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Theorem fedgmullem2 33682
Description: Lemma for fedgmul 33683. (Contributed by Thierry Arnoux, 20-Jul-2023.)
Hypotheses
Ref Expression
fedgmul.a 𝐴 = ((subringAlg ‘𝐸)‘𝑉)
fedgmul.b 𝐵 = ((subringAlg ‘𝐸)‘𝑈)
fedgmul.c 𝐶 = ((subringAlg ‘𝐹)‘𝑉)
fedgmul.f 𝐹 = (𝐸s 𝑈)
fedgmul.k 𝐾 = (𝐸s 𝑉)
fedgmul.1 (𝜑𝐸 ∈ DivRing)
fedgmul.2 (𝜑𝐹 ∈ DivRing)
fedgmul.3 (𝜑𝐾 ∈ DivRing)
fedgmul.4 (𝜑𝑈 ∈ (SubRing‘𝐸))
fedgmul.5 (𝜑𝑉 ∈ (SubRing‘𝐹))
fedgmullem.d 𝐷 = (𝑗𝑌, 𝑖𝑋 ↦ (𝑖(.r𝐸)𝑗))
fedgmullem.h 𝐻 = (𝑗𝑌, 𝑖𝑋 ↦ ((𝐺𝑗)‘𝑖))
fedgmullem.x (𝜑𝑋 ∈ (LBasis‘𝐶))
fedgmullem.y (𝜑𝑌 ∈ (LBasis‘𝐵))
fedgmullem2.1 (𝜑𝑊 ∈ (Base‘((Scalar‘𝐴) freeLMod (𝑌 × 𝑋))))
fedgmullem2.2 (𝜑 → (𝐴 Σg (𝑊f ( ·𝑠𝐴)𝐷)) = (0g𝐴))
Assertion
Ref Expression
fedgmullem2 (𝜑𝑊 = ((𝑌 × 𝑋) × {(0g‘(Scalar‘𝐴))}))
Distinct variable groups:   𝐴,𝑖,𝑗   𝜑,𝑖,𝑗   𝑖,𝐸,𝑗   𝐷,𝑖,𝑗   𝐶,𝑖   𝑗,𝑊,𝑖   𝑖,𝑌,𝑗   𝑖,𝑋,𝑗   𝐵,𝑖,𝑗   𝑈,𝑖
Allowed substitution hints:   𝐶(𝑗)   𝑈(𝑗)   𝐹(𝑖,𝑗)   𝐺(𝑖,𝑗)   𝐻(𝑖,𝑗)   𝐾(𝑖,𝑗)   𝑉(𝑖,𝑗)

Proof of Theorem fedgmullem2
Dummy variables 𝑏 𝑘 𝑙 𝑥 𝑦 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fedgmul.1 . . . . . . . . . . 11 (𝜑𝐸 ∈ DivRing)
2 fedgmul.3 . . . . . . . . . . 11 (𝜑𝐾 ∈ DivRing)
3 fedgmul.4 . . . . . . . . . . . . 13 (𝜑𝑈 ∈ (SubRing‘𝐸))
4 fedgmul.5 . . . . . . . . . . . . 13 (𝜑𝑉 ∈ (SubRing‘𝐹))
5 fedgmul.f . . . . . . . . . . . . . . 15 𝐹 = (𝐸s 𝑈)
65subsubrg 20599 . . . . . . . . . . . . . 14 (𝑈 ∈ (SubRing‘𝐸) → (𝑉 ∈ (SubRing‘𝐹) ↔ (𝑉 ∈ (SubRing‘𝐸) ∧ 𝑉𝑈)))
76biimpa 476 . . . . . . . . . . . . 13 ((𝑈 ∈ (SubRing‘𝐸) ∧ 𝑉 ∈ (SubRing‘𝐹)) → (𝑉 ∈ (SubRing‘𝐸) ∧ 𝑉𝑈))
83, 4, 7syl2anc 584 . . . . . . . . . . . 12 (𝜑 → (𝑉 ∈ (SubRing‘𝐸) ∧ 𝑉𝑈))
98simpld 494 . . . . . . . . . . 11 (𝜑𝑉 ∈ (SubRing‘𝐸))
10 fedgmul.a . . . . . . . . . . . 12 𝐴 = ((subringAlg ‘𝐸)‘𝑉)
11 fedgmul.k . . . . . . . . . . . 12 𝐾 = (𝐸s 𝑉)
1210, 11sralvec 33637 . . . . . . . . . . 11 ((𝐸 ∈ DivRing ∧ 𝐾 ∈ DivRing ∧ 𝑉 ∈ (SubRing‘𝐸)) → 𝐴 ∈ LVec)
131, 2, 9, 12syl3anc 1372 . . . . . . . . . 10 (𝜑𝐴 ∈ LVec)
14 lveclmod 21106 . . . . . . . . . 10 (𝐴 ∈ LVec → 𝐴 ∈ LMod)
1513, 14syl 17 . . . . . . . . 9 (𝜑𝐴 ∈ LMod)
16 fedgmullem.x . . . . . . . . . . 11 (𝜑𝑋 ∈ (LBasis‘𝐶))
17 eqid 2736 . . . . . . . . . . . 12 (Base‘𝐶) = (Base‘𝐶)
18 eqid 2736 . . . . . . . . . . . 12 (LBasis‘𝐶) = (LBasis‘𝐶)
1917, 18lbsss 21077 . . . . . . . . . . 11 (𝑋 ∈ (LBasis‘𝐶) → 𝑋 ⊆ (Base‘𝐶))
2016, 19syl 17 . . . . . . . . . 10 (𝜑𝑋 ⊆ (Base‘𝐶))
21 eqid 2736 . . . . . . . . . . . . . . . 16 (Base‘𝐸) = (Base‘𝐸)
2221subrgss 20573 . . . . . . . . . . . . . . 15 (𝑈 ∈ (SubRing‘𝐸) → 𝑈 ⊆ (Base‘𝐸))
233, 22syl 17 . . . . . . . . . . . . . 14 (𝜑𝑈 ⊆ (Base‘𝐸))
245, 21ressbas2 17284 . . . . . . . . . . . . . 14 (𝑈 ⊆ (Base‘𝐸) → 𝑈 = (Base‘𝐹))
2523, 24syl 17 . . . . . . . . . . . . 13 (𝜑𝑈 = (Base‘𝐹))
26 fedgmul.c . . . . . . . . . . . . . . 15 𝐶 = ((subringAlg ‘𝐹)‘𝑉)
2726a1i 11 . . . . . . . . . . . . . 14 (𝜑𝐶 = ((subringAlg ‘𝐹)‘𝑉))
28 eqid 2736 . . . . . . . . . . . . . . . 16 (Base‘𝐹) = (Base‘𝐹)
2928subrgss 20573 . . . . . . . . . . . . . . 15 (𝑉 ∈ (SubRing‘𝐹) → 𝑉 ⊆ (Base‘𝐹))
304, 29syl 17 . . . . . . . . . . . . . 14 (𝜑𝑉 ⊆ (Base‘𝐹))
3127, 30srabase 21178 . . . . . . . . . . . . 13 (𝜑 → (Base‘𝐹) = (Base‘𝐶))
3225, 31eqtrd 2776 . . . . . . . . . . . 12 (𝜑𝑈 = (Base‘𝐶))
3332, 23eqsstrrd 4018 . . . . . . . . . . 11 (𝜑 → (Base‘𝐶) ⊆ (Base‘𝐸))
3410a1i 11 . . . . . . . . . . . 12 (𝜑𝐴 = ((subringAlg ‘𝐸)‘𝑉))
3521subrgss 20573 . . . . . . . . . . . . 13 (𝑉 ∈ (SubRing‘𝐸) → 𝑉 ⊆ (Base‘𝐸))
369, 35syl 17 . . . . . . . . . . . 12 (𝜑𝑉 ⊆ (Base‘𝐸))
3734, 36srabase 21178 . . . . . . . . . . 11 (𝜑 → (Base‘𝐸) = (Base‘𝐴))
3833, 37sseqtrd 4019 . . . . . . . . . 10 (𝜑 → (Base‘𝐶) ⊆ (Base‘𝐴))
3920, 38sstrd 3993 . . . . . . . . 9 (𝜑𝑋 ⊆ (Base‘𝐴))
4034, 3, 36srasubrg 33636 . . . . . . . . . . . 12 (𝜑𝑈 ∈ (SubRing‘𝐴))
41 subrgsubg 20578 . . . . . . . . . . . 12 (𝑈 ∈ (SubRing‘𝐴) → 𝑈 ∈ (SubGrp‘𝐴))
4240, 41syl 17 . . . . . . . . . . 11 (𝜑𝑈 ∈ (SubGrp‘𝐴))
4310, 1, 9drgextvsca 33642 . . . . . . . . . . . . . 14 (𝜑 → (.r𝐸) = ( ·𝑠𝐴))
4443oveqdr 7460 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦𝑈)) → (𝑥(.r𝐸)𝑦) = (𝑥( ·𝑠𝐴)𝑦))
453adantr 480 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦𝑈)) → 𝑈 ∈ (SubRing‘𝐸))
468simprd 495 . . . . . . . . . . . . . . . 16 (𝜑𝑉𝑈)
4746adantr 480 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦𝑈)) → 𝑉𝑈)
48 simprl 770 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦𝑈)) → 𝑥 ∈ (Base‘(Scalar‘𝐴)))
49 ressabs 17295 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑈 ∈ (SubRing‘𝐸) ∧ 𝑉𝑈) → ((𝐸s 𝑈) ↾s 𝑉) = (𝐸s 𝑉))
503, 46, 49syl2anc 584 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → ((𝐸s 𝑈) ↾s 𝑉) = (𝐸s 𝑉))
515oveq1i 7442 . . . . . . . . . . . . . . . . . . . . 21 (𝐹s 𝑉) = ((𝐸s 𝑈) ↾s 𝑉)
5250, 51, 113eqtr4g 2801 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝐹s 𝑉) = 𝐾)
5327, 30srasca 21184 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝐹s 𝑉) = (Scalar‘𝐶))
5452, 53eqtr3d 2778 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐾 = (Scalar‘𝐶))
5554fveq2d 6909 . . . . . . . . . . . . . . . . . 18 (𝜑 → (Base‘𝐾) = (Base‘(Scalar‘𝐶)))
5611, 21ressbas2 17284 . . . . . . . . . . . . . . . . . . 19 (𝑉 ⊆ (Base‘𝐸) → 𝑉 = (Base‘𝐾))
5736, 56syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑𝑉 = (Base‘𝐾))
5834, 36srasca 21184 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝐸s 𝑉) = (Scalar‘𝐴))
5911, 58eqtrid 2788 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐾 = (Scalar‘𝐴))
6052, 53, 593eqtr3rd 2785 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (Scalar‘𝐴) = (Scalar‘𝐶))
6160fveq2d 6909 . . . . . . . . . . . . . . . . . 18 (𝜑 → (Base‘(Scalar‘𝐴)) = (Base‘(Scalar‘𝐶)))
6255, 57, 613eqtr4d 2786 . . . . . . . . . . . . . . . . 17 (𝜑𝑉 = (Base‘(Scalar‘𝐴)))
6362adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦𝑈)) → 𝑉 = (Base‘(Scalar‘𝐴)))
6448, 63eleqtrrd 2843 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦𝑈)) → 𝑥𝑉)
6547, 64sseldd 3983 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦𝑈)) → 𝑥𝑈)
66 simprr 772 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦𝑈)) → 𝑦𝑈)
67 eqid 2736 . . . . . . . . . . . . . . 15 (.r𝐸) = (.r𝐸)
6867subrgmcl 20585 . . . . . . . . . . . . . 14 ((𝑈 ∈ (SubRing‘𝐸) ∧ 𝑥𝑈𝑦𝑈) → (𝑥(.r𝐸)𝑦) ∈ 𝑈)
6945, 65, 66, 68syl3anc 1372 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦𝑈)) → (𝑥(.r𝐸)𝑦) ∈ 𝑈)
7044, 69eqeltrrd 2841 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦𝑈)) → (𝑥( ·𝑠𝐴)𝑦) ∈ 𝑈)
7170ralrimivva 3201 . . . . . . . . . . 11 (𝜑 → ∀𝑥 ∈ (Base‘(Scalar‘𝐴))∀𝑦𝑈 (𝑥( ·𝑠𝐴)𝑦) ∈ 𝑈)
72 eqid 2736 . . . . . . . . . . . . 13 (Scalar‘𝐴) = (Scalar‘𝐴)
73 eqid 2736 . . . . . . . . . . . . 13 (Base‘(Scalar‘𝐴)) = (Base‘(Scalar‘𝐴))
74 eqid 2736 . . . . . . . . . . . . 13 (Base‘𝐴) = (Base‘𝐴)
75 eqid 2736 . . . . . . . . . . . . 13 ( ·𝑠𝐴) = ( ·𝑠𝐴)
76 eqid 2736 . . . . . . . . . . . . 13 (LSubSp‘𝐴) = (LSubSp‘𝐴)
7772, 73, 74, 75, 76islss4 20961 . . . . . . . . . . . 12 (𝐴 ∈ LMod → (𝑈 ∈ (LSubSp‘𝐴) ↔ (𝑈 ∈ (SubGrp‘𝐴) ∧ ∀𝑥 ∈ (Base‘(Scalar‘𝐴))∀𝑦𝑈 (𝑥( ·𝑠𝐴)𝑦) ∈ 𝑈)))
7877biimpar 477 . . . . . . . . . . 11 ((𝐴 ∈ LMod ∧ (𝑈 ∈ (SubGrp‘𝐴) ∧ ∀𝑥 ∈ (Base‘(Scalar‘𝐴))∀𝑦𝑈 (𝑥( ·𝑠𝐴)𝑦) ∈ 𝑈)) → 𝑈 ∈ (LSubSp‘𝐴))
7915, 42, 71, 78syl12anc 836 . . . . . . . . . 10 (𝜑𝑈 ∈ (LSubSp‘𝐴))
8020, 32sseqtrrd 4020 . . . . . . . . . 10 (𝜑𝑋𝑈)
8118lbslinds 21854 . . . . . . . . . . . 12 (LBasis‘𝐶) ⊆ (LIndS‘𝐶)
8281, 16sselid 3980 . . . . . . . . . . 11 (𝜑𝑋 ∈ (LIndS‘𝐶))
8323, 37sseqtrd 4019 . . . . . . . . . . . . . 14 (𝜑𝑈 ⊆ (Base‘𝐴))
84 eqid 2736 . . . . . . . . . . . . . . 15 (𝐴s 𝑈) = (𝐴s 𝑈)
8584, 74ressbas2 17284 . . . . . . . . . . . . . 14 (𝑈 ⊆ (Base‘𝐴) → 𝑈 = (Base‘(𝐴s 𝑈)))
8683, 85syl 17 . . . . . . . . . . . . 13 (𝜑𝑈 = (Base‘(𝐴s 𝑈)))
8725, 86, 313eqtr3rd 2785 . . . . . . . . . . . 12 (𝜑 → (Base‘𝐶) = (Base‘(𝐴s 𝑈)))
8884, 72resssca 17388 . . . . . . . . . . . . . . 15 (𝑈 ∈ (SubRing‘𝐸) → (Scalar‘𝐴) = (Scalar‘(𝐴s 𝑈)))
893, 88syl 17 . . . . . . . . . . . . . 14 (𝜑 → (Scalar‘𝐴) = (Scalar‘(𝐴s 𝑈)))
9060, 89eqtr3d 2778 . . . . . . . . . . . . 13 (𝜑 → (Scalar‘𝐶) = (Scalar‘(𝐴s 𝑈)))
9190fveq2d 6909 . . . . . . . . . . . 12 (𝜑 → (Base‘(Scalar‘𝐶)) = (Base‘(Scalar‘(𝐴s 𝑈))))
9290fveq2d 6909 . . . . . . . . . . . 12 (𝜑 → (0g‘(Scalar‘𝐶)) = (0g‘(Scalar‘(𝐴s 𝑈))))
