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Theorem mplbas2 19518
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 19462 . . . 4 (𝜑𝑆 ∈ AssAlg)
5 mplbas2.p . . . . . 6 𝑃 = (𝐼 mPoly 𝑅)
6 eqid 2651 . . . . . 6 (Base‘𝑃) = (Base‘𝑃)
7 eqid 2651 . . . . . 6 (Base‘𝑆) = (Base‘𝑆)
85, 1, 6, 7mplbasss 19480 . . . . 5 (Base‘𝑃) ⊆ (Base‘𝑆)
98a1i 11 . . . 4 (𝜑 → (Base‘𝑃) ⊆ (Base‘𝑆))
10 mplbas2.v . . . . . . . 8 𝑉 = (𝐼 mVar 𝑅)
11 crngring 18604 . . . . . . . . 9 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
123, 11syl 17 . . . . . . . 8 (𝜑𝑅 ∈ Ring)
131, 10, 7, 2, 12mvrf 19472 . . . . . . 7 (𝜑𝑉:𝐼⟶(Base‘𝑆))
1413ffnd 6084 . . . . . 6 (𝜑𝑉 Fn 𝐼)
152adantr 480 . . . . . . . 8 ((𝜑𝑥𝐼) → 𝐼𝑊)
1612adantr 480 . . . . . . . 8 ((𝜑𝑥𝐼) → 𝑅 ∈ Ring)
17 simpr 476 . . . . . . . 8 ((𝜑𝑥𝐼) → 𝑥𝐼)
185, 10, 6, 15, 16, 17mvrcl 19497 . . . . . . 7 ((𝜑𝑥𝐼) → (𝑉𝑥) ∈ (Base‘𝑃))
1918ralrimiva 2995 . . . . . 6 (𝜑 → ∀𝑥𝐼 (𝑉𝑥) ∈ (Base‘𝑃))
20 ffnfv 6428 . . . . . 6 (𝑉:𝐼⟶(Base‘𝑃) ↔ (𝑉 Fn 𝐼 ∧ ∀𝑥𝐼 (𝑉𝑥) ∈ (Base‘𝑃)))
2114, 19, 20sylanbrc 699 . . . . 5 (𝜑𝑉:𝐼⟶(Base‘𝑃))
22 frn 6091 . . . . 5 (𝑉:𝐼⟶(Base‘𝑃) → ran 𝑉 ⊆ (Base‘𝑃))
2321, 22syl 17 . . . 4 (𝜑 → ran 𝑉 ⊆ (Base‘𝑃))
24 mplbas2.a . . . . 5 𝐴 = (AlgSpan‘𝑆)
2524, 7aspss 19380 . . . 4 ((𝑆 ∈ AssAlg ∧ (Base‘𝑃) ⊆ (Base‘𝑆) ∧ ran 𝑉 ⊆ (Base‘𝑃)) → (𝐴‘ran 𝑉) ⊆ (𝐴‘(Base‘𝑃)))
264, 9, 23, 25syl3anc 1366 . . 3 (𝜑 → (𝐴‘ran 𝑉) ⊆ (𝐴‘(Base‘𝑃)))
271, 5, 6, 2, 12mplsubrg 19488 . . . 4 (𝜑 → (Base‘𝑃) ∈ (SubRing‘𝑆))
281, 5, 6, 2, 12mpllss 19486 . . . 4 (𝜑 → (Base‘𝑃) ∈ (LSubSp‘𝑆))
29 eqid 2651 . . . . 5 (LSubSp‘𝑆) = (LSubSp‘𝑆)
3024, 7, 29aspid 19378 . . . 4 ((𝑆 ∈ AssAlg ∧ (Base‘𝑃) ∈ (SubRing‘𝑆) ∧ (Base‘𝑃) ∈ (LSubSp‘𝑆)) → (𝐴‘(Base‘𝑃)) = (Base‘𝑃))
314, 27, 28, 30syl3anc 1366 . . 3 (𝜑 → (𝐴‘(Base‘𝑃)) = (Base‘𝑃))
3226, 31sseqtrd 3674 . 2 (𝜑 → (𝐴‘ran 𝑉) ⊆ (Base‘𝑃))
33 eqid 2651 . . . . . 6 {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}
34 eqid 2651 . . . . . 6 (0g𝑅) = (0g𝑅)
35 eqid 2651 . . . . . 6 (1r𝑅) = (1r𝑅)
362adantr 480 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝑃)) → 𝐼𝑊)
37 eqid 2651 . . . . . 6 ( ·𝑠𝑃) = ( ·𝑠𝑃)
3812adantr 480 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝑃)) → 𝑅 ∈ Ring)
39 simpr 476 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝑃)) → 𝑥 ∈ (Base‘𝑃))
405, 33, 34, 35, 36, 6, 37, 38, 39mplcoe1 19513 . . . . 5 ((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)))))))
41 eqid 2651 . . . . . 6 (0g𝑃) = (0g𝑃)
425mplring 19500 . . . . . . . . 9 ((𝐼𝑊𝑅 ∈ Ring) → 𝑃 ∈ Ring)
432, 12, 42syl2anc 694 . . . . . . . 8 (𝜑𝑃 ∈ Ring)
44 ringabl 18626 . . . . . . . 8 (𝑃 ∈ Ring → 𝑃 ∈ Abel)
4543, 44syl 17 . . . . . . 7 (𝜑𝑃 ∈ Abel)
4645adantr 480 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝑃)) → 𝑃 ∈ Abel)
47 ovex 6718 . . . . . . . 8 (ℕ0𝑚 𝐼) ∈ V
