Theorem mplbas2 19470

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 19414	. . . 4 ⊢ (𝜑 → 𝑆 ∈ AssAlg)
5		mplbas2.p	. . . . . 6 ⊢ 𝑃 = (𝐼 mPoly 𝑅)
6		eqid 2622	. . . . . 6 ⊢ (Base‘𝑃) = (Base‘𝑃)
7		eqid 2622	. . . . . 6 ⊢ (Base‘𝑆) = (Base‘𝑆)
8	5, 1, 6, 7	mplbasss 19432	. . . . 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 18558	. . . . . . . . 9 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
12	3, 11	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ Ring)
13	1, 10, 7, 2, 12	mvrf 19424	. . . . . . 7 ⊢ (𝜑 → 𝑉:𝐼⟶(Base‘𝑆))
14	13	ffnd 6046	. . . . . 6 ⊢ (𝜑 → 𝑉 Fn 𝐼)
15	2	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐼 ∈ 𝑊)
16	12	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑅 ∈ Ring)
17		simpr 477	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑥 ∈ 𝐼)
18	5, 10, 6, 15, 16, 17	mvrcl 19449	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑉‘𝑥) ∈ (Base‘𝑃))
19	18	ralrimiva 2966	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 (𝑉‘𝑥) ∈ (Base‘𝑃))
20		ffnfv 6388	. . . . . 6 ⊢ (𝑉:𝐼⟶(Base‘𝑃) ↔ (𝑉 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝑉‘𝑥) ∈ (Base‘𝑃)))
21	14, 19, 20	sylanbrc 698	. . . . 5 ⊢ (𝜑 → 𝑉:𝐼⟶(Base‘𝑃))
22		frn 6053	. . . . 5 ⊢ (𝑉:𝐼⟶(Base‘𝑃) → ran 𝑉 ⊆ (Base‘𝑃))
23	21, 22	syl 17	. . . 4 ⊢ (𝜑 → ran 𝑉 ⊆ (Base‘𝑃))
24		mplbas2.a	. . . . 5 ⊢ 𝐴 = (AlgSpan‘𝑆)
25	24, 7	aspss 19332	. . . 4 ⊢ ((𝑆 ∈ AssAlg ∧ (Base‘𝑃) ⊆ (Base‘𝑆) ∧ ran 𝑉 ⊆ (Base‘𝑃)) → (𝐴‘ran 𝑉) ⊆ (𝐴‘(Base‘𝑃)))
26	4, 9, 23, 25	syl3anc 1326	. . 3 ⊢ (𝜑 → (𝐴‘ran 𝑉) ⊆ (𝐴‘(Base‘𝑃)))
27	1, 5, 6, 2, 12	mplsubrg 19440	. . . 4 ⊢ (𝜑 → (Base‘𝑃) ∈ (SubRing‘𝑆))
28	1, 5, 6, 2, 12	mpllss 19438	. . . 4 ⊢ (𝜑 → (Base‘𝑃) ∈ (LSubSp‘𝑆))
29		eqid 2622	. . . . 5 ⊢ (LSubSp‘𝑆) = (LSubSp‘𝑆)
30	24, 7, 29	aspid 19330	. . . 4 ⊢ ((𝑆 ∈ AssAlg ∧ (Base‘𝑃) ∈ (SubRing‘𝑆) ∧ (Base‘𝑃) ∈ (LSubSp‘𝑆)) → (𝐴‘(Base‘𝑃)) = (Base‘𝑃))
31	4, 27, 28, 30	syl3anc 1326	. . 3 ⊢ (𝜑 → (𝐴‘(Base‘𝑃)) = (Base‘𝑃))
32	26, 31	sseqtrd 3641	. 2 ⊢ (𝜑 → (𝐴‘ran 𝑉) ⊆ (Base‘𝑃))
33		eqid 2622	. . . . . 6 ⊢ {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}
34		eqid 2622	. . . . . 6 ⊢ (0g‘𝑅) = (0g‘𝑅)
35		eqid 2622	. . . . . 6 ⊢ (1r‘𝑅) = (1r‘𝑅)
36	2	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) → 𝐼 ∈ 𝑊)
37		eqid 2622	. . . . . 6 ⊢ ( ·𝑠 ‘𝑃) = ( ·𝑠 ‘𝑃)
38	12	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) → 𝑅 ∈ Ring)
39		simpr 477	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) → 𝑥 ∈ (Base‘𝑃))
40	5, 33, 34, 35, 36, 6, 37, 38, 39	mplcoe1 19465	. . . . 5 ⊢ ((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)))))))
41		eqid 2622	. . . . . 6 ⊢ (0g‘𝑃) = (0g‘𝑃)
42	5	mplring 19452	. . . . . . . . 9 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ Ring) → 𝑃 ∈ Ring)
43	2, 12, 42	syl2anc 693	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ Ring)
