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Theorem mvmulfval 20167
 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 20166 . . . 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 6113 . . . . 5 (1st𝑜) ∈ V
5 fvex 6113 . . . . 5 (2nd𝑜) ∈ V
6 xpeq12 5058 . . . . . . 7 ((𝑚 = (1st𝑜) ∧ 𝑛 = (2nd𝑜)) → (𝑚 × 𝑛) = ((1st𝑜) × (2nd𝑜)))
76oveq2d 6565 . . . . . 6 ((𝑚 = (1st𝑜) ∧ 𝑛 = (2nd𝑜)) → ((Base‘𝑟) ↑𝑚 (𝑚 × 𝑛)) = ((Base‘𝑟) ↑𝑚 ((1st𝑜) × (2nd𝑜))))
8 oveq2 6557 . . . . . . 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 4666 . . . . . . . 8 ((𝑚 = (1st𝑜) ∧ 𝑛 = (2nd𝑜)) → (𝑗𝑛 ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑦𝑗))) = (𝑗 ∈ (2nd𝑜) ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑦𝑗))))
1312oveq2d 6565 . . . . . . 7 ((𝑚 = (1st𝑜) ∧ 𝑛 = (2nd𝑜)) → (𝑟 Σg (𝑗𝑛 ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑦𝑗)))) = (𝑟 Σg (𝑗 ∈ (2nd𝑜) ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑦𝑗)))))
1410, 13mpteq12dv 4663 . . . . . 6 ((𝑚 = (1st𝑜) ∧ 𝑛 = (2nd𝑜)) → (𝑖𝑚 ↦ (𝑟 Σg (𝑗𝑛 ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑦𝑗))))) = (𝑖 ∈ (1st𝑜) ↦ (𝑟 Σg (𝑗 ∈ (2nd𝑜) ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑦𝑗))))))
157, 9, 14mpt2eq123dv 6615 . . . . 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 3529 . . . 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 790 . . . . . . . 8 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁⟩)) → 𝑟 = 𝑅)
1817fveq2d 6107 . . . . . . 7 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁⟩)) → (Base‘𝑟) = (Base‘𝑅))
19 mvmulfval.b . . . . . . 7 𝐵 = (Base‘𝑅)
2018, 19syl6eqr 2662 . . . . . 6 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁⟩)) → (Base‘𝑟) = 𝐵)
21 fveq2 6103 . . . . . . . . 9 (𝑜 = ⟨𝑀, 𝑁⟩ → (1st𝑜) = (1st ‘⟨𝑀, 𝑁⟩))
2221ad2antll 761 . . . . . . . 8 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁⟩)) → (1st𝑜) = (1st ‘⟨𝑀, 𝑁⟩))
23 mvmulfval.m . . . . . . . . . 10 (𝜑𝑀 ∈ Fin)
24 mvmulfval.n . . . . . . . . . 10 (𝜑𝑁 ∈ Fin)
25 op1stg 7071 . . . . . . . . . 10 ((𝑀 ∈ Fin ∧ 𝑁 ∈ Fin) → (1st ‘⟨𝑀, 𝑁⟩) = 𝑀)
2623, 24, 25syl2anc 691 . . . . . . . . 9 (𝜑 → (1st ‘⟨𝑀, 𝑁⟩) = 𝑀)
2726adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁⟩)) → (1st ‘⟨𝑀, 𝑁⟩) = 𝑀)
2822, 27eqtrd 2644 . . . . . . 7 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁⟩)) → (1st𝑜) = 𝑀)
29 fveq2 6103 . . . . . . . . 9 (𝑜 = ⟨𝑀, 𝑁⟩ → (2nd𝑜) = (2nd ‘⟨𝑀, 𝑁⟩))
3029ad2antll 761 . . . . . . . 8 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁⟩)) → (2nd𝑜) = (2nd ‘⟨𝑀, 𝑁⟩))
31 op2ndg 7072 . . . . . . . . . 10 ((𝑀 ∈ Fin ∧ 𝑁 ∈ Fin) → (2nd ‘⟨𝑀, 𝑁⟩) = 𝑁)
3223, 24, 31syl2anc 691 . . . . . . . . 9 (𝜑 → (2nd ‘⟨𝑀, 𝑁⟩) = 𝑁)