4847rabex 4845 . . . . . . 7 {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∈ V
4948a1i 11 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝑃)) → {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∈ V)
5023, 8syl6ss 3648 . . . . . . . . . 10 (𝜑 → ran 𝑉 ⊆ (Base‘𝑆))
5124, 7aspsubrg 19379 . . . . . . . . . 10 ((𝑆 ∈ AssAlg ∧ ran 𝑉 ⊆ (Base‘𝑆)) → (𝐴‘ran 𝑉) ∈ (SubRing‘𝑆))
524, 50, 51syl2anc 694 . . . . . . . . 9 (𝜑 → (𝐴‘ran 𝑉) ∈ (SubRing‘𝑆))
535, 1, 6mplval2 19479 . . . . . . . . . . 11 𝑃 = (𝑆s (Base‘𝑃))
5453subsubrg 18854 . . . . . . . . . 10 ((Base‘𝑃) ∈ (SubRing‘𝑆) → ((𝐴‘ran 𝑉) ∈ (SubRing‘𝑃) ↔ ((𝐴‘ran 𝑉) ∈ (SubRing‘𝑆) ∧ (𝐴‘ran 𝑉) ⊆ (Base‘𝑃))))
5527, 54syl 17 . . . . . . . . 9 (𝜑 → ((𝐴‘ran 𝑉) ∈ (SubRing‘𝑃) ↔ ((𝐴‘ran 𝑉) ∈ (SubRing‘𝑆) ∧ (𝐴‘ran 𝑉) ⊆ (Base‘𝑃))))
5652, 32, 55mpbir2and 977 . . . . . . . 8 (𝜑 → (𝐴‘ran 𝑉) ∈ (SubRing‘𝑃))
57 subrgsubg 18834 . . . . . . . 8 ((𝐴‘ran 𝑉) ∈ (SubRing‘𝑃) → (𝐴‘ran 𝑉) ∈ (SubGrp‘𝑃))
5856, 57syl 17 . . . . . . 7 (𝜑 → (𝐴‘ran 𝑉) ∈ (SubGrp‘𝑃))
5958adantr 480 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝑃)) → (𝐴‘ran 𝑉) ∈ (SubGrp‘𝑃))
605mpllmod 19499 . . . . . . . . . 10 ((𝐼𝑊𝑅 ∈ Ring) → 𝑃 ∈ LMod)
612, 12, 60syl2anc 694 . . . . . . . . 9 (𝜑𝑃 ∈ LMod)
6261ad2antrr 762 . . . . . . . 8 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → 𝑃 ∈ LMod)
6324, 7, 29asplss 19377 . . . . . . . . . . 11 ((𝑆 ∈ AssAlg ∧ ran 𝑉 ⊆ (Base‘𝑆)) → (𝐴‘ran 𝑉) ∈ (LSubSp‘𝑆))
644, 50, 63syl2anc 694 . . . . . . . . . 10 (𝜑 → (𝐴‘ran 𝑉) ∈ (LSubSp‘𝑆))
651, 2, 12psrlmod 19449 . . . . . . . . . . 11 (𝜑𝑆 ∈ LMod)
66 eqid 2651 . . . . . . . . . . . 12 (LSubSp‘𝑃) = (LSubSp‘𝑃)
6753, 29, 66lsslss 19009 . . . . . . . . . . 11 ((𝑆 ∈ LMod ∧ (Base‘𝑃) ∈ (LSubSp‘𝑆)) → ((𝐴‘ran 𝑉) ∈ (LSubSp‘𝑃) ↔ ((𝐴‘ran 𝑉) ∈ (LSubSp‘𝑆) ∧ (𝐴‘ran 𝑉) ⊆ (Base‘𝑃))))
6865, 28, 67syl2anc 694 . . . . . . . . . 10 (𝜑 → ((𝐴‘ran 𝑉) ∈ (LSubSp‘𝑃) ↔ ((𝐴‘ran 𝑉) ∈ (LSubSp‘𝑆) ∧ (𝐴‘ran 𝑉) ⊆ (Base‘𝑃))))
6964, 32, 68mpbir2and 977 . . . . . . . . 9 (𝜑 → (𝐴‘ran 𝑉) ∈ (LSubSp‘𝑃))
7069ad2antrr 762 . . . . . . . 8 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝐴‘ran 𝑉) ∈ (LSubSp‘𝑃))
71 eqid 2651 . . . . . . . . . . 11 (Base‘𝑅) = (Base‘𝑅)
725, 71, 6, 33, 39mplelf 19481 . . . . . . . . . 10 ((𝜑𝑥 ∈ (Base‘𝑃)) → 𝑥:{𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
7372ffvelrnda 6399 . . . . . . . . 9 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑥𝑘) ∈ (Base‘𝑅))
745, 36, 38mplsca 19493 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (Base‘𝑃)) → 𝑅 = (Scalar‘𝑃))
7574adantr 480 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → 𝑅 = (Scalar‘𝑃))
7675fveq2d 6233 . . . . . . . . 9 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → (Base‘𝑅) = (Base‘(Scalar‘𝑃)))
