Theorem mplbas2 22102

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 22031	. . . 4 ⊢ (𝜑 → 𝑆 ∈ AssAlg)
5		mplbas2.p	. . . . . 6 ⊢ 𝑃 = (𝐼 mPoly 𝑅)
6		eqid 2763	. . . . . 6 ⊢ (Base‘𝑃) = (Base‘𝑃)
7		eqid 2763	. . . . . 6 ⊢ (Base‘𝑆) = (Base‘𝑆)
8	5, 1, 6, 7	mplbasss 22055	. . . . 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 20305	. . . . . . . . 9 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
12	3, 11	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ Ring)
13	1, 10, 7, 2, 12	mvrf 22043	. . . . . . 7 ⊢ (𝜑 → 𝑉:𝐼⟶(Base‘𝑆))
14	13	ffnd 6692	. . . . . 6 ⊢ (𝜑 → 𝑉 Fn 𝐼)
15	2	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐼 ∈ 𝑊)
16	12	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑅 ∈ Ring)
17		simpr 488	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑥 ∈ 𝐼)
18	5, 10, 6, 15, 16, 17	mvrcl 22050	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑉‘𝑥) ∈ (Base‘𝑃))
19	18	ralrimiva 3155	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 (𝑉‘𝑥) ∈ (Base‘𝑃))
20		ffnfv 7100	. . . . . 6 ⊢ (𝑉:𝐼⟶(Base‘𝑃) ↔ (𝑉 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝑉‘𝑥) ∈ (Base‘𝑃)))
21	14, 19, 20	sylanbrc 592	. . . . 5 ⊢ (𝜑 → 𝑉:𝐼⟶(Base‘𝑃))
22	21	frnd 6700	. . . 4 ⊢ (𝜑 → ran 𝑉 ⊆ (Base‘𝑃))
23		mplbas2.a	. . . . 5 ⊢ 𝐴 = (AlgSpan‘𝑆)
24	23, 7	aspss 21935	. . . 4 ⊢ ((𝑆 ∈ AssAlg ∧ (Base‘𝑃) ⊆ (Base‘𝑆) ∧ ran 𝑉 ⊆ (Base‘𝑃)) → (𝐴‘ran 𝑉) ⊆ (𝐴‘(Base‘𝑃)))
25	4, 9, 22, 24	syl3anc 1392	. . 3 ⊢ (𝜑 → (𝐴‘ran 𝑉) ⊆ (𝐴‘(Base‘𝑃)))
26	1, 5, 6, 2, 12	mplsubrg 22063	. . . 4 ⊢ (𝜑 → (Base‘𝑃) ∈ (SubRing‘𝑆))
27	1, 5, 6, 2, 12	mpllss 22061	. . . 4 ⊢ (𝜑 → (Base‘𝑃) ∈ (LSubSp‘𝑆))
28		eqid 2763	. . . . 5 ⊢ (LSubSp‘𝑆) = (LSubSp‘𝑆)
29	23, 7, 28	aspid 21933	. . . 4 ⊢ ((𝑆 ∈ AssAlg ∧ (Base‘𝑃) ∈ (SubRing‘𝑆) ∧ (Base‘𝑃) ∈ (LSubSp‘𝑆)) → (𝐴‘(Base‘𝑃)) = (Base‘𝑃))
30	4, 26, 27, 29	syl3anc 1392	. . 3 ⊢ (𝜑 → (𝐴‘(Base‘𝑃)) = (Base‘𝑃))
31	25, 30	sseqtrd 3973	. 2 ⊢ (𝜑 → (𝐴‘ran 𝑉) ⊆ (Base‘𝑃))
32		eqid 2763	. . . 4 ⊢ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}
33		eqid 2763	. . . 4 ⊢ (0g‘𝑅) = (0g‘𝑅)
34		eqid 2763	. . . 4 ⊢ (1r‘𝑅) = (1r‘𝑅)
35	2	adantr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) → 𝐼 ∈ 𝑊)
36		eqid 2763	. . . 4 ⊢ ( ·𝑠 ‘𝑃) = ( ·𝑠 ‘𝑃)
37	12	adantr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) → 𝑅 ∈ Ring)
38		simpr 488	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) → 𝑥 ∈ (Base‘𝑃))
39	5, 32, 33, 34, 35, 6, 36, 37, 38	mplcoe1 22097	. . 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 2763	. . . 4 ⊢ (0g‘𝑃) = (0g‘𝑃)
41	5, 2, 12	mplringd 22081	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ Ring)
42		ringabl 20341	. . . . . 6 ⊢ (𝑃 ∈ Ring → 𝑃 ∈ Abel)
43	41, 42	syl 17	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ Abel)
44	43	adantr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) → 𝑃 ∈ Abel)
45		ovex 7429	. . . . . 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 6700	. . . . . . . 8 ⊢ (𝜑 → ran 𝑉 ⊆ (Base‘𝑆))
49	23, 7	aspsubrg 21934	. . . . . . . 8 ⊢ ((𝑆 ∈ AssAlg ∧ ran 𝑉 ⊆ (Base‘𝑆)) → (𝐴‘ran 𝑉) ∈ (SubRing‘𝑆))
50	4, 48, 49	syl2anc 593	. . . . . . 7 ⊢ (𝜑 → (𝐴‘ran 𝑉) ∈ (SubRing‘𝑆))
51	5, 1, 6	mplval2 22054	. . . . . . . . 9 ⊢ 𝑃 = (𝑆 ↾s (Base‘𝑃))
52	51	subsubrg 20658	. . . . . . . 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 723	. . . . . 6 ⊢ (𝜑 → (𝐴‘ran 𝑉) ∈ (SubRing‘𝑃))
