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Theorem mvmulfval 21079
Description: Functional value of the matrix vector multiplication operator. (Contributed by AV, 23-Feb-2019.)
Hypotheses
Ref Expression
mvmulfval.x × = (𝑅 maVecMul ⟨𝑀, 𝑁⟩)
mvmulfval.b 𝐵 = (Base‘𝑅)
mvmulfval.t · = (.r𝑅)
mvmulfval.r (𝜑𝑅𝑉)
mvmulfval.m (𝜑𝑀 ∈ Fin)
mvmulfval.n (𝜑𝑁 ∈ Fin)
Assertion
Ref Expression
mvmulfval (𝜑× = (𝑥 ∈ (𝐵m (𝑀 × 𝑁)), 𝑦 ∈ (𝐵m 𝑁) ↦ (𝑖𝑀 ↦ (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑥𝑗) · (𝑦𝑗)))))))
Distinct variable groups:   𝑖,𝑗,𝑥,𝑦,𝜑   𝑖,𝑀,𝑗,𝑥,𝑦   𝑖,𝑁,𝑗,𝑥,𝑦   𝑅,𝑖,𝑗,𝑥,𝑦   𝑥,𝐵,𝑦   𝑥, · ,𝑦,𝑖
Allowed substitution hints:   𝐵(𝑖,𝑗)   · (𝑗)   × (𝑥,𝑦,𝑖,𝑗)   𝑉(𝑥,𝑦,𝑖,𝑗)

Proof of Theorem mvmulfval
Dummy variables 𝑚 𝑛 𝑜 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mvmulfval.x . 2 × = (𝑅 maVecMul ⟨𝑀, 𝑁⟩)
2 df-mvmul 21078 . . . 4 maVecMul = (𝑟 ∈ V, 𝑜 ∈ V ↦ (1st𝑜) / 𝑚(2nd𝑜) / 𝑛(𝑥 ∈ ((Base‘𝑟) ↑m (𝑚 × 𝑛)), 𝑦 ∈ ((Base‘𝑟) ↑m 𝑛) ↦ (𝑖𝑚 ↦ (𝑟 Σg (𝑗𝑛 ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑦𝑗)))))))
32a1i 11 . . 3 (𝜑 → maVecMul = (𝑟 ∈ V, 𝑜 ∈ V ↦ (1st𝑜) / 𝑚(2nd𝑜) / 𝑛(𝑥 ∈ ((Base‘𝑟) ↑m (𝑚 × 𝑛)), 𝑦 ∈ ((Base‘𝑟) ↑m 𝑛) ↦ (𝑖𝑚 ↦ (𝑟 Σg (𝑗𝑛 ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑦𝑗))))))))
4 fvex 6676 . . . . 5 (1st𝑜) ∈ V
5 fvex 6676 . . . . 5 (2nd𝑜) ∈ V
6 xpeq12 5573 . . . . . . 7 ((𝑚 = (1st𝑜) ∧ 𝑛 = (2nd𝑜)) → (𝑚 × 𝑛) = ((1st𝑜) × (2nd𝑜)))
76oveq2d 7161 . . . . . 6 ((𝑚 = (1st𝑜) ∧ 𝑛 = (2nd𝑜)) → ((Base‘𝑟) ↑m (𝑚 × 𝑛)) = ((Base‘𝑟) ↑m ((1st𝑜) × (2nd𝑜))))
8 oveq2 7153 . . . . . . 7 (𝑛 = (2nd𝑜) → ((Base‘𝑟) ↑m 𝑛) = ((Base‘𝑟) ↑m (2nd𝑜)))
98adantl 482 . . . . . 6 ((𝑚 = (1st𝑜) ∧ 𝑛 = (2nd𝑜)) → ((Base‘𝑟) ↑m 𝑛) = ((Base‘𝑟) ↑m (2nd𝑜)))
10 simpl 483 . . . . . . 7 ((𝑚 = (1st𝑜) ∧ 𝑛 = (2nd𝑜)) → 𝑚 = (1st𝑜))
11 simpr 485 . . . . . . . . 9 ((𝑚 = (1st𝑜) ∧ 𝑛 = (2nd𝑜)) → 𝑛 = (2nd𝑜))
1211mpteq1d 5146 . . . . . . . 8 ((𝑚 = (1st𝑜) ∧ 𝑛 = (2nd𝑜)) → (𝑗𝑛 ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑦𝑗))) = (𝑗 ∈ (2nd𝑜) ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑦𝑗))))
1312oveq2d 7161 . . . . . . 7 ((𝑚 = (1st𝑜) ∧ 𝑛 = (2nd𝑜)) → (𝑟 Σg (𝑗𝑛 ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑦𝑗)))) = (𝑟 Σg (𝑗 ∈ (2nd𝑜) ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑦𝑗)))))
1410, 13mpteq12dv 5142 . . . . . 6 ((𝑚 = (1st𝑜) ∧ 𝑛 = (2nd𝑜)) → (𝑖𝑚 ↦ (𝑟 Σg (𝑗𝑛 ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑦𝑗))))) = (𝑖 ∈ (1st𝑜) ↦ (𝑟 Σg (𝑗 ∈ (2nd𝑜) ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑦𝑗))))))
157, 9, 14mpoeq123dv 7218 . . . . 5 ((𝑚 = (1st𝑜) ∧ 𝑛 = (2nd𝑜)) → (𝑥 ∈ ((Base‘𝑟) ↑m (𝑚 × 𝑛)), 𝑦 ∈ ((Base‘𝑟) ↑m 𝑛) ↦ (𝑖𝑚 ↦ (𝑟 Σg (𝑗𝑛 ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑦𝑗)))))) = (𝑥 ∈ ((Base‘𝑟) ↑m ((1st𝑜) × (2nd𝑜))), 𝑦 ∈ ((Base‘𝑟) ↑m (2nd𝑜)) ↦ (𝑖 ∈ (1st𝑜) ↦ (𝑟 Σg (𝑗 ∈ (2nd𝑜) ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑦𝑗)))))))