44		ringabl 18580	. . . . . . . 8 ⊢ (𝑃 ∈ Ring → 𝑃 ∈ Abel)
45	43, 44	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ Abel)
46	45	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) → 𝑃 ∈ Abel)
47		ovex 6678	. . . . . . . 8 ⊢ (ℕ0 ↑𝑚 𝐼) ∈ V
48	47	rabex 4813	. . . . . . 7 ⊢ {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∈ V
49	48	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) → {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∈ V)
50	23, 8	syl6ss 3615	. . . . . . . . . 10 ⊢ (𝜑 → ran 𝑉 ⊆ (Base‘𝑆))
51	24, 7	aspsubrg 19331	. . . . . . . . . 10 ⊢ ((𝑆 ∈ AssAlg ∧ ran 𝑉 ⊆ (Base‘𝑆)) → (𝐴‘ran 𝑉) ∈ (SubRing‘𝑆))
52	4, 50, 51	syl2anc 693	. . . . . . . . 9 ⊢ (𝜑 → (𝐴‘ran 𝑉) ∈ (SubRing‘𝑆))
53	5, 1, 6	mplval2 19431	. . . . . . . . . . 11 ⊢ 𝑃 = (𝑆 ↾s (Base‘𝑃))
54	53	subsubrg 18806	. . . . . . . . . 10 ⊢ ((Base‘𝑃) ∈ (SubRing‘𝑆) → ((𝐴‘ran 𝑉) ∈ (SubRing‘𝑃) ↔ ((𝐴‘ran 𝑉) ∈ (SubRing‘𝑆) ∧ (𝐴‘ran 𝑉) ⊆ (Base‘𝑃))))
55	27, 54	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ((𝐴‘ran 𝑉) ∈ (SubRing‘𝑃) ↔ ((𝐴‘ran 𝑉) ∈ (SubRing‘𝑆) ∧ (𝐴‘ran 𝑉) ⊆ (Base‘𝑃))))
56	52, 32, 55	mpbir2and 957	. . . . . . . 8 ⊢ (𝜑 → (𝐴‘ran 𝑉) ∈ (SubRing‘𝑃))
57		subrgsubg 18786	. . . . . . . 8 ⊢ ((𝐴‘ran 𝑉) ∈ (SubRing‘𝑃) → (𝐴‘ran 𝑉) ∈ (SubGrp‘𝑃))
58	56, 57	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝐴‘ran 𝑉) ∈ (SubGrp‘𝑃))
59	58	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) → (𝐴‘ran 𝑉) ∈ (SubGrp‘𝑃))
60	5	mpllmod 19451	. . . . . . . . . 10 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ Ring) → 𝑃 ∈ LMod)
61	2, 12, 60	syl2anc 693	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ LMod)
62	61	ad2antrr 762	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → 𝑃 ∈ LMod)
63	24, 7, 29	asplss 19329	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ AssAlg ∧ ran 𝑉 ⊆ (Base‘𝑆)) → (𝐴‘ran 𝑉) ∈ (LSubSp‘𝑆))
64	4, 50, 63	syl2anc 693	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴‘ran 𝑉) ∈ (LSubSp‘𝑆))
65	1, 2, 12	psrlmod 19401	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑆 ∈ LMod)
66		eqid 2622	. . . . . . . . . . . 12 ⊢ (LSubSp‘𝑃) = (LSubSp‘𝑃)
67	53, 29, 66	lsslss 18961	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ LMod ∧ (Base‘𝑃) ∈ (LSubSp‘𝑆)) → ((𝐴‘ran 𝑉) ∈ (LSubSp‘𝑃) ↔ ((𝐴‘ran 𝑉) ∈ (LSubSp‘𝑆) ∧ (𝐴‘ran 𝑉) ⊆ (Base‘𝑃))))
68	65, 28, 67	syl2anc 693	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐴‘ran 𝑉) ∈ (LSubSp‘𝑃) ↔ ((𝐴‘ran 𝑉) ∈ (LSubSp‘𝑆) ∧ (𝐴‘ran 𝑉) ⊆ (Base‘𝑃))))
69	64, 32, 68	mpbir2and 957	. . . . . . . . 9 ⊢ (𝜑 → (𝐴‘ran 𝑉) ∈ (LSubSp‘𝑃))
70	69	ad2antrr 762	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝐴‘ran 𝑉) ∈ (LSubSp‘𝑃))
71		eqid 2622	. . . . . . . . . . 11 ⊢ (Base‘𝑅) = (Base‘𝑅)