55		subrgsubg 20637	. . . . . 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 484	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) → (𝐴‘ran 𝑉) ∈ (SubGrp‘𝑃))
58	5, 2, 12	mpllmodd 22083	. . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ LMod)
59	58	ad2antrr 736	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → 𝑃 ∈ LMod)
60	23, 7, 28	asplss 21932	. . . . . . . . 9 ⊢ ((𝑆 ∈ AssAlg ∧ ran 𝑉 ⊆ (Base‘𝑆)) → (𝐴‘ran 𝑉) ∈ (LSubSp‘𝑆))
61	4, 48, 60	syl2anc 593	. . . . . . . 8 ⊢ (𝜑 → (𝐴‘ran 𝑉) ∈ (LSubSp‘𝑆))
62	1, 2, 12	psrlmod 22018	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ LMod)
63		eqid 2763	. . . . . . . . . 10 ⊢ (LSubSp‘𝑃) = (LSubSp‘𝑃)
64	51, 28, 63	lsslss 21035	. . . . . . . . 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 593	. . . . . . . 8 ⊢ (𝜑 → ((𝐴‘ran 𝑉) ∈ (LSubSp‘𝑃) ↔ ((𝐴‘ran 𝑉) ∈ (LSubSp‘𝑆) ∧ (𝐴‘ran 𝑉) ⊆ (Base‘𝑃))))
66	61, 31, 65	mpbir2and 723	. . . . . . 7 ⊢ (𝜑 → (𝐴‘ran 𝑉) ∈ (LSubSp‘𝑃))
67	66	ad2antrr 736	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝐴‘ran 𝑉) ∈ (LSubSp‘𝑃))
68		eqid 2763	. . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅)
69	5, 68, 6, 32, 38	mplelf 22056	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) → 𝑥:{𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
70	69	ffvelcdmda 7065	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝑥‘𝑘) ∈ (Base‘𝑅))
71	5, 35, 37	mplsca 22071	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) → 𝑅 = (Scalar‘𝑃))
72	71	adantr 484	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → 𝑅 = (Scalar‘𝑃))
73	72	fveq2d 6871	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (Base‘𝑅) = (Base‘(Scalar‘𝑃)))
74	70, 73	eleqtrd 2865	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝑥‘𝑘) ∈ (Base‘(Scalar‘𝑃)))
75	2	ad2antrr 736	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → 𝐼 ∈ 𝑊)
76		eqid 2763	. . . . . . . 8 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃)
77		eqid 2763	. . . . . . . 8 ⊢ (.g‘(mulGrp‘𝑃)) = (.g‘(mulGrp‘𝑃))
78	3	ad2antrr 736	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → 𝑅 ∈ CRing)
79		simpr 488	. . . . . . . 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 22101	. . . . . . 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 2763	. . . . . . . . 9 ⊢ (1r‘𝑃) = (1r‘𝑃)
82	76, 81	ringidval 20243	. . . . . . . 8 ⊢ (1r‘𝑃) = (0g‘(mulGrp‘𝑃))
83	5	mplcrng 22079	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ CRing) → 𝑃 ∈ CRing)
84	2, 3, 83	syl2anc 593	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ CRing)
85	76	crngmgp 20301	. . . . . . . . . 10 ⊢ (𝑃 ∈ CRing → (mulGrp‘𝑃) ∈ CMnd)
86	84, 85	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (mulGrp‘𝑃) ∈ CMnd)
87	86	ad2antrr 736	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (mulGrp‘𝑃) ∈ CMnd)
88	54	ad2antrr 736	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝐴‘ran 𝑉) ∈ (SubRing‘𝑃))
89	76	subrgsubm 20645	. . . . . . . . 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 784	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧 ∈ 𝐼) → 𝜑)
92	32	psrbag 21976	. . . . . . . . . . . . . 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 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝑘:𝐼⟶ℕ0 ∧ (◡𝑘 “ ℕ) ∈ Fin))
95	94	simpld 498	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → 𝑘:𝐼⟶ℕ0)
96	95	ffvelcdmda 7065	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧 ∈ 𝐼) → (𝑘‘𝑧) ∈ ℕ0)
97	23, 7	aspssid 21936	. . . . . . . . . . . . 13 ⊢ ((𝑆 ∈ AssAlg ∧ ran 𝑉 ⊆ (Base‘𝑆)) → ran 𝑉 ⊆ (𝐴‘ran 𝑉))