164, 5, 15csbie2 3919 . . . 4 (1st𝑜) / 𝑚(2nd𝑜) / 𝑛(𝑥 ∈ ((Base‘𝑟) ↑m (𝑚 × 𝑛)), 𝑦 ∈ ((Base‘𝑟) ↑m 𝑛) ↦ (𝑖𝑚 ↦ (𝑟 Σg (𝑗𝑛 ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑦𝑗)))))) = (𝑥 ∈ ((Base‘𝑟) ↑m ((1st𝑜) × (2nd𝑜))), 𝑦 ∈ ((Base‘𝑟) ↑m (2nd𝑜)) ↦ (𝑖 ∈ (1st𝑜) ↦ (𝑟 Σg (𝑗 ∈ (2nd𝑜) ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑦𝑗))))))
17 simprl 767 . . . . . . . 8 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁⟩)) → 𝑟 = 𝑅)
1817fveq2d 6667 . . . . . . 7 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁⟩)) → (Base‘𝑟) = (Base‘𝑅))
19 mvmulfval.b . . . . . . 7 𝐵 = (Base‘𝑅)
2018, 19syl6eqr 2871 . . . . . 6 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁⟩)) → (Base‘𝑟) = 𝐵)
21 fveq2 6663 . . . . . . . . 9 (𝑜 = ⟨𝑀, 𝑁⟩ → (1st𝑜) = (1st ‘⟨𝑀, 𝑁⟩))
2221ad2antll 725 . . . . . . . 8 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁⟩)) → (1st𝑜) = (1st ‘⟨𝑀, 𝑁⟩))
23 mvmulfval.m . . . . . . . . . 10 (𝜑𝑀 ∈ Fin)
24 mvmulfval.n . . . . . . . . . 10 (𝜑𝑁 ∈ Fin)
25 op1stg 7690 . . . . . . . . . 10 ((𝑀 ∈ Fin ∧ 𝑁 ∈ Fin) → (1st ‘⟨𝑀, 𝑁⟩) = 𝑀)
2623, 24, 25syl2anc 584 . . . . . . . . 9 (𝜑 → (1st ‘⟨𝑀, 𝑁⟩) = 𝑀)
2726adantr 481 . . . . . . . 8 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁⟩)) → (1st ‘⟨𝑀, 𝑁⟩) = 𝑀)
2822, 27eqtrd 2853 . . . . . . 7 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁⟩)) → (1st𝑜) = 𝑀)
29 fveq2 6663 . . . . . . . . 9 (𝑜 = ⟨𝑀, 𝑁⟩ → (2nd𝑜) = (2nd ‘⟨𝑀, 𝑁⟩))
3029ad2antll 725 . . . . . . . 8 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁⟩)) → (2nd𝑜) = (2nd ‘⟨𝑀, 𝑁⟩))
31 op2ndg 7691 . . . . . . . . . 10 ((𝑀 ∈ Fin ∧ 𝑁 ∈ Fin) → (2nd ‘⟨𝑀, 𝑁⟩) = 𝑁)
3223, 24, 31syl2anc 584 . . . . . . . . 9 (𝜑 → (2nd ‘⟨𝑀, 𝑁⟩) = 𝑁)
3332adantr 481 . . . . . . . 8 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁⟩)) → (2nd ‘⟨𝑀, 𝑁⟩) = 𝑁)
3430, 33eqtrd 2853 . . . . . . 7 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁⟩)) → (2nd𝑜) = 𝑁)
3528, 34xpeq12d 5579 . . . . . 6 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁⟩)) → ((1st𝑜) × (2nd𝑜)) = (𝑀 × 𝑁))
3620, 35oveq12d 7163 . . . . 5 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁⟩)) → ((Base‘𝑟) ↑m ((1st𝑜) × (2nd𝑜))) = (𝐵m (𝑀 × 𝑁)))
3720, 34oveq12d 7163 . . . . 5 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁⟩)) → ((Base‘𝑟) ↑m (2nd𝑜)) = (𝐵m 𝑁))
38 fveq2 6663 . . . . . . . . . . . 12 (𝑟 = 𝑅 → (.r𝑟) = (.r𝑅))
3938adantr 481 . . . . . . . . . . 11 ((𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁⟩) → (.r𝑟) = (.r𝑅))
4039adantl 482 . . . . . . . . . 10 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁⟩)) → (.r𝑟) = (.r𝑅))
41 mvmulfval.t . . . . . . . . . 10 · = (.r𝑅)
4240, 41syl6eqr 2871 . . . . . . . . 9 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁⟩)) → (.r𝑟) = · )
