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Theorem mplbas2 21588

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 21525	. . . 4 ⊢ (𝜑 → 𝑆 ∈ AssAlg)
5		mplbas2.p	. . . . . 6 ⊢ 𝑃 = (𝐼 mPoly 𝑅)
6		eqid 2732	. . . . . 6 ⊢ (Base‘𝑃) = (Base‘𝑃)
7		eqid 2732	. . . . . 6 ⊢ (Base‘𝑆) = (Base‘𝑆)
8	5, 1, 6, 7	mplbasss 21547	. . . . 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 20061	. . . . . . . . 9 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
12	3, 11	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ Ring)
13	1, 10, 7, 2, 12	mvrf 21535	. . . . . . 7 ⊢ (𝜑 → 𝑉:𝐼⟶(Base‘𝑆))
14	13	ffnd 6715	. . . . . 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 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑥 ∈ 𝐼)
18	5, 10, 6, 15, 16, 17	mvrcl 21542	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑉‘𝑥) ∈ (Base‘𝑃))
19	18	ralrimiva 3146	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 (𝑉‘𝑥) ∈ (Base‘𝑃))
20		ffnfv 7114	. . . . . 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 6722	. . . 4 ⊢ (𝜑 → ran 𝑉 ⊆ (Base‘𝑃))
23		mplbas2.a	. . . . 5 ⊢ 𝐴 = (AlgSpan‘𝑆)
24	23, 7	aspss 21422	. . . 4 ⊢ ((𝑆 ∈ AssAlg ∧ (Base‘𝑃) ⊆ (Base‘𝑆) ∧ ran 𝑉 ⊆ (Base‘𝑃)) → (𝐴‘ran 𝑉) ⊆ (𝐴‘(Base‘𝑃)))
25	4, 9, 22, 24	syl3anc 1371	. . 3 ⊢ (𝜑 → (𝐴‘ran 𝑉) ⊆ (𝐴‘(Base‘𝑃)))
26	1, 5, 6, 2, 12	mplsubrg 21555	. . . 4 ⊢ (𝜑 → (Base‘𝑃) ∈ (SubRing‘𝑆))
27	1, 5, 6, 2, 12	mpllss 21553	. . . 4 ⊢ (𝜑 → (Base‘𝑃) ∈ (LSubSp‘𝑆))
28		eqid 2732	. . . . 5 ⊢ (LSubSp‘𝑆) = (LSubSp‘𝑆)
29	23, 7, 28	aspid 21420	. . . 4 ⊢ ((𝑆 ∈ AssAlg ∧ (Base‘𝑃) ∈ (SubRing‘𝑆) ∧ (Base‘𝑃) ∈ (LSubSp‘𝑆)) → (𝐴‘(Base‘𝑃)) = (Base‘𝑃))
30	4, 26, 27, 29	syl3anc 1371	. . 3 ⊢ (𝜑 → (𝐴‘(Base‘𝑃)) = (Base‘𝑃))
31	25, 30	sseqtrd 4021	. 2 ⊢ (𝜑 → (𝐴‘ran 𝑉) ⊆ (Base‘𝑃))
32		eqid 2732	. . . 4 ⊢ {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}
33		eqid 2732	. . . 4 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
34		eqid 2732	. . . 4 ⊢ (1_r‘𝑅) = (1_r‘𝑅)
35	2	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) → 𝐼 ∈ 𝑊)
36		eqid 2732	. . . 4 ⊢ ( ·_𝑠 ‘𝑃) = ( ·_𝑠 ‘𝑃)
37	12	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) → 𝑅 ∈ Ring)
38		simpr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) → 𝑥 ∈ (Base‘𝑃))
39	5, 32, 33, 34, 35, 6, 36, 37, 38	mplcoe1 21583	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) → 𝑥 = (𝑃 Σ_g (𝑘 ∈ {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin} ↦ ((𝑥‘𝑘)( ·_𝑠 ‘𝑃)(𝑦 ∈ {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = 𝑘, (1_r‘𝑅), (0_g‘𝑅)))))))
40		eqid 2732	. . . 4 ⊢ (0_g‘𝑃) = (0_g‘𝑃)
41	5	mplring 21569	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ Ring) → 𝑃 ∈ Ring)
42	2, 12, 41	syl2anc 584	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ Ring)
43		ringabl 20091	. . . . . 6 ⊢ (𝑃 ∈ Ring → 𝑃 ∈ Abel)
44	42, 43	syl 17	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ Abel)