98	4, 48, 97	syl2anc 593	. . . . . . . . . . . 12 ⊢ (𝜑 → ran 𝑉 ⊆ (𝐴‘ran 𝑉))
99	98	ad3antrrr 740	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧 ∈ 𝐼) → ran 𝑉 ⊆ (𝐴‘ran 𝑉))
100	14	ad2antrr 736	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → 𝑉 Fn 𝐼)
101		fnfvelrn 7061	. . . . . . . . . . . 12 ⊢ ((𝑉 Fn 𝐼 ∧ 𝑧 ∈ 𝐼) → (𝑉‘𝑧) ∈ ran 𝑉)
102	100, 101	sylan 589	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧 ∈ 𝐼) → (𝑉‘𝑧) ∈ ran 𝑉)
103	99, 102	sseldd 3938	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧 ∈ 𝐼) → (𝑉‘𝑧) ∈ (𝐴‘ran 𝑉))
104	76, 6	mgpbas 20201	. . . . . . . . . . 11 ⊢ (Base‘𝑃) = (Base‘(mulGrp‘𝑃))
105		eqid 2763	. . . . . . . . . . . 12 ⊢ (.r‘𝑃) = (.r‘𝑃)
106	76, 105	mgpplusg 20200	. . . . . . . . . . 11 ⊢ (.r‘𝑃) = (+g‘(mulGrp‘𝑃))
107	105	subrgmcl 20644	. . . . . . . . . . . 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 1177	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐴‘ran 𝑉) ∧ 𝑣 ∈ (𝐴‘ran 𝑉)) → (𝑢(.r‘𝑃)𝑣) ∈ (𝐴‘ran 𝑉))
109	81	subrg1cl 20640	. . . . . . . . . . . 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 19139	. . . . . . . . . 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 1392	. . . . . . . . 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 7096	. . . . . . . 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 7208	. . . . . . . . . 10 ⊢ (𝜑 → (𝑧 ∈ 𝐼 ↦ ((𝑘‘𝑧)(.g‘(mulGrp‘𝑃))(𝑉‘𝑧))) ∈ V)
115	114	ad2antrr 736	. . . . . . . . 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 6559	. . . . . . . . . 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 6882	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (1r‘𝑃) ∈ V)
119	94	simprd 499	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (◡𝑘 “ ℕ) ∈ Fin)
120		elrabi 3647	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} → 𝑘 ∈ (ℕ0 ↑m 𝐼))
121		elmapi 8830	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (ℕ0 ↑m 𝐼) → 𝑘:𝐼⟶ℕ0)
122	121	adantl 485	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ (ℕ0 ↑m 𝐼)) → 𝑘:𝐼⟶ℕ0)
123	2	ad2antrr 736	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ (ℕ0 ↑m 𝐼)) → 𝐼 ∈ 𝑊)
124		fcdmnn0supp 12548	. . . . . . . . . . . . . . . 16 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑘:𝐼⟶ℕ0) → (𝑘 supp 0) = (◡𝑘 “ ℕ))
125	123, 122, 124	syl2anc 593	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ (ℕ0 ↑m 𝐼)) → (𝑘 supp 0) = (◡𝑘 “ ℕ))
126		eqimss 3995	. . . . . . . . . . . . . . 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 11184	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ V
129	128	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ (ℕ0 ↑m 𝐼)) → 0 ∈ V)
130	122, 127, 123, 129	suppssr 8175	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ (ℕ0 ↑m 𝐼)) ∧ 𝑧 ∈ (𝐼 ∖ (◡𝑘 “ ℕ))) → (𝑘‘𝑧) = 0)
131	120, 130	sylanl2 691	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧 ∈ (𝐼 ∖ (◡𝑘 “ ℕ))) → (𝑘‘𝑧) = 0)
132	131	oveq1d 7411	. . . . . . . . . . 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 740	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧 ∈ (𝐼 ∖ (◡𝑘 “ ℕ))) → 𝐼 ∈ 𝑊)
134	12	ad3antrrr 740	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧 ∈ (𝐼 ∖ (◡𝑘 “ ℕ))) → 𝑅 ∈ Ring)
135		eldifi 4085	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ (𝐼 ∖ (◡𝑘 “ ℕ)) → 𝑧 ∈ 𝐼)
