Theorem mplbas2 21907

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 21844	. . . 4 ⊢ (𝜑 → 𝑆 ∈ AssAlg)
5		mplbas2.p	. . . . . 6 ⊢ 𝑃 = (𝐼 mPoly 𝑅)
6		eqid 2724	. . . . . 6 ⊢ (Base‘𝑃) = (Base‘𝑃)
7		eqid 2724	. . . . . 6 ⊢ (Base‘𝑆) = (Base‘𝑆)
8	5, 1, 6, 7	mplbasss 21866	. . . . 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 20140	. . . . . . . . 9 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
12	3, 11	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ Ring)
13	1, 10, 7, 2, 12	mvrf 21854	. . . . . . 7 ⊢ (𝜑 → 𝑉:𝐼⟶(Base‘𝑆))
14	13	ffnd 6708	. . . . . 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 21861	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑉‘𝑥) ∈ (Base‘𝑃))
19	18	ralrimiva 3138	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 (𝑉‘𝑥) ∈ (Base‘𝑃))
20		ffnfv 7110	. . . . . 6 ⊢ (𝑉:𝐼⟶(Base‘𝑃) ↔ (𝑉 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝑉‘𝑥) ∈ (Base‘𝑃)))
21	14, 19, 20	sylanbrc 582	. . . . 5 ⊢ (𝜑 → 𝑉:𝐼⟶(Base‘𝑃))
22	21	frnd 6715	. . . 4 ⊢ (𝜑 → ran 𝑉 ⊆ (Base‘𝑃))
23		mplbas2.a	. . . . 5 ⊢ 𝐴 = (AlgSpan‘𝑆)
24	23, 7	aspss 21739	. . . 4 ⊢ ((𝑆 ∈ AssAlg ∧ (Base‘𝑃) ⊆ (Base‘𝑆) ∧ ran 𝑉 ⊆ (Base‘𝑃)) → (𝐴‘ran 𝑉) ⊆ (𝐴‘(Base‘𝑃)))
25	4, 9, 22, 24	syl3anc 1368	. . 3 ⊢ (𝜑 → (𝐴‘ran 𝑉) ⊆ (𝐴‘(Base‘𝑃)))
26	1, 5, 6, 2, 12	mplsubrg 21874	. . . 4 ⊢ (𝜑 → (Base‘𝑃) ∈ (SubRing‘𝑆))
27	1, 5, 6, 2, 12	mpllss 21872	. . . 4 ⊢ (𝜑 → (Base‘𝑃) ∈ (LSubSp‘𝑆))
28		eqid 2724	. . . . 5 ⊢ (LSubSp‘𝑆) = (LSubSp‘𝑆)
29	23, 7, 28	aspid 21737	. . . 4 ⊢ ((𝑆 ∈ AssAlg ∧ (Base‘𝑃) ∈ (SubRing‘𝑆) ∧ (Base‘𝑃) ∈ (LSubSp‘𝑆)) → (𝐴‘(Base‘𝑃)) = (Base‘𝑃))
30	4, 26, 27, 29	syl3anc 1368	. . 3 ⊢ (𝜑 → (𝐴‘(Base‘𝑃)) = (Base‘𝑃))
31	25, 30	sseqtrd 4014	. 2 ⊢ (𝜑 → (𝐴‘ran 𝑉) ⊆ (Base‘𝑃))
32		eqid 2724	. . . 4 ⊢ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}
33		eqid 2724	. . . 4 ⊢ (0g‘𝑅) = (0g‘𝑅)
34		eqid 2724	. . . 4 ⊢ (1r‘𝑅) = (1r‘𝑅)
35	2	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) → 𝐼 ∈ 𝑊)
36		eqid 2724	. . . 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 21902	. . 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 2724	. . . 4 ⊢ (0g‘𝑃) = (0g‘𝑃)
41	5	mplring 21888	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ Ring) → 𝑃 ∈ Ring)
42	2, 12, 41	syl2anc 583	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ Ring)
43		ringabl 20170	. . . . . 6 ⊢ (𝑃 ∈ Ring → 𝑃 ∈ Abel)
44	42, 43	syl 17	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ Abel)
45	44	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) → 𝑃 ∈ Abel)
46		ovex 7434	. . . . . 6 ⊢ (ℕ0 ↑m 𝐼) ∈ V
47	46	rabex 5322	. . . . 5 ⊢ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∈ V
48	47	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) → {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∈ V)
49	13	frnd 6715	. . . . . . . 8 ⊢ (𝜑 → ran 𝑉 ⊆ (Base‘𝑆))
50	23, 7	aspsubrg 21738	. . . . . . . 8 ⊢ ((𝑆 ∈ AssAlg ∧ ran 𝑉 ⊆ (Base‘𝑆)) → (𝐴‘ran 𝑉) ∈ (SubRing‘𝑆))
51	4, 49, 50	syl2anc 583	. . . . . . 7 ⊢ (𝜑 → (𝐴‘ran 𝑉) ∈ (SubRing‘𝑆))
52	5, 1, 6	mplval2 21865	. . . . . . . . 9 ⊢ 𝑃 = (𝑆 ↾s (Base‘𝑃))
