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Theorem cpmadugsumlemB 21483
Description: Lemma B for cpmadugsum 21487. (Contributed by AV, 2-Nov-2019.)
Hypotheses
Ref Expression
cpmadugsum.a 𝐴 = (𝑁 Mat 𝑅)
cpmadugsum.b 𝐵 = (Base‘𝐴)
cpmadugsum.p 𝑃 = (Poly1𝑅)
cpmadugsum.y 𝑌 = (𝑁 Mat 𝑃)
cpmadugsum.t 𝑇 = (𝑁 matToPolyMat 𝑅)
cpmadugsum.x 𝑋 = (var1𝑅)
cpmadugsum.e = (.g‘(mulGrp‘𝑃))
cpmadugsum.m · = ( ·𝑠𝑌)
cpmadugsum.r × = (.r𝑌)
cpmadugsum.1 1 = (1r𝑌)
Assertion
Ref Expression
cpmadugsumlemB (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ0𝑏 ∈ (𝐵m (0...𝑠)))) → ((𝑋 · 1 ) × (𝑌 Σg (𝑖 ∈ (0...𝑠) ↦ ((𝑖 𝑋) · (𝑇‘(𝑏𝑖)))))) = (𝑌 Σg (𝑖 ∈ (0...𝑠) ↦ (((𝑖 + 1) 𝑋) · (𝑇‘(𝑏𝑖))))))
Distinct variable groups:   𝐵,𝑖   𝑖,𝑀   𝑖,𝑁   𝑅,𝑖   𝑖,𝑋   𝑖,𝑌   × ,𝑖   · ,𝑖   1 ,𝑖   𝑖,𝑏   𝑖,𝑠
Allowed substitution hints:   𝐴(𝑖,𝑠,𝑏)   𝐵(𝑠,𝑏)   𝑃(𝑖,𝑠,𝑏)   𝑅(𝑠,𝑏)   𝑇(𝑖,𝑠,𝑏)   · (𝑠,𝑏)   × (𝑠,𝑏)   1 (𝑠,𝑏)   (𝑖,𝑠,𝑏)   𝑀(𝑠,𝑏)   𝑁(𝑠,𝑏)   𝑋(𝑠,𝑏)   𝑌(𝑠,𝑏)

Proof of Theorem cpmadugsumlemB
StepHypRef Expression
1 crngring 19306 . . . . . . . . . . . 12 (𝑅 ∈ CRing → 𝑅 ∈ Ring)
2 cpmadugsum.p . . . . . . . . . . . . 13 𝑃 = (Poly1𝑅)
32ply1ring 20881 . . . . . . . . . . . 12 (𝑅 ∈ Ring → 𝑃 ∈ Ring)
41, 3syl 17 . . . . . . . . . . 11 (𝑅 ∈ CRing → 𝑃 ∈ Ring)
543ad2ant2 1131 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑃 ∈ Ring)
6 eqid 2801 . . . . . . . . . . 11 (mulGrp‘𝑃) = (mulGrp‘𝑃)
76ringmgp 19300 . . . . . . . . . 10 (𝑃 ∈ Ring → (mulGrp‘𝑃) ∈ Mnd)
85, 7syl 17 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (mulGrp‘𝑃) ∈ Mnd)
98ad2antrr 725 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ0𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → (mulGrp‘𝑃) ∈ Mnd)
10 elfznn0 12999 . . . . . . . . 9 (𝑖 ∈ (0...𝑠) → 𝑖 ∈ ℕ0)
1110adantl 485 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ0𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → 𝑖 ∈ ℕ0)
12 1nn0 11905 . . . . . . . . 9 1 ∈ ℕ0
1312a1i 11 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ0𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → 1 ∈ ℕ0)
1413ad2ant2 1131 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑅 ∈ Ring)
15 cpmadugsum.x . . . . . . . . . . 11 𝑋 = (var1𝑅)
16 eqid 2801 . . . . . . . . . . 11 (Base‘𝑃) = (Base‘𝑃)
1715, 2, 16vr1cl 20850 . . . . . . . . . 10 (𝑅 ∈ Ring → 𝑋 ∈ (Base‘𝑃))
1814, 17syl 17 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑋 ∈ (Base‘𝑃))
1918ad2antrr 725 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ0𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → 𝑋 ∈ (Base‘𝑃))
206, 16mgpbas 19242 . . . . . . . . 9 (Base‘𝑃) = (Base‘(mulGrp‘𝑃))
21 cpmadugsum.e . . . . . . . . 9 = (.g‘(mulGrp‘𝑃))
22 eqid 2801 . . . . . . . . . 10 (.r𝑃) = (.r𝑃)
236, 22mgpplusg 19240 . . . . . . . . 9 (.r𝑃) = (+g‘(mulGrp‘𝑃))
2420, 21, 23mulgnn0dir 18253 . . . . . . . 8 (((mulGrp‘𝑃) ∈ Mnd ∧ (𝑖 ∈ ℕ0 ∧ 1 ∈ ℕ0𝑋 ∈ (Base‘𝑃))) → ((𝑖 + 1) 𝑋) = ((𝑖 𝑋)(.r𝑃)(1 𝑋)))
