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Theorem mplbas2 21907
Description: An alternative expression for the set of polynomials, as the smallest subalgebra of the set of power series that contains all the variable generators. (Contributed by Mario Carneiro, 10-Jan-2015.)
Hypotheses
Ref Expression
mplbas2.p 𝑃 = (𝐼 mPoly 𝑅)
mplbas2.s 𝑆 = (𝐼 mPwSer 𝑅)
mplbas2.v 𝑉 = (𝐼 mVar 𝑅)
mplbas2.a 𝐴 = (AlgSpan‘𝑆)
mplbas2.i (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 21844 . . . 4 (𝜑𝑆 ∈ AssAlg)
5 mplbas2.p . . . . . 6 𝑃 = (𝐼 mPoly 𝑅)
6 eqid 2724 . . . . . 6 (Base‘𝑃) = (Base‘𝑃)
7 eqid 2724 . . . . . 6 (Base‘𝑆) = (Base‘𝑆)
85, 1, 6, 7mplbasss 21866 . . . . 5 (Base‘𝑃) ⊆ (Base‘𝑆)
98a1i 11 . . . 4 (𝜑 → (Base‘𝑃) ⊆ (Base‘𝑆))
10 mplbas2.v . . . . . . . 8 𝑉 = (𝐼 mVar 𝑅)
11 crngring 20140 . . . . . . . . 9 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
123, 11syl 17 . . . . . . . 8 (𝜑𝑅 ∈ Ring)
131, 10, 7, 2, 12mvrf 21854 . . . . . . 7 (𝜑𝑉:𝐼⟶(Base‘𝑆))
1413ffnd 6708 . . . . . 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 21861 . . . . . . 7 ((𝜑𝑥𝐼) → (𝑉𝑥) ∈ (Base‘𝑃))
1918ralrimiva 3138 . . . . . 6 (𝜑 → ∀𝑥𝐼 (𝑉𝑥) ∈ (Base‘𝑃))
20 ffnfv 7110 . . . . . 6 (𝑉:𝐼⟶(Base‘𝑃) ↔ (𝑉 Fn 𝐼 ∧ ∀𝑥𝐼 (𝑉𝑥) ∈ (Base‘𝑃)))
2114, 19, 20sylanbrc 582 . . . . 5 (𝜑𝑉:𝐼⟶(Base‘𝑃))
2221frnd 6715 . . . 4 (𝜑 → ran 𝑉 ⊆ (Base‘𝑃))
23 mplbas2.a . . . . 5 𝐴 = (AlgSpan‘𝑆)
2423, 7aspss 21739 . . . 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 21874 . . . 4 (𝜑 → (Base‘𝑃) ∈ (SubRing‘𝑆))
271, 5, 6, 2, 12mpllss 21872 . . . 4 (𝜑 → (Base‘𝑃) ∈ (LSubSp‘𝑆))
28 eqid 2724 . . . . 5 (LSubSp‘𝑆) = (LSubSp‘𝑆)
2923, 7, 28aspid 21737 . . . 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 4014 . 2 (𝜑 → (𝐴‘ran 𝑉) ⊆ (Base‘𝑃))
32 eqid 2724 . . . 4 {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}
33 eqid 2724 . . . 4 (0g𝑅) = (0g𝑅)
34 eqid 2724 . . . 4 (1r𝑅) = (1r𝑅)
352adantr 480 . . . 4 ((𝜑𝑥 ∈ (Base‘𝑃)) → 𝐼𝑊)
36 eqid 2724 . . . 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 21902 . . 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 2724 . . . 4 (0g𝑃) = (0g𝑃)
415mplring 21888 . . . . . . 7 ((𝐼𝑊𝑅 ∈ Ring) → 𝑃 ∈ Ring)
422, 12, 41syl2anc 583 . . . . . 6 (𝜑𝑃 ∈ Ring)
43 ringabl 20170 . . . . . 6 (𝑃 ∈ Ring → 𝑃 ∈ Abel)
4442, 43syl 17 . . . . 5 (𝜑𝑃 ∈ Abel)
4544adantr 480 . . . 4 ((𝜑𝑥 ∈ (Base‘𝑃)) → 𝑃 ∈ Abel)
46 ovex 7434 . . . . . 6 (ℕ0m 𝐼) ∈ V
4746rabex 5322 . . . . 5 {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∈ V
4847a1i 11 . . . 4 ((𝜑𝑥 ∈ (Base‘𝑃)) → {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∈ V)
4913frnd 6715 . . . . . . . 8 (𝜑 → ran 𝑉 ⊆ (Base‘𝑆))
5023, 7aspsubrg 21738 . . . . . . . 8 ((𝑆 ∈ AssAlg ∧ ran 𝑉 ⊆ (Base‘𝑆)) → (𝐴‘ran 𝑉) ∈ (SubRing‘𝑆))
514, 49, 50syl2anc 583 . . . . . . 7 (𝜑 → (𝐴‘ran 𝑉) ∈ (SubRing‘𝑆))
525, 1, 6mplval2 21865 . . . . . . . . 9 𝑃 = (𝑆s (Base‘𝑃))
5352subsubrg 20490 . . . . . . . 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 710 . . . . . 6 (𝜑 → (𝐴‘ran 𝑉) ∈ (SubRing‘𝑃))
56 subrgsubg 20469 . . . . . 6 ((𝐴‘ran 𝑉) ∈ (SubRing‘𝑃) → (𝐴‘ran 𝑉) ∈ (SubGrp‘𝑃))
5755, 56syl 17 . . . . 5 (𝜑 → (𝐴‘ran 𝑉) ∈ (SubGrp‘𝑃))
5857adantr 480 . . . 4 ((𝜑𝑥 ∈ (Base‘𝑃)) → (𝐴‘ran 𝑉) ∈ (SubGrp‘𝑃))
