ran 𝑉) ∈ (SubRing‘𝑃) → (1r‘𝑃) ∈ (𝐴‘ran 𝑉))
110	54, 109	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (1r‘𝑃) ∈ (𝐴‘ran 𝑉))
111	104, 77, 106, 86, 31, 108, 82, 110	mulgnn0subcl 19025	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘‘𝑧) ∈ ℕ0 ∧ (𝑉‘𝑧) ∈ (𝐴‘ran 𝑉)) → ((𝑘‘𝑧)(.g‘(mulGrp‘𝑃))(𝑉‘𝑧)) ∈ (𝐴‘ran 𝑉))
112	91, 96, 103, 111	syl3anc 1373	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧 ∈ 𝐼) → ((𝑘‘𝑧)(.g‘(mulGrp‘𝑃))(𝑉‘𝑧)) ∈ (𝐴‘ran 𝑉))
113	112	fmpttd 7089	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝑧 ∈ 𝐼 ↦ ((𝑘‘𝑧)(.g‘(mulGrp‘𝑃))(𝑉‘𝑧))):𝐼⟶(𝐴‘ran 𝑉))
114	2	mptexd 7200	. . . . . . . . . 10 ⊢ (𝜑 → (𝑧 ∈ 𝐼 ↦ ((𝑘‘𝑧)(.g‘(mulGrp‘𝑃))(𝑉‘𝑧))) ∈ V)
115	114	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝑧 ∈ 𝐼 ↦ ((𝑘‘𝑧)(.g‘(mulGrp‘𝑃))(𝑉‘𝑧))) ∈ V)
116		funmpt 6556	. . . . . . . . . 10 ⊢ Fun (𝑧 ∈ 𝐼 ↦ ((𝑘‘𝑧)(.g‘(mulGrp‘𝑃))(𝑉‘𝑧)))
117	116	a1i 11	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → Fun (𝑧 ∈ 𝐼 ↦ ((𝑘‘𝑧)(.g‘(mulGrp‘𝑃))(𝑉‘𝑧))))
118		fvexd 6875	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (1r‘𝑃) ∈ V)
119	94	simprd 495	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (◡𝑘 “ ℕ) ∈ Fin)
120		elrabi 3656	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} → 𝑘 ∈ (ℕ0 ↑m 𝐼))
121		elmapi 8824	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (ℕ0 ↑m 𝐼) → 𝑘:𝐼⟶ℕ0)
122	121	adantl 481	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ (ℕ0 ↑m 𝐼)) → 𝑘:𝐼⟶ℕ0)
123	2	ad2antrr 726	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ (ℕ0 ↑m 𝐼)) → 𝐼 ∈ 𝑊)
124		fcdmnn0supp 12505	. . . . . . . . . . . . . . . 16 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑘:𝐼⟶ℕ0) → (𝑘 supp 0) = (◡𝑘 “ ℕ))
125	123, 122, 124	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ (ℕ0 ↑m 𝐼)) → (𝑘 supp 0) = (◡𝑘 “ ℕ))
126		eqimss 4007	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 supp 0) = (◡𝑘 “ ℕ) → (𝑘 supp 0) ⊆ (◡𝑘 “ ℕ))
127	125, 126	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ (ℕ0 ↑m 𝐼)) → (𝑘 supp 0) ⊆ (◡𝑘 “ ℕ))
128		c0ex 11174	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ V
129	128	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ (ℕ0 ↑m 𝐼)) → 0 ∈ V)
130	122, 127, 123, 129	suppssr 8176	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ (ℕ0 ↑m 𝐼)) ∧ 𝑧 ∈ (𝐼 ∖ (◡𝑘 “ ℕ))) → (𝑘‘𝑧) = 0)
131	120, 130	sylanl2 681	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧 ∈ (𝐼 ∖ (◡𝑘 “ ℕ))) → (𝑘‘𝑧) = 0)
132	131	oveq1d 7404	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧 ∈ (𝐼 ∖ (◡𝑘 “ ℕ))) → ((𝑘‘𝑧)(.g‘(mulGrp‘𝑃))(𝑉‘𝑧)) = (0(.g‘(mulGrp‘𝑃))(𝑉‘𝑧)))
133	2	ad3antrrr 730	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧 ∈ (𝐼 ∖ (◡𝑘 “ ℕ))) → 𝐼 ∈ 𝑊)
134	12	ad3antrrr 730	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧 ∈ (𝐼 ∖ (◡𝑘 “ ℕ))) → 𝑅 ∈ Ring)
135		eldifi 4096	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ (𝐼 ∖ (◡𝑘 “ ℕ)) → 𝑧 ∈ 𝐼)
136	135	adantl 481	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧 ∈ (𝐼 ∖ (◡𝑘 “ ℕ))) → 𝑧 ∈ 𝐼)
