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Theorem mplbas2 20226
 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 20169 . . . 4 (𝜑𝑆 ∈ AssAlg)
5 mplbas2.p . . . . . 6 𝑃 = (𝐼 mPoly 𝑅)
6 eqid 2821 . . . . . 6 (Base‘𝑃) = (Base‘𝑃)
7 eqid 2821 . . . . . 6 (Base‘𝑆) = (Base‘𝑆)
85, 1, 6, 7mplbasss 20187 . . . . 5 (Base‘𝑃) ⊆ (Base‘𝑆)
98a1i 11 . . . 4 (𝜑 → (Base‘𝑃) ⊆ (Base‘𝑆))
10 mplbas2.v . . . . . . . 8 𝑉 = (𝐼 mVar 𝑅)
11 crngring 19286 . . . . . . . . 9 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
123, 11syl 17 . . . . . . . 8 (𝜑𝑅 ∈ Ring)
131, 10, 7, 2, 12mvrf 20179 . . . . . . 7 (𝜑𝑉:𝐼⟶(Base‘𝑆))
1413ffnd 6488 . . . . . 6 (𝜑𝑉 Fn 𝐼)
152adantr 484 . . . . . . . 8 ((𝜑𝑥𝐼) → 𝐼𝑊)
1612adantr 484 . . . . . . . 8 ((𝜑𝑥𝐼) → 𝑅 ∈ Ring)
17 simpr 488 . . . . . . . 8 ((𝜑𝑥𝐼) → 𝑥𝐼)
185, 10, 6, 15, 16, 17mvrcl 20204 . . . . . . 7 ((𝜑𝑥𝐼) → (𝑉𝑥) ∈ (Base‘𝑃))
1918ralrimiva 3170 . . . . . 6 (𝜑 → ∀𝑥𝐼 (𝑉𝑥) ∈ (Base‘𝑃))
20 ffnfv 6855 . . . . . 6 (𝑉:𝐼⟶(Base‘𝑃) ↔ (𝑉 Fn 𝐼 ∧ ∀𝑥𝐼 (𝑉𝑥) ∈ (Base‘𝑃)))
2114, 19, 20sylanbrc 586 . . . . 5 (𝜑𝑉:𝐼⟶(Base‘𝑃))
2221frnd 6494 . . . 4 (𝜑 → ran 𝑉 ⊆ (Base‘𝑃))
23 mplbas2.a . . . . 5 𝐴 = (AlgSpan‘𝑆)
2423, 7aspss 20081 . . . 4 ((𝑆 ∈ AssAlg ∧ (Base‘𝑃) ⊆ (Base‘𝑆) ∧ ran 𝑉 ⊆ (Base‘𝑃)) → (𝐴‘ran 𝑉) ⊆ (𝐴‘(Base‘𝑃)))
254, 9, 22, 24syl3anc 1368 . . 3 (𝜑 → (𝐴‘ran 𝑉) ⊆ (𝐴‘(Base‘𝑃)))
261, 5, 6, 2, 12mplsubrg 20195 . . . 4 (𝜑 → (Base‘𝑃) ∈ (SubRing‘𝑆))
271, 5, 6, 2, 12mpllss 20193 . . . 4 (𝜑 → (Base‘𝑃) ∈ (LSubSp‘𝑆))
28 eqid 2821 . . . . 5 (LSubSp‘𝑆) = (LSubSp‘𝑆)
2923, 7, 28aspid 20079 . . . 4 ((𝑆 ∈ AssAlg ∧ (Base‘𝑃) ∈ (SubRing‘𝑆) ∧ (Base‘𝑃) ∈ (LSubSp‘𝑆)) → (𝐴‘(Base‘𝑃)) = (Base‘𝑃))
304, 26, 27, 29syl3anc 1368 . . 3 (𝜑 → (𝐴‘(Base‘𝑃)) = (Base‘𝑃))
3125, 30sseqtrd 3983 . 2 (𝜑 → (𝐴‘ran 𝑉) ⊆ (Base‘𝑃))
32 eqid 2821 . . . 4 {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}
33 eqid 2821 . . . 4 (0g𝑅) = (0g𝑅)
34 eqid 2821 . . . 4 (1r𝑅) = (1r𝑅)
352adantr 484 . . . 4 ((𝜑𝑥 ∈ (Base‘𝑃)) → 𝐼𝑊)
36 eqid 2821 . . . 4 ( ·𝑠𝑃) = ( ·𝑠𝑃)
3712adantr 484 . . . 4 ((𝜑𝑥 ∈ (Base‘𝑃)) → 𝑅 ∈ Ring)
38 simpr 488 . . . 4 ((𝜑𝑥 ∈ (Base‘𝑃)) → 𝑥 ∈ (Base‘𝑃))
395, 32, 33, 34, 35, 6, 36, 37, 38mplcoe1 20221 . . 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 2821 . . . 4 (0g𝑃) = (0g𝑃)
415mplring 20207 . . . . . . 7 ((𝐼𝑊𝑅 ∈ Ring) → 𝑃 ∈ Ring)
422, 12, 41syl2anc 587 . . . . . 6 (𝜑𝑃 ∈ Ring)
43 ringabl 19308 . . . . . 6 (𝑃 ∈ Ring → 𝑃 ∈ Abel)
4442, 43syl 17 . . . . 5 (𝜑𝑃 ∈ Abel)