53	52	subsubrg 20490	. . . . . . . 8 ⊢ ((Base‘𝑃) ∈ (SubRing‘𝑆) → ((𝐴‘ran 𝑉) ∈ (SubRing‘𝑃) ↔ ((𝐴‘ran 𝑉) ∈ (SubRing‘𝑆) ∧ (𝐴‘ran 𝑉) ⊆ (Base‘𝑃))))
54	26, 53	syl 17	. . . . . . 7 ⊢ (𝜑 → ((𝐴‘ran 𝑉) ∈ (SubRing‘𝑃) ↔ ((𝐴‘ran 𝑉) ∈ (SubRing‘𝑆) ∧ (𝐴‘ran 𝑉) ⊆ (Base‘𝑃))))
55	51, 31, 54	mpbir2and 710	. . . . . 6 ⊢ (𝜑 → (𝐴‘ran 𝑉) ∈ (SubRing‘𝑃))
56		subrgsubg 20469	. . . . . 6 ⊢ ((𝐴‘ran 𝑉) ∈ (SubRing‘𝑃) → (𝐴‘ran 𝑉) ∈ (SubGrp‘𝑃))
57	55, 56	syl 17	. . . . 5 ⊢ (𝜑 → (𝐴‘ran 𝑉) ∈ (SubGrp‘𝑃))
58	57	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) → (𝐴‘ran 𝑉) ∈ (SubGrp‘𝑃))
59	5	mpllmod 21887	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ Ring) → 𝑃 ∈ LMod)
60	2, 12, 59	syl2anc 583	. . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ LMod)
61	60	ad2antrr 723	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → 𝑃 ∈ LMod)
62	23, 7, 28	asplss 21736	. . . . . . . . 9 ⊢ ((𝑆 ∈ AssAlg ∧ ran 𝑉 ⊆ (Base‘𝑆)) → (𝐴‘ran 𝑉) ∈ (LSubSp‘𝑆))
63	4, 49, 62	syl2anc 583	. . . . . . . 8 ⊢ (𝜑 → (𝐴‘ran 𝑉) ∈ (LSubSp‘𝑆))
64	1, 2, 12	psrlmod 21831	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ LMod)
65		eqid 2724	. . . . . . . . . 10 ⊢ (LSubSp‘𝑃) = (LSubSp‘𝑃)
66	52, 28, 65	lsslss 20798	. . . . . . . . 9 ⊢ ((𝑆 ∈ LMod ∧ (Base‘𝑃) ∈ (LSubSp‘𝑆)) → ((𝐴‘ran 𝑉) ∈ (LSubSp‘𝑃) ↔ ((𝐴‘ran 𝑉) ∈ (LSubSp‘𝑆) ∧ (𝐴‘ran 𝑉) ⊆ (Base‘𝑃))))
67	64, 27, 66	syl2anc 583	. . . . . . . 8 ⊢ (𝜑 → ((𝐴‘ran 𝑉) ∈ (LSubSp‘𝑃) ↔ ((𝐴‘ran 𝑉) ∈ (LSubSp‘𝑆) ∧ (𝐴‘ran 𝑉) ⊆ (Base‘𝑃))))
68	63, 31, 67	mpbir2and 710	. . . . . . 7 ⊢ (𝜑 → (𝐴‘ran 𝑉) ∈ (LSubSp‘𝑃))
69	68	ad2antrr 723	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝐴‘ran 𝑉) ∈ (LSubSp‘𝑃))
70		eqid 2724	. . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅)
71	5, 70, 6, 32, 38	mplelf 21867	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) → 𝑥:{𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
72	71	ffvelcdmda 7076	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝑥‘𝑘) ∈ (Base‘𝑅))
73	5, 35, 37	mplsca 21882	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) → 𝑅 = (Scalar‘𝑃))
74	73	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → 𝑅 = (Scalar‘𝑃))
75	74	fveq2d 6885	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (Base‘𝑅) = (Base‘(Scalar‘𝑃)))
76	72, 75	eleqtrd 2827	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝑥‘𝑘) ∈ (Base‘(Scalar‘𝑃)))
77	2	ad2antrr 723	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → 𝐼 ∈ 𝑊)
78		eqid 2724	. . . . . . . 8 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃)
79		eqid 2724	. . . . . . . 8 ⊢ (.g‘(mulGrp‘𝑃)) = (.g‘(mulGrp‘𝑃))
80	3	ad2antrr 723	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → 𝑅 ∈ CRing)
81		simpr 484	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin})
82	5, 32, 33, 34, 77, 78, 79, 10, 80, 81	mplcoe2 21906	. . . . . . 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)))))
83		eqid 2724	. . . . . . . . 9 ⊢ (1r‘𝑃) = (1r‘𝑃)
84	78, 83	ringidval 20078	. . . . . . . 8 ⊢ (1r‘𝑃) = (0g‘(mulGrp‘𝑃))
85	5	mplcrng 21890	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ CRing) → 𝑃 ∈ CRing)
