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Theorem mplbas2 19867
 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 19811 . . . 4 (𝜑𝑆 ∈ AssAlg)
5 mplbas2.p . . . . . 6 𝑃 = (𝐼 mPoly 𝑅)
6 eqid 2777 . . . . . 6 (Base‘𝑃) = (Base‘𝑃)
7 eqid 2777 . . . . . 6 (Base‘𝑆) = (Base‘𝑆)
85, 1, 6, 7mplbasss 19829 . . . . 5 (Base‘𝑃) ⊆ (Base‘𝑆)
98a1i 11 . . . 4 (𝜑 → (Base‘𝑃) ⊆ (Base‘𝑆))
10 mplbas2.v . . . . . . . 8 𝑉 = (𝐼 mVar 𝑅)
11 crngring 18945 . . . . . . . . 9 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
123, 11syl 17 . . . . . . . 8 (𝜑𝑅 ∈ Ring)
131, 10, 7, 2, 12mvrf 19821 . . . . . . 7 (𝜑𝑉:𝐼⟶(Base‘𝑆))
1413ffnd 6292 . . . . . 6 (𝜑𝑉 Fn 𝐼)
152adantr 474 . . . . . . . 8 ((𝜑𝑥𝐼) → 𝐼𝑊)
1612adantr 474 . . . . . . . 8 ((𝜑𝑥𝐼) → 𝑅 ∈ Ring)
17 simpr 479 . . . . . . . 8 ((𝜑𝑥𝐼) → 𝑥𝐼)
185, 10, 6, 15, 16, 17mvrcl 19846 . . . . . . 7 ((𝜑𝑥𝐼) → (𝑉𝑥) ∈ (Base‘𝑃))
1918ralrimiva 3147 . . . . . 6 (𝜑 → ∀𝑥𝐼 (𝑉𝑥) ∈ (Base‘𝑃))
20 ffnfv 6652 . . . . . 6 (𝑉:𝐼⟶(Base‘𝑃) ↔ (𝑉 Fn 𝐼 ∧ ∀𝑥𝐼 (𝑉𝑥) ∈ (Base‘𝑃)))
2114, 19, 20sylanbrc 578 . . . . 5 (𝜑𝑉:𝐼⟶(Base‘𝑃))
2221frnd 6298 . . . 4 (𝜑 → ran 𝑉 ⊆ (Base‘𝑃))
23 mplbas2.a . . . . 5 𝐴 = (AlgSpan‘𝑆)
2423, 7aspss 19729 . . . 4 ((𝑆 ∈ AssAlg ∧ (Base‘𝑃) ⊆ (Base‘𝑆) ∧ ran 𝑉 ⊆ (Base‘𝑃)) → (𝐴‘ran 𝑉) ⊆ (𝐴‘(Base‘𝑃)))
254, 9, 22, 24syl3anc 1439 . . 3 (𝜑 → (𝐴‘ran 𝑉) ⊆ (𝐴‘(Base‘𝑃)))
261, 5, 6, 2, 12mplsubrg 19837 . . . 4 (𝜑 → (Base‘𝑃) ∈ (SubRing‘𝑆))
271, 5, 6, 2, 12mpllss 19835 . . . 4 (𝜑 → (Base‘𝑃) ∈ (LSubSp‘𝑆))
28 eqid 2777 . . . . 5 (LSubSp‘𝑆) = (LSubSp‘𝑆)
2923, 7, 28aspid 19727 . . . 4 ((𝑆 ∈ AssAlg ∧ (Base‘𝑃) ∈ (SubRing‘𝑆) ∧ (Base‘𝑃) ∈ (LSubSp‘𝑆)) → (𝐴‘(Base‘𝑃)) = (Base‘𝑃))
304, 26, 27, 29syl3anc 1439 . . 3 (𝜑 → (𝐴‘(Base‘𝑃)) = (Base‘𝑃))
3125, 30sseqtrd 3859 . 2 (𝜑 → (𝐴‘ran 𝑉) ⊆ (Base‘𝑃))
32 eqid 2777 . . . 4 {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}
33 eqid 2777 . . . 4 (0g𝑅) = (0g𝑅)
34 eqid 2777 . . . 4 (1r𝑅) = (1r𝑅)
352adantr 474 . . . 4 ((𝜑𝑥 ∈ (Base‘𝑃)) → 𝐼𝑊)
36 eqid 2777 . . . 4 ( ·𝑠𝑃) = ( ·𝑠𝑃)
3712adantr 474 . . . 4 ((𝜑𝑥 ∈ (Base‘𝑃)) → 𝑅 ∈ Ring)
38 simpr 479 . . . 4 ((𝜑𝑥 ∈ (Base‘𝑃)) → 𝑥 ∈ (Base‘𝑃))
395, 32, 33, 34, 35, 6, 36, 37, 38mplcoe1 19862 . . 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 2777 . . . 4 (0g𝑃) = (0g𝑃)
415mplring 19849 . . . . . . 7 ((𝐼𝑊𝑅 ∈ Ring) → 𝑃 ∈ Ring)
422, 12, 41syl2anc 579 . . . . . 6 (𝜑𝑃 ∈ Ring)
43 ringabl 18967 . . . . . 6 (𝑃 ∈ Ring → 𝑃 ∈ Abel)
4442, 43syl 17 . . . . 5 (𝜑𝑃 ∈ Abel)
4544adantr 474 . . . 4 ((𝜑𝑥 ∈ (Base‘𝑃)) → 𝑃 ∈ Abel)
46 ovex 6954 . . . . . 6 (ℕ0𝑚 𝐼) ∈ V
4746rabex 5049 . . . . 5 {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∈ V