136	135	adantl 485	. . . . . . . . . . . . 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 22050	. . . . . . . . . . . 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 19126	. . . . . . . . . . . 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 2798	. . . . . . . . . 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 8180	. . . . . . . . 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 9324	. . . . . . . . 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 1399	. . . . . . . 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 19969	. . . . . . 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 2863	. . . . . 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 2763	. . . . . . 7 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃)
147		eqid 2763	. . . . . . 7 ⊢ (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
148	146, 36, 147, 63	lssvscl 21029	. . . . . 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 849	. . . . 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 7096	. . . 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 7209	. . . . . . 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 6559	. . . . . . 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 6880	. . . . . . 7 ⊢ (0g‘𝑃) ∈ V
154	151, 152, 153	3pm3.2i 1354	. . . . . 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 22049	. . . . . . . 8 ⊢ (𝑥 ∈ (Base‘𝑃) ↔ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 finSupp (0g‘𝑅)))
157	156	simprbi 501	. . . . . . 7 ⊢ (𝑥 ∈ (Base‘𝑃) → 𝑥 finSupp (0g‘𝑅))
158	157	adantl 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) → 𝑥 finSupp (0g‘𝑅))
159	158	fsuppimpd 9313	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) → (𝑥 supp (0g‘𝑅)) ∈ Fin)
160		ssidd 3960	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) → (𝑥 supp (0g‘𝑅)) ⊆ (𝑥 supp (0g‘𝑅)))
161		fvexd 6882	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) → (0g‘𝑅) ∈ V)
162	69, 160, 47, 161	suppssr 8175	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ ({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∖ (𝑥 supp (0g‘𝑅)))) → (𝑥‘𝑘) = (0g‘𝑅))
163	71	fveq2d 6871	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) → (0g‘𝑅) = (0g‘(Scalar‘𝑃)))
164	163	adantr 484	. . . . . . . . 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 2798	. . . . . . . 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 7411	. . . . . . 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 4085	. . . . . . . 8 ⊢ (𝑘 ∈ ({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∖ (𝑥 supp (0g‘𝑅))) → 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin})
168	12	ad2antrr 736	. . . . . . . . . 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 22095	. . . . . . . . 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 2763	. . . . . . . . . 10 ⊢ (0g‘(Scalar‘𝑃)) = (0g‘(Scalar‘𝑃))
171	6, 146, 36, 170, 40	lmod0vs 20969	. . . . . . . . 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 593	. . . . . . . 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 602	. . . . . . 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 2798	. . . . . 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 8180	. . . . 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 9324	. . . . 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 847	. . . 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 19970	. . 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 2863	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) → 𝑥 ∈ (𝐴‘ran 𝑉))