86	2, 3, 85	syl2anc 583	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ CRing)
87	78	crngmgp 20136	. . . . . . . . . 10 ⊢ (𝑃 ∈ CRing → (mulGrp‘𝑃) ∈ CMnd)
88	86, 87	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (mulGrp‘𝑃) ∈ CMnd)
89	88	ad2antrr 723	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (mulGrp‘𝑃) ∈ CMnd)
90	55	ad2antrr 723	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝐴‘ran 𝑉) ∈ (SubRing‘𝑃))
91	78	subrgsubm 20477	. . . . . . . . 9 ⊢ ((𝐴‘ran 𝑉) ∈ (SubRing‘𝑃) → (𝐴‘ran 𝑉) ∈ (SubMnd‘(mulGrp‘𝑃)))
92	90, 91	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝐴‘ran 𝑉) ∈ (SubMnd‘(mulGrp‘𝑃)))
93		simplll 772	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧 ∈ 𝐼) → 𝜑)
94	32	psrbag 21779	. . . . . . . . . . . . . 14 ⊢ (𝐼 ∈ 𝑊 → (𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↔ (𝑘:𝐼⟶ℕ0 ∧ (◡𝑘 “ ℕ) ∈ Fin)))
95	35, 94	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) → (𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↔ (𝑘:𝐼⟶ℕ0 ∧ (◡𝑘 “ ℕ) ∈ Fin)))
96	95	biimpa 476	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝑘:𝐼⟶ℕ0 ∧ (◡𝑘 “ ℕ) ∈ Fin))
97	96	simpld 494	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → 𝑘:𝐼⟶ℕ0)
98	97	ffvelcdmda 7076	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧 ∈ 𝐼) → (𝑘‘𝑧) ∈ ℕ0)
99	23, 7	aspssid 21740	. . . . . . . . . . . . 13 ⊢ ((𝑆 ∈ AssAlg ∧ ran 𝑉 ⊆ (Base‘𝑆)) → ran 𝑉 ⊆ (𝐴‘ran 𝑉))
100	4, 49, 99	syl2anc 583	. . . . . . . . . . . 12 ⊢ (𝜑 → ran 𝑉 ⊆ (𝐴‘ran 𝑉))
101	100	ad3antrrr 727	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧 ∈ 𝐼) → ran 𝑉 ⊆ (𝐴‘ran 𝑉))
102	14	ad2antrr 723	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → 𝑉 Fn 𝐼)
103		fnfvelrn 7072	. . . . . . . . . . . 12 ⊢ ((𝑉 Fn 𝐼 ∧ 𝑧 ∈ 𝐼) → (𝑉‘𝑧) ∈ ran 𝑉)
104	102, 103	sylan 579	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧 ∈ 𝐼) → (𝑉‘𝑧) ∈ ran 𝑉)
105	101, 104	sseldd 3975	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧 ∈ 𝐼) → (𝑉‘𝑧) ∈ (𝐴‘ran 𝑉))
106	78, 6	mgpbas 20035	. . . . . . . . . . 11 ⊢ (Base‘𝑃) = (Base‘(mulGrp‘𝑃))
107		eqid 2724	. . . . . . . . . . . 12 ⊢ (.r‘𝑃) = (.r‘𝑃)
108	78, 107	mgpplusg 20033	. . . . . . . . . . 11 ⊢ (.r‘𝑃) = (+g‘(mulGrp‘𝑃))
109	107	subrgmcl 20476	. . . . . . . . . . . 12 ⊢ (((𝐴‘ran 𝑉) ∈ (SubRing‘𝑃) ∧ 𝑢 ∈ (𝐴‘ran 𝑉) ∧ 𝑣 ∈ (𝐴‘ran 𝑉)) → (𝑢(.r‘𝑃)𝑣) ∈ (𝐴‘ran 𝑉))
110	55, 109	syl3an1 1160	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐴‘ran 𝑉) ∧ 𝑣 ∈ (𝐴‘ran 𝑉)) → (𝑢(.r‘𝑃)𝑣) ∈ (𝐴‘ran 𝑉))
111	83	subrg1cl 20472	. . . . . . . . . . . 12 ⊢ ((𝐴‘ran 𝑉) ∈ (SubRing‘𝑃) → (1r‘𝑃) ∈ (𝐴‘ran 𝑉))
112	55, 111	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (1r‘𝑃) ∈ (𝐴‘ran 𝑉))
113	106, 79, 108, 88, 31, 110, 84, 112	mulgnn0subcl 19004	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘‘𝑧) ∈ ℕ0 ∧ (𝑉‘𝑧) ∈ (𝐴‘ran 𝑉)) → ((𝑘‘𝑧)(.g‘(mulGrp‘𝑃))(𝑉‘𝑧)) ∈ (𝐴‘ran 𝑉))
114	93, 98, 105, 113	syl3anc 1368	. . . . . . . . 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))
115	114	fmpttd 7106	. . . . . . . 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))
116	2	mptexd 7217	. . . . . . . . . 10 ⊢ (𝜑 → (𝑧 ∈ 𝐼 ↦ ((𝑘‘𝑧)(.g‘(mulGrp‘𝑃))(𝑉‘𝑧))) ∈ V)