4847a1i 11 . . . 4 ((𝜑𝑥 ∈ (Base‘𝑃)) → {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∈ V)
4913frnd 6298 . . . . . . . 8 (𝜑 → ran 𝑉 ⊆ (Base‘𝑆))
5023, 7aspsubrg 19728 . . . . . . . 8 ((𝑆 ∈ AssAlg ∧ ran 𝑉 ⊆ (Base‘𝑆)) → (𝐴‘ran 𝑉) ∈ (SubRing‘𝑆))
514, 49, 50syl2anc 579 . . . . . . 7 (𝜑 → (𝐴‘ran 𝑉) ∈ (SubRing‘𝑆))
525, 1, 6mplval2 19828 . . . . . . . . 9 𝑃 = (𝑆s (Base‘𝑃))
5352subsubrg 19198 . . . . . . . 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 703 . . . . . 6 (𝜑 → (𝐴‘ran 𝑉) ∈ (SubRing‘𝑃))
56 subrgsubg 19178 . . . . . 6 ((𝐴‘ran 𝑉) ∈ (SubRing‘𝑃) → (𝐴‘ran 𝑉) ∈ (SubGrp‘𝑃))
5755, 56syl 17 . . . . 5 (𝜑 → (𝐴‘ran 𝑉) ∈ (SubGrp‘𝑃))
5857adantr 474 . . . 4 ((𝜑𝑥 ∈ (Base‘𝑃)) → (𝐴‘ran 𝑉) ∈ (SubGrp‘𝑃))
595mpllmod 19848 . . . . . . . 8 ((𝐼𝑊𝑅 ∈ Ring) → 𝑃 ∈ LMod)
602, 12, 59syl2anc 579 . . . . . . 7 (𝜑𝑃 ∈ LMod)
6160ad2antrr 716 . . . . . 6 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → 𝑃 ∈ LMod)
6223, 7, 28asplss 19726 . . . . . . . . 9 ((𝑆 ∈ AssAlg ∧ ran 𝑉 ⊆ (Base‘𝑆)) → (𝐴‘ran 𝑉) ∈ (LSubSp‘𝑆))
634, 49, 62syl2anc 579 . . . . . . . 8 (𝜑 → (𝐴‘ran 𝑉) ∈ (LSubSp‘𝑆))
641, 2, 12psrlmod 19798 . . . . . . . . 9 (𝜑𝑆 ∈ LMod)
65 eqid 2777 . . . . . . . . . 10 (LSubSp‘𝑃) = (LSubSp‘𝑃)
6652, 28, 65lsslss 19356 . . . . . . . . 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 579 . . . . . . . 8 (𝜑 → ((𝐴‘ran 𝑉) ∈ (LSubSp‘𝑃) ↔ ((𝐴‘ran 𝑉) ∈ (LSubSp‘𝑆) ∧ (𝐴‘ran 𝑉) ⊆ (Base‘𝑃))))
6863, 31, 67mpbir2and 703 . . . . . . 7 (𝜑 → (𝐴‘ran 𝑉) ∈ (LSubSp‘𝑃))
6968ad2antrr 716 . . . . . 6 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝐴‘ran 𝑉) ∈ (LSubSp‘𝑃))
70 eqid 2777 . . . . . . . . 9 (Base‘𝑅) = (Base‘𝑅)
715, 70, 6, 32, 38mplelf 19830 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝑃)) → 𝑥:{𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
7271ffvelrnda 6623 . . . . . . 7 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑥𝑘) ∈ (Base‘𝑅))
735, 35, 37mplsca 19842 . . . . . . . . 9 ((𝜑𝑥 ∈ (Base‘𝑃)) → 𝑅 = (Scalar‘𝑃))
7473adantr 474 . . . . . . . 8 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → 𝑅 = (Scalar‘𝑃))
7574fveq2d 6450 . . . . . . 7 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → (Base‘𝑅) = (Base‘(Scalar‘𝑃)))
7672, 75eleqtrd 2860 . . . . . 6 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑥𝑘) ∈ (Base‘(Scalar‘𝑃)))
772ad2antrr 716 . . . . . . . 8 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → 𝐼𝑊)
78 eqid 2777 . . . . . . . 8 (mulGrp‘𝑃) = (mulGrp‘𝑃)
79 eqid 2777 . . . . . . . 8 (.g‘(mulGrp‘𝑃)) = (.g‘(mulGrp‘𝑃))
803ad2antrr 716 . . . . . . . 8 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → 𝑅 ∈ CRing)