72	5, 71, 6, 33, 39	mplelf 19433	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) → 𝑥:{𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
73	72	ffvelrnda 6359	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝑥‘𝑘) ∈ (Base‘𝑅))
74	5, 36, 38	mplsca 19445	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) → 𝑅 = (Scalar‘𝑃))
75	74	adantr 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → 𝑅 = (Scalar‘𝑃))
76	75	fveq2d 6195	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (Base‘𝑅) = (Base‘(Scalar‘𝑃)))
77	73, 76	eleqtrd 2703	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝑥‘𝑘) ∈ (Base‘(Scalar‘𝑃)))
78	2	ad2antrr 762	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → 𝐼 ∈ 𝑊)
79		eqid 2622	. . . . . . . . . 10 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃)
80		eqid 2622	. . . . . . . . . 10 ⊢ (.g‘(mulGrp‘𝑃)) = (.g‘(mulGrp‘𝑃))
81	3	ad2antrr 762	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → 𝑅 ∈ CRing)
82		simpr 477	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin})
83	5, 33, 34, 35, 78, 79, 80, 10, 81, 82	mplcoe2 19469	. . . . . . . . 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))) = ((mulGrp‘𝑃) Σg (𝑧 ∈ 𝐼 ↦ ((𝑘‘𝑧)(.g‘(mulGrp‘𝑃))(𝑉‘𝑧)))))
84		eqid 2622	. . . . . . . . . . 11 ⊢ (1r‘𝑃) = (1r‘𝑃)
85	79, 84	ringidval 18503	. . . . . . . . . 10 ⊢ (1r‘𝑃) = (0g‘(mulGrp‘𝑃))
86	5	mplcrng 19453	. . . . . . . . . . . . 13 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ CRing) → 𝑃 ∈ CRing)
87	2, 3, 86	syl2anc 693	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑃 ∈ CRing)
88	79	crngmgp 18555	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ CRing → (mulGrp‘𝑃) ∈ CMnd)
89	87, 88	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (mulGrp‘𝑃) ∈ CMnd)
90	89	ad2antrr 762	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (mulGrp‘𝑃) ∈ CMnd)
91	56	ad2antrr 762	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝐴‘ran 𝑉) ∈ (SubRing‘𝑃))
92	79	subrgsubm 18793	. . . . . . . . . . 11 ⊢ ((𝐴‘ran 𝑉) ∈ (SubRing‘𝑃) → (𝐴‘ran 𝑉) ∈ (SubMnd‘(mulGrp‘𝑃)))
93	91, 92	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝐴‘ran 𝑉) ∈ (SubMnd‘(mulGrp‘𝑃)))
94		simplll 798	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧 ∈ 𝐼) → 𝜑)
95	33	psrbag 19364	. . . . . . . . . . . . . . . 16 ⊢ (𝐼 ∈ 𝑊 → (𝑘 ∈ {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↔ (𝑘:𝐼⟶ℕ0 ∧ (◡𝑘 “ ℕ) ∈ Fin)))
96	36, 95	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) → (𝑘 ∈ {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ↔ (𝑘:𝐼⟶ℕ0 ∧ (◡𝑘 “ ℕ) ∈ Fin)))
97	96	biimpa 501	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝑘:𝐼⟶ℕ0 ∧ (◡𝑘 “ ℕ) ∈ Fin))
98	97	simpld 475	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → 𝑘:𝐼⟶ℕ0)
99	98	ffvelrnda 6359	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧 ∈ 𝐼) → (𝑘‘𝑧) ∈ ℕ0)