180	31, 179	eqelssd 3958	1 ⊢ (𝜑 → (𝐴‘ran 𝑉) = (Base‘𝑃))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1099   = wceq 1561   ∈ wcel 2143  ∀wral 3077  {crab 3415  Vcvv 3455   ∖ cdif 3902   ⊆ wss 3905  ifcif 4481   class class class wbr 5101   ↦ cmpt 5182  ^◡ccnv 5647  ran crn 5649   “ cima 5651  Fun wfun 6515   Fn wfn 6516  ⟶wf 6517  ‘cfv 6521  (class class class)co 7396   supp csupp 8140   ↑_m cmap 8808  Fincfn 8927   finSupp cfsupp 9305  0cc0 11084  ℕcn 12220  ℕ₀cn0 12491  Basecbs 17255  ._rcmulr 17297  Scalarcsca 17299   ·_𝑠 cvsca 17300  0_gc0g 17478   Σ_g cgsu 17479  SubMndcsubmnd 18826  ._gcmg 19119  SubGrpcsubg 19172  CMndccmn 19830  Abelcabl 19831  mulGrpcmgp 20196  1_rcur 20241  Ringcrg 20293  CRingccrg 20294  SubRingcsubrg 20629  LModclmod 20934  LSubSpclss 21005  AssAlgcasa 21909  AlgSpancasp 21910   mPwSer cmps 21963   mVar cmvr 21964   mPoly cmpl 21965

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5228  ax-sep 5247  ax-nul 5257  ax-pow 5323  ax-pr 5391  ax-un 7718  ax-cnex 11140  ax-resscn 11141  ax-1cn 11142  ax-icn 11143  ax-addcl 11144  ax-addrcl 11145  ax-mulcl 11146  ax-mulrcl 11147  ax-mulcom 11148  ax-addass 11149  ax-mulass 11150  ax-distr 11151  ax-i2m1 11152  ax-1ne0 11153  ax-1rid 11154  ax-rnegex 11155  ax-rrecex 11156  ax-cnre 11157  ax-pre-lttri 11158  ax-pre-lttrn 11159  ax-pre-ltadd 11160  ax-pre-mulgt0 11161

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3457  df-sbc 3746  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-tp 4588  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-iin 4953  df-br 5102  df-opab 5164  df-mpt 5183  df-tr 5209  df-id 5543  df-eprel 5548  df-po 5556  df-so 5557  df-fr 5601  df-se 5602  df-we 5603  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-pred 6288  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-isom 6530  df-riota 7353  df-ov 7399  df-oprab 7400  df-mpo 7401  df-of 7660  df-ofr 7661  df-om 7847  df-1st 7970  df-2nd 7971  df-supp 8141  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8381  df-1o 8437  df-2o 8438  df-er 8678  df-map 8810  df-pm 8811  df-ixp 8880  df-en 8928  df-dom 8929  df-sdom 8930  df-fin 8931  df-fsupp 9306  df-sup 9386  df-oi 9456  df-card 9909  df-pnf 11229  df-mnf 11230  df-xr 11231  df-ltxr 11232  df-le 11233  df-sub 11427  df-neg 11428  df-nn 12221  df-2 12290  df-3 12291  df-4 12292  df-5 12293  df-6 12294  df-7 12295  df-8 12296  df-9 12297  df-n0 12492  df-z 12579  df-dec 12699  df-uz 12850  df-fz 13523  df-fzo 13670  df-seq 14025  df-hash 14354  df-struct 17193  df-sets 17210  df-slot 17228  df-ndx 17240  df-base 17256  df-ress 17277  df-plusg 17309  df-mulr 17310  df-sca 17312  df-vsca 17313  df-ip 17314  df-tset 17315  df-ple 17316  df-ds 17318  df-hom 17320  df-cco 17321  df-0g 17480  df-gsum 17481  df-prds 17486  df-pws 17488  df-mre 17624  df-mrc 17625  df-acs 17627  df-mgm 18684  df-sgrp 18763  df-mnd 18779  df-mhm 18827  df-submnd 18828  df-grp 18988  df-minusg 18989  df-sbg 18990  df-mulg 19120  df-subg 19175  df-ghm 19264  df-cntz 19367  df-cmn 19832  df-abl 19833  df-mgp 20197  df-rng 20209  df-ur 20242  df-srg 20247  df-ring 20295  df-cring 20296  df-subrng 20606  df-subrg 20630  df-lmod 20936  df-lss 21006  df-assa 21912  df-asp 21913  df-psr 21968  df-mvr 21969  df-mpl 21970

This theorem is referenced by:  mplind  22130  evlseu  22143