4544adantr 484 . . . 4 ((𝜑𝑥 ∈ (Base‘𝑃)) → 𝑃 ∈ Abel)
46 ovex 7163 . . . . . 6 (ℕ0m 𝐼) ∈ V
4746rabex 5208 . . . . 5 {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∈ V
4847a1i 11 . . . 4 ((𝜑𝑥 ∈ (Base‘𝑃)) → {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∈ V)
4913frnd 6494 . . . . . . . 8 (𝜑 → ran 𝑉 ⊆ (Base‘𝑆))
5023, 7aspsubrg 20080 . . . . . . . 8 ((𝑆 ∈ AssAlg ∧ ran 𝑉 ⊆ (Base‘𝑆)) → (𝐴‘ran 𝑉) ∈ (SubRing‘𝑆))
514, 49, 50syl2anc 587 . . . . . . 7 (𝜑 → (𝐴‘ran 𝑉) ∈ (SubRing‘𝑆))
525, 1, 6mplval2 20186 . . . . . . . . 9 𝑃 = (𝑆s (Base‘𝑃))
5352subsubrg 19536 . . . . . . . 8 ((Base‘𝑃) ∈ (SubRing‘𝑆) → ((𝐴‘ran 𝑉) ∈ (SubRing‘𝑃) ↔ ((𝐴‘ran 𝑉) ∈ (SubRing‘𝑆) ∧ (𝐴‘ran 𝑉) ⊆ (Base‘𝑃))))
5426, 53syl 17 . . . . . . 7 (𝜑 → ((𝐴‘ran 𝑉) ∈ (SubRing‘𝑃) ↔ ((𝐴‘ran 𝑉) ∈ (SubRing‘𝑆) ∧ (𝐴‘ran 𝑉) ⊆ (Base‘𝑃))))
5551, 31, 54mpbir2and 712 . . . . . 6 (𝜑 → (𝐴‘ran 𝑉) ∈ (SubRing‘𝑃))
56 subrgsubg 19516 . . . . . 6 ((𝐴‘ran 𝑉) ∈ (SubRing‘𝑃) → (𝐴‘ran 𝑉) ∈ (SubGrp‘𝑃))
5755, 56syl 17 . . . . 5 (𝜑 → (𝐴‘ran 𝑉) ∈ (SubGrp‘𝑃))
5857adantr 484 . . . 4 ((𝜑𝑥 ∈ (Base‘𝑃)) → (𝐴‘ran 𝑉) ∈ (SubGrp‘𝑃))
595mpllmod 20206 . . . . . . . 8 ((𝐼𝑊𝑅 ∈ Ring) → 𝑃 ∈ LMod)
602, 12, 59syl2anc 587 . . . . . . 7 (𝜑𝑃 ∈ LMod)
6160ad2antrr 725 . . . . . 6 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → 𝑃 ∈ LMod)
6223, 7, 28asplss 20078 . . . . . . . . 9 ((𝑆 ∈ AssAlg ∧ ran 𝑉 ⊆ (Base‘𝑆)) → (𝐴‘ran 𝑉) ∈ (LSubSp‘𝑆))
634, 49, 62syl2anc 587 . . . . . . . 8 (𝜑 → (𝐴‘ran 𝑉) ∈ (LSubSp‘𝑆))
641, 2, 12psrlmod 20156 . . . . . . . . 9 (𝜑𝑆 ∈ LMod)
65 eqid 2821 . . . . . . . . . 10 (LSubSp‘𝑃) = (LSubSp‘𝑃)
6652, 28, 65lsslss 19708 . . . . . . . . 9 ((𝑆 ∈ LMod ∧ (Base‘𝑃) ∈ (LSubSp‘𝑆)) → ((𝐴‘ran 𝑉) ∈ (LSubSp‘𝑃) ↔ ((𝐴‘ran 𝑉) ∈ (LSubSp‘𝑆) ∧ (𝐴‘ran 𝑉) ⊆ (Base‘𝑃))))
6764, 27, 66syl2anc 587 . . . . . . . 8 (𝜑 → ((𝐴‘ran 𝑉) ∈ (LSubSp‘𝑃) ↔ ((𝐴‘ran 𝑉) ∈ (LSubSp‘𝑆) ∧ (𝐴‘ran 𝑉) ⊆ (Base‘𝑃))))
6863, 31, 67mpbir2and 712 . . . . . . 7 (𝜑 → (𝐴‘ran 𝑉) ∈ (LSubSp‘𝑃))
6968ad2antrr 725 . . . . . 6 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝐴‘ran 𝑉) ∈ (LSubSp‘𝑃))
70 eqid 2821 . . . . . . . . 9 (Base‘𝑅) = (Base‘𝑅)
715, 70, 6, 32, 38mplelf 20188 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝑃)) → 𝑥:{𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
7271ffvelrnda 6824 . . . . . . 7 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑥𝑘) ∈ (Base‘𝑅))
735, 35, 37mplsca 20200 . . . . . . . . 9 ((𝜑𝑥 ∈ (Base‘𝑃)) → 𝑅 = (Scalar‘𝑃))