137	5, 10, 6, 133, 134, 136	mvrcl 21907	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧 ∈ (𝐼 ∖ (◡𝑘 “ ℕ))) → (𝑉‘𝑧) ∈ (Base‘𝑃))
138	104, 82, 77	mulg0 19012	. . . . . . . . . . . 12 ⊢ ((𝑉‘𝑧) ∈ (Base‘𝑃) → (0(.g‘(mulGrp‘𝑃))(𝑉‘𝑧)) = (1r‘𝑃))
139	137, 138	syl 17	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧 ∈ (𝐼 ∖ (◡𝑘 “ ℕ))) → (0(.g‘(mulGrp‘𝑃))(𝑉‘𝑧)) = (1r‘𝑃))
140	132, 139	eqtrd 2765	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧 ∈ (𝐼 ∖ (◡𝑘 “ ℕ))) → ((𝑘‘𝑧)(.g‘(mulGrp‘𝑃))(𝑉‘𝑧)) = (1r‘𝑃))
141	140, 75	suppss2 8181	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → ((𝑧 ∈ 𝐼 ↦ ((𝑘‘𝑧)(.g‘(mulGrp‘𝑃))(𝑉‘𝑧))) supp (1r‘𝑃)) ⊆ (◡𝑘 “ ℕ))
142		suppssfifsupp 9337	. . . . . . . . 9 ⊢ ((((𝑧 ∈ 𝐼 ↦ ((𝑘‘𝑧)(.g‘(mulGrp‘𝑃))(𝑉‘𝑧))) ∈ V ∧ Fun (𝑧 ∈ 𝐼 ↦ ((𝑘‘𝑧)(.g‘(mulGrp‘𝑃))(𝑉‘𝑧))) ∧ (1r‘𝑃) ∈ V) ∧ ((◡𝑘 “ ℕ) ∈ Fin ∧ ((𝑧 ∈ 𝐼 ↦ ((𝑘‘𝑧)(.g‘(mulGrp‘𝑃))(𝑉‘𝑧))) supp (1r‘𝑃)) ⊆ (◡𝑘 “ ℕ))) → (𝑧 ∈ 𝐼 ↦ ((𝑘‘𝑧)(.g‘(mulGrp‘𝑃))(𝑉‘𝑧))) finSupp (1r‘𝑃))
143	115, 117, 118, 119, 141, 142	syl32anc 1380	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝑧 ∈ 𝐼 ↦ ((𝑘‘𝑧)(.g‘(mulGrp‘𝑃))(𝑉‘𝑧))) finSupp (1r‘𝑃))
144	82, 87, 75, 90, 113, 143	gsumsubmcl 19855	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → ((mulGrp‘𝑃) Σg (𝑧 ∈ 𝐼 ↦ ((𝑘‘𝑧)(.g‘(mulGrp‘𝑃))(𝑉‘𝑧)))) ∈ (𝐴‘ran 𝑉))
145	80, 144	eqeltrd 2829	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝑦 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = 𝑘, (1r‘𝑅), (0g‘𝑅))) ∈ (𝐴‘ran 𝑉))
146		eqid 2730	. . . . . . 7 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃)
147		eqid 2730	. . . . . . 7 ⊢ (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
148	146, 36, 147, 63	lssvscl 20867	. . . . . 6 ⊢ (((𝑃 ∈ LMod ∧ (𝐴‘ran 𝑉) ∈ (LSubSp‘𝑃)) ∧ ((𝑥‘𝑘) ∈ (Base‘(Scalar‘𝑃)) ∧ (𝑦 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = 𝑘, (1r‘𝑅), (0g‘𝑅))) ∈ (𝐴‘ran 𝑉))) → ((𝑥‘𝑘)( ·𝑠 ‘𝑃)(𝑦 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = 𝑘, (1r‘𝑅), (0g‘𝑅)))) ∈ (𝐴‘ran 𝑉))
149	59, 67, 74, 145, 148	syl22anc 838	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → ((𝑥‘𝑘)( ·𝑠 ‘𝑃)(𝑦 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = 𝑘, (1r‘𝑅), (0g‘𝑅)))) ∈ (𝐴‘ran 𝑉))
150	149	fmpttd 7089	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) → (𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ ((𝑥‘𝑘)( ·𝑠 ‘𝑃)(𝑦 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = 𝑘, (1r‘𝑅), (0g‘𝑅))))):{𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶(𝐴‘ran 𝑉))
151	45	mptrabex 7201	. . . . . . 7 ⊢ (𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ ((𝑥‘𝑘)( ·𝑠 ‘𝑃)(𝑦 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = 𝑘, (1r‘𝑅), (0g‘𝑅))))) ∈ V
152		funmpt 6556	. . . . . . 7 ⊢ Fun (𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ ((𝑥‘𝑘)( ·𝑠 ‘𝑃)(𝑦 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = 𝑘, (1r‘𝑅), (0g‘𝑅)))))
153		fvex 6873	. . . . . . 7 ⊢ (0g‘𝑃) ∈ V
