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Theorem mplbas2 22060
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 (𝜑𝐼𝑊)
mplbas2.r (𝜑𝑅 ∈ 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.
StepHypRef Expression
1 mplbas2.s . . . . 5 𝑆 = (𝐼 mPwSer 𝑅)
2 mplbas2.i . . . . 5 (𝜑𝐼𝑊)
3 mplbas2.r . . . . 5 (𝜑𝑅 ∈ CRing)
41, 2, 3psrassa 21993 . . . 4 (𝜑𝑆 ∈ AssAlg)
5 mplbas2.p . . . . . 6 𝑃 = (𝐼 mPoly 𝑅)
6 eqid 2737 . . . . . 6 (Base‘𝑃) = (Base‘𝑃)
7 eqid 2737 . . . . . 6 (Base‘𝑆) = (Base‘𝑆)
85, 1, 6, 7mplbasss 22017 . . . . 5 (Base‘𝑃) ⊆ (Base‘𝑆)
98a1i 11 . . . 4 (𝜑 → (Base‘𝑃) ⊆ (Base‘𝑆))
10 mplbas2.v . . . . . . . 8 𝑉 = (𝐼 mVar 𝑅)
11 crngring 20242 . . . . . . . . 9 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
123, 11syl 17 . . . . . . . 8 (𝜑𝑅 ∈ Ring)
131, 10, 7, 2, 12mvrf 22005 . . . . . . 7 (𝜑𝑉:𝐼⟶(Base‘𝑆))
1413ffnd 6737 . . . . . 6 (𝜑𝑉 Fn 𝐼)
152adantr 480 . . . . . . . 8 ((𝜑𝑥𝐼) → 𝐼𝑊)
1612adantr 480 . . . . . . . 8 ((𝜑𝑥𝐼) → 𝑅 ∈ Ring)
17 simpr 484 . . . . . . . 8 ((𝜑𝑥𝐼) → 𝑥𝐼)
185, 10, 6, 15, 16, 17mvrcl 22012 . . . . . . 7 ((𝜑𝑥𝐼) → (𝑉𝑥) ∈ (Base‘𝑃))
1918ralrimiva 3146 . . . . . 6 (𝜑 → ∀𝑥𝐼 (𝑉𝑥) ∈ (Base‘𝑃))
20 ffnfv 7139 . . . . . 6 (𝑉:𝐼⟶(Base‘𝑃) ↔ (𝑉 Fn 𝐼 ∧ ∀𝑥𝐼 (𝑉𝑥) ∈ (Base‘𝑃)))
2114, 19, 20sylanbrc 583 . . . . 5 (𝜑𝑉:𝐼⟶(Base‘𝑃))
2221frnd 6744 . . . 4 (𝜑 → ran 𝑉 ⊆ (Base‘𝑃))
23 mplbas2.a . . . . 5 𝐴 = (AlgSpan‘𝑆)
2423, 7aspss 21897 . . . 4 ((𝑆 ∈ AssAlg ∧ (Base‘𝑃) ⊆ (Base‘𝑆) ∧ ran 𝑉 ⊆ (Base‘𝑃)) → (𝐴‘ran 𝑉) ⊆ (𝐴‘(Base‘𝑃)))
254, 9, 22, 24syl3anc 1373 . . 3 (𝜑 → (𝐴‘ran 𝑉) ⊆ (𝐴‘(Base‘𝑃)))
261, 5, 6, 2, 12mplsubrg 22025 . . . 4 (𝜑 → (Base‘𝑃) ∈ (SubRing‘𝑆))
271, 5, 6, 2, 12mpllss 22023 . . . 4 (𝜑 → (Base‘𝑃) ∈ (LSubSp‘𝑆))
28 eqid 2737 . . . . 5 (LSubSp‘𝑆) = (LSubSp‘𝑆)
2923, 7, 28aspid 21895 . . . 4 ((𝑆 ∈ AssAlg ∧ (Base‘𝑃) ∈ (SubRing‘𝑆) ∧ (Base‘𝑃) ∈ (LSubSp‘𝑆)) → (𝐴‘(Base‘𝑃)) = (Base‘𝑃))
304, 26, 27, 29syl3anc 1373 . . 3 (𝜑 → (𝐴‘(Base‘𝑃)) = (Base‘𝑃))
3125, 30sseqtrd 4020 . 2 (𝜑 → (𝐴‘ran 𝑉) ⊆ (Base‘𝑃))
32 eqid 2737 . . . 4 {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}
33 eqid 2737 . . . 4 (0g𝑅) = (0g𝑅)
34 eqid 2737 . . . 4 (1r𝑅) = (1r𝑅)
352adantr 480 . . . 4 ((𝜑𝑥 ∈ (Base‘𝑃)) → 𝐼𝑊)
36 eqid 2737 . . . 4 ( ·𝑠𝑃) = ( ·𝑠𝑃)
3712adantr 480 . . . 4 ((𝜑𝑥 ∈ (Base‘𝑃)) → 𝑅 ∈ Ring)
38 simpr 484 . . . 4 ((𝜑𝑥 ∈ (Base‘𝑃)) → 𝑥 ∈ (Base‘𝑃))
395, 32, 33, 34, 35, 6, 36, 37, 38mplcoe1 22055 . . 3 ((𝜑𝑥 ∈ (Base‘𝑃)) → 𝑥 = (𝑃 Σg (𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ↦ ((𝑥𝑘)( ·𝑠𝑃)(𝑦 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = 𝑘, (1r𝑅), (0g𝑅)))))))
40 eqid 2737 . . . 4 (0g𝑃) = (0g𝑃)
415, 2, 12mplringd 22043 . . . . . 6 (𝜑𝑃 ∈ Ring)
42 ringabl 20278 . . . . . 6 (𝑃 ∈ Ring → 𝑃 ∈ Abel)
4341, 42syl 17 . . . . 5 (𝜑𝑃 ∈ Abel)