595mpllmod 21887 . . . . . . . 8 ((𝐼𝑊𝑅 ∈ Ring) → 𝑃 ∈ LMod)
602, 12, 59syl2anc 583 . . . . . . 7 (𝜑𝑃 ∈ LMod)
6160ad2antrr 723 . . . . . 6 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → 𝑃 ∈ LMod)
6223, 7, 28asplss 21736 . . . . . . . . 9 ((𝑆 ∈ AssAlg ∧ ran 𝑉 ⊆ (Base‘𝑆)) → (𝐴‘ran 𝑉) ∈ (LSubSp‘𝑆))
634, 49, 62syl2anc 583 . . . . . . . 8 (𝜑 → (𝐴‘ran 𝑉) ∈ (LSubSp‘𝑆))
641, 2, 12psrlmod 21831 . . . . . . . . 9 (𝜑𝑆 ∈ LMod)
65 eqid 2724 . . . . . . . . . 10 (LSubSp‘𝑃) = (LSubSp‘𝑃)
6652, 28, 65lsslss 20798 . . . . . . . . 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 583 . . . . . . . 8 (𝜑 → ((𝐴‘ran 𝑉) ∈ (LSubSp‘𝑃) ↔ ((𝐴‘ran 𝑉) ∈ (LSubSp‘𝑆) ∧ (𝐴‘ran 𝑉) ⊆ (Base‘𝑃))))
6863, 31, 67mpbir2and 710 . . . . . . 7 (𝜑 → (𝐴‘ran 𝑉) ∈ (LSubSp‘𝑃))
6968ad2antrr 723 . . . . . 6 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝐴‘ran 𝑉) ∈ (LSubSp‘𝑃))
70 eqid 2724 . . . . . . . . 9 (Base‘𝑅) = (Base‘𝑅)
715, 70, 6, 32, 38mplelf 21867 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝑃)) → 𝑥:{𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
7271ffvelcdmda 7076 . . . . . . 7 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑥𝑘) ∈ (Base‘𝑅))
735, 35, 37mplsca 21882 . . . . . . . . 9 ((𝜑𝑥 ∈ (Base‘𝑃)) → 𝑅 = (Scalar‘𝑃))
7473adantr 480 . . . . . . . 8 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → 𝑅 = (Scalar‘𝑃))
7574fveq2d 6885 . . . . . . 7 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → (Base‘𝑅) = (Base‘(Scalar‘𝑃)))
7672, 75eleqtrd 2827 . . . . . 6 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑥𝑘) ∈ (Base‘(Scalar‘𝑃)))
772ad2antrr 723 . . . . . . . 8 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → 𝐼𝑊)
78 eqid 2724 . . . . . . . 8 (mulGrp‘𝑃) = (mulGrp‘𝑃)
79 eqid 2724 . . . . . . . 8 (.g‘(mulGrp‘𝑃)) = (.g‘(mulGrp‘𝑃))
803ad2antrr 723 . . . . . . . 8 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → 𝑅 ∈ CRing)
81 simpr 484 . . . . . . . 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 21906 . . . . . . 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 2724 . . . . . . . . 9 (1r𝑃) = (1r𝑃)
8478, 83ringidval 20078 . . . . . . . 8 (1r𝑃) = (0g‘(mulGrp‘𝑃))
855mplcrng 21890 . . . . . . . . . . 11 ((𝐼𝑊𝑅 ∈ CRing) → 𝑃 ∈ CRing)
862, 3, 85syl2anc 583 . . . . . . . . . 10 (𝜑𝑃 ∈ CRing)
8778crngmgp 20136 . . . . . . . . . 10 (𝑃 ∈ CRing → (mulGrp‘𝑃) ∈ CMnd)
8886, 87syl 17 . . . . . . . . 9 (𝜑 → (mulGrp‘𝑃) ∈ CMnd)
8988ad2antrr 723 . . . . . . . 8 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → (mulGrp‘𝑃) ∈ CMnd)
9055ad2antrr 723 . . . . . . . . 9 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝐴‘ran 𝑉) ∈ (SubRing‘𝑃))
9178subrgsubm 20477 . . . . . . . . 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 772 . . . . . . . . . 10 ((((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧𝐼) → 𝜑)
9432psrbag 21779 . . . . . . . . . . . . . 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 476 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑘:𝐼⟶ℕ0 ∧ (𝑘 “ ℕ) ∈ Fin))
9796simpld 494 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → 𝑘:𝐼⟶ℕ0)
9897ffvelcdmda 7076 . . . . . . . . . 10 ((((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧𝐼) → (𝑘𝑧) ∈ ℕ0)