93 eqid 2736 . . . . . . . . . . . . . . . . 17 (+g𝐸) = (+g𝐸)
945, 93ressplusg 17335 . . . . . . . . . . . . . . . 16 (𝑈 ∈ (SubRing‘𝐸) → (+g𝐸) = (+g𝐹))
953, 94syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (+g𝐸) = (+g𝐹))
9634, 36sraaddg 21180 . . . . . . . . . . . . . . 15 (𝜑 → (+g𝐸) = (+g𝐴))
9727, 30sraaddg 21180 . . . . . . . . . . . . . . 15 (𝜑 → (+g𝐹) = (+g𝐶))
9895, 96, 973eqtr3rd 2785 . . . . . . . . . . . . . 14 (𝜑 → (+g𝐶) = (+g𝐴))
99 eqid 2736 . . . . . . . . . . . . . . . 16 (+g𝐴) = (+g𝐴)
10084, 99ressplusg 17335 . . . . . . . . . . . . . . 15 (𝑈 ∈ (SubRing‘𝐸) → (+g𝐴) = (+g‘(𝐴s 𝑈)))
1013, 100syl 17 . . . . . . . . . . . . . 14 (𝜑 → (+g𝐴) = (+g‘(𝐴s 𝑈)))
10298, 101eqtrd 2776 . . . . . . . . . . . . 13 (𝜑 → (+g𝐶) = (+g‘(𝐴s 𝑈)))
103102oveqdr 7460 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(+g𝐶)𝑦) = (𝑥(+g‘(𝐴s 𝑈))𝑦))
104 fedgmul.2 . . . . . . . . . . . . . . 15 (𝜑𝐹 ∈ DivRing)
10552, 2eqeltrd 2840 . . . . . . . . . . . . . . 15 (𝜑 → (𝐹s 𝑉) ∈ DivRing)
106 eqid 2736 . . . . . . . . . . . . . . . 16 (𝐹s 𝑉) = (𝐹s 𝑉)
10726, 106sralvec 33637 . . . . . . . . . . . . . . 15 ((𝐹 ∈ DivRing ∧ (𝐹s 𝑉) ∈ DivRing ∧ 𝑉 ∈ (SubRing‘𝐹)) → 𝐶 ∈ LVec)
108104, 105, 4, 107syl3anc 1372 . . . . . . . . . . . . . 14 (𝜑𝐶 ∈ LVec)
109 lveclmod 21106 . . . . . . . . . . . . . 14 (𝐶 ∈ LVec → 𝐶 ∈ LMod)
110108, 109syl 17 . . . . . . . . . . . . 13 (𝜑𝐶 ∈ LMod)
111 eqid 2736 . . . . . . . . . . . . . . 15 (Scalar‘𝐶) = (Scalar‘𝐶)
112 eqid 2736 . . . . . . . . . . . . . . 15 ( ·𝑠𝐶) = ( ·𝑠𝐶)
113 eqid 2736 . . . . . . . . . . . . . . 15 (Base‘(Scalar‘𝐶)) = (Base‘(Scalar‘𝐶))
11417, 111, 112, 113lmodvscl 20877 . . . . . . . . . . . . . 14 ((𝐶 ∈ LMod ∧ 𝑥 ∈ (Base‘(Scalar‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) → (𝑥( ·𝑠𝐶)𝑦) ∈ (Base‘𝐶))
1151143expb 1120 . . . . . . . . . . . . 13 ((𝐶 ∈ LMod ∧ (𝑥 ∈ (Base‘(Scalar‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥( ·𝑠𝐶)𝑦) ∈ (Base‘𝐶))
116110, 115sylan 580 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥( ·𝑠𝐶)𝑦) ∈ (Base‘𝐶))
117 fedgmul.b . . . . . . . . . . . . . . . 16 𝐵 = ((subringAlg ‘𝐸)‘𝑈)
118117, 1, 3drgextvsca 33642 . . . . . . . . . . . . . . 15 (𝜑 → (.r𝐸) = ( ·𝑠𝐵))
11943, 118eqtr3d 2778 . . . . . . . . . . . . . 14 (𝜑 → ( ·𝑠𝐴) = ( ·𝑠𝐵))
12084, 75ressvsca 17389 . . . . . . . . . . . . . . 15 (𝑈 ∈ (SubRing‘𝐸) → ( ·𝑠𝐴) = ( ·𝑠 ‘(𝐴s 𝑈)))
1213, 120syl 17 . . . . . . . . . . . . . 14 (𝜑 → ( ·𝑠𝐴) = ( ·𝑠 ‘(𝐴s 𝑈)))
1225, 67ressmulr 17352 . . . . . . . . . . . . . . . 16 (𝑈 ∈ (SubRing‘𝐸) → (.r𝐸) = (.r𝐹))
1233, 122syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (.r𝐸) = (.r𝐹))
12426, 104, 4drgextvsca 33642 . . . . . . . . . . . . . . 15 (𝜑 → (.r𝐹) = ( ·𝑠𝐶))
125123, 118, 1243eqtr3d 2784 . . . . . . . . . . . . . 14 (𝜑 → ( ·𝑠𝐵) = ( ·𝑠𝐶))
126119, 121, 1253eqtr3rd 2785 . . . . . . . . . . . . 13 (𝜑 → ( ·𝑠𝐶) = ( ·𝑠 ‘(𝐴s 𝑈)))
127126oveqdr 7460 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥( ·𝑠𝐶)𝑦) = (𝑥( ·𝑠 ‘(𝐴s 𝑈))𝑦))
128 ovexd 7467 . . . . . . . . . . . 12 (𝜑 → (𝐴s 𝑈) ∈ V)
12987, 91, 92, 103, 116, 127, 108, 128lindspropd 33412 . . . . . . . . . . 11 (𝜑 → (LIndS‘𝐶) = (LIndS‘(𝐴s 𝑈)))
13082, 129eleqtrd 2842 . . . . . . . . . 10 (𝜑𝑋 ∈ (LIndS‘(𝐴s 𝑈)))
13176, 84lsslinds 21852 . . . . . . . . . . 11 ((𝐴 ∈ LMod ∧ 𝑈 ∈ (LSubSp‘𝐴) ∧ 𝑋𝑈) → (𝑋 ∈ (LIndS‘(𝐴s 𝑈)) ↔ 𝑋 ∈ (LIndS‘𝐴)))
132131biimpa 476 . . . . . . . . . 10 (((𝐴 ∈ LMod ∧ 𝑈 ∈ (LSubSp‘𝐴) ∧ 𝑋𝑈) ∧ 𝑋 ∈ (LIndS‘(𝐴s 𝑈))) → 𝑋 ∈ (LIndS‘𝐴))
13315, 79, 80, 130, 132syl31anc 1374 . . . . . . . . 9 (𝜑𝑋 ∈ (LIndS‘𝐴))
134 eqid 2736 . . . . . . . . . . 11 (0g𝐴) = (0g𝐴)
135 eqid 2736 . . . . . . . . . . 11 (0g‘(Scalar‘𝐴)) = (0g‘(Scalar‘𝐴))
13674, 73, 72, 75, 134, 135islinds5 33396 . . . . . . . . . 10 ((𝐴 ∈ LMod ∧ 𝑋 ⊆ (Base‘𝐴)) → (𝑋 ∈ (LIndS‘𝐴) ↔ ∀𝑤 ∈ ((Base‘(Scalar‘𝐴)) ↑m 𝑋)((𝑤 finSupp (0g‘(Scalar‘𝐴)) ∧ (𝐴 Σg (𝑖𝑋 ↦ ((𝑤𝑖)( ·𝑠𝐴)𝑖))) = (0g𝐴)) → 𝑤 = (𝑋 × {(0g‘(Scalar‘𝐴))}))))
137136biimpa 476 . . . . . . . . 9 (((𝐴 ∈ LMod ∧ 𝑋 ⊆ (Base‘𝐴)) ∧ 𝑋 ∈ (LIndS‘𝐴)) → ∀𝑤 ∈ ((Base‘(Scalar‘𝐴)) ↑m 𝑋)((𝑤 finSupp (0g‘(Scalar‘𝐴)) ∧ (𝐴 Σg (𝑖𝑋 ↦ ((𝑤𝑖)( ·𝑠𝐴)𝑖))) = (0g𝐴)) → 𝑤 = (𝑋 × {(0g‘(Scalar‘𝐴))})))
13815, 39, 133, 137syl21anc 837 . . . . . . . 8 (𝜑 → ∀𝑤 ∈ ((Base‘(Scalar‘𝐴)) ↑m 𝑋)((𝑤 finSupp (0g‘(Scalar‘𝐴)) ∧ (𝐴 Σg (𝑖𝑋 ↦ ((𝑤𝑖)( ·𝑠𝐴)𝑖))) = (0g𝐴)) → 𝑤 = (𝑋 × {(0g‘(Scalar‘𝐴))})))
139138adantr 480 . . . . . . 7 ((𝜑𝑗𝑌) → ∀𝑤 ∈ ((Base‘(Scalar‘𝐴)) ↑m 𝑋)((𝑤 finSupp (0g‘(Scalar‘𝐴)) ∧ (𝐴 Σg (𝑖𝑋 ↦ ((𝑤𝑖)( ·𝑠𝐴)𝑖))) = (0g𝐴)) → 𝑤 = (𝑋 × {(0g‘(Scalar‘𝐴))})))
140 eqid 2736 . . . . . . . . . 10 (𝑖𝑋 ↦ (𝑗𝑊𝑖)) = (𝑖𝑋 ↦ (𝑗𝑊𝑖))
141 fvexd 6920 . . . . . . . . . 10 ((𝜑𝑗𝑌) → (0g𝐹) ∈ V)
142 fedgmullem.y . . . . . . . . . . 11 (𝜑𝑌 ∈ (LBasis‘𝐵))
143142adantr 480 . . . . . . . . . 10 ((𝜑𝑗𝑌) → 𝑌 ∈ (LBasis‘𝐵))
14416adantr 480 . . . . . . . . . 10 ((𝜑𝑗𝑌) → 𝑋 ∈ (LBasis‘𝐶))
145 fedgmullem2.1 . . . . . . . . . . . . . . 15 (𝜑𝑊 ∈ (Base‘((Scalar‘𝐴) freeLMod (𝑌 × 𝑋))))
146 fvexd 6920 . . . . . . . . . . . . . . . 16 (𝜑 → (Scalar‘𝐴) ∈ V)
147142, 16xpexd 7772 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑌 × 𝑋) ∈ V)
148 eqid 2736 . . . . . . . . . . . . . . . . 17 ((Scalar‘𝐴) freeLMod (𝑌 × 𝑋)) = ((Scalar‘𝐴) freeLMod (𝑌 × 𝑋))
149 eqid 2736 . . . . . . . . . . . . . . . . 17 (Base‘((Scalar‘𝐴) freeLMod (𝑌 × 𝑋))) = (Base‘((Scalar‘𝐴) freeLMod (𝑌 × 𝑋)))
150148, 73, 135, 149frlmelbas 21777 . . . . . . . . . . . . . . . 16 (((Scalar‘𝐴) ∈ V ∧ (𝑌 × 𝑋) ∈ V) → (𝑊 ∈ (Base‘((Scalar‘𝐴) freeLMod (𝑌 × 𝑋))) ↔ (𝑊 ∈ ((Base‘(Scalar‘𝐴)) ↑m (𝑌 × 𝑋)) ∧ 𝑊 finSupp (0g‘(Scalar‘𝐴)))))
151146, 147, 150syl2anc 584 . . . . . . . . . . . . . . 15 (𝜑 → (𝑊 ∈ (Base‘((Scalar‘𝐴) freeLMod (𝑌 × 𝑋))) ↔ (𝑊 ∈ ((Base‘(Scalar‘𝐴)) ↑m (𝑌 × 𝑋)) ∧ 𝑊 finSupp (0g‘(Scalar‘𝐴)))))
152145, 151mpbid 232 . . . . . . . . . . . . . 14 (𝜑 → (𝑊 ∈ ((Base‘(Scalar‘𝐴)) ↑m (𝑌 × 𝑋)) ∧ 𝑊 finSupp (0g‘(Scalar‘𝐴))))
153152simpld 494 . . . . . . . . . . . . 13 (𝜑𝑊 ∈ ((Base‘(Scalar‘𝐴)) ↑m (𝑌 × 𝑋)))
154 fvexd 6920 . . . . . . . . . . . . . 14 (𝜑 → (Base‘(Scalar‘𝐴)) ∈ V)
155154, 147elmapd 8881 . . . . . . . . . . . . 13 (𝜑 → (𝑊 ∈ ((Base‘(Scalar‘𝐴)) ↑m (𝑌 × 𝑋)) ↔ 𝑊:(𝑌 × 𝑋)⟶(Base‘(Scalar‘𝐴))))
156153, 155mpbid 232 . . . . . . . . . . . 12 (𝜑𝑊:(𝑌 × 𝑋)⟶(Base‘(Scalar‘𝐴)))
157156ffnd 6736 . . . . . . . . . . 11 (𝜑𝑊 Fn (𝑌 × 𝑋))
158157adantr 480 . . . . . . . . . 10 ((𝜑𝑗𝑌) → 𝑊 Fn (𝑌 × 𝑋))
159 simpr 484 . . . . . . . . . 10 ((𝜑𝑗𝑌) → 𝑗𝑌)
160152simprd 495 . . . . . . . . . . . 12 (𝜑𝑊 finSupp (0g‘(Scalar‘𝐴)))
161 drngring 20737 . . . . . . . . . . . . . . . 16 (𝐸 ∈ DivRing → 𝐸 ∈ Ring)
1621, 161syl 17 . . . . . . . . . . . . . . 15 (𝜑𝐸 ∈ Ring)
163 ringmnd 20241 . . . . . . . . . . . . . . 15 (𝐸 ∈ Ring → 𝐸 ∈ Mnd)
164162, 163syl 17 . . . . . . . . . . . . . 14 (𝜑𝐸 ∈ Mnd)
165 subrgsubg 20578 . . . . . . . . . . . . . . . . 17 (𝑉 ∈ (SubRing‘𝐸) → 𝑉 ∈ (SubGrp‘𝐸))
1669, 165syl 17 . . . . . . . . . . . . . . . 16 (𝜑𝑉 ∈ (SubGrp‘𝐸))
167 eqid 2736 . . . . . . . . . . . . . . . . 17 (0g𝐸) = (0g𝐸)
168167subg0cl 19153 . . . . . . . . . . . . . . . 16 (𝑉 ∈ (SubGrp‘𝐸) → (0g𝐸) ∈ 𝑉)
169166, 168syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (0g𝐸) ∈ 𝑉)
17046, 169sseldd 3983 . . . . . . . . . . . . . 14 (𝜑 → (0g𝐸) ∈ 𝑈)
1715, 21, 167ress0g 18776 . . . . . . . . . . . . . 14 ((𝐸 ∈ Mnd ∧ (0g𝐸) ∈ 𝑈𝑈 ⊆ (Base‘𝐸)) → (0g𝐸) = (0g𝐹))
172164, 170, 23, 171syl3anc 1372 . . . . . . . . . . . . 13 (𝜑 → (0g𝐸) = (0g𝐹))
17354fveq2d 6909 . . . . . . . . . . . . . 14 (𝜑 → (0g𝐾) = (0g‘(Scalar‘𝐶)))
17411, 167subrg0 20580 . . . . . . . . . . . . . . 15 (𝑉 ∈ (SubRing‘𝐸) → (0g𝐸) = (0g𝐾))
1759, 174syl 17 . . . . . . . . . . . . . 14 (𝜑 → (0g𝐸) = (0g𝐾))
17660fveq2d 6909 . . . . . . . . . . . . . 14 (𝜑 → (0g‘(Scalar‘𝐴)) = (0g‘(Scalar‘𝐶)))
177173, 175, 1763eqtr4d 2786 . . . . . . . . . . . . 13 (𝜑 → (0g𝐸) = (0g‘(Scalar‘𝐴)))
178172, 177eqtr3d 2778 . . . . . . . . . . . 12 (𝜑 → (0g𝐹) = (0g‘(Scalar‘𝐴)))
179160, 178breqtrrd 5170 . . . . . . . . . . 11 (𝜑𝑊 finSupp (0g𝐹))
180179adantr 480 . . . . . . . . . 10 ((𝜑𝑗𝑌) → 𝑊 finSupp (0g𝐹))
181140, 141, 143, 144, 158, 159, 180fsuppcurry1 32737 . . . . . . . . 9 ((𝜑𝑗𝑌) → (𝑖𝑋 ↦ (𝑗𝑊𝑖)) finSupp (0g𝐹))
182178adantr 480 . . . . . . . . 9 ((𝜑𝑗𝑌) → (0g𝐹) = (0g‘(Scalar‘𝐴)))
183181, 182breqtrd 5168 . . . . . . . 8 ((𝜑𝑗𝑌) → (𝑖𝑋 ↦ (𝑗𝑊𝑖)) finSupp (0g‘(Scalar‘𝐴)))
184 eqidd 2737 . . . . . . . . . . . . . . 15 ((𝜑𝑗𝑌) → (𝑖𝑋 ↦ (𝑗𝑊𝑖)) = (𝑖𝑋 ↦ (𝑗𝑊𝑖)))