4443adantr 480 . . . 4 ((𝜑𝑥 ∈ (Base‘𝑃)) → 𝑃 ∈ Abel)
45 ovex 7464 . . . . . 6 (ℕ0m 𝐼) ∈ V
4645rabex 5339 . . . . 5 {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∈ V
4746a1i 11 . . . 4 ((𝜑𝑥 ∈ (Base‘𝑃)) → {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∈ V)
4813frnd 6744 . . . . . . . 8 (𝜑 → ran 𝑉 ⊆ (Base‘𝑆))
4923, 7aspsubrg 21896 . . . . . . . 8 ((𝑆 ∈ AssAlg ∧ ran 𝑉 ⊆ (Base‘𝑆)) → (𝐴‘ran 𝑉) ∈ (SubRing‘𝑆))
504, 48, 49syl2anc 584 . . . . . . 7 (𝜑 → (𝐴‘ran 𝑉) ∈ (SubRing‘𝑆))
515, 1, 6mplval2 22016 . . . . . . . . 9 𝑃 = (𝑆s (Base‘𝑃))
5251subsubrg 20598 . . . . . . . 8 ((Base‘𝑃) ∈ (SubRing‘𝑆) → ((𝐴‘ran 𝑉) ∈ (SubRing‘𝑃) ↔ ((𝐴‘ran 𝑉) ∈ (SubRing‘𝑆) ∧ (𝐴‘ran 𝑉) ⊆ (Base‘𝑃))))
5326, 52syl 17 . . . . . . 7 (𝜑 → ((𝐴‘ran 𝑉) ∈ (SubRing‘𝑃) ↔ ((𝐴‘ran 𝑉) ∈ (SubRing‘𝑆) ∧ (𝐴‘ran 𝑉) ⊆ (Base‘𝑃))))
5450, 31, 53mpbir2and 713 . . . . . 6 (𝜑 → (𝐴‘ran 𝑉) ∈ (SubRing‘𝑃))
55 subrgsubg 20577 . . . . . 6 ((𝐴‘ran 𝑉) ∈ (SubRing‘𝑃) → (𝐴‘ran 𝑉) ∈ (SubGrp‘𝑃))
5654, 55syl 17 . . . . 5 (𝜑 → (𝐴‘ran 𝑉) ∈ (SubGrp‘𝑃))
5756adantr 480 . . . 4 ((𝜑𝑥 ∈ (Base‘𝑃)) → (𝐴‘ran 𝑉) ∈ (SubGrp‘𝑃))
585, 2, 12mpllmodd 22044 . . . . . . 7 (𝜑𝑃 ∈ LMod)
5958ad2antrr 726 . . . . . 6 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → 𝑃 ∈ LMod)
6023, 7, 28asplss 21894 . . . . . . . . 9 ((𝑆 ∈ AssAlg ∧ ran 𝑉 ⊆ (Base‘𝑆)) → (𝐴‘ran 𝑉) ∈ (LSubSp‘𝑆))
614, 48, 60syl2anc 584 . . . . . . . 8 (𝜑 → (𝐴‘ran 𝑉) ∈ (LSubSp‘𝑆))
621, 2, 12psrlmod 21980 . . . . . . . . 9 (𝜑𝑆 ∈ LMod)
63 eqid 2737 . . . . . . . . . 10 (LSubSp‘𝑃) = (LSubSp‘𝑃)
6451, 28, 63lsslss 20959 . . . . . . . . 9 ((𝑆 ∈ LMod ∧ (Base‘𝑃) ∈ (LSubSp‘𝑆)) → ((𝐴‘ran 𝑉) ∈ (LSubSp‘𝑃) ↔ ((𝐴‘ran 𝑉) ∈ (LSubSp‘𝑆) ∧ (𝐴‘ran 𝑉) ⊆ (Base‘𝑃))))
6562, 27, 64syl2anc 584 . . . . . . . 8 (𝜑 → ((𝐴‘ran 𝑉) ∈ (LSubSp‘𝑃) ↔ ((𝐴‘ran 𝑉) ∈ (LSubSp‘𝑆) ∧ (𝐴‘ran 𝑉) ⊆ (Base‘𝑃))))
6661, 31, 65mpbir2and 713 . . . . . . 7 (𝜑 → (𝐴‘ran 𝑉) ∈ (LSubSp‘𝑃))
6766ad2antrr 726 . . . . . 6 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝐴‘ran 𝑉) ∈ (LSubSp‘𝑃))
68 eqid 2737 . . . . . . . . 9 (Base‘𝑅) = (Base‘𝑅)
695, 68, 6, 32, 38mplelf 22018 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝑃)) → 𝑥:{𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
7069ffvelcdmda 7104 . . . . . . 7 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑥𝑘) ∈ (Base‘𝑅))
715, 35, 37mplsca 22033 . . . . . . . . 9 ((𝜑𝑥 ∈ (Base‘𝑃)) → 𝑅 = (Scalar‘𝑃))
7271adantr 480 . . . . . . . 8 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → 𝑅 = (Scalar‘𝑃))
7372fveq2d 6910 . . . . . . 7 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → (Base‘𝑅) = (Base‘(Scalar‘𝑃)))
7470, 73eleqtrd 2843 . . . . . 6 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑥𝑘) ∈ (Base‘(Scalar‘𝑃)))