7473adantr 484 . . . . . . . 8 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → 𝑅 = (Scalar‘𝑃))
7574fveq2d 6647 . . . . . . 7 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → (Base‘𝑅) = (Base‘(Scalar‘𝑃)))
7672, 75eleqtrd 2914 . . . . . 6 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑥𝑘) ∈ (Base‘(Scalar‘𝑃)))
772ad2antrr 725 . . . . . . . 8 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → 𝐼𝑊)
78 eqid 2821 . . . . . . . 8 (mulGrp‘𝑃) = (mulGrp‘𝑃)
79 eqid 2821 . . . . . . . 8 (.g‘(mulGrp‘𝑃)) = (.g‘(mulGrp‘𝑃))
803ad2antrr 725 . . . . . . . 8 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → 𝑅 ∈ CRing)
81 simpr 488 . . . . . . . 8 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin})
825, 32, 33, 34, 77, 78, 79, 10, 80, 81mplcoe2 20225 . . . . . . 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)))))
83 eqid 2821 . . . . . . . . 9 (1r𝑃) = (1r𝑃)
8478, 83ringidval 19231 . . . . . . . 8 (1r𝑃) = (0g‘(mulGrp‘𝑃))
855mplcrng 20209 . . . . . . . . . . 11 ((𝐼𝑊𝑅 ∈ CRing) → 𝑃 ∈ CRing)
862, 3, 85syl2anc 587 . . . . . . . . . 10 (𝜑𝑃 ∈ CRing)
8778crngmgp 19283 . . . . . . . . . 10 (𝑃 ∈ CRing → (mulGrp‘𝑃) ∈ CMnd)
8886, 87syl 17 . . . . . . . . 9 (𝜑 → (mulGrp‘𝑃) ∈ CMnd)
8988ad2antrr 725 . . . . . . . 8 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → (mulGrp‘𝑃) ∈ CMnd)
9055ad2antrr 725 . . . . . . . . 9 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝐴‘ran 𝑉) ∈ (SubRing‘𝑃))
9178subrgsubm 19523 . . . . . . . . 9 ((𝐴‘ran 𝑉) ∈ (SubRing‘𝑃) → (𝐴‘ran 𝑉) ∈ (SubMnd‘(mulGrp‘𝑃)))
9290, 91syl 17 . . . . . . . 8 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝐴‘ran 𝑉) ∈ (SubMnd‘(mulGrp‘𝑃)))
93 simplll 774 . . . . . . . . . 10 ((((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧𝐼) → 𝜑)
9432psrbag 20119 . . . . . . . . . . . . . 14 (𝐼𝑊 → (𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ↔ (𝑘:𝐼⟶ℕ0 ∧ (𝑘 “ ℕ) ∈ Fin)))
9535, 94syl 17 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (Base‘𝑃)) → (𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ↔ (𝑘:𝐼⟶ℕ0 ∧ (𝑘 “ ℕ) ∈ Fin)))
9695biimpa 480 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑘:𝐼⟶ℕ0 ∧ (𝑘 “ ℕ) ∈ Fin))
9796simpld 498 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → 𝑘:𝐼⟶ℕ0)
9897ffvelrnda 6824 . . . . . . . . . 10 ((((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧𝐼) → (𝑘𝑧) ∈ ℕ0)