154	151, 152, 153	3pm3.2i 1340	. . . . . 6 ⊢ ((𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ ((𝑥‘𝑘)( ·𝑠 ‘𝑃)(𝑦 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = 𝑘, (1r‘𝑅), (0g‘𝑅))))) ∈ V ∧ Fun (𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ ((𝑥‘𝑘)( ·𝑠 ‘𝑃)(𝑦 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = 𝑘, (1r‘𝑅), (0g‘𝑅))))) ∧ (0g‘𝑃) ∈ V)
155	154	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) → ((𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ ((𝑥‘𝑘)( ·𝑠 ‘𝑃)(𝑦 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = 𝑘, (1r‘𝑅), (0g‘𝑅))))) ∈ V ∧ Fun (𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ ((𝑥‘𝑘)( ·𝑠 ‘𝑃)(𝑦 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = 𝑘, (1r‘𝑅), (0g‘𝑅))))) ∧ (0g‘𝑃) ∈ V))
156	5, 1, 7, 33, 6	mplelbas 21906	. . . . . . . 8 ⊢ (𝑥 ∈ (Base‘𝑃) ↔ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 finSupp (0g‘𝑅)))
157	156	simprbi 496	. . . . . . 7 ⊢ (𝑥 ∈ (Base‘𝑃) → 𝑥 finSupp (0g‘𝑅))
158	157	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) → 𝑥 finSupp (0g‘𝑅))
159	158	fsuppimpd 9326	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) → (𝑥 supp (0g‘𝑅)) ∈ Fin)
160		ssidd 3972	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) → (𝑥 supp (0g‘𝑅)) ⊆ (𝑥 supp (0g‘𝑅)))
161		fvexd 6875	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) → (0g‘𝑅) ∈ V)
162	69, 160, 47, 161	suppssr 8176	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ ({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∖ (𝑥 supp (0g‘𝑅)))) → (𝑥‘𝑘) = (0g‘𝑅))
163	71	fveq2d 6864	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) → (0g‘𝑅) = (0g‘(Scalar‘𝑃)))
164	163	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ ({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∖ (𝑥 supp (0g‘𝑅)))) → (0g‘𝑅) = (0g‘(Scalar‘𝑃)))
165	162, 164	eqtrd 2765	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ ({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∖ (𝑥 supp (0g‘𝑅)))) → (𝑥‘𝑘) = (0g‘(Scalar‘𝑃)))
166	165	oveq1d 7404	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ ({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∖ (𝑥 supp (0g‘𝑅)))) → ((𝑥‘𝑘)( ·𝑠 ‘𝑃)(𝑦 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = 𝑘, (1r‘𝑅), (0g‘𝑅)))) = ((0g‘(Scalar‘𝑃))( ·𝑠 ‘𝑃)(𝑦 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = 𝑘, (1r‘𝑅), (0g‘𝑅)))))
167		eldifi 4096	. . . . . . . 8 ⊢ (𝑘 ∈ ({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∖ (𝑥 supp (0g‘𝑅))) → 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin})
168	12	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → 𝑅 ∈ Ring)
169	5, 6, 33, 34, 32, 75, 168, 79	mplmon 21948	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝑦 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = 𝑘, (1r‘𝑅), (0g‘𝑅))) ∈ (Base‘𝑃))
170		eqid 2730	. . . . . . . . . 10 ⊢ (0g‘(Scalar‘𝑃)) = (0g‘(Scalar‘𝑃))
171	6, 146, 36, 170, 40	lmod0vs 20807	. . . . . . . . 9 ⊢ ((𝑃 ∈ LMod ∧ (𝑦 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = 𝑘, (1r‘𝑅), (0g‘𝑅))) ∈ (Base‘𝑃)) → ((0g‘(Scalar‘𝑃))( ·𝑠 ‘𝑃)(𝑦 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = 𝑘, (1r‘𝑅), (0g‘𝑅)))) = (0g‘𝑃))
