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Theorem fedgmullem2 30630
Description: Lemma for fedgmul 30631 (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   𝐵,𝑖,𝑗   𝐶,𝑖   𝐷,𝑖,𝑗   𝑖,𝐸,𝑗   𝑈,𝑖   𝑖,𝑊,𝑗   𝑖,𝑋,𝑗   𝑖,𝑌,𝑗   𝜑,𝑖,𝑗
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 𝑤 𝑘 𝑥 𝑙 𝑦 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 19251 . . . . . . . . . . . . . 14 (𝑈 ∈ (SubRing‘𝐸) → (𝑉 ∈ (SubRing‘𝐹) ↔ (𝑉 ∈ (SubRing‘𝐸) ∧ 𝑉𝑈)))
76biimpa 477 . . . . . . . . . . . . 13 ((𝑈 ∈ (SubRing‘𝐸) ∧ 𝑉 ∈ (SubRing‘𝐹)) → (𝑉 ∈ (SubRing‘𝐸) ∧ 𝑉𝑈))
83, 4, 7syl2anc 584 . . . . . . . . . . . 12 (𝜑 → (𝑉 ∈ (SubRing‘𝐸) ∧ 𝑉𝑈))
98simpld 495 . . . . . . . . . . 11 (𝜑𝑉 ∈ (SubRing‘𝐸))
10 fedgmul.a . . . . . . . . . . . 12 𝐴 = ((subringAlg ‘𝐸)‘𝑉)
11 fedgmul.k . . . . . . . . . . . 12 𝐾 = (𝐸s 𝑉)
1210, 11sralvec 30594 . . . . . . . . . . 11 ((𝐸 ∈ DivRing ∧ 𝐾 ∈ DivRing ∧ 𝑉 ∈ (SubRing‘𝐸)) → 𝐴 ∈ LVec)
131, 2, 9, 12syl3anc 1364 . . . . . . . . . 10 (𝜑𝐴 ∈ LVec)
14 lveclmod 19568 . . . . . . . . . 10 (𝐴 ∈ LVec → 𝐴 ∈ LMod)
1513, 14syl 17 . . . . . . . . 9 (𝜑𝐴 ∈ LMod)
16 fedgmullem.x . . . . . . . . . . 11 (𝜑𝑋 ∈ (LBasis‘𝐶))
17 eqid 2795 . . . . . . . . . . . 12 (Base‘𝐶) = (Base‘𝐶)
18 eqid 2795 . . . . . . . . . . . 12 (LBasis‘𝐶) = (LBasis‘𝐶)
1917, 18lbsss 19539 . . . . . . . . . . 11 (𝑋 ∈ (LBasis‘𝐶) → 𝑋 ⊆ (Base‘𝐶))
2016, 19syl 17 . . . . . . . . . 10 (𝜑𝑋 ⊆ (Base‘𝐶))
21 eqid 2795 . . . . . . . . . . . . . . . 16 (Base‘𝐸) = (Base‘𝐸)
2221subrgss 19226 . . . . . . . . . . . . . . 15 (𝑈 ∈ (SubRing‘𝐸) → 𝑈 ⊆ (Base‘𝐸))
233, 22syl 17 . . . . . . . . . . . . . 14 (𝜑𝑈 ⊆ (Base‘𝐸))
245, 21ressbas2 16384 . . . . . . . . . . . . . 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 2795 . . . . . . . . . . . . . . . 16 (Base‘𝐹) = (Base‘𝐹)
2928subrgss 19226 . . . . . . . . . . . . . . 15 (𝑉 ∈ (SubRing‘𝐹) → 𝑉 ⊆ (Base‘𝐹))
304, 29syl 17 . . . . . . . . . . . . . 14 (𝜑𝑉 ⊆ (Base‘𝐹))
3127, 30srabase 19640 . . . . . . . . . . . . 13 (𝜑 → (Base‘𝐹) = (Base‘𝐶))
3225, 31eqtrd 2831 . . . . . . . . . . . 12 (𝜑𝑈 = (Base‘𝐶))
3332, 23eqsstrrd 3927 . . . . . . . . . . 11 (𝜑 → (Base‘𝐶) ⊆ (Base‘𝐸))
3410a1i 11 . . . . . . . . . . . 12 (𝜑𝐴 = ((subringAlg ‘𝐸)‘𝑉))
3521subrgss 19226 . . . . . . . . . . . . 13 (𝑉 ∈ (SubRing‘𝐸) → 𝑉 ⊆ (Base‘𝐸))
369, 35syl 17 . . . . . . . . . . . 12 (𝜑𝑉 ⊆ (Base‘𝐸))
3734, 36srabase 19640 . . . . . . . . . . 11 (𝜑 → (Base‘𝐸) = (Base‘𝐴))
3833, 37sseqtrd 3928 . . . . . . . . . 10 (𝜑 → (Base‘𝐶) ⊆ (Base‘𝐴))
3920, 38sstrd 3899 . . . . . . . . 9 (𝜑𝑋 ⊆ (Base‘𝐴))
4034, 3, 36srasubrg 30593 . . . . . . . . . . . 12 (𝜑𝑈 ∈ (SubRing‘𝐴))
41 subrgsubg 19231 . . . . . . . . . . . 12 (𝑈 ∈ (SubRing‘𝐴) → 𝑈 ∈ (SubGrp‘𝐴))
4240, 41syl 17 . . . . . . . . . . 11 (𝜑𝑈 ∈ (SubGrp‘𝐴))
4310, 1, 9drgextvsca 30597 . . . . . . . . . . . . . 14 (𝜑 → (.r𝐸) = ( ·𝑠𝐴))
4443oveqdr 7044 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦𝑈)) → (𝑥(.r𝐸)𝑦) = (𝑥( ·𝑠𝐴)𝑦))
453adantr 481 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦𝑈)) → 𝑈 ∈ (SubRing‘𝐸))
468simprd 496 . . . . . . . . . . . . . . . 16 (𝜑𝑉𝑈)
4746adantr 481 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦𝑈)) → 𝑉𝑈)
48 simprl 767 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦𝑈)) → 𝑥 ∈ (Base‘(Scalar‘𝐴)))
49 ressabs 16392 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑈 ∈ (SubRing‘𝐸) ∧ 𝑉𝑈) → ((𝐸s 𝑈) ↾s 𝑉) = (𝐸s 𝑉))
503, 46, 49syl2anc 584 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → ((𝐸s 𝑈) ↾s 𝑉) = (𝐸s 𝑉))
515oveq1i 7026 . . . . . . . . . . . . . . . . . . . . 21 (𝐹s 𝑉) = ((𝐸s 𝑈) ↾s 𝑉)
5250, 51, 113eqtr4g 2856 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝐹s 𝑉) = 𝐾)
5327, 30srasca 19643 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝐹s 𝑉) = (Scalar‘𝐶))
5452, 53eqtr3d 2833 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐾 = (Scalar‘𝐶))
5554fveq2d 6542 . . . . . . . . . . . . . . . . . 18 (𝜑 → (Base‘𝐾) = (Base‘(Scalar‘𝐶)))
5611, 21ressbas2 16384 . . . . . . . . . . . . . . . . . . 19 (𝑉 ⊆ (Base‘𝐸) → 𝑉 = (Base‘𝐾))
5736, 56syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑𝑉 = (Base‘𝐾))
5834, 36srasca 19643 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝐸s 𝑉) = (Scalar‘𝐴))
5911, 58syl5eq 2843 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐾 = (Scalar‘𝐴))
6059, 54eqtr3d 2833 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (Scalar‘𝐴) = (Scalar‘𝐶))
6160fveq2d 6542 . . . . . . . . . . . . . . . . . 18 (𝜑 → (Base‘(Scalar‘𝐴)) = (Base‘(Scalar‘𝐶)))
6255, 57, 613eqtr4d 2841 . . . . . . . . . . . . . . . . 17 (𝜑𝑉 = (Base‘(Scalar‘𝐴)))
6362adantr 481 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦𝑈)) → 𝑉 = (Base‘(Scalar‘𝐴)))
6448, 63eleqtrrd 2886 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦𝑈)) → 𝑥𝑉)
6547, 64sseldd 3890 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦𝑈)) → 𝑥𝑈)
66 simprr 769 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦𝑈)) → 𝑦𝑈)
67 eqid 2795 . . . . . . . . . . . . . . 15 (.r𝐸) = (.r𝐸)
6867subrgmcl 19237 . . . . . . . . . . . . . 14 ((𝑈 ∈ (SubRing‘𝐸) ∧ 𝑥𝑈𝑦𝑈) → (𝑥(.r𝐸)𝑦) ∈ 𝑈)
6945, 65, 66, 68syl3anc 1364 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦𝑈)) → (𝑥(.r𝐸)𝑦) ∈ 𝑈)
7044, 69eqeltrrd 2884 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐴)) ∧ 𝑦𝑈)) → (𝑥( ·𝑠𝐴)𝑦) ∈ 𝑈)
7170ralrimivva 3158 . . . . . . . . . . 11 (𝜑 → ∀𝑥 ∈ (Base‘(Scalar‘𝐴))∀𝑦𝑈 (𝑥( ·𝑠𝐴)𝑦) ∈ 𝑈)
72 eqid 2795 . . . . . . . . . . . . 13 (Scalar‘𝐴) = (Scalar‘𝐴)
73 eqid 2795 . . . . . . . . . . . . 13 (Base‘(Scalar‘𝐴)) = (Base‘(Scalar‘𝐴))
74 eqid 2795 . . . . . . . . . . . . 13 (Base‘𝐴) = (Base‘𝐴)
75 eqid 2795 . . . . . . . . . . . . 13 ( ·𝑠𝐴) = ( ·𝑠𝐴)
76 eqid 2795 . . . . . . . . . . . . 13 (LSubSp‘𝐴) = (LSubSp‘𝐴)
7772, 73, 74, 75, 76islss4 19424 . . . . . . . . . . . 12 (𝐴 ∈ LMod → (𝑈 ∈ (LSubSp‘𝐴) ↔ (𝑈 ∈ (SubGrp‘𝐴) ∧ ∀𝑥 ∈ (Base‘(Scalar‘𝐴))∀𝑦𝑈 (𝑥( ·𝑠𝐴)𝑦) ∈ 𝑈)))
7877biimpar 478 . . . . . . . . . . 11 ((𝐴 ∈ LMod ∧ (𝑈 ∈ (SubGrp‘𝐴) ∧ ∀𝑥 ∈ (Base‘(Scalar‘𝐴))∀𝑦𝑈 (𝑥( ·𝑠𝐴)𝑦) ∈ 𝑈)) → 𝑈 ∈ (LSubSp‘𝐴))
7915, 42, 71, 78syl12anc 833 . . . . . . . . . 10 (𝜑𝑈 ∈ (LSubSp‘𝐴))
8020, 32sseqtr4d 3929 . . . . . . . . . 10 (𝜑𝑋𝑈)
8118lbslinds 20659 . . . . . . . . . . . 12 (LBasis‘𝐶) ⊆ (LIndS‘𝐶)
8281, 16sseldi 3887 . . . . . . . . . . 11 (𝜑𝑋 ∈ (LIndS‘𝐶))
8323, 37sseqtrd 3928 . . . . . . . . . . . . . 14 (𝜑𝑈 ⊆ (Base‘𝐴))
84 eqid 2795 . . . . . . . . . . . . . . 15 (𝐴s 𝑈) = (𝐴s 𝑈)
8584, 74ressbas2 16384 . . . . . . . . . . . . . 14 (𝑈 ⊆ (Base‘𝐴) → 𝑈 = (Base‘(𝐴s 𝑈)))
8683, 85syl 17 . . . . . . . . . . . . 13 (𝜑𝑈 = (Base‘(𝐴s 𝑈)))
8732, 86eqtr3d 2833 . . . . . . . . . . . 12 (𝜑 → (Base‘𝐶) = (Base‘(𝐴s 𝑈)))
8884, 72resssca 16479 . . . . . . . . . . . . . . 15 (𝑈 ∈ (SubRing‘𝐸) → (Scalar‘𝐴) = (Scalar‘(𝐴s 𝑈)))
893, 88syl 17 . . . . . . . . . . . . . 14 (𝜑 → (Scalar‘𝐴) = (Scalar‘(𝐴s 𝑈)))
9060, 89eqtr3d 2833 . . . . . . . . . . . . 13 (𝜑 → (Scalar‘𝐶) = (Scalar‘(𝐴s 𝑈)))
9190fveq2d 6542 . . . . . . . . . . . 12 (𝜑 → (Base‘(Scalar‘𝐶)) = (Base‘(Scalar‘(𝐴s 𝑈))))
9290fveq2d 6542 . . . . . . . . . . . 12 (𝜑 → (0g‘(Scalar‘𝐶)) = (0g‘(Scalar‘(𝐴s 𝑈))))
93 eqid 2795 . . . . . . . . . . . . . . . . 17 (+g𝐸) = (+g𝐸)
945, 93ressplusg 16441 . . . . . . . . . . . . . . . 16 (𝑈 ∈ (SubRing‘𝐸) → (+g𝐸) = (+g𝐹))
953, 94syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (+g𝐸) = (+g𝐹))
9634, 36sraaddg 19641 . . . . . . . . . . . . . . 15 (𝜑 → (+g𝐸) = (+g𝐴))
9727, 30sraaddg 19641 . . . . . . . . . . . . . . 15 (𝜑 → (+g𝐹) = (+g𝐶))
9895, 96, 973eqtr3rd 2840 . . . . . . . . . . . . . 14 (𝜑 → (+g𝐶) = (+g𝐴))
99 eqid 2795 . . . . . . . . . . . . . . . 16 (+g𝐴) = (+g𝐴)
10084, 99ressplusg 16441 . . . . . . . . . . . . . . 15 (𝑈 ∈ (SubRing‘𝐸) → (+g𝐴) = (+g‘(𝐴s 𝑈)))