45	44	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) → 𝑃 ∈ Abel)
46		ovex 7438	. . . . . 6 ⊢ (ℕ₀ ↑_m 𝐼) ∈ V
47	46	rabex 5331	. . . . 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 6722	. . . . . . . 8 ⊢ (𝜑 → ran 𝑉 ⊆ (Base‘𝑆))
50	23, 7	aspsubrg 21421	. . . . . . . 8 ⊢ ((𝑆 ∈ AssAlg ∧ ran 𝑉 ⊆ (Base‘𝑆)) → (𝐴‘ran 𝑉) ∈ (SubRing‘𝑆))
51	4, 49, 50	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → (𝐴‘ran 𝑉) ∈ (SubRing‘𝑆))
52	5, 1, 6	mplval2 21546	. . . . . . . . 9 ⊢ 𝑃 = (𝑆 ↾_s (Base‘𝑃))
53	52	subsubrg 20382	. . . . . . . 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 711	. . . . . 6 ⊢ (𝜑 → (𝐴‘ran 𝑉) ∈ (SubRing‘𝑃))
56		subrgsubg 20361	. . . . . 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 481	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) → (𝐴‘ran 𝑉) ∈ (SubGrp‘𝑃))
59	5	mpllmod 21568	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ Ring) → 𝑃 ∈ LMod)
60	2, 12, 59	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ LMod)
61	60	ad2antrr 724	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}) → 𝑃 ∈ LMod)
62	23, 7, 28	asplss 21419	. . . . . . . . 9 ⊢ ((𝑆 ∈ AssAlg ∧ ran 𝑉 ⊆ (Base‘𝑆)) → (𝐴‘ran 𝑉) ∈ (LSubSp‘𝑆))
63	4, 49, 62	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → (𝐴‘ran 𝑉) ∈ (LSubSp‘𝑆))
64	1, 2, 12	psrlmod 21512	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ LMod)
65		eqid 2732	. . . . . . . . . 10 ⊢ (LSubSp‘𝑃) = (LSubSp‘𝑃)
66	52, 28, 65	lsslss 20564	. . . . . . . . 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 584	. . . . . . . 8 ⊢ (𝜑 → ((𝐴‘ran 𝑉) ∈ (LSubSp‘𝑃) ↔ ((𝐴‘ran 𝑉) ∈ (LSubSp‘𝑆) ∧ (𝐴‘ran 𝑉) ⊆ (Base‘𝑃))))
68	63, 31, 67	mpbir2and 711	. . . . . . 7 ⊢ (𝜑 → (𝐴‘ran 𝑉) ∈ (LSubSp‘𝑃))
69	68	ad2antrr 724	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}) → (𝐴‘ran 𝑉) ∈ (LSubSp‘𝑃))
70		eqid 2732	. . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅)
71	5, 70, 6, 32, 38	mplelf 21548	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) → 𝑥:{𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
72	71	ffvelcdmda 7083	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}) → (𝑥‘𝑘) ∈ (Base‘𝑅))
73	5, 35, 37	mplsca 21563	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) → 𝑅 = (Scalar‘𝑃))
74	73	adantr 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}) → 𝑅 = (Scalar‘𝑃))
75	74	fveq2d 6892	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}) → (Base‘𝑅) = (Base‘(Scalar‘𝑃)))
76	72, 75	eleqtrd 2835	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}) → (𝑥‘𝑘) ∈ (Base‘(Scalar‘𝑃)))
77	2	ad2antrr 724	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}) → 𝐼 ∈ 𝑊)
78		eqid 2732	. . . . . . . 8 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃)
79		eqid 2732	. . . . . . . 8 ⊢ (._g‘(mulGrp‘𝑃)) = (._g‘(mulGrp‘𝑃))