7773, 76eleqtrd 2732 . . . . . . . 8 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑥𝑘) ∈ (Base‘(Scalar‘𝑃)))
782ad2antrr 762 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → 𝐼𝑊)
79 eqid 2651 . . . . . . . . . 10 (mulGrp‘𝑃) = (mulGrp‘𝑃)
80 eqid 2651 . . . . . . . . . 10 (.g‘(mulGrp‘𝑃)) = (.g‘(mulGrp‘𝑃))
813ad2antrr 762 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → 𝑅 ∈ CRing)
82 simpr 476 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin})
835, 33, 34, 35, 78, 79, 80, 10, 81, 82mplcoe2 19517 . . . . . . . . 9 (((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)))))
84 eqid 2651 . . . . . . . . . . 11 (1r𝑃) = (1r𝑃)
8579, 84ringidval 18549 . . . . . . . . . 10 (1r𝑃) = (0g‘(mulGrp‘𝑃))
865mplcrng 19501 . . . . . . . . . . . . 13 ((𝐼𝑊𝑅 ∈ CRing) → 𝑃 ∈ CRing)
872, 3, 86syl2anc 694 . . . . . . . . . . . 12 (𝜑𝑃 ∈ CRing)
8879crngmgp 18601 . . . . . . . . . . . 12 (𝑃 ∈ CRing → (mulGrp‘𝑃) ∈ CMnd)
8987, 88syl 17 . . . . . . . . . . 11 (𝜑 → (mulGrp‘𝑃) ∈ CMnd)
9089ad2antrr 762 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → (mulGrp‘𝑃) ∈ CMnd)
9156ad2antrr 762 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝐴‘ran 𝑉) ∈ (SubRing‘𝑃))
9279subrgsubm 18841 . . . . . . . . . . 11 ((𝐴‘ran 𝑉) ∈ (SubRing‘𝑃) → (𝐴‘ran 𝑉) ∈ (SubMnd‘(mulGrp‘𝑃)))
9391, 92syl 17 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝐴‘ran 𝑉) ∈ (SubMnd‘(mulGrp‘𝑃)))
94 simplll 813 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧𝐼) → 𝜑)
9533psrbag 19412 . . . . . . . . . . . . . . . 16 (𝐼𝑊 → (𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ↔ (𝑘:𝐼⟶ℕ0 ∧ (𝑘 “ ℕ) ∈ Fin)))
9636, 95syl 17 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ (Base‘𝑃)) → (𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ↔ (𝑘:𝐼⟶ℕ0 ∧ (𝑘 “ ℕ) ∈ Fin)))
9796biimpa 500 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑘:𝐼⟶ℕ0 ∧ (𝑘 “ ℕ) ∈ Fin))
9897simpld 474 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → 𝑘:𝐼⟶ℕ0)
9998ffvelrnda 6399 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧𝐼) → (𝑘𝑧) ∈ ℕ0)
10024, 7aspssid 19381 . . . . . . . . . . . . . . 15 ((𝑆 ∈ AssAlg ∧ ran 𝑉 ⊆ (Base‘𝑆)) → ran 𝑉 ⊆ (𝐴‘ran 𝑉))
1014, 50, 100syl2anc 694 . . . . . . . . . . . . . 14 (𝜑 → ran 𝑉 ⊆ (𝐴‘ran 𝑉))
102101ad3antrrr 766 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧𝐼) → ran 𝑉 ⊆ (𝐴‘ran 𝑉))
10314ad2antrr 762 . . . . . . . . . . . . . 14 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → 𝑉 Fn 𝐼)
104 fnfvelrn 6396 . . . . . . . . . . . . . 14 ((𝑉 Fn 𝐼𝑧𝐼) → (𝑉𝑧) ∈ ran 𝑉)
105103, 104sylan 487 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧𝐼) → (𝑉𝑧) ∈ ran 𝑉)
106102, 105sseldd 3637 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧𝐼) → (𝑉𝑧) ∈ (𝐴‘ran 𝑉))
10779, 6mgpbas 18541 . . . . . . . . . . . . 13 (Base‘𝑃) = (Base‘(mulGrp‘𝑃))