259, 11, 13, 19, 24syl13anc 1369 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ0𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → ((𝑖 + 1) 𝑋) = ((𝑖 𝑋)(.r𝑃)(1 𝑋)))
262ply1crng 20831 . . . . . . . . . . . . 13 (𝑅 ∈ CRing → 𝑃 ∈ CRing)
2726anim2i 619 . . . . . . . . . . . 12 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑃 ∈ CRing))
28273adant3 1129 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝑁 ∈ Fin ∧ 𝑃 ∈ CRing))
29 cpmadugsum.y . . . . . . . . . . . 12 𝑌 = (𝑁 Mat 𝑃)
3029matsca2 21029 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑃 ∈ CRing) → 𝑃 = (Scalar‘𝑌))
3128, 30syl 17 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑃 = (Scalar‘𝑌))
3231ad2antrr 725 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ0𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → 𝑃 = (Scalar‘𝑌))
3332fveq2d 6653 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ0𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → (.r𝑃) = (.r‘(Scalar‘𝑌)))
34 eqidd 2802 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ0𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → (𝑖 𝑋) = (𝑖 𝑋))
3520, 21mulg1 18231 . . . . . . . . . 10 (𝑋 ∈ (Base‘𝑃) → (1 𝑋) = 𝑋)
3618, 35syl 17 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (1 𝑋) = 𝑋)
3736ad2antrr 725 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ0𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → (1 𝑋) = 𝑋)
3833, 34, 37oveq123d 7160 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ0𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → ((𝑖 𝑋)(.r𝑃)(1 𝑋)) = ((𝑖 𝑋)(.r‘(Scalar‘𝑌))𝑋))
3925, 38eqtrd 2836 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ0𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → ((𝑖 + 1) 𝑋) = ((𝑖 𝑋)(.r‘(Scalar‘𝑌))𝑋))
404anim2i 619 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑁 ∈ Fin ∧ 𝑃 ∈ Ring))
41403adant3 1129 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝑁 ∈ Fin ∧ 𝑃 ∈ Ring))
4229matring 21052 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑃 ∈ Ring) → 𝑌 ∈ Ring)
4341, 42syl 17 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑌 ∈ Ring)
4443ad2antrr 725 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ0𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → 𝑌 ∈ Ring)
45 simpll1 1209 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ0𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → 𝑁 ∈ Fin)
4614ad2antrr 725 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ0𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → 𝑅 ∈ Ring)
47 simplrl 776 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ0𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → 𝑠 ∈ ℕ0)
48 simprr 772 . . . . . . . . . 10 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ0𝑏 ∈ (𝐵m (0...𝑠)))) → 𝑏 ∈ (𝐵m (0...𝑠)))
4948anim1i 617 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ0𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → (𝑏 ∈ (𝐵m (0...𝑠)) ∧ 𝑖 ∈ (0...𝑠)))