100	24, 7	aspssid 19333	. . . . . . . . . . . . . . 15 ⊢ ((𝑆 ∈ AssAlg ∧ ran 𝑉 ⊆ (Base‘𝑆)) → ran 𝑉 ⊆ (𝐴‘ran 𝑉))
101	4, 50, 100	syl2anc 693	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ran 𝑉 ⊆ (𝐴‘ran 𝑉))
102	101	ad3antrrr 766	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧 ∈ 𝐼) → ran 𝑉 ⊆ (𝐴‘ran 𝑉))
103	14	ad2antrr 762	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → 𝑉 Fn 𝐼)
104		fnfvelrn 6356	. . . . . . . . . . . . . 14 ⊢ ((𝑉 Fn 𝐼 ∧ 𝑧 ∈ 𝐼) → (𝑉‘𝑧) ∈ ran 𝑉)
105	103, 104	sylan 488	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧 ∈ 𝐼) → (𝑉‘𝑧) ∈ ran 𝑉)
106	102, 105	sseldd 3604	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧 ∈ 𝐼) → (𝑉‘𝑧) ∈ (𝐴‘ran 𝑉))
107	79, 6	mgpbas 18495	. . . . . . . . . . . . 13 ⊢ (Base‘𝑃) = (Base‘(mulGrp‘𝑃))
108		eqid 2622	. . . . . . . . . . . . . 14 ⊢ (.r‘𝑃) = (.r‘𝑃)
109	79, 108	mgpplusg 18493	. . . . . . . . . . . . 13 ⊢ (.r‘𝑃) = (+g‘(mulGrp‘𝑃))
110	108	subrgmcl 18792	. . . . . . . . . . . . . 14 ⊢ (((𝐴‘ran 𝑉) ∈ (SubRing‘𝑃) ∧ 𝑢 ∈ (𝐴‘ran 𝑉) ∧ 𝑣 ∈ (𝐴‘ran 𝑉)) → (𝑢(.r‘𝑃)𝑣) ∈ (𝐴‘ran 𝑉))
111	56, 110	syl3an1 1359	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐴‘ran 𝑉) ∧ 𝑣 ∈ (𝐴‘ran 𝑉)) → (𝑢(.r‘𝑃)𝑣) ∈ (𝐴‘ran 𝑉))
112	84	subrg1cl 18788	. . . . . . . . . . . . . 14 ⊢ ((𝐴‘ran 𝑉) ∈ (SubRing‘𝑃) → (1r‘𝑃) ∈ (𝐴‘ran 𝑉))
113	56, 112	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (1r‘𝑃) ∈ (𝐴‘ran 𝑉))
114	107, 80, 109, 89, 32, 111, 85, 113	mulgnn0subcl 17554	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑘‘𝑧) ∈ ℕ0 ∧ (𝑉‘𝑧) ∈ (𝐴‘ran 𝑉)) → ((𝑘‘𝑧)(.g‘(mulGrp‘𝑃))(𝑉‘𝑧)) ∈ (𝐴‘ran 𝑉))
115	94, 99, 106, 114	syl3anc 1326	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧 ∈ 𝐼) → ((𝑘‘𝑧)(.g‘(mulGrp‘𝑃))(𝑉‘𝑧)) ∈ (𝐴‘ran 𝑉))
116		eqid 2622	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝐼 ↦ ((𝑘‘𝑧)(.g‘(mulGrp‘𝑃))(𝑉‘𝑧))) = (𝑧 ∈ 𝐼 ↦ ((𝑘‘𝑧)(.g‘(mulGrp‘𝑃))(𝑉‘𝑧)))
117	115, 116	fmptd 6385	. . . . . . . . . 10 ⊢ (((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))
118		mptexg 6484	. . . . . . . . . . . . 13 ⊢ (𝐼 ∈ 𝑊 → (𝑧 ∈ 𝐼 ↦ ((𝑘‘𝑧)(.g‘(mulGrp‘𝑃))(𝑉‘𝑧))) ∈ V)
119	2, 118	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑧 ∈ 𝐼 ↦ ((𝑘‘𝑧)(.g‘(mulGrp‘𝑃))(𝑉‘𝑧))) ∈ V)
120	119	ad2antrr 762	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝑧 ∈ 𝐼 ↦ ((𝑘‘𝑧)(.g‘(mulGrp‘𝑃))(𝑉‘𝑧))) ∈ V)
121		funmpt 5926	. . . . . . . . . . . 12 ⊢ Fun (𝑧 ∈ 𝐼 ↦ ((𝑘‘𝑧)(.g‘(mulGrp‘𝑃))(𝑉‘𝑧)))
122	121	a1i 11	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → Fun (𝑧 ∈ 𝐼 ↦ ((𝑘‘𝑧)(.g‘(mulGrp‘𝑃))(𝑉‘𝑧))))
123		fvexd 6203	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (1r‘𝑃) ∈ V)
124	97	simprd 479	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (◡𝑘 “ ℕ) ∈ Fin)
125		elrabi 3359	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} → 𝑘 ∈ (ℕ0 ↑𝑚 𝐼))
126		elmapi 7879	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ (ℕ0 ↑𝑚 𝐼) → 𝑘:𝐼⟶ℕ0)