108 eqid 2651 . . . . . . . . . . . . . 14 (.r𝑃) = (.r𝑃)
10979, 108mgpplusg 18539 . . . . . . . . . . . . 13 (.r𝑃) = (+g‘(mulGrp‘𝑃))
110108subrgmcl 18840 . . . . . . . . . . . . . 14 (((𝐴‘ran 𝑉) ∈ (SubRing‘𝑃) ∧ 𝑢 ∈ (𝐴‘ran 𝑉) ∧ 𝑣 ∈ (𝐴‘ran 𝑉)) → (𝑢(.r𝑃)𝑣) ∈ (𝐴‘ran 𝑉))
11156, 110syl3an1 1399 . . . . . . . . . . . . 13 ((𝜑𝑢 ∈ (𝐴‘ran 𝑉) ∧ 𝑣 ∈ (𝐴‘ran 𝑉)) → (𝑢(.r𝑃)𝑣) ∈ (𝐴‘ran 𝑉))
11284subrg1cl 18836 . . . . . . . . . . . . . 14 ((𝐴‘ran 𝑉) ∈ (SubRing‘𝑃) → (1r𝑃) ∈ (𝐴‘ran 𝑉))
11356, 112syl 17 . . . . . . . . . . . . 13 (𝜑 → (1r𝑃) ∈ (𝐴‘ran 𝑉))
114107, 80, 109, 89, 32, 111, 85, 113mulgnn0subcl 17601 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑘𝑧) ∈ ℕ0 ∧ (𝑉𝑧) ∈ (𝐴‘ran 𝑉)) → ((𝑘𝑧)(.g‘(mulGrp‘𝑃))(𝑉𝑧)) ∈ (𝐴‘ran 𝑉))
11594, 99, 106, 114syl3anc 1366 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧𝐼) → ((𝑘𝑧)(.g‘(mulGrp‘𝑃))(𝑉𝑧)) ∈ (𝐴‘ran 𝑉))
116 eqid 2651 . . . . . . . . . . 11 (𝑧𝐼 ↦ ((𝑘𝑧)(.g‘(mulGrp‘𝑃))(𝑉𝑧))) = (𝑧𝐼 ↦ ((𝑘𝑧)(.g‘(mulGrp‘𝑃))(𝑉𝑧)))
117115, 116fmptd 6425 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑧𝐼 ↦ ((𝑘𝑧)(.g‘(mulGrp‘𝑃))(𝑉𝑧))):𝐼⟶(𝐴‘ran 𝑉))
118 mptexg 6525 . . . . . . . . . . . . 13 (𝐼𝑊 → (𝑧𝐼 ↦ ((𝑘𝑧)(.g‘(mulGrp‘𝑃))(𝑉𝑧))) ∈ V)
1192, 118syl 17 . . . . . . . . . . . 12 (𝜑 → (𝑧𝐼 ↦ ((𝑘𝑧)(.g‘(mulGrp‘𝑃))(𝑉𝑧))) ∈ V)
120119ad2antrr 762 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑧𝐼 ↦ ((𝑘𝑧)(.g‘(mulGrp‘𝑃))(𝑉𝑧))) ∈ V)
121 funmpt 5964 . . . . . . . . . . . 12 Fun (𝑧𝐼 ↦ ((𝑘𝑧)(.g‘(mulGrp‘𝑃))(𝑉𝑧)))
122121a1i 11 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → Fun (𝑧𝐼 ↦ ((𝑘𝑧)(.g‘(mulGrp‘𝑃))(𝑉𝑧))))
123 fvexd 6241 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → (1r𝑃) ∈ V)
12497simprd 478 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑘 “ ℕ) ∈ Fin)
125 elrabi 3391 . . . . . . . . . . . . . . 15 (𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} → 𝑘 ∈ (ℕ0𝑚 𝐼))
126 elmapi 7921 . . . . . . . . . . . . . . . . 17 (𝑘 ∈ (ℕ0𝑚 𝐼) → 𝑘:𝐼⟶ℕ0)
127126adantl 481 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ (ℕ0𝑚 𝐼)) → 𝑘:𝐼⟶ℕ0)
1282ad2antrr 762 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ (ℕ0𝑚 𝐼)) → 𝐼𝑊)
129 frnnn0supp 11387 . . . . . . . . . . . . . . . . . 18 ((𝐼𝑊𝑘:𝐼⟶ℕ0) → (𝑘 supp 0) = (𝑘 “ ℕ))
130128, 127, 129syl2anc 694 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ (ℕ0𝑚 𝐼)) → (𝑘 supp 0) = (𝑘 “ ℕ))
131 eqimss 3690 . . . . . . . . . . . . . . . . 17 ((𝑘 supp 0) = (𝑘 “ ℕ) → (𝑘 supp 0) ⊆ (𝑘 “ ℕ))
132130, 131syl 17 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ (ℕ0𝑚 𝐼)) → (𝑘 supp 0) ⊆ (𝑘 “ ℕ))
133 c0ex 10072 . . . . . . . . . . . . . . . . 17 0 ∈ V
134133a1i 11 . . . . . . . . . . . . . . . 16 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ (ℕ0𝑚 𝐼)) → 0 ∈ V)