9923, 7aspssid 20082 . . . . . . . . . . . . 13 ((𝑆 ∈ AssAlg ∧ ran 𝑉 ⊆ (Base‘𝑆)) → ran 𝑉 ⊆ (𝐴‘ran 𝑉))
1004, 49, 99syl2anc 587 . . . . . . . . . . . 12 (𝜑 → ran 𝑉 ⊆ (𝐴‘ran 𝑉))
101100ad3antrrr 729 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧𝐼) → ran 𝑉 ⊆ (𝐴‘ran 𝑉))
10214ad2antrr 725 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → 𝑉 Fn 𝐼)
103 fnfvelrn 6821 . . . . . . . . . . . 12 ((𝑉 Fn 𝐼𝑧𝐼) → (𝑉𝑧) ∈ ran 𝑉)
104102, 103sylan 583 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧𝐼) → (𝑉𝑧) ∈ ran 𝑉)
105101, 104sseldd 3944 . . . . . . . . . 10 ((((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧𝐼) → (𝑉𝑧) ∈ (𝐴‘ran 𝑉))
10678, 6mgpbas 19223 . . . . . . . . . . 11 (Base‘𝑃) = (Base‘(mulGrp‘𝑃))
107 eqid 2821 . . . . . . . . . . . 12 (.r𝑃) = (.r𝑃)
10878, 107mgpplusg 19221 . . . . . . . . . . 11 (.r𝑃) = (+g‘(mulGrp‘𝑃))
109107subrgmcl 19522 . . . . . . . . . . . 12 (((𝐴‘ran 𝑉) ∈ (SubRing‘𝑃) ∧ 𝑢 ∈ (𝐴‘ran 𝑉) ∧ 𝑣 ∈ (𝐴‘ran 𝑉)) → (𝑢(.r𝑃)𝑣) ∈ (𝐴‘ran 𝑉))
11055, 109syl3an1 1160 . . . . . . . . . . 11 ((𝜑𝑢 ∈ (𝐴‘ran 𝑉) ∧ 𝑣 ∈ (𝐴‘ran 𝑉)) → (𝑢(.r𝑃)𝑣) ∈ (𝐴‘ran 𝑉))
11183subrg1cl 19518 . . . . . . . . . . . 12 ((𝐴‘ran 𝑉) ∈ (SubRing‘𝑃) → (1r𝑃) ∈ (𝐴‘ran 𝑉))
11255, 111syl 17 . . . . . . . . . . 11 (𝜑 → (1r𝑃) ∈ (𝐴‘ran 𝑉))
113106, 79, 108, 88, 31, 110, 84, 112mulgnn0subcl 18219 . . . . . . . . . 10 ((𝜑 ∧ (𝑘𝑧) ∈ ℕ0 ∧ (𝑉𝑧) ∈ (𝐴‘ran 𝑉)) → ((𝑘𝑧)(.g‘(mulGrp‘𝑃))(𝑉𝑧)) ∈ (𝐴‘ran 𝑉))
11493, 98, 105, 113syl3anc 1368 . . . . . . . . 9 ((((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧𝐼) → ((𝑘𝑧)(.g‘(mulGrp‘𝑃))(𝑉𝑧)) ∈ (𝐴‘ran 𝑉))
115114fmpttd 6852 . . . . . . . 8 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑧𝐼 ↦ ((𝑘𝑧)(.g‘(mulGrp‘𝑃))(𝑉𝑧))):𝐼⟶(𝐴‘ran 𝑉))
1162mptexd 6960 . . . . . . . . . 10 (𝜑 → (𝑧𝐼 ↦ ((𝑘𝑧)(.g‘(mulGrp‘𝑃))(𝑉𝑧))) ∈ V)
117116ad2antrr 725 . . . . . . . . 9 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑧𝐼 ↦ ((𝑘𝑧)(.g‘(mulGrp‘𝑃))(𝑉𝑧))) ∈ V)
118 funmpt 6366 . . . . . . . . . 10 Fun (𝑧𝐼 ↦ ((𝑘𝑧)(.g‘(mulGrp‘𝑃))(𝑉𝑧)))
119118a1i 11 . . . . . . . . 9 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → Fun (𝑧𝐼 ↦ ((𝑘𝑧)(.g‘(mulGrp‘𝑃))(𝑉𝑧))))
120 fvexd 6658 . . . . . . . . 9 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → (1r𝑃) ∈ V)
12196simprd 499 . . . . . . . . 9 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑘 “ ℕ) ∈ Fin)
122 elrabi 3652 . . . . . . . . . . . . 13 (𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} → 𝑘 ∈ (ℕ0m 𝐼))