50 cpmadugsum.a . . . . . . . . . 10 𝐴 = (𝑁 Mat 𝑅)
51 cpmadugsum.b . . . . . . . . . 10 𝐵 = (Base‘𝐴)
52 cpmadugsum.t . . . . . . . . . 10 𝑇 = (𝑁 matToPolyMat 𝑅)
5350, 51, 2, 29, 52m2pmfzmap 21356 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑠 ∈ ℕ0) ∧ (𝑏 ∈ (𝐵m (0...𝑠)) ∧ 𝑖 ∈ (0...𝑠))) → (𝑇‘(𝑏𝑖)) ∈ (Base‘𝑌))
5445, 46, 47, 49, 53syl31anc 1370 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ0𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → (𝑇‘(𝑏𝑖)) ∈ (Base‘𝑌))
55 eqid 2801 . . . . . . . . 9 (Base‘𝑌) = (Base‘𝑌)
56 cpmadugsum.r . . . . . . . . 9 × = (.r𝑌)
57 cpmadugsum.1 . . . . . . . . 9 1 = (1r𝑌)
5855, 56, 57ringlidm 19321 . . . . . . . 8 ((𝑌 ∈ Ring ∧ (𝑇‘(𝑏𝑖)) ∈ (Base‘𝑌)) → ( 1 × (𝑇‘(𝑏𝑖))) = (𝑇‘(𝑏𝑖)))
5944, 54, 58syl2anc 587 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ0𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → ( 1 × (𝑇‘(𝑏𝑖))) = (𝑇‘(𝑏𝑖)))
6059eqcomd 2807 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ0𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → (𝑇‘(𝑏𝑖)) = ( 1 × (𝑇‘(𝑏𝑖))))
6139, 60oveq12d 7157 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ0𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → (((𝑖 + 1) 𝑋) · (𝑇‘(𝑏𝑖))) = (((𝑖 𝑋)(.r‘(Scalar‘𝑌))𝑋) · ( 1 × (𝑇‘(𝑏𝑖)))))
6229matassa 21053 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑃 ∈ CRing) → 𝑌 ∈ AssAlg)
6327, 62syl 17 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑌 ∈ AssAlg)
64633adant3 1129 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑌 ∈ AssAlg)
6564ad2antrr 725 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ0𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → 𝑌 ∈ AssAlg)
6631eqcomd 2807 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (Scalar‘𝑌) = 𝑃)
6766fveq2d 6653 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (Base‘(Scalar‘𝑌)) = (Base‘𝑃))
6818, 67eleqtrrd 2896 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑋 ∈ (Base‘(Scalar‘𝑌)))
6968ad2antrr 725 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ0𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → 𝑋 ∈ (Base‘(Scalar‘𝑌)))
7020, 21mulgnn0cl 18240 . . . . . . . . 9 (((mulGrp‘𝑃) ∈ Mnd ∧ 𝑖 ∈ ℕ0𝑋 ∈ (Base‘𝑃)) → (𝑖 𝑋) ∈ (Base‘𝑃))
719, 11, 19, 70syl3anc 1368 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ0𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → (𝑖 𝑋) ∈ (Base‘𝑃))
7267ad2antrr 725 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ0𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → (Base‘(Scalar‘𝑌)) = (Base‘𝑃))
7371, 72eleqtrrd 2896 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ0𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → (𝑖 𝑋) ∈ (Base‘(Scalar‘𝑌)))
7440, 42syl 17 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑌 ∈ Ring)
75743adant3 1129 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑌 ∈ Ring)
7655, 57ringidcl 19318 . . . . . . . . 9 (𝑌 ∈ Ring → 1 ∈ (Base‘𝑌))