1013, 100syl 17 . . . . . . . . . . . . . 14 (𝜑 → (+g𝐴) = (+g‘(𝐴s 𝑈)))
10298, 101eqtrd 2831 . . . . . . . . . . . . 13 (𝜑 → (+g𝐶) = (+g‘(𝐴s 𝑈)))
103102oveqdr 7044 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ (Base‘𝐶) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥(+g𝐶)𝑦) = (𝑥(+g‘(𝐴s 𝑈))𝑦))
104 fedgmul.2 . . . . . . . . . . . . . . 15 (𝜑𝐹 ∈ DivRing)
10552, 2eqeltrd 2883 . . . . . . . . . . . . . . 15 (𝜑 → (𝐹s 𝑉) ∈ DivRing)
106 eqid 2795 . . . . . . . . . . . . . . . 16 (𝐹s 𝑉) = (𝐹s 𝑉)
10726, 106sralvec 30594 . . . . . . . . . . . . . . 15 ((𝐹 ∈ DivRing ∧ (𝐹s 𝑉) ∈ DivRing ∧ 𝑉 ∈ (SubRing‘𝐹)) → 𝐶 ∈ LVec)
108104, 105, 4, 107syl3anc 1364 . . . . . . . . . . . . . 14 (𝜑𝐶 ∈ LVec)
109 lveclmod 19568 . . . . . . . . . . . . . 14 (𝐶 ∈ LVec → 𝐶 ∈ LMod)
110108, 109syl 17 . . . . . . . . . . . . 13 (𝜑𝐶 ∈ LMod)
111 eqid 2795 . . . . . . . . . . . . . . 15 (Scalar‘𝐶) = (Scalar‘𝐶)
112 eqid 2795 . . . . . . . . . . . . . . 15 ( ·𝑠𝐶) = ( ·𝑠𝐶)
113 eqid 2795 . . . . . . . . . . . . . . 15 (Base‘(Scalar‘𝐶)) = (Base‘(Scalar‘𝐶))
11417, 111, 112, 113lmodvscl 19341 . . . . . . . . . . . . . 14 ((𝐶 ∈ LMod ∧ 𝑥 ∈ (Base‘(Scalar‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶)) → (𝑥( ·𝑠𝐶)𝑦) ∈ (Base‘𝐶))
1151143expb 1113 . . . . . . . . . . . . 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 . . . . . . . . . . . . . . . . 17 𝐵 = ((subringAlg ‘𝐸)‘𝑈)
118117, 1, 3drgextvsca 30597 . . . . . . . . . . . . . . . 16 (𝜑 → (.r𝐸) = ( ·𝑠𝐵))
11943, 118eqtr3d 2833 . . . . . . . . . . . . . . 15 (𝜑 → ( ·𝑠𝐴) = ( ·𝑠𝐵))
1205, 67ressmulr 16454 . . . . . . . . . . . . . . . . 17 (𝑈 ∈ (SubRing‘𝐸) → (.r𝐸) = (.r𝐹))
1213, 120syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → (.r𝐸) = (.r𝐹))
12226, 104, 4drgextvsca 30597 . . . . . . . . . . . . . . . 16 (𝜑 → (.r𝐹) = ( ·𝑠𝐶))
123121, 118, 1223eqtr3d 2839 . . . . . . . . . . . . . . 15 (𝜑 → ( ·𝑠𝐵) = ( ·𝑠𝐶))
124119, 123eqtr2d 2832 . . . . . . . . . . . . . 14 (𝜑 → ( ·𝑠𝐶) = ( ·𝑠𝐴))
12584, 75ressvsca 16480 . . . . . . . . . . . . . . 15 (𝑈 ∈ (SubRing‘𝐸) → ( ·𝑠𝐴) = ( ·𝑠 ‘(𝐴s 𝑈)))
1263, 125syl 17 . . . . . . . . . . . . . 14 (𝜑 → ( ·𝑠𝐴) = ( ·𝑠 ‘(𝐴s 𝑈)))
127124, 126eqtrd 2831 . . . . . . . . . . . . 13 (𝜑 → ( ·𝑠𝐶) = ( ·𝑠 ‘(𝐴s 𝑈)))
128127oveqdr 7044 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ (Base‘(Scalar‘𝐶)) ∧ 𝑦 ∈ (Base‘𝐶))) → (𝑥( ·𝑠𝐶)𝑦) = (𝑥( ·𝑠 ‘(𝐴s 𝑈))𝑦))
129 ovexd 7050 . . . . . . . . . . . 12 (𝜑 → (𝐴s 𝑈) ∈ V)
13087, 91, 92, 103, 116, 128, 108, 129lindspropd 30589 . . . . . . . . . . 11 (𝜑 → (LIndS‘𝐶) = (LIndS‘(𝐴s 𝑈)))
13182, 130eleqtrd 2885 . . . . . . . . . 10 (𝜑𝑋 ∈ (LIndS‘(𝐴s 𝑈)))
13276, 84lsslinds 20657 . . . . . . . . . . 11 ((𝐴 ∈ LMod ∧ 𝑈 ∈ (LSubSp‘𝐴) ∧ 𝑋𝑈) → (𝑋 ∈ (LIndS‘(𝐴s 𝑈)) ↔ 𝑋 ∈ (LIndS‘𝐴)))
133132biimpa 477 . . . . . . . . . 10 (((𝐴 ∈ LMod ∧ 𝑈 ∈ (LSubSp‘𝐴) ∧ 𝑋𝑈) ∧ 𝑋 ∈ (LIndS‘(𝐴s 𝑈))) → 𝑋 ∈ (LIndS‘𝐴))
13415, 79, 80, 131, 133syl31anc 1366 . . . . . . . . 9 (𝜑𝑋 ∈ (LIndS‘𝐴))
135 eqid 2795 . . . . . . . . . . 11 (0g𝐴) = (0g𝐴)
136 eqid 2795 . . . . . . . . . . 11 (0g‘(Scalar‘𝐴)) = (0g‘(Scalar‘𝐴))
13774, 73, 72, 75, 135, 136islinds5 30580 . . . . . . . . . 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))}))))
138137biimpa 477 . . . . . . . . 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))})))
13915, 39, 134, 138syl21anc 834 . . . . . . . 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))})))
140139adantr 481 . . . . . . 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))})))
141 eqid 2795 . . . . . . . . . 10 (𝑖𝑋 ↦ (𝑗𝑊𝑖)) = (𝑖𝑋 ↦ (𝑗𝑊𝑖))
142 fvexd 6553 . . . . . . . . . 10 ((𝜑𝑗𝑌) → (0g𝐹) ∈ V)
143 fedgmullem.y . . . . . . . . . . 11 (𝜑𝑌 ∈ (LBasis‘𝐵))
144143adantr 481 . . . . . . . . . 10 ((𝜑𝑗𝑌) → 𝑌 ∈ (LBasis‘𝐵))
14516adantr 481 . . . . . . . . . 10 ((𝜑𝑗𝑌) → 𝑋 ∈ (LBasis‘𝐶))
146 fedgmullem2.1 . . . . . . . . . . . . . . 15 (𝜑𝑊 ∈ (Base‘((Scalar‘𝐴) freeLMod (𝑌 × 𝑋))))
147 fvexd 6553 . . . . . . . . . . . . . . . 16 (𝜑 → (Scalar‘𝐴) ∈ V)
148143, 16xpexd 7331 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑌 × 𝑋) ∈ V)
149 eqid 2795 . . . . . . . . . . . . . . . . 17 ((Scalar‘𝐴) freeLMod (𝑌 × 𝑋)) = ((Scalar‘𝐴) freeLMod (𝑌 × 𝑋))
150 eqid 2795 . . . . . . . . . . . . . . . . 17 (Base‘((Scalar‘𝐴) freeLMod (𝑌 × 𝑋))) = (Base‘((Scalar‘𝐴) freeLMod (𝑌 × 𝑋)))
151149, 73, 136, 150frlmelbas 20582 . . . . . . . . . . . . . . . 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)))))
152147, 148, 151syl2anc 584 . . . . . . . . . . . . . . 15 (𝜑 → (𝑊 ∈ (Base‘((Scalar‘𝐴) freeLMod (𝑌 × 𝑋))) ↔ (𝑊 ∈ ((Base‘(Scalar‘𝐴)) ↑𝑚 (𝑌 × 𝑋)) ∧ 𝑊 finSupp (0g‘(Scalar‘𝐴)))))
153146, 152mpbid 233 . . . . . . . . . . . . . 14 (𝜑 → (𝑊 ∈ ((Base‘(Scalar‘𝐴)) ↑𝑚 (𝑌 × 𝑋)) ∧ 𝑊 finSupp (0g‘(Scalar‘𝐴))))
154153simpld 495 . . . . . . . . . . . . 13 (𝜑𝑊 ∈ ((Base‘(Scalar‘𝐴)) ↑𝑚 (𝑌 × 𝑋)))
155 fvexd 6553 . . . . . . . . . . . . . 14 (𝜑 → (Base‘(Scalar‘𝐴)) ∈ V)
156155, 148elmapd 8270 . . . . . . . . . . . . 13 (𝜑 → (𝑊 ∈ ((Base‘(Scalar‘𝐴)) ↑𝑚 (𝑌 × 𝑋)) ↔ 𝑊:(𝑌 × 𝑋)⟶(Base‘(Scalar‘𝐴))))
157154, 156mpbid 233 . . . . . . . . . . . 12 (𝜑𝑊:(𝑌 × 𝑋)⟶(Base‘(Scalar‘𝐴)))
158157ffnd 6383 . . . . . . . . . . 11 (𝜑𝑊 Fn (𝑌 × 𝑋))
159158adantr 481 . . . . . . . . . 10 ((𝜑𝑗𝑌) → 𝑊 Fn (𝑌 × 𝑋))
160 simpr 485 . . . . . . . . . 10 ((𝜑𝑗𝑌) → 𝑗𝑌)
161153simprd 496 . . . . . . . . . . . 12 (𝜑𝑊 finSupp (0g‘(Scalar‘𝐴)))
162 drngring 19199 . . . . . . . . . . . . . . . 16 (𝐸 ∈ DivRing → 𝐸 ∈ Ring)
1631, 162syl 17 . . . . . . . . . . . . . . 15 (𝜑𝐸 ∈ Ring)
164 ringmnd 18996 . . . . . . . . . . . . . . 15 (𝐸 ∈ Ring → 𝐸 ∈ Mnd)
165163, 164syl 17 . . . . . . . . . . . . . 14 (𝜑𝐸 ∈ Mnd)
166 subrgsubg 19231 . . . . . . . . . . . . . . . . 17 (𝑉 ∈ (SubRing‘𝐸) → 𝑉 ∈ (SubGrp‘𝐸))
1679, 166syl 17 . . . . . . . . . . . . . . . 16 (𝜑𝑉 ∈ (SubGrp‘𝐸))
168 eqid 2795 . . . . . . . . . . . . . . . . 17 (0g𝐸) = (0g𝐸)
169168subg0cl 18041 . . . . . . . . . . . . . . . 16 (𝑉 ∈ (SubGrp‘𝐸) → (0g𝐸) ∈ 𝑉)
170167, 169syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (0g𝐸) ∈ 𝑉)
17146, 170sseldd 3890 . . . . . . . . . . . . . 14 (𝜑 → (0g𝐸) ∈ 𝑈)
1725, 21, 168ress0g 17758 . . . . . . . . . . . . . 14 ((𝐸 ∈ Mnd ∧ (0g𝐸) ∈ 𝑈𝑈 ⊆ (Base‘𝐸)) → (0g𝐸) = (0g𝐹))
173165, 171, 23, 172syl3anc 1364 . . . . . . . . . . . . 13 (𝜑 → (0g𝐸) = (0g𝐹))
17454fveq2d 6542 . . . . . . . . . . . . . 14 (𝜑 → (0g𝐾) = (0g‘(Scalar‘𝐶)))
17511, 168subrg0 19232 . . . . . . . . . . . . . . 15 (𝑉 ∈ (SubRing‘𝐸) → (0g𝐸) = (0g𝐾))
1769, 175syl 17 . . . . . . . . . . . . . 14 (𝜑 → (0g𝐸) = (0g𝐾))
17760fveq2d 6542 . . . . . . . . . . . . . 14 (𝜑 → (0g‘(Scalar‘𝐴)) = (0g‘(Scalar‘𝐶)))
178174, 176, 1773eqtr4d 2841 . . . . . . . . . . . . 13 (𝜑 → (0g𝐸) = (0g‘(Scalar‘𝐴)))
179173, 178eqtr3d 2833 . . . . . . . . . . . 12 (𝜑 → (0g𝐹) = (0g‘(Scalar‘𝐴)))
180161, 179breqtrrd 4990 . . . . . . . . . . 11 (𝜑𝑊 finSupp (0g𝐹))
181180adantr 481 . . . . . . . . . 10 ((𝜑𝑗𝑌) → 𝑊 finSupp (0g𝐹))
182141, 142, 144, 145, 159, 160, 181fsuppcurry1 30149 . . . . . . . . 9 ((𝜑𝑗𝑌) → (𝑖𝑋 ↦ (𝑗𝑊𝑖)) finSupp (0g𝐹))
183179adantr 481 . . . . . . . . 9 ((𝜑𝑗𝑌) → (0g𝐹) = (0g‘(Scalar‘𝐴)))
184182, 183breqtrd 4988 . . . . . . . 8 ((𝜑𝑗𝑌) → (𝑖𝑋 ↦ (𝑗𝑊𝑖)) finSupp (0g‘(Scalar‘𝐴)))
185 eqidd 2796 . . . . . . . . . . . . . . 15 ((𝜑𝑗𝑌) → (𝑖𝑋 ↦ (𝑗𝑊𝑖)) = (𝑖𝑋 ↦ (𝑗𝑊𝑖)))
186157fovrnda 7175 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑗𝑌𝑖𝑋)) → (𝑗𝑊𝑖) ∈ (Base‘(Scalar‘𝐴)))
187186anassrs 468 . . . . . . . . . . . . . . 15 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → (𝑗𝑊𝑖) ∈ (Base‘(Scalar‘𝐴)))
188185, 187fvmpt2d 6647 . . . . . . . . . . . . . 14 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → ((𝑖𝑋 ↦ (𝑗𝑊𝑖))‘𝑖) = (𝑗𝑊𝑖))
189188oveq1d 7031 . . . . . . . . . . . . 13 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → (((𝑖𝑋 ↦ (𝑗𝑊𝑖))‘𝑖)( ·𝑠𝐴)𝑖) = ((𝑗𝑊𝑖)( ·𝑠𝐴)𝑖))