123 elmapi 8403 . . . . . . . . . . . . . . 15 (𝑘 ∈ (ℕ0m 𝐼) → 𝑘:𝐼⟶ℕ0)
124123adantl 485 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ (ℕ0m 𝐼)) → 𝑘:𝐼⟶ℕ0)
1252ad2antrr 725 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ (ℕ0m 𝐼)) → 𝐼𝑊)
126 frnnn0supp 11931 . . . . . . . . . . . . . . . 16 ((𝐼𝑊𝑘:𝐼⟶ℕ0) → (𝑘 supp 0) = (𝑘 “ ℕ))
127125, 124, 126syl2anc 587 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ (ℕ0m 𝐼)) → (𝑘 supp 0) = (𝑘 “ ℕ))
128 eqimss 3999 . . . . . . . . . . . . . . 15 ((𝑘 supp 0) = (𝑘 “ ℕ) → (𝑘 supp 0) ⊆ (𝑘 “ ℕ))
129127, 128syl 17 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ (ℕ0m 𝐼)) → (𝑘 supp 0) ⊆ (𝑘 “ ℕ))
130 c0ex 10612 . . . . . . . . . . . . . . 15 0 ∈ V
131130a1i 11 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ (ℕ0m 𝐼)) → 0 ∈ V)
132124, 129, 125, 131suppssr 7836 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ (ℕ0m 𝐼)) ∧ 𝑧 ∈ (𝐼 ∖ (𝑘 “ ℕ))) → (𝑘𝑧) = 0)
133122, 132sylanl2 680 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧 ∈ (𝐼 ∖ (𝑘 “ ℕ))) → (𝑘𝑧) = 0)
134133oveq1d 7145 . . . . . . . . . . 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)))
1352ad3antrrr 729 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧 ∈ (𝐼 ∖ (𝑘 “ ℕ))) → 𝐼𝑊)
13612ad3antrrr 729 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧 ∈ (𝐼 ∖ (𝑘 “ ℕ))) → 𝑅 ∈ Ring)
137 eldifi 4079 . . . . . . . . . . . . . 14 (𝑧 ∈ (𝐼 ∖ (𝑘 “ ℕ)) → 𝑧𝐼)
138137adantl 485 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧 ∈ (𝐼 ∖ (𝑘 “ ℕ))) → 𝑧𝐼)
1395, 10, 6, 135, 136, 138mvrcl 20204 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧 ∈ (𝐼 ∖ (𝑘 “ ℕ))) → (𝑉𝑧) ∈ (Base‘𝑃))
140106, 84, 79mulg0 18209 . . . . . . . . . . . 12 ((𝑉𝑧) ∈ (Base‘𝑃) → (0(.g‘(mulGrp‘𝑃))(𝑉𝑧)) = (1r𝑃))
141139, 140syl 17 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧 ∈ (𝐼 ∖ (𝑘 “ ℕ))) → (0(.g‘(mulGrp‘𝑃))(𝑉𝑧)) = (1r𝑃))
142134, 141eqtrd 2856 . . . . . . . . . 10 ((((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧 ∈ (𝐼 ∖ (𝑘 “ ℕ))) → ((𝑘𝑧)(.g‘(mulGrp‘𝑃))(𝑉𝑧)) = (1r𝑃))
143142, 77suppss2 7839 . . . . . . . . 9 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → ((𝑧𝐼 ↦ ((𝑘𝑧)(.g‘(mulGrp‘𝑃))(𝑉𝑧))) supp (1r𝑃)) ⊆ (𝑘 “ ℕ))
144 suppssfifsupp 8824 . . . . . . . . 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))
145117, 119, 120, 121, 143, 144syl32anc 1375 . . . . . . . 8 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑧𝐼 ↦ ((𝑘𝑧)(.g‘(mulGrp‘𝑃))(𝑉𝑧))) finSupp (1r𝑃))