7775, 76syl 17 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 1 ∈ (Base‘𝑌))
7877ad2antrr 725 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ0𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → 1 ∈ (Base‘𝑌))
79 eqid 2801 . . . . . . . 8 (Scalar‘𝑌) = (Scalar‘𝑌)
80 eqid 2801 . . . . . . . 8 (Base‘(Scalar‘𝑌)) = (Base‘(Scalar‘𝑌))
81 eqid 2801 . . . . . . . 8 (.r‘(Scalar‘𝑌)) = (.r‘(Scalar‘𝑌))
82 cpmadugsum.m . . . . . . . 8 · = ( ·𝑠𝑌)
8355, 79, 80, 81, 82, 56assa2ass 20556 . . . . . . 7 ((𝑌 ∈ AssAlg ∧ (𝑋 ∈ (Base‘(Scalar‘𝑌)) ∧ (𝑖 𝑋) ∈ (Base‘(Scalar‘𝑌))) ∧ ( 1 ∈ (Base‘𝑌) ∧ (𝑇‘(𝑏𝑖)) ∈ (Base‘𝑌))) → ((𝑋 · 1 ) × ((𝑖 𝑋) · (𝑇‘(𝑏𝑖)))) = (((𝑖 𝑋)(.r‘(Scalar‘𝑌))𝑋) · ( 1 × (𝑇‘(𝑏𝑖)))))
8465, 69, 73, 78, 54, 83syl122anc 1376 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ0𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → ((𝑋 · 1 ) × ((𝑖 𝑋) · (𝑇‘(𝑏𝑖)))) = (((𝑖 𝑋)(.r‘(Scalar‘𝑌))𝑋) · ( 1 × (𝑇‘(𝑏𝑖)))))
8584eqcomd 2807 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ0𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → (((𝑖 𝑋)(.r‘(Scalar‘𝑌))𝑋) · ( 1 × (𝑇‘(𝑏𝑖)))) = ((𝑋 · 1 ) × ((𝑖 𝑋) · (𝑇‘(𝑏𝑖)))))
8661, 85eqtrd 2836 . . . 4 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ0𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → (((𝑖 + 1) 𝑋) · (𝑇‘(𝑏𝑖))) = ((𝑋 · 1 ) × ((𝑖 𝑋) · (𝑇‘(𝑏𝑖)))))
8786mpteq2dva 5128 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ0𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑖 ∈ (0...𝑠) ↦ (((𝑖 + 1) 𝑋) · (𝑇‘(𝑏𝑖)))) = (𝑖 ∈ (0...𝑠) ↦ ((𝑋 · 1 ) × ((𝑖 𝑋) · (𝑇‘(𝑏𝑖))))))
8887oveq2d 7155 . 2 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ0𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ (0...𝑠) ↦ (((𝑖 + 1) 𝑋) · (𝑇‘(𝑏𝑖))))) = (𝑌 Σg (𝑖 ∈ (0...𝑠) ↦ ((𝑋 · 1 ) × ((𝑖 𝑋) · (𝑇‘(𝑏𝑖)))))))
89 eqid 2801 . . 3 (0g𝑌) = (0g𝑌)
90 eqid 2801 . . 3 (+g𝑌) = (+g𝑌)
9175adantr 484 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ0𝑏 ∈ (𝐵m (0...𝑠)))) → 𝑌 ∈ Ring)
92 ovexd 7174 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ0𝑏 ∈ (𝐵m (0...𝑠)))) → (0...𝑠) ∈ V)
9329matlmod 21038 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑃 ∈ Ring) → 𝑌 ∈ LMod)
9440, 93syl 17 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑌 ∈ LMod)
95943adant3 1129 . . . . 5 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑌 ∈ LMod)
961adantl 485 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑅 ∈ Ring)
9796, 17syl 17 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑋 ∈ (Base‘𝑃))
9827, 30syl 17 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑃 = (Scalar‘𝑌))
9998eqcomd 2807 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (Scalar‘𝑌) = 𝑃)
10099fveq2d 6653 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (Base‘(Scalar‘𝑌)) = (Base‘𝑃))
10197, 100eleqtrrd 2896 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝑋 ∈ (Base‘(Scalar‘𝑌)))
1021013adant3 1129 . . . . 5 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 𝑋 ∈ (Base‘(Scalar‘𝑌)))