752ad2antrr 726 . . . . . . . 8 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → 𝐼𝑊)
76 eqid 2737 . . . . . . . 8 (mulGrp‘𝑃) = (mulGrp‘𝑃)
77 eqid 2737 . . . . . . . 8 (.g‘(mulGrp‘𝑃)) = (.g‘(mulGrp‘𝑃))
783ad2antrr 726 . . . . . . . 8 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → 𝑅 ∈ CRing)
79 simpr 484 . . . . . . . 8 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin})
805, 32, 33, 34, 75, 76, 77, 10, 78, 79mplcoe2 22059 . . . . . . 7 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑦 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = 𝑘, (1r𝑅), (0g𝑅))) = ((mulGrp‘𝑃) Σg (𝑧𝐼 ↦ ((𝑘𝑧)(.g‘(mulGrp‘𝑃))(𝑉𝑧)))))
81 eqid 2737 . . . . . . . . 9 (1r𝑃) = (1r𝑃)
8276, 81ringidval 20180 . . . . . . . 8 (1r𝑃) = (0g‘(mulGrp‘𝑃))
835mplcrng 22041 . . . . . . . . . . 11 ((𝐼𝑊𝑅 ∈ CRing) → 𝑃 ∈ CRing)
842, 3, 83syl2anc 584 . . . . . . . . . 10 (𝜑𝑃 ∈ CRing)
8576crngmgp 20238 . . . . . . . . . 10 (𝑃 ∈ CRing → (mulGrp‘𝑃) ∈ CMnd)
8684, 85syl 17 . . . . . . . . 9 (𝜑 → (mulGrp‘𝑃) ∈ CMnd)
8786ad2antrr 726 . . . . . . . 8 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → (mulGrp‘𝑃) ∈ CMnd)
8854ad2antrr 726 . . . . . . . . 9 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝐴‘ran 𝑉) ∈ (SubRing‘𝑃))
8976subrgsubm 20585 . . . . . . . . 9 ((𝐴‘ran 𝑉) ∈ (SubRing‘𝑃) → (𝐴‘ran 𝑉) ∈ (SubMnd‘(mulGrp‘𝑃)))
9088, 89syl 17 . . . . . . . 8 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝐴‘ran 𝑉) ∈ (SubMnd‘(mulGrp‘𝑃)))
91 simplll 775 . . . . . . . . . 10 ((((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧𝐼) → 𝜑)
9232psrbag 21937 . . . . . . . . . . . . . 14 (𝐼𝑊 → (𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ↔ (𝑘:𝐼⟶ℕ0 ∧ (𝑘 “ ℕ) ∈ Fin)))
9335, 92syl 17 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (Base‘𝑃)) → (𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ↔ (𝑘:𝐼⟶ℕ0 ∧ (𝑘 “ ℕ) ∈ Fin)))
9493biimpa 476 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑘:𝐼⟶ℕ0 ∧ (𝑘 “ ℕ) ∈ Fin))
9594simpld 494 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → 𝑘:𝐼⟶ℕ0)
9695ffvelcdmda 7104 . . . . . . . . . 10 ((((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧𝐼) → (𝑘𝑧) ∈ ℕ0)
9723, 7aspssid 21898 . . . . . . . . . . . . 13 ((𝑆 ∈ AssAlg ∧ ran 𝑉 ⊆ (Base‘𝑆)) → ran 𝑉 ⊆ (𝐴‘ran 𝑉))
984, 48, 97syl2anc 584 . . . . . . . . . . . 12 (𝜑 → ran 𝑉 ⊆ (𝐴‘ran 𝑉))
9998ad3antrrr 730 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧𝐼) → ran 𝑉 ⊆ (𝐴‘ran 𝑉))
10014ad2antrr 726 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → 𝑉 Fn 𝐼)
101 fnfvelrn 7100 . . . . . . . . . . . 12 ((𝑉 Fn 𝐼𝑧𝐼) → (𝑉𝑧) ∈ ran 𝑉)
102100, 101sylan 580 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧𝐼) → (𝑉𝑧) ∈ ran 𝑉)
10399, 102sseldd 3984 . . . . . . . . . 10 ((((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧𝐼) → (𝑉𝑧) ∈ (𝐴‘ran 𝑉))