172	59, 169, 171	syl2anc 584	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → ((0g‘(Scalar‘𝑃))( ·𝑠 ‘𝑃)(𝑦 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = 𝑘, (1r‘𝑅), (0g‘𝑅)))) = (0g‘𝑃))
173	167, 172	sylan2 593	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ ({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∖ (𝑥 supp (0g‘𝑅)))) → ((0g‘(Scalar‘𝑃))( ·𝑠 ‘𝑃)(𝑦 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = 𝑘, (1r‘𝑅), (0g‘𝑅)))) = (0g‘𝑃))
174	166, 173	eqtrd 2765	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ ({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∖ (𝑥 supp (0g‘𝑅)))) → ((𝑥‘𝑘)( ·𝑠 ‘𝑃)(𝑦 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = 𝑘, (1r‘𝑅), (0g‘𝑅)))) = (0g‘𝑃))
175	174, 47	suppss2 8181	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) → ((𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ ((𝑥‘𝑘)( ·𝑠 ‘𝑃)(𝑦 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = 𝑘, (1r‘𝑅), (0g‘𝑅))))) supp (0g‘𝑃)) ⊆ (𝑥 supp (0g‘𝑅)))
176		suppssfifsupp 9337	. . . . 5 ⊢ ((((𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ ((𝑥‘𝑘)( ·𝑠 ‘𝑃)(𝑦 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = 𝑘, (1r‘𝑅), (0g‘𝑅))))) ∈ V ∧ Fun (𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ ((𝑥‘𝑘)( ·𝑠 ‘𝑃)(𝑦 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = 𝑘, (1r‘𝑅), (0g‘𝑅))))) ∧ (0g‘𝑃) ∈ V) ∧ ((𝑥 supp (0g‘𝑅)) ∈ Fin ∧ ((𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ ((𝑥‘𝑘)( ·𝑠 ‘𝑃)(𝑦 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = 𝑘, (1r‘𝑅), (0g‘𝑅))))) supp (0g‘𝑃)) ⊆ (𝑥 supp (0g‘𝑅)))) → (𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ ((𝑥‘𝑘)( ·𝑠 ‘𝑃)(𝑦 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = 𝑘, (1r‘𝑅), (0g‘𝑅))))) finSupp (0g‘𝑃))
177	155, 159, 175, 176	syl12anc 836	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) → (𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ ((𝑥‘𝑘)( ·𝑠 ‘𝑃)(𝑦 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = 𝑘, (1r‘𝑅), (0g‘𝑅))))) finSupp (0g‘𝑃))
178	40, 44, 47, 57, 150, 177	gsumsubgcl 19856	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) → (𝑃 Σg (𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ ((𝑥‘𝑘)( ·𝑠 ‘𝑃)(𝑦 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = 𝑘, (1r‘𝑅), (0g‘𝑅)))))) ∈ (𝐴‘ran 𝑉))
179	39, 178	eqeltrd 2829	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) → 𝑥 ∈ (𝐴‘ran 𝑉))
180	31, 179	eqelssd 3970	1 ⊢ (𝜑 → (𝐴‘ran 𝑉) = (Base‘𝑃))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3045  {crab 3408  Vcvv 3450   ∖ cdif 3913   ⊆ wss 3916  ifcif 4490   class class class wbr 5109   ↦ cmpt 5190  ^◡ccnv 5639  ran crn 5641   “ cima 5643  Fun wfun 6507   Fn wfn 6508  ⟶wf 6509  ‘cfv 6513  (class class class)co 7389   supp csupp 8141   ↑_m cmap 8801  Fincfn 8920   finSupp cfsupp 9318  0cc0 11074  ℕcn 12187  ℕ₀cn0 12448  Basecbs 17185  ._rcmulr 17227  Scalarcsca 17229   ·_𝑠 cvsca 17230  0_gc0g 17408   Σ_g cgsu 17409  SubMndcsubmnd 18715  ._gcmg 19005  SubGrpcsubg 19058  CMndccmn 19716  Abelcabl 19717  mulGrpcmgp 20055  1_rcur 20096  Ringcrg 20148  CRingccrg 20149  SubRingcsubrg 20484  LModclmod 20772  LSubSpclss 20843  AssAlgcasa 21765  AlgSpancasp 21766   mPwSer cmps 21819   mVar cmvr 21820   mPoly cmpl 21821