81 simpr 479 . . . . . . . 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 19866 . . . . . . 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 2777 . . . . . . . . 9 (1r𝑃) = (1r𝑃)
8478, 83ringidval 18890 . . . . . . . 8 (1r𝑃) = (0g‘(mulGrp‘𝑃))
855mplcrng 19850 . . . . . . . . . . 11 ((𝐼𝑊𝑅 ∈ CRing) → 𝑃 ∈ CRing)
862, 3, 85syl2anc 579 . . . . . . . . . 10 (𝜑𝑃 ∈ CRing)
8778crngmgp 18942 . . . . . . . . . 10 (𝑃 ∈ CRing → (mulGrp‘𝑃) ∈ CMnd)
8886, 87syl 17 . . . . . . . . 9 (𝜑 → (mulGrp‘𝑃) ∈ CMnd)
8988ad2antrr 716 . . . . . . . 8 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → (mulGrp‘𝑃) ∈ CMnd)
9055ad2antrr 716 . . . . . . . . 9 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝐴‘ran 𝑉) ∈ (SubRing‘𝑃))
9178subrgsubm 19185 . . . . . . . . 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 765 . . . . . . . . . 10 ((((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧𝐼) → 𝜑)
9432psrbag 19761 . . . . . . . . . . . . . 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 470 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑘:𝐼⟶ℕ0 ∧ (𝑘 “ ℕ) ∈ Fin))
9796simpld 490 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → 𝑘:𝐼⟶ℕ0)
9897ffvelrnda 6623 . . . . . . . . . 10 ((((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧𝐼) → (𝑘𝑧) ∈ ℕ0)
9923, 7aspssid 19730 . . . . . . . . . . . . 13 ((𝑆 ∈ AssAlg ∧ ran 𝑉 ⊆ (Base‘𝑆)) → ran 𝑉 ⊆ (𝐴‘ran 𝑉))
1004, 49, 99syl2anc 579 . . . . . . . . . . . 12 (𝜑 → ran 𝑉 ⊆ (𝐴‘ran 𝑉))
101100ad3antrrr 720 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧𝐼) → ran 𝑉 ⊆ (𝐴‘ran 𝑉))
10214ad2antrr 716 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → 𝑉 Fn 𝐼)
103 fnfvelrn 6620 . . . . . . . . . . . 12 ((𝑉 Fn 𝐼𝑧𝐼) → (𝑉𝑧) ∈ ran 𝑉)
104102, 103sylan 575 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧𝐼) → (𝑉𝑧) ∈ ran 𝑉)
105101, 104sseldd 3821 . . . . . . . . . 10 ((((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧𝐼) → (𝑉𝑧) ∈ (𝐴‘ran 𝑉))
10678, 6mgpbas 18882 . . . . . . . . . . 11 (Base‘𝑃) = (Base‘(mulGrp‘𝑃))
107 eqid 2777 . . . . . . . . . . . 12 (.r𝑃) = (.r𝑃)
10878, 107mgpplusg 18880 . . . . . . . . . . 11 (.r𝑃) = (+g‘(mulGrp‘𝑃))
109107subrgmcl 19184 . . . . . . . . . . . 12 (((𝐴‘ran 𝑉) ∈ (SubRing‘𝑃) ∧ 𝑢 ∈ (𝐴‘ran 𝑉) ∧ 𝑣 ∈ (𝐴‘ran 𝑉)) → (𝑢(.r𝑃)𝑣) ∈ (𝐴‘ran 𝑉))
11055, 109syl3an1 1163 . . . . . . . . . . 11 ((𝜑𝑢 ∈ (𝐴‘ran 𝑉) ∧ 𝑣 ∈ (𝐴‘ran 𝑉)) → (𝑢(.r𝑃)𝑣) ∈ (𝐴‘ran 𝑉))
11183subrg1cl 19180 . . . . . . . . . . . 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 17941 . . . . . . . . . 10 ((𝜑 ∧ (𝑘𝑧) ∈ ℕ0 ∧ (𝑉𝑧) ∈ (𝐴‘ran 𝑉)) → ((𝑘𝑧)(.g‘(mulGrp‘𝑃))(𝑉𝑧)) ∈ (𝐴‘ran 𝑉))
11493, 98, 105, 113syl3anc 1439 . . . . . . . . 9 ((((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧𝐼) → ((𝑘𝑧)(.g‘(mulGrp‘𝑃))(𝑉𝑧)) ∈ (𝐴‘ran 𝑉))