10476, 6mgpbas 20142 . . . . . . . . . . 11 (Base‘𝑃) = (Base‘(mulGrp‘𝑃))
105 eqid 2737 . . . . . . . . . . . 12 (.r𝑃) = (.r𝑃)
10676, 105mgpplusg 20141 . . . . . . . . . . 11 (.r𝑃) = (+g‘(mulGrp‘𝑃))
107105subrgmcl 20584 . . . . . . . . . . . 12 (((𝐴‘ran 𝑉) ∈ (SubRing‘𝑃) ∧ 𝑢 ∈ (𝐴‘ran 𝑉) ∧ 𝑣 ∈ (𝐴‘ran 𝑉)) → (𝑢(.r𝑃)𝑣) ∈ (𝐴‘ran 𝑉))
10854, 107syl3an1 1164 . . . . . . . . . . 11 ((𝜑𝑢 ∈ (𝐴‘ran 𝑉) ∧ 𝑣 ∈ (𝐴‘ran 𝑉)) → (𝑢(.r𝑃)𝑣) ∈ (𝐴‘ran 𝑉))
10981subrg1cl 20580 . . . . . . . . . . . 12 ((𝐴‘ran 𝑉) ∈ (SubRing‘𝑃) → (1r𝑃) ∈ (𝐴‘ran 𝑉))
11054, 109syl 17 . . . . . . . . . . 11 (𝜑 → (1r𝑃) ∈ (𝐴‘ran 𝑉))
111104, 77, 106, 86, 31, 108, 82, 110mulgnn0subcl 19105 . . . . . . . . . 10 ((𝜑 ∧ (𝑘𝑧) ∈ ℕ0 ∧ (𝑉𝑧) ∈ (𝐴‘ran 𝑉)) → ((𝑘𝑧)(.g‘(mulGrp‘𝑃))(𝑉𝑧)) ∈ (𝐴‘ran 𝑉))
11291, 96, 103, 111syl3anc 1373 . . . . . . . . 9 ((((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧𝐼) → ((𝑘𝑧)(.g‘(mulGrp‘𝑃))(𝑉𝑧)) ∈ (𝐴‘ran 𝑉))
113112fmpttd 7135 . . . . . . . 8 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑧𝐼 ↦ ((𝑘𝑧)(.g‘(mulGrp‘𝑃))(𝑉𝑧))):𝐼⟶(𝐴‘ran 𝑉))
1142mptexd 7244 . . . . . . . . . 10 (𝜑 → (𝑧𝐼 ↦ ((𝑘𝑧)(.g‘(mulGrp‘𝑃))(𝑉𝑧))) ∈ V)
115114ad2antrr 726 . . . . . . . . 9 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑧𝐼 ↦ ((𝑘𝑧)(.g‘(mulGrp‘𝑃))(𝑉𝑧))) ∈ V)
116 funmpt 6604 . . . . . . . . . 10 Fun (𝑧𝐼 ↦ ((𝑘𝑧)(.g‘(mulGrp‘𝑃))(𝑉𝑧)))
117116a1i 11 . . . . . . . . 9 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → Fun (𝑧𝐼 ↦ ((𝑘𝑧)(.g‘(mulGrp‘𝑃))(𝑉𝑧))))
118 fvexd 6921 . . . . . . . . 9 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → (1r𝑃) ∈ V)
11994simprd 495 . . . . . . . . 9 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑘 “ ℕ) ∈ Fin)
120 elrabi 3687 . . . . . . . . . . . . 13 (𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} → 𝑘 ∈ (ℕ0m 𝐼))
121 elmapi 8889 . . . . . . . . . . . . . . 15 (𝑘 ∈ (ℕ0m 𝐼) → 𝑘:𝐼⟶ℕ0)
122121adantl 481 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ (ℕ0m 𝐼)) → 𝑘:𝐼⟶ℕ0)
1232ad2antrr 726 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ (ℕ0m 𝐼)) → 𝐼𝑊)
124 fcdmnn0supp 12583 . . . . . . . . . . . . . . . 16 ((𝐼𝑊𝑘:𝐼⟶ℕ0) → (𝑘 supp 0) = (𝑘 “ ℕ))
125123, 122, 124syl2anc 584 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ (ℕ0m 𝐼)) → (𝑘 supp 0) = (𝑘 “ ℕ))
126 eqimss 4042 . . . . . . . . . . . . . . 15 ((𝑘 supp 0) = (𝑘 “ ℕ) → (𝑘 supp 0) ⊆ (𝑘 “ ℕ))
127125, 126syl 17 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ (ℕ0m 𝐼)) → (𝑘 supp 0) ⊆ (𝑘 “ ℕ))
128 c0ex 11255 . . . . . . . . . . . . . . 15 0 ∈ V
129128a1i 11 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ (ℕ0m 𝐼)) → 0 ∈ V)
130122, 127, 123, 129suppssr 8220 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ (ℕ0m 𝐼)) ∧ 𝑧 ∈ (𝐼 ∖ (𝑘 “ ℕ))) → (𝑘𝑧) = 0)