190119ad2antrr 722 . . . . . . . . . . . . . 14 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → ( ·𝑠𝐴) = ( ·𝑠𝐵))
191190oveqd 7033 . . . . . . . . . . . . 13 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → ((𝑗𝑊𝑖)( ·𝑠𝐴)𝑖) = ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖))
192189, 191eqtrd 2831 . . . . . . . . . . . 12 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → (((𝑖𝑋 ↦ (𝑗𝑊𝑖))‘𝑖)( ·𝑠𝐴)𝑖) = ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖))
193192mpteq2dva 5055 . . . . . . . . . . 11 ((𝜑𝑗𝑌) → (𝑖𝑋 ↦ (((𝑖𝑋 ↦ (𝑗𝑊𝑖))‘𝑖)( ·𝑠𝐴)𝑖)) = (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)))
194193oveq2d 7032 . . . . . . . . . 10 ((𝜑𝑗𝑌) → (𝐴 Σg (𝑖𝑋 ↦ (((𝑖𝑋 ↦ (𝑗𝑊𝑖))‘𝑖)( ·𝑠𝐴)𝑖))) = (𝐴 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖))))
1951adantr 481 . . . . . . . . . . . 12 ((𝜑𝑗𝑌) → 𝐸 ∈ DivRing)
1969adantr 481 . . . . . . . . . . . 12 ((𝜑𝑗𝑌) → 𝑉 ∈ (SubRing‘𝐸))
1972adantr 481 . . . . . . . . . . . 12 ((𝜑𝑗𝑌) → 𝐾 ∈ DivRing)
19810, 195, 196, 11, 197, 145drgextgsum 30601 . . . . . . . . . . 11 ((𝜑𝑗𝑌) → (𝐸 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖))) = (𝐴 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖))))
1993adantr 481 . . . . . . . . . . . 12 ((𝜑𝑗𝑌) → 𝑈 ∈ (SubRing‘𝐸))
200104adantr 481 . . . . . . . . . . . 12 ((𝜑𝑗𝑌) → 𝐹 ∈ DivRing)
201117, 195, 199, 5, 200, 145drgextgsum 30601 . . . . . . . . . . 11 ((𝜑𝑗𝑌) → (𝐸 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖))) = (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖))))
202198, 201eqtr3d 2833 . . . . . . . . . 10 ((𝜑𝑗𝑌) → (𝐴 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖))) = (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖))))
203194, 202eqtrd 2831 . . . . . . . . 9 ((𝜑𝑗𝑌) → (𝐴 Σg (𝑖𝑋 ↦ (((𝑖𝑋 ↦ (𝑗𝑊𝑖))‘𝑖)( ·𝑠𝐴)𝑖))) = (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖))))
204143mptexd 6853 . . . . . . . . . . . . . 14 (𝜑 → (𝑗𝑌 ↦ (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)))) ∈ V)
205 eqid 2795 . . . . . . . . . . . . . . . . . 18 (0g𝐵) = (0g𝐵)
206117, 5sralvec 30594 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐸 ∈ DivRing ∧ 𝐹 ∈ DivRing ∧ 𝑈 ∈ (SubRing‘𝐸)) → 𝐵 ∈ LVec)
2071, 104, 3, 206syl3anc 1364 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝐵 ∈ LVec)
208 lveclmod 19568 . . . . . . . . . . . . . . . . . . . . 21 (𝐵 ∈ LVec → 𝐵 ∈ LMod)
209207, 208syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐵 ∈ LMod)
210209adantr 481 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗𝑌) → 𝐵 ∈ LMod)
211 lmodabl 19371 . . . . . . . . . . . . . . . . . . 19 (𝐵 ∈ LMod → 𝐵 ∈ Abel)
212210, 211syl 17 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗𝑌) → 𝐵 ∈ Abel)
213117a1i 11 . . . . . . . . . . . . . . . . . . . . 21 (𝜑𝐵 = ((subringAlg ‘𝐸)‘𝑈))
214213, 3, 23srasubrg 30593 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑈 ∈ (SubRing‘𝐵))
215 subrgsubg 19231 . . . . . . . . . . . . . . . . . . . 20 (𝑈 ∈ (SubRing‘𝐵) → 𝑈 ∈ (SubGrp‘𝐵))
216214, 215syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑈 ∈ (SubGrp‘𝐵))
217216adantr 481 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗𝑌) → 𝑈 ∈ (SubGrp‘𝐵))
218110ad2antrr 722 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → 𝐶 ∈ LMod)
21961ad2antrr 722 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → (Base‘(Scalar‘𝐴)) = (Base‘(Scalar‘𝐶)))
220187, 219eleqtrd 2885 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → (𝑗𝑊𝑖) ∈ (Base‘(Scalar‘𝐶)))
22120ad2antrr 722 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → 𝑋 ⊆ (Base‘𝐶))
222 simpr 485 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → 𝑖𝑋)
223221, 222sseldd 3890 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → 𝑖 ∈ (Base‘𝐶))
22417, 111, 112, 113lmodvscl 19341 . . . . . . . . . . . . . . . . . . . . 21 ((𝐶 ∈ LMod ∧ (𝑗𝑊𝑖) ∈ (Base‘(Scalar‘𝐶)) ∧ 𝑖 ∈ (Base‘𝐶)) → ((𝑗𝑊𝑖)( ·𝑠𝐶)𝑖) ∈ (Base‘𝐶))
225218, 220, 223, 224syl3anc 1364 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → ((𝑗𝑊𝑖)( ·𝑠𝐶)𝑖) ∈ (Base‘𝐶))
226123oveqd 7033 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖) = ((𝑗𝑊𝑖)( ·𝑠𝐶)𝑖))
227226ad2antrr 722 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖) = ((𝑗𝑊𝑖)( ·𝑠𝐶)𝑖))
22832ad2antrr 722 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → 𝑈 = (Base‘𝐶))
229225, 227, 2283eltr4d 2898 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖) ∈ 𝑈)
230229fmpttd 6742 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗𝑌) → (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)):𝑋𝑈)
231213, 23srasca 19643 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝐸s 𝑈) = (Scalar‘𝐵))
2325, 231syl5eq 2843 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝐹 = (Scalar‘𝐵))
233232adantr 481 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗𝑌) → 𝐹 = (Scalar‘𝐵))
234 eqid 2795 . . . . . . . . . . . . . . . . . . 19 (Base‘𝐵) = (Base‘𝐵)
235 ovexd 7050 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → (𝑗𝑊𝑖) ∈ V)
23620, 33sstrd 3899 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝑋 ⊆ (Base‘𝐸))
237236adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑗𝑌𝑖𝑋)) → 𝑋 ⊆ (Base‘𝐸))
238 simprr 769 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑗𝑌𝑖𝑋)) → 𝑖𝑋)
239237, 238sseldd 3890 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑗𝑌𝑖𝑋)) → 𝑖 ∈ (Base‘𝐸))
240239anassrs 468 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → 𝑖 ∈ (Base‘𝐸))
241213, 23srabase 19640 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (Base‘𝐸) = (Base‘𝐵))
242241ad2antrr 722 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → (Base‘𝐸) = (Base‘𝐵))
243240, 242eleqtrd 2885 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → 𝑖 ∈ (Base‘𝐵))
244 eqid 2795 . . . . . . . . . . . . . . . . . . 19 (0g𝐹) = (0g𝐹)
245 eqid 2795 . . . . . . . . . . . . . . . . . . 19 ( ·𝑠𝐵) = ( ·𝑠𝐵)
246145, 210, 233, 234, 235, 243, 205, 244, 245, 182mptscmfsupp0 19389 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗𝑌) → (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)) finSupp (0g𝐵))
247205, 212, 145, 217, 230, 246gsumsubgcl 18760 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗𝑌) → (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖))) ∈ 𝑈)
248232fveq2d 6542 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (Base‘𝐹) = (Base‘(Scalar‘𝐵)))
24925, 248eqtrd 2831 . . . . . . . . . . . . . . . . . 18 (𝜑𝑈 = (Base‘(Scalar‘𝐵)))
250249adantr 481 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗𝑌) → 𝑈 = (Base‘(Scalar‘𝐵)))
251247, 250eleqtrd 2885 . . . . . . . . . . . . . . . 16 ((𝜑𝑗𝑌) → (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖))) ∈ (Base‘(Scalar‘𝐵)))
252251fmpttd 6742 . . . . . . . . . . . . . . 15 (𝜑 → (𝑗𝑌 ↦ (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)))):𝑌⟶(Base‘(Scalar‘𝐵)))
253252ffund 6386 . . . . . . . . . . . . . 14 (𝜑 → Fun (𝑗𝑌 ↦ (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)))))
254 fvexd 6553 . . . . . . . . . . . . . 14 (𝜑 → (0g‘(Scalar‘𝐵)) ∈ V)
255 fconstmpt 5500 . . . . . . . . . . . . . . . . . . . . 21 (𝑋 × {(0g‘(Scalar‘𝐴))}) = (𝑖𝑋 ↦ (0g‘(Scalar‘𝐴)))
256255eqeq2i 2807 . . . . . . . . . . . . . . . . . . . 20 ((𝑖𝑋 ↦ (𝑘𝑊𝑖)) = (𝑋 × {(0g‘(Scalar‘𝐴))}) ↔ (𝑖𝑋 ↦ (𝑘𝑊𝑖)) = (𝑖𝑋 ↦ (0g‘(Scalar‘𝐴))))
257 ovex 7048 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘𝑊𝑖) ∈ V
258257rgenw 3117 . . . . . . . . . . . . . . . . . . . . 21 𝑖𝑋 (𝑘𝑊𝑖) ∈ V
259 mpteqb 6653 . . . . . . . . . . . . . . . . . . . . 21 (∀𝑖𝑋 (𝑘𝑊𝑖) ∈ V → ((𝑖𝑋 ↦ (𝑘𝑊𝑖)) = (𝑖𝑋 ↦ (0g‘(Scalar‘𝐴))) ↔ ∀𝑖𝑋 (𝑘𝑊𝑖) = (0g‘(Scalar‘𝐴))))
260258, 259ax-mp 5 . . . . . . . . . . . . . . . . . . . 20 ((𝑖𝑋 ↦ (𝑘𝑊𝑖)) = (𝑖𝑋 ↦ (0g‘(Scalar‘𝐴))) ↔ ∀𝑖𝑋 (𝑘𝑊𝑖) = (0g‘(Scalar‘𝐴)))