14684, 89, 77, 92, 115, 145gsumsubmcl 19017 . . . . . . 7 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → ((mulGrp‘𝑃) Σg (𝑧𝐼 ↦ ((𝑘𝑧)(.g‘(mulGrp‘𝑃))(𝑉𝑧)))) ∈ (𝐴‘ran 𝑉))
14782, 146eqeltrd 2912 . . . . . 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))
148 eqid 2821 . . . . . . 7 (Scalar‘𝑃) = (Scalar‘𝑃)
149 eqid 2821 . . . . . . 7 (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
150148, 36, 149, 65lssvscl 19702 . . . . . 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))
15161, 69, 76, 147, 150syl22anc 837 . . . . 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))
152151fmpttd 6852 . . . 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))
15346mptrabex 6961 . . . . . . 7 (𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ↦ ((𝑥𝑘)( ·𝑠𝑃)(𝑦 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = 𝑘, (1r𝑅), (0g𝑅))))) ∈ V
154 funmpt 6366 . . . . . . 7 Fun (𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ↦ ((𝑥𝑘)( ·𝑠𝑃)(𝑦 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = 𝑘, (1r𝑅), (0g𝑅)))))
155 fvex 6656 . . . . . . 7 (0g𝑃) ∈ V
156153, 154, 1553pm3.2i 1336 . . . . . 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)
157156a1i 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))
1585, 1, 7, 33, 6mplelbas 20185 . . . . . . . 8 (𝑥 ∈ (Base‘𝑃) ↔ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 finSupp (0g𝑅)))
159158simprbi 500 . . . . . . 7 (𝑥 ∈ (Base‘𝑃) → 𝑥 finSupp (0g𝑅))
160159adantl 485 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝑃)) → 𝑥 finSupp (0g𝑅))
161160fsuppimpd 8816 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝑃)) → (𝑥 supp (0g𝑅)) ∈ Fin)
162 ssidd 3966 . . . . . . . . . 10 ((𝜑𝑥 ∈ (Base‘𝑃)) → (𝑥 supp (0g𝑅)) ⊆ (𝑥 supp (0g𝑅)))
163 fvexd 6658 . . . . . . . . . 10 ((𝜑𝑥 ∈ (Base‘𝑃)) → (0g𝑅) ∈ V)
16471, 162, 48, 163suppssr 7836 . . . . . . . . 9 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ ({𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∖ (𝑥 supp (0g𝑅)))) → (𝑥𝑘) = (0g𝑅))
16573fveq2d 6647 . . . . . . . . . 10 ((𝜑𝑥 ∈ (Base‘𝑃)) → (0g𝑅) = (0g‘(Scalar‘𝑃)))
166165adantr 484 . . . . . . . . 9 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ ({𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∖ (𝑥 supp (0g𝑅)))) → (0g𝑅) = (0g‘(Scalar‘𝑃)))