131120, 130sylanl2 681 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧 ∈ (𝐼 ∖ (𝑘 “ ℕ))) → (𝑘𝑧) = 0)
132131oveq1d 7446 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧 ∈ (𝐼 ∖ (𝑘 “ ℕ))) → ((𝑘𝑧)(.g‘(mulGrp‘𝑃))(𝑉𝑧)) = (0(.g‘(mulGrp‘𝑃))(𝑉𝑧)))
1332ad3antrrr 730 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧 ∈ (𝐼 ∖ (𝑘 “ ℕ))) → 𝐼𝑊)
13412ad3antrrr 730 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧 ∈ (𝐼 ∖ (𝑘 “ ℕ))) → 𝑅 ∈ Ring)
135 eldifi 4131 . . . . . . . . . . . . . 14 (𝑧 ∈ (𝐼 ∖ (𝑘 “ ℕ)) → 𝑧𝐼)
136135adantl 481 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧 ∈ (𝐼 ∖ (𝑘 “ ℕ))) → 𝑧𝐼)
1375, 10, 6, 133, 134, 136mvrcl 22012 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧 ∈ (𝐼 ∖ (𝑘 “ ℕ))) → (𝑉𝑧) ∈ (Base‘𝑃))
138104, 82, 77mulg0 19092 . . . . . . . . . . . 12 ((𝑉𝑧) ∈ (Base‘𝑃) → (0(.g‘(mulGrp‘𝑃))(𝑉𝑧)) = (1r𝑃))
139137, 138syl 17 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧 ∈ (𝐼 ∖ (𝑘 “ ℕ))) → (0(.g‘(mulGrp‘𝑃))(𝑉𝑧)) = (1r𝑃))
140132, 139eqtrd 2777 . . . . . . . . . 10 ((((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧 ∈ (𝐼 ∖ (𝑘 “ ℕ))) → ((𝑘𝑧)(.g‘(mulGrp‘𝑃))(𝑉𝑧)) = (1r𝑃))
141140, 75suppss2 8225 . . . . . . . . 9 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → ((𝑧𝐼 ↦ ((𝑘𝑧)(.g‘(mulGrp‘𝑃))(𝑉𝑧))) supp (1r𝑃)) ⊆ (𝑘 “ ℕ))
142 suppssfifsupp 9420 . . . . . . . . 9 ((((𝑧𝐼 ↦ ((𝑘𝑧)(.g‘(mulGrp‘𝑃))(𝑉𝑧))) ∈ V ∧ Fun (𝑧𝐼 ↦ ((𝑘𝑧)(.g‘(mulGrp‘𝑃))(𝑉𝑧))) ∧ (1r𝑃) ∈ V) ∧ ((𝑘 “ ℕ) ∈ Fin ∧ ((𝑧𝐼 ↦ ((𝑘𝑧)(.g‘(mulGrp‘𝑃))(𝑉𝑧))) supp (1r𝑃)) ⊆ (𝑘 “ ℕ))) → (𝑧𝐼 ↦ ((𝑘𝑧)(.g‘(mulGrp‘𝑃))(𝑉𝑧))) finSupp (1r𝑃))
143115, 117, 118, 119, 141, 142syl32anc 1380 . . . . . . . 8 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑧𝐼 ↦ ((𝑘𝑧)(.g‘(mulGrp‘𝑃))(𝑉𝑧))) finSupp (1r𝑃))
14482, 87, 75, 90, 113, 143gsumsubmcl 19937 . . . . . . 7 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → ((mulGrp‘𝑃) Σg (𝑧𝐼 ↦ ((𝑘𝑧)(.g‘(mulGrp‘𝑃))(𝑉𝑧)))) ∈ (𝐴‘ran 𝑉))
14580, 144eqeltrd 2841 . . . . . 6 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑦 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = 𝑘, (1r𝑅), (0g𝑅))) ∈ (𝐴‘ran 𝑉))
146 eqid 2737 . . . . . . 7 (Scalar‘𝑃) = (Scalar‘𝑃)
147 eqid 2737 . . . . . . 7 (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
148146, 36, 147, 63lssvscl 20953 . . . . . 6 (((𝑃 ∈ LMod ∧ (𝐴‘ran 𝑉) ∈ (LSubSp‘𝑃)) ∧ ((𝑥𝑘) ∈ (Base‘(Scalar‘𝑃)) ∧ (𝑦 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = 𝑘, (1r𝑅), (0g𝑅))) ∈ (𝐴‘ran 𝑉))) → ((𝑥𝑘)( ·𝑠𝑃)(𝑦 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = 𝑘, (1r𝑅), (0g𝑅)))) ∈ (𝐴‘ran 𝑉))
14959, 67, 74, 145, 148syl22anc 839 . . . . 5 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → ((𝑥𝑘)( ·𝑠𝑃)(𝑦 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = 𝑘, (1r𝑅), (0g𝑅)))) ∈ (𝐴‘ran 𝑉))