127	126	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ (ℕ0 ↑𝑚 𝐼)) → 𝑘:𝐼⟶ℕ0)
128	2	ad2antrr 762	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ (ℕ0 ↑𝑚 𝐼)) → 𝐼 ∈ 𝑊)
129		frnnn0supp 11349	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑘:𝐼⟶ℕ0) → (𝑘 supp 0) = (◡𝑘 “ ℕ))
130	128, 127, 129	syl2anc 693	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ (ℕ0 ↑𝑚 𝐼)) → (𝑘 supp 0) = (◡𝑘 “ ℕ))
131		eqimss 3657	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘 supp 0) = (◡𝑘 “ ℕ) → (𝑘 supp 0) ⊆ (◡𝑘 “ ℕ))
132	130, 131	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ (ℕ0 ↑𝑚 𝐼)) → (𝑘 supp 0) ⊆ (◡𝑘 “ ℕ))
133		c0ex 10034	. . . . . . . . . . . . . . . . 17 ⊢ 0 ∈ V
134	133	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ (ℕ0 ↑𝑚 𝐼)) → 0 ∈ V)
135	127, 132, 128, 134	suppssr 7326	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ (ℕ0 ↑𝑚 𝐼)) ∧ 𝑧 ∈ (𝐼 ∖ (◡𝑘 “ ℕ))) → (𝑘‘𝑧) = 0)
136	125, 135	sylanl2 683	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧 ∈ (𝐼 ∖ (◡𝑘 “ ℕ))) → (𝑘‘𝑧) = 0)
137	136	oveq1d 6665	. . . . . . . . . . . . 13 ⊢ ((((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)))
138	2	ad3antrrr 766	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧 ∈ (𝐼 ∖ (◡𝑘 “ ℕ))) → 𝐼 ∈ 𝑊)
139	12	ad3antrrr 766	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧 ∈ (𝐼 ∖ (◡𝑘 “ ℕ))) → 𝑅 ∈ Ring)
140		eldifi 3732	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ (𝐼 ∖ (◡𝑘 “ ℕ)) → 𝑧 ∈ 𝐼)
141	140	adantl 482	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧 ∈ (𝐼 ∖ (◡𝑘 “ ℕ))) → 𝑧 ∈ 𝐼)
142	5, 10, 6, 138, 139, 141	mvrcl 19449	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧 ∈ (𝐼 ∖ (◡𝑘 “ ℕ))) → (𝑉‘𝑧) ∈ (Base‘𝑃))
143	107, 85, 80	mulg0 17546	. . . . . . . . . . . . . 14 ⊢ ((𝑉‘𝑧) ∈ (Base‘𝑃) → (0(.g‘(mulGrp‘𝑃))(𝑉‘𝑧)) = (1r‘𝑃))
144	142, 143	syl 17	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧 ∈ (𝐼 ∖ (◡𝑘 “ ℕ))) → (0(.g‘(mulGrp‘𝑃))(𝑉‘𝑧)) = (1r‘𝑃))
145	137, 144	eqtrd 2656	. . . . . . . . . . . 12 ⊢ ((((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))
146	145, 78	suppss2 7329	. . . . . . . . . . 11 ⊢ (((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))
147		suppssfifsupp 8290	. . . . . . . . . . 11 ⊢ ((((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))
148	120, 122, 123, 124, 146, 147	syl32anc 1334	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → (𝑧 ∈ 𝐼 ↦ ((𝑘‘𝑧)(.g‘(mulGrp‘𝑃))(𝑉‘𝑧))) finSupp (1r‘𝑃))
149	85, 90, 78, 93, 117, 148	gsumsubmcl 18319	. . . . . . . . 9 ⊢ (((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))
150	83, 149	eqeltrd 2701	. . . . . . . 8 ⊢ (((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))
151		eqid 2622	. . . . . . . . 9 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃)