117	116	ad2antrr 723	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝑧 ∈ 𝐼 ↦ ((𝑘‘𝑧)(.g‘(mulGrp‘𝑃))(𝑉‘𝑧))) ∈ V)
118		funmpt 6576	. . . . . . . . . 10 ⊢ Fun (𝑧 ∈ 𝐼 ↦ ((𝑘‘𝑧)(.g‘(mulGrp‘𝑃))(𝑉‘𝑧)))
119	118	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))))
120		fvexd 6896	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (1r‘𝑃) ∈ V)
121	96	simprd 495	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (◡𝑘 “ ℕ) ∈ Fin)
122		elrabi 3669	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} → 𝑘 ∈ (ℕ0 ↑m 𝐼))
123		elmapi 8839	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (ℕ0 ↑m 𝐼) → 𝑘:𝐼⟶ℕ0)
124	123	adantl 481	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ (ℕ0 ↑m 𝐼)) → 𝑘:𝐼⟶ℕ0)
125	2	ad2antrr 723	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ (ℕ0 ↑m 𝐼)) → 𝐼 ∈ 𝑊)
126		fcdmnn0supp 12525	. . . . . . . . . . . . . . . 16 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑘:𝐼⟶ℕ0) → (𝑘 supp 0) = (◡𝑘 “ ℕ))
127	125, 124, 126	syl2anc 583	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ (ℕ0 ↑m 𝐼)) → (𝑘 supp 0) = (◡𝑘 “ ℕ))
128		eqimss 4032	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 supp 0) = (◡𝑘 “ ℕ) → (𝑘 supp 0) ⊆ (◡𝑘 “ ℕ))
129	127, 128	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ (ℕ0 ↑m 𝐼)) → (𝑘 supp 0) ⊆ (◡𝑘 “ ℕ))
130		c0ex 11205	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ V
131	130	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ (ℕ0 ↑m 𝐼)) → 0 ∈ V)
132	124, 129, 125, 131	suppssr 8175	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ (ℕ0 ↑m 𝐼)) ∧ 𝑧 ∈ (𝐼 ∖ (◡𝑘 “ ℕ))) → (𝑘‘𝑧) = 0)
133	122, 132	sylanl2 678	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧 ∈ (𝐼 ∖ (◡𝑘 “ ℕ))) → (𝑘‘𝑧) = 0)
134	133	oveq1d 7416	. . . . . . . . . . 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)))
135	2	ad3antrrr 727	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧 ∈ (𝐼 ∖ (◡𝑘 “ ℕ))) → 𝐼 ∈ 𝑊)
136	12	ad3antrrr 727	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧 ∈ (𝐼 ∖ (◡𝑘 “ ℕ))) → 𝑅 ∈ Ring)
137		eldifi 4118	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ (𝐼 ∖ (◡𝑘 “ ℕ)) → 𝑧 ∈ 𝐼)
138	137	adantl 481	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧 ∈ (𝐼 ∖ (◡𝑘 “ ℕ))) → 𝑧 ∈ 𝐼)
139	5, 10, 6, 135, 136, 138	mvrcl 21861	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧 ∈ (𝐼 ∖ (◡𝑘 “ ℕ))) → (𝑉‘𝑧) ∈ (Base‘𝑃))
140	106, 84, 79	mulg0 18992	. . . . . . . . . . . 12 ⊢ ((𝑉‘𝑧) ∈ (Base‘𝑃) → (0(.g‘(mulGrp‘𝑃))(𝑉‘𝑧)) = (1r‘𝑃))
141	139, 140	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))
142	134, 141	eqtrd 2764	. . . . . . . . . 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))
143	142, 77	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))
144		suppssfifsupp 9374	. . . . . . . . 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))
145	117, 119, 120, 121, 143, 144	syl32anc 1375	. . . . . . . 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))
146	84, 89, 77, 92, 115, 145	gsumsubmcl 19829	. . . . . . 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))