185156fovcdmda 7605 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑗𝑌𝑖𝑋)) → (𝑗𝑊𝑖) ∈ (Base‘(Scalar‘𝐴)))
186185anassrs 467 . . . . . . . . . . . . . . 15 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → (𝑗𝑊𝑖) ∈ (Base‘(Scalar‘𝐴)))
187184, 186fvmpt2d 7028 . . . . . . . . . . . . . 14 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → ((𝑖𝑋 ↦ (𝑗𝑊𝑖))‘𝑖) = (𝑗𝑊𝑖))
188187oveq1d 7447 . . . . . . . . . . . . 13 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → (((𝑖𝑋 ↦ (𝑗𝑊𝑖))‘𝑖)( ·𝑠𝐴)𝑖) = ((𝑗𝑊𝑖)( ·𝑠𝐴)𝑖))
189119ad2antrr 726 . . . . . . . . . . . . . 14 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → ( ·𝑠𝐴) = ( ·𝑠𝐵))
190189oveqd 7449 . . . . . . . . . . . . 13 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → ((𝑗𝑊𝑖)( ·𝑠𝐴)𝑖) = ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖))
191188, 190eqtrd 2776 . . . . . . . . . . . 12 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → (((𝑖𝑋 ↦ (𝑗𝑊𝑖))‘𝑖)( ·𝑠𝐴)𝑖) = ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖))
192191mpteq2dva 5241 . . . . . . . . . . 11 ((𝜑𝑗𝑌) → (𝑖𝑋 ↦ (((𝑖𝑋 ↦ (𝑗𝑊𝑖))‘𝑖)( ·𝑠𝐴)𝑖)) = (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)))
193192oveq2d 7448 . . . . . . . . . 10 ((𝜑𝑗𝑌) → (𝐴 Σg (𝑖𝑋 ↦ (((𝑖𝑋 ↦ (𝑗𝑊𝑖))‘𝑖)( ·𝑠𝐴)𝑖))) = (𝐴 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖))))
1941adantr 480 . . . . . . . . . . . 12 ((𝜑𝑗𝑌) → 𝐸 ∈ DivRing)
1959adantr 480 . . . . . . . . . . . 12 ((𝜑𝑗𝑌) → 𝑉 ∈ (SubRing‘𝐸))
1962adantr 480 . . . . . . . . . . . 12 ((𝜑𝑗𝑌) → 𝐾 ∈ DivRing)
19710, 194, 195, 11, 196, 144drgextgsum 33646 . . . . . . . . . . 11 ((𝜑𝑗𝑌) → (𝐸 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖))) = (𝐴 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖))))
1983adantr 480 . . . . . . . . . . . 12 ((𝜑𝑗𝑌) → 𝑈 ∈ (SubRing‘𝐸))
199104adantr 480 . . . . . . . . . . . 12 ((𝜑𝑗𝑌) → 𝐹 ∈ DivRing)
200117, 194, 198, 5, 199, 144drgextgsum 33646 . . . . . . . . . . 11 ((𝜑𝑗𝑌) → (𝐸 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖))) = (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖))))
201197, 200eqtr3d 2778 . . . . . . . . . 10 ((𝜑𝑗𝑌) → (𝐴 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖))) = (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖))))
202193, 201eqtrd 2776 . . . . . . . . 9 ((𝜑𝑗𝑌) → (𝐴 Σg (𝑖𝑋 ↦ (((𝑖𝑋 ↦ (𝑗𝑊𝑖))‘𝑖)( ·𝑠𝐴)𝑖))) = (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖))))
203142mptexd 7245 . . . . . . . . . . . . . 14 (𝜑 → (𝑗𝑌 ↦ (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)))) ∈ V)
204 eqid 2736 . . . . . . . . . . . . . . . . . 18 (0g𝐵) = (0g𝐵)
205117, 5sralvec 33637 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐸 ∈ DivRing ∧ 𝐹 ∈ DivRing ∧ 𝑈 ∈ (SubRing‘𝐸)) → 𝐵 ∈ LVec)
2061, 104, 3, 205syl3anc 1372 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝐵 ∈ LVec)
207 lveclmod 21106 . . . . . . . . . . . . . . . . . . . . 21 (𝐵 ∈ LVec → 𝐵 ∈ LMod)
208206, 207syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐵 ∈ LMod)
209208adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗𝑌) → 𝐵 ∈ LMod)
210 lmodabl 20908 . . . . . . . . . . . . . . . . . . 19 (𝐵 ∈ LMod → 𝐵 ∈ Abel)
211209, 210syl 17 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗𝑌) → 𝐵 ∈ Abel)
212117a1i 11 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝐵 = ((subringAlg ‘𝐸)‘𝑈))
213212, 3, 23srasubrg 33636 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑈 ∈ (SubRing‘𝐵))
214 subrgsubg 20578 . . . . . . . . . . . . . . . . . . . 20 (𝑈 ∈ (SubRing‘𝐵) → 𝑈 ∈ (SubGrp‘𝐵))
215213, 214syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑈 ∈ (SubGrp‘𝐵))
216215adantr 480 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗𝑌) → 𝑈 ∈ (SubGrp‘𝐵))
217110ad2antrr 726 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → 𝐶 ∈ LMod)
21861ad2antrr 726 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → (Base‘(Scalar‘𝐴)) = (Base‘(Scalar‘𝐶)))
219186, 218eleqtrd 2842 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → (𝑗𝑊𝑖) ∈ (Base‘(Scalar‘𝐶)))
22020ad2antrr 726 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → 𝑋 ⊆ (Base‘𝐶))
221 simpr 484 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → 𝑖𝑋)
222220, 221sseldd 3983 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → 𝑖 ∈ (Base‘𝐶))
22317, 111, 112, 113lmodvscl 20877 . . . . . . . . . . . . . . . . . . . . 21 ((𝐶 ∈ LMod ∧ (𝑗𝑊𝑖) ∈ (Base‘(Scalar‘𝐶)) ∧ 𝑖 ∈ (Base‘𝐶)) → ((𝑗𝑊𝑖)( ·𝑠𝐶)𝑖) ∈ (Base‘𝐶))
224217, 219, 222, 223syl3anc 1372 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → ((𝑗𝑊𝑖)( ·𝑠𝐶)𝑖) ∈ (Base‘𝐶))
225125oveqd 7449 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖) = ((𝑗𝑊𝑖)( ·𝑠𝐶)𝑖))
226225ad2antrr 726 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖) = ((𝑗𝑊𝑖)( ·𝑠𝐶)𝑖))
22732ad2antrr 726 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → 𝑈 = (Base‘𝐶))
228224, 226, 2273eltr4d 2855 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖) ∈ 𝑈)
229228fmpttd 7134 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗𝑌) → (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)):𝑋𝑈)
230212, 23srasca 21184 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝐸s 𝑈) = (Scalar‘𝐵))
2315, 230eqtrid 2788 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐹 = (Scalar‘𝐵))
232231adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗𝑌) → 𝐹 = (Scalar‘𝐵))
233 eqid 2736 . . . . . . . . . . . . . . . . . . 19 (Base‘𝐵) = (Base‘𝐵)
234 ovexd 7467 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → (𝑗𝑊𝑖) ∈ V)
23520, 33sstrd 3993 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝑋 ⊆ (Base‘𝐸))
236235adantr 480 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑗𝑌𝑖𝑋)) → 𝑋 ⊆ (Base‘𝐸))
237 simprr 772 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑗𝑌𝑖𝑋)) → 𝑖𝑋)
238236, 237sseldd 3983 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑗𝑌𝑖𝑋)) → 𝑖 ∈ (Base‘𝐸))
239238anassrs 467 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → 𝑖 ∈ (Base‘𝐸))
240212, 23srabase 21178 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (Base‘𝐸) = (Base‘𝐵))
241240ad2antrr 726 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → (Base‘𝐸) = (Base‘𝐵))
242239, 241eleqtrd 2842 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → 𝑖 ∈ (Base‘𝐵))
243 eqid 2736 . . . . . . . . . . . . . . . . . . 19 (0g𝐹) = (0g𝐹)
244 eqid 2736 . . . . . . . . . . . . . . . . . . 19 ( ·𝑠𝐵) = ( ·𝑠𝐵)
245144, 209, 232, 233, 234, 242, 204, 243, 244, 181mptscmfsupp0 20926 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗𝑌) → (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)) finSupp (0g𝐵))
246204, 211, 144, 216, 229, 245gsumsubgcl 19939 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗𝑌) → (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖))) ∈ 𝑈)
247231fveq2d 6909 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (Base‘𝐹) = (Base‘(Scalar‘𝐵)))
24825, 247eqtrd 2776 . . . . . . . . . . . . . . . . . 18 (𝜑𝑈 = (Base‘(Scalar‘𝐵)))
249248adantr 480 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗𝑌) → 𝑈 = (Base‘(Scalar‘𝐵)))
250246, 249eleqtrd 2842 . . . . . . . . . . . . . . . 16 ((𝜑𝑗𝑌) → (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖))) ∈ (Base‘(Scalar‘𝐵)))
251250fmpttd 7134 . . . . . . . . . . . . . . 15 (𝜑 → (𝑗𝑌 ↦ (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)))):𝑌⟶(Base‘(Scalar‘𝐵)))
252251ffund 6739 . . . . . . . . . . . . . 14 (𝜑 → Fun (𝑗𝑌 ↦ (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)))))
253 fvexd 6920 . . . . . . . . . . . . . 14 (𝜑 → (0g‘(Scalar‘𝐵)) ∈ V)
254 fconstmpt 5746 . . . . . . . . . . . . . . . . . . . . 21 (𝑋 × {(0g‘(Scalar‘𝐴))}) = (𝑖𝑋 ↦ (0g‘(Scalar‘𝐴)))
255254eqeq2i 2749 . . . . . . . . . . . . . . . . . . . 20 ((𝑖𝑋 ↦ (𝑘𝑊𝑖)) = (𝑋 × {(0g‘(Scalar‘𝐴))}) ↔ (𝑖𝑋 ↦ (𝑘𝑊𝑖)) = (𝑖𝑋 ↦ (0g‘(Scalar‘𝐴))))
256 ovex 7465 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘𝑊𝑖) ∈ V
257256rgenw 3064 . . . . . . . . . . . . . . . . . . . . 21 𝑖𝑋 (𝑘𝑊𝑖) ∈ V
258 mpteqb 7034 . . . . . . . . . . . . . . . . . . . . 21 (∀𝑖𝑋 (𝑘𝑊𝑖) ∈ V → ((𝑖𝑋 ↦ (𝑘𝑊𝑖)) = (𝑖𝑋 ↦ (0g‘(Scalar‘𝐴))) ↔ ∀𝑖𝑋 (𝑘𝑊𝑖) = (0g‘(Scalar‘𝐴))))
259257, 258ax-mp 5 . . . . . . . . . . . . . . . . . . . 20 ((𝑖𝑋 ↦ (𝑘𝑊𝑖)) = (𝑖𝑋 ↦ (0g‘(Scalar‘𝐴))) ↔ ∀𝑖𝑋 (𝑘𝑊𝑖) = (0g‘(Scalar‘𝐴)))