261256, 260bitri 276 . . . . . . . . . . . . . . . . . . 19 ((𝑖𝑋 ↦ (𝑘𝑊𝑖)) = (𝑋 × {(0g‘(Scalar‘𝐴))}) ↔ ∀𝑖𝑋 (𝑘𝑊𝑖) = (0g‘(Scalar‘𝐴)))
262261necon3abii 3030 . . . . . . . . . . . . . . . . . 18 ((𝑖𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0g‘(Scalar‘𝐴))}) ↔ ¬ ∀𝑖𝑋 (𝑘𝑊𝑖) = (0g‘(Scalar‘𝐴)))
263 df-ov 7019 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘𝑊𝑖) = (𝑊‘⟨𝑘, 𝑖⟩)
264263eqcomi 2804 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑊‘⟨𝑘, 𝑖⟩) = (𝑘𝑊𝑖)
265264a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑘𝑌) ∧ 𝑖𝑋) → (𝑊‘⟨𝑘, 𝑖⟩) = (𝑘𝑊𝑖))
266265eqeq1d 2797 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑘𝑌) ∧ 𝑖𝑋) → ((𝑊‘⟨𝑘, 𝑖⟩) = (0g‘(Scalar‘𝐴)) ↔ (𝑘𝑊𝑖) = (0g‘(Scalar‘𝐴))))
267266necon3abid 3020 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑘𝑌) ∧ 𝑖𝑋) → ((𝑊‘⟨𝑘, 𝑖⟩) ≠ (0g‘(Scalar‘𝐴)) ↔ ¬ (𝑘𝑊𝑖) = (0g‘(Scalar‘𝐴))))
268267rexbidva 3259 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑘𝑌) → (∃𝑖𝑋 (𝑊‘⟨𝑘, 𝑖⟩) ≠ (0g‘(Scalar‘𝐴)) ↔ ∃𝑖𝑋 ¬ (𝑘𝑊𝑖) = (0g‘(Scalar‘𝐴))))
269 rexnal 3202 . . . . . . . . . . . . . . . . . . 19 (∃𝑖𝑋 ¬ (𝑘𝑊𝑖) = (0g‘(Scalar‘𝐴)) ↔ ¬ ∀𝑖𝑋 (𝑘𝑊𝑖) = (0g‘(Scalar‘𝐴)))
270268, 269syl6rbb 289 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑘𝑌) → (¬ ∀𝑖𝑋 (𝑘𝑊𝑖) = (0g‘(Scalar‘𝐴)) ↔ ∃𝑖𝑋 (𝑊‘⟨𝑘, 𝑖⟩) ≠ (0g‘(Scalar‘𝐴))))
271262, 270syl5bb 284 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝑌) → ((𝑖𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0g‘(Scalar‘𝐴))}) ↔ ∃𝑖𝑋 (𝑊‘⟨𝑘, 𝑖⟩) ≠ (0g‘(Scalar‘𝐴))))
272271rabbidva 3424 . . . . . . . . . . . . . . . 16 (𝜑 → {𝑘𝑌 ∣ (𝑖𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0g‘(Scalar‘𝐴))})} = {𝑘𝑌 ∣ ∃𝑖𝑋 (𝑊‘⟨𝑘, 𝑖⟩) ≠ (0g‘(Scalar‘𝐴))})
273 fveq2 6538 . . . . . . . . . . . . . . . . . 18 (𝑧 = ⟨𝑘, 𝑖⟩ → (𝑊𝑧) = (𝑊‘⟨𝑘, 𝑖⟩))
274273neeq1d 3043 . . . . . . . . . . . . . . . . 17 (𝑧 = ⟨𝑘, 𝑖⟩ → ((𝑊𝑧) ≠ (0g‘(Scalar‘𝐴)) ↔ (𝑊‘⟨𝑘, 𝑖⟩) ≠ (0g‘(Scalar‘𝐴))))
275274dmrab 29952 . . . . . . . . . . . . . . . 16 dom {𝑧 ∈ (𝑌 × 𝑋) ∣ (𝑊𝑧) ≠ (0g‘(Scalar‘𝐴))} = {𝑘𝑌 ∣ ∃𝑖𝑋 (𝑊‘⟨𝑘, 𝑖⟩) ≠ (0g‘(Scalar‘𝐴))}
276272, 275syl6eqr 2849 . . . . . . . . . . . . . . 15 (𝜑 → {𝑘𝑌 ∣ (𝑖𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0g‘(Scalar‘𝐴))})} = dom {𝑧 ∈ (𝑌 × 𝑋) ∣ (𝑊𝑧) ≠ (0g‘(Scalar‘𝐴))})
277 fvexd 6553 . . . . . . . . . . . . . . . . . 18 (𝜑 → (0g‘(Scalar‘𝐴)) ∈ V)
278 suppvalfn 7688 . . . . . . . . . . . . . . . . . 18 ((𝑊 Fn (𝑌 × 𝑋) ∧ (𝑌 × 𝑋) ∈ V ∧ (0g‘(Scalar‘𝐴)) ∈ V) → (𝑊 supp (0g‘(Scalar‘𝐴))) = {𝑧 ∈ (𝑌 × 𝑋) ∣ (𝑊𝑧) ≠ (0g‘(Scalar‘𝐴))})
279158, 148, 277, 278syl3anc 1364 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑊 supp (0g‘(Scalar‘𝐴))) = {𝑧 ∈ (𝑌 × 𝑋) ∣ (𝑊𝑧) ≠ (0g‘(Scalar‘𝐴))})
280161fsuppimpd 8686 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑊 supp (0g‘(Scalar‘𝐴))) ∈ Fin)
281279, 280eqeltrrd 2884 . . . . . . . . . . . . . . . 16 (𝜑 → {𝑧 ∈ (𝑌 × 𝑋) ∣ (𝑊𝑧) ≠ (0g‘(Scalar‘𝐴))} ∈ Fin)
282 dmfi 8648 . . . . . . . . . . . . . . . 16 ({𝑧 ∈ (𝑌 × 𝑋) ∣ (𝑊𝑧) ≠ (0g‘(Scalar‘𝐴))} ∈ Fin → dom {𝑧 ∈ (𝑌 × 𝑋) ∣ (𝑊𝑧) ≠ (0g‘(Scalar‘𝐴))} ∈ Fin)
283281, 282syl 17 . . . . . . . . . . . . . . 15 (𝜑 → dom {𝑧 ∈ (𝑌 × 𝑋) ∣ (𝑊𝑧) ≠ (0g‘(Scalar‘𝐴))} ∈ Fin)
284276, 283eqeltrd 2883 . . . . . . . . . . . . . 14 (𝜑 → {𝑘𝑌 ∣ (𝑖𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0g‘(Scalar‘𝐴))})} ∈ Fin)
285 nfv 1892 . . . . . . . . . . . . . . . . . . 19 𝑖𝜑
286 nfcv 2949 . . . . . . . . . . . . . . . . . . . . 21 𝑖𝑌
287 nfmpt1 5058 . . . . . . . . . . . . . . . . . . . . . . 23 𝑖(𝑖𝑋 ↦ (𝑘𝑊𝑖))
288 nfcv 2949 . . . . . . . . . . . . . . . . . . . . . . 23 𝑖(𝑋 × {(0g‘(Scalar‘𝐴))})
289287, 288nfne 3087 . . . . . . . . . . . . . . . . . . . . . 22 𝑖(𝑖𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0g‘(Scalar‘𝐴))})
290289, 286nfrab 3345 . . . . . . . . . . . . . . . . . . . . 21 𝑖{𝑘𝑌 ∣ (𝑖𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0g‘(Scalar‘𝐴))})}
291286, 290nfdif 4023 . . . . . . . . . . . . . . . . . . . 20 𝑖(𝑌 ∖ {𝑘𝑌 ∣ (𝑖𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0g‘(Scalar‘𝐴))})})
292291nfcriv 2939 . . . . . . . . . . . . . . . . . . 19 𝑖 𝑗 ∈ (𝑌 ∖ {𝑘𝑌 ∣ (𝑖𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0g‘(Scalar‘𝐴))})})
293285, 292nfan 1881 . . . . . . . . . . . . . . . . . 18 𝑖(𝜑𝑗 ∈ (𝑌 ∖ {𝑘𝑌 ∣ (𝑖𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0g‘(Scalar‘𝐴))})}))
294 simpr 485 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑗 ∈ (𝑌 ∖ {𝑘𝑌 ∣ (𝑖𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0g‘(Scalar‘𝐴))})})) → 𝑗 ∈ (𝑌 ∖ {𝑘𝑌 ∣ (𝑖𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0g‘(Scalar‘𝐴))})}))
295294eldifad 3871 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑗 ∈ (𝑌 ∖ {𝑘𝑌 ∣ (𝑖𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0g‘(Scalar‘𝐴))})})) → 𝑗𝑌)
296294eldifbd 3872 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑗 ∈ (𝑌 ∖ {𝑘𝑌 ∣ (𝑖𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0g‘(Scalar‘𝐴))})})) → ¬ 𝑗 ∈ {𝑘𝑌 ∣ (𝑖𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0g‘(Scalar‘𝐴))})})
297 oveq1 7023 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑘 = 𝑗 → (𝑘𝑊𝑖) = (𝑗𝑊𝑖))
298297mpteq2dv 5056 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑘 = 𝑗 → (𝑖𝑋 ↦ (𝑘𝑊𝑖)) = (𝑖𝑋 ↦ (𝑗𝑊𝑖)))
299298neeq1d 3043 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑘 = 𝑗 → ((𝑖𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0g‘(Scalar‘𝐴))}) ↔ (𝑖𝑋 ↦ (𝑗𝑊𝑖)) ≠ (𝑋 × {(0g‘(Scalar‘𝐴))})))
300299elrab 3618 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑗 ∈ {𝑘𝑌 ∣ (𝑖𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0g‘(Scalar‘𝐴))})} ↔ (𝑗𝑌 ∧ (𝑖𝑋 ↦ (𝑗𝑊𝑖)) ≠ (𝑋 × {(0g‘(Scalar‘𝐴))})))
301296, 300sylnib 329 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑗 ∈ (𝑌 ∖ {𝑘𝑌 ∣ (𝑖𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0g‘(Scalar‘𝐴))})})) → ¬ (𝑗𝑌 ∧ (𝑖𝑋 ↦ (𝑗𝑊𝑖)) ≠ (𝑋 × {(0g‘(Scalar‘𝐴))})))
302295, 301mpnanrd 410 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑗 ∈ (𝑌 ∖ {𝑘𝑌 ∣ (𝑖𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0g‘(Scalar‘𝐴))})})) → ¬ (𝑖𝑋 ↦ (𝑗𝑊𝑖)) ≠ (𝑋 × {(0g‘(Scalar‘𝐴))}))
303 nne 2988 . . . . . . . . . . . . . . . . . . . . . . . 24 (¬ (𝑖𝑋 ↦ (𝑗𝑊𝑖)) ≠ (𝑋 × {(0g‘(Scalar‘𝐴))}) ↔ (𝑖𝑋 ↦ (𝑗𝑊𝑖)) = (𝑋 × {(0g‘(Scalar‘𝐴))}))
304302, 303sylib 219 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑗 ∈ (𝑌 ∖ {𝑘𝑌 ∣ (𝑖𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0g‘(Scalar‘𝐴))})})) → (𝑖𝑋 ↦ (𝑗𝑊𝑖)) = (𝑋 × {(0g‘(Scalar‘𝐴))}))
305304, 255syl6eq 2847 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑗 ∈ (𝑌 ∖ {𝑘𝑌 ∣ (𝑖𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0g‘(Scalar‘𝐴))})})) → (𝑖𝑋 ↦ (𝑗𝑊𝑖)) = (𝑖𝑋 ↦ (0g‘(Scalar‘𝐴))))
306 ovex 7048 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗𝑊𝑖) ∈ V
307306rgenw 3117 . . . . . . . . . . . . . . . . . . . . . . 23 𝑖𝑋 (𝑗𝑊𝑖) ∈ V
308 mpteqb 6653 . . . . . . . . . . . . . . . . . . . . . . 23 (∀𝑖𝑋 (𝑗𝑊𝑖) ∈ V → ((𝑖𝑋 ↦ (𝑗𝑊𝑖)) = (𝑖𝑋 ↦ (0g‘(Scalar‘𝐴))) ↔ ∀𝑖𝑋 (𝑗𝑊𝑖) = (0g‘(Scalar‘𝐴))))
309307, 308ax-mp 5 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖𝑋 ↦ (𝑗𝑊𝑖)) = (𝑖𝑋 ↦ (0g‘(Scalar‘𝐴))) ↔ ∀𝑖𝑋 (𝑗𝑊𝑖) = (0g‘(Scalar‘𝐴)))
310305, 309sylib 219 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗 ∈ (𝑌 ∖ {𝑘𝑌 ∣ (𝑖𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0g‘(Scalar‘𝐴))})})) → ∀𝑖𝑋 (𝑗𝑊𝑖) = (0g‘(Scalar‘𝐴)))
311310r19.21bi 3175 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑗 ∈ (𝑌 ∖ {𝑘𝑌 ∣ (𝑖𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0g‘(Scalar‘𝐴))})})) ∧ 𝑖𝑋) → (𝑗𝑊𝑖) = (0g‘(Scalar‘𝐴)))
312311oveq1d 7031 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑗 ∈ (𝑌 ∖ {𝑘𝑌 ∣ (𝑖𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0g‘(Scalar‘𝐴))})})) ∧ 𝑖𝑋) → ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖) = ((0g‘(Scalar‘𝐴))( ·𝑠𝐵)𝑖))