80	3	ad2antrr 724	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}) → 𝑅 ∈ CRing)
81		simpr 485	. . . . . . . 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 21587	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}) → (𝑦 ∈ {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = 𝑘, (1_r‘𝑅), (0_g‘𝑅))) = ((mulGrp‘𝑃) Σ_g (𝑧 ∈ 𝐼 ↦ ((𝑘‘𝑧)(._g‘(mulGrp‘𝑃))(𝑉‘𝑧)))))
83		eqid 2732	. . . . . . . . 9 ⊢ (1_r‘𝑃) = (1_r‘𝑃)
84	78, 83	ringidval 20000	. . . . . . . 8 ⊢ (1_r‘𝑃) = (0_g‘(mulGrp‘𝑃))
85	5	mplcrng 21571	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ CRing) → 𝑃 ∈ CRing)
86	2, 3, 85	syl2anc 584	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ CRing)
87	78	crngmgp 20057	. . . . . . . . . 10 ⊢ (𝑃 ∈ CRing → (mulGrp‘𝑃) ∈ CMnd)
88	86, 87	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (mulGrp‘𝑃) ∈ CMnd)
89	88	ad2antrr 724	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}) → (mulGrp‘𝑃) ∈ CMnd)
90	55	ad2antrr 724	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}) → (𝐴‘ran 𝑉) ∈ (SubRing‘𝑃))
91	78	subrgsubm 20368	. . . . . . . . 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 773	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧 ∈ 𝐼) → 𝜑)
94	32	psrbag 21461	. . . . . . . . . . . . . 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 477	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}) → (𝑘:𝐼⟶ℕ₀ ∧ (^◡𝑘 “ ℕ) ∈ Fin))
97	96	simpld 495	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}) → 𝑘:𝐼⟶ℕ₀)
98	97	ffvelcdmda 7083	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧 ∈ 𝐼) → (𝑘‘𝑧) ∈ ℕ₀)
99	23, 7	aspssid 21423	. . . . . . . . . . . . 13 ⊢ ((𝑆 ∈ AssAlg ∧ ran 𝑉 ⊆ (Base‘𝑆)) → ran 𝑉 ⊆ (𝐴‘ran 𝑉))
100	4, 49, 99	syl2anc 584	. . . . . . . . . . . 12 ⊢ (𝜑 → ran 𝑉 ⊆ (𝐴‘ran 𝑉))
101	100	ad3antrrr 728	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧 ∈ 𝐼) → ran 𝑉 ⊆ (𝐴‘ran 𝑉))
102	14	ad2antrr 724	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}) → 𝑉 Fn 𝐼)
103		fnfvelrn 7079	. . . . . . . . . . . 12 ⊢ ((𝑉 Fn 𝐼 ∧ 𝑧 ∈ 𝐼) → (𝑉‘𝑧) ∈ ran 𝑉)
104	102, 103	sylan 580	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧 ∈ 𝐼) → (𝑉‘𝑧) ∈ ran 𝑉)
105	101, 104	sseldd 3982	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧 ∈ 𝐼) → (𝑉‘𝑧) ∈ (𝐴‘ran 𝑉))
106	78, 6	mgpbas 19987	. . . . . . . . . . 11 ⊢ (Base‘𝑃) = (Base‘(mulGrp‘𝑃))
107		eqid 2732	. . . . . . . . . . . 12 ⊢ (._r‘𝑃) = (._r‘𝑃)
108	78, 107	mgpplusg 19985	. . . . . . . . . . 11 ⊢ (._r‘𝑃) = (+_g‘(mulGrp‘𝑃))
109	107	subrgmcl 20367	. . . . . . . . . . . 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 1163	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑢 ∈ (𝐴‘ran 𝑉) ∧ 𝑣 ∈ (𝐴‘ran 𝑉)) → (𝑢(._r‘𝑃)𝑣) ∈ (𝐴‘ran 𝑉))
111	83	subrg1cl 20363	. . . . . . . . . . . 12 ⊢ ((𝐴‘ran 𝑉) ∈ (SubRing‘𝑃) → (1_r‘𝑃) ∈ (𝐴‘ran 𝑉))
112	55, 111	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (1_r‘𝑃) ∈ (𝐴‘ran 𝑉))