135127, 132, 128, 134suppssr 7371 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ (ℕ0𝑚 𝐼)) ∧ 𝑧 ∈ (𝐼 ∖ (𝑘 “ ℕ))) → (𝑘𝑧) = 0)
136125, 135sylanl2 684 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧 ∈ (𝐼 ∖ (𝑘 “ ℕ))) → (𝑘𝑧) = 0)
137136oveq1d 6705 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧 ∈ (𝐼 ∖ (𝑘 “ ℕ))) → ((𝑘𝑧)(.g‘(mulGrp‘𝑃))(𝑉𝑧)) = (0(.g‘(mulGrp‘𝑃))(𝑉𝑧)))
1382ad3antrrr 766 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧 ∈ (𝐼 ∖ (𝑘 “ ℕ))) → 𝐼𝑊)
13912ad3antrrr 766 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧 ∈ (𝐼 ∖ (𝑘 “ ℕ))) → 𝑅 ∈ Ring)
140 eldifi 3765 . . . . . . . . . . . . . . . 16 (𝑧 ∈ (𝐼 ∖ (𝑘 “ ℕ)) → 𝑧𝐼)
141140adantl 481 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧 ∈ (𝐼 ∖ (𝑘 “ ℕ))) → 𝑧𝐼)
1425, 10, 6, 138, 139, 141mvrcl 19497 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧 ∈ (𝐼 ∖ (𝑘 “ ℕ))) → (𝑉𝑧) ∈ (Base‘𝑃))
143107, 85, 80mulg0 17593 . . . . . . . . . . . . . 14 ((𝑉𝑧) ∈ (Base‘𝑃) → (0(.g‘(mulGrp‘𝑃))(𝑉𝑧)) = (1r𝑃))
144142, 143syl 17 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧 ∈ (𝐼 ∖ (𝑘 “ ℕ))) → (0(.g‘(mulGrp‘𝑃))(𝑉𝑧)) = (1r𝑃))
145137, 144eqtrd 2685 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) ∧ 𝑧 ∈ (𝐼 ∖ (𝑘 “ ℕ))) → ((𝑘𝑧)(.g‘(mulGrp‘𝑃))(𝑉𝑧)) = (1r𝑃))
146145, 78suppss2 7374 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → ((𝑧𝐼 ↦ ((𝑘𝑧)(.g‘(mulGrp‘𝑃))(𝑉𝑧))) supp (1r𝑃)) ⊆ (𝑘 “ ℕ))
147 suppssfifsupp 8331 . . . . . . . . . . 11 ((((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))
148120, 122, 123, 124, 146, 147syl32anc 1374 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑧𝐼 ↦ ((𝑘𝑧)(.g‘(mulGrp‘𝑃))(𝑉𝑧))) finSupp (1r𝑃))
14985, 90, 78, 93, 117, 148gsumsubmcl 18365 . . . . . . . . 9 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → ((mulGrp‘𝑃) Σg (𝑧𝐼 ↦ ((𝑘𝑧)(.g‘(mulGrp‘𝑃))(𝑉𝑧)))) ∈ (𝐴‘ran 𝑉))
15083, 149eqeltrd 2730 . . . . . . . 8 (((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))
151 eqid 2651 . . . . . . . . 9 (Scalar‘𝑃) = (Scalar‘𝑃)
152 eqid 2651 . . . . . . . . 9 (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
153151, 37, 152, 66lssvscl 19003 . . . . . . . 8 (((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))
15462, 70, 77, 150, 153syl22anc 1367 . . . . . . 7 (((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))
155 eqid 2651 . . . . . . 7 (𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ↦ ((𝑥𝑘)( ·𝑠𝑃)(𝑦 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = 𝑘, (1r𝑅), (0g𝑅))))) = (𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ↦ ((𝑥𝑘)( ·𝑠𝑃)(𝑦 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = 𝑘, (1r𝑅), (0g𝑅)))))
156154, 155fmptd 6425 . . . . . 6 ((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))