313117, 1, 3drgext0g 30596 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (0g𝐸) = (0g𝐵))
314117, 1, 3drgext0gsca 30598 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (0g𝐵) = (0g‘(Scalar‘𝐵)))
315313, 178, 3143eqtr3d 2839 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (0g‘(Scalar‘𝐴)) = (0g‘(Scalar‘𝐵)))
316315ad2antrr 722 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑗 ∈ (𝑌 ∖ {𝑘𝑌 ∣ (𝑖𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0g‘(Scalar‘𝐴))})})) ∧ 𝑖𝑋) → (0g‘(Scalar‘𝐴)) = (0g‘(Scalar‘𝐵)))
317316oveq1d 7031 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑗 ∈ (𝑌 ∖ {𝑘𝑌 ∣ (𝑖𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0g‘(Scalar‘𝐴))})})) ∧ 𝑖𝑋) → ((0g‘(Scalar‘𝐴))( ·𝑠𝐵)𝑖) = ((0g‘(Scalar‘𝐵))( ·𝑠𝐵)𝑖))
318209ad2antrr 722 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑗 ∈ (𝑌 ∖ {𝑘𝑌 ∣ (𝑖𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0g‘(Scalar‘𝐴))})})) ∧ 𝑖𝑋) → 𝐵 ∈ LMod)
319295, 243syldanl 601 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑗 ∈ (𝑌 ∖ {𝑘𝑌 ∣ (𝑖𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0g‘(Scalar‘𝐴))})})) ∧ 𝑖𝑋) → 𝑖 ∈ (Base‘𝐵))
320 eqid 2795 . . . . . . . . . . . . . . . . . . . . 21 (Scalar‘𝐵) = (Scalar‘𝐵)
321 eqid 2795 . . . . . . . . . . . . . . . . . . . . 21 (0g‘(Scalar‘𝐵)) = (0g‘(Scalar‘𝐵))
322234, 320, 245, 321, 205lmod0vs 19357 . . . . . . . . . . . . . . . . . . . 20 ((𝐵 ∈ LMod ∧ 𝑖 ∈ (Base‘𝐵)) → ((0g‘(Scalar‘𝐵))( ·𝑠𝐵)𝑖) = (0g𝐵))
323318, 319, 322syl2anc 584 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑗 ∈ (𝑌 ∖ {𝑘𝑌 ∣ (𝑖𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0g‘(Scalar‘𝐴))})})) ∧ 𝑖𝑋) → ((0g‘(Scalar‘𝐵))( ·𝑠𝐵)𝑖) = (0g𝐵))
324312, 317, 3233eqtrd 2835 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑗 ∈ (𝑌 ∖ {𝑘𝑌 ∣ (𝑖𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0g‘(Scalar‘𝐴))})})) ∧ 𝑖𝑋) → ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖) = (0g𝐵))
325293, 324mpteq2da 5054 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗 ∈ (𝑌 ∖ {𝑘𝑌 ∣ (𝑖𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0g‘(Scalar‘𝐴))})})) → (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)) = (𝑖𝑋 ↦ (0g𝐵)))
326325oveq2d 7032 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ (𝑌 ∖ {𝑘𝑌 ∣ (𝑖𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0g‘(Scalar‘𝐴))})})) → (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖))) = (𝐵 Σg (𝑖𝑋 ↦ (0g𝐵))))
327209, 211syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐵 ∈ Abel)
328 ablgrp 18638 . . . . . . . . . . . . . . . . . . 19 (𝐵 ∈ Abel → 𝐵 ∈ Grp)
329 grpmnd 17868 . . . . . . . . . . . . . . . . . . 19 (𝐵 ∈ Grp → 𝐵 ∈ Mnd)
330327, 328, 3293syl 18 . . . . . . . . . . . . . . . . . 18 (𝜑𝐵 ∈ Mnd)
331205gsumz 17813 . . . . . . . . . . . . . . . . . 18 ((𝐵 ∈ Mnd ∧ 𝑋 ∈ (LBasis‘𝐶)) → (𝐵 Σg (𝑖𝑋 ↦ (0g𝐵))) = (0g𝐵))
332330, 16, 331syl2anc 584 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐵 Σg (𝑖𝑋 ↦ (0g𝐵))) = (0g𝐵))
333332adantr 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ (𝑌 ∖ {𝑘𝑌 ∣ (𝑖𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0g‘(Scalar‘𝐴))})})) → (𝐵 Σg (𝑖𝑋 ↦ (0g𝐵))) = (0g𝐵))
334314adantr 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑗 ∈ (𝑌 ∖ {𝑘𝑌 ∣ (𝑖𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0g‘(Scalar‘𝐴))})})) → (0g𝐵) = (0g‘(Scalar‘𝐵)))
335326, 333, 3343eqtrd 2835 . . . . . . . . . . . . . . 15 ((𝜑𝑗 ∈ (𝑌 ∖ {𝑘𝑌 ∣ (𝑖𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0g‘(Scalar‘𝐴))})})) → (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖))) = (0g‘(Scalar‘𝐵)))
336335, 143suppss2 7715 . . . . . . . . . . . . . 14 (𝜑 → ((𝑗𝑌 ↦ (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)))) supp (0g‘(Scalar‘𝐵))) ⊆ {𝑘𝑌 ∣ (𝑖𝑋 ↦ (𝑘𝑊𝑖)) ≠ (𝑋 × {(0g‘(Scalar‘𝐴))})})
337 suppssfifsupp 8694 . . . . . . . . . . . . . 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)))
338204, 253, 254, 284, 336, 337syl32anc 1371 . . . . . . . . . . . . 13 (𝜑 → (𝑗𝑌 ↦ (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)))) finSupp (0g‘(Scalar‘𝐵)))
339 eqidd 2796 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑗𝑌 ↦ (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)))) = (𝑗𝑌 ↦ (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)))))
340 ovexd 7050 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗𝑌) → (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖))) ∈ V)
341339, 340fvmpt2d 6647 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗𝑌) → ((𝑗𝑌 ↦ (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖))))‘𝑗) = (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖))))
342341oveq1d 7031 . . . . . . . . . . . . . . . 16 ((𝜑𝑗𝑌) → (((𝑗𝑌 ↦ (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖))))‘𝑗)( ·𝑠𝐵)𝑗) = ((𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)))( ·𝑠𝐵)𝑗))
343342mpteq2dva 5055 . . . . . . . . . . . . . . 15 (𝜑 → (𝑗𝑌 ↦ (((𝑗𝑌 ↦ (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖))))‘𝑗)( ·𝑠𝐵)𝑗)) = (𝑗𝑌 ↦ ((𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)))( ·𝑠𝐵)𝑗)))
344343oveq2d 7032 . . . . . . . . . . . . . 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))))
345119adantr 481 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗𝑌) → ( ·𝑠𝐴) = ( ·𝑠𝐵))
34643ad2antrr 722 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → (.r𝐸) = ( ·𝑠𝐴))
347346oveqd 7033 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → ((𝑗𝑊𝑖)(.r𝐸)𝑖) = ((𝑗𝑊𝑖)( ·𝑠𝐴)𝑖))
348347mpteq2dva 5055 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗𝑌) → (𝑖𝑋 ↦ ((𝑗𝑊𝑖)(.r𝐸)𝑖)) = (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐴)𝑖)))
349118adantr 481 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑗𝑌) → (.r𝐸) = ( ·𝑠𝐵))
350349oveqd 7033 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑗𝑌) → ((𝑗𝑊𝑖)(.r𝐸)𝑖) = ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖))
351350mpteq2dv 5056 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗𝑌) → (𝑖𝑋 ↦ ((𝑗𝑊𝑖)(.r𝐸)𝑖)) = (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)))
352348, 351eqtr3d 2833 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗𝑌) → (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐴)𝑖)) = (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)))
353352oveq2d 7032 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗𝑌) → (𝐴 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐴)𝑖))) = (𝐴 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖))))
354 eqidd 2796 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗𝑌) → 𝑗 = 𝑗)
355345, 353, 354oveq123d 7037 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗𝑌) → ((𝐴 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐴)𝑖)))( ·𝑠𝐴)𝑗) = ((𝐴 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)))( ·𝑠𝐵)𝑗))
356202oveq1d 7031 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑗𝑌) → ((𝐴 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)))( ·𝑠𝐵)𝑗) = ((𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)))( ·𝑠𝐵)𝑗))
357355, 356eqtrd 2831 . . . . . . . . . . . . . . . . 17 ((𝜑𝑗𝑌) → ((𝐴 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐴)𝑖)))( ·𝑠𝐴)𝑗) = ((𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)))( ·𝑠𝐵)𝑗))
358357mpteq2dva 5055 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑗𝑌 ↦ ((𝐴 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐴)𝑖)))( ·𝑠𝐴)𝑗)) = (𝑗𝑌 ↦ ((𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)))( ·𝑠𝐵)𝑗)))
359358oveq2d 7032 . . . . . . . . . . . . . . 15 (𝜑 → (𝐴 Σg (𝑗𝑌 ↦ ((𝐴 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐴)𝑖)))( ·𝑠𝐴)𝑗))) = (𝐴 Σg (𝑗𝑌 ↦ ((𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)))( ·𝑠𝐵)𝑗))))