9923, 7aspssid 21740 . . . . . . . . . . . . 13 ((𝑆 ∈ AssAlg ∧ ran 𝑉 ⊆ (Base‘𝑆)) → ran 𝑉 ⊆ (𝐴‘ran 𝑉))
1004, 49, 99syl2anc 583 . . . . . . . . . . . 12 (𝜑 → ran 𝑉 ⊆ (𝐴‘ran 𝑉))
101100ad3antrrr 727 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧𝐼) → ran 𝑉 ⊆ (𝐴‘ran 𝑉))
10214ad2antrr 723 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → 𝑉 Fn 𝐼)
103 fnfvelrn 7072 . . . . . . . . . . . 12 ((𝑉 Fn 𝐼𝑧𝐼) → (𝑉𝑧) ∈ ran 𝑉)
104102, 103sylan 579 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧𝐼) → (𝑉𝑧) ∈ ran 𝑉)
105101, 104sseldd 3975 . . . . . . . . . 10 ((((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧𝐼) → (𝑉𝑧) ∈ (𝐴‘ran 𝑉))
10678, 6mgpbas 20035 . . . . . . . . . . 11 (Base‘𝑃) = (Base‘(mulGrp‘𝑃))
107 eqid 2724 . . . . . . . . . . . 12 (.r𝑃) = (.r𝑃)
10878, 107mgpplusg 20033 . . . . . . . . . . 11 (.r𝑃) = (+g‘(mulGrp‘𝑃))
109107subrgmcl 20476 . . . . . . . . . . . 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 20472 . . . . . . . . . . . 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 19004 . . . . . . . . . 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 7106 . . . . . . . 8 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑧𝐼 ↦ ((𝑘𝑧)(.g‘(mulGrp‘𝑃))(𝑉𝑧))):𝐼⟶(𝐴‘ran 𝑉))
1162mptexd 7217 . . . . . . . . . 10 (𝜑 → (𝑧𝐼 ↦ ((𝑘𝑧)(.g‘(mulGrp‘𝑃))(𝑉𝑧))) ∈ V)
117116ad2antrr 723 . . . . . . . . 9 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑧𝐼 ↦ ((𝑘𝑧)(.g‘(mulGrp‘𝑃))(𝑉𝑧))) ∈ V)
118 funmpt 6576 . . . . . . . . . 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 6896 . . . . . . . . 9 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → (1r𝑃) ∈ V)
12196simprd 495 . . . . . . . . 9 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑘 “ ℕ) ∈ Fin)
122 elrabi 3669 . . . . . . . . . . . . 13 (𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} → 𝑘 ∈ (ℕ0m 𝐼))
123 elmapi 8839 . . . . . . . . . . . . . . 15 (𝑘 ∈ (ℕ0m 𝐼) → 𝑘:𝐼⟶ℕ0)
124123adantl 481 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ (ℕ0m 𝐼)) → 𝑘:𝐼⟶ℕ0)
1252ad2antrr 723 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ (ℕ0m 𝐼)) → 𝐼𝑊)
126 fcdmnn0supp 12525 . . . . . . . . . . . . . . . 16 ((𝐼𝑊𝑘:𝐼⟶ℕ0) → (𝑘 supp 0) = (𝑘 “ ℕ))
127125, 124, 126syl2anc 583 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ (ℕ0m 𝐼)) → (𝑘 supp 0) = (𝑘 “ ℕ))
128 eqimss 4032 . . . . . . . . . . . . . . 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 11205 . . . . . . . . . . . . . . 15 0 ∈ V
131130a1i 11 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ (ℕ0m 𝐼)) → 0 ∈ V)
132124, 129, 125, 131suppssr 8175 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ (ℕ0m 𝐼)) ∧ 𝑧 ∈ (𝐼 ∖ (𝑘 “ ℕ))) → (𝑘𝑧) = 0)
133122, 132sylanl2 678 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧 ∈ (𝐼 ∖ (𝑘 “ ℕ))) → (𝑘𝑧) = 0)
134133oveq1d 7416 . . . . . . . . . . 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 727 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧 ∈ (𝐼 ∖ (𝑘 “ ℕ))) → 𝐼𝑊)
13612ad3antrrr 727 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧 ∈ (𝐼 ∖ (𝑘 “ ℕ))) → 𝑅 ∈ Ring)
137 eldifi 4118 . . . . . . . . . . . . . 14 (𝑧 ∈ (𝐼 ∖ (𝑘 “ ℕ)) → 𝑧𝐼)
138137adantl 481 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧 ∈ (𝐼 ∖ (𝑘 “ ℕ))) → 𝑧𝐼)