4342oveqd 7162 . . . . . . . 8 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁⟩)) → ((𝑖𝑥𝑗)(.r𝑟)(𝑦𝑗)) = ((𝑖𝑥𝑗) · (𝑦𝑗)))
4434, 43mpteq12dv 5142 . . . . . . 7 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁⟩)) → (𝑗 ∈ (2nd𝑜) ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑦𝑗))) = (𝑗𝑁 ↦ ((𝑖𝑥𝑗) · (𝑦𝑗))))
4517, 44oveq12d 7163 . . . . . 6 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁⟩)) → (𝑟 Σg (𝑗 ∈ (2nd𝑜) ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑦𝑗)))) = (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑥𝑗) · (𝑦𝑗)))))
4628, 45mpteq12dv 5142 . . . . 5 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁⟩)) → (𝑖 ∈ (1st𝑜) ↦ (𝑟 Σg (𝑗 ∈ (2nd𝑜) ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑦𝑗))))) = (𝑖𝑀 ↦ (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑥𝑗) · (𝑦𝑗))))))
4736, 37, 46mpoeq123dv 7218 . . . 4 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁⟩)) → (𝑥 ∈ ((Base‘𝑟) ↑m ((1st𝑜) × (2nd𝑜))), 𝑦 ∈ ((Base‘𝑟) ↑m (2nd𝑜)) ↦ (𝑖 ∈ (1st𝑜) ↦ (𝑟 Σg (𝑗 ∈ (2nd𝑜) ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑦𝑗)))))) = (𝑥 ∈ (𝐵m (𝑀 × 𝑁)), 𝑦 ∈ (𝐵m 𝑁) ↦ (𝑖𝑀 ↦ (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑥𝑗) · (𝑦𝑗)))))))
4816, 47syl5eq 2865 . . 3 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁⟩)) → (1st𝑜) / 𝑚(2nd𝑜) / 𝑛(𝑥 ∈ ((Base‘𝑟) ↑m (𝑚 × 𝑛)), 𝑦 ∈ ((Base‘𝑟) ↑m 𝑛) ↦ (𝑖𝑚 ↦ (𝑟 Σg (𝑗𝑛 ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑦𝑗)))))) = (𝑥 ∈ (𝐵m (𝑀 × 𝑁)), 𝑦 ∈ (𝐵m 𝑁) ↦ (𝑖𝑀 ↦ (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑥𝑗) · (𝑦𝑗)))))))
49 mvmulfval.r . . . 4 (𝜑𝑅𝑉)
5049elexd 3512 . . 3 (𝜑𝑅 ∈ V)
51 opex 5347 . . . 4 𝑀, 𝑁⟩ ∈ V
5251a1i 11 . . 3 (𝜑 → ⟨𝑀, 𝑁⟩ ∈ V)
53 ovex 7178 . . . . 5 (𝐵m (𝑀 × 𝑁)) ∈ V
54 ovex 7178 . . . . 5 (𝐵m 𝑁) ∈ V
5553, 54mpoex 7766 . . . 4 (𝑥 ∈ (𝐵m (𝑀 × 𝑁)), 𝑦 ∈ (𝐵m 𝑁) ↦ (𝑖𝑀 ↦ (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑥𝑗) · (𝑦𝑗)))))) ∈ V
5655a1i 11 . . 3 (𝜑 → (𝑥 ∈ (𝐵m (𝑀 × 𝑁)), 𝑦 ∈ (𝐵m 𝑁) ↦ (𝑖𝑀 ↦ (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑥𝑗) · (𝑦𝑗)))))) ∈ V)
573, 48, 50, 52, 56ovmpod 7291 . 2 (𝜑 → (𝑅 maVecMul ⟨𝑀, 𝑁⟩) = (𝑥 ∈ (𝐵m (𝑀 × 𝑁)), 𝑦 ∈ (𝐵m 𝑁) ↦ (𝑖𝑀 ↦ (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑥𝑗) · (𝑦𝑗)))))))
581, 57syl5eq 2865 1 (𝜑× = (𝑥 ∈ (𝐵m (𝑀 × 𝑁)), 𝑦 ∈ (𝐵m 𝑁) ↦ (𝑖𝑀 ↦ (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑥𝑗) · (𝑦𝑗)))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  Vcvv 3492  csb 3880  cop 4563  cmpt 5137   × cxp 5546  cfv 6348  (class class class)co 7145  cmpo 7147  1st c1st 7676  2nd c2nd 7677  m cmap 8395  Fincfn 8497  Basecbs 16471  .rcmulr 16554   Σg cgsu 16702   maVecMul cmvmul 21077
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7678  df-2nd 7679  df-mvmul 21078
This theorem is referenced by:  mvmulval  21080  mavmuldm  21087  mavmul0g  21090
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