36010, 21sraring 30591 . . . . . . . . . . . . . . . . . . . 20 ((𝐸 ∈ Ring ∧ 𝑉 ⊆ (Base‘𝐸)) → 𝐴 ∈ Ring)
361163, 36, 360syl2anc 584 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐴 ∈ Ring)
362 ringcmn 19021 . . . . . . . . . . . . . . . . . . 19 (𝐴 ∈ Ring → 𝐴 ∈ CMnd)
363361, 362syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑𝐴 ∈ CMnd)
364163adantr 481 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑗𝑌𝑖𝑋)) → 𝐸 ∈ Ring)
365 eqid 2795 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (LBasis‘𝐵) = (LBasis‘𝐵)
366234, 365lbsss 19539 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑌 ∈ (LBasis‘𝐵) → 𝑌 ⊆ (Base‘𝐵))
367143, 366syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑𝑌 ⊆ (Base‘𝐵))
368367, 241sseqtr4d 3929 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑𝑌 ⊆ (Base‘𝐸))
369368adantr 481 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ (𝑗𝑌𝑖𝑋)) → 𝑌 ⊆ (Base‘𝐸))
370 simprl 767 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ (𝑗𝑌𝑖𝑋)) → 𝑗𝑌)
371369, 370sseldd 3890 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑗𝑌𝑖𝑋)) → 𝑗 ∈ (Base‘𝐸))
37221, 67ringcl 19001 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐸 ∈ Ring ∧ 𝑖 ∈ (Base‘𝐸) ∧ 𝑗 ∈ (Base‘𝐸)) → (𝑖(.r𝐸)𝑗) ∈ (Base‘𝐸))
373364, 239, 371, 372syl3anc 1364 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑗𝑌𝑖𝑋)) → (𝑖(.r𝐸)𝑗) ∈ (Base‘𝐸))
37437adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑗𝑌𝑖𝑋)) → (Base‘𝐸) = (Base‘𝐴))
375373, 374eleqtrd 2885 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑗𝑌𝑖𝑋)) → (𝑖(.r𝐸)𝑗) ∈ (Base‘𝐴))
376375ralrimivva 3158 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ∀𝑗𝑌𝑖𝑋 (𝑖(.r𝐸)𝑗) ∈ (Base‘𝐴))
377 fedgmullem.d . . . . . . . . . . . . . . . . . . . . 21 𝐷 = (𝑗𝑌, 𝑖𝑋 ↦ (𝑖(.r𝐸)𝑗))
378377fmpo 7622 . . . . . . . . . . . . . . . . . . . 20 (∀𝑗𝑌𝑖𝑋 (𝑖(.r𝐸)𝑗) ∈ (Base‘𝐴) ↔ 𝐷:(𝑌 × 𝑋)⟶(Base‘𝐴))
379376, 378sylib 219 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐷:(𝑌 × 𝑋)⟶(Base‘𝐴))
38072, 73, 75, 74, 15, 157, 379, 148lcomf 19363 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑊𝑓 ( ·𝑠𝐴)𝐷):(𝑌 × 𝑋)⟶(Base‘𝐴))
38172, 73, 75, 74, 15, 157, 379, 148, 135, 136, 161lcomfsupp 19364 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑊𝑓 ( ·𝑠𝐴)𝐷) finSupp (0g𝐴))
38274, 135, 363, 143, 16, 380, 381gsumxp 18816 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐴 Σg (𝑊𝑓 ( ·𝑠𝐴)𝐷)) = (𝐴 Σg (𝑗𝑌 ↦ (𝐴 Σg (𝑖𝑋 ↦ (𝑗(𝑊𝑓 ( ·𝑠𝐴)𝐷)𝑖))))))
383 fedgmullem2.2 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐴 Σg (𝑊𝑓 ( ·𝑠𝐴)𝐷)) = (0g𝐴))
3841633ad2ant1 1126 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑𝑗𝑌𝑖𝑋) → 𝐸 ∈ Ring)
3851573ad2ant1 1126 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝜑𝑗𝑌𝑖𝑋) → 𝑊:(𝑌 × 𝑋)⟶(Base‘(Scalar‘𝐴)))
38657, 55eqtrd 2831 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝜑𝑉 = (Base‘(Scalar‘𝐶)))
387386, 36eqsstrrd 3927 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝜑 → (Base‘(Scalar‘𝐶)) ⊆ (Base‘𝐸))
38861, 387eqsstrd 3926 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝜑 → (Base‘(Scalar‘𝐴)) ⊆ (Base‘𝐸))
389388, 37sseqtrd 3928 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝜑 → (Base‘(Scalar‘𝐴)) ⊆ (Base‘𝐴))
3903893ad2ant1 1126 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝜑𝑗𝑌𝑖𝑋) → (Base‘(Scalar‘𝐴)) ⊆ (Base‘𝐴))
391385, 390fssd 6396 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝜑𝑗𝑌𝑖𝑋) → 𝑊:(𝑌 × 𝑋)⟶(Base‘𝐴))
392 simp2 1130 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝜑𝑗𝑌𝑖𝑋) → 𝑗𝑌)
393 simp3 1131 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝜑𝑗𝑌𝑖𝑋) → 𝑖𝑋)
394391, 392, 393fovrnd 7176 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝜑𝑗𝑌𝑖𝑋) → (𝑗𝑊𝑖) ∈ (Base‘𝐴))
395373ad2ant1 1126 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝜑𝑗𝑌𝑖𝑋) → (Base‘𝐸) = (Base‘𝐴))
396394, 395eleqtrrd 2886 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑𝑗𝑌𝑖𝑋) → (𝑗𝑊𝑖) ∈ (Base‘𝐸))
3972393impb 1108 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑𝑗𝑌𝑖𝑋) → 𝑖 ∈ (Base‘𝐸))
3983713impb 1108 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑𝑗𝑌𝑖𝑋) → 𝑗 ∈ (Base‘𝐸))
39921, 67ringass 19004 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝐸 ∈ Ring ∧ ((𝑗𝑊𝑖) ∈ (Base‘𝐸) ∧ 𝑖 ∈ (Base‘𝐸) ∧ 𝑗 ∈ (Base‘𝐸))) → (((𝑗𝑊𝑖)(.r𝐸)𝑖)(.r𝐸)𝑗) = ((𝑗𝑊𝑖)(.r𝐸)(𝑖(.r𝐸)𝑗)))
400384, 396, 397, 398, 399syl13anc 1365 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑𝑗𝑌𝑖𝑋) → (((𝑗𝑊𝑖)(.r𝐸)𝑖)(.r𝐸)𝑗) = ((𝑗𝑊𝑖)(.r𝐸)(𝑖(.r𝐸)𝑗)))
401400mpoeq3dva 7089 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑 → (𝑗𝑌, 𝑖𝑋 ↦ (((𝑗𝑊𝑖)(.r𝐸)𝑖)(.r𝐸)𝑗)) = (𝑗𝑌, 𝑖𝑋 ↦ ((𝑗𝑊𝑖)(.r𝐸)(𝑖(.r𝐸)𝑗))))
402 ovexd 7050 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑𝑗𝑌𝑖𝑋) → (𝑗𝑊𝑖) ∈ V)
403 ovexd 7050 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑𝑗𝑌𝑖𝑋) → (𝑖(.r𝐸)𝑗) ∈ V)
404 fnov 7138 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑊 Fn (𝑌 × 𝑋) ↔ 𝑊 = (𝑗𝑌, 𝑖𝑋 ↦ (𝑗𝑊𝑖)))
405158, 404sylib 219 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜑𝑊 = (𝑗𝑌, 𝑖𝑋 ↦ (𝑗𝑊𝑖)))
406377a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜑𝐷 = (𝑗𝑌, 𝑖𝑋 ↦ (𝑖(.r𝐸)𝑗)))
407143, 16, 402, 403, 405, 406offval22 7639 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑 → (𝑊𝑓 (.r𝐸)𝐷) = (𝑗𝑌, 𝑖𝑋 ↦ ((𝑗𝑊𝑖)(.r𝐸)(𝑖(.r𝐸)𝑗))))
408 ofeq 7269 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((.r𝐸) = ( ·𝑠𝐴) → ∘𝑓 (.r𝐸) = ∘𝑓 ( ·𝑠𝐴))
40943, 408syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜑 → ∘𝑓 (.r𝐸) = ∘𝑓 ( ·𝑠𝐴))
410409oveqd 7033 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑 → (𝑊𝑓 (.r𝐸)𝐷) = (𝑊𝑓 ( ·𝑠𝐴)𝐷))
411401, 407, 4103eqtr2rd 2838 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑 → (𝑊𝑓 ( ·𝑠𝐴)𝐷) = (𝑗𝑌, 𝑖𝑋 ↦ (((𝑗𝑊𝑖)(.r𝐸)𝑖)(.r𝐸)𝑗)))
412411ad2antrr 722 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → (𝑊𝑓 ( ·𝑠𝐴)𝐷) = (𝑗𝑌, 𝑖𝑋 ↦ (((𝑗𝑊𝑖)(.r𝐸)𝑖)(.r𝐸)𝑗)))
413412oveqd 7033 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → (𝑗(𝑊𝑓 ( ·𝑠𝐴)𝐷)𝑖) = (𝑗(𝑗𝑌, 𝑖𝑋 ↦ (((𝑗𝑊𝑖)(.r𝐸)𝑖)(.r𝐸)𝑗))𝑖))
414 simplr 765 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → 𝑗𝑌)
415 ovexd 7050 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → (((𝑗𝑊𝑖)(.r𝐸)𝑖)(.r𝐸)𝑗) ∈ V)
416 eqid 2795 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑗𝑌, 𝑖𝑋 ↦ (((𝑗𝑊𝑖)(.r𝐸)𝑖)(.r𝐸)𝑗)) = (𝑗𝑌, 𝑖𝑋 ↦ (((𝑗𝑊𝑖)(.r𝐸)𝑖)(.r𝐸)𝑗))
417416ovmpt4g 7153 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑗𝑌𝑖𝑋 ∧ (((𝑗𝑊𝑖)(.r𝐸)𝑖)(.r𝐸)𝑗) ∈ V) → (𝑗(𝑗𝑌, 𝑖𝑋 ↦ (((𝑗𝑊𝑖)(.r𝐸)𝑖)(.r𝐸)𝑗))𝑖) = (((𝑗𝑊𝑖)(.r𝐸)𝑖)(.r𝐸)𝑗))
418414, 222, 415, 417syl3anc 1364 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → (𝑗(𝑗𝑌, 𝑖𝑋 ↦ (((𝑗𝑊𝑖)(.r𝐸)𝑖)(.r𝐸)𝑗))𝑖) = (((𝑗𝑊𝑖)(.r𝐸)𝑖)(.r𝐸)𝑗))
419413, 418eqtrd 2831 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → (𝑗(𝑊𝑓 ( ·𝑠𝐴)𝐷)𝑖) = (((𝑗𝑊𝑖)(.r𝐸)𝑖)(.r𝐸)𝑗))
420419mpteq2dva 5055 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑗𝑌) → (𝑖𝑋 ↦ (𝑗(𝑊𝑓 ( ·𝑠𝐴)𝐷)𝑖)) = (𝑖𝑋 ↦ (((𝑗𝑊𝑖)(.r𝐸)𝑖)(.r𝐸)𝑗)))
421420oveq2d 7032 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗𝑌) → (𝐸 Σg (𝑖𝑋 ↦ (𝑗(𝑊𝑓 ( ·𝑠𝐴)𝐷)𝑖))) = (𝐸 Σg (𝑖𝑋 ↦ (((𝑗𝑊𝑖)(.r𝐸)𝑖)(.r𝐸)𝑗))))
422163adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑗𝑌) → 𝐸 ∈ Ring)
423368sselda 3889 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑗𝑌) → 𝑗 ∈ (Base‘𝐸))
424163ad2antrr 722 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → 𝐸 ∈ Ring)
425387ad2antrr 722 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → (Base‘(Scalar‘𝐶)) ⊆ (Base‘𝐸))
426425, 220sseldd 3890 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → (𝑗𝑊𝑖) ∈ (Base‘𝐸))
42721, 67ringcl 19001 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐸 ∈ Ring ∧ (𝑗𝑊𝑖) ∈ (Base‘𝐸) ∧ 𝑖 ∈ (Base‘𝐸)) → ((𝑗𝑊𝑖)(.r𝐸)𝑖) ∈ (Base‘𝐸))
428424, 426, 240, 427syl3anc 1364 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑗𝑌) ∧ 𝑖𝑋) → ((𝑗𝑊𝑖)(.r𝐸)𝑖) ∈ (Base‘𝐸))