10343, 76syl 17 . . . . 5 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → 1 ∈ (Base‘𝑌))
10455, 79, 82, 80lmodvscl 19648 . . . . 5 ((𝑌 ∈ LMod ∧ 𝑋 ∈ (Base‘(Scalar‘𝑌)) ∧ 1 ∈ (Base‘𝑌)) → (𝑋 · 1 ) ∈ (Base‘𝑌))
10595, 102, 103, 104syl3anc 1368 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝑋 · 1 ) ∈ (Base‘𝑌))
106105adantr 484 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ0𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑋 · 1 ) ∈ (Base‘𝑌))
10795ad2antrr 725 . . . 4 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ0𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → 𝑌 ∈ LMod)
10830eqcomd 2807 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑃 ∈ CRing) → (Scalar‘𝑌) = 𝑃)
109108fveq2d 6653 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑃 ∈ CRing) → (Base‘(Scalar‘𝑌)) = (Base‘𝑃))
11028, 109syl 17 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → (Base‘(Scalar‘𝑌)) = (Base‘𝑃))
111110eleq2d 2878 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) → ((𝑖 𝑋) ∈ (Base‘(Scalar‘𝑌)) ↔ (𝑖 𝑋) ∈ (Base‘𝑃)))
112111ad2antrr 725 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ0𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → ((𝑖 𝑋) ∈ (Base‘(Scalar‘𝑌)) ↔ (𝑖 𝑋) ∈ (Base‘𝑃)))
11371, 112mpbird 260 . . . 4 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ0𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → (𝑖 𝑋) ∈ (Base‘(Scalar‘𝑌)))
11455, 79, 82, 80lmodvscl 19648 . . . 4 ((𝑌 ∈ LMod ∧ (𝑖 𝑋) ∈ (Base‘(Scalar‘𝑌)) ∧ (𝑇‘(𝑏𝑖)) ∈ (Base‘𝑌)) → ((𝑖 𝑋) · (𝑇‘(𝑏𝑖))) ∈ (Base‘𝑌))
115107, 113, 54, 114syl3anc 1368 . . 3 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ0𝑏 ∈ (𝐵m (0...𝑠)))) ∧ 𝑖 ∈ (0...𝑠)) → ((𝑖 𝑋) · (𝑇‘(𝑏𝑖))) ∈ (Base‘𝑌))
116 simpl1 1188 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ0𝑏 ∈ (𝐵m (0...𝑠)))) → 𝑁 ∈ Fin)
11714adantr 484 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ0𝑏 ∈ (𝐵m (0...𝑠)))) → 𝑅 ∈ Ring)
118 simprl 770 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ0𝑏 ∈ (𝐵m (0...𝑠)))) → 𝑠 ∈ ℕ0)
119 eqid 2801 . . . . 5 (𝑖 ∈ (0...𝑠) ↦ ((𝑖 𝑋) · (𝑇‘(𝑏𝑖)))) = (𝑖 ∈ (0...𝑠) ↦ ((𝑖 𝑋) · (𝑇‘(𝑏𝑖))))
120 fzfid 13340 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑠 ∈ ℕ0) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) → (0...𝑠) ∈ Fin)
121 ovexd 7174 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑠 ∈ ℕ0) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) ∧ 𝑖 ∈ (0...𝑠)) → ((𝑖 𝑋) · (𝑇‘(𝑏𝑖))) ∈ V)
122 fvexd 6664 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑠 ∈ ℕ0) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) → (0g𝑌) ∈ V)
123119, 120, 121, 122fsuppmptdm 8832 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑠 ∈ ℕ0) ∧ 𝑏 ∈ (𝐵m (0...𝑠))) → (𝑖 ∈ (0...𝑠) ↦ ((𝑖 𝑋) · (𝑇‘(𝑏𝑖)))) finSupp (0g𝑌))
124116, 117, 118, 48, 123syl31anc 1370 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ0𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑖 ∈ (0...𝑠) ↦ ((𝑖 𝑋) · (𝑇‘(𝑏𝑖)))) finSupp (0g𝑌))