260255, 259bitri 275 . . . . . . . . . . . . . . . . . . 19 ((𝑖𝑋 ↦ (𝑘𝑊𝑖)) = (𝑋 × {(0g‘(Scalar‘𝐴))}) ↔ ∀𝑖𝑋 (𝑘𝑊𝑖) = (0g‘(Scalar‘𝐴)))
261260necon3abii 2986 . . . . . . . . . . . . . . . . . 18 ((𝑖𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0g‘(Scalar‘𝐴))}) ↔ ¬ ∀𝑖𝑋 (𝑘𝑊𝑖) = (0g‘(Scalar‘𝐴)))
262 df-ov 7435 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘𝑊𝑖) = (𝑊‘⟨𝑘, 𝑖⟩)
263262eqcomi 2745 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑊‘⟨𝑘, 𝑖⟩) = (𝑘𝑊𝑖)
264263a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑘𝑌) ∧ 𝑖𝑋) → (𝑊‘⟨𝑘, 𝑖⟩) = (𝑘𝑊𝑖))
265264eqeq1d 2738 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑘𝑌) ∧ 𝑖𝑋) → ((𝑊‘⟨𝑘, 𝑖⟩) = (0g‘(Scalar‘𝐴)) ↔ (𝑘𝑊𝑖) = (0g‘(Scalar‘𝐴))))
266265necon3abid 2976 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑘𝑌) ∧ 𝑖𝑋) → ((𝑊‘⟨𝑘, 𝑖⟩) ≠ (0g‘(Scalar‘𝐴)) ↔ ¬ (𝑘𝑊𝑖) = (0g‘(Scalar‘𝐴))))
267266rexbidva 3176 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘𝑌) → (∃𝑖𝑋 (𝑊‘⟨𝑘, 𝑖⟩) ≠ (0g‘(Scalar‘𝐴)) ↔ ∃𝑖𝑋 ¬ (𝑘𝑊𝑖) = (0g‘(Scalar‘𝐴))))
268 rexnal 3099 . . . . . . . . . . . . . . . . . . 19 (∃𝑖𝑋 ¬ (𝑘𝑊𝑖) = (0g‘(Scalar‘𝐴)) ↔ ¬ ∀𝑖𝑋 (𝑘𝑊𝑖) = (0g‘(Scalar‘𝐴)))
269267, 268bitr2di 288 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝑌) → (¬ ∀𝑖𝑋 (𝑘𝑊𝑖) = (0g‘(Scalar‘𝐴)) ↔ ∃𝑖𝑋 (𝑊‘⟨𝑘, 𝑖⟩) ≠ (0g‘(Scalar‘𝐴))))
270261, 269bitrid 283 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝑌) → ((𝑖𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0g‘(Scalar‘𝐴))}) ↔ ∃𝑖𝑋 (𝑊‘⟨𝑘, 𝑖⟩) ≠ (0g‘(Scalar‘𝐴))))
271270rabbidva 3442 . . . . . . . . . . . . . . . 16 (𝜑 → {𝑘𝑌 ∣ (𝑖𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0g‘(Scalar‘𝐴))})} = {𝑘𝑌 ∣ ∃𝑖𝑋 (𝑊‘⟨𝑘, 𝑖⟩) ≠ (0g‘(Scalar‘𝐴))})
272 fveq2 6905 . . . . . . . . . . . . . . . . . 18 (𝑧 = ⟨𝑘, 𝑖⟩ → (𝑊𝑧) = (𝑊‘⟨𝑘, 𝑖⟩))
273272neeq1d 2999 . . . . . . . . . . . . . . . . 17 (𝑧 = ⟨𝑘, 𝑖⟩ → ((𝑊𝑧) ≠ (0g‘(Scalar‘𝐴)) ↔ (𝑊‘⟨𝑘, 𝑖⟩) ≠ (0g‘(Scalar‘𝐴))))
274273dmrab 32517 . . . . . . . . . . . . . . . 16 dom {𝑧 ∈ (𝑌 × 𝑋) ∣ (𝑊𝑧) ≠ (0g‘(Scalar‘𝐴))} = {𝑘𝑌 ∣ ∃𝑖𝑋 (𝑊‘⟨𝑘, 𝑖⟩) ≠ (0g‘(Scalar‘𝐴))}
275271, 274eqtr4di 2794 . . . . . . . . . . . . . . 15 (𝜑 → {𝑘𝑌 ∣ (𝑖𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0g‘(Scalar‘𝐴))})} = dom {𝑧 ∈ (𝑌 × 𝑋) ∣ (𝑊𝑧) ≠ (0g‘(Scalar‘𝐴))})
276 fvexd 6920 . . . . . . . . . . . . . . . . . 18 (𝜑 → (0g‘(Scalar‘𝐴)) ∈ V)
277 suppvalfn 8194 . . . . . . . . . . . . . . . . . 18 ((𝑊 Fn (𝑌 × 𝑋) ∧ (𝑌 × 𝑋) ∈ V ∧ (0g‘(Scalar‘𝐴)) ∈ V) → (𝑊 supp (0g‘(Scalar‘𝐴))) = {𝑧 ∈ (𝑌 × 𝑋) ∣ (𝑊𝑧) ≠ (0g‘(Scalar‘𝐴))})
278157, 147, 276, 277syl3anc 1372 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑊 supp (0g‘(Scalar‘𝐴))) = {𝑧 ∈ (𝑌 × 𝑋) ∣ (𝑊𝑧) ≠ (0g‘(Scalar‘𝐴))})
279160fsuppimpd 9410 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑊 supp (0g‘(Scalar‘𝐴))) ∈ Fin)
280278, 279eqeltrrd 2841 . . . . . . . . . . . . . . . 16 (𝜑 → {𝑧 ∈ (𝑌 × 𝑋) ∣ (𝑊𝑧) ≠ (0g‘(Scalar‘𝐴))} ∈ Fin)
281 dmfi 9376 . . . . . . . . . . . . . . . 16 ({𝑧 ∈ (𝑌 × 𝑋) ∣ (𝑊𝑧) ≠ (0g‘(Scalar‘𝐴))} ∈ Fin → dom {𝑧 ∈ (𝑌 × 𝑋) ∣ (𝑊𝑧) ≠ (0g‘(Scalar‘𝐴))} ∈ Fin)
282280, 281syl 17 . . . . . . . . . . . . . . 15 (𝜑 → dom {𝑧 ∈ (𝑌 × 𝑋) ∣ (𝑊𝑧) ≠ (0g‘(Scalar‘𝐴))} ∈ Fin)
283275, 282eqeltrd 2840 . . . . . . . . . . . . . 14 (𝜑 → {𝑘𝑌 ∣ (𝑖𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0g‘(Scalar‘𝐴))})} ∈ Fin)
284 nfv 1913 . . . . . . . . . . . . . . . . . . 19 𝑖𝜑
285 nfcv 2904 . . . . . . . . . . . . . . . . . . . . 21 𝑖𝑌
286 nfmpt1 5249 . . . . . . . . . . . . . . . . . . . . . . 23 𝑖(𝑖𝑋 ↦ (𝑘𝑊𝑖))
287 nfcv 2904 . . . . . . . . . . . . . . . . . . . . . . 23 𝑖(𝑋 × {(0g‘(Scalar‘𝐴))})
288286, 287nfne 3042 . . . . . . . . . . . . . . . . . . . . . 22 𝑖(𝑖𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0g‘(Scalar‘𝐴))})
289288, 285nfrabw 3474 . . . . . . . . . . . . . . . . . . . . 21 𝑖{𝑘𝑌 ∣ (𝑖𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0g‘(Scalar‘𝐴))})}
290285, 289nfdif 4128 . . . . . . . . . . . . . . . . . . . 20 𝑖(𝑌 ∖ {𝑘𝑌 ∣ (𝑖𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0g‘(Scalar‘𝐴))})})
291290nfcri 2896 . . . . . . . . . . . . . . . . . . 19 𝑖 𝑗 ∈ (𝑌 ∖ {𝑘𝑌 ∣ (𝑖𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0g‘(Scalar‘𝐴))})})
292284, 291nfan 1898 . . . . . . . . . . . . . . . . . 18 𝑖(𝜑𝑗 ∈ (𝑌 ∖ {𝑘𝑌 ∣ (𝑖𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0g‘(Scalar‘𝐴))})}))
293 simpr 484 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑗 ∈ (𝑌 ∖ {𝑘𝑌 ∣ (𝑖𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0g‘(Scalar‘𝐴))})})) → 𝑗 ∈ (𝑌 ∖ {𝑘𝑌 ∣ (𝑖𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0g‘(Scalar‘𝐴))})}))
294293eldifad 3962 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑗 ∈ (𝑌 ∖ {𝑘𝑌 ∣ (𝑖𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0g‘(Scalar‘𝐴))})})) → 𝑗𝑌)
295293eldifbd 3963 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑗 ∈ (𝑌 ∖ {𝑘𝑌 ∣ (𝑖𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0g‘(Scalar‘𝐴))})})) → ¬ 𝑗 ∈ {𝑘𝑌 ∣ (𝑖𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0g‘(Scalar‘𝐴))})})
296 oveq1 7439 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑘 = 𝑗 → (𝑘𝑊𝑖) = (𝑗𝑊𝑖))
297296mpteq2dv 5243 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑘 = 𝑗 → (𝑖𝑋 ↦ (𝑘𝑊𝑖)) = (𝑖𝑋 ↦ (𝑗𝑊𝑖)))
298297neeq1d 2999 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑘 = 𝑗 → ((𝑖𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0g‘(Scalar‘𝐴))}) ↔ (𝑖𝑋 ↦ (𝑗𝑊𝑖)) ≠ (𝑋 × {(0g‘(Scalar‘𝐴))})))
299298elrab 3691 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑗 ∈ {𝑘𝑌 ∣ (𝑖𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0g‘(Scalar‘𝐴))})} ↔ (𝑗𝑌 ∧ (𝑖𝑋 ↦ (𝑗𝑊𝑖)) ≠ (𝑋 × {(0g‘(Scalar‘𝐴))})))
300295, 299sylnib 328 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑗 ∈ (𝑌 ∖ {𝑘𝑌 ∣ (𝑖𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0g‘(Scalar‘𝐴))})})) → ¬ (𝑗𝑌 ∧ (𝑖𝑋 ↦ (𝑗𝑊𝑖)) ≠ (𝑋 × {(0g‘(Scalar‘𝐴))})))
301294, 300mpnanrd 409 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑗 ∈ (𝑌 ∖ {𝑘𝑌 ∣ (𝑖𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0g‘(Scalar‘𝐴))})})) → ¬ (𝑖𝑋 ↦ (𝑗𝑊𝑖)) ≠ (𝑋 × {(0g‘(Scalar‘𝐴))}))
302 nne 2943 . . . . . . . . . . . . . . . . . . . . . . . 24 (¬ (𝑖𝑋 ↦ (𝑗𝑊𝑖)) ≠ (𝑋 × {(0g‘(Scalar‘𝐴))}) ↔ (𝑖𝑋 ↦ (𝑗𝑊𝑖)) = (𝑋 × {(0g‘(Scalar‘𝐴))}))
303301, 302sylib 218 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑗 ∈ (𝑌 ∖ {𝑘𝑌 ∣ (𝑖𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0g‘(Scalar‘𝐴))})})) → (𝑖𝑋 ↦ (𝑗𝑊𝑖)) = (𝑋 × {(0g‘(Scalar‘𝐴))}))
304303, 254eqtrdi 2792 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑗 ∈ (𝑌 ∖ {𝑘𝑌 ∣ (𝑖𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0g‘(Scalar‘𝐴))})})) → (𝑖𝑋 ↦ (𝑗𝑊𝑖)) = (𝑖𝑋 ↦ (0g‘(Scalar‘𝐴))))
305 ovex 7465 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗𝑊𝑖) ∈ V
306305rgenw 3064 . . . . . . . . . . . . . . . . . . . . . . 23 𝑖𝑋 (𝑗𝑊𝑖) ∈ V
307 mpteqb 7034 . . . . . . . . . . . . . . . . . . . . . . 23 (∀𝑖𝑋 (𝑗𝑊𝑖) ∈ V → ((𝑖𝑋 ↦ (𝑗𝑊𝑖)) = (𝑖𝑋 ↦ (0g‘(Scalar‘𝐴))) ↔ ∀𝑖𝑋 (𝑗𝑊𝑖) = (0g‘(Scalar‘𝐴))))
308306, 307ax-mp 5 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖𝑋 ↦ (𝑗𝑊𝑖)) = (𝑖𝑋 ↦ (0g‘(Scalar‘𝐴))) ↔ ∀𝑖𝑋 (𝑗𝑊𝑖) = (0g‘(Scalar‘𝐴)))
309304, 308sylib 218 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗 ∈ (𝑌 ∖ {𝑘𝑌 ∣ (𝑖𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0g‘(Scalar‘𝐴))})})) → ∀𝑖𝑋 (𝑗𝑊𝑖) = (0g‘(Scalar‘𝐴)))
310309r19.21bi 3250 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑗 ∈ (𝑌 ∖ {𝑘𝑌 ∣ (𝑖𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0g‘(Scalar‘𝐴))})})) ∧ 𝑖𝑋) → (𝑗𝑊𝑖) = (0g‘(Scalar‘𝐴)))
311310oveq1d 7447 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑗 ∈ (𝑌 ∖ {𝑘𝑌 ∣ (𝑖𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0g‘(Scalar‘𝐴))})})) ∧ 𝑖𝑋) → ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖) = ((0g‘(Scalar‘𝐴))( ·𝑠𝐵)𝑖))
312117, 1, 3drgext0g 33641 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (0g𝐸) = (0g𝐵))
313117, 1, 3drgext0gsca 33643 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (0g𝐵) = (0g‘(Scalar‘𝐵)))
314312, 177, 3133eqtr3d 2784 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (0g‘(Scalar‘𝐴)) = (0g‘(Scalar‘𝐵)))
315314ad2antrr 726 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑗 ∈ (𝑌 ∖ {𝑘𝑌 ∣ (𝑖𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0g‘(Scalar‘𝐴))})})) ∧ 𝑖𝑋) → (0g‘(Scalar‘𝐴)) = (0g‘(Scalar‘𝐵)))