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5236  ax-sep 5253  ax-nul 5263  ax-pow 5322  ax-pr 5389  ax-un 7713  ax-cnex 11130  ax-resscn 11131  ax-1cn 11132  ax-icn 11133  ax-addcl 11134  ax-addrcl 11135  ax-mulcl 11136  ax-mulrcl 11137  ax-mulcom 11138  ax-addass 11139  ax-mulass 11140  ax-distr 11141  ax-i2m1 11142  ax-1ne0 11143  ax-1rid 11144  ax-rnegex 11145  ax-rrecex 11146  ax-cnre 11147  ax-pre-lttri 11148  ax-pre-lttrn 11149  ax-pre-ltadd 11150  ax-pre-mulgt0 11151

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3756  df-csb 3865  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-pss 3936  df-nul 4299  df-if 4491  df-pw 4567  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4874  df-int 4913  df-iun 4959  df-iin 4960  df-br 5110  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5535  df-eprel 5540  df-po 5548  df-so 5549  df-fr 5593  df-se 5594  df-we 5595  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-pred 6276  df-ord 6337  df-on 6338  df-lim 6339  df-suc 6340  df-iota 6466  df-fun 6515  df-fn 6516  df-f 6517  df-f1 6518  df-fo 6519  df-f1o 6520  df-fv 6521  df-isom 6522  df-riota 7346  df-ov 7392  df-oprab 7393  df-mpo 7394  df-of 7655  df-ofr 7656  df-om 7845  df-1st 7970  df-2nd 7971  df-supp 8142  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8380  df-1o 8436  df-2o 8437  df-er 8673  df-map 8803  df-pm 8804  df-ixp 8873  df-en 8921  df-dom 8922  df-sdom 8923  df-fin 8924  df-fsupp 9319  df-sup 9399  df-oi 9469  df-card 9898  df-pnf 11216  df-mnf 11217  df-xr 11218  df-ltxr 11219  df-le 11220  df-sub 11413  df-neg 11414  df-nn 12188  df-2 12250  df-3 12251  df-4 12252  df-5 12253  df-6 12254  df-7 12255  df-8 12256  df-9 12257  df-n0 12449  df-z 12536  df-dec 12656  df-uz 12800  df-fz 13475  df-fzo 13622  df-seq 13973  df-hash 14302  df-struct 17123  df-sets 17140  df-slot 17158  df-ndx 17170  df-base 17186  df-ress 17207  df-plusg 17239  df-mulr 17240  df-sca 17242  df-vsca 17243  df-ip 17244  df-tset 17245  df-ple 17246  df-ds 17248  df-hom 17250  df-cco 17251  df-0g 17410  df-gsum 17411  df-prds 17416  df-pws 17418  df-mre 17553  df-mrc 17554  df-acs 17556  df-mgm 18573  df-sgrp 18652  df-mnd 18668  df-mhm 18716  df-submnd 18717  df-grp 18874  df-minusg 18875  df-sbg 18876  df-mulg 19006  df-subg 19061  df-ghm 19151  df-cntz 19255  df-cmn 19718  df-abl 19719  df-mgp 20056  df-rng 20068  df-ur 20097  df-srg 20102  df-ring 20150  df-cring 20151  df-subrng 20461  df-subrg 20485  df-lmod 20774  df-lss 20844  df-assa 21768  df-asp 21769  df-psr 21824  df-mvr 21825  df-mpl 21826

This theorem is referenced by:  mplind  21983  evlseu  21996