115114fmpttd 6649 . . . . . . . 8 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑧𝐼 ↦ ((𝑘𝑧)(.g‘(mulGrp‘𝑃))(𝑉𝑧))):𝐼⟶(𝐴‘ran 𝑉))
116 mptexg 6756 . . . . . . . . . . 11 (𝐼𝑊 → (𝑧𝐼 ↦ ((𝑘𝑧)(.g‘(mulGrp‘𝑃))(𝑉𝑧))) ∈ V)
1172, 116syl 17 . . . . . . . . . 10 (𝜑 → (𝑧𝐼 ↦ ((𝑘𝑧)(.g‘(mulGrp‘𝑃))(𝑉𝑧))) ∈ V)
118117ad2antrr 716 . . . . . . . . 9 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑧𝐼 ↦ ((𝑘𝑧)(.g‘(mulGrp‘𝑃))(𝑉𝑧))) ∈ V)
119 funmpt 6173 . . . . . . . . . 10 Fun (𝑧𝐼 ↦ ((𝑘𝑧)(.g‘(mulGrp‘𝑃))(𝑉𝑧)))
120119a1i 11 . . . . . . . . 9 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → Fun (𝑧𝐼 ↦ ((𝑘𝑧)(.g‘(mulGrp‘𝑃))(𝑉𝑧))))
121 fvexd 6461 . . . . . . . . 9 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → (1r𝑃) ∈ V)
12296simprd 491 . . . . . . . . 9 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑘 “ ℕ) ∈ Fin)
123 elrabi 3566 . . . . . . . . . . . . 13 (𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} → 𝑘 ∈ (ℕ0𝑚 𝐼))
124 elmapi 8162 . . . . . . . . . . . . . . 15 (𝑘 ∈ (ℕ0𝑚 𝐼) → 𝑘:𝐼⟶ℕ0)
125124adantl 475 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ (ℕ0𝑚 𝐼)) → 𝑘:𝐼⟶ℕ0)
1262ad2antrr 716 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ (ℕ0𝑚 𝐼)) → 𝐼𝑊)
127 frnnn0supp 11700 . . . . . . . . . . . . . . . 16 ((𝐼𝑊𝑘:𝐼⟶ℕ0) → (𝑘 supp 0) = (𝑘 “ ℕ))
128126, 125, 127syl2anc 579 . . . . . . . . . . . . . . 15 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ (ℕ0𝑚 𝐼)) → (𝑘 supp 0) = (𝑘 “ ℕ))
129 eqimss 3875 . . . . . . . . . . . . . . 15 ((𝑘 supp 0) = (𝑘 “ ℕ) → (𝑘 supp 0) ⊆ (𝑘 “ ℕ))
130128, 129syl 17 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ (ℕ0𝑚 𝐼)) → (𝑘 supp 0) ⊆ (𝑘 “ ℕ))
131 c0ex 10370 . . . . . . . . . . . . . . 15 0 ∈ V
132131a1i 11 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ (ℕ0𝑚 𝐼)) → 0 ∈ V)
133125, 130, 126, 132suppssr 7608 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ (ℕ0𝑚 𝐼)) ∧ 𝑧 ∈ (𝐼 ∖ (𝑘 “ ℕ))) → (𝑘𝑧) = 0)
134123, 133sylanl2 671 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧 ∈ (𝐼 ∖ (𝑘 “ ℕ))) → (𝑘𝑧) = 0)
135134oveq1d 6937 . . . . . . . . . . 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)))
1362ad3antrrr 720 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧 ∈ (𝐼 ∖ (𝑘 “ ℕ))) → 𝐼𝑊)
13712ad3antrrr 720 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧 ∈ (𝐼 ∖ (𝑘 “ ℕ))) → 𝑅 ∈ Ring)
138 eldifi 3954 . . . . . . . . . . . . . 14 (𝑧 ∈ (𝐼 ∖ (𝑘 “ ℕ)) → 𝑧𝐼)
139138adantl 475 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧 ∈ (𝐼 ∖ (𝑘 “ ℕ))) → 𝑧𝐼)
1405, 10, 6, 136, 137, 139mvrcl 19846 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧 ∈ (𝐼 ∖ (𝑘 “ ℕ))) → (𝑉𝑧) ∈ (Base‘𝑃))