12555, 89, 90, 56, 91, 92, 106, 115, 124gsummulc2 19357 . 2 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ0𝑏 ∈ (𝐵m (0...𝑠)))) → (𝑌 Σg (𝑖 ∈ (0...𝑠) ↦ ((𝑋 · 1 ) × ((𝑖 𝑋) · (𝑇‘(𝑏𝑖)))))) = ((𝑋 · 1 ) × (𝑌 Σg (𝑖 ∈ (0...𝑠) ↦ ((𝑖 𝑋) · (𝑇‘(𝑏𝑖)))))))
12688, 125eqtr2d 2837 1 (((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing ∧ 𝑀𝐵) ∧ (𝑠 ∈ ℕ0𝑏 ∈ (𝐵m (0...𝑠)))) → ((𝑋 · 1 ) × (𝑌 Σg (𝑖 ∈ (0...𝑠) ↦ ((𝑖 𝑋) · (𝑇‘(𝑏𝑖)))))) = (𝑌 Σg (𝑖 ∈ (0...𝑠) ↦ (((𝑖 + 1) 𝑋) · (𝑇‘(𝑏𝑖))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2112  Vcvv 3444   class class class wbr 5033  cmpt 5113  cfv 6328  (class class class)co 7139  m cmap 8393  Fincfn 8496   finSupp cfsupp 8821  0cc0 10530  1c1 10531   + caddc 10533  0cn0 11889  ...cfz 12889  Basecbs 16479  +gcplusg 16561  .rcmulr 16562  Scalarcsca 16564   ·𝑠 cvsca 16565  0gc0g 16709   Σg cgsu 16710  Mndcmnd 17907  .gcmg 18220  mulGrpcmgp 19236  1rcur 19248  Ringcrg 19294  CRingccrg 19295  LModclmod 19631  AssAlgcasa 20543  var1cv1 20809  Poly1cpl1 20810   Mat cmat 21016   matToPolyMat cmat2pmat 21313
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-ot 4537  df-uni 4804  df-int 4842  df-iun 4886  df-iin 4887  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-se 5483  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-isom 6337  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-of 7393  df-ofr 7394  df-om 7565  df-1st 7675  df-2nd 7676  df-supp 7818  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-2o 8090  df-oadd 8093  df-er 8276  df-map 8395  df-pm 8396  df-ixp 8449  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-fsupp 8822  df-sup 8894  df-oi 8962  df-card 9356  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-nn 11630  df-2 11692  df-3 11693  df-4 11694  df-5 11695  df-6 11696  df-7 11697  df-8 11698  df-9 11699  df-n0 11890  df-z 11974  df-dec 12091  df-uz 12236  df-fz 12890  df-fzo 13033  df-seq 13369  df-hash 13691  df-struct 16481  df-ndx 16482  df-slot 16483  df-base 16485  df-sets 16486  df-ress 16487  df-plusg 16574  df-mulr 16575  df-sca 16577  df-vsca 16578  df-ip 16579  df-tset 16580  df-ple 16581  df-ds 16583  df-hom 16585  df-cco 16586  df-0g 16711  df-gsum 16712  df-prds 16717  df-pws 16719  df-mre 16853  df-mrc 16854  df-acs 16856  df-mgm 17848  df-sgrp 17897  df-mnd 17908  df-mhm 17952  df-submnd 17953  df-grp 18102  df-minusg 18103  df-sbg 18104  df-mulg 18221  df-subg 18272  df-ghm 18352  df-cntz 18443  df-cmn 18904  df-abl 18905  df-mgp 19237  df-ur 19249  df-ring 19296  df-cring 19297  df-subrg 19530  df-lmod 19633  df-lss 19701  df-sra 19941  df-rgmod 19942  df-dsmm 20425  df-frlm 20440  df-assa 20546  df-ascl 20548  df-psr 20598  df-mvr 20599  df-mpl 20600  df-opsr 20602  df-psr1 20813  df-vr1 20814  df-ply1 20815  df-mamu 20995  df-mat 21017  df-mat2pmat 21316
This theorem is referenced by:  cpmadugsumlemF  21485
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