316315oveq1d 7447 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑗 ∈ (𝑌 ∖ {𝑘𝑌 ∣ (𝑖𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0g‘(Scalar‘𝐴))})})) ∧ 𝑖𝑋) → ((0g‘(Scalar‘𝐴))( ·𝑠𝐵)𝑖) = ((0g‘(Scalar‘𝐵))( ·𝑠𝐵)𝑖))
317208ad2antrr 726 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑗 ∈ (𝑌 ∖ {𝑘𝑌 ∣ (𝑖𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0g‘(Scalar‘𝐴))})})) ∧ 𝑖𝑋) → 𝐵 ∈ LMod)
318294, 242syldanl 602 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑗 ∈ (𝑌 ∖ {𝑘𝑌 ∣ (𝑖𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0g‘(Scalar‘𝐴))})})) ∧ 𝑖𝑋) → 𝑖 ∈ (Base‘𝐵))
319 eqid 2736 . . . . . . . . . . . . . . . . . . . . 21 (Scalar‘𝐵) = (Scalar‘𝐵)
320 eqid 2736 . . . . . . . . . . . . . . . . . . . . 21 (0g‘(Scalar‘𝐵)) = (0g‘(Scalar‘𝐵))
321233, 319, 244, 320, 204lmod0vs 20894 . . . . . . . . . . . . . . . . . . . 20 ((𝐵 ∈ LMod ∧ 𝑖 ∈ (Base‘𝐵)) → ((0g‘(Scalar‘𝐵))( ·𝑠𝐵)𝑖) = (0g𝐵))
322317, 318, 321syl2anc 584 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑗 ∈ (𝑌 ∖ {𝑘𝑌 ∣ (𝑖𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0g‘(Scalar‘𝐴))})})) ∧ 𝑖𝑋) → ((0g‘(Scalar‘𝐵))( ·𝑠𝐵)𝑖) = (0g𝐵))
323311, 316, 3223eqtrd 2780 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ (𝑌 ∖ {𝑘𝑌 ∣ (𝑖𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0g‘(Scalar‘𝐴))})})) ∧ 𝑖𝑋) → ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖) = (0g𝐵))
324292, 323mpteq2da 5239 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ (𝑌 ∖ {𝑘𝑌 ∣ (𝑖𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0g‘(Scalar‘𝐴))})})) → (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)) = (𝑖𝑋 ↦ (0g𝐵)))
325324oveq2d 7448 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ (𝑌 ∖ {𝑘𝑌 ∣ (𝑖𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0g‘(Scalar‘𝐴))})})) → (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖))) = (𝐵 Σg (𝑖𝑋 ↦ (0g𝐵))))
326 ablgrp 19804 . . . . . . . . . . . . . . . . . . 19 (𝐵 ∈ Abel → 𝐵 ∈ Grp)
327 grpmnd 18959 . . . . . . . . . . . . . . . . . . 19 (𝐵 ∈ Grp → 𝐵 ∈ Mnd)
328208, 210, 326, 3274syl 19 . . . . . . . . . . . . . . . . . 18 (𝜑𝐵 ∈ Mnd)
329204gsumz 18850 . . . . . . . . . . . . . . . . . 18 ((𝐵 ∈ Mnd ∧ 𝑋 ∈ (LBasis‘𝐶)) → (𝐵 Σg (𝑖𝑋 ↦ (0g𝐵))) = (0g𝐵))
330328, 16, 329syl2anc 584 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐵 Σg (𝑖𝑋 ↦ (0g𝐵))) = (0g𝐵))
331330adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ (𝑌 ∖ {𝑘𝑌 ∣ (𝑖𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0g‘(Scalar‘𝐴))})})) → (𝐵 Σg (𝑖𝑋 ↦ (0g𝐵))) = (0g𝐵))
332313adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ (𝑌 ∖ {𝑘𝑌 ∣ (𝑖𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0g‘(Scalar‘𝐴))})})) → (0g𝐵) = (0g‘(Scalar‘𝐵)))
333325, 331, 3323eqtrd 2780 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ (𝑌 ∖ {𝑘𝑌 ∣ (𝑖𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0g‘(Scalar‘𝐴))})})) → (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖))) = (0g‘(Scalar‘𝐵)))
334333, 142suppss2 8226 . . . . . . . . . . . . . 14 (𝜑 → ((𝑗𝑌 ↦ (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)))) supp (0g‘(Scalar‘𝐵))) ⊆ {𝑘𝑌 ∣ (𝑖𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0g‘(Scalar‘𝐴))})})
335 suppssfifsupp 9421 . . . . . . . . . . . . . 14 ((((𝑗𝑌 ↦ (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)))) ∈ V ∧ Fun (𝑗𝑌 ↦ (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)))) ∧ (0g‘(Scalar‘𝐵)) ∈ V) ∧ ({𝑘𝑌 ∣ (𝑖𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0g‘(Scalar‘𝐴))})} ∈ Fin ∧ ((𝑗𝑌 ↦ (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)))) supp (0g‘(Scalar‘𝐵))) ⊆ {𝑘𝑌 ∣ (𝑖𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0g‘(Scalar‘𝐴))})})) → (𝑗𝑌 ↦ (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)))) finSupp (0g‘(Scalar‘𝐵)))
336203, 252, 253, 283, 334, 335syl32anc 1379 . . . . . . . . . . . . 13 (𝜑 → (𝑗𝑌 ↦ (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)))) finSupp (0g‘(Scalar‘𝐵)))
337 eqidd 2737 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑗𝑌 ↦ (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)))) = (𝑗𝑌 ↦ (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)))))
338 ovexd 7467 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗𝑌) → (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖))) ∈ V)
339337, 338fvmpt2d 7028 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗𝑌) → ((𝑗𝑌 ↦ (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖))))‘𝑗) = (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖))))
340339oveq1d 7447 . . . . . . . . . . . . . . . 16 ((𝜑𝑗𝑌) → (((𝑗𝑌 ↦ (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖))))‘𝑗)( ·𝑠𝐵)𝑗) = ((𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)))( ·𝑠𝐵)𝑗))
341340mpteq2dva 5241 . . . . . . . . . . . . . . 15 (𝜑 → (𝑗𝑌 ↦ (((𝑗𝑌 ↦ (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖))))‘𝑗)( ·𝑠𝐵)𝑗)) = (𝑗𝑌 ↦ ((𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)))( ·𝑠𝐵)𝑗)))
342341oveq2d 7448 . . . . . . . . . . . . . 14 (𝜑 → (𝐵 Σg (𝑗𝑌 ↦ (((𝑗𝑌 ↦ (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖))))‘𝑗)( ·𝑠𝐵)𝑗))) = (𝐵 Σg (𝑗𝑌 ↦ ((𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)))( ·𝑠𝐵)𝑗))))
343119adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗𝑌) → ( ·𝑠𝐴) = ( ·𝑠𝐵))
34443ad2antrr 726 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → (.r𝐸) = ( ·𝑠𝐴))
345344oveqd 7449 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → ((𝑗𝑊𝑖)(.r𝐸)𝑖) = ((𝑗𝑊𝑖)( ·𝑠𝐴)𝑖))
346345mpteq2dva 5241 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗𝑌) → (𝑖𝑋 ↦ ((𝑗𝑊𝑖)(.r𝐸)𝑖)) = (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐴)𝑖)))
347118adantr 480 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑗𝑌) → (.r𝐸) = ( ·𝑠𝐵))
348347oveqd 7449 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑗𝑌) → ((𝑗𝑊𝑖)(.r𝐸)𝑖) = ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖))
349348mpteq2dv 5243 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗𝑌) → (𝑖𝑋 ↦ ((𝑗𝑊𝑖)(.r𝐸)𝑖)) = (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)))
350346, 349eqtr3d 2778 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗𝑌) → (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐴)𝑖)) = (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)))
351350oveq2d 7448 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗𝑌) → (𝐴 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐴)𝑖))) = (𝐴 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖))))
352 eqidd 2737 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗𝑌) → 𝑗 = 𝑗)
353343, 351, 352oveq123d 7453 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗𝑌) → ((𝐴 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐴)𝑖)))( ·𝑠𝐴)𝑗) = ((𝐴 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)))( ·𝑠𝐵)𝑗))
354201oveq1d 7447 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗𝑌) → ((𝐴 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)))( ·𝑠𝐵)𝑗) = ((𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)))( ·𝑠𝐵)𝑗))
355353, 354eqtrd 2776 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗𝑌) → ((𝐴 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐴)𝑖)))( ·𝑠𝐴)𝑗) = ((𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)))( ·𝑠𝐵)𝑗))
356355mpteq2dva 5241 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑗𝑌 ↦ ((𝐴 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐴)𝑖)))( ·𝑠𝐴)𝑗)) = (𝑗𝑌 ↦ ((𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)))( ·𝑠𝐵)𝑗)))
357356oveq2d 7448 . . . . . . . . . . . . . . 15 (𝜑 → (𝐴 Σg (𝑗𝑌 ↦ ((𝐴 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐴)𝑖)))( ·𝑠𝐴)𝑗))) = (𝐴 Σg (𝑗𝑌 ↦ ((𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)))( ·𝑠𝐵)𝑗))))
35810, 21sraring 21194 . . . . . . . . . . . . . . . . . . . 20 ((𝐸 ∈ Ring ∧ 𝑉 ⊆ (Base‘𝐸)) → 𝐴 ∈ Ring)
359162, 36, 358syl2anc 584 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐴 ∈ Ring)
360 ringcmn 20280 . . . . . . . . . . . . . . . . . . 19 (𝐴 ∈ Ring → 𝐴 ∈ CMnd)
361359, 360syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑𝐴 ∈ CMnd)
362162adantr 480 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑗𝑌𝑖𝑋)) → 𝐸 ∈ Ring)
363 eqid 2736 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (LBasis‘𝐵) = (LBasis‘𝐵)
