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Theorem mvmulfval 20396
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 20395 . . . 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 6239 . . . . 5 (1st𝑜) ∈ V
5 fvex 6239 . . . . 5 (2nd𝑜) ∈ V
6 xpeq12 5168 . . . . . . 7 ((𝑚 = (1st𝑜) ∧ 𝑛 = (2nd𝑜)) → (𝑚 × 𝑛) = ((1st𝑜) × (2nd𝑜)))
76oveq2d 6706 . . . . . 6 ((𝑚 = (1st𝑜) ∧ 𝑛 = (2nd𝑜)) → ((Base‘𝑟) ↑𝑚 (𝑚 × 𝑛)) = ((Base‘𝑟) ↑𝑚 ((1st𝑜) × (2nd𝑜))))
8 oveq2 6698 . . . . . . 7 (𝑛 = (2nd𝑜) → ((Base‘𝑟) ↑𝑚 𝑛) = ((Base‘𝑟) ↑𝑚 (2nd𝑜)))
98adantl 481 . . . . . 6 ((𝑚 = (1st𝑜) ∧ 𝑛 = (2nd𝑜)) → ((Base‘𝑟) ↑𝑚 𝑛) = ((Base‘𝑟) ↑𝑚 (2nd𝑜)))
10 simpl 472 . . . . . . 7 ((𝑚 = (1st𝑜) ∧ 𝑛 = (2nd𝑜)) → 𝑚 = (1st𝑜))
11 simpr 476 . . . . . . . . 9 ((𝑚 = (1st𝑜) ∧ 𝑛 = (2nd𝑜)) → 𝑛 = (2nd𝑜))
1211mpteq1d 4771 . . . . . . . 8 ((𝑚 = (1st𝑜) ∧ 𝑛 = (2nd𝑜)) → (𝑗𝑛 ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑦𝑗))) = (𝑗 ∈ (2nd𝑜) ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑦𝑗))))
1312oveq2d 6706 . . . . . . 7 ((𝑚 = (1st𝑜) ∧ 𝑛 = (2nd𝑜)) → (𝑟 Σg (𝑗𝑛 ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑦𝑗)))) = (𝑟 Σg (𝑗 ∈ (2nd𝑜) ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑦𝑗)))))
1410, 13mpteq12dv 4766 . . . . . 6 ((𝑚 = (1st𝑜) ∧ 𝑛 = (2nd𝑜)) → (𝑖𝑚 ↦ (𝑟 Σg (𝑗𝑛 ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑦𝑗))))) = (𝑖 ∈ (1st𝑜) ↦ (𝑟 Σg (𝑗 ∈ (2nd𝑜) ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑦𝑗))))))
157, 9, 14mpt2eq123dv 6759 . . . . 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 3596 . . . 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 809 . . . . . . . 8 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁⟩)) → 𝑟 = 𝑅)
1817fveq2d 6233 . . . . . . 7 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁⟩)) → (Base‘𝑟) = (Base‘𝑅))
19 mvmulfval.b . . . . . . 7 𝐵 = (Base‘𝑅)
2018, 19syl6eqr 2703 . . . . . 6 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁⟩)) → (Base‘𝑟) = 𝐵)
21 fveq2 6229 . . . . . . . . 9 (𝑜 = ⟨𝑀, 𝑁⟩ → (1st𝑜) = (1st ‘⟨𝑀, 𝑁⟩))
2221ad2antll 765 . . . . . . . 8 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁⟩)) → (1st𝑜) = (1st ‘⟨𝑀, 𝑁⟩))
23 mvmulfval.m . . . . . . . . . 10 (𝜑𝑀 ∈ Fin)
24 mvmulfval.n . . . . . . . . . 10 (𝜑𝑁 ∈ Fin)
25 op1stg 7222 . . . . . . . . . 10 ((𝑀 ∈ Fin ∧ 𝑁 ∈ Fin) → (1st ‘⟨𝑀, 𝑁⟩) = 𝑀)
2623, 24, 25syl2anc 694 . . . . . . . . 9 (𝜑 → (1st ‘⟨𝑀, 𝑁⟩) = 𝑀)
2726adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁⟩)) → (1st ‘⟨𝑀, 𝑁⟩) = 𝑀)
2822, 27eqtrd 2685 . . . . . . 7 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁⟩)) → (1st𝑜) = 𝑀)
29 fveq2 6229 . . . . . . . . 9 (𝑜 = ⟨𝑀, 𝑁⟩ → (2nd𝑜) = (2nd ‘⟨𝑀, 𝑁⟩))
3029ad2antll 765 . . . . . . . 8 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁⟩)) → (2nd𝑜) = (2nd ‘⟨𝑀, 𝑁⟩))
31 op2ndg 7223 . . . . . . . . . 10 ((𝑀 ∈ Fin ∧ 𝑁 ∈ Fin) → (2nd ‘⟨𝑀, 𝑁⟩) = 𝑁)
3223, 24, 31syl2anc 694 . . . . . . . . 9 (𝜑 → (2nd ‘⟨𝑀, 𝑁⟩) = 𝑁)
3332adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁⟩)) → (2nd ‘⟨𝑀, 𝑁⟩) = 𝑁)
3430, 33eqtrd 2685 . . . . . . 7 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁⟩)) → (2nd𝑜) = 𝑁)