152		eqid 2622	. . . . . . . . 9 ⊢ (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
153	151, 37, 152, 66	lssvscl 18955	. . . . . . . 8 ⊢ (((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))
154	62, 70, 77, 150, 153	syl22anc 1327	. . . . . . 7 ⊢ (((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))
155		eqid 2622	. . . . . . 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))))) = (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)))))
156	154, 155	fmptd 6385	. . . . . 6 ⊢ ((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))
157	47	mptrabex 6488	. . . . . . . . 9 ⊢ (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
158		funmpt 5926	. . . . . . . . 9 ⊢ 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)))))
159		fvex 6201	. . . . . . . . 9 ⊢ (0g‘𝑃) ∈ V
160	157, 158, 159	3pm3.2i 1239	. . . . . . . 8 ⊢ ((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)
161	160	a1i 11	. . . . . . 7 ⊢ ((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))
162	5, 1, 7, 34, 6	mplelbas 19430	. . . . . . . . . 10 ⊢ (𝑥 ∈ (Base‘𝑃) ↔ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 finSupp (0g‘𝑅)))
163	162	simprbi 480	. . . . . . . . 9 ⊢ (𝑥 ∈ (Base‘𝑃) → 𝑥 finSupp (0g‘𝑅))
164	163	adantl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) → 𝑥 finSupp (0g‘𝑅))
165	164	fsuppimpd 8282	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) → (𝑥 supp (0g‘𝑅)) ∈ Fin)
166		ssid 3624	. . . . . . . . . . . . 13 ⊢ (𝑥 supp (0g‘𝑅)) ⊆ (𝑥 supp (0g‘𝑅))
167	166	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) → (𝑥 supp (0g‘𝑅)) ⊆ (𝑥 supp (0g‘𝑅)))
168		fvexd 6203	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) → (0g‘𝑅) ∈ V)
169	72, 167, 49, 168	suppssr 7326	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ ({𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∖ (𝑥 supp (0g‘𝑅)))) → (𝑥‘𝑘) = (0g‘𝑅))
170	74	fveq2d 6195	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) → (0g‘𝑅) = (0g‘(Scalar‘𝑃)))
171	170	adantr 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ ({𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∖ (𝑥 supp (0g‘𝑅)))) → (0g‘𝑅) = (0g‘(Scalar‘𝑃)))
172	169, 171	eqtrd 2656	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ ({𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∖ (𝑥 supp (0g‘𝑅)))) → (𝑥‘𝑘) = (0g‘(Scalar‘𝑃)))
173	172	oveq1d 6665	. . . . . . . . 9 ⊢ (((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)))))
174		eldifi 3732	. . . . . . . . . 10 ⊢ (𝑘 ∈ ({𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∖ (𝑥 supp (0g‘𝑅))) → 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin})
175	12	ad2antrr 762	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) → 𝑅 ∈ Ring)
176	5, 6, 34, 35, 33, 78, 175, 82	mplmon 19463	. . . . . . . . . . 11 ⊢ (((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))