113	106, 79, 108, 88, 31, 110, 84, 112	mulgnn0subcl 18961	. . . . . . . . . 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 1371	. . . . . . . . 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 7111	. . . . . . . 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 7222	. . . . . . . . . 10 ⊢ (𝜑 → (𝑧 ∈ 𝐼 ↦ ((𝑘‘𝑧)(._g‘(mulGrp‘𝑃))(𝑉‘𝑧))) ∈ V)
117	116	ad2antrr 724	. . . . . . . . 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 6583	. . . . . . . . . 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 6903	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}) → (1_r‘𝑃) ∈ V)
121	96	simprd 496	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}) → (^◡𝑘 “ ℕ) ∈ Fin)
122		elrabi 3676	. . . . . . . . . . . . 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 482	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ (ℕ₀ ↑_m 𝐼)) → 𝑘:𝐼⟶ℕ₀)
125	2	ad2antrr 724	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ (ℕ₀ ↑_m 𝐼)) → 𝐼 ∈ 𝑊)
126		fcdmnn0supp 12524	. . . . . . . . . . . . . . . 16 ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑘:𝐼⟶ℕ₀) → (𝑘 supp 0) = (^◡𝑘 “ ℕ))
127	125, 124, 126	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ (ℕ₀ ↑_m 𝐼)) → (𝑘 supp 0) = (^◡𝑘 “ ℕ))
128		eqimss 4039	. . . . . . . . . . . . . . 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 11204	. . . . . . . . . . . . . . 15 ⊢ 0 ∈ V
131	130	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ (ℕ₀ ↑_m 𝐼)) → 0 ∈ V)
132	124, 129, 125, 131	suppssr 8177	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ (ℕ₀ ↑_m 𝐼)) ∧ 𝑧 ∈ (𝐼 ∖ (^◡𝑘 “ ℕ))) → (𝑘‘𝑧) = 0)
133	122, 132	sylanl2 679	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧 ∈ (𝐼 ∖ (^◡𝑘 “ ℕ))) → (𝑘‘𝑧) = 0)
134	133	oveq1d 7420	. . . . . . . . . . 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 728	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧 ∈ (𝐼 ∖ (^◡𝑘 “ ℕ))) → 𝐼 ∈ 𝑊)
136	12	ad3antrrr 728	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧 ∈ (𝐼 ∖ (^◡𝑘 “ ℕ))) → 𝑅 ∈ Ring)
137		eldifi 4125	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ (𝐼 ∖ (^◡𝑘 “ ℕ)) → 𝑧 ∈ 𝐼)
138	137	adantl 482	. . . . . . . . . . . . 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 21542	. . . . . . . . . . . 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 18951	. . . . . . . . . . . 12 ⊢ ((𝑉‘𝑧) ∈ (Base‘𝑃) → (0(._g‘(mulGrp‘𝑃))(𝑉‘𝑧)) = (1_r‘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)) = (1_r‘𝑃))
142	134, 141	eqtrd 2772	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧 ∈ (𝐼 ∖ (^◡𝑘 “ ℕ))) → ((𝑘‘𝑧)(._g‘(mulGrp‘𝑃))(𝑉‘𝑧)) = (1_r‘𝑃))
143	142, 77	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 (1_r‘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))) ∧ (1_r‘𝑃) ∈ V) ∧ ((^◡𝑘 “ ℕ) ∈ Fin ∧ ((𝑧 ∈ 𝐼 ↦ ((𝑘‘𝑧)(._g‘(mulGrp‘𝑃))(𝑉‘𝑧))) supp (1_r‘𝑃)) ⊆ (^◡𝑘 “ ℕ))) → (𝑧 ∈ 𝐼 ↦ ((𝑘‘𝑧)(._g‘(mulGrp‘𝑃))(𝑉‘𝑧))) finSupp (1_r‘𝑃))