167164, 166eqtrd 2856 . . . . . . . 8 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ ({𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∖ (𝑥 supp (0g𝑅)))) → (𝑥𝑘) = (0g‘(Scalar‘𝑃)))
168167oveq1d 7145 . . . . . . 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)))))
169 eldifi 4079 . . . . . . . 8 (𝑘 ∈ ({𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∖ (𝑥 supp (0g𝑅))) → 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin})
17012ad2antrr 725 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → 𝑅 ∈ Ring)
1715, 6, 33, 34, 32, 77, 170, 81mplmon 20219 . . . . . . . . 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))
172 eqid 2821 . . . . . . . . . 10 (0g‘(Scalar‘𝑃)) = (0g‘(Scalar‘𝑃))
1736, 148, 36, 172, 40lmod0vs 19642 . . . . . . . . 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))
17461, 171, 173syl2anc 587 . . . . . . . 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))
175169, 174sylan2 595 . . . . . . 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))
176168, 175eqtrd 2856 . . . . . 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))
177176, 48suppss2 7839 . . . . 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)))
178 suppssfifsupp 8824 . . . . 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))
179157, 161, 177, 178syl12anc 835 . . . 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))
18040, 45, 48, 58, 152, 179gsumsubgcl 19018 . . 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))
18139, 180eqeltrd 2912 . 2 ((𝜑𝑥 ∈ (Base‘𝑃)) → 𝑥 ∈ (𝐴‘ran 𝑉))
18231, 181eqelssd 3964 1 (𝜑 → (𝐴‘ran 𝑉) = (Base‘𝑃))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2115  ∀wral 3126  {crab 3130  Vcvv 3471   ∖ cdif 3907   ⊆ wss 3910  ifcif 4440   class class class wbr 5039   ↦ cmpt 5119  ◡ccnv 5527  ran crn 5529   “ cima 5531  Fun wfun 6322   Fn wfn 6323  ⟶wf 6324  ‘cfv 6328  (class class class)co 7130   supp csupp 7805   ↑m cmap 8381  Fincfn 8484   finSupp cfsupp 8809  0cc0 10514  ℕcn 11615  ℕ0cn0 11875  Basecbs 16461  .rcmulr 16544  Scalarcsca 16546   ·𝑠 cvsca 16547  0gc0g 16691   Σg cgsu 16692  SubMndcsubmnd 17933  .gcmg 18202  SubGrpcsubg 18251  CMndccmn 18884  Abelcabl 18885  mulGrpcmgp 19217  1rcur 19229  Ringcrg 19275  CRingccrg 19276  SubRingcsubrg 19506  LModclmod 19609  LSubSpclss 19678  AssAlgcasa 20057  AlgSpancasp 20058   mPwSer cmps 20106   mVar cmvr 20107   mPoly cmpl 20108 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-rep 