150149fmpttd 7135 . . . 4 ((𝜑𝑥 ∈ (Base‘𝑃)) → (𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ↦ ((𝑥𝑘)( ·𝑠𝑃)(𝑦 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = 𝑘, (1r𝑅), (0g𝑅))))):{𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}⟶(𝐴‘ran 𝑉))
15145mptrabex 7245 . . . . . . 7 (𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ↦ ((𝑥𝑘)( ·𝑠𝑃)(𝑦 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = 𝑘, (1r𝑅), (0g𝑅))))) ∈ V
152 funmpt 6604 . . . . . . 7 Fun (𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ↦ ((𝑥𝑘)( ·𝑠𝑃)(𝑦 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = 𝑘, (1r𝑅), (0g𝑅)))))
153 fvex 6919 . . . . . . 7 (0g𝑃) ∈ V
154151, 152, 1533pm3.2i 1340 . . . . . 6 ((𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ↦ ((𝑥𝑘)( ·𝑠𝑃)(𝑦 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = 𝑘, (1r𝑅), (0g𝑅))))) ∈ V ∧ Fun (𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ↦ ((𝑥𝑘)( ·𝑠𝑃)(𝑦 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = 𝑘, (1r𝑅), (0g𝑅))))) ∧ (0g𝑃) ∈ V)
155154a1i 11 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝑃)) → ((𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ↦ ((𝑥𝑘)( ·𝑠𝑃)(𝑦 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = 𝑘, (1r𝑅), (0g𝑅))))) ∈ V ∧ Fun (𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ↦ ((𝑥𝑘)( ·𝑠𝑃)(𝑦 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = 𝑘, (1r𝑅), (0g𝑅))))) ∧ (0g𝑃) ∈ V))
1565, 1, 7, 33, 6mplelbas 22011 . . . . . . . 8 (𝑥 ∈ (Base‘𝑃) ↔ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 finSupp (0g𝑅)))
157156simprbi 496 . . . . . . 7 (𝑥 ∈ (Base‘𝑃) → 𝑥 finSupp (0g𝑅))
158157adantl 481 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝑃)) → 𝑥 finSupp (0g𝑅))
159158fsuppimpd 9409 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝑃)) → (𝑥 supp (0g𝑅)) ∈ Fin)
160 ssidd 4007 . . . . . . . . . 10 ((𝜑𝑥 ∈ (Base‘𝑃)) → (𝑥 supp (0g𝑅)) ⊆ (𝑥 supp (0g𝑅)))
161 fvexd 6921 . . . . . . . . . 10 ((𝜑𝑥 ∈ (Base‘𝑃)) → (0g𝑅) ∈ V)
16269, 160, 47, 161suppssr 8220 . . . . . . . . 9 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ ({𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∖ (𝑥 supp (0g𝑅)))) → (𝑥𝑘) = (0g𝑅))
16371fveq2d 6910 . . . . . . . . . 10 ((𝜑𝑥 ∈ (Base‘𝑃)) → (0g𝑅) = (0g‘(Scalar‘𝑃)))
164163adantr 480 . . . . . . . . 9 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ ({𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∖ (𝑥 supp (0g𝑅)))) → (0g𝑅) = (0g‘(Scalar‘𝑃)))
165162, 164eqtrd 2777 . . . . . . . 8 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ ({𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∖ (𝑥 supp (0g𝑅)))) → (𝑥𝑘) = (0g‘(Scalar‘𝑃)))