147	82, 146	eqeltrd 2825	. . . . . 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))
148		eqid 2724	. . . . . . 7 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃)
149		eqid 2724	. . . . . . 7 ⊢ (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
150	148, 36, 149, 65	lssvscl 20792	. . . . . 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))
151	61, 69, 76, 147, 150	syl22anc 836	. . . . 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))
152	151	fmpttd 7106	. . . 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))
153	46	mptrabex 7218	. . . . . . 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
154		funmpt 6576	. . . . . . 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)))))
155		fvex 6894	. . . . . . 7 ⊢ (0g‘𝑃) ∈ V
156	153, 154, 155	3pm3.2i 1336	. . . . . 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)
157	156	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))
158	5, 1, 7, 33, 6	mplelbas 21860	. . . . . . . 8 ⊢ (𝑥 ∈ (Base‘𝑃) ↔ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 finSupp (0g‘𝑅)))
159	158	simprbi 496	. . . . . . 7 ⊢ (𝑥 ∈ (Base‘𝑃) → 𝑥 finSupp (0g‘𝑅))
160	159	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) → 𝑥 finSupp (0g‘𝑅))
161	160	fsuppimpd 9365	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) → (𝑥 supp (0g‘𝑅)) ∈ Fin)
162		ssidd 3997	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) → (𝑥 supp (0g‘𝑅)) ⊆ (𝑥 supp (0g‘𝑅)))
163		fvexd 6896	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) → (0g‘𝑅) ∈ V)
164	71, 162, 48, 163	suppssr 8175	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ ({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∖ (𝑥 supp (0g‘𝑅)))) → (𝑥‘𝑘) = (0g‘𝑅))
165	73	fveq2d 6885	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) → (0g‘𝑅) = (0g‘(Scalar‘𝑃)))
166	165	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ ({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∖ (𝑥 supp (0g‘𝑅)))) → (0g‘𝑅) = (0g‘(Scalar‘𝑃)))
167	164, 166	eqtrd 2764	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ ({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∖ (𝑥 supp (0g‘𝑅)))) → (𝑥‘𝑘) = (0g‘(Scalar‘𝑃)))
168	167	oveq1d 7416	. . . . . . 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)))))
169		eldifi 4118	. . . . . . . 8 ⊢ (𝑘 ∈ ({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∖ (𝑥 supp (0g‘𝑅))) → 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin})
170	12	ad2antrr 723	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → 𝑅 ∈ Ring)
171	5, 6, 33, 34, 32, 77, 170, 81	mplmon 21900	. . . . . . . . 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))
172		eqid 2724	. . . . . . . . . 10 ⊢ (0g‘(Scalar‘𝑃)) = (0g‘(Scalar‘𝑃))
173	6, 148, 36, 172, 40	lmod0vs 20731	. . . . . . . . 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))
174	61, 171, 173	syl2anc 583	. . . . . . . 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))
175	169, 174	sylan2 592	. . . . . . 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))
176	168, 175	eqtrd 2764	. . . . . 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))