5163  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436  ax-cnex 10570  ax-resscn 10571  ax-1cn 10572  ax-icn 10573  ax-addcl 10574  ax-addrcl 10575  ax-mulcl 10576  ax-mulrcl 10577  ax-mulcom 10578  ax-addass 10579  ax-mulass 10580  ax-distr 10581  ax-i2m1 10582  ax-1ne0 10583  ax-1rid 10584  ax-rnegex 10585  ax-rrecex 10586  ax-cnre 10587  ax-pre-lttri 10588  ax-pre-lttrn 10589  ax-pre-ltadd 10590  ax-pre-mulgt0 10591 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-nel 3112  df-ral 3131  df-rex 3132  df-reu 3133  df-rmo 3134  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-tp 4545  df-op 4547  df-uni 4812  df-int 4850  df-iun 4894  df-iin 4895  df-br 5040  df-opab 5102  df-mpt 5120  df-tr 5146  df-id 5433  df-eprel 5438  df-po 5447  df-so 5448  df-fr 5487  df-se 5488  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6121  df-ord 6167  df-on 6168  df-lim 6169  df-suc 6170  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-isom 6337  df-riota 7088  df-ov 7133  df-oprab 7134  df-mpo 7135  df-of 7384  df-ofr 7385  df-om 7556  df-1st 7664  df-2nd 7665  df-supp 7806  df-wrecs 7922  df-recs 7983  df-rdg 8021  df-1o 8077  df-2o 8078  df-oadd 8081  df-er 8264  df-map 8383  df-pm 8384  df-ixp 8437  df-en 8485  df-dom 8486  df-sdom 8487  df-fin 8488  df-fsupp 8810  df-oi 8950  df-card 9344  df-pnf 10654  df-mnf 10655  df-xr 10656  df-ltxr 10657  df-le 10658  df-sub 10849  df-neg 10850  df-nn 11616  df-2 11678  df-3 11679  df-4 11680  df-5 11681  df-6 11682  df-7 11683  df-8 11684  df-9 11685  df-n0 11876  df-z 11960  df-uz 12222  df-fz 12876  df-fzo 13017  df-seq 13353  df-hash 13675  df-struct 16463  df-ndx 16464  df-slot 16465  df-base 16467  df-sets 16468  df-ress 16469  df-plusg 16556  df-mulr 16557  df-sca 16559  df-vsca 16560  df-tset 16562  df-0g 16693  df-gsum 16694  df-mre 16835  df-mrc 16836  df-acs 16838  df-mgm 17830  df-sgrp 17879  df-mnd 17890  df-mhm 17934  df-submnd 17935  df-grp 18084  df-minusg 18085  df-sbg 18086  df-mulg 18203  df-subg 18254  df-ghm 18334  df-cntz 18425  df-cmn 18886  df-abl 18887  df-mgp 19218  df-ur 19230  df-srg 19234  df-ring 19277  df-cring 19278  df-subrg 19508  df-lmod 19611  df-lss 19679  df-assa 20060  df-asp 20061  df-psr 20111  df-mvr 20112  df-mpl 20113 This theorem is referenced by:  mplind  20257  evlseu  20271
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