166165oveq1d 7446 . . . . . . 7 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ ({𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∖ (𝑥 supp (0g𝑅)))) → ((𝑥𝑘)( ·𝑠𝑃)(𝑦 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = 𝑘, (1r𝑅), (0g𝑅)))) = ((0g‘(Scalar‘𝑃))( ·𝑠𝑃)(𝑦 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = 𝑘, (1r𝑅), (0g𝑅)))))
167 eldifi 4131 . . . . . . . 8 (𝑘 ∈ ({𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∖ (𝑥 supp (0g𝑅))) → 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin})
16812ad2antrr 726 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → 𝑅 ∈ Ring)
1695, 6, 33, 34, 32, 75, 168, 79mplmon 22053 . . . . . . . . 9 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑦 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = 𝑘, (1r𝑅), (0g𝑅))) ∈ (Base‘𝑃))
170 eqid 2737 . . . . . . . . . 10 (0g‘(Scalar‘𝑃)) = (0g‘(Scalar‘𝑃))
1716, 146, 36, 170, 40lmod0vs 20893 . . . . . . . . 9 ((𝑃 ∈ LMod ∧ (𝑦 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = 𝑘, (1r𝑅), (0g𝑅))) ∈ (Base‘𝑃)) → ((0g‘(Scalar‘𝑃))( ·𝑠𝑃)(𝑦 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = 𝑘, (1r𝑅), (0g𝑅)))) = (0g𝑃))
17259, 169, 171syl2anc 584 . . . . . . . 8 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → ((0g‘(Scalar‘𝑃))( ·𝑠𝑃)(𝑦 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = 𝑘, (1r𝑅), (0g𝑅)))) = (0g𝑃))
173167, 172sylan2 593 . . . . . . 7 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ ({𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∖ (𝑥 supp (0g𝑅)))) → ((0g‘(Scalar‘𝑃))( ·𝑠𝑃)(𝑦 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = 𝑘, (1r𝑅), (0g𝑅)))) = (0g𝑃))
174166, 173eqtrd 2777 . . . . . 6 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ ({𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∖ (𝑥 supp (0g𝑅)))) → ((𝑥𝑘)( ·𝑠𝑃)(𝑦 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = 𝑘, (1r𝑅), (0g𝑅)))) = (0g𝑃))
175174, 47suppss2 8225 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝑃)) → ((𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ↦ ((𝑥𝑘)( ·𝑠𝑃)(𝑦 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = 𝑘, (1r𝑅), (0g𝑅))))) supp (0g𝑃)) ⊆ (𝑥 supp (0g𝑅)))
176 suppssfifsupp 9420 . . . . 5 ((((𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ↦ ((𝑥𝑘)( ·𝑠𝑃)(𝑦 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = 𝑘, (1r𝑅), (0g𝑅))))) ∈ V ∧ Fun (𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ↦ ((𝑥𝑘)( ·𝑠𝑃)(𝑦 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = 𝑘, (1r𝑅), (0g𝑅))))) ∧ (0g𝑃) ∈ V) ∧ ((𝑥 supp (0g𝑅)) ∈ Fin ∧ ((𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ↦ ((𝑥𝑘)( ·𝑠𝑃)(𝑦 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = 𝑘, (1r𝑅), (0g𝑅))))) supp (0g𝑃)) ⊆ (𝑥 supp (0g𝑅)))) → (𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ↦ ((𝑥𝑘)( ·𝑠𝑃)(𝑦 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = 𝑘, (1r𝑅), (0g𝑅))))) finSupp (0g𝑃))