15747mptrabex 6529 . . . . . . . . 9 (𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ↦ ((𝑥𝑘)( ·𝑠𝑃)(𝑦 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = 𝑘, (1r𝑅), (0g𝑅))))) ∈ V
158 funmpt 5964 . . . . . . . . 9 Fun (𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ↦ ((𝑥𝑘)( ·𝑠𝑃)(𝑦 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = 𝑘, (1r𝑅), (0g𝑅)))))
159 fvex 6239 . . . . . . . . 9 (0g𝑃) ∈ V
160157, 158, 1593pm3.2i 1259 . . . . . . . 8 ((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)
161160a1i 11 . . . . . . 7 ((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))
1625, 1, 7, 34, 6mplelbas 19478 . . . . . . . . . 10 (𝑥 ∈ (Base‘𝑃) ↔ (𝑥 ∈ (Base‘𝑆) ∧ 𝑥 finSupp (0g𝑅)))
163162simprbi 479 . . . . . . . . 9 (𝑥 ∈ (Base‘𝑃) → 𝑥 finSupp (0g𝑅))
164163adantl 481 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝑃)) → 𝑥 finSupp (0g𝑅))
165164fsuppimpd 8323 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝑃)) → (𝑥 supp (0g𝑅)) ∈ Fin)
166 ssid 3657 . . . . . . . . . . . . 13 (𝑥 supp (0g𝑅)) ⊆ (𝑥 supp (0g𝑅))
167166a1i 11 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (Base‘𝑃)) → (𝑥 supp (0g𝑅)) ⊆ (𝑥 supp (0g𝑅)))
168 fvexd 6241 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (Base‘𝑃)) → (0g𝑅) ∈ V)
16972, 167, 49, 168suppssr 7371 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ ({𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∖ (𝑥 supp (0g𝑅)))) → (𝑥𝑘) = (0g𝑅))
17074fveq2d 6233 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (Base‘𝑃)) → (0g𝑅) = (0g‘(Scalar‘𝑃)))
171170adantr 480 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ ({𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∖ (𝑥 supp (0g𝑅)))) → (0g𝑅) = (0g‘(Scalar‘𝑃)))
172169, 171eqtrd 2685 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ ({𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∖ (𝑥 supp (0g𝑅)))) → (𝑥𝑘) = (0g‘(Scalar‘𝑃)))
173172oveq1d 6705 . . . . . . . . 9 (((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)))))
174 eldifi 3765 . . . . . . . . . 10 (𝑘 ∈ ({𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ∖ (𝑥 supp (0g𝑅))) → 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin})
17512ad2antrr 762 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → 𝑅 ∈ Ring)
1765, 6, 34, 35, 33, 78, 175, 82mplmon 19511 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑃)) ∧ 𝑘 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin}) → (𝑦 ∈ {𝑓 ∈ (ℕ0𝑚 𝐼) ∣ (𝑓 “ ℕ) ∈ Fin} ↦ if(𝑦 = 𝑘, (1r𝑅), (0g𝑅))) ∈ (Base‘𝑃))
177 eqid 2651 . . . . . . . . . . . 12 (0g‘(Scalar‘𝑃)) = (0g‘(Scalar‘𝑃))
1786, 151, 37, 177, 41lmod0vs 18944 . . . . . . . . . . 11 ((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))
17962, 176, 178syl2anc 694 . . . . . . . . . 10 (((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))