141106, 84, 79mulg0 17933 . . . . . . . . . . . 12 ((𝑉𝑧) ∈ (Base‘𝑃) → (0(.g‘(mulGrp‘𝑃))(𝑉𝑧)) = (1r𝑃))
142140, 141syl 17 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧 ∈ (𝐼 ∖ (𝑘 “ ℕ))) → (0(.g‘(mulGrp‘𝑃))(𝑉𝑧)) = (1r𝑃))
143135, 142eqtrd 2813 . . . . . . . . . 10 ((((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧 ∈ (𝐼 ∖ (𝑘 “ ℕ))) → ((𝑘𝑧)(.g‘(mulGrp‘𝑃))(𝑉𝑧)) = (1r𝑃))
144143, 77suppss2 7611 . . . . . . . . 9 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → ((𝑧𝐼 ↦ ((𝑘𝑧)(.g‘(mulGrp‘𝑃))(𝑉𝑧))) supp (1r𝑃)) ⊆ (𝑘 “ ℕ))
145 suppssfifsupp 8578 . . . . . . . . 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))
146118, 120, 121, 122, 144, 145syl32anc 1446 . . . . . . . 8 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑧𝐼 ↦ ((𝑘𝑧)(.g‘(mulGrp‘𝑃))(𝑉𝑧))) finSupp (1r𝑃))
14784, 89, 77, 92, 115, 146gsumsubmcl 18705 . . . . . . 7 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → ((mulGrp‘𝑃) Σg (𝑧𝐼 ↦ ((𝑘𝑧)(.g‘(mulGrp‘𝑃))(𝑉𝑧)))) ∈ (𝐴‘ran 𝑉))
14882, 147eqeltrd 2858 . . . . . 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))
149 eqid 2777 . . . . . . 7 (Scalar‘𝑃) = (Scalar‘𝑃)
150 eqid 2777 . . . . . . 7 (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
151149, 36, 150, 65lssvscl 19350 . . . . . 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))
15261, 69, 76, 148, 151syl22anc 829 . . . . 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))
153152fmpttd 6649 . . . 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))
15446mptrabex 6760 . . . . . . 7 (𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ↦ ((𝑥𝑘)( ·𝑠𝑃)(𝑦 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = 𝑘, (1r𝑅), (0g𝑅))))) ∈ V
155 funmpt 6173 . . . . . . 7 Fun (𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ↦ ((𝑥𝑘)( ·𝑠𝑃)(𝑦 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = 𝑘, (1r𝑅), (0g𝑅)))))
156 fvex 6459 . . . . . . 7 (0g𝑃) ∈ V
157154, 155, 1563pm3.2i 1395 . . . . . 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)
158157a1i 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))
1595, 1, 7, 33, 6mplelbas 19827 . . . . . . . 8 (𝑥 ∈ (Base‘𝑃) ↔ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 finSupp (0g𝑅)))
160159simprbi 492 . . . . . . 7 (𝑥 ∈ (Base‘𝑃) → 𝑥 finSupp (0g𝑅))
161160adantl 475 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝑃)) → 𝑥 finSupp (0g𝑅))
162161fsuppimpd 8570 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝑃)) → (𝑥 supp (0g𝑅)) ∈ Fin)
163 ssidd 3842 . . . . . . . . . 10 ((𝜑𝑥 ∈ (Base‘𝑃)) → (𝑥 supp (0g𝑅)) ⊆ (𝑥 supp (0g𝑅)))
164 fvexd 6461 . . . . . . . . . 10 ((𝜑𝑥 ∈ (Base‘𝑃)) → (0g𝑅) ∈ V)