364233, 363lbsss 21077 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑌 ∈ (LBasis‘𝐵) → 𝑌 ⊆ (Base‘𝐵))
365142, 364syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑𝑌 ⊆ (Base‘𝐵))
366365, 240sseqtrrd 4020 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑𝑌 ⊆ (Base‘𝐸))
367366adantr 480 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ (𝑗𝑌𝑖𝑋)) → 𝑌 ⊆ (Base‘𝐸))
368 simprl 770 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ (𝑗𝑌𝑖𝑋)) → 𝑗𝑌)
369367, 368sseldd 3983 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑗𝑌𝑖𝑋)) → 𝑗 ∈ (Base‘𝐸))
37021, 67ringcl 20248 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐸 ∈ Ring ∧ 𝑖 ∈ (Base‘𝐸) ∧ 𝑗 ∈ (Base‘𝐸)) → (𝑖(.r𝐸)𝑗) ∈ (Base‘𝐸))
371362, 238, 369, 370syl3anc 1372 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑗𝑌𝑖𝑋)) → (𝑖(.r𝐸)𝑗) ∈ (Base‘𝐸))
37237adantr 480 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑗𝑌𝑖𝑋)) → (Base‘𝐸) = (Base‘𝐴))
373371, 372eleqtrd 2842 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑗𝑌𝑖𝑋)) → (𝑖(.r𝐸)𝑗) ∈ (Base‘𝐴))
374373ralrimivva 3201 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ∀𝑗𝑌𝑖𝑋 (𝑖(.r𝐸)𝑗) ∈ (Base‘𝐴))
375 fedgmullem.d . . . . . . . . . . . . . . . . . . . . 21 𝐷 = (𝑗𝑌, 𝑖𝑋 ↦ (𝑖(.r𝐸)𝑗))
376375fmpo 8094 . . . . . . . . . . . . . . . . . . . 20 (∀𝑗𝑌𝑖𝑋 (𝑖(.r𝐸)𝑗) ∈ (Base‘𝐴) ↔ 𝐷:(𝑌 × 𝑋)⟶(Base‘𝐴))
377374, 376sylib 218 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐷:(𝑌 × 𝑋)⟶(Base‘𝐴))
37872, 73, 75, 74, 15, 156, 377, 147lcomf 20900 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑊f ( ·𝑠𝐴)𝐷):(𝑌 × 𝑋)⟶(Base‘𝐴))
37972, 73, 75, 74, 15, 156, 377, 147, 134, 135, 160lcomfsupp 20901 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑊f ( ·𝑠𝐴)𝐷) finSupp (0g𝐴))
38074, 134, 361, 142, 16, 378, 379gsumxp 19995 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐴 Σg (𝑊f ( ·𝑠𝐴)𝐷)) = (𝐴 Σg (𝑗𝑌 ↦ (𝐴 Σg (𝑖𝑋 ↦ (𝑗(𝑊f ( ·𝑠𝐴)𝐷)𝑖))))))
381 fedgmullem2.2 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐴 Σg (𝑊f ( ·𝑠𝐴)𝐷)) = (0g𝐴))
3821623ad2ant1 1133 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑𝑗𝑌𝑖𝑋) → 𝐸 ∈ Ring)
3831563ad2ant1 1133 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝜑𝑗𝑌𝑖𝑋) → 𝑊:(𝑌 × 𝑋)⟶(Base‘(Scalar‘𝐴)))
38457, 55eqtrd 2776 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝜑𝑉 = (Base‘(Scalar‘𝐶)))
385384, 36eqsstrrd 4018 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝜑 → (Base‘(Scalar‘𝐶)) ⊆ (Base‘𝐸))
38661, 385eqsstrd 4017 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝜑 → (Base‘(Scalar‘𝐴)) ⊆ (Base‘𝐸))
387386, 37sseqtrd 4019 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝜑 → (Base‘(Scalar‘𝐴)) ⊆ (Base‘𝐴))
3883873ad2ant1 1133 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝜑𝑗𝑌𝑖𝑋) → (Base‘(Scalar‘𝐴)) ⊆ (Base‘𝐴))
389383, 388fssd 6752 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝜑𝑗𝑌𝑖𝑋) → 𝑊:(𝑌 × 𝑋)⟶(Base‘𝐴))
390 simp2 1137 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝜑𝑗𝑌𝑖𝑋) → 𝑗𝑌)
391 simp3 1138 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝜑𝑗𝑌𝑖𝑋) → 𝑖𝑋)
392389, 390, 391fovcdmd 7606 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝜑𝑗𝑌𝑖𝑋) → (𝑗𝑊𝑖) ∈ (Base‘𝐴))
393373ad2ant1 1133 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝜑𝑗𝑌𝑖𝑋) → (Base‘𝐸) = (Base‘𝐴))
394392, 393eleqtrrd 2843 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑𝑗𝑌𝑖𝑋) → (𝑗𝑊𝑖) ∈ (Base‘𝐸))
3952383impb 1114 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑𝑗𝑌𝑖𝑋) → 𝑖 ∈ (Base‘𝐸))
3963693impb 1114 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑𝑗𝑌𝑖𝑋) → 𝑗 ∈ (Base‘𝐸))
39721, 67ringass 20251 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝐸 ∈ Ring ∧ ((𝑗𝑊𝑖) ∈ (Base‘𝐸) ∧ 𝑖 ∈ (Base‘𝐸) ∧ 𝑗 ∈ (Base‘𝐸))) → (((𝑗𝑊𝑖)(.r𝐸)𝑖)(.r𝐸)𝑗) = ((𝑗𝑊𝑖)(.r𝐸)(𝑖(.r𝐸)𝑗)))
398382, 394, 395, 396, 397syl13anc 1373 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑𝑗𝑌𝑖𝑋) → (((𝑗𝑊𝑖)(.r𝐸)𝑖)(.r𝐸)𝑗) = ((𝑗𝑊𝑖)(.r𝐸)(𝑖(.r𝐸)𝑗)))
399398mpoeq3dva 7511 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑 → (𝑗𝑌, 𝑖𝑋 ↦ (((𝑗𝑊𝑖)(.r𝐸)𝑖)(.r𝐸)𝑗)) = (𝑗𝑌, 𝑖𝑋 ↦ ((𝑗𝑊𝑖)(.r𝐸)(𝑖(.r𝐸)𝑗))))
400 ovexd 7467 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑𝑗𝑌𝑖𝑋) → (𝑗𝑊𝑖) ∈ V)
401 ovexd 7467 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑𝑗𝑌𝑖𝑋) → (𝑖(.r𝐸)𝑗) ∈ V)
402 fnov 7565 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑊 Fn (𝑌 × 𝑋) ↔ 𝑊 = (𝑗𝑌, 𝑖𝑋 ↦ (𝑗𝑊𝑖)))
403157, 402sylib 218 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜑𝑊 = (𝑗𝑌, 𝑖𝑋 ↦ (𝑗𝑊𝑖)))
404375a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜑𝐷 = (𝑗𝑌, 𝑖𝑋 ↦ (𝑖(.r𝐸)𝑗)))
405142, 16, 400, 401, 403, 404offval22 8114 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑 → (𝑊f (.r𝐸)𝐷) = (𝑗𝑌, 𝑖𝑋 ↦ ((𝑗𝑊𝑖)(.r𝐸)(𝑖(.r𝐸)𝑗))))
40643ofeqd 7700 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜑 → ∘f (.r𝐸) = ∘f ( ·𝑠𝐴))
407406oveqd 7449 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑 → (𝑊f (.r𝐸)𝐷) = (𝑊f ( ·𝑠𝐴)𝐷))
408399, 405, 4073eqtr2rd 2783 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → (𝑊f ( ·𝑠𝐴)𝐷) = (𝑗𝑌, 𝑖𝑋 ↦ (((𝑗𝑊𝑖)(.r𝐸)𝑖)(.r𝐸)𝑗)))
409408ad2antrr 726 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → (𝑊f ( ·𝑠𝐴)𝐷) = (𝑗𝑌, 𝑖𝑋 ↦ (((𝑗𝑊𝑖)(.r𝐸)𝑖)(.r𝐸)𝑗)))
410409oveqd 7449 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → (𝑗(𝑊f ( ·𝑠𝐴)𝐷)𝑖) = (𝑗(𝑗𝑌, 𝑖𝑋 ↦ (((𝑗𝑊𝑖)(.r𝐸)𝑖)(.r𝐸)𝑗))𝑖))
411 simplr 768 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → 𝑗𝑌)
412 ovexd 7467 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → (((𝑗𝑊𝑖)(.r𝐸)𝑖)(.r𝐸)𝑗) ∈ V)
413 eqid 2736 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑗𝑌, 𝑖𝑋 ↦ (((𝑗𝑊𝑖)(.r𝐸)𝑖)(.r𝐸)𝑗)) = (𝑗𝑌, 𝑖𝑋 ↦ (((𝑗𝑊𝑖)(.r𝐸)𝑖)(.r𝐸)𝑗))
414413ovmpt4g 7581 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑗𝑌𝑖𝑋 ∧ (((𝑗𝑊𝑖)(.r𝐸)𝑖)(.r𝐸)𝑗) ∈ V) → (𝑗(𝑗𝑌, 𝑖𝑋 ↦ (((𝑗𝑊𝑖)(.r𝐸)𝑖)(.r𝐸)𝑗))𝑖) = (((𝑗𝑊𝑖)(.r𝐸)𝑖)(.r𝐸)𝑗))
415411, 221, 412, 414syl3anc 1372 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → (𝑗(𝑗𝑌, 𝑖𝑋 ↦ (((𝑗𝑊𝑖)(.r𝐸)𝑖)(.r𝐸)𝑗))𝑖) = (((𝑗𝑊𝑖)(.r𝐸)𝑖)(.r𝐸)𝑗))
416410, 415eqtrd 2776 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → (𝑗(𝑊f ( ·𝑠𝐴)𝐷)𝑖) = (((𝑗𝑊𝑖)(.r𝐸)𝑖)(.r𝐸)𝑗))
417416mpteq2dva 5241 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑗𝑌) → (𝑖𝑋 ↦ (𝑗(𝑊f ( ·𝑠𝐴)𝐷)𝑖)) = (𝑖𝑋 ↦ (((𝑗𝑊𝑖)(.r𝐸)𝑖)(.r𝐸)𝑗)))
418417oveq2d 7448 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗𝑌) → (𝐸 Σg (𝑖𝑋 ↦ (𝑗(𝑊f ( ·𝑠𝐴)𝐷)𝑖))) = (𝐸 Σg (𝑖𝑋 ↦ (((𝑗𝑊𝑖)(.r𝐸)𝑖)(.r𝐸)𝑗))))
419162adantr 480 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑗𝑌) → 𝐸 ∈ Ring)
420366sselda 3982 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑗𝑌) → 𝑗 ∈ (Base‘𝐸))
421162ad2antrr 726 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → 𝐸 ∈ Ring)
422385ad2antrr 726 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → (Base‘(Scalar‘𝐶)) ⊆ (Base‘𝐸))
423422, 219sseldd 3983 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → (𝑗𝑊𝑖) ∈ (Base‘𝐸))
42421, 67ringcl 20248 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐸 ∈ Ring ∧ (𝑗𝑊𝑖) ∈ (Base‘𝐸) ∧ 𝑖 ∈ (Base‘𝐸)) → ((𝑗𝑊𝑖)(.r𝐸)𝑖) ∈ (Base‘𝐸))
425421, 423, 239, 424syl3anc 1372 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → ((𝑗𝑊𝑖)(.r𝐸)𝑖) ∈ (Base‘𝐸))
426312adantr 480 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑗𝑌) → (0g𝐸) = (0g𝐵))
427245, 349, 4263brtr4d 5174 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑗𝑌) → (𝑖𝑋 ↦ ((𝑗𝑊𝑖)(.r𝐸)𝑖)) finSupp (0g𝐸))
42821, 167, 67, 419, 144, 420, 425, 427gsummulc1 20314 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗𝑌) → (𝐸 Σg (𝑖𝑋 ↦ (((𝑗𝑊𝑖)(.r𝐸)𝑖)(.r𝐸)𝑗))) = ((𝐸 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)(.r𝐸)𝑖)))(.r𝐸)𝑗))
429418, 428eqtrd 2776 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗𝑌) → (𝐸 Σg (𝑖𝑋 ↦ (𝑗(𝑊f ( ·𝑠𝐴)𝐷)𝑖))) = ((𝐸 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)(.r𝐸)𝑖)))(.r𝐸)𝑗))
430144mptexd 7245 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗𝑌) → (𝑖𝑋 ↦ (𝑗(𝑊f ( ·𝑠𝐴)𝐷)𝑖)) ∈ V)
43115adantr 480 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗𝑌) → 𝐴 ∈ LMod)
43236adantr 480 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗𝑌) → 𝑉 ⊆ (Base‘𝐸))
43310, 430, 194, 431, 432gsumsra 33051 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗𝑌) → (𝐸 Σg (𝑖𝑋 ↦ (𝑗(𝑊f ( ·𝑠𝐴)𝐷)𝑖))) = (𝐴 Σg (𝑖𝑋 ↦ (𝑗(𝑊f ( ·𝑠𝐴)𝐷)𝑖))))
434144mptexd 7245 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑗𝑌) → (𝑖𝑋 ↦ ((𝑗𝑊𝑖)(.r𝐸)𝑖)) ∈ V)