177	176, 48	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)))
178		suppssfifsupp 9374	. . . . 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))
179	157, 161, 177, 178	syl12anc 834	. . . 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))
180	40, 45, 48, 58, 152, 179	gsumsubgcl 19830	. . 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))
181	39, 180	eqeltrd 2825	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) → 𝑥 ∈ (𝐴‘ran 𝑉))
182	31, 181	eqelssd 3995	1 ⊢ (𝜑 → (𝐴‘ran 𝑉) = (Base‘𝑃))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  ∀wral 3053  {crab 3424  Vcvv 3466   ∖ cdif 3937   ⊆ wss 3940  ifcif 4520   class class class wbr 5138   ↦ cmpt 5221  ^◡ccnv 5665  ran crn 5667   “ cima 5669  Fun wfun 6527   Fn wfn 6528  ⟶wf 6529  ‘cfv 6533  (class class class)co 7401   supp csupp 8140   ↑_m cmap 8816  Fincfn 8935   finSupp cfsupp 9357  0cc0 11106  ℕcn 12209  ℕ₀cn0 12469  Basecbs 17143  ._rcmulr 17197  Scalarcsca 17199   ·_𝑠 cvsca 17200  0_gc0g 17384   Σ_g cgsu 17385  SubMndcsubmnd 18702  ._gcmg 18985  SubGrpcsubg 19037  CMndccmn 19690  Abelcabl 19691  mulGrpcmgp 20029  1_rcur 20076  Ringcrg 20128  CRingccrg 20129  SubRingcsubrg 20459  LModclmod 20696  LSubSpclss 20768  AssAlgcasa 21713  AlgSpancasp 21714   mPwSer cmps 21766   mVar cmvr 21767   mPoly cmpl 21768

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-tp 4625  df-op 4627  df-uni 4900  df-int 4941  df-iun 4989  df-iin 4990  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-se 5622  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-isom 6542  df-riota 7357  df-ov 7404  df-oprab 7405  df-mpo 7406  df-of 7663  df-ofr 7664  df-om 7849  df-1st 7968  df-2nd 7969  df-supp 8141  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-1o 8461  df-er 8699  df-map 8818  df-pm 8819  df-ixp 8888  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-fsupp 9358  df-sup 9433  df-oi 9501  df-card 9930  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11443  df-neg 11444  df-nn 12210  df-2 12272  df-3 12273  df-4 12274  df-5 12275  df-6 12276  df-7 12277  df-8 12278  df-9 12279  df-n0 12470  df-z 12556  df-dec 12675  df-uz 12820  df-fz 13482  df-fzo 13625  df-seq 13964  df-hash 14288  df-struct 17079  df-sets 17096  df-slot 17114  df-ndx 17126  df-base 17144  df-ress 17173  df-plusg 17209  df-mulr 17210  df-sca 17212  df-vsca 17213  df-ip 17214  df-tset 17215  df-ple 17216  df-ds 17218  df-hom 17220  df-cco 17221  df-0g 17386  df-gsum 17387  df-prds 17392  df-pws 17394  df-mre 17529  df-mrc 17530  df-acs 17532  df-mgm 18563  df-sgrp 18642  df-mnd 18658  df-mhm 18703  df-submnd 18704  df-grp 18856  df-minusg 18857  df-sbg 18858  df-mulg 18986  df-subg 19040  df-ghm 19129  df-cntz 19223  df-cmn 19692  df-abl 19693  df-mgp 20030  df-rng 20048  df-ur 20077  df-srg 20082  df-ring 20130  df-cring 20131  df-subrng 20436  df-subrg 20461  df-lmod 20698  df-lss 20769  df-assa 21716  df-asp 21717  df-psr 21771  df-mvr 21772  df-mpl 21773

This theorem is referenced by:  mplind  21941  evlseu  21956