180174, 179sylan2 490 . . . . . . . . 9 (((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))
181173, 180eqtrd 2685 . . . . . . . 8 (((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))
182181, 49suppss2 7374 . . . . . . 7 ((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)))
183 suppssfifsupp 8331 . . . . . . 7 ((((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))
184161, 165, 182, 183syl12anc 1364 . . . . . 6 ((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))
18541, 46, 49, 59, 156, 184gsumsubgcl 18366 . . . . 5 ((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))
18640, 185eqeltrd 2730 . . . 4 ((𝜑𝑥 ∈ (Base‘𝑃)) → 𝑥 ∈ (𝐴‘ran 𝑉))
187186ex 449 . . 3 (𝜑 → (𝑥 ∈ (Base‘𝑃) → 𝑥 ∈ (𝐴‘ran 𝑉)))
188187ssrdv 3642 . 2 (𝜑 → (Base‘𝑃) ⊆ (𝐴‘ran 𝑉))
18932, 188eqssd 3653 1 (𝜑 → (𝐴‘ran 𝑉) = (Base‘𝑃))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  {crab 2945  Vcvv 3231  cdif 3604  wss 3607  ifcif 4119   class class class wbr 4685  cmpt 4762  ccnv 5142  ran crn 5144  cima 5146  Fun wfun 5920   Fn wfn 5921  wf 5922  cfv 5926  (class class class)co 6690   supp csupp 7340  𝑚 cmap 7899  Fincfn 7997   finSupp cfsupp 8316  0cc0 9974  cn 11058  0cn0 11330  Basecbs 15904  .rcmulr 15989  Scalarcsca 15991   ·𝑠 cvsca 15992  0gc0g 16147   Σg cgsu 16148  SubMndcsubmnd 17381  .gcmg 17587  SubGrpcsubg 17635  CMndccmn 18239  Abelcabl 18240  mulGrpcmgp 18535  1rcur 18547  Ringcrg 18593  CRingccrg 18594  SubRingcsubrg 18824  LModclmod 18911  LSubSpclss 18980  AssAlgcasa 19357  AlgSpancasp 19358   mPwSer cmps 19399   mVar cmvr 19400   mPoly cmpl 19401
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-ofr 6940  df-om 7108  df-1st 7210  df-2nd 7211  df-supp 7341  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-ixp 7951  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-fsupp 8317  df-oi 8456  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-7 11122  df-8 11123  df-9 11124  df-n0 11331  df-z 11416  df-uz 11726  df-fz 12365  df-fzo 12505  df-seq 12842  df-hash 13158  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-ress 15912  df-plusg 16001  df-mulr 16002  df-sca 16004  df-vsca 16005  df-tset 16007  df-0g 16149  df-gsum 16150  df-mre 16293  df-mrc 16294  df-acs 16296  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-mhm 17382  df-submnd 17383  df-grp 17472  df-minusg 17473  df-sbg 17474  df-mulg 17588  df-subg 17638  df-ghm 17705  df-cntz 17796  df-cmn 18241  df-abl 18242  df-mgp 18536  df-ur 18548  df-srg 18552  df-ring 18595  df-cring 18596  df-subrg 18826  df-lmod 18913  df-lss 18981  df-assa 19360  df-asp 19361  df-psr 19404  df-mvr 19405  df-mpl 19406
This theorem is referenced by:  mplind  19550  evlseu  19564
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