145	117, 119, 120, 121, 143, 144	syl32anc 1378	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}) → (𝑧 ∈ 𝐼 ↦ ((𝑘‘𝑧)(._g‘(mulGrp‘𝑃))(𝑉‘𝑧))) finSupp (1_r‘𝑃))
146	84, 89, 77, 92, 115, 145	gsumsubmcl 19781	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}) → ((mulGrp‘𝑃) Σ_g (𝑧 ∈ 𝐼 ↦ ((𝑘‘𝑧)(._g‘(mulGrp‘𝑃))(𝑉‘𝑧)))) ∈ (𝐴‘ran 𝑉))
147	82, 146	eqeltrd 2833	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}) → (𝑦 ∈ {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = 𝑘, (1_r‘𝑅), (0_g‘𝑅))) ∈ (𝐴‘ran 𝑉))
148		eqid 2732	. . . . . . 7 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃)
149		eqid 2732	. . . . . . 7 ⊢ (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
150	148, 36, 149, 65	lssvscl 20558	. . . . . 6 ⊢ (((𝑃 ∈ LMod ∧ (𝐴‘ran 𝑉) ∈ (LSubSp‘𝑃)) ∧ ((𝑥‘𝑘) ∈ (Base‘(Scalar‘𝑃)) ∧ (𝑦 ∈ {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = 𝑘, (1_r‘𝑅), (0_g‘𝑅))) ∈ (𝐴‘ran 𝑉))) → ((𝑥‘𝑘)( ·_𝑠 ‘𝑃)(𝑦 ∈ {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = 𝑘, (1_r‘𝑅), (0_g‘𝑅)))) ∈ (𝐴‘ran 𝑉))
151	61, 69, 76, 147, 150	syl22anc 837	. . . . 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, (1_r‘𝑅), (0_g‘𝑅)))) ∈ (𝐴‘ran 𝑉))
152	151	fmpttd 7111	. . . 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, (1_r‘𝑅), (0_g‘𝑅))))):{𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}⟶(𝐴‘ran 𝑉))
153	46	mptrabex 7223	. . . . . . 7 ⊢ (𝑘 ∈ {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin} ↦ ((𝑥‘𝑘)( ·_𝑠 ‘𝑃)(𝑦 ∈ {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = 𝑘, (1_r‘𝑅), (0_g‘𝑅))))) ∈ V
154		funmpt 6583	. . . . . . 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, (1_r‘𝑅), (0_g‘𝑅)))))
155		fvex 6901	. . . . . . 7 ⊢ (0_g‘𝑃) ∈ V
156	153, 154, 155	3pm3.2i 1339	. . . . . 6 ⊢ ((𝑘 ∈ {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin} ↦ ((𝑥‘𝑘)( ·_𝑠 ‘𝑃)(𝑦 ∈ {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = 𝑘, (1_r‘𝑅), (0_g‘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, (1_r‘𝑅), (0_g‘𝑅))))) ∧ (0_g‘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, (1_r‘𝑅), (0_g‘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, (1_r‘𝑅), (0_g‘𝑅))))) ∧ (0_g‘𝑃) ∈ V))
158	5, 1, 7, 33, 6	mplelbas 21541	. . . . . . . 8 ⊢ (𝑥 ∈ (Base‘𝑃) ↔ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 finSupp (0_g‘𝑅)))
159	158	simprbi 497	. . . . . . 7 ⊢ (𝑥 ∈ (Base‘𝑃) → 𝑥 finSupp (0_g‘𝑅))
160	159	adantl 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) → 𝑥 finSupp (0_g‘𝑅))
161	160	fsuppimpd 9365	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) → (𝑥 supp (0_g‘𝑅)) ∈ Fin)
162		ssidd 4004	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) → (𝑥 supp (0_g‘𝑅)) ⊆ (𝑥 supp (0_g‘𝑅)))
163		fvexd 6903	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) → (0_g‘𝑅) ∈ V)