429313adantr 481 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑗𝑌) → (0g𝐸) = (0g𝐵))
430246, 351, 4293brtr4d 4994 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑗𝑌) → (𝑖𝑋 ↦ ((𝑗𝑊𝑖)(.r𝐸)𝑖)) finSupp (0g𝐸))
43121, 168, 93, 67, 422, 145, 423, 428, 430gsummulc1 19046 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗𝑌) → (𝐸 Σg (𝑖𝑋 ↦ (((𝑗𝑊𝑖)(.r𝐸)𝑖)(.r𝐸)𝑗))) = ((𝐸 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)(.r𝐸)𝑖)))(.r𝐸)𝑗))
432421, 431eqtrd 2831 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗𝑌) → (𝐸 Σg (𝑖𝑋 ↦ (𝑗(𝑊𝑓 ( ·𝑠𝐴)𝐷)𝑖))) = ((𝐸 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)(.r𝐸)𝑖)))(.r𝐸)𝑗))
433145mptexd 6853 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗𝑌) → (𝑖𝑋 ↦ (𝑗(𝑊𝑓 ( ·𝑠𝐴)𝐷)𝑖)) ∈ V)
43415adantr 481 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗𝑌) → 𝐴 ∈ LMod)
43536adantr 481 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗𝑌) → 𝑉 ⊆ (Base‘𝐸))
43610, 433, 195, 434, 435gsumsra 30493 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗𝑌) → (𝐸 Σg (𝑖𝑋 ↦ (𝑗(𝑊𝑓 ( ·𝑠𝐴)𝐷)𝑖))) = (𝐴 Σg (𝑖𝑋 ↦ (𝑗(𝑊𝑓 ( ·𝑠𝐴)𝐷)𝑖))))
437145mptexd 6853 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑗𝑌) → (𝑖𝑋 ↦ ((𝑗𝑊𝑖)(.r𝐸)𝑖)) ∈ V)
43810, 437, 195, 434, 435gsumsra 30493 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑗𝑌) → (𝐸 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)(.r𝐸)𝑖))) = (𝐴 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)(.r𝐸)𝑖))))
439438oveq1d 7031 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗𝑌) → ((𝐸 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)(.r𝐸)𝑖)))(.r𝐸)𝑗) = ((𝐴 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)(.r𝐸)𝑖)))(.r𝐸)𝑗))
44043adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑗𝑌) → (.r𝐸) = ( ·𝑠𝐴))
441348oveq2d 7032 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑𝑗𝑌) → (𝐴 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)(.r𝐸)𝑖))) = (𝐴 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐴)𝑖))))
442440, 441, 354oveq123d 7037 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑𝑗𝑌) → ((𝐴 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)(.r𝐸)𝑖)))(.r𝐸)𝑗) = ((𝐴 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐴)𝑖)))( ·𝑠𝐴)𝑗))
443439, 442eqtrd 2831 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑗𝑌) → ((𝐸 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)(.r𝐸)𝑖)))(.r𝐸)𝑗) = ((𝐴 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐴)𝑖)))( ·𝑠𝐴)𝑗))
444432, 436, 4433eqtr3d 2839 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑗𝑌) → (𝐴 Σg (𝑖𝑋 ↦ (𝑗(𝑊𝑓 ( ·𝑠𝐴)𝐷)𝑖))) = ((𝐴 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐴)𝑖)))( ·𝑠𝐴)𝑗))
445444mpteq2dva 5055 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑗𝑌 ↦ (𝐴 Σg (𝑖𝑋 ↦ (𝑗(𝑊𝑓 ( ·𝑠𝐴)𝐷)𝑖)))) = (𝑗𝑌 ↦ ((𝐴 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐴)𝑖)))( ·𝑠𝐴)𝑗)))
446445oveq2d 7032 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝐴 Σg (𝑗𝑌 ↦ (𝐴 Σg (𝑖𝑋 ↦ (𝑗(𝑊𝑓 ( ·𝑠𝐴)𝐷)𝑖))))) = (𝐴 Σg (𝑗𝑌 ↦ ((𝐴 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐴)𝑖)))( ·𝑠𝐴)𝑗))))
447382, 383, 4463eqtr3rd 2840 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐴 Σg (𝑗𝑌 ↦ ((𝐴 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐴)𝑖)))( ·𝑠𝐴)𝑗))) = (0g𝐴))
44810, 1, 9drgext0g 30596 . . . . . . . . . . . . . . . 16 (𝜑 → (0g𝐸) = (0g𝐴))
449447, 448, 3133eqtr2d 2837 . . . . . . . . . . . . . . 15 (𝜑 → (𝐴 Σg (𝑗𝑌 ↦ ((𝐴 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐴)𝑖)))( ·𝑠𝐴)𝑗))) = (0g𝐵))
45010, 1, 9, 11, 2, 143drgextgsum 30601 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐸 Σg (𝑗𝑌 ↦ ((𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)))( ·𝑠𝐵)𝑗))) = (𝐴 Σg (𝑗𝑌 ↦ ((𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)))( ·𝑠𝐵)𝑗))))
451117, 1, 3, 5, 104, 143drgextgsum 30601 . . . . . . . . . . . . . . . 16 (𝜑 → (𝐸 Σg (𝑗𝑌 ↦ ((𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)))( ·𝑠𝐵)𝑗))) = (𝐵 Σg (𝑗𝑌 ↦ ((𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)))( ·𝑠𝐵)𝑗))))
452450, 451eqtr3d 2833 . . . . . . . . . . . . . . 15 (𝜑 → (𝐴 Σg (𝑗𝑌 ↦ ((𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)))( ·𝑠𝐵)𝑗))) = (𝐵 Σg (𝑗𝑌 ↦ ((𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)))( ·𝑠𝐵)𝑗))))
453359, 449, 4523eqtr3rd 2840 . . . . . . . . . . . . . 14 (𝜑 → (𝐵 Σg (𝑗𝑌 ↦ ((𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)))( ·𝑠𝐵)𝑗))) = (0g𝐵))
454344, 453eqtrd 2831 . . . . . . . . . . . . 13 (𝜑 → (𝐵 Σg (𝑗𝑌 ↦ (((𝑗𝑌 ↦ (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖))))‘𝑗)( ·𝑠𝐵)𝑗))) = (0g𝐵))
455 breq1 4965 . . . . . . . . . . . . . . . 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))))
456 nfmpt1 5058 . . . . . . . . . . . . . . . . . . . 20 𝑗(𝑗𝑌 ↦ (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖))))
457456nfeq2 2964 . . . . . . . . . . . . . . . . . . 19 𝑗 𝑏 = (𝑗𝑌 ↦ (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖))))
458 fveq1 6537 . . . . . . . . . . . . . . . . . . . . 21 (𝑏 = (𝑗𝑌 ↦ (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)))) → (𝑏𝑗) = ((𝑗𝑌 ↦ (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖))))‘𝑗))
459458oveq1d 7031 . . . . . . . . . . . . . . . . . . . 20 (𝑏 = (𝑗𝑌 ↦ (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)))) → ((𝑏𝑗)( ·𝑠𝐵)𝑗) = (((𝑗𝑌 ↦ (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖))))‘𝑗)( ·𝑠𝐵)𝑗))
460459adantr 481 . . . . . . . . . . . . . . . . . . 19 ((𝑏 = (𝑗𝑌 ↦ (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)))) ∧ 𝑗𝑌) → ((𝑏𝑗)( ·𝑠𝐵)𝑗) = (((𝑗𝑌 ↦ (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖))))‘𝑗)( ·𝑠𝐵)𝑗))
461457, 460mpteq2da 5054 . . . . . . . . . . . . . . . . . 18 (𝑏 = (𝑗𝑌 ↦ (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)))) → (𝑗𝑌 ↦ ((𝑏𝑗)( ·𝑠𝐵)𝑗)) = (𝑗𝑌 ↦ (((𝑗𝑌 ↦ (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖))))‘𝑗)( ·𝑠𝐵)𝑗)))
462461oveq2d 7032 . . . . . . . . . . . . . . . . 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))))
463462eqeq1d 2797 . . . . . . . . . . . . . . . 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)))
464455, 463anbi12d 630 . . . . . . . . . . . . . . 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))))
465 eqeq1 2799 . . . . . . . . . . . . . . 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))})))
466464, 465imbi12d 346 . . . . . . . . . . . . . 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))}))))
467365lbslinds 20659 . . . . . . . . . . . . . . . 16 (LBasis‘𝐵) ⊆ (LIndS‘𝐵)
468467, 143sseldi 3887 . . . . . . . . . . . . . . 15 (𝜑𝑌 ∈ (LIndS‘𝐵))
469 eqid 2795 . . . . . . . . . . . . . . . . 17 (Base‘(Scalar‘𝐵)) = (Base‘(Scalar‘𝐵))
470234, 469, 320, 245, 205, 321islinds5 30580 . . . . . . . . . . . . . . . 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))}))))
471470biimpa 477 . . . . . . . . . . . . . . 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))})))
472209, 367, 468, 471syl21anc 834 . . . . . . . . . . . . . 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))})))
473 fvexd 6553 . . . . . . . . . . . . . . 15 (𝜑 → (Base‘(Scalar‘𝐵)) ∈ V)
474 elmapg 8269 . . . . . . . . . . . . . . . 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))))
475474biimpar 478 . . . . . . . . . . . . . . 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))
476473, 143, 252, 475syl21anc 834 . . . . . . . . . . . . . 14 (𝜑 → (𝑗𝑌 ↦ (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)))) ∈ ((Base‘(Scalar‘𝐵)) ↑𝑚 𝑌))
477466, 472, 476rspcdva 3565 . . . . . . . . . . . . 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))})))
478338, 454, 477mp2and 695 . . . . . . . . . . . 12 (𝜑 → (𝑗𝑌 ↦ (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)))) = (𝑌 × {(0g‘(Scalar‘𝐵))}))