3332adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁⟩)) → (2nd ‘⟨𝑀, 𝑁⟩) = 𝑁)
3430, 33eqtrd 2644 . . . . . . 7 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁⟩)) → (2nd𝑜) = 𝑁)
3528, 34xpeq12d 5064 . . . . . 6 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁⟩)) → ((1st𝑜) × (2nd𝑜)) = (𝑀 × 𝑁))
3620, 35oveq12d 6567 . . . . 5 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁⟩)) → ((Base‘𝑟) ↑𝑚 ((1st𝑜) × (2nd𝑜))) = (𝐵𝑚 (𝑀 × 𝑁)))
3720, 34oveq12d 6567 . . . . 5 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁⟩)) → ((Base‘𝑟) ↑𝑚 (2nd𝑜)) = (𝐵𝑚 𝑁))
38 fveq2 6103 . . . . . . . . . . . 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 2662 . . . . . . . . 9 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁⟩)) → (.r𝑟) = · )
4342oveqd 6566 . . . . . . . 8 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁⟩)) → ((𝑖𝑥𝑗)(.r𝑟)(𝑦𝑗)) = ((𝑖𝑥𝑗) · (𝑦𝑗)))
4434, 43mpteq12dv 4663 . . . . . . 7 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁⟩)) → (𝑗 ∈ (2nd𝑜) ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑦𝑗))) = (𝑗𝑁 ↦ ((𝑖𝑥𝑗) · (𝑦𝑗))))
4517, 44oveq12d 6567 . . . . . 6 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁⟩)) → (𝑟 Σg (𝑗 ∈ (2nd𝑜) ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑦𝑗)))) = (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑥𝑗) · (𝑦𝑗)))))
4628, 45mpteq12dv 4663 . . . . 5 ((𝜑 ∧ (𝑟 = 𝑅𝑜 = ⟨𝑀, 𝑁⟩)) → (𝑖 ∈ (1st𝑜) ↦ (𝑟 Σg (𝑗 ∈ (2nd𝑜) ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑦𝑗))))) = (𝑖𝑀 ↦ (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑥𝑗) · (𝑦𝑗))))))
4736, 37, 46mpt2eq123dv 6615 . . . 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 2656 . . 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 3185 . . . 4 (𝑅𝑉𝑅 ∈ V)
5149, 50syl 17 . . 3 (𝜑𝑅 ∈ V)
52 opex 4859 . . . 4 𝑀, 𝑁⟩ ∈ V
5352a1i 11 . . 3 (𝜑 → ⟨𝑀, 𝑁⟩ ∈ V)
54 ovex 6577 . . . . 5 (𝐵𝑚 (𝑀 × 𝑁)) ∈ V
55 ovex 6577 . . . . 5 (𝐵𝑚 𝑁) ∈ V
5654, 55mpt2ex 7136 . . . 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 6686 . 2 (𝜑 → (𝑅 maVecMul ⟨𝑀, 𝑁⟩) = (𝑥 ∈ (𝐵𝑚 (𝑀 × 𝑁)), 𝑦 ∈ (𝐵𝑚 𝑁) ↦ (𝑖𝑀 ↦ (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑥𝑗) · (𝑦𝑗)))))))
591, 58syl5eq 2656 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 1475   ∈ wcel 1977  Vcvv 3173  ⦋csb 3499  ⟨cop 4131   ↦ cmpt 4643   × cxp 5036  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  1st c1st 7057  2nd c2nd 7058   ↑𝑚 cmap 7744  Fincfn 7841  Basecbs 15695  .rcmulr 15769   Σg cgsu 15924   maVecMul cmvmul 20165 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-mvmul 20166 This theorem is referenced by:  mvmulval  20168  mavmuldm  20175  mavmul0g  20178
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