164	71, 162, 48, 163	suppssr 8177	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ ({𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin} ∖ (𝑥 supp (0_g‘𝑅)))) → (𝑥‘𝑘) = (0_g‘𝑅))
165	73	fveq2d 6892	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) → (0_g‘𝑅) = (0_g‘(Scalar‘𝑃)))
166	165	adantr 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ ({𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin} ∖ (𝑥 supp (0_g‘𝑅)))) → (0_g‘𝑅) = (0_g‘(Scalar‘𝑃)))
167	164, 166	eqtrd 2772	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ ({𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin} ∖ (𝑥 supp (0_g‘𝑅)))) → (𝑥‘𝑘) = (0_g‘(Scalar‘𝑃)))
168	167	oveq1d 7420	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ ({𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin} ∖ (𝑥 supp (0_g‘𝑅)))) → ((𝑥‘𝑘)( ·_𝑠 ‘𝑃)(𝑦 ∈ {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = 𝑘, (1_r‘𝑅), (0_g‘𝑅)))) = ((0_g‘(Scalar‘𝑃))( ·_𝑠 ‘𝑃)(𝑦 ∈ {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = 𝑘, (1_r‘𝑅), (0_g‘𝑅)))))
169		eldifi 4125	. . . . . . . 8 ⊢ (𝑘 ∈ ({𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin} ∖ (𝑥 supp (0_g‘𝑅))) → 𝑘 ∈ {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin})
170	12	ad2antrr 724	. . . . . . . . . 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 21581	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}) → (𝑦 ∈ {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = 𝑘, (1_r‘𝑅), (0_g‘𝑅))) ∈ (Base‘𝑃))
172		eqid 2732	. . . . . . . . . 10 ⊢ (0_g‘(Scalar‘𝑃)) = (0_g‘(Scalar‘𝑃))
173	6, 148, 36, 172, 40	lmod0vs 20497	. . . . . . . . 9 ⊢ ((𝑃 ∈ LMod ∧ (𝑦 ∈ {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = 𝑘, (1_r‘𝑅), (0_g‘𝑅))) ∈ (Base‘𝑃)) → ((0_g‘(Scalar‘𝑃))( ·_𝑠 ‘𝑃)(𝑦 ∈ {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = 𝑘, (1_r‘𝑅), (0_g‘𝑅)))) = (0_g‘𝑃))
174	61, 171, 173	syl2anc 584	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin}) → ((0_g‘(Scalar‘𝑃))( ·_𝑠 ‘𝑃)(𝑦 ∈ {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = 𝑘, (1_r‘𝑅), (0_g‘𝑅)))) = (0_g‘𝑃))
175	169, 174	sylan2 593	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ ({𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin} ∖ (𝑥 supp (0_g‘𝑅)))) → ((0_g‘(Scalar‘𝑃))( ·_𝑠 ‘𝑃)(𝑦 ∈ {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = 𝑘, (1_r‘𝑅), (0_g‘𝑅)))) = (0_g‘𝑃))
176	168, 175	eqtrd 2772	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ ({𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin} ∖ (𝑥 supp (0_g‘𝑅)))) → ((𝑥‘𝑘)( ·_𝑠 ‘𝑃)(𝑦 ∈ {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = 𝑘, (1_r‘𝑅), (0_g‘𝑅)))) = (0_g‘𝑃))