43510, 434, 194, 431, 432gsumsra 33051 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑗𝑌) → (𝐸 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)(.r𝐸)𝑖))) = (𝐴 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)(.r𝐸)𝑖))))
436435oveq1d 7447 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗𝑌) → ((𝐸 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)(.r𝐸)𝑖)))(.r𝐸)𝑗) = ((𝐴 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)(.r𝐸)𝑖)))(.r𝐸)𝑗))
43743adantr 480 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑗𝑌) → (.r𝐸) = ( ·𝑠𝐴))
438346oveq2d 7448 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑗𝑌) → (𝐴 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)(.r𝐸)𝑖))) = (𝐴 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐴)𝑖))))
439437, 438, 352oveq123d 7453 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗𝑌) → ((𝐴 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)(.r𝐸)𝑖)))(.r𝐸)𝑗) = ((𝐴 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐴)𝑖)))( ·𝑠𝐴)𝑗))
440436, 439eqtrd 2776 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗𝑌) → ((𝐸 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)(.r𝐸)𝑖)))(.r𝐸)𝑗) = ((𝐴 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐴)𝑖)))( ·𝑠𝐴)𝑗))
441429, 433, 4403eqtr3d 2784 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗𝑌) → (𝐴 Σg (𝑖𝑋 ↦ (𝑗(𝑊f ( ·𝑠𝐴)𝐷)𝑖))) = ((𝐴 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐴)𝑖)))( ·𝑠𝐴)𝑗))
442441mpteq2dva 5241 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑗𝑌 ↦ (𝐴 Σg (𝑖𝑋 ↦ (𝑗(𝑊f ( ·𝑠𝐴)𝐷)𝑖)))) = (𝑗𝑌 ↦ ((𝐴 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐴)𝑖)))( ·𝑠𝐴)𝑗)))
443442oveq2d 7448 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐴 Σg (𝑗𝑌 ↦ (𝐴 Σg (𝑖𝑋 ↦ (𝑗(𝑊f ( ·𝑠𝐴)𝐷)𝑖))))) = (𝐴 Σg (𝑗𝑌 ↦ ((𝐴 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐴)𝑖)))( ·𝑠𝐴)𝑗))))
444380, 381, 4433eqtr3rd 2785 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐴 Σg (𝑗𝑌 ↦ ((𝐴 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐴)𝑖)))( ·𝑠𝐴)𝑗))) = (0g𝐴))
44510, 1, 9drgext0g 33641 . . . . . . . . . . . . . . . 16 (𝜑 → (0g𝐸) = (0g𝐴))
446444, 445, 3123eqtr2d 2782 . . . . . . . . . . . . . . 15 (𝜑 → (𝐴 Σg (𝑗𝑌 ↦ ((𝐴 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐴)𝑖)))( ·𝑠𝐴)𝑗))) = (0g𝐵))
44710, 1, 9, 11, 2, 142drgextgsum 33646 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐸 Σg (𝑗𝑌 ↦ ((𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)))( ·𝑠𝐵)𝑗))) = (𝐴 Σg (𝑗𝑌 ↦ ((𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)))( ·𝑠𝐵)𝑗))))
448117, 1, 3, 5, 104, 142drgextgsum 33646 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐸 Σg (𝑗𝑌 ↦ ((𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)))( ·𝑠𝐵)𝑗))) = (𝐵 Σg (𝑗𝑌 ↦ ((𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)))( ·𝑠𝐵)𝑗))))
449447, 448eqtr3d 2778 . . . . . . . . . . . . . . 15 (𝜑 → (𝐴 Σg (𝑗𝑌 ↦ ((𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)))( ·𝑠𝐵)𝑗))) = (𝐵 Σg (𝑗𝑌 ↦ ((𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)))( ·𝑠𝐵)𝑗))))
450357, 446, 4493eqtr3rd 2785 . . . . . . . . . . . . . 14 (𝜑 → (𝐵 Σg (𝑗𝑌 ↦ ((𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)))( ·𝑠𝐵)𝑗))) = (0g𝐵))
451342, 450eqtrd 2776 . . . . . . . . . . . . 13 (𝜑 → (𝐵 Σg (𝑗𝑌 ↦ (((𝑗𝑌 ↦ (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖))))‘𝑗)( ·𝑠𝐵)𝑗))) = (0g𝐵))
452 breq1 5145 . . . . . . . . . . . . . . . 16 (𝑏 = (𝑗𝑌 ↦ (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)))) → (𝑏 finSupp (0g‘(Scalar‘𝐵)) ↔ (𝑗𝑌 ↦ (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)))) finSupp (0g‘(Scalar‘𝐵))))
453 nfmpt1 5249 . . . . . . . . . . . . . . . . . . . 20 𝑗(𝑗𝑌 ↦ (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖))))
454453nfeq2 2922 . . . . . . . . . . . . . . . . . . 19 𝑗 𝑏 = (𝑗𝑌 ↦ (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖))))
455 fveq1 6904 . . . . . . . . . . . . . . . . . . . . 21 (𝑏 = (𝑗𝑌 ↦ (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)))) → (𝑏𝑗) = ((𝑗𝑌 ↦ (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖))))‘𝑗))
456455oveq1d 7447 . . . . . . . . . . . . . . . . . . . 20 (𝑏 = (𝑗𝑌 ↦ (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)))) → ((𝑏𝑗)( ·𝑠𝐵)𝑗) = (((𝑗𝑌 ↦ (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖))))‘𝑗)( ·𝑠𝐵)𝑗))
457456adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝑏 = (𝑗𝑌 ↦ (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)))) ∧ 𝑗𝑌) → ((𝑏𝑗)( ·𝑠𝐵)𝑗) = (((𝑗𝑌 ↦ (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖))))‘𝑗)( ·𝑠𝐵)𝑗))
458454, 457mpteq2da 5239 . . . . . . . . . . . . . . . . . 18 (𝑏 = (𝑗𝑌 ↦ (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)))) → (𝑗𝑌 ↦ ((𝑏𝑗)( ·𝑠𝐵)𝑗)) = (𝑗𝑌 ↦ (((𝑗𝑌 ↦ (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖))))‘𝑗)( ·𝑠𝐵)𝑗)))
459458oveq2d 7448 . . . . . . . . . . . . . . . . 17 (𝑏 = (𝑗𝑌 ↦ (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)))) → (𝐵 Σg (𝑗𝑌 ↦ ((𝑏𝑗)( ·𝑠𝐵)𝑗))) = (𝐵 Σg (𝑗𝑌 ↦ (((𝑗𝑌 ↦ (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖))))‘𝑗)( ·𝑠𝐵)𝑗))))
460459eqeq1d 2738 . . . . . . . . . . . . . . . 16 (𝑏 = (𝑗𝑌 ↦ (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)))) → ((𝐵 Σg (𝑗𝑌 ↦ ((𝑏𝑗)( ·𝑠𝐵)𝑗))) = (0g𝐵) ↔ (𝐵 Σg (𝑗𝑌 ↦ (((𝑗𝑌 ↦ (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖))))‘𝑗)( ·𝑠𝐵)𝑗))) = (0g𝐵)))
461452, 460anbi12d 632 . . . . . . . . . . . . . . 15 (𝑏 = (𝑗𝑌 ↦ (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)))) → ((𝑏 finSupp (0g‘(Scalar‘𝐵)) ∧ (𝐵 Σg (𝑗𝑌 ↦ ((𝑏𝑗)( ·𝑠𝐵)𝑗))) = (0g𝐵)) ↔ ((𝑗𝑌 ↦ (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)))) finSupp (0g‘(Scalar‘𝐵)) ∧ (𝐵 Σg (𝑗𝑌 ↦ (((𝑗𝑌 ↦ (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖))))‘𝑗)( ·𝑠𝐵)𝑗))) = (0g𝐵))))
462 eqeq1 2740 . . . . . . . . . . . . . . 15 (𝑏 = (𝑗𝑌 ↦ (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)))) → (𝑏 = (𝑌 × {(0g‘(Scalar‘𝐵))}) ↔ (𝑗𝑌 ↦ (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)))) = (𝑌 × {(0g‘(Scalar‘𝐵))})))
463461, 462imbi12d 344 . . . . . . . . . . . . . 14 (𝑏 = (𝑗𝑌 ↦ (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)))) → (((𝑏 finSupp (0g‘(Scalar‘𝐵)) ∧ (𝐵 Σg (𝑗𝑌 ↦ ((𝑏𝑗)( ·𝑠𝐵)𝑗))) = (0g𝐵)) → 𝑏 = (𝑌 × {(0g‘(Scalar‘𝐵))})) ↔ (((𝑗𝑌 ↦ (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)))) finSupp (0g‘(Scalar‘𝐵)) ∧ (𝐵 Σg (𝑗𝑌 ↦ (((𝑗𝑌 ↦ (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖))))‘𝑗)( ·𝑠𝐵)𝑗))) = (0g𝐵)) → (𝑗𝑌 ↦ (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)))) = (𝑌 × {(0g‘(Scalar‘𝐵))}))))
464363lbslinds 21854 . . . . . . . . . . . . . . . 16 (LBasis‘𝐵) ⊆ (LIndS‘𝐵)
465464, 142sselid 3980 . . . . . . . . . . . . . . 15 (𝜑𝑌 ∈ (LIndS‘𝐵))
466 eqid 2736 . . . . . . . . . . . . . . . . 17 (Base‘(Scalar‘𝐵)) = (Base‘(Scalar‘𝐵))
467233, 466, 319, 244, 204, 320islinds5 33396 . . . . . . . . . . . . . . . 16 ((𝐵 ∈ LMod ∧ 𝑌 ⊆ (Base‘𝐵)) → (𝑌 ∈ (LIndS‘𝐵) ↔ ∀𝑏 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑌)((𝑏 finSupp (0g‘(Scalar‘𝐵)) ∧ (𝐵 Σg (𝑗𝑌 ↦ ((𝑏𝑗)( ·𝑠𝐵)𝑗))) = (0g𝐵)) → 𝑏 = (𝑌 × {(0g‘(Scalar‘𝐵))}))))
468467biimpa 476 . . . . . . . . . . . . . . 15 (((𝐵 ∈ LMod ∧ 𝑌 ⊆ (Base‘𝐵)) ∧ 𝑌 ∈ (LIndS‘𝐵)) → ∀𝑏 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑌)((𝑏 finSupp (0g‘(Scalar‘𝐵)) ∧ (𝐵 Σg (𝑗𝑌 ↦ ((𝑏𝑗)( ·𝑠𝐵)𝑗))) = (0g𝐵)) → 𝑏 = (𝑌 × {(0g‘(Scalar‘𝐵))})))
469208, 365, 465, 468syl21anc 837 . . . . . . . . . . . . . 14 (𝜑 → ∀𝑏 ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑌)((𝑏 finSupp (0g‘(Scalar‘𝐵)) ∧ (𝐵 Σg (𝑗𝑌 ↦ ((𝑏𝑗)( ·𝑠𝐵)𝑗))) = (0g𝐵)) → 𝑏 = (𝑌 × {(0g‘(Scalar‘𝐵))})))
470 fvexd 6920 . . . . . . . . . . . . . . 15 (𝜑 → (Base‘(Scalar‘𝐵)) ∈ V)
471 elmapg 8880 . . . . . . . . . . . . . . . 16 (((Base‘(Scalar‘𝐵)) ∈ V ∧ 𝑌 ∈ (LBasis‘𝐵)) → ((𝑗𝑌 ↦ (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)))) ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑌) ↔ (𝑗𝑌 ↦ (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)))):𝑌⟶(Base‘(Scalar‘𝐵))))
472471biimpar 477 . . . . . . . . . . . . . . 15 ((((Base‘(Scalar‘𝐵)) ∈ V ∧ 𝑌 ∈ (LBasis‘𝐵)) ∧ (𝑗𝑌 ↦ (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)))):𝑌⟶(Base‘(Scalar‘𝐵))) → (𝑗𝑌 ↦ (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)))) ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑌))