479 fconstmpt 5500 . . . . . . . . . . . 12 (𝑌 × {(0g‘(Scalar‘𝐵))}) = (𝑗𝑌 ↦ (0g‘(Scalar‘𝐵)))
480478, 479syl6eq 2847 . . . . . . . . . . 11 (𝜑 → (𝑗𝑌 ↦ (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)))) = (𝑗𝑌 ↦ (0g‘(Scalar‘𝐵))))
481 ovex 7048 . . . . . . . . . . . . 13 (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖))) ∈ V
482481rgenw 3117 . . . . . . . . . . . 12 𝑗𝑌 (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖))) ∈ V
483 mpteqb 6653 . . . . . . . . . . . 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))))
484482, 483ax-mp 5 . . . . . . . . . . 11 ((𝑗𝑌 ↦ (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖)))) = (𝑗𝑌 ↦ (0g‘(Scalar‘𝐵))) ↔ ∀𝑗𝑌 (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖))) = (0g‘(Scalar‘𝐵)))
485480, 484sylib 219 . . . . . . . . . 10 (𝜑 → ∀𝑗𝑌 (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖))) = (0g‘(Scalar‘𝐵)))
486485r19.21bi 3175 . . . . . . . . 9 ((𝜑𝑗𝑌) → (𝐵 Σg (𝑖𝑋 ↦ ((𝑗𝑊𝑖)( ·𝑠𝐵)𝑖))) = (0g‘(Scalar‘𝐵)))
487313, 448, 3143eqtr3rd 2840 . . . . . . . . . 10 (𝜑 → (0g‘(Scalar‘𝐵)) = (0g𝐴))
488487adantr 481 . . . . . . . . 9 ((𝜑𝑗𝑌) → (0g‘(Scalar‘𝐵)) = (0g𝐴))
489203, 486, 4883eqtrd 2835 . . . . . . . 8 ((𝜑𝑗𝑌) → (𝐴 Σg (𝑖𝑋 ↦ (((𝑖𝑋 ↦ (𝑗𝑊𝑖))‘𝑖)( ·𝑠𝐴)𝑖))) = (0g𝐴))
490184, 489jca 512 . . . . . . 7 ((𝜑𝑗𝑌) → ((𝑖𝑋 ↦ (𝑗𝑊𝑖)) finSupp (0g‘(Scalar‘𝐴)) ∧ (𝐴 Σg (𝑖𝑋 ↦ (((𝑖𝑋 ↦ (𝑗𝑊𝑖))‘𝑖)( ·𝑠𝐴)𝑖))) = (0g𝐴)))
491187fmpttd 6742 . . . . . . . . 9 ((𝜑𝑗𝑌) → (𝑖𝑋 ↦ (𝑗𝑊𝑖)):𝑋⟶(Base‘(Scalar‘𝐴)))
492 fvexd 6553 . . . . . . . . . 10 ((𝜑𝑗𝑌) → (Base‘(Scalar‘𝐴)) ∈ V)
493492, 145elmapd 8270 . . . . . . . . 9 ((𝜑𝑗𝑌) → ((𝑖𝑋 ↦ (𝑗𝑊𝑖)) ∈ ((Base‘(Scalar‘𝐴)) ↑𝑚 𝑋) ↔ (𝑖𝑋 ↦ (𝑗𝑊𝑖)):𝑋⟶(Base‘(Scalar‘𝐴))))
494491, 493mpbird 258 . . . . . . . 8 ((𝜑𝑗𝑌) → (𝑖𝑋 ↦ (𝑗𝑊𝑖)) ∈ ((Base‘(Scalar‘𝐴)) ↑𝑚 𝑋))
495 simpr 485 . . . . . . . . . . 11 (((𝜑𝑗𝑌) ∧ 𝑤 = (𝑖𝑋 ↦ (𝑗𝑊𝑖))) → 𝑤 = (𝑖𝑋 ↦ (𝑗𝑊𝑖)))
496495breq1d 4972 . . . . . . . . . 10 (((𝜑𝑗𝑌) ∧ 𝑤 = (𝑖𝑋 ↦ (𝑗𝑊𝑖))) → (𝑤 finSupp (0g‘(Scalar‘𝐴)) ↔ (𝑖𝑋 ↦ (𝑗𝑊𝑖)) finSupp (0g‘(Scalar‘𝐴))))
497 nfv 1892 . . . . . . . . . . . . . 14 𝑖(𝜑𝑗𝑌)
498 nfmpt1 5058 . . . . . . . . . . . . . . 15 𝑖(𝑖𝑋 ↦ (𝑗𝑊𝑖))
499498nfeq2 2964 . . . . . . . . . . . . . 14 𝑖 𝑤 = (𝑖𝑋 ↦ (𝑗𝑊𝑖))
500497, 499nfan 1881 . . . . . . . . . . . . 13 𝑖((𝜑𝑗𝑌) ∧ 𝑤 = (𝑖𝑋 ↦ (𝑗𝑊𝑖)))
501 simplr 765 . . . . . . . . . . . . . . 15 ((((𝜑𝑗𝑌) ∧ 𝑤 = (𝑖𝑋 ↦ (𝑗𝑊𝑖))) ∧ 𝑖𝑋) → 𝑤 = (𝑖𝑋 ↦ (𝑗𝑊𝑖)))
502501fveq1d 6540 . . . . . . . . . . . . . 14 ((((𝜑𝑗𝑌) ∧ 𝑤 = (𝑖𝑋 ↦ (𝑗𝑊𝑖))) ∧ 𝑖𝑋) → (𝑤𝑖) = ((𝑖𝑋 ↦ (𝑗𝑊𝑖))‘𝑖))
503502oveq1d 7031 . . . . . . . . . . . . 13 ((((𝜑𝑗𝑌) ∧ 𝑤 = (𝑖𝑋 ↦ (𝑗𝑊𝑖))) ∧ 𝑖𝑋) → ((𝑤𝑖)( ·𝑠𝐴)𝑖) = (((𝑖𝑋 ↦ (𝑗𝑊𝑖))‘𝑖)( ·𝑠𝐴)𝑖))
504500, 503mpteq2da 5054 . . . . . . . . . . . 12 (((𝜑𝑗𝑌) ∧ 𝑤 = (𝑖𝑋 ↦ (𝑗𝑊𝑖))) → (𝑖𝑋 ↦ ((𝑤𝑖)( ·𝑠𝐴)𝑖)) = (𝑖𝑋 ↦ (((𝑖𝑋 ↦ (𝑗𝑊𝑖))‘𝑖)( ·𝑠𝐴)𝑖)))
505504oveq2d 7032 . . . . . . . . . . 11 (((𝜑𝑗𝑌) ∧ 𝑤 = (𝑖𝑋 ↦ (𝑗𝑊𝑖))) → (𝐴 Σg (𝑖𝑋 ↦ ((𝑤𝑖)( ·𝑠𝐴)𝑖))) = (𝐴 Σg (𝑖𝑋 ↦ (((𝑖𝑋 ↦ (𝑗𝑊𝑖))‘𝑖)( ·𝑠𝐴)𝑖))))
506505eqeq1d 2797 . . . . . . . . . 10 (((𝜑𝑗𝑌) ∧ 𝑤 = (𝑖𝑋 ↦ (𝑗𝑊𝑖))) → ((𝐴 Σg (𝑖𝑋 ↦ ((𝑤𝑖)( ·𝑠𝐴)𝑖))) = (0g𝐴) ↔ (𝐴 Σg (𝑖𝑋 ↦ (((𝑖𝑋 ↦ (𝑗𝑊𝑖))‘𝑖)( ·𝑠𝐴)𝑖))) = (0g𝐴)))
507496, 506anbi12d 630 . . . . . . . . 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))))
508495eqeq1d 2797 . . . . . . . . 9 (((𝜑𝑗𝑌) ∧ 𝑤 = (𝑖𝑋 ↦ (𝑗𝑊𝑖))) → (𝑤 = (𝑋 × {(0g‘(Scalar‘𝐴))}) ↔ (𝑖𝑋 ↦ (𝑗𝑊𝑖)) = (𝑋 × {(0g‘(Scalar‘𝐴))})))
509507, 508imbi12d 346 . . . . . . . 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))}))))
510494, 509rspcdv 3562 . . . . . . 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))}))))
511140, 490, 510mp2d 49 . . . . . 6 ((𝜑𝑗𝑌) → (𝑖𝑋 ↦ (𝑗𝑊𝑖)) = (𝑋 × {(0g‘(Scalar‘𝐴))}))
512511, 255syl6eq 2847 . . . . 5 ((𝜑𝑗𝑌) → (𝑖𝑋 ↦ (𝑗𝑊𝑖)) = (𝑖𝑋 ↦ (0g‘(Scalar‘𝐴))))
513512, 309sylib 219 . . . 4 ((𝜑𝑗𝑌) → ∀𝑖𝑋 (𝑗𝑊𝑖) = (0g‘(Scalar‘𝐴)))
514513ralrimiva 3149 . . 3 (𝜑 → ∀𝑗𝑌𝑖𝑋 (𝑗𝑊𝑖) = (0g‘(Scalar‘𝐴)))
515 eqidd 2796 . . . 4 ((𝑗 = 𝑘𝑖 = 𝑙) → (0g‘(Scalar‘𝐴)) = (0g‘(Scalar‘𝐴)))
516 fvexd 6553 . . . 4 ((𝜑𝑗𝑌𝑖𝑋) → (0g‘(Scalar‘𝐴)) ∈ V)
517 fvexd 6553 . . . 4 ((𝜑𝑘𝑌𝑙𝑋) → (0g‘(Scalar‘𝐴)) ∈ V)
518158, 515, 516, 517fnmpoovd 7638 . . 3 (𝜑 → (𝑊 = (𝑘𝑌, 𝑙𝑋 ↦ (0g‘(Scalar‘𝐴))) ↔ ∀𝑗𝑌𝑖𝑋 (𝑗𝑊𝑖) = (0g‘(Scalar‘𝐴))))
519514, 518mpbird 258 . 2 (𝜑𝑊 = (𝑘𝑌, 𝑙𝑋 ↦ (0g‘(Scalar‘𝐴))))
520 fconstmpo 7125 . 2 ((𝑌 × 𝑋) × {(0g‘(Scalar‘𝐴))}) = (𝑘𝑌, 𝑙𝑋 ↦ (0g‘(Scalar‘𝐴)))
521519, 520syl6eqr 2849 1 (𝜑𝑊 = ((𝑌 × 𝑋) × {(0g‘(Scalar‘𝐴))}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  w3a 1080   = wceq 1522  wcel 2081  wne 2984  wral 3105  wrex 3106  {crab 3109  Vcvv 3437  cdif 3856  wss 3859  {csn 4472  cop 4478   class class class wbr 4962  cmpt 5041   × cxp 5441  dom cdm 5443  Fun wfun 6219   Fn wfn 6220  wf 6221  cfv 6225  (class class class)co 7016  cmpo 7018  𝑓 cof 7265   supp csupp 7681  𝑚 cmap 8256  Fincfn 8357   finSupp cfsupp 8679  Basecbs 16312  s cress 16313  +gcplusg 16394  .rcmulr 16395  Scalarcsca 16397   ·𝑠 cvsca 16398  0gc0g 16542   Σg cgsu 16543  Mndcmnd 17733  Grpcgrp 17861  SubGrpcsubg 18027  CMndccmn 18633  Abelcabl 18634  Ringcrg 18987  DivRingcdr 19192  SubRingcsubrg 19221  LModclmod 19324  LSubSpclss 19393  LBasisclbs 19536  LVecclvec 19564  subringAlg csra 19630   freeLMod cfrlm 20572  LIndSclinds 20631
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319  ax-cnex 10439  ax-resscn 10440  ax-1cn 10441  ax-icn 10442  ax-addcl 10443  ax-addrcl 10444  ax-mulcl 10445  ax-mulrcl 10446  ax-mulcom 10447  ax-addass 10448  ax-mulass 10449  ax-distr 10450  ax-i2m1 10451  ax-1ne0 10452  ax-1rid 10453  ax-rnegex 10454  ax-rrecex 10455  ax-cnre 10456  ax-pre-lttri 10457  ax-pre-lttrn 10458  ax-pre-ltadd 10459  ax-pre-mulgt0 10460
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-fal 1535  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-nel 3091  df-ral 3110  df-rex 3111  df-reu 3112  df-rmo 3113  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-int 4783  df-iun 4827  df-iin 4828  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-se 5403  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-pred 6023  df-ord 6069  df-on 6070  df-lim 6071  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-isom 6234  df-riota 6977  df-ov 7019  df-oprab 7020  df-mpo 7021  df-of 7267  df-om 7437  df-1st 7545  df-2nd 7546  df-supp 7682  df-wrecs 7798  df-recs 7860  df-rdg 7898  df-1o 7953  df-oadd 7957  df-er 8139  df-map 8258  df-ixp 8311  df-en 8358  df-dom 8359  df-sdom 8360  df-fin 8361  df-fsupp 8680  df-sup 8752  df-oi 8820  df-card 9214  df-pnf 10523  df-mnf 10524  df-xr 10525  df-ltxr 10526  df-le 10527  df-sub 10719  df-neg 10720  df-nn 11487  df-2 11548  df-3 11549  df-4 11550  df-5 11551  df-6 11552  df-7 11553  df-8 11554  df-9 11555  df-n0 11746  df-z 11830  df-dec 11948  df-uz 12094  df-fz 12743  df-fzo 12884  df-seq 13220  df-hash 13541  df-struct 16314  df-ndx 16315  df-slot 16316  df-base 16318  df-sets 16319  df-ress 16320  df-plusg 16407  df-mulr 16408  df-sca 16410  df-vsca 16411  df-ip 16412  df-tset 16413  df-ple 16414  df-ds 16416  df-hom 16418  df-cco 16419  df-0g 16544  df-gsum 16545  df-prds 16550  df-pws 16552  df-mre 16686  df-mrc 16687  df-acs 16689  df-mgm 17681  df-sgrp 17723  df-mnd 17734  df-mhm 17774  df-submnd 17775  df-grp 17864  df-minusg 17865  df-sbg 17866  df-mulg 17982  df-subg 18030  df-ghm 18097  df-cntz 18188  df-cmn 18635  df-abl 18636  df-mgp 18930  df-ur 18942  df-ring 18989  df-drng 19194  df-subrg 19223  df-lmod 19326  df-lss 19394  df-lsp 19434  df-lmhm 19484  df-lbs 19537  df-lvec 19565  df-sra 19634  df-rgmod 19635  df-nzr 19720  df-dsmm 20558  df-frlm 20573  df-uvc 20609  df-lindf 20632  df-linds 20633
This theorem is referenced by:  fedgmul  30631
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