177155, 159, 175, 176syl12anc 837 . . . 4 ((𝜑𝑥 ∈ (Base‘𝑃)) → (𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ↦ ((𝑥𝑘)( ·𝑠𝑃)(𝑦 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = 𝑘, (1r𝑅), (0g𝑅))))) finSupp (0g𝑃))
17840, 44, 47, 57, 150, 177gsumsubgcl 19938 . . 3 ((𝜑𝑥 ∈ (Base‘𝑃)) → (𝑃 Σg (𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ↦ ((𝑥𝑘)( ·𝑠𝑃)(𝑦 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = 𝑘, (1r𝑅), (0g𝑅)))))) ∈ (𝐴‘ran 𝑉))
17939, 178eqeltrd 2841 . 2 ((𝜑𝑥 ∈ (Base‘𝑃)) → 𝑥 ∈ (𝐴‘ran 𝑉))
18031, 179eqelssd 4005 1 (𝜑 → (𝐴‘ran 𝑉) = (Base‘𝑃))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  {crab 3436  Vcvv 3480  cdif 3948  wss 3951  ifcif 4525   class class class wbr 5143  cmpt 5225  ccnv 5684  ran crn 5686  cima 5688  Fun wfun 6555   Fn wfn 6556  wf 6557  cfv 6561  (class class class)co 7431   supp csupp 8185  m cmap 8866  Fincfn 8985   finSupp cfsupp 9401  0cc0 11155  cn 12266  0cn0 12526  Basecbs 17247  .rcmulr 17298  Scalarcsca 17300   ·𝑠 cvsca 17301  0gc0g 17484   Σg cgsu 17485  SubMndcsubmnd 18795  .gcmg 19085  SubGrpcsubg 19138  CMndccmn 19798  Abelcabl 19799  mulGrpcmgp 20137  1rcur 20178  Ringcrg 20230  CRingccrg 20231  SubRingcsubrg 20569  LModclmod 20858  LSubSpclss 20929  AssAlgcasa 21870  AlgSpancasp 21871   mPwSer cmps 21924   mVar cmvr 21925   mPoly cmpl 21926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-ofr 7698  df-om 7888  df-1st 8014  df-2nd 8015  df-supp 8186  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-map 8868  df-pm 8869  df-ixp 8938  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fsupp 9402  df-sup 9482  df-oi 9550  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-uz 12879  df-fz 13548  df-fzo 13695  df-seq 14043  df-hash 14370  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-mulr 17311  df-sca 17313  df-vsca 17314  df-ip 17315  df-tset 17316  df-ple 17317  df-ds 17319  df-hom 17321  df-cco 17322  df-0g 17486  df-gsum 17487  df-prds 17492  df-pws 17494  df-mre 17629  df-mrc 17630  df-acs 17632  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-mhm 18796  df-submnd 18797  df-grp 18954  df-minusg 18955  df-sbg 18956  df-mulg 19086  df-subg 19141  df-ghm 19231  df-cntz 19335  df-cmn 19800  df-abl 19801  df-mgp 20138  df-rng 20150  df-ur 20179  df-srg 20184  df-ring 20232  df-cring 20233  df-subrng 20546  df-subrg 20570  df-lmod 20860  df-lss 20930  df-assa 21873  df-asp 21874  df-psr 21929  df-mvr 21930  df-mpl 21931
This theorem is referenced by:  mplind  22094  evlseu  22107
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