3528, 34xpeq12d 5174 . . . . . 6 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁⟩)) → ((1st𝑜) × (2nd𝑜)) = (𝑀 × 𝑁))
3620, 35oveq12d 6708 . . . . 5 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁⟩)) → ((Base‘𝑟) ↑𝑚 ((1st𝑜) × (2nd𝑜))) = (𝐵𝑚 (𝑀 × 𝑁)))
3720, 34oveq12d 6708 . . . . 5 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁⟩)) → ((Base‘𝑟) ↑𝑚 (2nd𝑜)) = (𝐵𝑚 𝑁))
38 fveq2 6229 . . . . . . . . . . . 12 (𝑟 = 𝑅 → (.r𝑟) = (.r𝑅))
3938adantr 480 . . . . . . . . . . 11 ((𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁⟩) → (.r𝑟) = (.r𝑅))
4039adantl 481 . . . . . . . . . 10 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁⟩)) → (.r𝑟) = (.r𝑅))
41 mvmulfval.t . . . . . . . . . 10 · = (.r𝑅)
4240, 41syl6eqr 2703 . . . . . . . . 9 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁⟩)) → (.r𝑟) = · )
4342oveqd 6707 . . . . . . . 8 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁⟩)) → ((𝑖𝑥𝑗)(.r𝑟)(𝑦𝑗)) = ((𝑖𝑥𝑗) · (𝑦𝑗)))
4434, 43mpteq12dv 4766 . . . . . . 7 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁⟩)) → (𝑗 ∈ (2nd𝑜) ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑦𝑗))) = (𝑗𝑁 ↦ ((𝑖𝑥𝑗) · (𝑦𝑗))))
4517, 44oveq12d 6708 . . . . . 6 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁⟩)) → (𝑟 Σg (𝑗 ∈ (2nd𝑜) ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑦𝑗)))) = (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑥𝑗) · (𝑦𝑗)))))
4628, 45mpteq12dv 4766 . . . . 5 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁⟩)) → (𝑖 ∈ (1st𝑜) ↦ (𝑟 Σg (𝑗 ∈ (2nd𝑜) ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑦𝑗))))) = (𝑖𝑀 ↦ (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑥𝑗) · (𝑦𝑗))))))
4736, 37, 46mpt2eq123dv 6759 . . . 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 2697 . . 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)
50 elex 3243 . . . 4 (𝑅𝑉𝑅 ∈ V)
5149, 50syl 17 . . 3 (𝜑𝑅 ∈ V)
52 opex 4962 . . . 4 𝑀, 𝑁⟩ ∈ V
5352a1i 11 . . 3 (𝜑 → ⟨𝑀, 𝑁⟩ ∈ V)
54 ovex 6718 . . . . 5 (𝐵𝑚 (𝑀 × 𝑁)) ∈ V
55 ovex 6718 . . . . 5 (𝐵𝑚 𝑁) ∈ V
5654, 55mpt2ex 7292 . . . 4 (𝑥 ∈ (𝐵𝑚 (𝑀 × 𝑁)), 𝑦 ∈ (𝐵𝑚 𝑁) ↦ (𝑖𝑀 ↦ (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑥𝑗) · (𝑦𝑗)))))) ∈ V
5756a1i 11 . . 3 (𝜑 → (𝑥 ∈ (𝐵𝑚 (𝑀 × 𝑁)), 𝑦 ∈ (𝐵𝑚 𝑁) ↦ (𝑖𝑀 ↦ (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑥𝑗) · (𝑦𝑗)))))) ∈ V)
583, 48, 51, 53, 57ovmpt2d 6830 . 2 (𝜑 → (𝑅 maVecMul ⟨𝑀, 𝑁⟩) = (𝑥 ∈ (𝐵𝑚 (𝑀 × 𝑁)), 𝑦 ∈ (𝐵𝑚 𝑁) ↦ (𝑖𝑀 ↦ (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑥𝑗) · (𝑦𝑗)))))))
591, 58syl5eq 2697 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 383   = wceq 1523  wcel 2030  Vcvv 3231  csb 3566  cop 4216  cmpt 4762   × cxp 5141  cfv 5926  (class class class)co 6690  cmpt2 6692  1st c1st 7208  2nd c2nd 7209  𝑚 cmap 7899  Fincfn 7997  Basecbs 15904  .rcmulr 15989   Σg cgsu 16148   maVecMul cmvmul 20394
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-mvmul 20395
This theorem is referenced by:  mvmulval  20397  mavmuldm  20404  mavmul0g  20407
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