1395, 10, 6, 135, 136, 138mvrcl 21861 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧 ∈ (𝐼 ∖ (𝑘 “ ℕ))) → (𝑉𝑧) ∈ (Base‘𝑃))
140106, 84, 79mulg0 18992 . . . . . . . . . . . 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 2764 . . . . . . . . . 10 ((((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧 ∈ (𝐼 ∖ (𝑘 “ ℕ))) → ((𝑘𝑧)(.g‘(mulGrp‘𝑃))(𝑉𝑧)) = (1r𝑃))
143142, 77suppss2 8180 . . . . . . . . 9 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → ((𝑧𝐼 ↦ ((𝑘𝑧)(.g‘(mulGrp‘𝑃))(𝑉𝑧))) supp (1r𝑃)) ⊆ (𝑘 “ ℕ))
144 suppssfifsupp 9374 . . . . . . . . 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 19829 . . . . . . 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 2825 . . . . . 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 2724 . . . . . . 7 (Scalar‘𝑃) = (Scalar‘𝑃)
149 eqid 2724 . . . . . . 7 (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
150148, 36, 149, 65lssvscl 20792 . . . . . 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 836 . . . . 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 7106 . . . 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 7218 . . . . . . 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 6576 . . . . . . 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 6894 . . . . . . 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 21860 . . . . . . . 8 (𝑥 ∈ (Base‘𝑃) ↔ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 finSupp (0g𝑅)))
159158simprbi 496 . . . . . . 7 (𝑥 ∈ (Base‘𝑃) → 𝑥 finSupp (0g𝑅))
160159adantl 481 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝑃)) → 𝑥 finSupp (0g𝑅))
161160fsuppimpd 9365 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝑃)) → (𝑥 supp (0g𝑅)) ∈ Fin)
162 ssidd 3997 . . . . . . . . . 10 ((𝜑𝑥 ∈ (Base‘𝑃)) → (𝑥 supp (0g𝑅)) ⊆ (𝑥 supp (0g𝑅)))
163 fvexd 6896 . . . . . . . . . 10 ((𝜑𝑥 ∈ (Base‘𝑃)) → (0g𝑅) ∈ V)
16471, 162, 48, 163suppssr 8175 . . . . . . . . 9 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ ({𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∖ (𝑥 supp (0g𝑅)))) → (𝑥𝑘) = (0g𝑅))
16573fveq2d 6885 . . . . . . . . . 10 ((𝜑𝑥 ∈ (Base‘𝑃)) → (0g𝑅) = (0g‘(Scalar‘𝑃)))
166165adantr 480 . . . . . . . . 9 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ ({𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∖ (𝑥 supp (0g𝑅)))) → (0g𝑅) = (0g‘(Scalar‘𝑃)))
167164, 166eqtrd 2764 . . . . . . . 8 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ ({𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∖ (𝑥 supp (0g𝑅)))) → (𝑥𝑘) = (0g‘(Scalar‘𝑃)))
168167oveq1d 7416 . . . . . . 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 4118 . . . . . . . 8 (𝑘 ∈ ({𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∖ (𝑥 supp (0g𝑅))) → 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin})
17012ad2antrr 723 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0m 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → 𝑅 ∈ Ring)
1715, 6, 33, 34, 32, 77, 170, 81mplmon 21900 . . . . . . . . 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 2724 . . . . . . . . . 10 (0g‘(Scalar‘𝑃)) = (0g‘(Scalar‘𝑃))