16571, 163, 48, 164suppssr 7608 . . . . . . . . 9 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ ({𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∖ (𝑥 supp (0g𝑅)))) → (𝑥𝑘) = (0g𝑅))
16673fveq2d 6450 . . . . . . . . . 10 ((𝜑𝑥 ∈ (Base‘𝑃)) → (0g𝑅) = (0g‘(Scalar‘𝑃)))
167166adantr 474 . . . . . . . . 9 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ ({𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∖ (𝑥 supp (0g𝑅)))) → (0g𝑅) = (0g‘(Scalar‘𝑃)))
168165, 167eqtrd 2813 . . . . . . . 8 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ ({𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∖ (𝑥 supp (0g𝑅)))) → (𝑥𝑘) = (0g‘(Scalar‘𝑃)))
169168oveq1d 6937 . . . . . . 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)))))
170 eldifi 3954 . . . . . . . 8 (𝑘 ∈ ({𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∖ (𝑥 supp (0g𝑅))) → 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin})
17112ad2antrr 716 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → 𝑅 ∈ Ring)
1725, 6, 33, 34, 32, 77, 171, 81mplmon 19860 . . . . . . . . 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))
173 eqid 2777 . . . . . . . . . 10 (0g‘(Scalar‘𝑃)) = (0g‘(Scalar‘𝑃))
1746, 149, 36, 173, 40lmod0vs 19288 . . . . . . . . 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))
17561, 172, 174syl2anc 579 . . . . . . . 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))
176170, 175sylan2 586 . . . . . . 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))
177169, 176eqtrd 2813 . . . . . 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))
178177, 48suppss2 7611 . . . . 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)))
179 suppssfifsupp 8578 . . . . 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))
180158, 162, 178, 179syl12anc 827 . . . 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))
18140, 45, 48, 58, 153, 180gsumsubgcl 18706 . . 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))
18239, 181eqeltrd 2858 . 2 ((𝜑𝑥 ∈ (Base‘𝑃)) → 𝑥 ∈ (𝐴‘ran 𝑉))
18331, 182eqelssd 3840 1 (𝜑 → (𝐴‘ran 𝑉) = (Base‘𝑃))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   ∧ w3a 1071   = wceq 1601   ∈ wcel 2106  ∀wral 3089  {crab 3093  Vcvv 3397   ∖ cdif 3788   ⊆ wss 3791  ifcif 4306   class class class wbr 4886   ↦ cmpt 4965  ◡ccnv 5354  ran crn 5356   “ cima 5358  Fun wfun 6129   Fn wfn 6130  ⟶wf 6131  ‘cfv 6135  (class class class)co 6922   supp csupp 7576   ↑𝑚 cmap 8140  Fincfn 8241   finSupp cfsupp 8563  0cc0 10272  ℕcn 11374  ℕ0cn0 11642  Basecbs 16255  .rcmulr 16339  Scalarcsca 16341   ·𝑠 cvsca 16342  0gc0g 16486   Σg cgsu 16487  