473470, 142, 251, 472syl21anc 837 . . . . . . . . . . . . . 14 (𝜑 → (𝑗𝑌 ↦ (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)))) ∈ ((Base‘(Scalar‘𝐵)) ↑m 𝑌))
474463, 469, 473rspcdva 3622 . . . . . . . . . . . . 13 (𝜑 → (((𝑗𝑌 ↦ (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)))) finSupp (0g‘(Scalar‘𝐵)) ∧ (𝐵 Σg (𝑗𝑌 ↦ (((𝑗𝑌 ↦ (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖))))‘𝑗)( ·𝑠𝐵)𝑗))) = (0g𝐵)) → (𝑗𝑌 ↦ (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)))) = (𝑌 × {(0g‘(Scalar‘𝐵))})))
475336, 451, 474mp2and 699 . . . . . . . . . . . 12 (𝜑 → (𝑗𝑌 ↦ (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)))) = (𝑌 × {(0g‘(Scalar‘𝐵))}))
476 fconstmpt 5746 . . . . . . . . . . . 12 (𝑌 × {(0g‘(Scalar‘𝐵))}) = (𝑗𝑌 ↦ (0g‘(Scalar‘𝐵)))
477475, 476eqtrdi 2792 . . . . . . . . . . 11 (𝜑 → (𝑗𝑌 ↦ (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)))) = (𝑗𝑌 ↦ (0g‘(Scalar‘𝐵))))
478 ovex 7465 . . . . . . . . . . . . 13 (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖))) ∈ V
479478rgenw 3064 . . . . . . . . . . . 12 𝑗𝑌 (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖))) ∈ V
480 mpteqb 7034 . . . . . . . . . . . 12 (∀𝑗𝑌 (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖))) ∈ V → ((𝑗𝑌 ↦ (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)))) = (𝑗𝑌 ↦ (0g‘(Scalar‘𝐵))) ↔ ∀𝑗𝑌 (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖))) = (0g‘(Scalar‘𝐵))))
481479, 480ax-mp 5 . . . . . . . . . . 11 ((𝑗𝑌 ↦ (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)))) = (𝑗𝑌 ↦ (0g‘(Scalar‘𝐵))) ↔ ∀𝑗𝑌 (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖))) = (0g‘(Scalar‘𝐵)))
482477, 481sylib 218 . . . . . . . . . 10 (𝜑 → ∀𝑗𝑌 (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖))) = (0g‘(Scalar‘𝐵)))
483482r19.21bi 3250 . . . . . . . . 9 ((𝜑𝑗𝑌) → (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖))) = (0g‘(Scalar‘𝐵)))
484312, 445, 3133eqtr3rd 2785 . . . . . . . . . 10 (𝜑 → (0g‘(Scalar‘𝐵)) = (0g𝐴))
485484adantr 480 . . . . . . . . 9 ((𝜑𝑗𝑌) → (0g‘(Scalar‘𝐵)) = (0g𝐴))
486202, 483, 4853eqtrd 2780 . . . . . . . 8 ((𝜑𝑗𝑌) → (𝐴 Σg (𝑖𝑋 ↦ (((𝑖𝑋 ↦ (𝑗𝑊𝑖))‘𝑖)( ·𝑠𝐴)𝑖))) = (0g𝐴))
487183, 486jca 511 . . . . . . 7 ((𝜑𝑗𝑌) → ((𝑖𝑋 ↦ (𝑗𝑊𝑖)) finSupp (0g‘(Scalar‘𝐴)) ∧ (𝐴 Σg (𝑖𝑋 ↦ (((𝑖𝑋 ↦ (𝑗𝑊𝑖))‘𝑖)( ·𝑠𝐴)𝑖))) = (0g𝐴)))
488186fmpttd 7134 . . . . . . . . 9 ((𝜑𝑗𝑌) → (𝑖𝑋 ↦ (𝑗𝑊𝑖)):𝑋⟶(Base‘(Scalar‘𝐴)))
489 fvexd 6920 . . . . . . . . . 10 ((𝜑𝑗𝑌) → (Base‘(Scalar‘𝐴)) ∈ V)
490489, 144elmapd 8881 . . . . . . . . 9 ((𝜑𝑗𝑌) → ((𝑖𝑋 ↦ (𝑗𝑊𝑖)) ∈ ((Base‘(Scalar‘𝐴)) ↑m 𝑋) ↔ (𝑖𝑋 ↦ (𝑗𝑊𝑖)):𝑋⟶(Base‘(Scalar‘𝐴))))
491488, 490mpbird 257 . . . . . . . 8 ((𝜑𝑗𝑌) → (𝑖𝑋 ↦ (𝑗𝑊𝑖)) ∈ ((Base‘(Scalar‘𝐴)) ↑m 𝑋))
492 simpr 484 . . . . . . . . . . 11 (((𝜑𝑗𝑌) ∧ 𝑤 = (𝑖𝑋 ↦ (𝑗𝑊𝑖))) → 𝑤 = (𝑖𝑋 ↦ (𝑗𝑊𝑖)))
493492breq1d 5152 . . . . . . . . . 10 (((𝜑𝑗𝑌) ∧ 𝑤 = (𝑖𝑋 ↦ (𝑗𝑊𝑖))) → (𝑤 finSupp (0g‘(Scalar‘𝐴)) ↔ (𝑖𝑋 ↦ (𝑗𝑊𝑖)) finSupp (0g‘(Scalar‘𝐴))))
494 nfv 1913 . . . . . . . . . . . . . 14 𝑖(𝜑𝑗𝑌)
495 nfmpt1 5249 . . . . . . . . . . . . . . 15 𝑖(𝑖𝑋 ↦ (𝑗𝑊𝑖))
496495nfeq2 2922 . . . . . . . . . . . . . 14 𝑖 𝑤 = (𝑖𝑋 ↦ (𝑗𝑊𝑖))
497494, 496nfan 1898 . . . . . . . . . . . . 13 𝑖((𝜑𝑗𝑌) ∧ 𝑤 = (𝑖𝑋 ↦ (𝑗𝑊𝑖)))
498 simplr 768 . . . . . . . . . . . . . . 15 ((((𝜑𝑗𝑌) ∧ 𝑤 = (𝑖𝑋 ↦ (𝑗𝑊𝑖))) ∧ 𝑖𝑋) → 𝑤 = (𝑖𝑋 ↦ (𝑗𝑊𝑖)))
499498fveq1d 6907 . . . . . . . . . . . . . 14 ((((𝜑𝑗𝑌) ∧ 𝑤 = (𝑖𝑋 ↦ (𝑗𝑊𝑖))) ∧ 𝑖𝑋) → (𝑤𝑖) = ((𝑖𝑋 ↦ (𝑗𝑊𝑖))‘𝑖))
500499oveq1d 7447 . . . . . . . . . . . . 13 ((((𝜑𝑗𝑌) ∧ 𝑤 = (𝑖𝑋 ↦ (𝑗𝑊𝑖))) ∧ 𝑖𝑋) → ((𝑤𝑖)( ·𝑠𝐴)𝑖) = (((𝑖𝑋 ↦ (𝑗𝑊𝑖))‘𝑖)( ·𝑠𝐴)𝑖))
501497, 500mpteq2da 5239 . . . . . . . . . . . 12 (((𝜑𝑗𝑌) ∧ 𝑤 = (𝑖𝑋 ↦ (𝑗𝑊𝑖))) → (𝑖𝑋 ↦ ((𝑤𝑖)( ·𝑠𝐴)𝑖)) = (𝑖𝑋 ↦ (((𝑖𝑋 ↦ (𝑗𝑊𝑖))‘𝑖)( ·𝑠𝐴)𝑖)))
502501oveq2d 7448 . . . . . . . . . . 11 (((𝜑𝑗𝑌) ∧ 𝑤 = (𝑖𝑋 ↦ (𝑗𝑊𝑖))) → (𝐴 Σg (𝑖𝑋 ↦ ((𝑤𝑖)( ·𝑠𝐴)𝑖))) = (𝐴 Σg (𝑖𝑋 ↦ (((𝑖𝑋 ↦ (𝑗𝑊𝑖))‘𝑖)( ·𝑠𝐴)𝑖))))
503502eqeq1d 2738 . . . . . . . . . 10 (((𝜑𝑗𝑌) ∧ 𝑤 = (𝑖𝑋 ↦ (𝑗𝑊𝑖))) → ((𝐴 Σg (𝑖𝑋 ↦ ((𝑤𝑖)( ·𝑠𝐴)𝑖))) = (0g𝐴) ↔ (𝐴 Σg (𝑖𝑋 ↦ (((𝑖𝑋 ↦ (𝑗𝑊𝑖))‘𝑖)( ·𝑠𝐴)𝑖))) = (0g𝐴)))
504493, 503anbi12d 632 . . . . . . . . 9 (((𝜑𝑗𝑌) ∧ 𝑤 = (𝑖𝑋 ↦ (𝑗𝑊𝑖))) → ((𝑤 finSupp (0g‘(Scalar‘𝐴)) ∧ (𝐴 Σg (𝑖𝑋 ↦ ((𝑤𝑖)( ·𝑠𝐴)𝑖))) = (0g𝐴)) ↔ ((𝑖𝑋 ↦ (𝑗𝑊𝑖)) finSupp (0g‘(Scalar‘𝐴)) ∧ (𝐴 Σg (𝑖𝑋 ↦ (((𝑖𝑋 ↦ (𝑗𝑊𝑖))‘𝑖)( ·𝑠𝐴)𝑖))) = (0g𝐴))))
505492eqeq1d 2738 . . . . . . . . 9 (((𝜑𝑗𝑌) ∧ 𝑤 = (𝑖𝑋 ↦ (𝑗𝑊𝑖))) → (𝑤 = (𝑋 × {(0g‘(Scalar‘𝐴))}) ↔ (𝑖𝑋 ↦ (𝑗𝑊𝑖)) = (𝑋 × {(0g‘(Scalar‘𝐴))})))
506504, 505imbi12d 344 . . . . . . . 8 (((𝜑𝑗𝑌) ∧ 𝑤 = (𝑖𝑋 ↦ (𝑗𝑊𝑖))) → (((𝑤 finSupp (0g‘(Scalar‘𝐴)) ∧ (𝐴 Σg (𝑖𝑋 ↦ ((𝑤𝑖)( ·𝑠𝐴)𝑖))) = (0g𝐴)) → 𝑤 = (𝑋 × {(0g‘(Scalar‘𝐴))})) ↔ (((𝑖𝑋 ↦ (𝑗𝑊𝑖)) finSupp (0g‘(Scalar‘𝐴)) ∧ (𝐴 Σg (𝑖𝑋 ↦ (((𝑖𝑋 ↦ (𝑗𝑊𝑖))‘𝑖)( ·𝑠𝐴)𝑖))) = (0g𝐴)) → (𝑖𝑋 ↦ (𝑗𝑊𝑖)) = (𝑋 × {(0g‘(Scalar‘𝐴))}))))
507491, 506rspcdv 3613 . . . . . . 7 ((𝜑𝑗𝑌) → (∀𝑤 ∈ ((Base‘(Scalar‘𝐴)) ↑m 𝑋)((𝑤 finSupp (0g‘(Scalar‘𝐴)) ∧ (𝐴 Σg (𝑖𝑋 ↦ ((𝑤𝑖)( ·𝑠𝐴)𝑖))) = (0g𝐴)) → 𝑤 = (𝑋 × {(0g‘(Scalar‘𝐴))})) → (((𝑖𝑋 ↦ (𝑗𝑊𝑖)) finSupp (0g‘(Scalar‘𝐴)) ∧ (𝐴 Σg (𝑖𝑋 ↦ (((𝑖𝑋 ↦ (𝑗𝑊𝑖))‘𝑖)( ·𝑠𝐴)𝑖))) = (0g𝐴)) → (𝑖𝑋 ↦ (𝑗𝑊𝑖)) = (𝑋 × {(0g‘(Scalar‘𝐴))}))))
508139, 487, 507mp2d 49 . . . . . 6 ((𝜑𝑗𝑌) → (𝑖𝑋 ↦ (𝑗𝑊𝑖)) = (𝑋 × {(0g‘(Scalar‘𝐴))}))
509508, 254eqtrdi 2792 . . . . 5 ((𝜑𝑗𝑌) → (𝑖𝑋 ↦ (𝑗𝑊𝑖)) = (𝑖𝑋 ↦ (0g‘(Scalar‘𝐴))))
510509, 308sylib 218 . . . 4 ((𝜑𝑗𝑌) → ∀𝑖𝑋 (𝑗𝑊𝑖) = (0g‘(Scalar‘𝐴)))
511510ralrimiva 3145 . . 3 (𝜑 → ∀𝑗𝑌𝑖𝑋 (𝑗𝑊𝑖) = (0g‘(Scalar‘𝐴)))
512 eqidd 2737 . . . 4 ((𝑗 = 𝑘𝑖 = 𝑙) → (0g‘(Scalar‘𝐴)) = (0g‘(Scalar‘𝐴)))
513 fvexd 6920 . . . 4 ((𝜑𝑗𝑌𝑖𝑋) → (0g‘(Scalar‘𝐴)) ∈ V)
514 fvexd 6920 . . . 4 ((𝜑𝑘𝑌𝑙𝑋) → (0g‘(Scalar‘𝐴)) ∈ V)
515157, 512, 513, 514fnmpoovd 8113 . . 3 (𝜑 → (𝑊 = (𝑘𝑌, 𝑙𝑋 ↦ (0g‘(Scalar‘𝐴))) ↔ ∀𝑗𝑌𝑖𝑋 (𝑗𝑊𝑖) = (0g‘(Scalar‘𝐴))))
516511, 515mpbird 257 . 2 (𝜑𝑊 = (𝑘𝑌, 𝑙𝑋 ↦ (0g‘(Scalar‘𝐴))))
517 fconstmpo 7551 . 2 ((𝑌 × 𝑋) × {(0g‘(Scalar‘𝐴))}) = (𝑘𝑌, 𝑙𝑋 ↦ (0g‘(Scalar‘𝐴)))
518516, 517eqtr4di 2794 1 (𝜑𝑊 = ((𝑌 × 𝑋) × {(0g‘(Scalar‘𝐴))}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1086   = wceq 1539  wcel 2107  wne 2939  wral 3060  wrex 3069  {crab 3435  Vcvv 3479  cdif 3947  wss 3950  {csn 4625  cop 4631   class class class wbr 5142  cmpt 5224   × cxp 5682  dom cdm 5684  Fun wfun 6554   Fn wfn 6555  wf 6556  cfv 6560  (class class class)co 7432  cmpo 7434  f cof 7696   supp csupp 8186  m cmap 8867  Fincfn 8986   finSupp cfsupp 9402  Basecbs 17248  s cress 17275  +gcplusg 17298  .rcmulr 17299  Scalarcsca 17301   ·𝑠 cvsca 17302  0gc0g 17485   Σg cgsu 17486  Mndcmnd 18748  Grpcgrp 18952  SubGrpcsubg 19139  CMndccmn 19799  Abelcabl 19800  Ringcrg 20231  SubRingcsubrg 20570  DivRingcdr 20730  LModclmod 20859  LSubSpclss 20930  LBasisclbs 21074  LVecclvec 21102  subringAlg csra 21171   freeLMod cfrlm 21767  LIndSclinds 21826
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-iin 4993  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-se 5637  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-isom 6569  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-of 7698  df-om 7889  df-1st 8015  df-2nd 8016  df-supp 8187  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-2o 8508  df-er 8746  df-map 8869  df-ixp 8939  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-fsupp 9403  df-sup 9483  df-oi 9551  df-card 9980  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-nn 12268  df-2 12330  df-3 12331  df-4 12332  df-5 12333  df-6 12334  df-7 12335  df-8 12336  df-9 12337  df-n0 12529  df-z 12616  df-dec 12736  df-uz 12880  df-fz 13549  df-fzo 13696  df-seq 14044  df-hash 14371  df-struct 17185  df-sets 17202  df-slot 17220  df-ndx 17232  df-base 17249  df-ress 17276  df-plusg 17311  df-mulr 17312  df-sca 17314  df-vsca 17315  df-ip 17316  df-tset 17317  df-ple 17318  df-ds 17320  df-hom 17322  df-cco 17323  df-0g 17487  df-gsum 17488  df-prds 17493  df-pws 17495  df-mre 17630  df-mrc 17631  df-acs 17633  df-mgm 18654  df-sgrp 18733  df-mnd 18749  df-mhm 18797  df-submnd 18798  df-grp 18955  df-minusg 18956  df-sbg 18957  df-mulg 19087  df-subg 19142  df-ghm 19232  df-cntz 19336  df-cmn 19801  df-abl 19802  df-mgp 20139  df-rng 20151  df-ur 20180  df-ring 20233  df-nzr 20514  df-subrng 20547  df-subrg 20571  df-drng 20732  df-lmod 20861  df-lss 20931  df-lsp 20971  df-lmhm 21022  df-lbs 21075  df-lvec 21103  df-sra 21173  df-rgmod 21174  df-dsmm 21753  df-frlm 21768  df-uvc 21804  df-lindf 21827  df-linds 21828
This theorem is referenced by:  fedgmul  33683
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