177		eqid 2622	. . . . . . . . . . . 12 ⊢ (0g‘(Scalar‘𝑃)) = (0g‘(Scalar‘𝑃))
178	6, 151, 37, 177, 41	lmod0vs 18896	. . . . . . . . . . 11 ⊢ ((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))
179	62, 176, 178	syl2anc 693	. . . . . . . . . 10 ⊢ (((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))
180	174, 179	sylan2 491	. . . . . . . . 9 ⊢ (((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))
181	173, 180	eqtrd 2656	. . . . . . . 8 ⊢ (((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))
182	181, 49	suppss2 7329	. . . . . . 7 ⊢ ((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)))
183		suppssfifsupp 8290	. . . . . . 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 ∧ 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))
184	161, 165, 182, 183	syl12anc 1324	. . . . . 6 ⊢ ((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))
185	41, 46, 49, 59, 156, 184	gsumsubgcl 18320	. . . . 5 ⊢ ((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))
186	40, 185	eqeltrd 2701	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) → 𝑥 ∈ (𝐴‘ran 𝑉))
187	186	ex 450	. . 3 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝑃) → 𝑥 ∈ (𝐴‘ran 𝑉)))
188	187	ssrdv 3609	. 2 ⊢ (𝜑 → (Base‘𝑃) ⊆ (𝐴‘ran 𝑉))
189	32, 188	eqssd 3620	1 ⊢ (𝜑 → (𝐴‘ran 𝑉) = (Base‘𝑃))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912  {crab 2916  Vcvv 3200   ∖ cdif 3571   ⊆ wss 3574  ifcif 4086   class class class wbr 4653   ↦ cmpt 4729  ^◡ccnv 5113  ran crn 5115   “ cima 5117  Fun wfun 5882   Fn wfn 5883  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650   supp csupp 7295   ↑_𝑚 cmap 7857  Fincfn 7955   finSupp cfsupp 8275  0cc0 9936  ℕcn 11020  ℕ₀cn0 11292  Basecbs 15857  ._rcmulr 15942  Scalarcsca 15944   ·_𝑠 cvsca 15945  0_gc0g 16100   Σ_g cgsu 16101  SubMndcsubmnd 17334  ._gcmg 17540  SubGrpcsubg 17588  CMndccmn 18193  Abelcabl 18194  mulGrpcmgp 18489  1_rcur 18501  Ringcrg 18547  CRingccrg 18548  SubRingcsubrg 18776  LModclmod 18863  LSubSpclss 18932  AssAlgcasa 19309  AlgSpancasp 19310   mPwSer cmps 19351   mVar cmvr 19352   mPoly cmpl 19353

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-ofr 6898  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-tset 15960  df-0g 16102  df-gsum 16103  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-submnd 17336  df-grp 17425  df-minusg 17426  df-sbg 17427  df-mulg 17541  df-subg 17591  df-ghm 17658  df-cntz 17750  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-srg 18506  df-ring 18549  df-cring 18550  df-subrg 18778  df-lmod 18865  df-lss 18933  df-assa 19312  df-asp 19313  df-psr 19356  df-mvr 19357  df-mpl 19358

This theorem is referenced by:  mplind  19502  evlseu  19516