SubMndcsubmnd 17720  .gcmg 17927  SubGrpcsubg 17972  CMndccmn 18579  Abelcabl 18580  mulGrpcmgp 18876  1rcur 18888  Ringcrg 18934  CRingccrg 18935  SubRingcsubrg 19168  LModclmod 19255  LSubSpclss 19324  AssAlgcasa 19706  AlgSpancasp 19707   mPwSer cmps 19748   mVar cmvr 19749   mPoly cmpl 19750 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-inf2 8835  ax-cnex 10328  ax-resscn 10329  ax-1cn 10330  ax-icn 10331  ax-addcl 10332  ax-addrcl 10333  ax-mulcl 10334  ax-mulrcl 10335  ax-mulcom 10336  ax-addass 10337  ax-mulass 10338  ax-distr 10339  ax-i2m1 10340  ax-1ne0 10341  ax-1rid 10342  ax-rnegex 10343  ax-rrecex 10344  ax-cnre 10345  ax-pre-lttri 10346  ax-pre-lttrn 10347  ax-pre-ltadd 10348  ax-pre-mulgt0 10349 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-nel 3075  df-ral 3094  df-rex 3095  df-reu 3096  df-rmo 3097  df-rab 3098  df-v 3399  df-sbc 3652  df-csb 3751  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-pss 3807  df-nul 4141  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4672  df-int 4711  df-iun 4755  df-iin 4756  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-se 5315  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-isom 6144  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-of 7174  df-ofr 7175  df-om 7344  df-1st 7445  df-2nd 7446  df-supp 7577  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-1o 7843  df-2o 7844  df-oadd 7847  df-er 8026  df-map 8142  df-pm 8143  df-ixp 8195  df-en 8242  df-dom 8243  df-sdom 8244  df-fin 8245  df-fsupp 8564  df-oi 8704  df-card 9098  df-pnf 10413  df-mnf 10414  df-xr 10415  df-ltxr 10416  df-le 10417  df-sub 10608  df-neg 10609  df-nn 11375  df-2 11438  df-3 11439  df-4 11440  df-5 11441  df-6 11442  df-7 11443  df-8 11444  df-9 11445  df-n0 11643  df-z 11729  df-uz 11993  df-fz 12644  df-fzo 12785  df-seq 13120  df-hash 13436  df-struct 16257  df-ndx 16258  df-slot 16259  df-base 16261  df-sets 16262  df-ress 16263  df-plusg 16351  df-mulr 16352  df-sca 16354  df-vsca 16355  df-tset 16357  df-0g 16488  df-gsum 16489  df-mre 16632  df-mrc 16633  df-acs 16635  df-mgm 17628  df-sgrp 17670  df-mnd 17681  df-mhm 17721  df-submnd 17722  df-grp 17812  df-minusg 17813  df-sbg 17814  df-mulg 17928  df-subg 17975  df-ghm 18042  df-cntz 18133  df-cmn 18581  df-abl 18582  df-mgp 18877  df-ur 18889  df-srg 18893  df-ring 18936  df-cring 18937  df-subrg 19170  df-lmod 19257  df-lss 19325  df-assa 19709  df-asp 19710  df-psr 19753  df-mvr 19754  df-mpl 19755 This theorem is referenced by:  mplind  19898  evlseu  19912
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