177	176, 48	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, (1_r‘𝑅), (0_g‘𝑅))))) supp (0_g‘𝑃)) ⊆ (𝑥 supp (0_g‘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, (1_r‘𝑅), (0_g‘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, (1_r‘𝑅), (0_g‘𝑅))))) ∧ (0_g‘𝑃) ∈ V) ∧ ((𝑥 supp (0_g‘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, (1_r‘𝑅), (0_g‘𝑅))))) supp (0_g‘𝑃)) ⊆ (𝑥 supp (0_g‘𝑅)))) → (𝑘 ∈ {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin} ↦ ((𝑥‘𝑘)( ·_𝑠 ‘𝑃)(𝑦 ∈ {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = 𝑘, (1_r‘𝑅), (0_g‘𝑅))))) finSupp (0_g‘𝑃))
179	157, 161, 177, 178	syl12anc 835	. . . 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, (1_r‘𝑅), (0_g‘𝑅))))) finSupp (0_g‘𝑃))
180	40, 45, 48, 58, 152, 179	gsumsubgcl 19782	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) → (𝑃 Σ_g (𝑘 ∈ {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin} ↦ ((𝑥‘𝑘)( ·_𝑠 ‘𝑃)(𝑦 ∈ {𝑓 ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = 𝑘, (1_r‘𝑅), (0_g‘𝑅)))))) ∈ (𝐴‘ran 𝑉))
181	39, 180	eqeltrd 2833	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑃)) → 𝑥 ∈ (𝐴‘ran 𝑉))
182	31, 181	eqelssd 4002	1 ⊢ (𝜑 → (𝐴‘ran 𝑉) = (Base‘𝑃))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∀wral 3061 {crab 3432 Vcvv 3474 ∖ cdif 3944 ⊆ wss 3947 ifcif 4527 class class class wbr 5147 ↦ cmpt 5230 ^◡ccnv 5674 ran crn 5676 “ cima 5678 Fun wfun 6534 Fn wfn 6535 ⟶wf 6536 ‘cfv 6540 (class class class)co 7405 supp csupp 8142 ↑_m cmap 8816 Fincfn 8935 finSupp cfsupp 9357 0cc0 11106 ℕcn 12208 ℕ₀cn0 12468 Basecbs 17140 ._rcmulr 17194 Scalarcsca 17196 ·_𝑠 cvsca 17197 0_gc0g 17381 Σ_g cgsu 17382 SubMndcsubmnd 18666 ._gcmg 18944 SubGrpcsubg 18994 CMndccmn 19642 Abelcabl 19643 mulGrpcmgp 19981 1_rcur 19998 Ringcrg 20049 CRingccrg 20050 SubRingcsubrg 20351 LModclmod 20463 LSubSpclss 20534 AssAlgcasa 21396 AlgSpancasp 21397 mPwSer cmps 21448 mVar cmvr 21449 mPoly cmpl 21450

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 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 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7666 df-ofr 7667 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8143 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 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 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-fz 13481 df-fzo 13624 df-seq 13963 df-hash 14287 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-hom 17217 df-cco 17218 df-0g 17383 df-gsum 17384 df-prds 17389 df-pws 17391 df-mre 17526 df-mrc 17527 df-acs 17529 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-mhm 18667 df-submnd 18668 df-grp 18818 df-minusg 18819 df-sbg 18820 df-mulg 18945 df-subg 18997 df-ghm 19084 df-cntz 19175 df-cmn 19644 df-abl 19645 df-mgp 19982 df-ur 19999 df-srg 20003 df-ring 20051 df-cring 20052 df-subrg 20353 df-lmod 20465 df-lss 20535 df-assa 21399 df-asp 21400 df-psr 21453 df-mvr 21454 df-mpl 21455

This theorem is referenced by: mplind 21622 evlseu 21637

W3C validator