1736, 148, 36, 172, 40lmod0vs 20731 . . . . . . . . 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 583 . . . . . . . 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 592 . . . . . . 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 2764 . . . . . 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 8180 . . . . 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 9374 . . . . 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 834 . . . 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 19830 . . 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 2825 . 2 ((𝜑𝑥 ∈ (Base‘𝑃)) → 𝑥 ∈ (𝐴‘ran 𝑉))
18231, 181eqelssd 3995 1 (𝜑 → (𝐴‘ran 𝑉) = (Base‘𝑃))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1084   = wceq 1533  wcel 2098  wral 3053  {crab 3424  Vcvv 3466  cdif 3937  wss 3940  ifcif 4520   class class class wbr 5138  cmpt 5221  ccnv 5665  ran crn 5667  cima 5669  Fun wfun 6527   Fn wfn 6528  wf 6529  cfv 6533  (class class class)co 7401   supp csupp 8140  m cmap 8816  Fincfn 8935   finSupp cfsupp 9357  0cc0 11106  cn 12209  0cn0 12469  Basecbs 17143  .rcmulr 17197  Scalarcsca 17199   ·𝑠 cvsca 17200  0gc0g 17384   Σg cgsu 17385  SubMndcsubmnd 18702  .gcmg 18985  SubGrpcsubg 19037  CMndccmn 19690  Abelcabl 19691  mulGrpcmgp 20029  1rcur 20076  Ringcrg 20128  CRingccrg 20129  SubRingcsubrg 20459  LModclmod 20696  LSubSpclss 20768  AssAlgcasa 21713  AlgSpancasp 21714   mPwSer cmps 21766   mVar cmvr 21767   mPoly cmpl 21768
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-tp 4625  df-op 4627  df-uni 4900  df-int 4941  df-iun 4989  df-iin 4990  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-se 5622  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-isom 6542  df-riota 7357  df-ov 7404  df-oprab 7405  df-mpo 7406  df-of 7663  df-ofr 7664  df-om 7849  df-1st 7968  df-2nd 7969  df-supp 8141  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-1o 8461  df-er 8699  df-map 8818  df-pm 8819  df-ixp 8888  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-fsupp 9358  df-sup 9433  df-oi 9501  df-card 9930  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11443  df-neg 11444  df-nn 12210  df-2 12272  df-3 12273  df-4 12274  df-5 12275  df-6 12276  df-7 12277  df-8 12278  df-9 12279  df-n0 12470  df-z 12556  df-dec 12675  df-uz 12820  df-fz 13482  df-fzo 13625  df-seq 13964  df-hash 14288  df-struct 17079  df-sets 17096  df-slot 17114  df-ndx 17126  df-base 17144  df-ress 17173  df-plusg 17209  df-mulr 17210  df-sca 17212  df-vsca 17213  df-ip 17214  df-tset 17215  df-ple 17216  df-ds 17218  df-hom 17220  df-cco 17221  df-0g 17386  df-gsum 17387  df-prds 17392  df-pws 17394  df-mre 17529  df-mrc 17530  df-acs 17532  df-mgm 18563  df-sgrp 18642  df-mnd 18658  df-mhm 18703  df-submnd 18704  df-grp 18856  df-minusg 18857  df-sbg 18858  df-mulg 18986  df-subg 19040  df-ghm 19129  df-cntz 19223  df-cmn 19692  df-abl 19693  df-mgp 20030  df-rng 20048  df-ur 20077  df-srg 20082  df-ring 20130  df-cring 20131  df-subrng 20436  df-subrg 20461  df-lmod 20698  df-lss 20769  df-assa 21716  df-asp 21717  df-psr 21771  df-mvr 21772  df-mpl 21773
This theorem is referenced by:  mplind  21941  evlseu  21956
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